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Upadhyay +Abstract—In this paper, for the first time, we present a direct +construction of multiple zero-correlation zone (ZCZ) sequence +sets with inter-set zero-cross correlation zone (ZCCZ) from +generalised Boolean function. The presented ZCZ sequence sets +are optimal and their union is near-optimal ZCZ sequence set. +This work partially settles the open problem introduced by Tang +et al. in their 2010 paper using direct construction. The proposed +construction is presented by two layer graphical representation. +Finally, the construction is compared with existing state-of-the- +art. +Index Terms—Generalised Boolean function (GBF), zero-cross +correlation zone (ZCCZ), zero-correlation zone (ZCZ), multiple +ZCZ sequence sets. +I. INTRODUCTION +Z +-complementary pairs (ZCPs) were introduced by Fan et +al. [1] to overcome the limitation on the lengths of Golay +complementary pairs (GCPs) [2]–[5]. The idea of ZCPs was +generalized to Z-complementary code set (ZCCS) by Feng +et al. in [6]. A ZCCS refers to a set of K codes, each of +which consists of M constituent sequences of identical length +L, having ideal aperiodic auto- and cross-correlation properties +inside the ZCZ width (Z) [7], [8]. When Z = L and K = M, +the set is called complete complementary code CCC [9]– +[11]. To reduce the “near-far effect” and ensure interference- +free communication in asynchronous CDMA systems, ZCZ +sequences were introduced in the late 1990s [12]. When the +received signal delays within ZCZ, ZCZ sequences can be +employed to remove or reduce MAI and multipath interference +(MPI) in quasi synchronous CDMA (QS-CDMA) systems +[13], [14]. Although the ZCZ spreading sequences prevent co- +channel interference within each cell, inter-cell interference +across neighbouring cells is unavoidable [15]. +To address the aforementioned shortcoming, the idea of +multiple ZCZ sequence sets with inter set zero-cross corre- +lation zone (ZCCZ) has recently been proposed [16]–[23]. +A multiple ZCZ sequence set comprises ZCZ sequence sets +as its subsets and the cross-correlation function between two +arbitrary sequences from different subsets has either ZCCZ +or low cross-correlation zone (LCCZ). Authors in [24] and +[25] used generalised bent function and perfectly non-linear +functions respectively to construct multiple ZCZ sequence +sets. But they tend to achieve only multiple ZCZ sequence +set with interset LCCZ instead of ZCCZ. In [26], authors +presented construction of multiple ZCZ sequence sets using +discrete Fourier transform (DFT) matrices. Furthermore, an +asymmetric ZCZ (A-ZCZ) sequence set is a multiple ZCZ +sequence set and the ZCCZ between two arbitrary sequences +from distinct subsets has a large ZCCZ [20]. To obtain A-ZCZ +sequence sets, interleaving techniques on perfect sequences +are presented in the literature [21]–[23]. Since, perfect se- +quences are available only for very few lengths therefore these +constructions also have very limited lengths. Additionally, +the DFT matrices [18]–[20] and Hadamard matrices [16] +are also used to construct A-ZCZ sequences. But, all these +constructions are indirect. The limitation of A-ZCZ sequence +set is that the large ZCCZ is obtained at the cost of optimality +of ZCZ sequence sets. +In [17], Tang et al. proposed a method for constructing +multiple binary ZCZ sequence sets from mutually orthog- +onal Golay complementary set (MOGCS) with good inter- +set cross-correlation property and provided an open problem +as* “we propose the following open problem: Construct N +ZCZ sequence sets Zi, 0 ≤ i < N, satisfy: 1. Each Zi is +an (K, Z, L)-ZCZ sequence set with KZ/L = 1/2; 2. The +sets have a common zero correlation zone of length Zc with +Zc = Z/N”. +Motivated by the above open problem given in [17], in this +letter, we propose a direct construction of near-optimal mul- +tiple ZCZ sequence sets using generalised Boolean function +(GBF). Since, proposed construction is based on GBFs, there- +fore it is suitable for rapid hardware generation. A graphical +analysis of our proposed construction has also been provided. +Also, it is the first time that a direct construction of multiple +ZCZ sequence sets with ZCCZ is presented. The proposed +construction generalizes construction given in [13] and it is +optimal over several constructions of A-ZCZ sequence sets +presented in [16], [18]–[23], [27]. +II. NOTATIONS AND DEFINITIONS +A. Definition and Correlation Functions +Let a1 = (a10, a11, . . . , a1(L−1)) and a2 = (a20, a21, . . . , +a2(L−1)) be two sequences of equal length L, having entries +from complex numbers. For an integer u, we define aperiodic +cross-correlation function (ACCF) of sequences a1 and a2 as +γ(a1, a2)(u) = +��L−1−u +i=0 +a1ia∗ +2(i+u), +0 ≤ u < L, +�L+u−1 +i=0 +a1(i−u)a∗ +2i, +−L < u < 0. +(1) +Moreover, ACCF is termed as aperiodic auto-correlation func- +tion (AACF) if a1 = a2 and denoted as γ(a1)(u). Next, we +define periodic cross-correlation function (PCCF) in terms of +ACCF as +φ(a1, a2)(u) = γ(a1, a2)(u) + γ∗(a2, a1)(L − u). +(2) +*The notations has been changed as per this work. +arXiv:2301.02144v1 [cs.IT] 5 Jan 2023 + +2 +Definition 1: Let C = {C0, C1, . . . , CP −1} be a collection +of P codes (matrices) having M rows and L columns. Define +Cη = [aη +0 aη +1 . . . aη +M−1]T +M×L, +(3) +where aη +ν (0 ≤ ν ≤ M − 1, 0 ≤ η ≤ P − 1) is the νth row +sequence or νth constituent sequence and [·]T represents trans- +pose of matrix [·]. Then the ACCF of two codes Cη1, Cη2 ∈ C +is defined as +γ(Cη1, Cη2)(u) = +M−1 +� +ν=0 +γ(aη1 +ν , aη2 +ν )(u). +(4) +Definition 2: Let C be a code set as defined in (3) which +satisfies following correlation properties +γ(Cη1, Cη2)(u) = +� +� +� +� +� +LM, +η1 = η2 and u = 0, +0, +η1 = η2 and 0 < |u| < L, +0, +η1 ̸= η2 and |u| < L. +(5) +Then C is known as (P, M, L)-MOGCS and each code in C +is called GCS. Moreover, if P = M then C is known as CCC +set and denoted by (P, P, L)-CCC. +Definition 3: Let Zl = {zl +0, zl +1, . . . , zl +K−1} be a collection +of K L-length sequences, i.e., +zl +i = (zl +i0, zl +i1, . . . , zl +iL−1), +0 ≤ i ≤ K − 1. +Then, Z is called (K, Z, L)-ZCZ sequence set if it satisfies +following, +φ(zl +i, zl +j)(u) = +� +� +� +� +� +0, +i = j and 1 ≤ |u|≤Z, +0, +i ̸= j and 0 ≤ |u|≤Z, +L, +i = j and u = 0, +(6) +where 0 ≤ i, j ≤ K − 1 and Z is termed as ZCZ width. +Definition 4: Let Z be a collection of N, (K, Z, L)-ZCZ +sequence sets then Z = {Z1, Z2, . . . , ZN} is known as a +multiple ZCZ sequence set with ZCCZ equal to Zc, if for +0 ≤ |u| < Zc, φ(zl +i, zl′ +j )(u) = 0, ∀1 ≤ l ̸= l′ ≤ N and +0 ≤ i, j ≤ K − 1. +Definition 5 (Tang-Fan-Matsufuji Bound [28]): Let Z be +any (K, Z, L)-ZCZ sequence set. Then, KZ ≤ L. If for any +Z, KZ = L (or K(Z + 1) = L) then Z is called optimal (or +near-optimal) ZCZ sequence set. However, in case of binary +ZCZ sequence set the bound is reduced to 2KZ ≤ L. +B. Generalised Boolean Function (GBF) [29] +We define a complex valued sequence corresponding to a +GBF, f : {0, 1}m −→ Zq of m variables as +Ψ(f) = +� +ωf0, ωf1, . . . , ωf2m−1� +, +(7) +where fj = f(j0, j1, . . . , jm−1), ω = exp +� +2π√−1/q +� +, and +(j0, j1, . . . , jm−1) is the binary vector representation of j, +where as in the remainder of this letter, q is an even integer +not less than 2. Corresponding to a GBF f with m variables +the sequence Ψ(f) is of length 2m. +Definition 6: Let J = {j0, j1, . . . , jk−1} ⊂ {0, 1, . . . , n−1} +and xJ = [xj0, xj1, . . . , xjk−1]. For a constant e ∈ {0, 1}k, +f|xJ=e is known as restriction of f over e and is obtained by +substituting xjβ = eβ (β = 0, 1, ..., k − 1) in the function f. +Moreover, the sequence Ψ(f|xJ=e) is the same as sequence +Ψ(f) of length 2m except for the positions ijβ ̸= eβ for each +0 ≤ β < k, at these positions Ψ(f|xJ=e) has the zero entries. +C. Quadratic Forms and Graphs [30] +Let f be GBF of order r over m variables. If f|xJ=e +is a quadratic GBF, then graph of f|xJ=e, i.e., G(f|xJ=e) +has vertex set V , where V = {x0, x1, . . . , xm−1}\{xj0, xj1, +. . . , xjk−1}. If there is a term qβ1β2xβ1xβ2 (0 ≤ β1 < β2 < +m, xβ1, xβ2 ∈ V ) in the GBF f|xJ=e with qβ1β2 ̸= 0 (qβ1β2 ∈ +Zq) then by connecting the vertices xβ1 and xβ2 by an edge, +the graph G(f|xJ=e) can be obtained. For k = 0, the graph +of f|xJ=e is the same as that of f. +D. Generalized Reed-Muller Codes +Definition 7: Let q ≥ 2 and 0 ≤ r ≤ m,, then a linear code +over Zq generated by the Zq-valued sequences corresponding +to the monomials of degree at most r in x0, x1, . . . , xm−1 is +said to be rth order generalised Reed-Muller (RM) code and +denoted as RMq(r; m). +E. The Existing Construction of Multiple CCCs +Lemma 1 ( [31]): Let m, k, and s are integers with +0 ≤ s ≤ k ≤ m−2. Define Js = {jk−s, jk−s+1, . . . , jk−1} = +{m − s, m − s + 1, . . . , m − 1}, J = {j0, j1, . . . , jk−1−s} ⊂ +Zm−s, I += +{i0, i1, . . . , im−k−1} += +Zm−s\J, x += +� +xj0, xj1, . . . , xjk−s−1 +� +, xs = +� +xjk−s, xjk−s+1, . . . , xjk−1 +� +. Let +π be a permutation on symbols {0, 1, . . . , m − k − 1}. Let f +be a quadratic GBF over the m variables x0, x1, . . . , xm−1, +such that for e ∈ {0, 1}k−s, +f|x=e = Q + +m−k−1 +� +β=0 +uβxiβ + +s−1 +� +β=0 +vβxjk−s+β + v, +(8) +where +Q = q +2 +m−k−2 +� +β=0 +xiπ(β)xiπ(β+1), +(9) +uβ ∈ Zq ∀ 0 ≤ β ≤ m−k −1, vβ ∈ Zq ∀ 0 ≤ β ≤ s−1, and +v ∈ Zq Let γ1 and γ2 be two end vertices of the path G(Q), +t1 = �s−1 +β=0 bk+1+β2β, t2 = �k +β=0 bβ2β, where bβ ∈ {0, 1} +for 0 ≤ β ≤ k +s. For the natural order generated by (t1, t2), +Define the set S(t1,t2) by +� +� +�f + q +2 +� +� +k−1 +� +β=0 +dβxjβ + dxγ1 + +k−1 +� +β=0 +bβxjβ ++bkxγ2 + +k−1 +� +β=k−s +dβbs+1+β +� +� : dβ, d ∈ {0, 1} +� +� +� . +(10) +Let St1 = +� +S(t1,t2) : 0 ≤ t2 ≤ 2k+1 − 1 +� +, 0 ≤ t1 ≤ 2s − 1. +Then {St1 : 0 ≤ t1 ≤ 2s − 1} is a collection of 2s CCCs, and +any two GCSs from different CCCs St1 and St′ +1 with 0 ≤ +t1 ̸= t′ +1 ≤ 2s − 1 have a ZCCZ of width 2m−s. +For the fixed values of t1 and t2, S(t1,t2) is a GCS. Let us +denote, +S(t1,t2) = +� +s(t1,t2) +0 +s(t1,t2) +1 +. . . s(t1,t2) +2k+1−1 +�T +, +(11) + +3 +where s(t1,t2) +ν +(0 ≤ ν ≤ 2k+1 − 1) is νth row sequence of +S(t1,t2). +Lemma 2 ( [13]): Let q = 2 and x0, x1, . . . , xk, xk+1 be k+ +2 binary variables. Also, let h be a Boolean function defined +on x0, x1, . . . , xk, xk+1 as follow +h = +k+1 +� +β=1 +cβxβx0 + +� +1≤µ<ν≤k +dµνxµxν + +k+1 +� +α=0 +eαxα + e′, (14) +where ck+1 = 1, cβ ∈ Z2 for 1 ≤ β ≤ k, dµν, eα, e′ ∈ Z2. Let +h denotes the binary vector corresponding to function h, i.e., +h = [h0, h1, . . . , h2k+2−1] . +For 0 ≤ τ ≤ 2k+1 − 1, we have +(−1)hτ +hτ+1 + (−1)hτ+2k+1+hτ+1+2k+1 = 0, +(15) +where the operation in the subscripts is done in modulo 2k+2. +III. PROPOSED CONSTRUCTION +In this section, we provide a GBF which generates the +required multiple ZCZ sequence sets. +Theorem 1: Let x0, x1, . . . , xm+k+1 are m + k + 2 binary +variables. Define a GBF f(x0, x1, . . . , xm−1) on m variables +same as in Lemma 1, i.e., removing J = {j0, j1, . . . , jk−1−s} +having k − s vertices from the graph of f results in s isolated +vertices in Js and a path on m−k vertices in I. Define another +GBF h(xm, xm+1, . . . , xm+k+1) on k + 2 variables as +h = +k+1 +� +r=1 +crxm+rxm + +� +2≤µ<ν≤t +dµνxm+µxm+ν ++ +k+1 +� +β=1 +eβxm+β + e′, +(16) +where ck+1 ̸= 0, cr ∈ Z2 for 1 ≤ r ≤ k, dµν, eβ ∈ Z2. For +a fixed value of t1, define the set Zt1 = {Ψ(zt1 +t2) : 0 ≤ t2 ≤ +2k+1 − 1} by +� +� +�f + h + q +2 +� +� +k−1 +� +β=0 +xm+βxjβ + xm+kxγ1 + +k−1 +� +β=0 +bβxjβ ++bkxγ2 + +k−1 +� +β=k−s +xm+βbs+1+β +� +� +� +� +� . +(17) +Then Z += +� +Zt1 +: 0 ≤ t1 ≤ 2s − 1} is a collection of +2s (2k+1, 2m ,2m+k+2)-ZCZ sequence sets having ZCCZ +equals to 2m−s − 1. +Proof: Using (10), (11), (17) and taking natural order +generated by t2, we get Zt1 = +� +Zt1 +0 , Zt1 +1 +� +, where +� +Zt1 +0 , Zt1 +1 +� +is horizontal concatenation of matrices Zt1 +0 and Zt1 +1 and these +matrices are defined as, +Zt1 +0 = +� +������ +s(t1,0) +0 +ωk0 +s(t1,0) +1 +ωk1 +. . . +s(t1,0) +l−1 ωkl−1 +s(t1,1) +0 +ωk0 +s(t1,1) +1 +ωk1 +. . . +s(t1,1) +l−1 ωkl−1 +... +... +... +... +s(t1,l−1) +0 +ωk0 +s(t1,l−1) +1 +ωk1 +. . . +s(t1,l−1) +l−1 +ωkl−1 +� +������ +, +Zt1 +1 = +� +������ +s(t1,0) +0 +ωkl +s(t1,0) +1 +ωkl+1 +. . . +s(t1,0) +l−1 ωk2l−1 +s(t1,1) +0 +ωkl +s(t1,1) +1 +ωkl+1 +. . . +s(t1,1) +l−1 ωk2l−1 +... +... +... +... +s(t1,l−1) +0 +ωkl +s(t1,l−1) +1 +ωkl+1 +. . . +s(t1,l−1) +l−1 +ωk2l−1 +� +������ +, +where l = 2k. Now, we need to prove that Zt1 is a (2k+1, 2m +, 2m+k+2)-ZCZ sequence set. For 0 ≤ i, j ≤ 2k+1−1, periodic +correlation of Ψ(zt1 +i ) and Ψ(zt1 +j ) at any time shift 0 ≤ τ ≤ +2m is given by (12). Next, by (15), (12) and aperiodic sum +property of CCCs, we get, +φ(Ψ(zt1 +i ), Ψ(zt1 +j ))(τ) = 2 · +2k+1−1 +� +m=0 +γ +� +ci +m, cj +m +� +(τ) +(18) += +� +2k+m+2, +if τ = 0 and i = j, +0, +otherwise. +Which proves that Zt1 is a (2k+1, 2m, 2m+k+2)-ZCZ sequence +sets ∀ 0 ≤ t1 ≤ 2s − 1. Now, let 0 ≤ t1 ̸= t′ +1 < 2s and +0 ≤ i, j ≤ 2k+1 − 1 then for 0 ≤ τ ≤ 2m−s − 1, the value +of φ(Ψ(zt1 +i ), Ψ(zt′ +1 +j ))(τ) is given by (13). Now, by (15), (13) +and ZCCZ property of CCCs in Lemma 1, we get +φ(Ψ(zt1 +i ), Ψ(zt′ +1 +j ))(τ) = 0, +∀ 0 ≤ τ ≤ 2m−s − 1. +Remark 1: Theorem 1 constructed 2s ZCZ sequence sets +with parameter (2k+1, 2m, 2m+k+2) having common ZCZ +equals to 2m−s − 1. Since 2k+1 · 2m/2m+k+2 = 1/2 and +Zc = 2m−s − 1 = (Z + 1)/N. +φ(Ψ(zt1 +i ), Ψ(zt1 +j ))(τ) =2 · +2k+1−1 +� +m=0 +γ +� +s(t1,i) +m +, s(t1,j) +m +� +(τ) + [(−1)hl−1+hl + (−1)h2l−1+h0]γ∗ � +s(t1,j) +0 +, s(t1,i) +2l−1 +� +(L − τ) ++ +2l−2 +� +m=0 +[(−1)hm+hm+1 + (−1)hm+2l+hm+1+2l]γ∗ � +s(t1,j) +m+1 , s(t1,i) +m +� +(L − τ). +(12) +φ(Ψ(zt1 +i ), Ψ(zt′ +1 +j ))(τ) =2 · +l−1 +� +m=0 +γ +� +s(t1,i) +m +, s(t′ +1,j) +m +� +(τ) + [(−1)hl−1+hl + (−1)h2l−1+h0]γ∗ � +s(t′ +1,j) +0 +, s(t1,i) +2l−1 +� +(L − τ) ++ +l−2 +� +m=0 +[(−1)hm+hm+1 + (−1)hm+l+hm+1+l]γ∗ � +s(t′ +1,j) +m+1 , s(t1,i) +m +� +(L − τ). +(13) + +4 +Remark 2: Since the set of isolated vertices in Theorem +1 contribute to multipleness of constructed multiple ZCZ +sequence set. Hence, if we put s = 0, i.e., Js = φ in Theorem +1 then our construction reduces to construction presented in +[13]. Therefore, construction provided in [13] is a special case +of the proposed construction. +Corollary 1: Collection of all the ZCZ sequences in Theo- +rem 1, i.e., {Ψ(zt1 +t2) : 0 ≤ t2 ≤ 2k+1 − 1, 0 ≤ t1 ≤ 2s − 1} +is a near-optimal (2k+s+1, 2m−s − 1, 2m+k+2)-ZCZ sequence +set. +Proof: Directly follows from Theorem 1. +Remark 3: It is the first time in the literature that the +direct construction of optimal multiple ZCZ sequence sets is +provided such that their union is a near-optimal ZCZ sequence +set. Which makes our construction advantageous over several +constructions of A-ZCZ sequence sets which are presented in +the literature [16], [18]–[23], [27]. The detailed comparison of +the proposed work is provided in Table I. +Remark 4: From equation (17), it can be seen that the +proposed multiple ZCZ sequence sets are obtained from sec- +ond order cosets of generalised RM code. Since, RM codes +have efficient encoding, good error correction properties and +important practical advantage of being easy to decode [32]. +Hence, our proposed construction has advantage over any other +non-GBF based construction. +IV. GRAPHICAL INTERPRETATION OF THE PROPOSED +CONSTRUCTION +This section interprets the proposed construction with +graphical point of view. +Fig. 1 depicts a graphical repre- + +. +. +. +. +. +. +. . . +𝑖𝜋(0) +𝑖𝜋(𝑚−𝑘−1) +𝑖𝜋(1) +𝑖𝜋(2) +𝑗𝑘−1−𝑠 +𝑗0 +m-s +m−1 + +m +m+k-1-s +I +J +J s +Lower Layer +Upper Layer +. +. +. +. +. +. +m+k +m+k-1 +m+k-s +m+k+1 +Fig. 1: Graphical representation of (17). +sentation of (17). The graph has a two-layered structure with +a horizontal straight line which is separating the upper and +bottom layers. The upper layer and lower layer correspond to +graphs of Boolean functions f and h respectively. These layers +are interconnected through the set of edges +{xj0xm, xj1xm+1, . . . , xjk−1−sxm+k−1−s, xm−sxm+k−s, +xm−s+1xm+k−s+1, . . . , xm−1xm+k−1}, +and the vertex xm+k is connected to any of the end vertices +of the path in I. Interestingly, the ZCZ of each ZCZ sequence +set is equals to the power of number of vertices in the upper +layer of the graph and ZCCZ of ZCZ sequence sets equals to +one less than the power of number of vertices in the upper +layer of graph except isolated vertices. +Example 1: Let m = 4, q = 2, s = 1, and k = 2. Assume +J = {0}, Js = {3}, I = {1, 2} and GBFs +f = x0x1 + x0x2 + x0x3 + x1x2 + x1 + x2, +h = x4x5 + x4x6 + x4x7 + x4. +(19) +Generate two sequence sets Z0 and Z1 as +Z0 = {Ψ(f+h+x0x4+x2x6+x3x5+b0·x0+b1·x3+b2·x1 ++ 0 · x5) : b0, b1, b2 ∈ Z2}, +Z1 = {Ψ(f +h+x0x4+x2x6+x3x5+b0·x0+b1·x3+b2·x1 ++ 1 · x5) : b0, b1, b2 ∈ Z2}. +(20) +Then Z0 and Z1 are two optimal (8, 16, 256)-ZCZ sequence +sets having inter-set ZCCZ equals to 8. Moreover, Z = Z0∪Z1 +is also an optimal (16, 7, 256)-ZCZ sequence set. In Fig. 2, a +graph corresponding to quadratic form, i.e., f + h + x0x4 + +x2x6 + x3x5 of Example 1 is presented. +5 +6 +4 +7 +2 +1 +0 +3 +I +J +J +s +Upper Layer +Lower Layer +Fig. 2: Graphical representation of f +h+x0x4+x2x6+x3x5. +V. CONCLUSION +In this paper, we partially answered the open problem +provided by Tang et al. [17]. For the first time in the +literature, we proposed a direct construction of multiple +(2k+1, 2m, 2m+k+2)-ZCZ sequence sets having ZCCZ equals +to (Z + 1)/N = 2m−s using GBF. +TABLE I: Comparison of the proposed construction with [19], [20], [22], [23], [26]. +Ref. +Method +Parameter1 +Optimality2 +ZCCZ +No. of sets +Constraints +[20, Th. 1] +Indirect +(L, M − 1, LP) +No +2M − 1 +N +N = ⌊ T +M ⌋ > 1, L = KM, M > 1, K > 1 +[20, Th. 2] +Indirect +(T, M, TL) +No +TL +N +N = ⌊ T +M ⌋ > 1, L = KM, M > 1, K > 1 +[19] +Indirect +(M, M − 1, PM) +Yes +PM − 1 +N +N = ⌊ T +M ⌋, N > 1, M > 1 +[22] +Indirect +(L, P, TLP) +No +TLP +T +gcd(T, P) = 1, gcd(L, P) = 1(orL|PorP|L) +[23] +Indirect +(2M, Z, 2TP) +No +2TP +T +⌊ P −2 +Z ⌋ = M or ⌊ P −1 +Z ⌋ = M, Z ≤ 2 +[26] +Indirect +(N 2, N, N) +Yes +Z + 1 +M +N is order of DFT matrix, N = M(Z + 1) +This paper +Direct +(2k+1, 2m, 2m+k+2) +Yes +2m−s − 1 +2s +0 ≤ s ≤ k ≤ m − 2 +1 Parameter of each ZCZ sequence set. +2 Optimality of each ZCZ sequence set. + +5 +REFERENCES +[1] P. 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Xu, “Multiple complete complementary codes +with inter-set zero cross-correlation zone,” IEEE Trans. on Commun., +vol. 68, no. 3, pp. 1925–1936, 2020. +[32] J. A. Davis and J. Jedwab, “Peak-to-mean power control in OFDM, +Golay complementary sequences, and Reed-Muller codes,” IEEE Trans. +on inf. theory, vol. 45, no. 7, pp. 2397–2417, 1999. + diff --git a/3dA0T4oBgHgl3EQfNP-W/content/tmp_files/load_file.txt b/3dA0T4oBgHgl3EQfNP-W/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e874c6695837da6307f351d56d24fab6c593ef17 --- /dev/null +++ b/3dA0T4oBgHgl3EQfNP-W/content/tmp_files/load_file.txt @@ -0,0 +1,595 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf,len=594 +page_content='1 A Direct Construction of Near-Optimal Multiple ZCZ Sequence Sets Nishant Kumar, Sudhan Majhi, Senior Member, IEEE, and Ashish K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Upadhyay Abstract—In this paper, for the first time, we present a direct construction of multiple zero-correlation zone (ZCZ) sequence sets with inter-set zero-cross correlation zone (ZCCZ) from generalised Boolean function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' The presented ZCZ sequence sets are optimal and their union is near-optimal ZCZ sequence set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' This work partially settles the open problem introduced by Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' in their 2010 paper using direct construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' The proposed construction is presented by two layer graphical representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Finally, the construction is compared with existing state-of-the- art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Index Terms—Generalised Boolean function (GBF), zero-cross correlation zone (ZCCZ), zero-correlation zone (ZCZ), multiple ZCZ sequence sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' INTRODUCTION Z complementary pairs (ZCPs) were introduced by Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' [1] to overcome the limitation on the lengths of Golay complementary pairs (GCPs) [2]–[5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' The idea of ZCPs was generalized to Z-complementary code set (ZCCS) by Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' A ZCCS refers to a set of K codes, each of which consists of M constituent sequences of identical length L, having ideal aperiodic auto- and cross-correlation properties inside the ZCZ width (Z) [7], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' When Z = L and K = M, the set is called complete complementary code CCC [9]– [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' To reduce the “near-far effect” and ensure interference- free communication in asynchronous CDMA systems, ZCZ sequences were introduced in the late 1990s [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' When the received signal delays within ZCZ, ZCZ sequences can be employed to remove or reduce MAI and multipath interference (MPI) in quasi synchronous CDMA (QS-CDMA) systems [13], [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Although the ZCZ spreading sequences prevent co- channel interference within each cell, inter-cell interference across neighbouring cells is unavoidable [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' To address the aforementioned shortcoming, the idea of multiple ZCZ sequence sets with inter set zero-cross corre- lation zone (ZCCZ) has recently been proposed [16]–[23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' A multiple ZCZ sequence set comprises ZCZ sequence sets as its subsets and the cross-correlation function between two arbitrary sequences from different subsets has either ZCCZ or low cross-correlation zone (LCCZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Authors in [24] and [25] used generalised bent function and perfectly non-linear functions respectively to construct multiple ZCZ sequence sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' But they tend to achieve only multiple ZCZ sequence set with interset LCCZ instead of ZCCZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' In [26], authors presented construction of multiple ZCZ sequence sets using discrete Fourier transform (DFT) matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Furthermore, an asymmetric ZCZ (A-ZCZ) sequence set is a multiple ZCZ sequence set and the ZCCZ between two arbitrary sequences from distinct subsets has a large ZCCZ [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' To obtain A-ZCZ sequence sets, interleaving techniques on perfect sequences are presented in the literature [21]–[23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Since, perfect se- quences are available only for very few lengths therefore these constructions also have very limited lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Additionally, the DFT matrices [18]–[20] and Hadamard matrices [16] are also used to construct A-ZCZ sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' But, all these constructions are indirect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' The limitation of A-ZCZ sequence set is that the large ZCCZ is obtained at the cost of optimality of ZCZ sequence sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' In [17], Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' proposed a method for constructing multiple binary ZCZ sequence sets from mutually orthog- onal Golay complementary set (MOGCS) with good inter- set cross-correlation property and provided an open problem as* “we propose the following open problem: Construct N ZCZ sequence sets Zi, 0 ≤ i < N, satisfy: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Each Zi is an (K, Z, L)-ZCZ sequence set with KZ/L = 1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' The sets have a common zero correlation zone of length Zc with Zc = Z/N”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Motivated by the above open problem given in [17], in this letter, we propose a direct construction of near-optimal mul- tiple ZCZ sequence sets using generalised Boolean function (GBF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Since, proposed construction is based on GBFs, there- fore it is suitable for rapid hardware generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' A graphical analysis of our proposed construction has also been provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Also, it is the first time that a direct construction of multiple ZCZ sequence sets with ZCCZ is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' The proposed construction generalizes construction given in [13] and it is optimal over several constructions of A-ZCZ sequence sets presented in [16], [18]–[23], [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' NOTATIONS AND DEFINITIONS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Definition and Correlation Functions Let a1 = (a10, a11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , a1(L−1)) and a2 = (a20, a21, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , a2(L−1)) be two sequences of equal length L, having entries from complex numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' For an integer u, we define aperiodic cross-correlation function (ACCF) of sequences a1 and a2 as γ(a1, a2)(u) = ��L−1−u i=0 a1ia∗ 2(i+u), 0 ≤ u < L, �L+u−1 i=0 a1(i−u)a∗ 2i, −L < u < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' (1) Moreover, ACCF is termed as aperiodic auto-correlation func- tion (AACF) if a1 = a2 and denoted as γ(a1)(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Next, we define periodic cross-correlation function (PCCF) in terms of ACCF as φ(a1, a2)(u) = γ(a1, a2)(u) + γ∗(a2, a1)(L − u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' (2) The notations has been changed as per this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content='02144v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content='IT] 5 Jan 2023 2 Definition 1: Let C = {C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , CP −1} be a collection of P codes (matrices) having M rows and L columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Define Cη = [aη 0 aη 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' aη M−1]T M×L, (3) where aη ν (0 ≤ ν ≤ M − 1, 0 ≤ η ≤ P − 1) is the νth row sequence or νth constituent sequence and [·]T represents trans- pose of matrix [·].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Then the ACCF of two codes Cη1, Cη2 ∈ C is defined as γ(Cη1, Cη2)(u) = M−1 � ν=0 γ(aη1 ν , aη2 ν )(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' (4) Definition 2: Let C be a code set as defined in (3) which satisfies following correlation properties γ(Cη1, Cη2)(u) = � � � � � LM, η1 = η2 and u = 0, 0, η1 = η2 and 0 < |u| < L, 0, η1 ̸= η2 and |u| < L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' (5) Then C is known as (P, M, L)-MOGCS and each code in C is called GCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Moreover, if P = M then C is known as CCC set and denoted by (P, P, L)-CCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Definition 3: Let Zl = {zl 0, zl 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , zl K−1} be a collection of K L-length sequences, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=', zl i = (zl i0, zl i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , zl iL−1), 0 ≤ i ≤ K − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Then, Z is called (K, Z, L)-ZCZ sequence set if it satisfies following, φ(zl i, zl j)(u) = � � � � � 0, i = j and 1 ≤ |u|≤Z, 0, i ̸= j and 0 ≤ |u|≤Z, L, i = j and u = 0, (6) where 0 ≤ i, j ≤ K − 1 and Z is termed as ZCZ width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Definition 4: Let Z be a collection of N, (K, Z, L)-ZCZ sequence sets then Z = {Z1, Z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , ZN} is known as a multiple ZCZ sequence set with ZCCZ equal to Zc, if for 0 ≤ |u| < Zc, φ(zl i, zl′ j )(u) = 0, ∀1 ≤ l ̸= l′ ≤ N and 0 ≤ i, j ≤ K − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Definition 5 (Tang-Fan-Matsufuji Bound [28]): Let Z be any (K, Z, L)-ZCZ sequence set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Then, KZ ≤ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' If for any Z, KZ = L (or K(Z + 1) = L) then Z is called optimal (or near-optimal) ZCZ sequence set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' However, in case of binary ZCZ sequence set the bound is reduced to 2KZ ≤ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Generalised Boolean Function (GBF) [29] We define a complex valued sequence corresponding to a GBF, f : {0, 1}m −→ Zq of m variables as Ψ(f) = � ωf0, ωf1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , ωf2m−1� , (7) where fj = f(j0, j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , jm−1), ω = exp � 2π√−1/q � , and (j0, j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , jm−1) is the binary vector representation of j, where as in the remainder of this letter, q is an even integer not less than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Corresponding to a GBF f with m variables the sequence Ψ(f) is of length 2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Definition 6: Let J = {j0, j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , jk−1} ⊂ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , n−1} and xJ = [xj0, xj1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , xjk−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' For a constant e ∈ {0, 1}k, f|xJ=e is known as restriction of f over e and is obtained by substituting xjβ = eβ (β = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=', k − 1) in the function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Moreover, the sequence Ψ(f|xJ=e) is the same as sequence Ψ(f) of length 2m except for the positions ijβ ̸= eβ for each 0 ≤ β < k, at these positions Ψ(f|xJ=e) has the zero entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Quadratic Forms and Graphs [30] Let f be GBF of order r over m variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' If f|xJ=e is a quadratic GBF, then graph of f|xJ=e, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=', G(f|xJ=e) has vertex set V , where V = {x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , xm−1}\\{xj0, xj1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , xjk−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' If there is a term qβ1β2xβ1xβ2 (0 ≤ β1 < β2 < m, xβ1, xβ2 ∈ V ) in the GBF f|xJ=e with qβ1β2 ̸= 0 (qβ1β2 ∈ Zq) then by connecting the vertices xβ1 and xβ2 by an edge, the graph G(f|xJ=e) can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' For k = 0, the graph of f|xJ=e is the same as that of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Generalized Reed-Muller Codes Definition 7: Let q ≥ 2 and 0 ≤ r ≤ m,, then a linear code over Zq generated by the Zq-valued sequences corresponding to the monomials of degree at most r in x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , xm−1 is said to be rth order generalised Reed-Muller (RM) code and denoted as RMq(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' The Existing Construction of Multiple CCCs Lemma 1 ( [31]): Let m, k, and s are integers with 0 ≤ s ≤ k ≤ m−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Define Js = {jk−s, jk−s+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , jk−1} = {m − s, m − s + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , m − 1}, J = {j0, j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , jk−1−s} ⊂ Zm−s, I = {i0, i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , im−k−1} = Zm−s\\J, x = � xj0, xj1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , xjk−s−1 � , xs = � xjk−s, xjk−s+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , xjk−1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Let π be a permutation on symbols {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , m − k − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Let f be a quadratic GBF over the m variables x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , xm−1, such that for e ∈ {0, 1}k−s, f|x=e = Q + m−k−1 � β=0 uβxiβ + s−1 � β=0 vβxjk−s+β + v, (8) where Q = q 2 m−k−2 � β=0 xiπ(β)xiπ(β+1), (9) uβ ∈ Zq ∀ 0 ≤ β ≤ m−k −1, vβ ∈ Zq ∀ 0 ≤ β ≤ s−1, and v ∈ Zq Let γ1 and γ2 be two end vertices of the path G(Q), t1 = �s−1 β=0 bk+1+β2β, t2 = �k β=0 bβ2β, where bβ ∈ {0, 1} for 0 ≤ β ≤ k +s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' For the natural order generated by (t1, t2), Define the set S(t1,t2) by � � �f + q 2 � � k−1 � β=0 dβxjβ + dxγ1 + k−1 � β=0 bβxjβ +bkxγ2 + k−1 � β=k−s dβbs+1+β � � : dβ, d ∈ {0, 1} � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' (10) Let St1 = � S(t1,t2) : 0 ≤ t2 ≤ 2k+1 − 1 � , 0 ≤ t1 ≤ 2s − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Then {St1 : 0 ≤ t1 ≤ 2s − 1} is a collection of 2s CCCs, and any two GCSs from different CCCs St1 and St′ 1 with 0 ≤ t1 ̸= t′ 1 ≤ 2s − 1 have a ZCCZ of width 2m−s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' For the fixed values of t1 and t2, S(t1,t2) is a GCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Let us denote, S(t1,t2) = � s(t1,t2) 0 s(t1,t2) 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' s(t1,t2) 2k+1−1 �T , (11) 3 where s(t1,t2) ν (0 ≤ ν ≤ 2k+1 − 1) is νth row sequence of S(t1,t2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Lemma 2 ( [13]): Let q = 2 and x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , xk, xk+1 be k+ 2 binary variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Also, let h be a Boolean function defined on x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , xk, xk+1 as follow h = k+1 � β=1 cβxβx0 + � 1≤µ<ν≤k dµνxµxν + k+1 � α=0 eαxα + e′, (14) where ck+1 = 1, cβ ∈ Z2 for 1 ≤ β ≤ k, dµν, eα, e′ ∈ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Let h denotes the binary vector corresponding to function h, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=', h = [h0, h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , h2k+2−1] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' For 0 ≤ τ ≤ 2k+1 − 1, we have (−1)hτ +hτ+1 + (−1)hτ+2k+1+hτ+1+2k+1 = 0, (15) where the operation in the subscripts is done in modulo 2k+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' PROPOSED CONSTRUCTION In this section, we provide a GBF which generates the required multiple ZCZ sequence sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Theorem 1: Let x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , xm+k+1 are m + k + 2 binary variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Define a GBF f(x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , xm−1) on m variables same as in Lemma 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=', removing J = {j0, j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , jk−1−s} having k − s vertices from the graph of f results in s isolated vertices in Js and a path on m−k vertices in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Define another GBF h(xm, xm+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , xm+k+1) on k + 2 variables as h = k+1 � r=1 crxm+rxm + � 2≤µ<ν≤t dµνxm+µxm+ν + k+1 � β=1 eβxm+β + e′, (16) where ck+1 ̸= 0, cr ∈ Z2 for 1 ≤ r ≤ k, dµν, eβ ∈ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' For a fixed value of t1, define the set Zt1 = {Ψ(zt1 t2) : 0 ≤ t2 ≤ 2k+1 − 1} by � � �f + h + q 2 � � k−1 � β=0 xm+βxjβ + xm+kxγ1 + k−1 � β=0 bβxjβ +bkxγ2 + k−1 � β=k−s xm+βbs+1+β � � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' (17) Then Z = � Zt1 : 0 ≤ t1 ≤ 2s − 1} is a collection of 2s (2k+1, 2m ,2m+k+2)-ZCZ sequence sets having ZCCZ equals to 2m−s − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Proof: Using (10), (11), (17) and taking natural order generated by t2, we get Zt1 = � Zt1 0 , Zt1 1 � , where � Zt1 0 , Zt1 1 � is horizontal concatenation of matrices Zt1 0 and Zt1 1 and these matrices are defined as, Zt1 0 = � ������ s(t1,0) 0 ωk0 s(t1,0) 1 ωk1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' s(t1,0) l−1 ωkl−1 s(t1,1) 0 ωk0 s(t1,1) 1 ωk1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' s(t1,1) l−1 ωkl−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' s(t1,l−1) 0 ωk0 s(t1,l−1) 1 ωk1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' s(t1,l−1) l−1 ωkl−1 � ������ , Zt1 1 = � ������ s(t1,0) 0 ωkl s(t1,0) 1 ωkl+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' s(t1,0) l−1 ωk2l−1 s(t1,1) 0 ωkl s(t1,1) 1 ωkl+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' s(t1,1) l−1 ωk2l−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' s(t1,l−1) 0 ωkl s(t1,l−1) 1 ωkl+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' s(t1,l−1) l−1 ωk2l−1 � ������ , where l = 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Now, we need to prove that Zt1 is a (2k+1, 2m , 2m+k+2)-ZCZ sequence set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' For 0 ≤ i, j ≤ 2k+1−1, periodic correlation of Ψ(zt1 i ) and Ψ(zt1 j ) at any time shift 0 ≤ τ ≤ 2m is given by (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Next, by (15), (12) and aperiodic sum property of CCCs, we get, φ(Ψ(zt1 i ), Ψ(zt1 j ))(τ) = 2 · 2k+1−1 � m=0 γ � ci m, cj m � (τ) (18) = � 2k+m+2, if τ = 0 and i = j, 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Which proves that Zt1 is a (2k+1, 2m, 2m+k+2)-ZCZ sequence sets ∀ 0 ≤ t1 ≤ 2s − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Now, let 0 ≤ t1 ̸= t′ 1 < 2s and 0 ≤ i, j ≤ 2k+1 − 1 then for 0 ≤ τ ≤ 2m−s − 1, the value of φ(Ψ(zt1 i ), Ψ(zt′ 1 j ))(τ) is given by (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Now, by (15), (13) and ZCCZ property of CCCs in Lemma 1, we get φ(Ψ(zt1 i ), Ψ(zt′ 1 j ))(τ) = 0, ∀ 0 ≤ τ ≤ 2m−s − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Remark 1: Theorem 1 constructed 2s ZCZ sequence sets with parameter (2k+1, 2m, 2m+k+2) having common ZCZ equals to 2m−s − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Since 2k+1 · 2m/2m+k+2 = 1/2 and Zc = 2m−s − 1 = (Z + 1)/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' φ(Ψ(zt1 i ), Ψ(zt1 j ))(τ) =2 · 2k+1−1 � m=0 γ � s(t1,i) m , s(t1,j) m � (τ) + [(−1)hl−1+hl + (−1)h2l−1+h0]γ∗ � s(t1,j) 0 , s(t1,i) 2l−1 � (L − τ) + 2l−2 � m=0 [(−1)hm+hm+1 + (−1)hm+2l+hm+1+2l]γ∗ � s(t1,j) m+1 , s(t1,i) m � (L − τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' (12) φ(Ψ(zt1 i ), Ψ(zt′ 1 j ))(τ) =2 · l−1 � m=0 γ � s(t1,i) m , s(t′ 1,j) m � (τ) + [(−1)hl−1+hl + (−1)h2l−1+h0]γ∗ � s(t′ 1,j) 0 , s(t1,i) 2l−1 � (L − τ) + l−2 � m=0 [(−1)hm+hm+1 + (−1)hm+l+hm+1+l]γ∗ � s(t′ 1,j) m+1 , s(t1,i) m � (L − τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' (13) 4 Remark 2: Since the set of isolated vertices in Theorem 1 contribute to multipleness of constructed multiple ZCZ sequence set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Hence, if we put s = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=', Js = φ in Theorem 1 then our construction reduces to construction presented in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Therefore, construction provided in [13] is a special case of the proposed construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Corollary 1: Collection of all the ZCZ sequences in Theo- rem 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=', {Ψ(zt1 t2) : 0 ≤ t2 ≤ 2k+1 − 1, 0 ≤ t1 ≤ 2s − 1} is a near-optimal (2k+s+1, 2m−s − 1, 2m+k+2)-ZCZ sequence set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Proof: Directly follows from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Remark 3: It is the first time in the literature that the direct construction of optimal multiple ZCZ sequence sets is provided such that their union is a near-optimal ZCZ sequence set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Which makes our construction advantageous over several constructions of A-ZCZ sequence sets which are presented in the literature [16], [18]–[23], [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' The detailed comparison of the proposed work is provided in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Remark 4: From equation (17), it can be seen that the proposed multiple ZCZ sequence sets are obtained from sec- ond order cosets of generalised RM code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Since, RM codes have efficient encoding, good error correction properties and important practical advantage of being easy to decode [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Hence, our proposed construction has advantage over any other non-GBF based construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' GRAPHICAL INTERPRETATION OF THE PROPOSED CONSTRUCTION This section interprets the proposed construction with graphical point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' 1 depicts a graphical repre- .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' 𝑖𝜋(0) 𝑖𝜋(𝑚−𝑘−1) 𝑖𝜋(1) 𝑖𝜋(2) 𝑗𝑘−1−𝑠 𝑗0 m-s m−1 m m+k-1-s I J J s Lower Layer Upper Layer .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' m+k m+k-1 m+k-s m+k+1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' 1: Graphical representation of (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' sentation of (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' The graph has a two-layered structure with a horizontal straight line which is separating the upper and bottom layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' The upper layer and lower layer correspond to graphs of Boolean functions f and h respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' These layers are interconnected through the set of edges {xj0xm, xj1xm+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , xjk−1−sxm+k−1−s, xm−sxm+k−s, xm−s+1xm+k−s+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , xm−1xm+k−1}, and the vertex xm+k is connected to any of the end vertices of the path in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Interestingly, the ZCZ of each ZCZ sequence set is equals to the power of number of vertices in the upper layer of the graph and ZCCZ of ZCZ sequence sets equals to one less than the power of number of vertices in the upper layer of graph except isolated vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Example 1: Let m = 4, q = 2, s = 1, and k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Assume J = {0}, Js = {3}, I = {1, 2} and GBFs f = x0x1 + x0x2 + x0x3 + x1x2 + x1 + x2, h = x4x5 + x4x6 + x4x7 + x4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' (19) Generate two sequence sets Z0 and Z1 as Z0 = {Ψ(f+h+x0x4+x2x6+x3x5+b0·x0+b1·x3+b2·x1 + 0 · x5) : b0, b1, b2 ∈ Z2}, Z1 = {Ψ(f +h+x0x4+x2x6+x3x5+b0·x0+b1·x3+b2·x1 + 1 · x5) : b0, b1, b2 ∈ Z2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' (20) Then Z0 and Z1 are two optimal (8, 16, 256)-ZCZ sequence sets having inter-set ZCCZ equals to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Moreover, Z = Z0∪Z1 is also an optimal (16, 7, 256)-ZCZ sequence set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' 2, a graph corresponding to quadratic form, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=', f + h + x0x4 + x2x6 + x3x5 of Example 1 is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' 5 6 4 7 2 1 0 3 I J J s Upper Layer Lower Layer Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' 2: Graphical representation of f +h+x0x4+x2x6+x3x5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' CONCLUSION In this paper, we partially answered the open problem provided by Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' For the first time in the literature, we proposed a direct construction of multiple (2k+1, 2m, 2m+k+2)-ZCZ sequence sets having ZCCZ equals to (Z + 1)/N = 2m−s using GBF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' TABLE I: Comparison of the proposed construction with [19], [20], [22], [23], [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Method Parameter1 Optimality2 ZCCZ No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' of sets Constraints [20, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' 1] Indirect (L, M − 1, LP) No 2M − 1 N N = ⌊ T M ⌋ > 1, L = KM, M > 1, K > 1 [20, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' 2] Indirect (T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' TL) No TL N N = ⌊ T M ⌋ > 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' L = KM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' M > 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' K > 1 [19] Indirect (M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' M − 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' PM) Yes PM − 1 N N = ⌊ T M ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' N > 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' M > 1 [22] Indirect (L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' TLP) No TLP T gcd(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' P) = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' gcd(L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' P) = 1(orL|PorP|L) [23] Indirect (2M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' 2TP) No 2TP T ⌊ P −2 Z ⌋ = M or ⌊ P −1 Z ⌋ = M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Z ≤ 2 [26] Indirect (N 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' N) Yes Z + 1 M N is order of DFT matrix,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' N = M(Z + 1) This paper Direct (2k+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' 2m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' 2m+k+2) Yes 2m−s − 1 2s 0 ≤ s ≤ k ≤ m − 2 1 Parameter of each ZCZ sequence set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' 2 Optimality of each ZCZ sequence set.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Davis and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Jedwab, “Peak-to-mean power control in OFDM, Golay complementary sequences, and Reed-Muller codes,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' on inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' 45, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' 2397–2417, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} diff --git a/4NE1T4oBgHgl3EQfAgLJ/content/tmp_files/load_file.txt b/4NE1T4oBgHgl3EQfAgLJ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2fedb5949cb549640e4e4b7deb640d71f1b80e55 --- /dev/null +++ b/4NE1T4oBgHgl3EQfAgLJ/content/tmp_files/load_file.txt @@ -0,0 +1,961 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf,len=960 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='02841v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='DS] 7 Jan 2023 LEVEL-2 LARGE DEVIATION PRINCIPLE FOR COUNTABLE MARKOV SHIFTS WITHOUT GIBBS STATES HIROKI TAKAHASI Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We consider level-2 large deviations for the one-sided countable full shift without assuming the existence of Bowen’s Gibbs state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' To deal with non- compact closed sets, we provide a sufficient condition in terms of inducing which ensures the exponential tightness of a sequence of Borel probability measures constructed from periodic configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Under this condition we establish the level-2 Large Deviation Principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We apply our results to the continued fraction expansion of real numbers in [0, 1) generated by the R´enyi map, and obtain the level-2 Large Deviation Principle, as well as a weighted equidistribution of a set of quadratic irrationals to equilibrium states of the R´enyi map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Introduction Dynamical systems (iterated maps) equipped with finite Markov partitions are represented as finite Markov shifts, and the construction of relevant invariant mea- sures and the investigation of their statistical properties are done on the symbolic level, with adaptations of ideas in statistical mechanics (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=', [4, 5, 29, 32, 39]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' This thermodynamic formalism initiated in the 60s has been successfully extended to maps with infinite Markov partitions and shift spaces with countably infinite number of states (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=', [1, 6, 10, 11, 18, 30, 31, 41]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' This paper is concerned with level-2 large deviations for such countable Markov shifts, and its application to a dynamical system related to number theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The theory of large deviations aims to characterize limit behaviors of probability measures in terms of rate functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let X be a topological space, and let M(X ) denote the space of Borel probability measures on X endowed with the weak* topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We say a sequence {˜µn}∞ n=1 in M(X ) satisfies the Large Deviation Prin- ciple (LDP) if there exists a lower semicontinuous function I : X → [0, ∞] such that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1) lim inf n→∞ 1 n log ˜µn(G) ≥ − inf G I for any open set G ⊂ X , and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='2) lim sup n→∞ 1 n log ˜µn(C) ≤ − inf C I for any closed set C ⊂ X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We call x ∈ X a minimizer if I(x) = 0 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The set of minimizers is a closed set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The LDP means that in the limit n → ∞ the measure ˜µn assigns all but Date: January 10, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' 37A44, 37A50, 37A60, 60F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Keywords: thermodynamic formalism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Gibbs state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Large Deviation Principle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' periodic points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' equidistribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' 1 2 HIROKI TAKAHASI exponentially small mass to the set of minimizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The function I is called a rate function, and called a good rate function if its level set {x ∈ X : I(x) ≤ α} is compact for any α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' If X is a metric space and {˜µn}∞ n=1 satisfies the LDP, the rate function is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The setup in our mind is that X is the space of Borel probability measures on a topological space X on which a Borel map σ: X → X acts, and each ˜µn ∈ M(X ) is given in terms of empirical measures δn x = (1/n) �n−1 k=0 δσkx, where δσkx ∈ X denotes the unit point mass at σkx ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We refer to the LDP in this setup as level-2 [8, Chapter 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For topologically mixing finite Markov shifts together with H¨older continuous potentials, the level-2 LDP for empirical distributions and that for sequences con- structed from empirical measures on periodic orbits were established in [15, 22, 33] and [16] respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' A key ingredient in these classical cases is the existence of Bowen’s Gibbs states [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' With the aid of Bowen’s Gibbs states, one can deduce the lower bound (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1) by combining Birkhoff’s and Shannon-McMillan-Breiman’s theorems, and the upper bound (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='2) by modifying the standard proof of the variational principle [40] (see [33]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For countable Markov shifts, Bowen’s Gibbs states were constructed under the assumption of a good regularity of potentials and a strong connectivity of transition matrices defining the shift spaces (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=', [1, 6, 11, 18, 30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Several level-2 LDPs were established in [34] under the existence of Bowen’s Gibbs states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' It has been realized that not all dynamically relevant invariant probability mea- sures correspond to Bowen’s Gibbs states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' One of the best known examples is the absolutely continuous invariant probability measure of an interval map of Manneville-Pomeau type, with finitely many branches and a neutral fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Such a measure still retains a weak form of Bowen’s Gibbs state [41], and has the weak Gibbsian property in statistical mechanics sense [9, 17, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For a ther- modynamic formalism and level-2 large deviations for a class of this map, see [12, 28, 41] and [25, 26] respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' With these historical developments and the abundance of interesting dynamical systems modeled by countable Markov shifts without Bowen’s Gibbs states (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=', [13, 43]), it is important to establish the level-2 LDP for countable Markov shifts without assuming the existence of Bowen’s Gibbs states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' A main new difficulty for countable Markov shifts is a treatment of non-compact closed sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We say a sequence {˜µn}∞ n=1 of Borel probability measures on a non- compact space X is exponentially tight if for any L > 0 there exists a compact set K ⊂ X such that lim sup n→∞ 1 n log ˜µn(X \\ K) ≤ −L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' If {˜µn}∞ n=1 is exponentially tight, then the upper bound (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='2) for any compact closed set implies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='2) for any closed set which is not necessarily compact, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=', [7] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The proof of the level-2 LDPs in [34] relies on the existence of Bowen’s Gibbs states in order to verify the exponential tightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Our strategy for countable Markov shifts without Bowen’s Gibbs states is to use inducing to verify the exponential tightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Inducing is a familiar procedure in ergodic theory originally considered in works by Kakutani, Rohlin and others, and LEVEL-2 LDP FOR COUNTABLE MARKOV SHIFTS WITHOUT GIBBS STATES 3 was used in the construction of absolutely continuous invariant measures or Gibbs- equilibrium states (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=', [2, 3, 23, 24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' An inducing scheme we use here is given by the first return map to an a priori fixed domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' In terms of this inducing, we will formulate a sufficient condition which ensures the exponential tightness for the original system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' A key concept is that of local Gibbs states introduced in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Statements of results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Throughout the rest of this paper, let N denote the discrete set of positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let X denote the one-sided infinite Cartesian product topological space of N, called a countable full shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The topology of X has a base that consists of cylinders [p1 · · · pn] = {x = (xn)∞ n=1 ∈ X : xk = pk for every k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' , n}}, where n ≥ 1 and p1 · · ·pn ∈ Nn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' This topology is metrizable with the metric d(x, y) = exp (− inf{n ≥ 1: xn ̸= yn}) where exp(−∞) = 0 by convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let σ denote the left shift acting on X: (σx)n = xn+1 for n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let φ: X → R be a function, called a potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We say φ is acceptable if it is uniformly continuous and satisfies sup p∈N � sup [p] φ − inf [p] φ � < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We say φ is locally H¨older continuous if there exist C > 0 and α ∈ (0, 1] such that for any p ∈ N and all x, y ∈ [p], |φ(x) − φ(y)| ≤ Cd(x, y)α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Clearly, if φ is locally H¨older continuous then it is acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For each n ≥ 1 we write Snφ for the Birkhoff sum �n−1 k=0 φ ◦ σk, and introduce a pressure (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='3) P(φ) = lim n→∞ 1 n log � p1···pn∈Nn sup [p1···pn] exp Snφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' This limit exists by the sub-additivity [4, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='18], which is never −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let φ: X → R be acceptable and satisfy P(φ) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We consider a sequence {˜µn}∞ n=1 of Borel probability measures on M(X) given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='4) ˜µn = 1 Zn(φ) � x∈En exp Snφ(x)δδn x, where En = {x ∈ X : σnx = x}, and δδn x denotes the unit point mass at δn x, and Zn(φ) the normalizing constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' In dynamical systems terms, En is the set of periodic points of period n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' In statistical mechanics terms, the measure ˜µn is closely related to the canonical ensemble subject to a periodic boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' An inducing scheme consists of a subset X∗ of X of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='5) X∗ = X \\ � p∈N∩[1,p∗−1] [p], 4 HIROKI TAKAHASI where p∗ ≥ 2, and a function R: X∗ → N ∪ {∞} given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='6) R(x) = inf{n ≥ 1: σnx ∈ X∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Given an inducing scheme (X∗, R) we define an induced map (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='7) τ : X∗ ∩ ∞ � k=1 σ−kX∗ �→ σR(x)x ∈ X∗, and an inducing domain (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='8) Σ = ∞ � n=0 τ −n � X∗ ∩ ∞ � k=1 σ−kX∗ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' In other words, τ is the first return map to X∗ and Σ is the domain on which τ can be iterated infinitely many times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We call (Σ, τ|Σ) an induced system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Given a potential φ: X → R, we introduce a parametrized family of induced potentials Φγ : Σ → R (γ ∈ R) by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='9) Φγ(x) = SR(x)φ(x) − γR(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' As shown in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1, the induced system has a countably infinite partition that conjugates the system to the countable full shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The local H¨older continuity of the induced potential Φγ and its pressure P(Φγ) are well-defined in terms of this conjugacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Our main result is stated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Theorem A (the level-2 Large Deviation Principle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let φ: X → R be acceptable and satisfy P(φ) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Assume there exists an induced system for which the induced potentials Φγ, γ ∈ R are locally H¨older continuous, and there exists γ0 ∈ R such that P(Φγ0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Then {˜µn}∞ n=1 is exponentially tight and satisfies the LDP with the good rate function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let us define the rate function in Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let M(X, σ) denote the set of σ- invariant elements of M(X) and let Mφ(X, σ) = {µ ∈ M(X, σ): � φdµ > −∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Define Fφ : M(X) → [−∞, 0] by Fφ(µ) = � h(µ) + � φdµ − P(φ) if µ ∈ Mφ(X, σ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' −∞ otherwise, where h(µ) ∈ [0, ∞] denotes the measure-theoretic entropy of µ with respect to σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since φ is acceptable and P(φ) < ∞, sup φ < ∞ is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For each µ ∈ Mφ(X, σ), � φdµ is finite [18, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='9] and we have h(µ) + � φdµ ≤ P(φ) < ∞, and so h(µ) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' If φ is acceptable, then we have P(φ) = sup � h(µ) + � φdµ: µ ∈ Mφ(X, σ) � , known as the variational principle [18, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' A measure µ ∈ Mφ(X, σ) which attains this supremum is called an equilibrium state for the potential φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The rate function Iφ : M(X) → [0, ∞] in Theorem A is given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='10) Iφ(µ) = − inf G∋µ sup G Fφ, LEVEL-2 LDP FOR COUNTABLE MARKOV SHIFTS WITHOUT GIBBS STATES 5 where the infimum is taken over all open subsets G of M(X) containing µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since the entropy is not upper semicontinuous on M(X, σ), Iφ may not be equal to −Fφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since the sequence {˜µn}∞ n=1 in Theorem A is exponentially tight, it is tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' By Prohorov’s theorem, it has a limit point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since the rate function Iφ in Theorem A is the good rate function, there exists at least one minimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' If the minimizer is unique, we obtain a “level-2 weighted equidistribution of elements of �∞ n=1 En toward minimizers”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Theorem B (level-2 weighted equidistribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let φ: X → R be acceptable and satisfy P(φ) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Assume there exists an induced system for which the induced potentials Φγ, γ ∈ R are locally H¨older continuous, and there exists γ0 ∈ R such that P(Φγ0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Assume that the minimizer of the rate function Iφ is unique, denoted by µmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For any bounded continuous function ˜ϕ: M(X) → R, lim n→∞ 1 Zn(φ) � x∈En exp Snφ(x) ˜ϕ(δn x) = ˜ϕ(µmin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Under the assumption of Theorem A, minimizers are not always unique, and not always an equilibrium state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' A sufficient condition was given in [35] which ensures that minimizers are equilibrium states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Taking various continuous functions ˜ϕ in Theorem B, we obtain convergences of various time averages over the elements of En.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let C(X) denote the set of real-valued bounded continuous functions on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Corollary (Inspired by Olsen [21, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Under the assumption of Theo- rem B, assume moreover the minimizer is unique, denoted by µmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' (a) For all ϕ, ψ ∈ C(X), lim n→∞ 1 Zn(φ) � x∈En exp Snφ(x) 1 n2Snϕ(x)Snψ(x) = � ϕdµmin � ψdµmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' (b) For ϕ, ψ ∈ C(X) with inf ψ > 0, lim n→∞ 1 Zn(φ) � x∈En exp Snφ(x)Snϕ(x) Snψ(x) = � ϕdµmin � ψdµmin .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' (c) For π1, π2 ∈ C(X) and a bounded continuous function f : R → R, lim n→∞ 1 Zn(φ) � x∈En exp Snφ(x) 1 n2 n−1 � k1,k2=0 f(π1(σk1x) + π2(σk2x)) = � fd(µmin ◦ π−1 1 ⊗ µmin ◦ π−1 2 ), where ⊗ denotes the convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Apply Theorem B to the bounded continuous functions µ ∈ M(X) �→ � ϕdµ � ψdµ, µ ∈ M(X) �→ � ϕdµ/ � ψdµ, µ ∈ M(X) �→ � fd(µ ◦ π−1 1 ⊗ µ ◦ π−1 2 ) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' □ 6 HIROKI TAKAHASI 1 0 1/2 2/3 1 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The graph of the R´enyi map T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Our results can be applied to dynamical systems modeled by the countable full shift without Bowen’s Gibbs state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The assumption in Theo- rem A can be verified, for example, for the infinite Manneville-Pomeau map [13, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='2], and the two-dimensional conformal maps in [43, Section 5] related to number theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Minimizers of the associated rate functions are not unique, and so Theorem B does not apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Further applications of different taste will be given in our forthcoming paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' A prime example to which our results apply is the R´enyi map T : [0, 1) → [0, 1) given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='11) T(ξ) = 1 1 − ξ − � 1 1 − ξ � , where ⌊·⌋ denotes the floor function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The graph of T is obtained by reversing the graph of the well-known Gauss map ξ ∈ (0, 1] → 1/ξ − ⌊1/ξ⌋ ∈ [0, 1) around the axis {ξ = 1/2}, as shown in FIGURE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The map T leaves invariant the absolutely continuous infinite measure dx/x, and x = 0 is its neutral fixed point:T(0) = 0, T ′(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The asymptotic distribution of typical orbits, in the Lebesgue measure sense, are concentrated on this neutral fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The iteration of T generates an infinite continued fraction expansion of each number ξ ∈ [0, 1) of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='12) ξ = 1 − 1 d1(ξ) − 1 d2(ξ) − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' , where dn(ξ) = ⌊1/(1 − T n−1(ξ))⌋ + 1 ≥ 2 for n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Using the infinite Markov partition {Jp}p∈N, Jp = [1 − 1/p, 1 − 1/(p + 1)) of [0, 1), one can represent T as the left shift acting on X [13, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The map (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='13) π: (xn)∞ n=1 ∈ X �→ π((xn)∞ n=1) ∈ ∞ � n=1 T −n+1(J(xn)) ⊂ [0, 1) is a well-defined homeomorphism onto its image satisfying T ◦ π = π ◦ σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We consider the potential φ = − log |T ′ ◦ π|, where T ′ denotes the derivative of T which is one-sided at boundary points of the Markov partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' From the mean LEVEL-2 LDP FOR COUNTABLE MARKOV SHIFTS WITHOUT GIBBS STATES 7 value theorem applied to the inverse branches of T, for any p ≥ 1 and all ξ, η ∈ Jp we have log |T ′(ξ)| |T ′(η)| ≤ 2|T(ξ) − T(η)| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' In particular, φ is acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since sup[p] eφ is comparable to p−2, P(βφ) < ∞ holds if and only if � p∈N p−2β is finite, which is equivalent to β > 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' It is easy to see that for n ≥ 1, T n maps [0, 1/(n + 1)) diffeomorphically onto [0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The mean value theorem implies limn→∞ sup[0,1/(n+1)) |(T n)′| = ∞, while |(T n)′(0)| = 1 for n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' It follows that for any β > 1/2 there is no Bowen’s Gibbs state for the potential βφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Meanwhile, it is known [13] that for β > 1/2, the equilibrium state for βφ is unique, which we denote by µβφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For 1/2 < β < 1, µβφ has positive entropy and fully supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For β ≥ 1, µβφ is the unit point mass at π−1(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let I denote the set of irrational numbers in (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The set En corresponds to the set of numbers in I ∪ {0} for which the continued fraction (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='12) is periodic of period n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' As in the proof of [14, Theorem 28], one can show that any number in �∞ n=1{ξ ∈ I: T n(ξ) = ξ} is a quadratic irrational, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=', an irrational root of a quadratic polynomial with integer coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Conversely, any quadratic irrational in I has an eventually periodic continued fraction of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='12), see [20, Theorem 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' An induced system as in Theorem A is obtained from the first return map to the interval (1/2, 1) not containing the neutral fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' From Theorems A and B we obtain the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For ξ ∈ [0, 1) and n ≥ 1, let δn ξ denote the empirical measure (1/n) �n−1 k=0 δT k(ξ) on [0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For any 1/2 < β ≤ 1, the sequence of Borel probability measures on M(π(X)) given by 1 Zn(βφ) � ξ∈I∪{0} T n(ξ)=ξ |(T n)′(ξ)|−βδδn ξ for n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' satisfies the LDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The minimizer is unique and it is the unit point mass at µβφ◦π−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Moreover, for any bounded continuous function ˜ϕ: M(π(X)) → R we have lim n→∞ 1 Zn(βφ) � ξ∈I∪{0} T n(ξ)=ξ |(T n)′(ξ)|−β ˜ϕ(δn ξ ) = ˜ϕ(µβφ ◦ π−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Structure of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The rest of this paper consists of two sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' In Section 2 we verify the exponential tightness of the sequence in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='4) under the assumption of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' In Section 3 we complete proofs of all the theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We close with a remark on possible generalizations of the main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Exponential tightness The aim of this section is to verify the exponential tightness of the sequence in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1 we start with a symbolic representation of the induced system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='2 we introduce the notion of local Gibbs states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='3 we prove a main technical estimate assuming the existence of a local Gibbs state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Using this 8 HIROKI TAKAHASI estimate, we verify the exponential tightness in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='5 we show that the assumption of Theorem A implies the existence of a local Gibbs state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Symbolic representation of the induced system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For a set S and an integer j ≥ 1, let Sj denote the set of words of elements of S of word length j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We introduce an empty word ∅ and set S0 = {∅}, a∅ = a = ∅a, a∅b = ab for a, b ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We set W(S) = � j≥1 Sj, N0 = N ∪ {0} and W0(S) = W(S) ∪ S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let (X∗, R) be an inducing scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let N∗ = N ∩ [p∗, ∞) and N∗ = N ∩ [1, p∗ − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For each p ∈ N∗ and ω ∈ W0(N∗), the set � q∈N∗[pωq] is mapped by the induced map τ in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='7) bijectively onto X∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since the domain X∗∩�∞ k=1 σ−kX∗ of definition of τ is partitioned into countably infinite sets of this form, the induced system τ|Σ is represented as the countable full shift over the infinite alphabet (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1) A = � � q∈N∗ [pωq]: p ∈ N∗ and ω ∈ W0(N∗) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' To make this statement into a rigorous one, we endow A with the discrete topology, and consider the countable full shift AN = {z = (zn)∞ n=1: zn ∈ A for every n ≥ 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We use bold letters to denote elements of W(A), and a double square bracket [[·]] to denote cylinders in AN: the n-cylinder (n ≥ 1) spanned by a = a1 · · · an ∈ An is [[a]] = {z = (zk)∞ k=1 ∈ AN : zk = ak for every k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' , n}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' By definition, for each k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' , n} we have ak = � q∈N∗[pkωjkq] where pk ∈ N∗, jk ∈ N0, ωjk ∈ Njk ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let ∥a∥ denote the word length of p1ωj1p2ωj2 · · · pnωjn in W(N), namely ∥a∥ = n + j1 + · · · + jn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let [a] denote the corresponding ∥a∥-cylinder in X, namely [a] = [p1ωj1p2ωj2 · · · pnωjn] ⊂ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' It is easy to check that a coding map Π: AN → Σ given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='2) Π: (zn)∞ n=1 ∈ AN �→ Π((zn)∞ n=1) ∈ ∞ � n=1 [z1 · · · zn] ⊂ Σ is well-defined, and is a homeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let θ denote the left shift acting on AN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Clearly we have Π ◦ θ = τ|Σ ◦ Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The following notation will be frequently used later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For a = a1 · · · an ∈ An as above with [a] = [p1ωj1p2ωj2 · · · pnωjn] and q ∈ N∗, let aq = p1ωj1p2ωj2 · · · pnωjnq ∈ W(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let (Σ, τ|Σ) be an induced system and let Π be the coding map in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' LEVEL-2 LDP FOR COUNTABLE MARKOV SHIFTS WITHOUT GIBBS STATES 9 (a) For every a ∈ W(A), Π[[a]] = Σ ∩ � q∈N∗ [aq].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' (b) If a, b ∈ W(A) satisfy ∥a∥ = ∥b∥, then a = b or [[a]] ∩ [[b]] = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' If n ≥ 1, a ∈ An then �n−1 k=0 R ◦ τ k equals ∥a∥ on Π[[a]], which implies (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' If ∥a∥ = ∥b∥ and a ̸= b then (a) implies Π[[a]] ∩ Π[[b]] = ∅, and so [[a]] ∩ [[b]] = ∅, which verifies (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Local Gibbs states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let φ: X → R satisfy P(φ) < ∞ and let (Σ, τ|Σ) be an induced system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' A Borel probability measure λφ on AN is called a local Gibbs state for the potential φ associated with (Σ, τ|Σ), if there exist constants C ≥ 1, γ0 ∈ R such that for any a ∈ W(A) and any x ∈ Π[[a]] we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='3) C−1 ≤ λφ[[a]] exp � S∥a∥φ(x) − γ0∥a∥ � ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We do not require the θ-invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' If the context is clear, we simply call λφ a local Gibbs state (associated with (Σ, τ|Σ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' If λφ is a local Gibbs state, then for any a ∈ W(A), the λφ-measure of the cylinder [[a]] in AN is given (up to multiplicative constants) by the Birkhoff sum of φ along the orbit of x of length ∥a∥ and the word length ∥a∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since the word length of a as a word in W(A) does not appear in the formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='3), λφ well captures part of the original dynamics (X, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' If λφ is a local Gibbs state, the Borel probability measure λφ ◦ Π−1 on Σ is τ|Σ-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' These two measures are related as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let φ: X → R satisfy P(φ) < ∞, let (Σ, τ|Σ) be an induced system and let λφ be a local Gibbs state associated with (Σ, τ|Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For any a ∈ W(A) we have λφ[[a]] = λφ ◦ Π−1(Σ ∩ [a]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We write νφ for λφ ◦ Π−1, and {R = n} for {z ∈ Σ: R(z) = n} for each n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We have νφ{R = n} > 0 for every n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let νφ|{R=n} denote the restriction of νφ to {R = n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The measure µφ = ∞ � n=1 n−1 � k=0 νφ|{R=n} ◦ σ−k is a finite measure if and only if � Rdνφ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since {R = n} is disjoint from �n−1 k=1 σ−k(Σ) for n ≥ 2, we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='4) µφ|Σ = ∞ � n=1 νφ|{R=n} = νφ = λφ ◦ Π−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For any a ∈ W(A) we have Π[[a]] ⊂ Σ, and so (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='5) λφ[[a]] = µφΠ[[a]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' 10 HIROKI TAKAHASI By µφ|Σ = νφ in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='4) and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1(a), for any a ∈ W(A) we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='6) µφΠ[[a]] = νφΠ[[a]] = � q∈N∗ νφ(Σ ∩ [aq]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let j ≥ 1 be such that a ∈ Aj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Then σ∥a∥ and τ j coincide on Σ ∩ [a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since � q∈N∗(Σ∩[aq]) ⊂ (τ|Σ)−j(� q∈N∗[q]) = ∅ by τ(Σ) ⊂ Σ, we have νφ(� q∈N∗ Σ∩[aq]) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Combining this with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='5), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='6) we obtain λφ[[a]] = � q∈N∗ νφ(Σ ∩ [aq]) + � q∈N∗ νφ(Σ ∩ [aq]) = νφ(Σ ∩ [a]), as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Exponential decay on partition functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The next proposition pro- vides a main technical estimate under the existence of a local Gibbs state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let φ: X → R be acceptable and satisfy P(φ) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Assume there exist an induced system (Σ, τ|Σ) and an associated local Gibbs state λφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' There exist δ′ ∈ (0, 1/5] and n0 ≥ 1 such that if δ ∈ (0, δ′] and {Ni}∞ i=1 is a non-decreasing integer sequence such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='7) max N∗ ≤ N1 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='8) ∞ � k=Ni+1 � a∈A Π[[a]]⊂[k] λφ[[a]] ≤ δ2i for every i ≥ 1, then for every n ≥ n0 and every m ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' , n} we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='9) � x∈En δn x (X\\Γ)=m/n exp Snφ(x) ≤ eγ0n2nn(4δ)m 1 − 4δ , where (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='10) Γ = {x = (xi)∞ i=1 ∈ X : xi ≤ Ni for every i ≥ 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='3 asserts that contributions of elements of En to Zn(φ) whose orbit escape from the compact set Γ exactly m times within period n is exponentially small in m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Similar estimates were obtained in [34] under the existence of Bowen’s Gibbs states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since λφ is a local Gibbs state, there exist constants C ≥ 1 and γ0 ∈ R such that for any a ∈ W(A) and any x ∈ Π[[a]] we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='11) C−1 ≤ λφ[[a]] exp � S∥a∥φ(x) − γ0∥a∥ � ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For the rest of the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='3, we shall use the notation a ≪ b for two positive reals a, b to indicate that a/b is bounded from infinity by a constant which depends only on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' If a ≪ b and b ≪ a, we shall write a ≍ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' LEVEL-2 LDP FOR COUNTABLE MARKOV SHIFTS WITHOUT GIBBS STATES 11 The first inequality in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='11) will be used to bound a partial sum of Zn(φ) from above by a sum of λφ-measures of cylinders in AN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Further, we will bound this sum using the product property which is a consequence of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='11): (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='12) λφ[[ab]] ≍ λφ[[a]]λφ[[b]] for a, b ∈ W(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let δ ∈ (0, 1/5] and let {Ni}∞ i=1 be a non-decreasing integer sequence satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='7) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let Γ = Γ({Ni}∞ i=1) be the compact subset of X given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' If x ∈ En \\ �n−1 i=0 σ−i(Σ) then xi ≤ max N∗ = N1 ≤ Ni for 1 ≤ i ≤ n, and so δn x(Γ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' By this and the periodicity of elements of En, for 1 ≤ m ≤ n we have � x∈En δn x (X\\Γ)=m/n exp Snφ(x) = � x∈En∩�n−1 i=0 σ−i(Σ) δn x (X\\Γ)=m/n exp Snφ(x) ≤ n � x∈En∩Σ δn x (X\\Γ)=m/n exp Snφ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='13) To bound the last sum in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='13), we decompose the set {x ∈ En ∩ Σ: δn x(X \\ Γ) = m/n} into subsets sharing the same itinerary up to time n, and estimate a contribution from each subset separately, and finally unify all these estimates counting the total number of possible itineraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Define a function r: X \\ Γ → N by r(x) = min{i ≥ 1: xi > Ni}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='7) we have Σ ⊂ X \\ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Hence, for each x ∈ Σ there are infinitely many i ≥ 0 with σix /∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' By an itinerary of x ∈ Σ we mean two sequences {nj(x)}∞ j=1, {rj(x)}∞ j=1 in N0 given by the recursion formulas n1(x) = min{i ≥ 0: σix /∈ Γ}, and rj(x) = r(σnj(x)x), nj+1(x) = min{i ≥ nj(x) + rj(x): σix /∈ Γ} for j ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let x ∈ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' (a) {i ≥ 0: σix /∈ Γ} = �∞ j=1[nj(x), nj(x) + rj(x) − 1] ∩ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' (b) xnj(x)+rj(x) ∈ N∗ for every j ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since {Ni}∞ i=1 is non-decreasing, if x /∈ Γ then σix /∈ Γ for 0 ≤ i ≤ r(x) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' This implies (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since σnj(x)x = xnj(x)+1xnj(x)+2 · · · and σnj(x)x /∈ Γ with r(σnj(x)x) = rj(x), we obtain xnj(x)+rj(x) > Nrj(x) ≥ N1, which together with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='7) yields xnj(x)+rj(x) ∈ N∗ as in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' □ For each j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' , m} and n1 · · · nj ∈ Nj 0, r1 · · · rj ∈ Nj with n1 < · · · < nj ≤ n, we put (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='14) ∆ r1···rj n1···nj = {x ∈ En ∩ Σ: (ni(x), ri(x)) = (ni, ri) for every i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' , j}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' If δ > 0 is sufficiently small, then for j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' , m}, n1 · · · nj ∈ Nj 0 and r1 · · · rj ∈ Nj such that ∆ r1···rj n1···nj ̸= ∅ we have � x∈∆ r1···rj n1···nj exp Snφ(x) ≤ eγ0nδr1+···+rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' 12 HIROKI TAKAHASI Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We start with the case j = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We introduce two sets of induced words B0 = {b ∈ W(A): ∥b∥ = n − n1 − r1 + 1, Π[[b]] ⊂ ∪∞ k=Nr1+1[k]} and D0 = {d ∈ W(A): ∥d∥ = n1 + r1 − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For each x ∈ ∆r1 n1 we have xn+1 = x1 ∈ N∗ by the definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='14), and xn1+r1 ∈ N∗ by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Hence there exist d ∈ D0 and b ∈ B0 such that [d] = [x1 · · ·xn1+r1−1] and [b] = [xn1+r1 · · · xn], and so x ∈ Π[[db]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='11) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='12), exp Snφ(x) ≪ eγ0nλφ[[db]] ≪ eγ0nλφ[[d]]λφ[[b]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Summing this inequality over all x ∈ ∆r1 n1 and then using � b∈B0 λφ[[b]] ≤ δ2r1 from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='8) and � b∈D0 λφ[[d]] ≤ 1 which follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1(b), we obtain � x∈∆r1 n1 exp Snφ(x) ≪eγ0n � d∈D0 λφ[[d]] � b∈B0 λφ[[b]] ≤ eγ0nδ2r1 ≤ eγ0nδr1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='15) provided δ is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' In case m = 1 we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' To proceed, suppose m ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' let j, j+1 ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' , m} and let n1 · · · njnj+1 ∈ Nj+1 0 , r1 · · · rjrj+1 ∈ Nj+1 be such that ∆ r1···rjrj+1 n1···njnj+1 ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Define Aj = {a ∈ W(A): ∥a∥ = nj + rj, Π[[a]] ∩ ∆ r1···rj+1 n1···nj+1 ̸= ∅}, Bj = {b ∈ W(A): ∥b∥ = n − nj+1 − rj+1 + 1, Π[[b]] ⊂ ∪∞ k=Nrj+1+1[k]}, Cj = {c ∈ W(A): ∥c∥ = n − nj − rj} , Dj = {d ∈ W(A): ∥d∥ = nj+1 + rj+1 − 1} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let a ∈ Aj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For each c ∈ Cj we have ∥ac∥ = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since X is the full shift, Π[[ac]] contains a unique element of En ∩ Σ which we denote by ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Then we have ac ∈ Π[[a]] ∩ ∆ r1···rj n1···nj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='11) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='12), exp Snφ(ac) ≫ eγ0nλφ[[a]]λφ[[c]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' This implies � x∈Π[[a]]∩∆ r1···rj n1···nj exp Snφ(x) ≫ eγ0nλφ[[a]] � c∈Cj λφ[[c]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='16) By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='2, for each c ∈ Cj we have λφ[[c]] = λφ ◦ Π−1(Σ ∩ [c]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since the sets Σ ∩ [c], c ∈ Cj are pairwise disjoint and their union equals Σ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='17) � c∈Cj λφ[[c]] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Combining (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='16) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='17) yields � x∈Π[[a]]∩∆ r1···rj n1···nj exp Snφ(x) ≫ eγ0nλφ[[a]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='18) Similarly to the case j = 1, for each x ∈ Π[[a]]∩∆ r1···rj+1 n1···nj+1 we have xn+1 = x1 ∈ N∗ by the definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='14) and xnj+1+rj+1 ∈ N∗ by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Hence there exist LEVEL-2 LDP FOR COUNTABLE MARKOV SHIFTS WITHOUT GIBBS STATES 13 d ∈ Dj, b ∈ Bj such that [d] = [x1 · · · xnj+1+rj+1−1] and [b] = [xnj+1+rj+1 · · ·xn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We have [[d]] ⊂ [[a]] and x ∈ Π[[db]], and by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='11) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='12), exp Snφ(x) ≪ eγ0nλφ[[d]]λφ[[b]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Summing this inequality over all x ∈ Π[[a]]∩∆ r1···rj+1 n1···nj+1, and then using � b∈Bj λφ[[b]] ≤ δ2rj+1 from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='8) and � d∈Dj [[d]]⊂[[a]] λφ[[d]] ≤ λφ[[a]] from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1(b), we obtain � x∈Π[[a]]∩∆ r1···rj+1 n1···nj+1 exp Snφ(x) ≪ eγ0n � d∈Dj [[d]]⊂[[a]] λφ[[d]] � b∈Bj λφ[[b]] ≤ eγ0nλφ[[a]]δ2rj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='19) The two estimates in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='18) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='19) yield � x∈Π[[a]]∩∆ r1···rj+1 n1···nj+1 exp Snφ(x) � x∈Π[[a]]∩∆ r1···rj n1···nj exp Snφ(x) ≤ δrj+1, provided δ is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Rearranging this inequality and summing the result over all a ∈ Aj yields � x∈∆ r1···rj+1 n1···nj+1 exp Snφ(x) ≤ δrj+1 � x∈∆ r1···rj n1···nj exp Snφ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Applying this inequality recursively and combining the final result with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='15) yields the desired inequality in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' □ For integers L ≥ m and s ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' , m}, we denote by KL,s the set of elements (n1 · · · ns, r1 · · · rs) of Ns 0×Ns such that 0 ≤ n1 < · · · < ns ≤ n and r1+· · ·+rs = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The number of ways of locating n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' , ns in [0, n] does not exceed ( n s ), and for each location (n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' , ns) the number of all feasible combinations of (r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' , rs) with r1 +· · ·+rs = L is bounded by the number of ways of dividing L objects into s groups, not exceeding � L+s−1 s−1 � ≤ 2L+s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' This yields #KL,s ≤ ( n s ) � L+s−1 s−1 � ≤ 2n2L+s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Clearly, for each x ∈ En ∩ Σ satisfying δn x(X \\ Γ) = m/n there exist L ≥ m, s ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' , m} and (n1 · · · ns, r1 · · · rs) ∈ KL,s such that x ∈ ∆r1···rs n1···ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' If δ ∈ (0, 1/5] is sufficiently small, then together with Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='5 we obtain � x∈En∩Σ δn x (X\\Γ)=m/n exp Snφ(x) ≤ m � s=1 ∞ � L=m � (n1···ns,r1···rs)∈KL,s � x∈∆r1···rs n1···ns exp Snφ(x) ≤ 2n m � s=1 ∞ � L=m 2L+s−1δL ≤ 2n ∞ � L=m (4δ)L = 2n(4δ)m 1 − 4δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' From this and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='13), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='9) follows and the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='3 is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' □ 14 HIROKI TAKAHASI 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Verifying exponential tightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We now use Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='3 to show the desired exponential tightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let φ: X → R be acceptable such that P(φ) < ∞ and let (Σ, τ|Σ) be an induced system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' If there exists a local Gibbs state for the potential φ associated with (Σ, τ|Σ), then {˜µn}∞ n=1 is exponentially tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The argument below is an adaptation of the proof of Sanov’s theorem (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=', [7]) to our setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For each integer ℓ ≥ 1, we fix δℓ ∈ (0, 1/5] such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='20) 1 1 − 4δℓ ∞ � m=0 e2ℓ2m(4δℓ)m ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We apply Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='3 and fix a non-decreasing integer sequence {Ni}∞ i=1 such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='7) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='8) with δ = δℓ hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We define a compact subset Γℓ = Γ({Ni}∞ i=1) of X by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='10), and set Kℓ = � ν ∈ M(X): ν(Γℓ) ≥ 1 − 1 ℓ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since M(X) is a Polish space and Γℓ is a closed set, the weak* convergence µk → µ for a sequence {µk}∞ k=1 in Kℓ implies lim supk→∞ µk(Γℓ) ≤ µ(Γℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Hence, Kℓ is a closed set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For an integer L ≥ 1 we define KL = ∞ � ℓ=L Kℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' This set is tight, and by Prohorov’s theorem any sequence in KL has a limit point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Hence it is sequentially compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since the weak* topology is metrizable with the bounded Lipschitz metric, KL is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' By Chebyshev’s inequality, for n ≥ 1 we have � x∈En exp(ℓ2nδn x (X\\Γℓ))≥eℓn exp Snφ(x) ≤ e−2ℓn � x∈En δn x (X\\Γℓ)≥1/n exp � 2ℓ2nδn x(X \\ Γℓ) � exp Snφ(x) = e−2ℓn n � m=1 e2ℓ2m � x∈En δn x (X\\Γ)=m/n exp Snφ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Combining this inequality with Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='3 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='20), we have � x∈En δn x /∈Kℓ exp Snφ(x) = � x∈En δn x (X\\Γℓ)≥ 1 ℓ exp Snφ(x) = � x∈En exp(ℓ2nδn x (X\\Γℓ))≥eℓn exp Snφ(x) ≤ 2nn 1 − 4δℓ eγ0ne−2ℓn n � m=0 e2ℓ2m(4δℓ)m ≤ 10 · 2nneγ0ne−2ℓn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' LEVEL-2 LDP FOR COUNTABLE MARKOV SHIFTS WITHOUT GIBBS STATES 15 If L ≥ 1 is large enough, then ˜µn(M(X) \\ KL) ≤ ∞ � ℓ=L ˜µn(M(X) \\ Kℓ) = ∞ � ℓ=L 1 Zn(φ) � x∈En δn x /∈Kℓ exp Snφ(x) ≤ 10 · 2nneγ0n Zn(φ) ∞ � ℓ=L e−2ℓn ≤ 2neγ0n Zn(φ) e−Ln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Combining this with the equality limn→∞(1/n) log Zn(φ) = P(φ) < ∞ which fol- lows from the uniform continuity of φ, we obtain lim sup n→∞ 1 n log ˜µn(M(X) \\ KL) ≤ −L + log 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since L ≥ 1 is an arbitrary large integer, {˜µn}∞ n=1 is exponentially tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Existence of a local Gibbs state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The next proposition ensures the exis- tence of a local Gibbs state under the assumption of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let φ: X → R satisfy P(φ) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Assume there exists an induced system (Σ, τ|Σ) for which the induced potentials Φγ, γ ∈ R associated with φ are locally H¨older continuous, and there exists γ0 ∈ R such that P(Φγ0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Then there exists a local Gibbs state for the potential φ associated with (Σ, τ|Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Note that P(Φγ0 ◦ Π) = P(Φγ0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since AN is the countable full shift, the finiteness of P(Φγ0 ◦ Π) implies the summability of the potential Φγ0 ◦ Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' By [18, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='5] together with the summability and the local H¨older continuity of Φγ0 ◦ Π, there exists a unique θ-invariant Bowen’s Gibbs state for the potential Φγ0 ◦ Π, which we denote by λφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' There exists C ≥ 1 such that for every m ≥ 1, any a ∈ Am and any z ∈ [[a]] we have C−1 ≤ λφ[[a]] exp � −P(Φγ0 ◦ Π)m + �m−1 k=0 Φγ0 ◦ Π(θkz) � ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For the series in the denominator of the fraction, for x ∈ Π[[a]] we have m−1 � k=0 Φγ0 ◦ Π(θkΠ−1(x)) = S�m−1 k=0 R(τ kx)φ(x) − γ0 m−1 � k=0 R(τ kx) = S∥a∥φ(x) − γ0∥a∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Substituting this and P(Φγ0 ◦ Π) = 0 into the denominator of the fraction implies that λφ is a local Gibbs state for the potential φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Under the assumption and notation of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='7 and its proof, if γ0 = P(φ), λφ is θ-invariant and � Rd(λφ ◦ Π−1) < ∞, then the measure 1 � Rd(λφ ◦ Π−1) ∞ � n=0 (λφ ◦ Π−1)|{R>n} ◦ σ−n is in Mφ(X, σ), and it is an equilibrium state for the potential φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The normalized restriction of this measure to Σ is λφ ◦ Π−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' 16 HIROKI TAKAHASI 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Proofs of the main results In this section we complete the proofs of all the theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' In Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='2, we prove lower and upper bounds for certain fundamental open and closed subsets of M(X) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='3, we combine these bounds and the exponential tightness verified in Section 2 to complete the proof of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' In Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='4 we complete the proof of Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' In view of applications, in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='5 we give a sufficient condition for the vanishing of the pressure of the induced potential that is assumed in Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Using this, we complete the proof of Theorem C in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Lower bound for fundamental open sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We introduce notations in this and the next two subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let Cu(X) denote the set of real-valued bounded uniformly continuous functions on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For an integer ℓ ≥ 1 we define Cu(X)ℓ = {⃗ϕ = (ϕ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' , ϕℓ): ϕj ∈ Cu(X) for every j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' , ℓ}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For ⃗ϕ = (ϕ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' , ϕℓ) ∈ Cu(X)ℓ, ⃗α = (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' , αℓ) ∈ Rℓ and µ ∈ M(X), the expression � ⃗ϕdµ > ⃗α indicates that � ϕjdµ > αj holds for all j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' , ℓ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The meaning of � ⃗ϕdµ ≥ ⃗α is analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Put ∥⃗α∥ = max1≤j≤ℓ |αj|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For ε ∈ R we write ⃗ε = (ε, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' , ε) ∈ Rℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For n ≥ 1 and p1 · · · pn ∈ Nn, let p1 · · · pn denote the element of En that is contained in [p1 · · · pn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let φ: X → R be acceptable and satisfy P(φ) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let ℓ ≥ 1, ⃗ϕ ∈ Cu(X)ℓ and ⃗α ∈ Rℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let G ⊂ M(X) be an open set of the form G = � µ ∈ M(X): � ⃗ϕdµ > ⃗α � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For any measure µ ∈ Mφ(X, σ) ∩ G, we have lim inf n→∞ 1 n log ˜µn(G) ≥ Fφ(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' By virtue of the definition of the pressure P(φ), it suffices to show that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1) lim inf n→∞ 1 n log � x∈En δn x ∈G exp Snφ(x) ≥ h(µ) + � φdµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The proof of [36, Main Theorem] works verbatim to show the next lemma that approximates non-ergodic measures with ergodic ones in a particular sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For any µ ∈ Mφ(X, σ) and any ε > 0 there exists an ergodic measure µ′ ∈ Mφ(X, σ) which is supported on a compact set and satisfies |h(µ) − h(µ′)| < ε, ���� � ⃗ϕdµ − � ⃗ϕdµ′ ���� < ε and ���� � φdµ − � φdµ′ ���� < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='2, it suffices to show (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1) for all µ ∈ Mφ(X, σ) which is ergodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let ε > 0 be such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='2) � ⃗ϕdµ > ⃗α + ⃗ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' LEVEL-2 LDP FOR COUNTABLE MARKOV SHIFTS WITHOUT GIBBS STATES 17 From the uniform continuity of each component of ⃗ϕ and that of φ, from Birkhoff’s ergodic theorem and Shannon-McMillan-Breiman’s theorem, for any sufficiently large n ≥ 1 there is a finite subset Gn of Nn such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='3) ���� 1 n log #Gn − h(µ) ���� < ε 2, and for every p1 · · · pn ∈ Gn, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='4) sup x∈[p1···pn] ���� � ⃗ϕdδn x − � ⃗ϕdµ ���� < ε 2 and sup x∈[p1···pn] ���� 1 nSnφ(x) − � φdµ ���� < ε 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='2) and the first inequality in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='4) yield � ⃗ϕdδn p1···pn > ⃗α, and the second inequality in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='4) yields (1/n)Snφ(p1 · · · pn) > � φdµ − ε/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Therefore � x∈En δn x ∈G exp Snφ(x) ≥ � p1···pn∈Gn exp Snφ(p1 · · · pn) ≥ #Gn exp � n � φdµ − εn 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Taking logarithms and dividing by a sufficiently large n we have 1 n log � x∈En δn x ∈G exp Snφ(x) ≥ 1 n log #Gn + � φdµ − ε 2 > h(µ) + � φdµ − ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Letting n → ∞ and then ε → 0 yields (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Upper bound for fundamental closed sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We proceed to upper bounds on fundamental closed sets, which are not necessarily compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let φ: X → R be acceptable and satisfy P(φ) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let ℓ ≥ 1, ⃗ϕ ∈ Cu(X)ℓ, ⃗α ∈ Rℓ and let C ⊂ M(X) be a non-empty closed set of the form C = � µ ∈ M(X): � ⃗ϕdµ ≥ ⃗α � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For any ε > 0 there exists µ ∈ Mφ(X, σ) such that � ⃗ϕdµ > ⃗α − ⃗ε and lim sup n→∞ 1 n log ˜µn(C) ≤ Fφ(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' A main ingredient is the next lemma, the proof of which is analogous to the standard proof of the variational principle [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For n ≥ 1 we put Dn(φ) = sup p1···pn∈Nn sup x,y∈[p1···pn] Snφ(x) − Snφ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For any ε > 0 there exists n0 ≥ 1 such that if n ≥ n0 then for any non-empty finite subset Cn of Nn satisfying δn p1···pn ∈ C for every p1 · · · pn ∈ Cn, there exists a measure µ0 ∈ Mφ(X, σ) such that log � p1···pn∈Cn sup [p1···pn] exp Snφ ≤ � h(µ0) + � φdµ0 � n+Dn(φ) and � ⃗ϕdµ0 > ⃗α−⃗ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' 18 HIROKI TAKAHASI Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since all components of ⃗ϕ are bounded uniformly continuous, for any ε > 0 there exists n0 ≥ 1 such that if n ≥ n0 then for any p1 · · · pn ∈ Nn satisfying δn p1···pn ∈ C, � ⃗ϕdδn x ≥ ⃗α − (1/2)⃗ε holds for any x ∈ [p1 · · · pn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' In what follows we assume n ≥ n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Set Λ = �∞ k=0 σ−nk(� p1···pn∈Cn[p1 · · · pn]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Then σn|Λ : Λ → Λ is topologically conjugate to the left shift acting on the finite full shift space CN n = {(ˆpm)∞ m=1 : ˆpm ∈ Cn for every m ≥ 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since the function ˆφ = Snφ induces a continuous potential on CN n , the variational principle [4] yields sup ˆµ∈M(Λ,σn|Λ) � h(ˆµ) + � ˆφdˆµ � = lim m→∞ 1 m log � ˆp1···ˆpm∈Cm n sup [ˆp1···ˆpm] � exp m−1 � k=0 ˆφ ◦ σnk � , where M(Λ, σn|Λ) denotes the space of σn|Λ-invariant Borel probability measures endowed with the weak* topology, and h(ˆµ) denotes the measure-theoretic entropy of ˆµ ∈ M(Λ, σn|Λ) with respect to σn|Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For the series in the right-hand side, we have � ˆp1···ˆpm∈Cm n sup [ˆp1···ˆpm] exp �m−1 � k=0 ˆφ ◦ σnk � ≥ � � p1···pn∈Cn inf [p1···pn] exp Snφ �m ≥ � exp(−Dn(φ)) � p1···pn∈Cn sup [p1···pn] exp Snφ �m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Taking logarithms of both sides, dividing by m and then letting m → ∞ gives lim m→∞ 1 m log � ˆp1···ˆpm∈Cm n sup [ˆp1···ˆpm] exp �m−1 � k=0 ˆφ ◦ σnk � ≥ log � p1···pn∈Cn sup [p1···pn] exp Snφ−Dn(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Plugging this into the previous inequality yields sup ˆµ∈M(Λ,σn|Λ) � h(ˆµ) + � ˆφdˆµ � ≥ log � p1···pn∈Cn sup [p1···pn] exp Snφ − Dn(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' By the compactness of the space M(Λ, σn|Λ) and the upper semicontinuity of the map ˆµ �→ h(ˆµ) + � ˆφdˆµ on this space, the supremum is attained, say by ˆµ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The measure µ0 = (1/n) �n−1 j=0 ˆµ0 ◦ σ−j is in Mφ(X, σ) and satisfies � h(µ0) + � φdµ0 � n = sup ˆµ∈M(Λ,σn|Λ) � h(ˆµ) + � ˆφdˆµ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since the support of µ0 is contained in set {x ∈ X : � ⃗ϕdδn x > ⃗α − ⃗ε/2} by the choice of n0 and the assumption n ≥ n0, we obtain � ⃗ϕdµ0 > ⃗α−⃗ε as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' □ Continuing the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='3, we note that P(φ) < ∞ implies Zn(φ) < ∞ for every n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Hence it is possible to choose a finite subset Cn of the countable LEVEL-2 LDP FOR COUNTABLE MARKOV SHIFTS WITHOUT GIBBS STATES 19 set � p1 · · · pn ∈ Nn : δn p1···pn ∈ C � such that � p1···pn∈Nn δn p1···pn∈C exp Snφ(p1 · · · pn) ≤ 2 � p1···pn∈Cn exp Snφ(p1 · · · pn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' By this inequality and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='4, there exists µ0 ∈ Mφ(X, σ) such that � ⃗ϕdµ0 > ⃗α − ⃗ε and log � x∈En δn x ∈C exp Snφ(x) = log � p1···pn∈Nn δn p1···pn∈C exp Snφ(p1 · · · pn) ≤ log � p1···pn∈Cn exp Snφ(p1 · · · pn) + log 2 ≤ log � p1···pn∈Cn sup [p1···pn] exp Snφ + log 2 ≤ � h(µ0) + � φdµ0 � n + Dn(φ) + log 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since φ is acceptable, it is uniformly continuous and so Dn(φ) = o(n) (n → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Dividing both sides of the above last displayed inequality by n, letting n → ∞ and combining the result with P(φ) = limn→∞(1/n) log Zn(φ) yields the desired inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Proof of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let φ: X → R be acceptable and satisfy P(φ) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Assume there exists an induced system for which the induced potentials Φγ, γ ∈ R associated with φ are locally H¨older continuous, and there exists γ0 ∈ R such that P(Φγ0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let G be a non-empty open subset of M(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since subsets of M(X) of the form � µ ∈ M(X): � ⃗ϕdµ > ⃗α � with ℓ ≥ 1, ⃗ϕ ∈ Cu(X)ℓ, ⃗α ∈ Rℓ constitute a base of the weak* topology of M(X), G is written as the union G = � λ Gλ of sets Gλ of this form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For each Gλ, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1 gives lim inf n→∞ 1 n log ˜µn(Gλ) ≥ sup Gλ Fφ, and hence lim inf n→∞ 1 n log ˜µn(G) ≥ sup λ sup Gλ Fφ = sup G Fφ = − inf G Iφ, as required in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let C be a compact closed subset of M(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let G be an arbitrary open set containing C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since M(X) is metrizable by the bounded Lipschitz metric and C is compact, we can choose ε > 0 and finitely many closed sets C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' , Cs of the form Ck = � µ ∈ M(X): � ⃗ϕkdµ ≥ ⃗αk � with ℓk ≥ 1, ⃗ϕk ∈ Cu(X)ℓk, ⃗αk ∈ Rℓk, so that C ⊂ �s k=1 Ck ⊂ �s k=1 Ck(ε) ⊂ G where Ck(ε) = {µ ∈ M(X): � ⃗ϕkdµ > ⃗αk − ⃗ε}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='4 and Fφ ≤ −Iφ, for 1 ≤ k ≤ s we have lim sup n→∞ 1 n log ˜µn(Ck) ≤ − inf Ck(ε) Iφ + ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' 20 HIROKI TAKAHASI Then we have lim sup n→∞ 1 n log ˜µn(C) ≤ max 1≤k≤s � − inf Ck(ε) Iφ � + ε ≤ − inf G Iφ + ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since ε > 0 is arbitrary and G is an arbitrary open set containing C, it follows that lim sup n→∞ 1 n log ˜µn(C) ≤ inf G⊃C � − inf G Iφ � = − inf C Iφ, as required in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The last equality is due to the lower semicontinuity of Iφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since {˜µn}∞ n=1 is exponentially tight by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='6, the standard arguments as in [7] show the upper bound (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='2) for any non-compact closed subset of M(X), and that Iφ is a good rate function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' This completes the proof of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Proof of Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let φ: X → R be acceptable and satisfy P(φ) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Assume there exists an induced system for which the associated induced potentials Φγ, γ ∈ R are locally H¨older continuous, and there exists γ0 ∈ R such that P(Φγ0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Assume that the minimizer of the rate function Iφ in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='10) is unique, denoted by µmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let {˜µnj}∞ j=1 be an arbitrary convergent subsequence of {˜µn}∞ n=1 with the limit measure ˜µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' It suffices to show that ˜µ is the unit point mass at µmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We fix a metric which generates the weak* topology on M(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since the rate function Iφ in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='10) is the good rate function by Theorem A, for any α > 0 the level set L(α) = {µ ∈ M(X): Iφ(µ) ≤ α} is a compact set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let µ ∈ M(X) \\ {µmin}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' By the lower semicontinuity of the rate function and Iφ(µ) > 0, it is possible to take r > 0 such that the closure of the open ball Br(µ) of radius r about µ does not intersect L(Iφ(µ)/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The weak* convergence ˜µnj → ˜µ gives ˜µ(Br(µ)) ≤ lim inf j→∞ ˜µnj(Br(µ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' By this and the large deviations upper bound for closed sets (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='2), we have ˜µ(Br(µ)) ≤ lim sup j→∞ ˜µnj(Br(µ)) ≤ lim sup j→∞ exp � −Iφ(µ)nj 2 � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Hence, the support of ˜µ does not contain µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since µ is an arbitrary element of M(X) \\ {µmin}, it follows that ˜µ is the unit point mass at µmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' This completes the proof of Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Sufficient condition for vanishing of pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' A direct check of the condition P(Φγ0) = 0 in Theorem A may be cumbersome, while checking the finiteness of induced pressures is considered to be easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' In view of applications, we give a sufficient condition for the second assumption in Theorem A on the induced potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let φ: X → R be acceptable and satisfy P(φ) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let (Σ, τ|Σ) be an induced system and let Φγ : Σ → R (γ ∈ R) be the associated family of induced potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' If there exists δ ∈ R such that 0 < P(Φδ) < ∞, then there exists γ0 ∈ R such that P(Φγ0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' LEVEL-2 LDP FOR COUNTABLE MARKOV SHIFTS WITHOUT GIBBS STATES 21 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let γ ∈ R and suppose P(Φγ) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since ∥a∥ ≥ n for any a ∈ An and P(Φγ) is finite, for any γ′ > γ we have � a∈An sup [a] exp(S∥a∥φ − γ′∥a∥) ≤ exp(−(γ′ − γ)n) � a∈An sup [a] exp(S∥a∥φ − γ∥a∥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Taking logarithms, dividing by n and letting n → ∞ yields P(Φγ′) ≤ −γ′ + γ + P(Φγ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' This and the assumption in the lemma together imply that both γ∞ = inf{γ ∈ R: P(Φγ) < ∞} and γ0 = inf{γ > γ∞: P(Φγ) ≥ 0} are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' By the variational principle [18, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='8], γ ∈ (γ∞, ∞) �→ P(Φγ) is convex and so continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Hence P(Φγ0) = 0 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Proof of Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Recall that T : [0, 1) → [0, 1) denotes the R´enyi map (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For a bounded interval J ⊂ R let |J| denote its Euclidean length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' In consideration of the neutral fixed point 0 of T, we set N∗ = N \\ {1}, N∗ = {1} and define an inducing scheme (X∗, R) by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='5), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='6), the induced system (Σ, τ|Σ) by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='7), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='8), and define an infinite alphabet A and a coding map Π by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='2) respectively, keeping the notation in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We have Σ = (1/2, 1)∩I and A = ��∞ q=2[p1n−1q]: p ≥ 2 and n ≥ 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For simplicity we will denote by C various positive constants which depend only on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For each a = �∞ q=2[p1n−1q] ∈ A, we put J(a) = T −1([1/(∥a∥ + 1), 1/∥a∥)) ∩ Jp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Then R equals ∥a∥ on the set Π[[a]] = π−1(J(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' There exists C ≥ 1 such that for any a ∈ W(A) we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='5) C−1 ≤ |J(a)| · ∥a∥2 ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Define an induced map U : � a∈A J(a) → [0, 1) by U|J(a) = T ∥a∥|J(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Recall that φ = − log |T ′ ◦ π|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For β, γ ∈ R define Φβ,γ : Σ → R by Φβ,γ(x) = βSR(x)φ(x) − γR(x), which is the induced potential associated with βφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For all β, γ ∈ R, Φβ,γ is locally H¨older continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' From the bounded distortion near the neutral fixed point [19, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='2], there exists C > 0 such that for any a ∈ A and all x, y ∈ [a] we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='6) Φβ,γ(x) − Φβ,γ(y) ≤ Cβ|U(ξ) − U(η)| ≤ Cβ, where ξ = π(x) and η = π(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' If x ̸= y then d(x, y) = e−n holds for some n ≥ 2, and there exists a1 · · ·an ∈ An such that x, y ∈ [[a1 · · · an]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since there is ρ > 1 such that inf[0,1)\\J1 |T ′| ≥ ρ, if n ≥ 3 then we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='7) |U(ξ) − U(η)| ≤ |Un−1(ξ) − Un−1(η)| inf�n−1 k=2 U−k(J(ak)) |(Un−2)′| ≤ ρ2−n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The local H¨older continuity of Φβ,γ follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='6) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For any β ∈ (1/2, 1] there exists γ ∈ R such that 0 ≤ P(Φβ,γ) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' 22 HIROKI TAKAHASI Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' From Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='6, |T(Jp)| = 1 for p ≥ 2 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='5), there exists C ≥ 1 such that for a = �∞ q=2[p1n−1q] ∈ A and all β, γ ∈ R we have 1 |Jp|β sup [a] exp Φβ,γ = e−γn sup Π[a] exp(βSnφ) |Jp|β ≤ Ce−γnn−2β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Summing the result over all a ∈ A, we have ∞ � n=1 � a∈A ∥a∥=n sup [a] exp Φβ,γ ≤ C ∞ � n=1 e−γnn−2β ∞ � p=2 |Jp|β ≤ C ∞ � n=1 e−γnn−2β ∞ � p=2 p−2β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let β ∈ (1/2, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Then we have P(βφ) > 0 [13], and the above series is finite for all γ ∈ (0, P(βφ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' In particular, P(Φβ,P (βφ)) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since any measure in M(X, σ) other than the unit point mass at 1∞ = 111 · · · charges Σ, the equilibrium state µβφ for the potential βφ satisfies µβφ(Σ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Let ˆµβφ denote the normalized restriction of µβφ to Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since τ is the first return map to Σ, ˆµβφ is τ|Σ-invariant and satisfies � Rdˆµβφ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' By the variational principle for Φβ,P (βφ) and Abramov- Kac’s formula [44, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1], we obtain ∞ > P(Φβ,P (βφ)) ≥ h(ˆµβφ) + � (Φβ − P(βφ)R)dˆµβφ = (Fβφ(µβφ) + P(βφ) − P(βφ)) � Rdˆµβφ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We have verified1 that 0 ≤ P(Φβ,P (βφ)) < ∞ as reqiured in the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For the remaining case β = 1, we have P(φ) = 0 [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' From Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='6 there is C ≥ 1 such that for n ≥ 1 and a = a1 · · · an ∈ An, C−1 ≤ ���n k=1 U−k(J(ak)) �� sup[a] exp ��n−1 k=0 Φ1,0 ◦ τ k� ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Summing this double inequalities over all a ∈ An, taking logarithms, dividing by n and then letting n → ∞ yields P(Φ1,0) = 0 as required in the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='6 and Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='5, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='7 together verify the assumption in Theorem A for the potential βφ, β ∈ (1/2, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' It follows from [35] that for any β ∈ (1/2, 1], any minimizer of the rate function Iβφ is an equilibrium state for βφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since the equilibrium state for βφ is unique [13], the minimizer of the rate function Iβφ is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Since the map π in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='13) is continuous, the assertions in Theorem C follow from Theorems A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Some generalizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' We have worked on two full shift spaces X and Σ (or AN), the latter obtained from the former via inducing (recall Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The assumption that X is the full shift has been used to construct sets of periodic points of the same period, in the proofs of exponential tightness (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='5) and the lower large deviation bound (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' For the induced system (Σ, τ|Σ), we have effected the thermodynamic formalism for countable Markov shifts [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' 1In fact, one can show P(Φβ,P (βφ)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' See [23] for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' LEVEL-2 LDP FOR COUNTABLE MARKOV SHIFTS WITHOUT GIBBS STATES 23 The setup in this paper can be slightly generalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' The above-mentioned con- structions of sets of periodic points can be done even if X is replaced by a finitely primitive shift (see [18] for the definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Then the induced shift space becomes finitely irreducible, for which the thermodynamic formalism works too [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' This research was partially supported by the JSPS KAK- ENHI 19K21835 and 20H01811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' References [1] Aaronson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=', Denker, M.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' 133 (2005) 2283–2295.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content=' Keio Institute of Pure and Applied Sciences (KiPAS), Department of Mathe- matics, Keio University, Yokohama, 223-8522, JAPAN Email address: hiroki@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='keio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} +page_content='jp' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfAgLJ/content/2301.02841v1.pdf'} diff --git a/4tE1T4oBgHgl3EQfSwM3/content/2301.03069v1.pdf b/4tE1T4oBgHgl3EQfSwM3/content/2301.03069v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..9e8e9791f4917c4a560becd18cd9c03e3ac0dc68 --- /dev/null +++ b/4tE1T4oBgHgl3EQfSwM3/content/2301.03069v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:1152e20d057ffb5918ca3904658efc8ffadae2c8d40d47975fa7138a6a138d4c +size 6056704 diff --git 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Maksymov 1* +and Andrey Pototsky 2 +Maksymov, I.S.; Pototsky, A. +Experimental and theoretical study of +solitary-like wave dynamics of liquid +film flows over a vibrated inclined +plane. Preprints 2022, 1, 0. +https://doi.org/ +Received: +Accepted: +Published: +1 +Optical Sciences Centre, Swinburne University of Technology, Hawthorn, VIC 3122, Australia; +imaksymov@swin.edu.au; Tel.: +61-3-3921-4805 +2 +Department of Mathematics, Swinburne University of Technology, Hawthorn, VIC 3122, Australia; +apototskyy@swin.edu.au; Tel.: +61-3-9214-4653 +Abstract: Solitary-like surface waves that originate from the spatio-temporal evolution of falling liquid +films have been the subject of theoretical and experimental research due to their unique properties that +are not readily observed in the physical system of other nature. Here we investigate, experimentally and +theoretically, the dynamics of solitary-like surface waves in a liquid layer on an inclined plane that is +subjected to a harmonic low-frequency vibration in the range from 30 to 50 Hz. We demonstrate that the +vibration results in a decrease in the average and peak amplitude of the long solitary-like surface waves. +However, the speed of these waves remains largely unaffected by the vibration, implying that they may +propagate over large distances almost without changing their amplitude, thus rendering them suitable +for a number of practical applications, where the immunity of pulses that carry information to external +vibrations is required. +Keywords: falling liquid films; solitary waves; surface waves; vibrations +1. Introduction +Solitary waves—physical waves that maintain their shape and move with a constant +velocity due to a cancellation of nonlinear effects and dispersive processes in the medium +[1]—have been a long-term subject of fundamental and applied research studies in the fields +of optics [2], fluid dynamics [3], magnetism [4], acoustics [5], electronics [6] and biology +[7,8]. However, despite a good understanding of the physical properties of solitary waves of +different kinds, their experimental studies often involve expensive and difficult to operate +equipment such as intense laser beams and nonlinear-optical materials in the field of optics +[2] and sources of high-power microwave radiation in the field of magnetism [4], respectively. +Yet, in some systems such as biological nerve fibres [7,8] the observation of solitary-like waves +requires significant preparatory works and is possible mostly when a number of specific +experimental conditions are satisfied. Such technical challenges complicate both fundamental +studies and verification of numerous theoretical works predicting that solitary waves could +be used in communication [9,10], sensing [11] and data processing [12] devices and systems. +There also exists a class of material solitary-like surface waves that originate from spatio- +temporal evolution of falling liquid films [13,14]. Since the equipment needed to create falling +liquid films is, in general, simpler than that used in experiments in the fields of optics and +magnetism, the waves of this kind have attracted attention of many scientists [15–28] following +the pioneering experiments conducted by the Kapitzas [29]. In fact, while such solitary-like +surface waves share many physical features with the other known types of solitary waves, +they can exhibit unique physical properties not observed in other systems [24,30,31]. For +instance, they can merge instead of passing through each other without significant change, +with the latter being the case of two solitary waves governed by the well-known KdV equation +[3,32]. The analysis of solitary-like surface waves in flowing liquid films is also important +because liquid films, as well as similar physical systems [33–35], are often encountered in +arXiv:2301.03300v1 [physics.flu-dyn] 9 Jan 2023 + +2 of 15 +the fields of earth and planetary sciences [36,37] and in technological processes [38], where +the liquids of interest can also experience temperature gradients [14,28] and vibrations [39– +41]. Given this, the effect of vibrations on the wave dynamics of film flows has become +an independent subject of fundamental and applied research [42–46]. In particular, it has +been shown that vibrations can suppress certain waves on the surface of flowing liquid films +[42] but in a relevant experiment [47] it has been demonstrated that vibrations can promote +unusual regimes of spontaneous drop movement. Speaking broadly, the study of the effect of +vibrations should also help develop communication, sensing and data processing systems +that are immune to undesirable mechanical impacts on devices that use liquids as a medium +that provides the critical functionality (see, e.g., [48–50]). +Although, traditionally, greater attention has been paid to the wave dynamics on free- +falling vertical liquid films [13,14], studies of surface waves on liquid films flowing over +slightly inclined planes have also been conducted given an essentially the same physics as +in the case of vertical systems [24,42]. However, reports on experimental results involving +the effect of vibrations are rather scarce and scattered in the literature sources [39,40,45]. In +particular, in [39] it has been shown that the vibration of a horizontal tube with a liquid thin +film flowing over it results in the appearance of ripple waves at the vibration frequency. The +amplitude of the so-created waves depends on the vibration amplitude and can reach the +amplitude of periodic waves existing on the film surface without vibration. Subsequently, +high-amplitude vibrations result in an increase in the film thickness and a concomitant increase +in the speed of the waves. However, the opposite conclusions were drawn in [40], which +is, most likely, a result of the differences in the system (a liquid film under two-phase flow +conditions) investigated in that paper. It is also well-known that in horizontal liquid layers a +harmonic vibration excites two different types of standing surface waves: harmonic waves +that oscillate at the vibration frequency and subharmonic waves that oscillate at the half of the +vibration frequency [51]. However, the presence of a mean flow across the layer changes the +response frequency of the excited waves [42–44]. Surface waves excited by harmonic vibration +in a liquid film flowing over a vertical plane were investigated experimentally in [45] and the +results obtained in that work validated the linear theory developed in [42–44]. +Thus, mostly the experimental work [45] represents an attempt to systematically study +the physics of wave motion on a vibrated plane. However, in general, building a setup +involving liquids flowing down a vibrated vertical surface requires non-standard equipment +built according to demanding technical specifications. In particular, the liquid should be +supplied to the inlet located at the upper part of the plane so that the flow rate is not affected +by the vibration. This is because the thickness of the liquid film is known to be very sensitive +to external disturbances, including vibrations caused by the pump used to deliver the liquid +from a reservoir to the inlet [22]. Moreover, the shaker producing the vibration should be +connected to the vertically positioned surface via a vibration transmission structure. Some of +the engineering challenges of creating such a structure are the need to move a considerable +total mass of the supporting structure and liquid with high precision, and to ensure that the +amplitude of the vibration across the plane area is uniform [45]. To resolve the problem of +non-uniform vibration amplitude, in Ref. [45] it is was suggested that qualitatively similar +results could be obtained vibrating just one side of the plane, i.e. vibrating just a portion of the +liquid, thus also significantly reducing the total mass that needs to be moved by the shaker. +In this paper, we present and discuss a technically simple and compact experimental +setup for the investigation of solitary-like surface waves on a slightly inclined plane positioned +on top of a vibrating table and equipped with an auxiliary channel that recycles the liquid +used in experiment, thus decreasing the chance of spills of the liquid and its unwanted contact +with the measurement and imaging equipment, and also decreasing the total mass that needs +to be moved by the shaker. We employ this setup to demonstrate that the instabilities of + +3 of 15 +the thin liquid film caused by the vibrations result in a decrease in the peak amplitude of +the solitary-like surface waves. We conclude that, despite these changes, the speed of the +solitary-like waves does not appreciably change due to vibration. As a result, these waves can +propagate for long distances without changing their shape and, therefore, can be used in the +practical applications discussed in this work. We also demonstrate the advantage of using +frequency-wavevector dispersion maps for the analysis of the properties of rolling waves, thus +extending the toolbox of experimentalists working on this class of wave motion phenomena. +Our experimental results are validated using the Shkadov model [52,53]—a boundary-layer +hydrodynamic model derived from the Navier-Stokes equation under the assumption of +self-similar parabolic longitudinal velocity flow field across the layer. +Figure 1. Sketch of the experimental setup used to observe the solitary-like surface waves. For the sake +of clarity, only the main constructive features are shown, including the inclined plate, where the waves +are observed, the pathway for recycling of the used liquid and the vibrating table. The dimensions and +relative positions of the components in this sketch are not to scale. +2. Background and Experimental Methods +When a single-layer liquid film flows down an inclined plane with a no-slip boundary, +the resulting Nusselt flat film flow profile assumes a parabolic longitudinal velocity shape +having the largest velocity at the free surface [13,14,24]. In this flow regime, a long-wavelength +surface instability develops when the average flow rate exceeds a certain critical value [15]. +When the disturbances are excited naturally, in general four regimes of different wave be- +haviour can be observed in the downstream regions of the inclined plane [13]. The first +regime is observed in a section of the plane that is adjacent to the inlet of the liquid, where +small disturbances caused by the inlet structure are amplified while moving downstream +and forming predominantly monochromatic waves. The second regime is observed in the +following downstream region, where the monochromatic waves grow in amplitude and then +develop higher-order frequency harmonics due to nonlinear effects. Then, as a result of com- +plex nonlinear interactions, two-dimensional solitary-like waves are formed, and then they +propagate further downstream exhibiting unique properties that, in part, coincide with those +of other known solitary waves but, in general, are unique [24]. Finally, three-dimensional +waves start to form due to transverse variations [13,14]. +It is noteworthy that not all aforementioned regimes can necessarily be observed in +practice [13]. Yet, it is well-known that when the initial natural disturbance at the inlet is +nearly monochromatic, the waves emerging in the region located immediately after the inlet +can first inherit the frequency of the disturbance and then evolve into a solitary-like wave far + +UV light +digital camera +inlet +rolling waves +dwnd +inclined plane +H +recycled liquid +vibrating table +shaker +accelerometer +fluorescent +liquid4 of 15 +downstream [13,14,24]. However, when either the thickness of the liquid film or the fluid flow +is periodically modulated at the inlet, solitary-like surface waves develop almost immediately +after leaving the inlet area [14,24], which indicates that the nonlinear evolution of the flow +over an inclined plane is dominated by solitary-like waves independently of whether their +formation was deliberately forced or resulted naturally. +Figure 1 shows a sketch of the setup that enables observing the formation of both forced +and natural (unforced) solitary-like surface waves. The setup is assembled on a vibrating +table that is driven by a shaker (35 W, 20–80 Hz response, Dayton Audio, USA) and where +the vibration amplitude is controlled by an analog accelerometer (ADXL326, Analog Devices, +USA). The inclined plane, where the waves are observed, is a 5-cm-wide and 50-cm-long rigid +aluminium plate. The surface of the plate was chemically treated to improve the formation +of the liquid film. The inclination angle is θ = 3 o. The inclined plate was mounted on top +of wider open channel used to recycle the liquid by redirecting it to the main reservoir. A +low-vibration DC voltage pump driven via a customised electronic circuit was used to supply +water from the reservoir to the inlet. The electronic modulation of the pump flow rate enabled +controlling the thickness of the liquid film and creating solitary-like waves. At the stage of +preparation to the experiments, an organic fluorescent dye (Tintex, Australia) was added to +tap water in the concentration of 1 g per litre, thus leading to the emission of bright green +fluorescence light when the surface of the inclined plate was illuminated with UV-A light. All +experiments were conducted in a darkened room using an overhead digital camera capable +of recording videos in a slow motion regime. The resulting videos were post-processed in +Octave software using customised computational procedures enabling the extraction of the +wave amplitude from the intensity profile of the fluorescence images. All experiments were +conducted in an acoustically isolated room with environmental humidity and temperature +levels. +Figure 2. Instantaneous profiles the surface waves in a liquid film flowing over the inclined plate. The +profiles were obtained from the fluorescence images. The frequency of the flow forcing resulting in the +formation of solitary-like waves is 2 Hz. (a) No vibration. (b) The inclined plate is vibrated with the +frequency of 48 Hz and the peak vibration amplitude of 1g. +3. Experimental Results +Figure 2 shows the representative images of the solitatry-like surface waves propagating +over a downstream section of the inclined plate, when the vibration is turned off (Fig. 2a) and +when the plate is vibrated with the frequency of 48 Hz and the peak amplitude of 1g (Fig. +2b), being g the gravitational acceleration. The frequency of the forcing of the solitary-like + +Amplitude (arb. units) +(a) +Amplitude (arb. units) +(b) +1 +0 +0.05 +0.05 +0.1 +0.1 +0.15 +0.15 +0.2 +X +0.2 +Downstream distance (m) +0.25 +0.25 +X +0.3 +0.3 +0.06 +0.04 +0.04 +0.02 +X +0.02 +Plate width (m) +0.35 +0 +Plate width (m5 of 15 +waves is 2 Hz in both panels of Fig. 2. The images were obtained from the selected individual +fluorescence frames of the recorded videos of the propagating waves. Without vibration (Fig. +2a), we can observe a train of downstream-propagating solitary pulses. A closer inspection +also reveals the existence of periodic waves with an amplitude that is much smaller than +that of solitary-like waves. When the plate is subjected to vibration (Fig. 2b), we continue +observing a train of solitary pulses with an approximately the same pulse periodicity as in Fig. +2a. However, the peak amplitude of the pulses is lower than in the case without vibration. Yet, +in agreement with the relevant theory [42,44] and experiment on the vertical plane [45], we +also observe the short wavelength ripples arising due to the onset of the Faraday instability. +Using our fluorescence intensity analysis software, we register the profiles of the waves at +the points located on the centreline of the inclined plate along the downstream direction, and +we plot the so-obtained data as a function of time. The resulting spatiotemporal false-colour +maps are plotted in Fig. 3 with the observation period of 2 s for the scenario of no vibration +(Fig. 3a) and with the 48 Hz vibration (Fig. 3b). In those figures, we can see the traces of +several solitary-like waves that propagate in the downstream direction. The traces are more +distinguishable and have a higher false-colour amplitudes in the case of no vibration than +with the vibration, which confirms our observation of a decrease in the peak amplitude of the +solitary pulses due to the vibration in Fig. 2. The ripple waves caused by the vibration-induced +Faraday instabilities can also be seen in Fig. 3b. It is noteworthy that the separation between +the traces and the relative position of the traces in the time-downstream coordinate space +are very similar without and with the vibration. This indicates that, even though the peak +amplitude of the solitary-like waves is affected by the vibration, in general the vibration does +not change the shape of the soliton pulses. +Figure 3. False-colour maps showing the spatiotemporal traces of solitary-like waves forced at the +frequency of 2 Hz. (a) No vibration. (b) The inclined plate is vibrated with the frequency of 48 Hz and +the peak vibration amplitude of 1g. +Then, we apply a two-dimensional Fourier transformation to the spatiotemporal data to +obtain the dispersion maps as a function of frequency f and wavevector k. Since the speed of +a wave is given by u = ω/k = 2π f /k, using the resulting dispersion maps we can identify +the bands of constant f /k ratio that correspond to waves travelling along the inclines plate at + +(a) 2 +(b) 2 +1.5 +0.5 +1.5 +0.5 +S +S +0 +0 +0.5 +-0.5 +0.5 +-0.5 +0 +OF +0 +0.1 +0.2 +0.3 +0.4 +0 +0.1 +0.2 +0.3 +0.4 +Downstream distance (m) +Downstream distance (m)6 of 15 +a constant speed. Yet, we apply the standard f-k filtering procedures to remove noise from the +dispersion characteristics [54]. Figure 4 shows the dispersion maps for the case of no vibration +(Fig. 4a) and with the 48 Hz vibration (Fig. 4b). While the negative frequency regions of the +dispersion maps originate from the mathematical properties of the Fourier transformation, +the sign of the wavevector has the physical meaning as it determines the direction of the wave +propagation. +We first analyse the dispersion map in Fig. 4a and its zoomed image presented in Fig. 5a, +where we can see a set of high-magnitude discrete bands that are superimposed on a broader +continuum band of a lower magnitude. The frequencies of the discrete bands correspond to +the forcing frequency of the solitary-like waves 2 Hz and its higher-order harmonics of 4, 6 +and 8 Hz and so forth. The origin of the harmonics is due to the nonlinear effects as discussed +below. The spectrum of the discrete bands changes as the frequency of forcing of the solitary- +like waves is changed. When the modulation of the pump flow was turned off, i.e. with no +wave forcing, the discrete bands completely disappeared. However, a continuum band was +always observed independently of whether the forcing was turned on or off. Subsequently, we +associate the continuum band with natural periodic rolling waves propagating on the surface +of the liquid film flowing over the inclined surface. According to the frequency-wavevector +spectral analysis theory [54], a fit of the observed bands with a straight line produces the +velocity of the solitary-like wave of 0.27 ± 0.02 m/s. +When the plate is vibrated (Fig. 4b), in addition to the dispersion bands discussed in Fig. +4a we observe two new isolated bands that can be associated with the Faraday instability. +Moreover, the close-up of the dispersion map (Fig. 5b) shows that the magnitude of the +discrete modes decreased due to the vibration, which is an observation that is consistent with +our conclusions made earlier in the text. Yet, the bands in Fig. 5b can also be fitted with a +straight line that corresponds to the wave velocity of 0.27 ± 0.02 m/s. +Figure 4. Dispersion maps of the solitary-like waves forced at the frequency of 2 Hz. (a) No vibration. +(b) The inclined plate is vibrated with the frequency of 48 Hz and the peak vibration amplitude of 1g. +The plots are slightly oversaturated for the sake of a better visual presentation. +Empirically, the presence of the discrete bands at the forcing frequency of 2 Hz and +its higher-order harmonics can be explained using the well-established analogy between +the rolling waves and acoustic waves [55]. Indeed, the solitary-like surface waves in Fig. +2 can be regarded as large-amplitude shock-like disturbances (in the sub-field of physically + +(a) +0.1 +(b) +0.1 +40 +40 +0.08 +0.08 +20 +20 +Frequency (Hz) +0.06 +Frequency (Hz) +0.06 +0 +0 +0.04 +0.04 +20 +-20 +0.02 +0.02 +-40 +-40 +0 +-1.5 +-1 +-0.5 +0 +0.5 +1.5 +-1.5 +-1 +-0.5 +0 +0.5 +1.5 +Wavevector (mm-1) +Wavevector (mm-1)7 of 15 +similar roll waves in open channel such a discontinuity is called the hydraulic jump [14,21,34]). +Shock waves are also well-known in the field of nonlinear acoustic, where their formation +is accompanied by strong nonlinear effects such as the generation of higher-order harmonic +frequencies [50]. Considering longitudinal acoustic waves that can be described as alternating +areas of compression and rarefaction in the medium, we can show that the points of the crests +of an acoustic wave travel faster than the speed of sound in the medium, but the points of +the wave troughs travel slower [50]. This physical process underpins the formation of an +acoustic shock wave [55]. In turn, in the field of rolling waves, the crest of a large-amplitude +solitary-like wave is connected to its trough by a discontinuity, where the flow regime abruptly +changes from a supercritical condition and where the fluid moves faster than the wave, to a +subcritical one, where the fluid moves slower [14,34]. As a result, the spectrum of the wave +becomes enriched by higher-order harmonic of the frequency of forcing. +Qualitatively similar results were obtained at the vibration frequencies in the range from +30 Hz to 50 Hz, and they were validated by our theoretical analysis, the results of which are +presented in the following section. +Figure 5. Close-ups of the dispersion maps presented in Fig. 4. The linear fits of the dispersion bands +and the corresponding wave velocities are shown. +4. Theory +The theoretical description of a non-steady flow in the presence of deformable interfaces, +such as the flow in a thin liquid layer down a vibrated incline, is a notoriously difficult +hydrodynamic problem. The exact analysis is only available in the linear case, i.e. when +the deformation amplitude of the liquid-air interface is much smaller than the average film +thickness [42]. In the general case, when a harmonic vibration is applied both in the perpen- +dicular and parallel directions with respect to the the inclined plate, the unperturbed base +flow is given by a superposition of a steady Nusselt flow and of an additional harmonically +oscillating flow parallel to the incline with a flat free surface [42]. The Navier-Stokes equation +for an incompressible fluid can be linearised about the base flow and the Floquet theory-based +stability analysis determines if the flow is stable or unstable. +The full nonlinear problem with large amplitude deformation of the film surface can +only be studied approximately using simplified hydrodynamic models. Here we use the +well-known Shkadov model [52,53], which can be derived from the Navier-Stokes equation + +0.27 m/s +0.27 m/s +(a) +10 +0.1 +(b) +10 +0.1 +0.08 +0.08 +5 +5 +Frequency (Hz) +0.06 +Frequency (Hz) +0.06 +0 +0.04 +0.04 +-5 +-5 +0.02 +0.02 +-10 +-10 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +Wavevector (mm-1) +Wavevector (mm1)8 of 15 +by assuming a self-similar parabolic longitudinal velocity profile. The model developed by +Shkadov has been used earlier to study nonlinear solitary waves in falling liquid films in +the absence of vibration [22,23] and to investigate the onset of Faraday waves in vertically +vibrated isolated liquid drops [56]. +Thus, we consider a liquid film with the local film thickness h(x, t) flowing down an +inclined solid plate that makes an angle θ with the horizontal, as shown in Fig. 1. In our model, +the x-axis is chosen to be parallel to the plate with the positive direction pointing down the +incline. To capture rolling waves, we use a one-dimensional version of the Shkadov model, +which is formulated as a set of two coupled nonlinear equations for h(x, t) and the local flux +across the layer q(x, t) = � h(x,t) +0 +u(x, z, t) dz, where u(x, z, t) is the longitudinal flow velocity +and z-axis is perpendicular to the incline +ρ +� +∂tq + 6 +5∂x +�q2 +h +�� += +−3µq +h2 + σh∂3 +xh − ρg(t) cos(θ)h∂xh + ρg(t) sin(θ)h, +∂th + ∂xq += +0, +(1) +where µ is the dynamic viscosity, σ is the liquid-air surface tension and the time-dependent +gravity acceleration due to vibration is g(t) = g(1 + a cos(ωt)). The inclination of the plate +leads to a re-distribution of the vertical vibration into a longitudinal g(t) sin(θ) and an orthog- +onal g(t) cos(θ) components, respectively. +The base flow corresponds to a time-periodic spatially homogeneous flux q0(t) and a +flat film surface h0 = const. From Eqs. (1) we obtain the following expression by setting +∂xq0(t) = ∂xh0 = 0: +q0(t) = g sin(θ)h3 +0 +3ν +� +1 + +a cos(ωt) +(2h2 +0/3l2ac)2 + 1 + 2h2 +0 +3l2ac +a sin(ωt) +(2h2 +0/3l2ac)2 + 1 +� +, +(2) +where lac = √ +2ν/ω represents the length of the acoustic boundary layer. +In the absence of vibration, i.e. when a = 0, the base flow is the time-independent Nusselt +flow, where the linear stability is well-known in the case of a falling film, i.e. at θ = 90o +[22,23]. For an arbitrary inclination angle θ, the instability sets in when Re > cot(θ), where +Re = q0/ν = g sin(θ)h3 +0 +3ν2 +is the Reynolds number. To put this condition into perspective, for a +water film on a θ = 3o incline, the flow is unstable when h0 > 0.48 mm. The corresponding +instability is called gravitational instability and it leads to the onset of long surface waves +propagating downstream. The wavelength of unstable waves is longer than λc = 2π/kc, +where kc is the critical wave vector of the gravitational instability +kc = +� +ρg sin(θ) +σ +� +g sin(θ)h3 +0 +3ν2 +− cot(θ) +��1/2 +. +(3) +Neutrally stable waves with the wavelength λc = 2π/kc propagate downstream with a speed +c, which is twice as large as the surface speed in the Nusselt flow, i.e. c = g sin(θ)h2 +0/ν. +When the vibration is switched on, the Faraday instability mode develops and it competes +with the gravitational instability mode. The linear stability of a flat film flowing down an +incline under the combined action of the longitudinal and orthogonal vibration has been +investigated in Ref. [43] using a theoretical approach based on the exact linearisation of the +Navier-Stokes equation [42]. In the relevant work Ref. [57], an integral boundary layer model +has been formulated and applied to study nonlinear Faraday waves in liquid films on a +horizontal plate subjected to horizontal and vertical vibrations. In Refs. [44,45], the nonlinear +dynamics of a liquid film falling down a vertical vibrated plate is investigated theoretically + +9 of 15 +and experimentally. However, it should be emphasised that large-amplitude surface waves in +a liquid film flowing down an incline in the presence of both the longitudinal and orthogonal +vibrations have not been studied earlier. +To study the stability of the base flow Eqs. (2) we use the standard plane-wave ansatz +q(x, t) = q0(t) + ˜q(t)eikx and h(x, t) = h0 + ˜h(t)eikx, where k is the wavevector of the small- +amplitude perturbation. By differentiating the second equation in Eqs. (1) with respect to time +and the first equation with respect to x, the flux perturbation ˜q can be eliminated to yield a +complex-valued Mathieu-like equation for the film thickness perturbation ˜h +∂tt ˜h + A(t)∂t ˜h + B(t)˜h = 0, +(4) +with A(t) = 3ν +h2 +0 + 12 +5 ik q0(t) +h0 +and B(t) = ikg(t) sin(θ) + σ +ρ h0k4 + g(t) cos(θ)h0k2 + ik6ν q0(t) +h3 +0 +− +6 +5k2 q0(t)2 +h2 +0 . +According to the Floquet theory, the solution of Eq. (4) is given by ˜h(t) = H(t)eλt, +where H(t) is some bounded periodic function with the period T = 2π/ω and λ is the +Floquet exponent. The solution is stable when the real part of the largest Floquet exponent +is negative, i.e. Re(λ) < 0 and it is unstable otherwise. The Floquet exponents are related to +the monodromy matrix M via Re(λ) = 1 +T ln(|Λ|), where Λ is the eigenvalue of M. The 2 × 2 +complex-valued monodromy matrix M is given by the fundamental solution matrix that is +obtained by writing Eq. (4) as a system of two first-order equations and integrating it over one +period T with the unit 2 × 2 matrix as the initial condition. +For the inclination angle θ = 30, we choose the thickness of the water film h0 = 0.6 mm, +which is slightly above the critical value for the gravitational instability of h0 = 0.48 mm. The +marginal stability curves that correspond to Re(λ) = 0 are shown in Fig. 6 for four different +vibration frequencies f = 18, 20, 25, 48 Hz. The critical wavevector of the gravitational +instability Eq. (3) is marked by kc in Fig. 6d and it remains unaffected by the vibration. The +shaded regions in Fig. 6d indicate the unstable areas. The Faraday instability sets in at the +vibration amplitude ac that corresponds to the tip of the lowest Faraday tongue. The value +of ac, as extracted from Fig. 6, slightly increases with f, namely: ac = 0.33 for f = 18 Hz, +ac = 0.35 for f = 20 Hz, ac = 0.38 for f = 25 Hz and ac = 0.48 for f = 48 Hz. This observation +confirms the earlier statement that, for the range of frequencies between 30 Hz and 50 Hz, the +surface waves are much more sensitive to the changes of the vibration amplitude a than to the +changes of the vibration frequency f. Indeed, comparing Figs. 6c,d we see only a marginal +difference in the critical amplitude ac when the frequency is doubled. On the other hand, +increasing the value of a from a = 0.5 to a = 1 will significantly broaden the band of unstable +wavevectors of the Faraday instability, thus significantly changing the dynamics of the surface +waves. + +10 of 15 +0.2 +0.4 +0.6 +0.8 +1 +k (mm +-1) +0 +1 +2 +3 +4 +5 +a +0.2 +0.4 +0.6 +0.8 +1 +k (mm +-1) +0 +1 +2 +3 +4 +5 +a +0.2 +0.4 +0.6 +0.8 +1 +k (mm +-1) +0 +1 +2 +3 +4 +5 +a +0 +0.5 +1 +1.5 +2 +k (mm +-1) +0 +1 +2 +3 +4 +5 +a +(a) +(b) +(c) +(d) +f=18 Hz +f=20 Hz +f=25 Hz +f=48 Hz +kc +ac +Figure 6. Marginal stability curves of the base flow Eq. (2) for a 0.6 mm water film on a θ = 30 incline +vibrated at (a) f = 18 Hz, (b) f = 20 Hz, (c) f = 25 Hz and (d) f = 48 Hz. The shaded regions in +panel (d) highlights the unstable areas, kc indicates the critical wave vector of the gravitational long- +wave instability Eq. (3) and ac marks the critical vibration amplitude when the Faraday instability sets in. +To better understand the temporal signature of the surface waves in response to vibration, +we compute the imaginary part of the Floquet exponent Im(λ) = ω +2π (arg(Λ)) + ωn, where, +as before, Λ is the eigenvalue of the monodromy matrix and n is an arbitrary integer. Any +neutrally stable wave, i.e. Re(λ) = 0, can be represented in the form ˜h(x, t) = eikxH(t)eλt = +H(t)eikx+iIm(λ)t, where H(t) is a bounded 2π/ω-periodic function. Therefore, the temporal +spectrum of such a neutrally stable wave contains delta-peaks located at ω +2π (arg(Λ)) + ωn. +The temporal spectrum of a growing wave with Re(λ) > 0 contains the same delta peaks that +will appear slightly smeared. +At this stage, it is important to emphasise that the temporal response of the surface waves +that develop on the surface of a liquid layer on a vibrated incline is not necessarily harmonic +(frequencies ωn) or subharmonic (frequencies ω/2 + ωn). This feature is in stark contrast to +the standard Faraday instability in horizontal liquid layers, when the neutrally stable waves +are always harmonic or subharmonic standing waves [51]. In some special cases, however, +such as discussed in Ref. [45] for transversally vibrated falling liquid films, the magnitude +of ω +2π (arg(Λ)) may be close to zero or ω/2, leading to an almost harmonic or subharmonic +response. For the fluid parameters used in the present study, the frequency of the Faraday +mode is significantly shifted from ω or ω/2, as shown in Fig. 4b. + +11 of 15 +Next, we simulate the experimental conditions, at which the results shown in Fig. 4b +was obtained, to gain a better understanding of how the vibration changes the dynamics of +the waves in the early stages of evolution. Thus, we numerically integrate Eqs. (1) over the +time interval of 3 seconds in the system of length of 60 cm with periodic boundaries. The +vibration amplitude is a = 0.8 and the other parameters are the same as in Fig. 6d. As the +initial conditions, we use zero flux and random initial perturbation of the flat film surface +with the amplitude of 10−3 mm. The dispersion map is obtained taking the two-dimensional +Fourier transformation of the solution h(x, t). The contour lines of the dispersion map that +correspond to the level of 3% of its maximum are shown by the thick lines in Fig. 7. The thin +solid lines in Fig. 7 correspond to the dispersion curves Im(λ)(k), computed from Eq. (4) for +a = 0.8 and f = 48 Hz. It can be seen that the results of the direct simulation of the full system +Eq. (1) are in perfect agreement with the dispersion curves of the small-amplitude surface +waves. +-1 +-0.5 +0 +0.5 +1 +k (mm +-1) +-40 +-20 +0 +20 +40 +f (Hz) +Figure 7. Contour plot (thick lines) of the dispersion map of the solution of Eqs. (1) computed over the +time interval of 3 seconds with a random initial perturbation of the flat film surface. The thickness of the +water film is h = 0.6 mm and the vibration parameters are a = 0.8 and f = 48 Hz, similarly to Fig. 4b. +The thin solid lines correspond to the imaginary part of the Floquet exponent Im(λ) of the most unstable +mode. +The dispersion map in Fig. 7 is dominated by the delta-peaks located at f = ±14 and f = +±34 Hz, thus confirming that the primary response of the liquid film to a harmonic vibration +is neither harmonic, nor subharmonic. Qualitatively, the shift of the response frequency away +from the standard for the Faraday instability subharmonic mode can be explained as follows. +In a horizontal layer vibrated at frequency ω, the Faraday instability sets in the form of a +standing wave oscillating at the subharmonic frequency ω/2. Any standing wave can be +represented as a superposition of two plane waves travelling at the phase speed of c = ω/(2k) +in the opposite directions, i.e. h(x, t) = Aeiωt/2+ikx + Aeiωt/2−ikx + cc. When the layer is +slightly inclined with the positive direction pointing downstream, it would be reasonable to + +12 of 15 +assume that the plane wave propagating downstream will increase its phase speed by some +amount δc, but the wave propagating upstream will decrease its phase speed by the same +amount δc. Assuming that the wavevector remains unaffected by a small inclination angle, +the resulting solution is represented by h(x, t) = Aei(ω/2−kδc)t/2+ikx + Aei(ω/2+kδc)t/2−ikx + cc. +Therefore, the temporal spectrum of h(x, t) will contain delta-peaks located at ±(ω/2 + kδc) +and (±(ω/2 − kδc)), in agreement with Fig. 7. +Alongside the delta-peaks, the dispersion map in Fig. 7 also contains a band of linearly +unstable plane waves with the wavevectors k < kc. These long waves are amplified as the +result of the gravitational instability mode. It can be seen that the gravitational band falls +perfectly on the central dispersion curve that passes through the origin. The central dispersion +branch in Fig. 7 is almost indistinguishable from the dispersion curve in the absence of +vibration (not shown). This allows us to conclude that a relatively strong vibration (sufficiently +strong to excite Faraday waves) has almost no effect on the phase speed c = Im(λ)/k of the +long gravitational surface waves. +To study the interaction between the Faraday waves and gravitational surface waves in +the nonlinear regime, we solve Eqs. (1) over the time interval of 15 seconds with and without +vibration and compare the respective dispersion maps in Fig. 8. +Figure 8. (a) Dispersion map obtained from the solution of Eqs. (1) in the absence of vibration for a +0.6 mm water film on a θ = 3o incline. (b) Dispersion map of the solution of Eqs. (1) when the inclined +plane vibrated at f = 48 Hz with the amplitude a = 0.8. The scaling for the vertical axis is in arbitrary +logarithmic units. +It is evident from Fig. 8 that vibration leads to a suppression of the long surface waves. +Indeed, the magnitude of the dispersion band that corresponds to the gravitational waves is +significantly smaller when the film is vibrated. This result is in qualitative agreement with +Fig. 2. +5. Conclusions +In conclusion, our experiments with a sub-millimetre thick water layer on a slightly +inclined vertically vibrated plate demonstrate that low-frequency vibration in the range +between 30 and 50 Hz suppresses the development of long rolling surface waves propagating +downstream. These surface waves appear as the result of the long-scale gravitational instability +of the base flow in the absence of vibration [15,16] and may also be excited by mechanically +perturbing the flow at the inlet. A relatively small thickness of the water layer (under 1 mm) +is required to suppress the three-dimensional instability of the rolling waves that is known +to develop at large flow rates. Experimental findings are verified using a boundary-layer + +15 +(a) +15 +(b) +10 +10 +5 +5 +0 +0 +-5 +-5 +-10 +-10 +-15 +-15 +50 +50 +40 +40 +30F +30 +20 +20 +10 +10 +f (Hz) +f (Hz) +-10 +k (mm-1) +-10 +k (mm-1) +-20F +-20 +-30 +0.5 +-30 +0.5 +0 +40 +0.5 +-40 +0 +-0.5 +-50 +50 +-113 of 15 +hydrodynamic model [52,53] obtained from the Navier-Stokes equation by assuming a self- +similar parabolic longitudinal flow velocity. Linear stability and nonlinear dynamics of the +surface waves obtained with the model qualitatively confirm the main experimental findings. +Without vibration, the Fourier content of surface waves is represented by a broad band +of unstable wave vectors k < kc, where kc is a critical cut-off wave vector of the gravitational +instability (see Eq. 3). As the instability unfolds, the wavelength of the dominant wave quickly +increases until it develops into a solitary-like wave [24]. For fluids with a relatively small +viscosity, such as water, the characteristic time required for solitary rolling waves to develop +on a 3o incline is of the order of several seconds. In the nonlinear regime, the Fourier content +of the surface waves is dominated by solitary-like waves characterised by a small wavevector +with a background of smaller amplitude shorter waves, which is shown in Fig. 8a and Fig. 4a. +We observe that the properties of the surface waves change dramatically when the layer +is vibrated. Thus, a relatively weak vibration (the vibration amplitude a < g) leads to the +onset of the secondary Faraday instability in the form of short waves with a wavelength +of λ ≈ 5 . . . 10 mm when vibrated at f = 48 Hz. In agreement with the earlier theoretical +studies [42–44], the temporal frequency of the Faraday waves is shifted away from the +harmonic (48 Hz) and subharmonic (24 Hz) response that is typical of Faraday instability +in horizontal liquid layers. In fact, the inclination angle of the plate acts as a wave filter, +splitting a standing Faraday wave into two plane waves: one propagating upstream and +one propagating downstream. Similarly to the Doppler effect, the wave that propagates +downstream increases its speed and, therefore, increases its temporal frequency, while the +wave that propagates upstream decreases its speed and frequency. For water layers vibrated +at 48 Hz, we observe the following shifts in frequency away from the subharmonic response: +from 24 Hz to approximately 40 Hz for the downstream wave and from 24 Hz to approximately +8 Hz for the upstream wave. +In the nonlinear regime, the interaction between shorter Faraday waves and longer +gravitational waves leads to the broadening of their respective bands in the f-k dispersion +map. Most importantly, we find that the average and peak amplitudes of the long-scale +gravitational waves are significantly reduced when vibration is applied. This result is rather +intriguing since the total influx of energy is larger in the vibrated system when both gravity +and vibration together drive the flow, unlike in the non-vibrated case, where the only source +of energy is due to gravity. Yet, nonlinear wave interaction leads to an uneven re-distribution +of energy amongst the Faraday and gravitational waves in favour of the former. The physical +mechanism responsible for the suppression of gravitational waves remains an open question; +however, it is plausible to assume that the fast-oscillating fluid flow in the form of circulation +patterns [58] in pulsating Faraday waves may slow down the redistribution of fluid on the +large scale, required for the growth and development of the gravitational waves. This result is +even more surprising since we did not observe any noticeable change in the velocity of the +gravitational waves induced by vibration. +Apart from a contribution of the fundamental knowledge, the results presented in this +work may be used to better understand and further improve certain technological processes +that rely on falling liquid films. 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E 2022, +105, 044206. + diff --git a/59E1T4oBgHgl3EQfmwQa/content/tmp_files/load_file.txt b/59E1T4oBgHgl3EQfmwQa/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3917071d03ffced2eec423b07b6211178b8b19fe --- /dev/null +++ b/59E1T4oBgHgl3EQfmwQa/content/tmp_files/load_file.txt @@ -0,0 +1,1150 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf,len=1149 +page_content='Article Experimental and Theoretical Study of Solitary-like Wave Dynamics of Liquid Film Flows over a Vibrated Inclined Plane Ivan S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Maksymov 1* and Andrey Pototsky 2 Maksymov, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Pototsky, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Experimental and theoretical study of solitary-like wave dynamics of liquid film flows over a vibrated inclined plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Preprints 2022, 1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='org/ Received: Accepted: Published: 1 Optical Sciences Centre, Swinburne University of Technology, Hawthorn, VIC 3122, Australia;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' imaksymov@swin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='au;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Tel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' : +61-3-3921-4805 2 Department of Mathematics, Swinburne University of Technology, Hawthorn, VIC 3122, Australia;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' apototskyy@swin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='au;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Tel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' : +61-3-9214-4653 Abstract: Solitary-like surface waves that originate from the spatio-temporal evolution of falling liquid films have been the subject of theoretical and experimental research due to their unique properties that are not readily observed in the physical system of other nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Here we investigate, experimentally and theoretically, the dynamics of solitary-like surface waves in a liquid layer on an inclined plane that is subjected to a harmonic low-frequency vibration in the range from 30 to 50 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' We demonstrate that the vibration results in a decrease in the average and peak amplitude of the long solitary-like surface waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' However, the speed of these waves remains largely unaffected by the vibration, implying that they may propagate over large distances almost without changing their amplitude, thus rendering them suitable for a number of practical applications, where the immunity of pulses that carry information to external vibrations is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Keywords: falling liquid films;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' solitary waves;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' surface waves;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' vibrations 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Introduction Solitary waves—physical waves that maintain their shape and move with a constant velocity due to a cancellation of nonlinear effects and dispersive processes in the medium [1]—have been a long-term subject of fundamental and applied research studies in the fields of optics [2], fluid dynamics [3], magnetism [4], acoustics [5], electronics [6] and biology [7,8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' However, despite a good understanding of the physical properties of solitary waves of different kinds, their experimental studies often involve expensive and difficult to operate equipment such as intense laser beams and nonlinear-optical materials in the field of optics [2] and sources of high-power microwave radiation in the field of magnetism [4], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Yet, in some systems such as biological nerve fibres [7,8] the observation of solitary-like waves requires significant preparatory works and is possible mostly when a number of specific experimental conditions are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Such technical challenges complicate both fundamental studies and verification of numerous theoretical works predicting that solitary waves could be used in communication [9,10], sensing [11] and data processing [12] devices and systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' There also exists a class of material solitary-like surface waves that originate from spatio- temporal evolution of falling liquid films [13,14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Since the equipment needed to create falling liquid films is, in general, simpler than that used in experiments in the fields of optics and magnetism, the waves of this kind have attracted attention of many scientists [15–28] following the pioneering experiments conducted by the Kapitzas [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' In fact, while such solitary-like surface waves share many physical features with the other known types of solitary waves, they can exhibit unique physical properties not observed in other systems [24,30,31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' For instance, they can merge instead of passing through each other without significant change, with the latter being the case of two solitary waves governed by the well-known KdV equation [3,32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The analysis of solitary-like surface waves in flowing liquid films is also important because liquid films, as well as similar physical systems [33–35], are often encountered in arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='03300v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='flu-dyn] 9 Jan 2023 2 of 15 the fields of earth and planetary sciences [36,37] and in technological processes [38], where the liquids of interest can also experience temperature gradients [14,28] and vibrations [39– 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Given this, the effect of vibrations on the wave dynamics of film flows has become an independent subject of fundamental and applied research [42–46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' In particular, it has been shown that vibrations can suppress certain waves on the surface of flowing liquid films [42] but in a relevant experiment [47] it has been demonstrated that vibrations can promote unusual regimes of spontaneous drop movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Speaking broadly, the study of the effect of vibrations should also help develop communication, sensing and data processing systems that are immune to undesirable mechanical impacts on devices that use liquids as a medium that provides the critical functionality (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=', [48–50]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Although, traditionally, greater attention has been paid to the wave dynamics on free- falling vertical liquid films [13,14], studies of surface waves on liquid films flowing over slightly inclined planes have also been conducted given an essentially the same physics as in the case of vertical systems [24,42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' However, reports on experimental results involving the effect of vibrations are rather scarce and scattered in the literature sources [39,40,45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' In particular, in [39] it has been shown that the vibration of a horizontal tube with a liquid thin film flowing over it results in the appearance of ripple waves at the vibration frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The amplitude of the so-created waves depends on the vibration amplitude and can reach the amplitude of periodic waves existing on the film surface without vibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Subsequently, high-amplitude vibrations result in an increase in the film thickness and a concomitant increase in the speed of the waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' However, the opposite conclusions were drawn in [40], which is, most likely, a result of the differences in the system (a liquid film under two-phase flow conditions) investigated in that paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' It is also well-known that in horizontal liquid layers a harmonic vibration excites two different types of standing surface waves: harmonic waves that oscillate at the vibration frequency and subharmonic waves that oscillate at the half of the vibration frequency [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' However, the presence of a mean flow across the layer changes the response frequency of the excited waves [42–44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Surface waves excited by harmonic vibration in a liquid film flowing over a vertical plane were investigated experimentally in [45] and the results obtained in that work validated the linear theory developed in [42–44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Thus, mostly the experimental work [45] represents an attempt to systematically study the physics of wave motion on a vibrated plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' However, in general, building a setup involving liquids flowing down a vibrated vertical surface requires non-standard equipment built according to demanding technical specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' In particular, the liquid should be supplied to the inlet located at the upper part of the plane so that the flow rate is not affected by the vibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' This is because the thickness of the liquid film is known to be very sensitive to external disturbances, including vibrations caused by the pump used to deliver the liquid from a reservoir to the inlet [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Moreover, the shaker producing the vibration should be connected to the vertically positioned surface via a vibration transmission structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Some of the engineering challenges of creating such a structure are the need to move a considerable total mass of the supporting structure and liquid with high precision, and to ensure that the amplitude of the vibration across the plane area is uniform [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' To resolve the problem of non-uniform vibration amplitude, in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' [45] it is was suggested that qualitatively similar results could be obtained vibrating just one side of the plane, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' vibrating just a portion of the liquid, thus also significantly reducing the total mass that needs to be moved by the shaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' In this paper, we present and discuss a technically simple and compact experimental setup for the investigation of solitary-like surface waves on a slightly inclined plane positioned on top of a vibrating table and equipped with an auxiliary channel that recycles the liquid used in experiment, thus decreasing the chance of spills of the liquid and its unwanted contact with the measurement and imaging equipment, and also decreasing the total mass that needs to be moved by the shaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' We employ this setup to demonstrate that the instabilities of 3 of 15 the thin liquid film caused by the vibrations result in a decrease in the peak amplitude of the solitary-like surface waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' We conclude that, despite these changes, the speed of the solitary-like waves does not appreciably change due to vibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' As a result, these waves can propagate for long distances without changing their shape and, therefore, can be used in the practical applications discussed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' We also demonstrate the advantage of using frequency-wavevector dispersion maps for the analysis of the properties of rolling waves, thus extending the toolbox of experimentalists working on this class of wave motion phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Our experimental results are validated using the Shkadov model [52,53]—a boundary-layer hydrodynamic model derived from the Navier-Stokes equation under the assumption of self-similar parabolic longitudinal velocity flow field across the layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Sketch of the experimental setup used to observe the solitary-like surface waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' For the sake of clarity, only the main constructive features are shown, including the inclined plate, where the waves are observed, the pathway for recycling of the used liquid and the vibrating table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The dimensions and relative positions of the components in this sketch are not to scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Background and Experimental Methods When a single-layer liquid film flows down an inclined plane with a no-slip boundary, the resulting Nusselt flat film flow profile assumes a parabolic longitudinal velocity shape having the largest velocity at the free surface [13,14,24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' In this flow regime, a long-wavelength surface instability develops when the average flow rate exceeds a certain critical value [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' When the disturbances are excited naturally, in general four regimes of different wave be- haviour can be observed in the downstream regions of the inclined plane [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The first regime is observed in a section of the plane that is adjacent to the inlet of the liquid, where small disturbances caused by the inlet structure are amplified while moving downstream and forming predominantly monochromatic waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The second regime is observed in the following downstream region, where the monochromatic waves grow in amplitude and then develop higher-order frequency harmonics due to nonlinear effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Then, as a result of com- plex nonlinear interactions, two-dimensional solitary-like waves are formed, and then they propagate further downstream exhibiting unique properties that, in part, coincide with those of other known solitary waves but, in general, are unique [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Finally, three-dimensional waves start to form due to transverse variations [13,14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' It is noteworthy that not all aforementioned regimes can necessarily be observed in practice [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Yet, it is well-known that when the initial natural disturbance at the inlet is nearly monochromatic, the waves emerging in the region located immediately after the inlet can first inherit the frequency of the disturbance and then evolve into a solitary-like wave far UV light digital camera inlet rolling waves dwnd inclined plane H recycled liquid vibrating table shaker accelerometer fluorescent liquid4 of 15 downstream [13,14,24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' However, when either the thickness of the liquid film or the fluid flow is periodically modulated at the inlet, solitary-like surface waves develop almost immediately after leaving the inlet area [14,24], which indicates that the nonlinear evolution of the flow over an inclined plane is dominated by solitary-like waves independently of whether their formation was deliberately forced or resulted naturally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Figure 1 shows a sketch of the setup that enables observing the formation of both forced and natural (unforced) solitary-like surface waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The setup is assembled on a vibrating table that is driven by a shaker (35 W, 20–80 Hz response, Dayton Audio, USA) and where the vibration amplitude is controlled by an analog accelerometer (ADXL326, Analog Devices, USA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The inclined plane, where the waves are observed, is a 5-cm-wide and 50-cm-long rigid aluminium plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The surface of the plate was chemically treated to improve the formation of the liquid film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The inclination angle is θ = 3 o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The inclined plate was mounted on top of wider open channel used to recycle the liquid by redirecting it to the main reservoir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' A low-vibration DC voltage pump driven via a customised electronic circuit was used to supply water from the reservoir to the inlet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The electronic modulation of the pump flow rate enabled controlling the thickness of the liquid film and creating solitary-like waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' At the stage of preparation to the experiments, an organic fluorescent dye (Tintex, Australia) was added to tap water in the concentration of 1 g per litre, thus leading to the emission of bright green fluorescence light when the surface of the inclined plate was illuminated with UV-A light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' All experiments were conducted in a darkened room using an overhead digital camera capable of recording videos in a slow motion regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The resulting videos were post-processed in Octave software using customised computational procedures enabling the extraction of the wave amplitude from the intensity profile of the fluorescence images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' All experiments were conducted in an acoustically isolated room with environmental humidity and temperature levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Instantaneous profiles the surface waves in a liquid film flowing over the inclined plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The profiles were obtained from the fluorescence images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The frequency of the flow forcing resulting in the formation of solitary-like waves is 2 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' (a) No vibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' (b) The inclined plate is vibrated with the frequency of 48 Hz and the peak vibration amplitude of 1g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Experimental Results Figure 2 shows the representative images of the solitatry-like surface waves propagating over a downstream section of the inclined plate, when the vibration is turned off (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 2a) and when the plate is vibrated with the frequency of 48 Hz and the peak amplitude of 1g (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 2b), being g the gravitational acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The frequency of the forcing of the solitary-like Amplitude (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' units) (a) Amplitude (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' units) (b) 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='2 X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='2 Downstream distance (m) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='25 X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='02 X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='02 Plate width (m) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='35 0 Plate width (m5 of 15 waves is 2 Hz in both panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The images were obtained from the selected individual fluorescence frames of the recorded videos of the propagating waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Without vibration (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 2a), we can observe a train of downstream-propagating solitary pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' A closer inspection also reveals the existence of periodic waves with an amplitude that is much smaller than that of solitary-like waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' When the plate is subjected to vibration (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 2b), we continue observing a train of solitary pulses with an approximately the same pulse periodicity as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' However, the peak amplitude of the pulses is lower than in the case without vibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Yet, in agreement with the relevant theory [42,44] and experiment on the vertical plane [45], we also observe the short wavelength ripples arising due to the onset of the Faraday instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Using our fluorescence intensity analysis software, we register the profiles of the waves at the points located on the centreline of the inclined plate along the downstream direction, and we plot the so-obtained data as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The resulting spatiotemporal false-colour maps are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 3 with the observation period of 2 s for the scenario of no vibration (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 3a) and with the 48 Hz vibration (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' In those figures, we can see the traces of several solitary-like waves that propagate in the downstream direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The traces are more distinguishable and have a higher false-colour amplitudes in the case of no vibration than with the vibration, which confirms our observation of a decrease in the peak amplitude of the solitary pulses due to the vibration in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The ripple waves caused by the vibration-induced Faraday instabilities can also be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' It is noteworthy that the separation between the traces and the relative position of the traces in the time-downstream coordinate space are very similar without and with the vibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' This indicates that, even though the peak amplitude of the solitary-like waves is affected by the vibration, in general the vibration does not change the shape of the soliton pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' False-colour maps showing the spatiotemporal traces of solitary-like waves forced at the frequency of 2 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' (a) No vibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' (b) The inclined plate is vibrated with the frequency of 48 Hz and the peak vibration amplitude of 1g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Then, we apply a two-dimensional Fourier transformation to the spatiotemporal data to obtain the dispersion maps as a function of frequency f and wavevector k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Since the speed of a wave is given by u = ω/k = 2π f /k, using the resulting dispersion maps we can identify the bands of constant f /k ratio that correspond to waves travelling along the inclines plate at (a) 2 (b) 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='5 S S 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='5 0 OF 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='4 Downstream distance (m) Downstream distance (m)6 of 15 a constant speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Yet, we apply the standard f-k filtering procedures to remove noise from the dispersion characteristics [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Figure 4 shows the dispersion maps for the case of no vibration (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 4a) and with the 48 Hz vibration (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' While the negative frequency regions of the dispersion maps originate from the mathematical properties of the Fourier transformation, the sign of the wavevector has the physical meaning as it determines the direction of the wave propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' We first analyse the dispersion map in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 4a and its zoomed image presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 5a, where we can see a set of high-magnitude discrete bands that are superimposed on a broader continuum band of a lower magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The frequencies of the discrete bands correspond to the forcing frequency of the solitary-like waves 2 Hz and its higher-order harmonics of 4, 6 and 8 Hz and so forth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The origin of the harmonics is due to the nonlinear effects as discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The spectrum of the discrete bands changes as the frequency of forcing of the solitary- like waves is changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' When the modulation of the pump flow was turned off, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' with no wave forcing, the discrete bands completely disappeared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' However, a continuum band was always observed independently of whether the forcing was turned on or off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Subsequently, we associate the continuum band with natural periodic rolling waves propagating on the surface of the liquid film flowing over the inclined surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' According to the frequency-wavevector spectral analysis theory [54], a fit of the observed bands with a straight line produces the velocity of the solitary-like wave of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='02 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' When the plate is vibrated (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 4b), in addition to the dispersion bands discussed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 4a we observe two new isolated bands that can be associated with the Faraday instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Moreover, the close-up of the dispersion map (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 5b) shows that the magnitude of the discrete modes decreased due to the vibration, which is an observation that is consistent with our conclusions made earlier in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Yet, the bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 5b can also be fitted with a straight line that corresponds to the wave velocity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='02 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Dispersion maps of the solitary-like waves forced at the frequency of 2 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' (a) No vibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' (b) The inclined plate is vibrated with the frequency of 48 Hz and the peak vibration amplitude of 1g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The plots are slightly oversaturated for the sake of a better visual presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Empirically, the presence of the discrete bands at the forcing frequency of 2 Hz and its higher-order harmonics can be explained using the well-established analogy between the rolling waves and acoustic waves [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Indeed, the solitary-like surface waves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 2 can be regarded as large-amplitude shock-like disturbances (in the sub-field of physically (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='1 (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='1 40 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='08 20 20 Frequency (Hz) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='06 Frequency (Hz) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='06 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='04 20 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='02 40 40 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='5 Wavevector (mm-1) Wavevector (mm-1)7 of 15 similar roll waves in open channel such a discontinuity is called the hydraulic jump [14,21,34]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Shock waves are also well-known in the field of nonlinear acoustic, where their formation is accompanied by strong nonlinear effects such as the generation of higher-order harmonic frequencies [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Considering longitudinal acoustic waves that can be described as alternating areas of compression and rarefaction in the medium, we can show that the points of the crests of an acoustic wave travel faster than the speed of sound in the medium, but the points of the wave troughs travel slower [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' This physical process underpins the formation of an acoustic shock wave [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' In turn, in the field of rolling waves, the crest of a large-amplitude solitary-like wave is connected to its trough by a discontinuity, where the flow regime abruptly changes from a supercritical condition and where the fluid moves faster than the wave, to a subcritical one, where the fluid moves slower [14,34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' As a result, the spectrum of the wave becomes enriched by higher-order harmonic of the frequency of forcing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Qualitatively similar results were obtained at the vibration frequencies in the range from 30 Hz to 50 Hz, and they were validated by our theoretical analysis, the results of which are presented in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Close-ups of the dispersion maps presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The linear fits of the dispersion bands and the corresponding wave velocities are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Theory The theoretical description of a non-steady flow in the presence of deformable interfaces, such as the flow in a thin liquid layer down a vibrated incline, is a notoriously difficult hydrodynamic problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The exact analysis is only available in the linear case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' when the deformation amplitude of the liquid-air interface is much smaller than the average film thickness [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' In the general case, when a harmonic vibration is applied both in the perpen- dicular and parallel directions with respect to the the inclined plate, the unperturbed base flow is given by a superposition of a steady Nusselt flow and of an additional harmonically oscillating flow parallel to the incline with a flat free surface [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The Navier-Stokes equation for an incompressible fluid can be linearised about the base flow and the Floquet theory-based stability analysis determines if the flow is stable or unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The full nonlinear problem with large amplitude deformation of the film surface can only be studied approximately using simplified hydrodynamic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Here we use the well-known Shkadov model [52,53], which can be derived from the Navier-Stokes equation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='27 m/s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='27 m/s (a) 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='1 (b) 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='08 5 5 Frequency (Hz) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='06 Frequency (Hz) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='06 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='04 5 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='02 10 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='6 Wavevector (mm-1) Wavevector (mm1)8 of 15 by assuming a self-similar parabolic longitudinal velocity profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The model developed by Shkadov has been used earlier to study nonlinear solitary waves in falling liquid films in the absence of vibration [22,23] and to investigate the onset of Faraday waves in vertically vibrated isolated liquid drops [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Thus, we consider a liquid film with the local film thickness h(x, t) flowing down an inclined solid plate that makes an angle θ with the horizontal, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' In our model, the x-axis is chosen to be parallel to the plate with the positive direction pointing down the incline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' To capture rolling waves,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' we use a one-dimensional version of the Shkadov model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' which is formulated as a set of two coupled nonlinear equations for h(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' t) and the local flux across the layer q(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' t) = � h(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='t) 0 u(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' t) dz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' where u(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' t) is the longitudinal flow velocity and z-axis is perpendicular to the incline ρ � ∂tq + 6 5∂x �q2 h �� = −3µq h2 + σh∂3 xh − ρg(t) cos(θ)h∂xh + ρg(t) sin(θ)h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' ∂th + ∂xq = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' (1) where µ is the dynamic viscosity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' σ is the liquid-air surface tension and the time-dependent gravity acceleration due to vibration is g(t) = g(1 + a cos(ωt)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The inclination of the plate leads to a re-distribution of the vertical vibration into a longitudinal g(t) sin(θ) and an orthog- onal g(t) cos(θ) components, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The base flow corresponds to a time-periodic spatially homogeneous flux q0(t) and a flat film surface h0 = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' (1) we obtain the following expression by setting ∂xq0(t) = ∂xh0 = 0: q0(t) = g sin(θ)h3 0 3ν � 1 + a cos(ωt) (2h2 0/3l2ac)2 + 1 + 2h2 0 3l2ac a sin(ωt) (2h2 0/3l2ac)2 + 1 � , (2) where lac = √ 2ν/ω represents the length of the acoustic boundary layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' In the absence of vibration, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' when a = 0, the base flow is the time-independent Nusselt flow, where the linear stability is well-known in the case of a falling film, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' at θ = 90o [22,23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' For an arbitrary inclination angle θ, the instability sets in when Re > cot(θ), where Re = q0/ν = g sin(θ)h3 0 3ν2 is the Reynolds number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' To put this condition into perspective, for a water film on a θ = 3o incline, the flow is unstable when h0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='48 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The corresponding instability is called gravitational instability and it leads to the onset of long surface waves propagating downstream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The wavelength of unstable waves is longer than λc = 2π/kc, where kc is the critical wave vector of the gravitational instability kc = � ρg sin(θ) σ � g sin(θ)h3 0 3ν2 − cot(θ) ��1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' (3) Neutrally stable waves with the wavelength λc = 2π/kc propagate downstream with a speed c, which is twice as large as the surface speed in the Nusselt flow, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' c = g sin(θ)h2 0/ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' When the vibration is switched on, the Faraday instability mode develops and it competes with the gravitational instability mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The linear stability of a flat film flowing down an incline under the combined action of the longitudinal and orthogonal vibration has been investigated in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' [43] using a theoretical approach based on the exact linearisation of the Navier-Stokes equation [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' In the relevant work Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' [57], an integral boundary layer model has been formulated and applied to study nonlinear Faraday waves in liquid films on a horizontal plate subjected to horizontal and vertical vibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' In Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' [44,45], the nonlinear dynamics of a liquid film falling down a vertical vibrated plate is investigated theoretically 9 of 15 and experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' However, it should be emphasised that large-amplitude surface waves in a liquid film flowing down an incline in the presence of both the longitudinal and orthogonal vibrations have not been studied earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' To study the stability of the base flow Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' (2) we use the standard plane-wave ansatz q(x, t) = q0(t) + ˜q(t)eikx and h(x, t) = h0 + ˜h(t)eikx, where k is the wavevector of the small- amplitude perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' By differentiating the second equation in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' (1) with respect to time and the first equation with respect to x, the flux perturbation ˜q can be eliminated to yield a complex-valued Mathieu-like equation for the film thickness perturbation ˜h ∂tt ˜h + A(t)∂t ˜h + B(t)˜h = 0, (4) with A(t) = 3ν h2 0 + 12 5 ik q0(t) h0 and B(t) = ikg(t) sin(θ) + σ ρ h0k4 + g(t) cos(θ)h0k2 + ik6ν q0(t) h3 0 − 6 5k2 q0(t)2 h2 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' According to the Floquet theory, the solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' (4) is given by ˜h(t) = H(t)eλt, where H(t) is some bounded periodic function with the period T = 2π/ω and λ is the Floquet exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The solution is stable when the real part of the largest Floquet exponent is negative, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Re(λ) < 0 and it is unstable otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The Floquet exponents are related to the monodromy matrix M via Re(λ) = 1 T ln(|Λ|), where Λ is the eigenvalue of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The 2 × 2 complex-valued monodromy matrix M is given by the fundamental solution matrix that is obtained by writing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' (4) as a system of two first-order equations and integrating it over one period T with the unit 2 × 2 matrix as the initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' For the inclination angle θ = 30, we choose the thickness of the water film h0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='6 mm, which is slightly above the critical value for the gravitational instability of h0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='48 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The marginal stability curves that correspond to Re(λ) = 0 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 6 for four different vibration frequencies f = 18, 20, 25, 48 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The critical wavevector of the gravitational instability Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' (3) is marked by kc in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 6d and it remains unaffected by the vibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The shaded regions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 6d indicate the unstable areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The Faraday instability sets in at the vibration amplitude ac that corresponds to the tip of the lowest Faraday tongue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The value of ac, as extracted from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 6, slightly increases with f, namely: ac = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='33 for f = 18 Hz, ac = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='35 for f = 20 Hz, ac = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='38 for f = 25 Hz and ac = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='48 for f = 48 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' This observation confirms the earlier statement that, for the range of frequencies between 30 Hz and 50 Hz, the surface waves are much more sensitive to the changes of the vibration amplitude a than to the changes of the vibration frequency f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Indeed, comparing Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 6c,d we see only a marginal difference in the critical amplitude ac when the frequency is doubled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' On the other hand, increasing the value of a from a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='5 to a = 1 will significantly broaden the band of unstable wavevectors of the Faraday instability, thus significantly changing the dynamics of the surface waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 10 of 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='8 1 k (mm 1) 0 1 2 3 4 5 a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='8 1 k (mm 1) 0 1 2 3 4 5 a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='8 1 k (mm 1) 0 1 2 3 4 5 a 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='5 2 k (mm 1) 0 1 2 3 4 5 a (a) (b) (c) (d) f=18 Hz f=20 Hz f=25 Hz f=48 Hz kc ac Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Marginal stability curves of the base flow Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' (2) for a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='6 mm water film on a θ = 30 incline vibrated at (a) f = 18 Hz, (b) f = 20 Hz, (c) f = 25 Hz and (d) f = 48 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The shaded regions in panel (d) highlights the unstable areas, kc indicates the critical wave vector of the gravitational long- wave instability Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' (3) and ac marks the critical vibration amplitude when the Faraday instability sets in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' To better understand the temporal signature of the surface waves in response to vibration, we compute the imaginary part of the Floquet exponent Im(λ) = ω 2π (arg(Λ)) + ωn, where, as before, Λ is the eigenvalue of the monodromy matrix and n is an arbitrary integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Any neutrally stable wave, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Re(λ) = 0, can be represented in the form ˜h(x, t) = eikxH(t)eλt = H(t)eikx+iIm(λ)t, where H(t) is a bounded 2π/ω-periodic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Therefore, the temporal spectrum of such a neutrally stable wave contains delta-peaks located at ω 2π (arg(Λ)) + ωn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The temporal spectrum of a growing wave with Re(λ) > 0 contains the same delta peaks that will appear slightly smeared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' At this stage, it is important to emphasise that the temporal response of the surface waves that develop on the surface of a liquid layer on a vibrated incline is not necessarily harmonic (frequencies ωn) or subharmonic (frequencies ω/2 + ωn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' This feature is in stark contrast to the standard Faraday instability in horizontal liquid layers, when the neutrally stable waves are always harmonic or subharmonic standing waves [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' In some special cases, however, such as discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' [45] for transversally vibrated falling liquid films, the magnitude of ω 2π (arg(Λ)) may be close to zero or ω/2, leading to an almost harmonic or subharmonic response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' For the fluid parameters used in the present study, the frequency of the Faraday mode is significantly shifted from ω or ω/2, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 11 of 15 Next, we simulate the experimental conditions, at which the results shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 4b was obtained, to gain a better understanding of how the vibration changes the dynamics of the waves in the early stages of evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Thus, we numerically integrate Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' (1) over the time interval of 3 seconds in the system of length of 60 cm with periodic boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The vibration amplitude is a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='8 and the other parameters are the same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 6d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' As the initial conditions, we use zero flux and random initial perturbation of the flat film surface with the amplitude of 10−3 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The dispersion map is obtained taking the two-dimensional Fourier transformation of the solution h(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The contour lines of the dispersion map that correspond to the level of 3% of its maximum are shown by the thick lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The thin solid lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 7 correspond to the dispersion curves Im(λ)(k), computed from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' (4) for a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='8 and f = 48 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' It can be seen that the results of the direct simulation of the full system Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' (1) are in perfect agreement with the dispersion curves of the small-amplitude surface waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='5 1 k (mm 1) 40 20 0 20 40 f (Hz) Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Contour plot (thick lines) of the dispersion map of the solution of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' (1) computed over the time interval of 3 seconds with a random initial perturbation of the flat film surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The thickness of the water film is h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='6 mm and the vibration parameters are a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='8 and f = 48 Hz, similarly to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The thin solid lines correspond to the imaginary part of the Floquet exponent Im(λ) of the most unstable mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The dispersion map in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 7 is dominated by the delta-peaks located at f = ±14 and f = ±34 Hz, thus confirming that the primary response of the liquid film to a harmonic vibration is neither harmonic, nor subharmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Qualitatively, the shift of the response frequency away from the standard for the Faraday instability subharmonic mode can be explained as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' In a horizontal layer vibrated at frequency ω, the Faraday instability sets in the form of a standing wave oscillating at the subharmonic frequency ω/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Any standing wave can be represented as a superposition of two plane waves travelling at the phase speed of c = ω/(2k) in the opposite directions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' h(x, t) = Aeiωt/2+ikx + Aeiωt/2−ikx + cc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' When the layer is slightly inclined with the positive direction pointing downstream, it would be reasonable to 12 of 15 assume that the plane wave propagating downstream will increase its phase speed by some amount δc, but the wave propagating upstream will decrease its phase speed by the same amount δc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Assuming that the wavevector remains unaffected by a small inclination angle, the resulting solution is represented by h(x, t) = Aei(ω/2−kδc)t/2+ikx + Aei(ω/2+kδc)t/2−ikx + cc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Therefore, the temporal spectrum of h(x, t) will contain delta-peaks located at ±(ω/2 + kδc) and (±(ω/2 − kδc)), in agreement with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Alongside the delta-peaks, the dispersion map in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 7 also contains a band of linearly unstable plane waves with the wavevectors k < kc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' These long waves are amplified as the result of the gravitational instability mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' It can be seen that the gravitational band falls perfectly on the central dispersion curve that passes through the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The central dispersion branch in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 7 is almost indistinguishable from the dispersion curve in the absence of vibration (not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' This allows us to conclude that a relatively strong vibration (sufficiently strong to excite Faraday waves) has almost no effect on the phase speed c = Im(λ)/k of the long gravitational surface waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' To study the interaction between the Faraday waves and gravitational surface waves in the nonlinear regime, we solve Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' (1) over the time interval of 15 seconds with and without vibration and compare the respective dispersion maps in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' (a) Dispersion map obtained from the solution of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' (1) in the absence of vibration for a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='6 mm water film on a θ = 3o incline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' (b) Dispersion map of the solution of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' (1) when the inclined plane vibrated at f = 48 Hz with the amplitude a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The scaling for the vertical axis is in arbitrary logarithmic units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' It is evident from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 8 that vibration leads to a suppression of the long surface waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Indeed, the magnitude of the dispersion band that corresponds to the gravitational waves is significantly smaller when the film is vibrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' This result is in qualitative agreement with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Conclusions In conclusion, our experiments with a sub-millimetre thick water layer on a slightly inclined vertically vibrated plate demonstrate that low-frequency vibration in the range between 30 and 50 Hz suppresses the development of long rolling surface waves propagating downstream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' These surface waves appear as the result of the long-scale gravitational instability of the base flow in the absence of vibration [15,16] and may also be excited by mechanically perturbing the flow at the inlet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' A relatively small thickness of the water layer (under 1 mm) is required to suppress the three-dimensional instability of the rolling waves that is known to develop at large flow rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Experimental findings are verified using a boundary-layer 15 (a) 15 (b) 10 10 5 5 0 0 5 5 10 10 15 15 50 50 40 40 30F 30 20 20 10 10 f (Hz) f (Hz) 10 k (mm-1) 10 k (mm-1) 20F 20 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='5 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='5 0 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='5 40 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='5 50 50 113 of 15 hydrodynamic model [52,53] obtained from the Navier-Stokes equation by assuming a self- similar parabolic longitudinal flow velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Linear stability and nonlinear dynamics of the surface waves obtained with the model qualitatively confirm the main experimental findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Without vibration, the Fourier content of surface waves is represented by a broad band of unstable wave vectors k < kc, where kc is a critical cut-off wave vector of the gravitational instability (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' As the instability unfolds, the wavelength of the dominant wave quickly increases until it develops into a solitary-like wave [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' For fluids with a relatively small viscosity, such as water, the characteristic time required for solitary rolling waves to develop on a 3o incline is of the order of several seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' In the nonlinear regime, the Fourier content of the surface waves is dominated by solitary-like waves characterised by a small wavevector with a background of smaller amplitude shorter waves, which is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 8a and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' We observe that the properties of the surface waves change dramatically when the layer is vibrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Thus, a relatively weak vibration (the vibration amplitude a < g) leads to the onset of the secondary Faraday instability in the form of short waves with a wavelength of λ ≈ 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 10 mm when vibrated at f = 48 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' In agreement with the earlier theoretical studies [42–44], the temporal frequency of the Faraday waves is shifted away from the harmonic (48 Hz) and subharmonic (24 Hz) response that is typical of Faraday instability in horizontal liquid layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' In fact, the inclination angle of the plate acts as a wave filter, splitting a standing Faraday wave into two plane waves: one propagating upstream and one propagating downstream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Similarly to the Doppler effect, the wave that propagates downstream increases its speed and, therefore, increases its temporal frequency, while the wave that propagates upstream decreases its speed and frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' For water layers vibrated at 48 Hz, we observe the following shifts in frequency away from the subharmonic response: from 24 Hz to approximately 40 Hz for the downstream wave and from 24 Hz to approximately 8 Hz for the upstream wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' In the nonlinear regime, the interaction between shorter Faraday waves and longer gravitational waves leads to the broadening of their respective bands in the f-k dispersion map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Most importantly, we find that the average and peak amplitudes of the long-scale gravitational waves are significantly reduced when vibration is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' This result is rather intriguing since the total influx of energy is larger in the vibrated system when both gravity and vibration together drive the flow, unlike in the non-vibrated case, where the only source of energy is due to gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Yet, nonlinear wave interaction leads to an uneven re-distribution of energy amongst the Faraday and gravitational waves in favour of the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' The physical mechanism responsible for the suppression of gravitational waves remains an open question;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' however, it is plausible to assume that the fast-oscillating fluid flow in the form of circulation patterns [58] in pulsating Faraday waves may slow down the redistribution of fluid on the large scale, required for the growth and development of the gravitational waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' This result is even more surprising since we did not observe any noticeable change in the velocity of the gravitational waves induced by vibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Apart from a contribution of the fundamental knowledge, the results presented in this work may be used to better understand and further improve certain technological processes that rely on falling liquid films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Yet, the demonstrated immunity of the solitary-like waves to external vibration and their intriguing nonlinear dynamical behaviour will be of interest to researchers working on emergent technologies, where both solitary waves and fluidic systems play an important role [7,8,12,48,59–61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Author Contributions: I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' conducted the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' conducted the theoretical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Both authors wrote the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Conflicts of Interest: The authors declare no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 14 of 15 References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Remoissenet, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Waves Called Solitons: Concepts and Experiments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Springer, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Boes, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Wu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Nguyen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' ;' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 2020, 11, 2568.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Kulikov, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Zak, M.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' New J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 2021, 23, 023013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Chang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Wave evolution 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Physics 1966, 45, 150–155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Esmail, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Shkadov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Nonlinear theory of waves in a viscous liquid layer.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' An experimental study of wave inception on falling liquid films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 1972, 27, 1257–1265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 19.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Michelson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' On Irregular Wavy Flow of a Liquid Film Down a Vertical Plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 1980, 63, 2112–2114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Pumir, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Manneville, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Pomeau, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' On solitary waves running down an inclined plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 1983, 135, 27—-50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Alekseenko, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Nakoryakov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Pokusaev, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Wave formation on vertical falling liquid films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Multiph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Flow 1985, 11, 607–627.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Trifonov, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Tsvelodub, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Nonlinear waves on the surface of a falling liquid film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Part 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Waves of the first family and their stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 1991, 229, 531—-554.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Liu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Gollub, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Solitary wave dynamics of film flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Fluids 1994, 6, 1702–1712.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Yu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Wasden, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Dukler, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Balakotaiah, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Nonlinear evolution of waves on falling films at high Reynolds numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Fluids 1995, 7, 1886–1902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Oron, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Davis, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Bankoff, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Long-scale evolution of thin liquid films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 1997, 69, 931–980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Nguyen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Balakotaiah, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Modeling and experimental studies of wave evolution on free falling viscous films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Fluids 2000, 12, 2236–2256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Thiele, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Knobloch, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Thin liquid films on a slightly inclined heated plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' D: Nonlinear Phenomena 2004, 190, 213–248.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Kapitza, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Kapitza, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Wave flow of thin liquid layers of fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Zh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Eksp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Teor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Fiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 1949, 19, 105–120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Kerchman, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Frenkel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Bontozoglou, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Observations of solitary wave dynamics of film flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 2001, 435, 191–215.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 32.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Akinaga, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Itoh, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Nokami, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Kobayashi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Kinoshita, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Dynamic nonlinear behavior of ionic liquid-based reservoir computing devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' ACS Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Interfaces 2022, 14, 36890–36901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Maksymov, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Pototsky, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Suslov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Neural echo state network using oscillations of gas bubbles in water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} +page_content=' E 2022, 105, 044206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfmwQa/content/2301.03300v1.pdf'} diff --git a/6NAzT4oBgHgl3EQfEfqj/vector_store/index.faiss b/6NAzT4oBgHgl3EQfEfqj/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..fcd9049eee184de7abbffdf3b987a8df0e72b4ec --- /dev/null +++ b/6NAzT4oBgHgl3EQfEfqj/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:6e5f854ebf9fc0609ac8c2b7f52cca3357f8fe29c6c338de2f71418919260960 +size 6291501 diff --git a/6dE4T4oBgHgl3EQfcQxJ/content/tmp_files/2301.05081v1.pdf.txt b/6dE4T4oBgHgl3EQfcQxJ/content/tmp_files/2301.05081v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1b1f560ff54b993d4c6aa7b10038aee52c7de330 --- /dev/null +++ b/6dE4T4oBgHgl3EQfcQxJ/content/tmp_files/2301.05081v1.pdf.txt @@ -0,0 +1,786 @@ +Reconfigurable magnetic-field-free superconducting +diode effect in multi-terminal Josephson junctions +Fan Zhang,1 Mostafa Tanhayi Ahari,2 +Asmaul Smitha Rashid,3 George J. de Coster,4 +Takashi Taniguchi,5 Kenji Watanabe,6 Matthew J. Gilbert,7,2 +Nitin Samarth,1∗ Morteza Kayyalha3∗ +1Department of Physics, The Pennsylvania State University, University Park, PA 16802, USA +2Materials Research Laboratory, The Grainger College of Engineering, University of Illinois, +Urbana-Champaign, IL 61801, USA +3Department of Electrical Engineering, The Pennsylvania State University, University Park, +PA 16802, USA +4DEVCOM Army Research Laboratory, 2800 Powder Mill Rd, Adelphi, MD, 20783, USA +5International Center for Materials, Nanoarchitectonics +National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan +6Research Center for Functional Materials +National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan +7Department of Electrical Engineering, University of Illinois, Urbana-Champaign, IL 61801, USA +∗Correspondding author: nsamarth@psu.edu. +∗Correspondding author: mzk463@psu.edu. +The superconducting diode effect (SDE) has attracted growing interest in re- +cent years as it potentially enables dissipationless and directional charge trans- +port for applications in superconducting quantum circuits. Here, we demon- +strate a materials-agnostic and magnetic-field-free approach based on four- +terminal Josephson junctions (JJs) to engineer a superconducting diode with +1 +arXiv:2301.05081v1 [cond-mat.supr-con] 12 Jan 2023 + +a record-high efficiency (∼ 100%). We show that the SDE is reconfigurable +by applying control currents to different terminals. We attribute the observed +SDE to the asymmetry of the effective current-phase relation (CPR), which we +derive from a circuit-network model. Our findings demonstrate the emergence +of a new form of the CPR in multi-terminal JJs that can emulate macroscopic +transport signatures of superconducting systems with broken inversion and +time-reversal symmetries. +Introduction +In linear electrical networks, the concept of reciprocity implies a symmetric relationship be- +tween the applied current and measured voltage. In other words, the voltage magnitude remains +the same if the polarity of the current source is reversed from positive to negative (1). Vio- +lating this fundamental symmetry in semiconductor technology has led to a plethora of new +devices including diodes, transistors, rectifiers, and photodetectors (2,3,4,5). In superconduc- +tors, engineering non-reciprocity requires simultaneous breaking of time-reversal and inversion +symmetries, known collectively as chiral symmetry (6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16). The +superconducting diode effect (SDE) is defined as an asymmetry in the critical current when the +current sweep direction is reversed. +Macroscopic transport signatures of the SDE are determined by the current-phase relation +(CPR), which typically depends on the band structure and microscopic details of the underlying +material system. Therefore, careful engineering of the interplay between spin-orbit interactions, +topological phases, and magnetic fields can lead to the observation of the SDE. To this point, the +SDE has been experimentally reported in a multitude of systems including but not limited to: +non-centrosymmetric superconductors with the magneto-chiral anisotropy (17, 18, 19, 20, 21), +Josephson junctions (JJs) based on Dirac semimetals with finite-momentum Cooper pairing +2 + +(22), two-dimensional (2D) van der Waals heterostructures (23, 21, 24, 25, 26), three-terminal +JJs based on InAs in the presence of a magnetic field (27), and a network of graphene JJs (28). +In this work, we develop a reconfigurable, materials-agnostic, and magnetic-field-free method +to engineer a synthetic CPR that emulates macroscopic transport signatures of systems with bro- +ken inversion and time-reversal symmetries. We consider a multi-terminal Josephson junction +(MTJJ) fabricated in graphene, which has a symmetric Fermi surface with no spin-orbit cou- +pling (both inversion and time-reversal symmetries are preserved). We show that the MTJJ +can emulate the SDE, which is a macroscopic transport signature predicted to emerge in sys- +tems with broken inversion and time-reversal symmetries. We further show that the SDE is +reconfigurable: the MTJJ exhibits the typical Josephson effect with no SDE under a symmetric +current-bias configuration. However, it manifests the SDE under an asymmetric current-bias +configuration (see Fig. 1 and fig. S1 for details of asymmetric and symmetric bias configura- +tions, respectively). The observed SDE is also magnetic-field-free and reversible with efficien- +cies as large as ∼ 100%. We explain the experimental observations by modeling our system +using a circuit network of coupled resistively-shunted junctions (RSJs). We find that the semi- +classical RSJ model accurately captures the non-reciprocal transport effect in the system. We +calculate an effective non-sinusoidal CPR arising from circuit network effects among the super- +conducting terminals. We show that this effective CPR is not symmetric under sign-reversal of +the superconducting phase, resulting in the observation of the SDE. Our combined experimental +and theoretical findings establish a new materials-agnostic platform based on MTJJs to engineer +novel forms of CPRs and non-reciprocal superconducting properties for potential applications +in superconducting cryogenics and quantum technology. +3 + +Results and Discussion +We fabricate four-terminal JJs on hBN/graphene/hBN van der Waals heterostructures which +are edge-contacted by Ti(10 nm)/Al(100 nm) superconducting electrodes. Figure 1A shows an +atomic force microscope (AFM) image of a representative four-terminal JJ. We characterize +the junction in two asymmetric bias-current configurations (see our prior work (29) and fig. +S1 in the Supplementary Materials for results obtained from symmetric configurations). In +configuration 1 (2), we ground terminal 2 (3) and apply a constant control current to terminal 3 +(2). In both configurations, we measure the SDE by sweeping the current I1 of terminal 1 and +applying a control current I4 to terminal 4. Figure 1B shows V12 vs I1 measured in Config. 1 +at I3 = 0 nA, I4 = 10 nA, Vg = 30 V, B = 0 G, and T = 12 mK. We observe that the critical +current (I+ +c ) for positive sweep direction, marked by red arrows, is around 4 nA, whereas the +critical current (I− +c ) for the negative sweep direction, marked by black arrows, is around −24 +nA. We also observe different critical currents (I+ +c or I− +c ) and return currents (I+ +r or I− +r ) for +each sweep direction, likely due to the Joule heating effect (30). We note that increasing the +temperature reduces the impact of Joule heating, thereby resulting in similar critical and return +currents (see fig. S4 in the Supplementary Materials). +To elucidate the origin of the observed SDE, we consider a circuit-network model of coupled +RSJs (Fig. 1C). The RSJ model represents individual junctions by a two-fluid system in which +the total junction current is the sum of a pair current ip +jk(t) and a dissipative quasiparticle current +iq +jk(t) (31,32). Here, we assume diffusive transport in which the pair current is given by the first +Josephson relation ip +jk(t) = Ijk +c sin(φjk(t)), where Ijk +c +is the critical current between terminals +j and k and φjk(t) ≡ φj(t) − φk(t) is the gauge-invariant phase difference satisfying the sec- +ond Josephson relation dφjk(t)/dt = (2e/ℏ)Vjk(t). The quasiparticle current is due to a finite +voltage Vjk(t) across the junction, iq +jk(t) = GjkVjk(t), where Gjk is a constant phenomenolog- +4 + +ical conductance tensor (see the Supplementary Materials for the RSJ parameters). A circuit +network model of JJs typically includes a parallel capacitance as well. In our graphene-based +JJs, however, the junction capacitance is negligible and, hence, we only consider the resistance +of the junction (33). Imposing current conservation (Kirchhoff’s current law) at terminal j, we +obtain +Ij = +� +k +(ip +jk + iq +jk). +(1) +In this study, we are interested in the emergent non-reciprocal superconducting properties in +the four-terminal JJ. Therefore, we only consider small bias currents such that no quasiparticle +current flows between the terminals, i.e., iq +jk = 0. In this case, starting from Eq. 1 and assuming +I3 = 0 nA for Config. 1, we may analytically obtain expressions for the other terminal currents +as +I1 = I14 +c sin φ14 + I13 +c sin φ13 + I12 +c sin φ1, +I4 = I41 +c sin φ41 + I43 +c sin φ43 + I42 +c sin φ4, +0 = I32 +c sin φ3 + I31 +c sin φ31 + I34 +c sin φ34. +(2) +For fixed (φ1, I4) in Eq. 2, we find φ3, φ4, and, consequently, the following effective CPR: +I1(φ1, I4) = +� +n +an sin nφ1 + bn cos nφ1, +(3) +where n is an integer and (an, bn) are the amplitudes of the nth harmonic for the sine and cosine +functions, respectively. Comparing our numerical simulation to Eq. 3, we find that b0, a1, and +a2 are the only dominant factors in our device (Fig. 1D). This leads to +I1(φ1, I4) ≈ b0 + a1 sin φ1 + a2 sin 2φ1, +(4) +where b0 ∝ I4 and (a1, a2) are independent of I4 (see Supplementary Materials for more de- +tails). +5 + +The symmetry of this effective CPR depends on the current-bias configuration. +For ex- +ample, Eq. 4 is symmetric under simultaneous sign reversal of φ1 and I4, i.e., I1(φ1, I4) = +−I1(−φ1, −I4). However, for a fixed I4 ̸= 0, Eq. 4 represents a CPR which is asymmetric +under sign reversal of φ1 , i.e., I1(φ1, I4) ̸= −I1(−φ1, I4). This is because for fixed I4 ̸= 0, b0 +does not change sign when φ1 → −φ1. In general, the non-reciprocity arises from an asymme- +try in the free energy of the system for opposite current directions ±I1 and fixed I4. To see this, +we consider the expression for the free energy of the system +F(I1, I4, φ1, φ3, φ4) = (ℏ/2e) +� +I1φ1 + I4φ4 + +� +j0 a constant, u : R≥0 → R≥1 +a sublinear function. A map f : X → Y is an (L, u)-SBE if the following conditions are satisfied: +1. f(x0) = y0, +2. ∀x1, x2 ∈ X, +1 +L|x1 − x2|X − u +� +|x1|X ∨ |x2|X +� +≤ |f(x1) − f(x2)|Y ≤ L|x1 − x2|X + u +� +|x1|X ∨ |x2|X +� +, +3. ∀y ∈ Y ∃x ∈ X such that |y − f(x)|Y ≤ u(|y|Y ). +1 + +A map is an SBE if it is an (L, u)-SBE for some L and u as above. +I prove SBE-completeness for irreducible lattices in groups without R-rank 1 factors. +Theorem 1.3 (SBE-Completeness). Let G be a real centre-free semisimple Lie group without compact or +R-rank 1 factors. +Let Γ ≤ G be an irreducible lattice and Λ an abstract finitely generated group, both +considered as metric spaces with some word metric. Assume there is an (L, u)-SBE f : Λ → Γ with u a +subadditive sublinear function. Then there is a group homomorphism Φ : Λ → G with finite kernel whose +image Λ′ := Φ(Λ) is a lattice in G. Moreover, Λ′ is uniform if and only if Γ is uniform. +The main ingredients in the proof are of independent interest: the first is geometric rigidity for the +corresponding symmetric space, stating that every self SBE of such a space is sublinearly close to an isom- +etry. This generalizes Kleiner and Leeb’s result [32] on self quasi-isometries of symmetric spaces, as well as +Eskin’s [22] and Drutu’s [16] results on self quasi-isometries of non-uniform lattices. For the definition of the +‘compact core’ of a lattice see Section 2.2.2. +Theorem 1.4 (Sublinear Geometric Rigidity). Let X be a symmetric space of noncompact type without +R-rank 1 factors, Γ ≤ Isom(X) an irreducible lattice and X0 ⊂ X the compact core of Γ in X. For any +(L, u)-SBE map f : X0 → X0 there is a sublinear function v = v(L, u) and an isometry g : X → X such +that d +� +q(x), g(x) +� +≤ v(|x|) for all x ∈ X0. +The proof of Theorem 1.3 actually requires a stronger version of Theorem 1.4, formulated in Lemma 6.24. +Notice that Theorem 1.3 and Theorem 1.4 both hold for uniform as well as non-uniform lattices. I remark +that ‘generalized quasi-isometries’ already appeared in the context of geometric rigidity, as a technical tool +in Eskin and Farb’s work [21] [22] on quasi-isometries. Indeed much of their work is carried for maps which +are even more general than SBE. It seems however that their approach cannot yield a sublinear bound as in +Theorem 1.4, which is necessary for the proof of Theorem 1.3. +The second ingredient is a property I call sublinear rigidity, stating that a discrete subgroup Λ ≤ G which +sublinearly covers a lattice is itself a lattice. Sublinear rigidity holds for groups of any R-rank, and is the +cornerstone of this work. Its proof contains the bulk of the original ideas that appear in this paper. +Definition 1.5. For a function u : R≥0 → R>0 and a subset Y ⊂ X, define the u-neighbourhood of Y to be +Nu(Y ) := {x ∈ X | d(x, Y ) ≤ u(|x|)} +A subset Y ⊂ X is said to sublinearly cover Z ⊂ X if Z ⊂ Nu(Y ) for some sublinear function u. +For the definition of a Q-rank 1 lattice, see Section 2.1 and Theorem 2.20. +Theorem 1.6 (Sublinear Rigidity). Let G be a real centre-free semisimple Lie group without compact factors. +Let Γ ≤ G be an irreducible lattice, Λ ≤ G a discrete subgroup that sublinearly covers Γ. If Γ is of Q-rank 1, +assume further that Λ is irreducible. Then Λ is a lattice in G. +The notion of irreducibility in the non-standard context of a general (non-lattice) subgroup is explained +Section 4.4.2 (see Definition 4.47). Sublinear neighbourhoods arise naturally in the presence of SBE maps: +the essential difference between a quasi-isometry and an SBE is that ‘far away in the space’, the ‘additive’ +error term of an SBE gets larger and larger. One is led to consider metric neighbourhoods that grow - +sublinearly, yet unboundedly - with the distance to some (arbitrary) fixed base point. +The hypothesis that u is sublinear is optimal in the sense that u could not be taken to be an arbitrary +linear function. Indeed, the geometric meaning of Γ ⊂ Nu(Λ) is that for every element γ ∈ Γ, the ball +BG +� +γ, u(|γ|) +� +‘of sublinear radius about γ’ must intersect Λ. +Observe that if f is the identity function +f(r) = r, then by definition f(|g|) = f +� +d(g, eG) +� += d(g, eG). In particular G lies in the f-neighbourhood of +the trivial subgroup which is, after all, not a lattice in G. For uniform lattices and for lattices with Kazhdan’s +property (T) one can however relax the assumption of sublinearity: +2 + +Theorem 1.7 (Theorems 3.3 and 5.2). Let G be a real centre-free semisimple Lie group without compact +factors, Γ ≤ G an irreducible lattice, Λ ≤ G a discrete subgroup. Fix ε > 0 and assume Γ ⊂ Nu(Λ) for the +function u(r) = ε · r. +1. If Γ is uniform and ε < 1, then Λ is a uniform lattice. +2. If Γ has Kazhdan’s property (T) then there is a constant ε(G) such that if ε < ε(G) then Λ is a lattice. +Theorem 1.6 generalizes the case where Γ lies in a bounded neighbourhood ND(Λ) for some D > 0, proved +by Eskin and Schwartz in a slightly modified version (see Section 4.3 below). In the bounded case, the result +is much stronger and states that Λ must be commensurable to Γ (except in groups locally isomorphic to +SL2(R), see [52]). In the sublinear setting I can only prove a limited commensurability result, which stems +from a reduction to the bounded setting: +Theorem 1.8. Let G, Γ and Λ be as in Theorem 1.6. If Γ is uniform then so is Λ. If Γ is of Q-rank 1 then: +1. If Γ ̸⊂ ND(Λ) for any D > 0, then also Λ is of Q-rank 1. . +2. If Γ ⊂ ND(Λ) for some D > 0 and in addition Γ sublinearly covers Λ, then Λ is commensurable to Γ. +I stress that the case where Γ and Λ each sublinearly covers the other arises naturally in the context of +SBE-completeness, see Theorem 6.4. +The most interesting case in the proof of Theorem 1.6 is when Γ is of Q-rank 1. In that case, the proof +is entirely geometric, relying on the following key proposition which might be of independent interest: +Proposition 1.9. In the setting of Theorem 1.6, assume that Γ is a Q-rank 1 lattice which does not lie +in any bounded neighbourhood of Λ. Then there exists a horosphere H based at the rational Tits building +associated to Γ such that +� +Λ ∩ StabG(H) +� +· x0 intersects H in a cocompact metric lattice. Moreover, the +bounded horoball HB does not intersect the orbit Λ · x0. +1.1 +Outline of Proof +My proofs rely and draw on the works on the quasi-isometric rigidity for non-uniform lattices, due to +Schwartz [52] in R-rank 1, and to Drut¸u [16] and Eskin [22] independently in groups of R-rank greater than +1, often called higher rank groups. The geometric proof of Theorem 1.6 for Q-rank 1 lattices is quite delicate +and involved. For this reason I give here a detailed sketch of the arguments and of the ideas one should have +in mind when reading the proof. What is written here is a good enough account if one wishes to understand +the main ideas while avoiding the technical details. I end the section with a brief sketch of the proofs for +SBE-completeness and for sublinear rigidity in the case of property (T) groups. +Strategy for Q-rank 1 Lattices. +Denote dγ := d(γ, Λ). The novel case is when {dγ}γ∈Γ is unbounded. +The rationale for the proof comes from a conjecture of Margulis, recently proved in full generality by Benoist +and Miquel ([5], see Theorem 4.4 below). Their result states that a discrete subgroup in a higher rank Lie +group is a lattice as soon as it intersects a horospherical subgroup in a lattice. This result could be seen +as an algebraic converse to the geometric structure of a Q-rank 1 lattice, whose orbit in X intersects some +parabolic horospheres in a cocompact (metric) lattice (see Section 2.2 for details). Proposition 1.9 is the +geometric analogue of the Benoist-Miquel criterion, and basically completes the proof in the higher R-rank +case (some non-trivial translation work is needed, see Section 4.4). Also, it easily follows from Proposition 1.9 +that every Γ-conical limit point is also Λ-conical (Corollary 4.27). The proof for R-rank 1 groups is then a +simple use of a criterion of Kapovich-Liu for geometrically finite groups ([30], see Theorem 4.6). I now give +a detailed description of the proof of Proposition 1.9. +3 + +The ABC of Sublinear Constraints. +Fix a point x0 ∈ X = G/K, identify Γ and Λ with Γ · x0 and +Λ · x0 respectively. Observe that by definition of dγ the interior of balls of the form B(γx0, dγ) does not +intersect Λ · x0. I call such balls (or general metric sets) Λ-free. Moreover, these balls intersect Λ · x0 (only) +in the bounding sphere: call such balls (sets) tangent to Λ. The Λ-free and, respectively, Γ-free regions in X +are the main objects of interest in this work. Since Theorem 1.6 is known in the case of bounded {dγ}γ∈Γ, +it makes sense to think about large Λ-free regions as ‘problematic’. The state of mind of the proof relies on +two easy observations that complete each other. +1. The sublinear constraint implies that dγn → ∞ forces |γn| → ∞, suggesting that ‘problematic’ Λ-free +regions should appear only ‘far away’ in the space. +2. On the other hand Λ is a group, and being Λ-free is a Λ-invariant property. In particular any metric +situation that can be described in terms of the Λ-orbit (e.g. B(γ, dγ) is a Λ-free ball tangent to Λ) can +be translated back to x0. This means that ‘problematic’ regions could actually be found near x0. +The moral of these observations can be formulated into a general principle that lies in the heart of the +argument. The sublinear constraint dγ ≤ u(|γ|) gives rise to many other constraints of ‘sublinear’ nature. +Each such constraint actually yields a uniform constraint inside any fixed bounded neighbourhood of x0. +Since Λ and Γ are groups, many of these uniform bounds which are produced ‘near’ x0 turn out to be global +bounds that depend only on the group and not on a specific orbit point. Put differently: the trick is to +describe metric situations in terms of the Γ and Λ orbits. One then uses the group invariance in order to +move these metric situations around the space to a place where the sublinear constraint can be exploited. +The Argument. +The above paragraph should more or less suffice the reader to produce a complete proof +for uniform lattices. For non-uniform lattices, denote by λγ the closest Λ-orbit point to the point γ ∈ Γ. For +H a cusp horosphere of Γ, let ΓH := {γ ∈ Γ | γx0 ∈ H}. The ultimate goal is to show that the metric lattice +ΓH · x0 yields a metric lattice that is more or less {λγx0}γ∈ΓH. One proceeds by the following steps. +1. Finding Λ-free horoballs (Section 4.2.1): The arbitrarily large Λ-free balls B(γn, u(|γn|)) are translated +to x0, and the compactness of the unit tangent space at x0 yields a converging direction which is the +base point at infinity of a Λ-free horoball. Translating by Λ, this yields Λ-free horoballs tangent to +every Λ-orbit point. +2. Controlling angles (Section 4.2.2): For every γ ∈ Γ with dγ uniformly large enough, one associates a +point ξ at X(∞) such that ξ is the base point of a Λ-free horoball tangent to λγx0. The angle between +the geodesics [λγx0, γx0] and [λγX0, ξ) is shown to be small as dγ grows large. This is used to show +that arbitrarily large dγ give rise to arbitrarily deep Γ-orbit points inside Λ-free horoballs. A key step +is Lemma 3.8, producing a Γ-free Euclidean cylinder between λγx0 and γx0. +3. Λ-cocompact horospheres (Section 4.2.3): One uses uniform bounds near x0 to prove that every Λ-free +horoball that is (almost) tangent to Λ must lie in a uniformly bounded neighbourhood of Λ · x0. If +dγ is large enough for some γ that lies on a horosphere HΓ of a cusp of Γ, then any γ′ ∈ ΓHΓ also +admits large dγ′. Since the bounds from the previous steps only depend on dγ′, all λγ′ are forced to +lie on the same horosphere HΛ parallel to HΓ. One concludes that Λ · x0 intersects HΛ on the nose in +a cocompact metric lattice. +Property (T). +Sublinear rigidity for groups with property (T) is established by the criterion that a discrete +subgroup there is a lattice if and only if it has the same exponential growth rate as a lattice (Leuzinger [36]). +It is quite straightforward that sublinear distortion cannot affect this growth rate (Corollary 5.10). +SBE Rigidity. +The general scheme for SBE-completeness is parallel to the quasi-isometry case: each +λ ∈ Λ naturally gives rise to an SBE X0 → X0 of the compact core of Γ. Each self SBE is close to an +isometry by Theorem 1.4, allowing to embed Λ as a discrete subgroup of isometries in G that sublinearly +4 + +covers Γ. The proof for Theorem 1.4 heavily relies on Drut¸u’s argument for quasi-isometries [16], which uses +the properties of the induced biLipschitz map on the asymptotic cone. As SBE also induce such a biLipschitz +map - indeed that was a main motivation for Cornulier to study SBE [13] - it is possible to generally follow +Drut¸u’s argument also in the SBE setting. +1.2 +Possible Improvements, Related and Future Work +The proof suggests three natural improvements to the statement of Theorem 1.6. One could probably relax +the assumption of trivial centre and allow finite centre, if the same relaxation is applicable in Leuzinger’s +work on property (T) groups [36] and in Prasad’s work [49]. In particular, the proofs for uniform lattices and +for Q-rank 1 lattices hold also for groups G with finite centre. The irreducibility of Λ may be derived directly +from the irreducibility of Γ. Lastly, in view of the geometric characterization of Q-rank (see Corollary D in +[35]), it is reasonable that the Q-rank of Λ should equal that of Γ. +There are problems that arise naturally from this work which seem to require new ideas: +Question. Let G be a real finite-centre semisimple Lie group without compact factors that admits a R- +rank 1 factor. Are the classes of uniform and non-uniform lattices of G SBE-complete? In particular, is this +true when G is of R-rank 1? +See Section 6.3.4 below for a discussion on the case of R-rank 1 factors. SBE of R-rank 1 symmetric spaces +is the main focus in Pallier’s work [44]. He investigated the sublinearly large scale geometry of hyperbolic +spaces, and proved that two R-rank 1 symmetric spaces that are SBE are homothetic, answering a question +of Drut¸u (see Remarks 1.16 and 1.17 in [13]). Also in this context Pallier and Qing [47] recently showed that +the sublinear Morse boundary is an SBE invariant. +Another problem is to find a non-trivial example of the setting of Theorem 1.6: +Question. Let G be a real finite-centre semisimple Lie group without compact factors, Γ ≤ G a lattice. +1. Does there exist a finitely generated group that is SBE to Γ but not quasi-isometric to it? Or, at least, +not known to be quasi-isometric to one? +2. Does there exist Λ ≤ G discrete that ε-linearly covers Γ but which does not sublinearly cover it? +On the side of the proof, it would be very interesting if the geometric ideas that prove Theorem 1.6 for +Q-rank 1 lattices could be applied to any Q-rank. While there are apparent places where the proof uses the +unique geometry of Q-rank 1 lattices, most of the geometric arguments leading to Proposition 1.9 seem to +be susceptible to the higher Q-rank setting. Such a generalization of the proof would definitely shed more +light on the mysterious lattice arising from growth considerations in property (T) groups, and in particular +on the question of commensurability of Λ and Γ in that case. It would also be interesting to see whether +one can push the geometric argument forward in order to establish a complete geometric analogue of the +Benoist-Miquel criterion. Namely, can one find a direct geometric proof that Λ admits finite co-volume +(perhaps similarly to Schwartz’s argument in the bounded case, see Section 4.3 below). +Lastly, one could possibly relate this work to the work of Fraczyk and Gelander [24], who proved that +a discrete subgroup (of a higher rank simple Lie group) is a lattice if and only if it has bounded injectivity +radius. While their result seem very much related to the condition Γ ⊂ Nu(Λ), the nature of their work does +not give explicit bounds on the injectivity radius. Specifically, given r > 0 one cannot tell directly from their +results how ‘far’ one must wander in X in order to find a point with injectivity radius r. Perhaps one could +use the sublinear results of this work to say something about the relation of |x|X and InjRad(x). +1.3 +Acknowledgments +This paper is based on my DPhil thesis, supervised by Cornelia Drut¸u. +I thank her for suggesting the +question of SBE-completeness and for guiding me in my first steps in the theory of Lie groups. I thank Uri +Bader, Tsachik Gelander and the Midrasha on groups at the Weizmann institute, where I learned the basics +5 + +of symmetric spaces. I thank my thesis examiners Emmanuel Breuillard and Yves Cornulier for their careful +inspection and numerous remarks. I thank Elon Lindenstrauss for telling me about Leuzinger’s result [36] +on property (T), and Or Landesberg, Omri Solan, Elyasheev Leibtag, Itamar Vigdorovich and Tal Cohen +for many discussions on different aspects of this paper. Finally I thank Gabriel Pallier for explaining his +examples of some unusual SBE in R-rank 1, and for his interest in this work. +2 +Preliminaries +For standard definitions and facts about fundamental domains, see Chapter 5.6.4 in [17]. The facts about +fundamental domains for Q-rank 1 lattices appear in Raghunathan’s book [50] and in Prasad’s work on +rigidity of Q-rank 1 lattices [49]. In notations and generalities I follow: Borel’s book on algebraic groups [6], +Helgason’s books on Lie groups and symmetric spaces [27, 28], and Eberline’s book on the geometry of +symmetric spaces of noncompact type [20]. +2.1 +Generalities on Semisimple Lie Groups and their Lattices +Let G be a real centre-free semisimple Lie group without compact factors. A discrete subgroup Γ ≤ G is +a lattice if Γ\G carries a finite volume G-invariant measure. Equivalently, Γ is a lattice if Γ\X is a finite +volume Riemannian manifold, where X = G/K is the symmetric space of noncompact type corresponding +to G. A lattice is irreducible if its projection to every simple factor of G is dense. The group G can be +viewed as an algebraic group via the adjoint representation. If G is of R-rank greater than 1, then by the +Margulis arithmeticity theorem every irreducible lattice of G is arithmetic. The Q-rank Γ is the Q-rank of +the Q-structure associated to (G, Γ) be the arithmeticity theorem. A result of Prasad [49] states that if G +admits a R-rank 1 factor, then a non-uniform irreducible lattice of G is of Q-rank 1 . +The group G has Kazhdan’s property (T) if and only if it does not admit an SO(n, 1) or an SU(n, 1) +factor, and an irreducible lattice Γ ≤ G has property (T) if and only if G has property (T). Together with +Prasad’s result, I may conclude: +Lemma 2.1. Let G be a real centre-free semisimple Lie group without compact factors, and Γ ≤ G an +irreducible lattice. Then at least one of the following occurs: (a) G has property (T) (b) Γ is a non-uniform +Q-rank 1 lattice (c) Γ is uniform. +Theorem 1.6 is therefore an immediate result of Theorem 3.1, Theorem 4.1 and Theorem 5.1. +2.2 +Cusps and the Rational Tits Building +The facts about symmetric spaces of noncompact type can be found in Eberline’s book [20]. +Since the +geometry of Q-rank 1 lattices resembles that of lattices in R-rank 1, the reader could for the most part +simply have the image of the hyperbolic plane in mind. If one wishes to see flats that are not geodesics, then +a product of two hyperbolic planes is enough. Even the product of the hyperbolic plane and R is helpful, +albeit this space has a Euclidean factor. +2.2.1 +Basic Geometry of Symmetric Spaces of Noncompact Type +Visual Boundary. +The visual boundary X(∞) of X is the set of equivalence classes of geodesic rays, +where two geodesic rays are equivalent if their Hausdorff distance is finite. For a ray η : [0, ∞) → X, η(∞) +denotes the equivalence class of η in X(∞). There are two natural topologies on X(∞) that will be of use. +The cone topology is the one given by viewing X(∞) as the set of all geodesic rays emanating from some +fixed base point x0, with topology induced by the unit tangent space at x0. There is a natural topology on +X := X ∪ X(∞) such that X is the compactification of X and the induced topology on X(∞) is the cone +topology. A well known fact about geodesic rays in nonpositively curved spaces, stating that two ‘close’ +geodesic rays fellow travel: +6 + +Lemma 2.2. Given time T and ε > 0, there is an angle α = α(T, ε) so that if η1, η2 are two geodesic rays +with η1(0) = η2(0) = x for some x ∈ X and ∡x(η1, η2) ≤ α then dX +� +η1(t), η2(t) +� +< ε for all t ≤ T . +The Tits metric on X(∞) is defined as follows. +Given a totally geodesic submanifold Y ⊂ X, let +Y (∞) ⊂ X(∞) be the subset of all points that admit a geodesic ray η lying inside Y (or, equivalently, those +points that admit a ray lying at bounded Hausdorff distance to Y ). For any ξ1, ξ2 ∈ X(∞) there exists a +flat F ⊂ X such that ξ1, ξ2 ∈ F(∞). Define dT (ξ1, ξ2) ∈ [0, π] to be the angle between two geodesic rays +η1, η2 ⊂ F emanating from some point x ∈ F and with η1(∞) = ξ1, η2(∞) = ξ2. This is a well defined metric +on X(∞), called the Tits metric. The pair +� +X(∞), dT +� +is a geodesic metric space, and isometries of X act +on it by isometries. I will use the following relation between the cone and the Tits topologies: +Proposition 2.3 (Section 3.1 in [20]). Let X be a symmetric space of noncompact type. The Tits metric on +X(∞) is semicontinuous with respect to the cone topology: for any ξ, ζ ∈ X(∞) and every ε > 0, there exists +neighbourhoods of the cone topology U, V ⊂ X(∞) of ξ and ζ respectively such that for all ξ′ ∈ U, ζ′ ∈ V one +has +∡(ξ′, ζ′) ≥ ∡(ξ, ζ) − ε +Moreover, for any flat F ⊂ X, the cone topology and the Tits topology coincide on F(∞). +Busemann Functions, Horoballs and Horospheres. +Horoballs and horospheres play a crucial role in +the proof, a role which stems from their role in the geometric description of the compact core of non-uniform +lattices (see Theorem 2.20 below). A Busemann function on X is any function of the form +fη(x) = lim +t→∞ d +� +x, η(t) +� +− t +for some geodesic ray η of X. A horoball HB ⊂ X is an open sublevel set of a Busemann function. A +horosphere H ⊂ X is a level set of a Busemann function. Two equivalent geodesic rays η1, η2 give rise to +Busemann function which differ by a constant, i.e. fη1 − fη2 = C for some C ∈ R. If HB is the sublevel set +of fη, then η(∞) is called the base point of the horoball HB (and respectively of the horosphere H = ∂HB). +The base point of a horoball is well defined, i.e. it depends only on η(∞) and not on η. For every choice of +x ∈ X, ξ ∈ X(∞) there is a unique horosphere H based at ξ with x ∈ H. I denote this horosphere by H(x, ξ) +and the bounded horoball HB(x, ξ). The following proposition collects some basic properties that will be of +use. +Proposition 2.4 (Proposition 1.10.5 in [20]). Let x ∈ X, ξ ∈ X(∞), and let H = H(x, ξ), HB the horoball +bounded by H and f the Busemann function based at ξ with f(x) = 0. +1. For any point y ∈ X, let η be the bi-infinite geodesic determined by the geodesic [y, ξ). Then PH(y) = +η ∩ H, where PH(y) is the unique point closest to y on H. +2. For any point y ∈ X, f(y) = ±d +� +y, PH(y) +� +. Moreover, f(y) is negative if and only if y ∈ HB. +3. If x′ ∈ X, then the horospheres H = H(x, ξ) and H′ = H(x′, ξ) are equidistant: if y ∈ H, y′ ∈ H′, then +d(y, H′) = d(y′, H). Such horospheres are called parallel. +A Busemann function fη thus naturally determines a filtration of X by the co-dimension 1 manifolds +{Ht}t ∈ R. By convention I usually assume that Ht := {x ∈ X | fη(x) = −t}. +Remark 2.5. The stabilizer of a point ξ ∈ X(∞) acts transitively on the set of horospheres based at ξ, so +every two such horospheres are isometric. Moreover, there is a close relation between the induced metrics on +horospheres with the same base point. Briefly, if dH denotes the induced distance on a horosphere H ⊂ X, +then dH +� +PH(x), PH(y) +� +for any two points x, y ∈ H′ can be bounded uniformly below and above as a function +of the distance dH′(x, y) and the curvature bounds on X. See Heintze-Im hof [26] for precise statements. +7 + +Using the above properties one can show that two horospheres are parallel if and only if they are based +at the same point. In particular for every x ∈ X, ξ ∈ X(∞) it holds that StabG +� +H(x, ξ) +� +⊂ StabG(ξ). In +addition, If H, H′ are two parallel horospheres based at the same ξ ∈ X(∞) and A ⊂ H is a cocompact +metric lattice in H, then πH′(A) is a cocompact metric lattice in H′. +For a point ξ ∈ X(∞) and a flat F with ξ ∈ F(∞), one readily observes that every horoball HB based +at ξ intersects F in a Euclidean half space. In particular for every ζ ∈ X(∞) with dT (ξ, ζ) < π +2 , for every +geodesic ray η with η(∞) = ζ and every horoball HB based at ξ there is some T for which η↾t>T ⊂ HB. +Parabolic and Horospherical Subgroups. +The isometries of X are classified into elliptic, hyperbolic, +and parabolic isometries. Most significant for this paper are the parabolic isometries, i.e. those g ∈ G whose +displacement function x �→ gx does not attain a minimum in X. Every such isometry fixes (at least) one +point in X(∞). A group P ≤ Isom(X) is called geometrically parabolic if it is of the form Gξ := StabG(ξ) for +some ξ ∈ X(∞). Such groups act transitively on X, and in particular act transitively on the set of geodesic +rays in the equivalence class of ξ. The same holds also for the identity component G◦ +ξ. An element g ∈ Gξ +acts by permutation on the set of horoballs based at ξ. This permutation is a translation with respect to +the filtration of the space X by horospheres based at ξ. Put differently, if {Ht}t∈R is a filtration of X by +horospheres based at ξ, then for every g ∈ Gξ there is l(g) ∈ R such that gHt = Ht+l(g). It is quite clear +from all of the above that for every horosphere based on ξ, the group GH := StabG(H) acts transitively on +H, and the same holds for G◦ +H. +I now present a fundamental structure theorem for geometrically parabolic groups. Denote g := Lie(G), +and let g = k ⊕ p be a Cartan decomposition defined using the maximal compact subgroup K ≤ G. Recall +that the Lie exponential map exp : g → G gives rise to a family of 1-parameter subgroups of the form +exp(tX) for each X ∈ p. +Proposition 2.6 (Proposition 2.17.3 in [20]). Let x ∈ X(∞), and let X ∈ p be the tangent vector of the +unit speed geodesic [x0, ξ). Let ht +ξ be the 1-parameter subgroup defined by t �→ exp(tX). Then an element +g ∈ G fixes ξ if and only if limt→∞ h−t +ξ ght +ξ exists. +Proposition 2.7 (Langlands Decomposition, Propositions 2.17.5 and 2.17.25 in [20]). Let ξ ∈ X(∞) and +ht +ξ as in Proposition 2.6. Let F be a flat containing [x0, ξ) and A ≤ G the maximal abelian subgroup such +that Ax0 = F. Denote Gξ := StabG(ξ), and define Tξ : Gξ → G by g �→ limn→∞ h−n +ξ +ghn +ξ . Then Tξ is a +homomorphism, and there are subgroups Nξ, Aξ, Kξ ≤ Gξ such that: +1. Aξ = exp +� +Z(X) ∩ p +� +, where Z(X) is the centralizer of X in g. Moreover, every element a ∈ Aξ lies in +some conjugate Ag = gAg−1 with the property that [x0, ξ) ⊂ F g := Agx0. +2. Kξ ≤ K = StabG(x0) is the compact subgroup fixing the bi-infinite geodesic determined by [x0, ξ). +3. KξAξ = AξKξ. +4. Nξ = Ker(Tξ). It is a connected normal subgroup of Gξ. +5. Gξ = NξAξKξ, and the indicated decomposition of an element is unique. +6. G = NξAξK, and the indicated decomposition of an element is unique. In case ξ is a regular point at +X(∞), this decomposition is the Iwasawa decomposition. +7. Gξ has finitely many connected components, and G◦ +ξ = (KξAξ)◦Nξ. +8. Gξ is self normalizing. +Viewing G as an algebraic group, the geometrically parabolic subgroups are exactly the (algebraically) +non-trivial parabolic subgroups, i.e. proper subgroups of G that contain a normalizer of a maximal unipotent +subgroup. Proposition 2.7 is a geometric formulation of the algebraic Langlands decomposition of parabolic +groups. Recall that a horospherical subgroup is the unipotent radical of a non-trivial parabolic group, or +equivalently groups of the form Ug := {u ∈ G | limn→∞ g−nugn = idG}. The latter implies that Nξ a +horospherical subgroup of G. +8 + +Limit Set. +An important set associated to a discrete group ∆ ≤ G acting by isometries on X is the limit +set L∆. By definition L∆ := ∆ · x ∩ X(∞), i.e. it is the intersection with X(∞) of the closure, in the +compactification X = X ∪ X(∞), of an orbit ∆ · x. It is clear that L∆ does not depend on the choice of +x ∈ X. The limit set of any lattice is always the entire X(∞). In fact, much more is true: +Definition 2.8. Let ∆ ≤ G = Isom(X), and SX the unit tangent bundle. A vector v ∈ SX is ∆-periodic +if there is δ ∈ ∆ and s > 0 such that δη(t) = η(t + s) for all t ∈ R, where η is the bi-infinite geodesic +determined by the vector v. +For a flat F ⊂ X (including geodesics), denote ∆F := {δ | δF = F}. The flat F is called a ∆-periodic +flat if there exists a compact set C ⊂ F such that ∆F C = F. +Proposition 2.9 (Propositions 4.7.3., 4.7.5, 4.7.7 in [20], Lemma 8.3′ in [41]). If Γ ≤ G = Isom(X) is a +lattice, then: +1. The subset in SX of Γ-periodic vectors is dense. +2. Let F ⊂ X be any flat, η any bi-infinite geodesic in F, and denote v = ˙η(0) ∈ SX the initial velocity +vector. There is a sequence vn ∈ SX of regular vectors such that +(a) limn→∞ vn = v. +(b) The bi-infinite geodesics ηn determined by vn are all Γ-periodic. +(c) Denote by Fn the (unique) flat containing ηn. Each Fn is Γ-periodic. +Put differently, the set of Γ-periodic flats is dense in the set of flats of X. +Remark 2.10. See Definition 2.25 for the notion of regular tangent vectors. +Points in the limit set of a group are classified according to how the orbit approaches them. +Definition 2.11. Let ∆ ≤ G be a discrete subgroup, ξ ∈ L∆. +1. The point ξ is called conical if for some (hence any) x and some (hence every) geodesic ray η with +η(∞) = ξ there is a number D = D(x, η) such that for every T ∈ R>0 there is t > T for which +B +� +η(t), D +� +∩ ∆ · x ̸= ∅. Since ∆ is discrete, this is equivalent to ∆ · x ∩ ND(η) being infinite. +2. The point ξ is called horospherical if for every horoball HB based at ξ and every x ∈ X, ∆ · x ∩ HB is +non-empty. In particular, a conical limit point is horospherical. +3. The point ξ is non-horospherical if it is not horospherical. +As a corollary of Proposition 2.9, one has: +Corollary 2.12. If Γ ≤ Isom(X) is a lattice, then +1. The set of Γ-conical limit points is dense in X(∞) with the cone topology. +2. LΓ = X(∞). +I finish this section with some results on geometrically finite subgroups of isometries in R-rank 1. +Definition 2.13. Let X be a R-rank 1 symmetric space and ∆ ≤ Isom(X) a discrete subgroup. Denote by +Hull(∆) the closed convex hull in X = X ∪X(∞) of the limit set L∆, and Hull(∆) = X ∩Hull(∆). By virtue +of negative curvature, Hull(∆) is the union of all geodesics η such that η(∞), η(−∞) ∈ L∆. The convex core +of ∆ is defined to be ∆\Hull(∆) ⊂ ∆\X, i.e., the quotient of Hull(∆) by the ∆-action. +Definition 2.14 (Bowditch [9], see Theorem 1.4 in [30]). Let X be a R-rank 1 symmetric space, i.e. a +symmetric space of pinched negative curvature. A discrete group ∆ ≤ G = Isom(X) is geometrically finite if +for some δ > 0, the uniform δ-neighbourhood in ∆\X of the convex core Nδ +� +∆\Hull(∆) +� +, has finite volume +and there is a bound on the orders of finite subgroups of ∆. A group is geometrically infinite if it is not +geometrically finite. +9 + +Immediately from the definition of geometrical finiteness, one gets a simple criterion for a subgroup to +be a lattice: +Corollary 2.15. Let X be a R-rank 1 symmetric space. If ∆ ≤ Isom(X) is geometrically finite and admits +L∆ = X(∞), then ∆ is a lattice in Isom(X). +Sublinear distortion does not effect the limit set, as the following lemma shows. +Lemma 2.16. Let Γ, Λ ≤ G be discrete subgroups and u : R≥0 → R>0 a sublinear function. If Γ ⊂ Nu(Λ), +then LΓ ⊂ LΛ, i.e. every Γ-limit point is a Λ-limit point. In particular, if Γ is a lattice then LΛ = X(∞). +Proof. By definition, one has to show that given a point ξ ∈ X(∞) and a sequence γn ∈ Γ such that γnx0 → ξ +(in the cone topology on X), there is a corresponding sequence λn ∈ Λ with λnx0 → ξ. Define λn := λγn to +be the closest point to γn in Λ, and to ease notation denote xn := γnx0, x′ +n = λnx0. Let also ηn := [x0, xn] +and η′ +n := [x0, x′ +n] be unit speed geodesics. Finally, let Tn denote the time in which ηn terminates, i.e. +ηn(Tn) = xn. +Convergence in the cone topology xn → ξ is equivalent to the fact that the geodesics ηn converge to +η := [x0, ξ) uniformly on compact sets. This means in particular that Tn → ∞. Non-positive curvature +guarantees that the functions Fn(t) := d +� +η(t), ηn(t) +� +, F ′ +n(t) := d +� +η(t), η′ +n(t) +� +and Gn := d +� +ηn(t), η′ +n(t) +� +are +convex (F ′ +n is just a notation, completely unrelated to the derivative of Fn). Since Gn(0) = Fn(0) = F ′ +n(0) = 0 +any of these functions is either constant 0 or monotonically increasing, so proving uniform convergence of +η′ +n to η amounts to proving limn F ′ +n(T ) = 0 for every T ∈ R≥0. +Triangle inequality gives F ′ +n(T ) ≤ Fn(T ) + Gn(T ), and by assumption limn Fn(T ) = 0. Notice that +Gn(Tn) = d(xn, x′ +n) ≤ u(|γn|) = u(Tn). Writing T = +T +Tn · Tn, convexity of Gn implies +Gn(T ) ≤ (1 − T +Tn +)Gn(0) + T +Tn +Gn(Tn) ≤ 0 + T +Tn +u(Tn) = T · u(Tn) +Tn +As limn Tn = ∞ it follows from sublinearity that limn Gn(T ) = 0. I conclude that η′ +n converge to η +uniformly on compact sets, therefore ξ lies in the limit set of Λ. +Remark 2.17. In Section 4.2 I prove that in the setting of Proposition 4.12, the set of Λ-conical limit points +contains the set of Γ-conical limit points (Corollary 4.27). It holds that the conical limit points of Γ are +dense in X(∞) (in the cone topology, see Corollary 2.12) and therefore LΛ = LΓ = X(∞). In particular, +every Γ-limit point is a Λ-limit point. The strength of Lemma 2.16 is that it does not assume anything on Γ +other than that it is sublinearly covered by Λ. In particular, Lemma 2.16 does not require Γ to be a lattice. +2.2.2 +Cusps, Compact Core, and the Rational Tits Building +In this section I present some of the structure theory of non-compact quotients of X. The focus is on the +structure of ‘cusps’ in noncompact finite volume quotients of symmetric spaces, and the ‘location’ of cusps +on the visual boundary. +Cusps and Compact Core. +Consider V = Γ\X, for Γ ≤ G a non-uniform lattice. This is a locally +symmetric space of finite volume. The term ‘cusps’ is an informal name given to those areas in a locally +symmetric space through which one can ‘escape to infinity’. Another description is that cusps are the ends +of the complement of a large enough compact set in V. In strictly negative curvature, i.e. in R-rank 1 locally +symmetric spaces, these cusps have a precise description as submanifolds of the form C × R≥0 for a compact +manifold C, and metrically (C, t) gets narrower as t → ∞. There are finitely many cusps, each corresponding +to a point at X(∞) called a ‘parabolic point’. See e.g. Introduction in [3] or [19]. A fundamental feature +of the cusps is that one can ‘chop’ them out of the quotient manifold V and get a compact manifold. This +could be done in such a way so that: +1. The lifts of the chopped parts to the universal cover X are disjoint. +10 + +2. Each cusp is covered in X by the Γ-orbit of a horoball, that is, the lift of a cusp is the Γ-orbit of a +horoball. The respective base points are called parabolic points of Γ in X(∞). +3. Γ acts on X \ +� � +i∈I HBi +� +cocompactly, where {HBi}i∈I is the set of horoballs coming from the lifts +of cusps. +Since there are only finitely many cusps and Γ discrete, there are exactly countably many such horoballs. +See for example Section 12.6 in [17] where this is illustrated in the case of the real hyperbolic spaces Hn. +Formally, one has; +Theorem 2.18 (Theorem 3.1 in [19], see also Introduction therein). Assume X is of R-rank 1, and Γ ≤ G a +non-uniform lattice. The space V = Γ\X has only finitely many (topological) ends and each end is parabolic +and Riemannian collared. In particular, each cusp is a quotient of a horoball HB based at a parabolic limit +point ξ such that Γ ∩ Gξ acts cocompactly on H = ∂HB. +For symmetric spaces of higher rank, a similar construction is available (see [37]). By removing a countable +family of horoballs from X, one obtains a subspace on which Γ acts cocompactly. There are two main +differences from the situation in R-rank 1. One is that an orbit map γ �→ γx is a quasi-isometric embedding +of Γ (with the word metric) into X. +Theorem 2.19 (Lubozki-Mozes-Raghunathan, Theorem A in [38]). Let G be a semisimple Lie group of +higher R-rank, dG a left invariant metric induced from some Riemannian metric on G. Let Γ an irreducible +lattice, dΓ the corresponding word metric on Γ. Then dG↾Γ×Γ and dΓ are Lipschitz equivalent. +This result plays a significant preliminary role in the proofs of quasi-isometric rigidity for non-uniform +lattices in higher rank symmetric spaces in both [16] and [22]. The second difference in higher rank spaces +is that the horoballs could not in general be taken to be disjoint. However, in the special case of Q-rank 1 +lattices the horoballs can be taken to be disjoint. Recall that the Q-structure of (G, Γ) is a Q-structure on +G = G(R) in which Γ is an arithmetic lattice. The following theorem sums up the relevant properties for +Q-rank 1 lattices. +Theorem 2.20 (Theorem 4.2 and Proposition 2.1 in [34], see also Remarks 3 and 4 in [37], Section 13 +in [50], and Proposition 2.1 in [49]). Assume X is of higher rank, and Γ ≤ G an irreducible torsion-free +non-uniform lattice. On the locally symmetric space V = Γ\X there exists a continuous and piece-wise real +analytic exhaustion function h : V → [0, ∞) such that, for any s > 0, the sublevel set V(s) := {h < s} is a +compact submanifold with corners of V. Moreover the boundary of V(s), which is a level set of h, consists of +projections of subsets of horospheres in X. +The Γ-action on the above set of horospheres has finitely many orbits, and the following conditions are +equivalent: +1. The corresponding horoballs bounded by these horospheres can be taken to be disjoint. +2. For each such horosphere H the action of Γ ∩ StabG(H) on H is cocompact. +3. The Q-structure of (G, Γ) is of Q-rank 1. +4. The lattice Γ is of Q-rank 1. +Definition 2.21. In the setting of Theorem 2.18 and Theorem 2.20, the horoballs and horospheres that +appear in the statement are called (global) +horoballs (horospheres) of Γ. Base points of these are called +parabolic limit points of Γ. +The above geometric characterization of Q-rank 1 lattices is all that I use in order to prove the key +Proposition 4.12. The compact core of Γ is the complement in X of the horoballs of Γ. The group Γ acts on +it cocompactly. The following corollary describes the orbit of Q-rank 1 lattices in X, and especially some +finiteness properties which I will use. +11 + +Corollary 2.22. Let Γ ≤ G be a Q-rank 1 lattice, x ∈ X, and ξ a parabolic limit point. There is a unique +horosphere H based at ξ such that both following conditions hold: +1. Γ · x ∩ H is a cocompact metric lattice in H. +2. Γ · x ∩ HB = ∅, where HB is the horoball bounded by H. +Call H an x-horosphere of Γ at ξ, and the corresponding bounded horoball an x-horoball of Γ. Moreover, +one has: +1. For every C there exists a bound K = K(x, C) so that B(x, C) intersects at most K x-horospheres of +Γ. +2. There is D = D(x) > 0 such that H ⊂ ND(Γ · x ∩ H) for any x-horosphere H of Γ. The constant D is +called the compactness number of (Γ, x). +3. There is a number N = N(Γ) such that every point x ∈ X admits exactly N x-horospheres of Γ that +intersect x. These are called the horospheres of (Γ, x). +Proof. Let x ∈ X. Since Γ is a Q-rank 1 lattice, the stabilizer in Γ of a parabolic limit point ξ ∈ X(∞) acts +cocompactly on each horosphere based at ξ, and in particular on Hx := H(x, ξ). Let D(x, H) be such that +Hx ⊂ ND(Γ · x ∩ Hx). A priori D(x) depends on H, but the fact that Γ acts by isometries implies that for +every horosphere of the form H′ := γH = H(γx, γξ) for some γ ∈ Γ, one has H′ ⊂ ND(Γ · x ∩ H′). So D(x) +depends only on the Γ-orbit of H. Since Γ acts on the set of parabolic limit points with finitely many orbits +(that is to say Γ\X has finitely many cusps) one may take D = D(x) to be the maximum of the respective +bounds on each orbit. This gives the required compactness number. +Thinking of horoballs of γ as lifts of ends of the complement of some compact subset of V = Γ\X, one +sees that there is some horoball HB based at ξ so that Γ · x /∈ HB. Denote H = ∂HB, and y = PHB(x) ∈ H +be the projection on the closed convex set that is the closure of the horoball HB. Finally, Let η = [x, y] +and denote l = d(x, y). The existence of the required horosphere is equivalent to the fact that the following +non-empty set admits a maximum: +Hx,ξ := {t ∈ [0, l] | Γ · x ∩ H +� +η(t), ξ +� +̸= ∅} +Indeed 0 ∈ Hx,ξ and it is a bounded set, so it admits a supremum T . Moreover, this set is discrete. +If t were an accumulation point, then for any small ε > 0 the geodesic segment η↾(t−ε,t+ε) would intersect +D-cocompactly infinitely many horospheres of (Γ, x). Orbit points on different horospheres are in particular +different points, therefore the set B +� +η(t), D + ε +� +∩ Γ · x would be infinite, contradicting discreteness of Γ. I +conclude that T is a maximum, and that H +� +η(T ), ξ +� +is the unique desired horosphere. +The argument above generally shows that there cannot be an accumulation point in X of x-horospheres +of Γ. In particular, for every C, the ball B(x, C) intersects only finitely many x-horospheres of Γ, say K(C), +proving item 1 in the ‘moreover’ statement. +For the last statement, simply note that the horoballs of Γ are the Γ-translates of finitely many horoballs. +In the terminology of the statement, infinitely many horospheres of (Γ, x) imply that infinitely many of +them are in the same Γ-orbit. Suppose these are {Hn}n∈N, with base points ξn that are evidently pairwise +different. Finally let γn ∈ Γ for which γnH1 = Hn. Since Γ ∩ StabG(Hn) acts cocompactly on Hn, there +is γ′ +n ∈ Γ ∩ StabG(Hn) that maps γnx to B(x, D). Discreteness of Γ implies that the set γ′ +nγnx is finite, +hence infinitely many of these points are the same point. However γ′ +nγnξ1 = ξn and therefore γ′ +nγn ̸= IdX, +contradicting the fact that Γ is torsion-free. +Remark 2.23. For the most part, I am interested in a fixed base point x0 and the x0-horospheres and +horoballs. By a slight abuse of terminology I omit x0 and call these objects ‘horospheres of Γ’ and ‘horoballs +of Γ’, respectively, denoting the associated cocompactness number DΓ. +Corollary 2.22 allows to upgrade a metric lattice of H to a metric lattice coming from StabG(H) only. +12 + +Lemma 2.24. Assume that a torsion-free discrete group ∆ ≤ G = Isom(X) admits the following: +1. There is a bound N such that at each point x ∈ ∆ · x0 there are at most N horoballs that are tangent +to x and do not intersect ∆ · x0. +2. There is a horosphere H ⊂ X such that +(a) The set ∆ · x0 ∩ H is a cocompact metric lattice in H. +(b) The horoball HB bounded by H is ∆-free, i.e. ∆ · x0 ∩ HB ⊂ H. +Then +� +∆ ∩ StabG(H) +� +· x0 is also a cocompact metric lattice in H. +Proof. This is the Pigeonhole Principle. Fix x ∈ ∆ · x0 ∩ H, and let {HBi}i∈{1,...,N} be the finite set of +horoballs tangent to x that do not intersect ∆ · x0. Assume w.l.o.g that H is the bounding horosphere of +HB1. Fix a ∆-orbit point δ0x0 ∈ ∆ · x0 ∩ H. Up to translating by some element of ∆, I may assume x = x0. +For any other ∆-orbit point δx0 ∈ ∆ · x0 ∩ H let i(δ) ∈ {1, . . . , N} be the index of the horoball δ−1HB1. +Notice that from hypothesis 1 it indeed follows that δ−1HB1 ∈ {HBi}i∈{1,...,N}, because the action of ∆ is +by isometries. Define δi to be the element in ∆ for which: +1. i(δi) = i, i.e. δ−1 +i +(HB1) = HBi +2. d(δix0, x0) is minimal among all such δ ∈ ∆. +For some i ∈ {1, . . . , N} there is a δ ∈ ∆ with i = i(δ), while for others there might not be. I will only +care about those i for which there is such δ. Assume w.l.o.g that these are i ∈ {1, . . . M} for M ≤ N, and let +L := max1≤i≤M{d(x0, δix0)} < ∞. For such an index i0, the ∆-orbit points that share the same i(δ) = i0 +are in the same ∆ ∩ StabG(H) orbit, namely +{δx0 | i(δ) = i0} ⊂ +� +∆ ∩ StabG(H) +� +· δi0x0 +Indeed, if δ−1HB1 = HBi0 then by definition δδ−1 +i0 HB1 = δHBi0 = HB1, hence δδ−1 +i0 +∈ StabG(H) is +an element mapping δi0x0 to δx0. Let now δx0 ∈ H. Its distance from the orbit +� +∆ ∩ StabG(H) +� +· x0 is +� +∆ ∩ StabG(H) +� +-invariant, therefore +d +� +δx0, +� +∆ ∩ StabG(H) +� +x0 +� += d +� +δi(δ)x0, +� +∆ ∩ StabG(H) +� +x0 +� +≤ d(δi(δ), x0) +The right-hand side is uniformly bounded by L, proving that +(∆ · x0 ∩ H) ⊂ NL +�� +∆ ∩ StabG(H) +� +· x0 +� +The fact that ∆ · x0 ∩ H is a cocompact metric lattice in H renders (∆ ∩ StabG(H) +� +· x0 a cocompact +metric lattice in h as well, as claimed. +Real and Rational Tits Buildings. +The location of the parabolic points in X(∞) also plays an important +role in the geometry of X. In case Γ is an arithmetic lattice, the natural framework to consider these points +is the so called rational Tits building. +This is a building structure on the subset of parabolic points at +X(∞), sometimes referred to as ‘rational points’ in this case. They are exactly those points in X(∞) whose +stabilizers are Q-defined (algebraic) parabolic groups of G (see Section 4.4 for more details). I present this +object, denoted WQ(Γ), together with the more familiar real Tits building structure on X(∞) with the Tits +metric. The main goal is to present the results of Hattori [25], that give a good description of the rational +Tits building in terms of conical and horospherical limit points. In case G is of R-rank 1, by WQ(Γ) I mean +the (countable) set of parabolic limit points of Γ (so that X(∞) \ WQ(Γ) is comprised of conical limit points +only, see Theorem 2.29). +13 + +Definition 2.25. A geodesic η is said to be regular if it is contained in a unique maximal flat F ⊂ X. The +point η(∞) ∈ X(∞) is called a regular point of X(∞). A point ξ ∈ X(∞) is singular if it is not regular. +Regularity does not depend on the choice of representative geodesic ray η for ξ. +A Weyl chamber of X(∞), or an open spherical chamber, is any connected component in the Tits topology +of X(∞) \ S, where S ⊂ X(∞) is the subset of singular points at X(∞). +Proposition 2.26 (Propositions 2.2 and 3.2 in [29] and Section 8 in [3]). The Weyl chambers induce a +simplicial complex structure on X(∞) that is a spherical Tits building. The apartments of the building are +exactly the sets of the form F(∞) ⊂ X(∞) for all flats F ⊂ X, and the chambers are exactly the Weyl +chambers at X(∞). Moreover, the Tits metric completely determines the building structure, and vice versa, +and +� +X(∞), dT +� +is a metric realization of the Tits building at X(∞). +None of the rich theory of buildings is used directly in this paper. Given a non-uniform lattice of Γ ≤ G +the rational Tits building WQ(Γ) is a building structure on the subset of parabolic points. It is not in general +a sub-building of the real spherical building. Flats of X correspond to real maximal split tori in G. Since +G is an algebraic group defined over Q, one can consider the maximal Q-split tori. The rational flats of X +are then the G(Q)-orbits of maximal Q-split tori of G, and the rational boundary are all points ξ ∈ X(∞) +such that ξ ∈ FQ(∞) for some rational flat FQ(∞). One defines regular rational directions and rational Weyl +chambers in an analogous way to the real case, this time taking only rational flats into account. For further +details details see [29], and Section 2 in [25]. +Theorem 2.27 (Theorem A in [25]). Let X = G/K be a symmetric space of non-compact type and of higher +rank, and let Γ ≤ Isom(X) be an irreducible non-uniform lattice. Then WQ(Γ) does not include horospherical +limit points. The π +2 -neighbourhood +N π +2 +� +WQ(Γ) +� +:= {ξ ∈ X(∞) | dT +� +ξ, WQ(Γ) +� +< π +2 } +does not include conical limit points. +In Q-rank 1 , the converse statement also holds: +Theorem 2.28 (Theorem B in [25]). Let X = G/K be a symmetric space of non-compact type of higher +rank and Γ ≤ Isom(X) be an irreducible non-uniform lattice. Let +V = {ξ ∈ X(∞) | dT +� +ξ, WQ(Γ) +� +≥ π +2 } +Suppose that Γ is of Q-rank 1 . Then V consists of conical limit points only. +In groups of R-rank 1 one has the following well known fact: +Theorem 2.29 (Proposition 5.4.2 and Theorem 6.1 in [10], see also Theorem 12.29 in [17]). Let X be a +symmetric space of noncompact type and of R-rank 1, Γ ≤ Isom(X) a lattice. Then every ξ ∈ X(∞) is either +conical or non-horospherical. +Corollary 2.30. When Γ is of Q-rank 1 , the following holds: +1. WQ(Γ) = {ξ ∈ X(∞) | N π +2 (ξ) does not contain conical limit points} +2. Any two points ξ, ξ′ ∈ WQ(Γ) are at Tits distance = π. +Proof. In view of Theorem 2.29, for R-rank 1 both statements hold trivially. In higher rank, both follow +from the following observation: for any point ξ′ ∈ WQ(Γ) and any point ζ ∈ X(∞) with d(ζ, ξ′) = π +2 , ζ is +conical. To see this notice that ζ lies on the boundary of a horosphere based at ξ′: take a flat F ⊂ X with +ξ′, ζ ∈ F(∞). Any geodesic with limit ζ is contained in (a Euclidean) horosphere based at ξ′. The fact that +Γ is cocompact on the horospheres based at WQ(Γ) implies that ζ is conical. +14 + +The second item of the corollary follows: let ξ, ξ′ ∈ WQ(Γ) and c : [0, α] a Tits geodesic joining them. +There is a flat F ⊂ X containing both ξ, ξ′ as well as c ⊂ F(∞). Every point ζ that is at Tits distance +π +2 from either ξ or ξ′ is conical, and no point inside the π +2 neighbourhood of either ξ or ξ′ is conical. In +F(∞) the Tits metric is the same as the Tits metric on the Euclidean space of an equal rank. Therefore +one may prolong the geodesic c so that c(0) = ξ, c(α) = ξ′ and c(π) = ξ′′. If dT (ξ, ξ′) < π, then there is +a point along this prolonged geodesic that is at Tits distance exactly π +2 from ξ (so it is conical by the first +paragraph), but at Tits distance strictly less than π +2 from ξ′ (so it cannot be conical by Theorem 2.27). +Therefore dT (ξ, ξ′) = π. +For the first item, one containment is just Hattori’s Theorem 2.27. For the other containment, pretty +much the same argument from above works. Assume for some ξ ∈ X(∞) that N π +2 (ξ) consists of non-conical +limit points. In particular ξ itself is not conical, and by Theorem 2.28 it holds that d(ξ, WQ(Γ)) < π +2 . Let +ξ′ ∈ WQ(Γ) be a point realizing this distance. As above, this gives rise to a flat F containing ξ, ξ′ and +another point ζ that is at Tits distance π +2 from ξ′ but at Tits distance strictly less than π +2 from ξ. The first +forces ζ to be conical, and the latter forces it to be non-conical, a contradiction. +Hattori’s characterization relies on a simple lemma which will also be of use in the sequel. It relates the +(linear) penetration rate of a geodesic into a horoball to the Tits distance. +Lemma 2.31 (Lemma 3.4 in [25]). Let X be a symmetric space of higher rank and of noncompact type. +Let η1, η2 : [0, ∞) → X be two geodesic rays, α := dT +� +η1(∞), η2(∞) +� +and b2 the Busemann function +corresponding to η2. Then there exists a positive constant C1, depending only on η1 and η2, such that: +1. If α > π +2 , then for all t ≥ 0 +b2 +� +η1(t) +� +≥ −t · cos α − C1 +2. If α = π +2 , then b2 +� +η1(t) +� +is monotone non-increasing in t and −C1 ≤ b2 +� +η1(t) +� +. +3. If α < π +2 , then for all t ≥ 0 +b2 +� +η1(t) +� +≤ −t · cos α − C1 +Remark 2.32. If X is a symmetric spaces of R-rank 1, maximal flats are geodesics. Therefore every two +points ξ, ζ ∈ X(∞) admit dT (ξ, ζ) = π, and it is clear from the strict negative curvature that Lemma 2.31 +is true also in this case. +3 +Uniform Lattices +In this section I prove: +Theorem 3.1. Let G be a semisimple Lie group without compact factors and with finite centre. Let Γ ≤ G +be a lattice, Λ ≤ G a discrete subgroup such that Γ ⊂ Nu(Λ) for some sublinear function u. If Γ is uniform, +then Λ is a uniform lattice. +The focus of this paper is on sublinear distortion, however for uniform lattices (and also for lattices that +have property (T), see Section 5), a slightly stronger result holds. I call this ε-linear rigidity. +Definition 3.2. Let f, g : R≥0 → R>0 be two monotonically increasing functions. Call f asymptotically +smaller than g if lim sup f +g ≤ 1. Denote this relation by f ⪯∞ g. +Theorem 3.3. In the setting of Theorem 3.1, the conclusion holds also under the relaxed assumption that +u(r) ⪯∞ εr for any 0 < ε < 1. +Clearly Theorem 3.3 implies Theorem 3.1. From now and until the end of this section, the standing +assumptions are those of Theorem 3.3. +15 + +Lattice Criterion. +A discrete group is a uniform lattice if and only if it admits a relatively compact +fundamental domain. The criterion I use is the immediate consequence that if Γ is uniform and u is bounded +(i.e. Γ ⊂ ND(Λ) for some D > 0), then Λ is a uniform lattice. +Outline of Proof and Use of ε-Linearity. +The goal is to show that the ε-linearity of u forces Γ ⊂ ND(Λ) +for some D > 0, i.e. that Γ actually lies inside a bounded neighbourhood of Λ. The proof is by way of +contradiction. If there is no such D > 0 then there are arbitrarily large balls that do not intersect Λ. The +proof goes by finding such large Λ-free balls that are all tangent to some fixed arbitrary point x ∈ X (see +Figure 1). The ε-linearity then gives rise to concentric Γ-free balls that are arbitrarily large, contradicting +the fact that Γ is a uniform lattice. +Remark 3.4. The main difference from the non-uniform case is that for a non-uniform lattice Γ, the space +X does admit arbitrarily large Γ-free balls. This situation requires different lattice criteria and much extra +work. Still the proof for the uniform case, though essentially no more than a few lines, lies the foundations +for and presents the logic of the much more involved case of Q-rank 1 lattices. +3.1 +Notations and Terminology +Definition 3.5. Let X be a metric space, Y, Z ⊂ X two closed subsets of X. The closest point projection +of Y to Z is the set theoretic map pZ : Y → Z defined by pZ(y) := zy, where zy ∈ Z is any point realizing +the distance d(y, Z) = d(y, z). If X is a proper metric space and Z discrete, then there are at most finitely +many such points. In any case of multiple points, pZ chooses one arbitrarily. +The particular case of interest from now on is where the metric space is the pointed symmetric space +(X, x0), the two subsets are the orbits Γ · x0 and Λ · x0, and the projection is pΛ·x0 : Γ · x0 → Λ · x0. To ease +notation I often denote this projection by pΛ (there is no risk of ambiguity since the subgroups Λ and Γ are +always considered in the context of their respective orbits in X and not in G). +The following definitions will be used repeatedly in both this section and in Section 4. It mainly fixes +terminology and notation of the geometric situation illustrated in Figure 1. +Definition 3.6. Let H ≤ G = Isom(X)◦. A set U ⊂ X is called H-free if H · x0 ∩ Int(U) = ∅, where Int(U) +is the topological interior of U. That is, U is called H-free if its interior does not intersect the H-orbit H ·x0. +Definition 3.7. Denote PΛ(γx0) = PΛ(γ) = λγx0. +1. dγ := d(γx0, λγx0). +2. Bγ := B(γx0, dγ). It is a Λ-free ball centred at γx0 and tangent to λγx0. +3. x′ +γ := λ−1 +γ γx0. Notice |x′ +γ| = dγ. +4. B′ +γ := λ−1 +γ Bγ = B(x′ +γ, dγ). It is Λ-free as a Λ-translate of the Λ-free ball Bγ, and is tangent to x0. +5. For s ∈ R>0 and a ball B = B(x, r), denote sB := B(x, sr), the rescaled ball with same centre and +radius sr. +6. For a sequence γn, denote by λn, dn, Bn, B′ +n, x′ +n the respective λγn, dγn, etc. +3.2 +Proof of Theorem 3.3 +Lemma 3.8. Let x ∈ X. There exists S = S(x, u) ∈ (0, 1) such that for every s ∈ (0, S) there is R = R(s, S) +such that if r > R and B = B(y, r) is a Λ-free ball tangent to x, then sB is Γ-free. +In particular, the existence of arbitrarily large Λ-free balls that are all tangent to a fixed point x ∈ X +implies the existence of arbitrarily large Γ-free balls. +16 + +x0 +x′ +γ += dγ += s · dγ +Γ − free +Λ − free +γx0 += dγ +Λ − free +λγx0 +Lλ−1 +γ +Figure 1: Basic Setting and Lemma 3.8. A Λ-free ball about γx0 of radius dγ, translated by λ−1 +γ +to a ball +tangent to x0. The linear ratio between |x′ +γ| = dγ and the Λ-free radius forces the red ball to be Γ-free. +There is a slightly stronger version of Lemma 3.8 if u is sublinear: +Lemma 3.9. Let G, Γ, Λ and u be as in Theorem 3.1 (in particular, u is a sublinear function and Γ ⊂ Nu(Γ)). +For every x ∈ X and every s ∈ (0, 1) there exists R = R(x, s) > 0 such that for every r > R, if B = B(y, r) +is a Λ-free ball tangent to x then sB is Γ-free. +I omit the proof of Lemma 3.9, which is a slightly simpler version of the proof of Lemma 3.8. +Proof of Lemma 3.8. The proof is more easily read if one assumes x = x0 and u(r) = εr so I begin with this +case. Assume B = B(y, R) is Λ-free for some y ∈ X. The assumption x = x0 means that |y| = d(y, x0) = r. +Assume that for a given s ∈ (0, 1), the ball sB intersects Γ · x0. This gives rise to an element γ ∈ Γ such +that: +1. |γ| = d(γx0, x0) ≤ d(y, x0) + d(γx0, y) = (1 + s)r (triangle inequality). In particular, d(γx0, Λ · x0) ≤ +ε(1 + s)r +2. B +� +γx0, (1 − s)r +� +⊂ B(y, r), so it is Λ-free. +I conclude that for s for which sB ∩ Γ · x0 ̸= ∅, one has the inequality (1 − s)r ≤ ε(1 + s)r, i.e. 1−s +1+s ≤ ε. +The number ε is fixed and smaller than 1, while 1−s +1+s limit to 1 monotonically from below as s > 0 tend to +0. I conclude that there is a segment (0, S) ⊂ (0, 1) such that for all s ∈ (0, S), sB is Γ-free. +Assume now that x ̸= x0 and u(r) ⪯∞ εr. As above, if γx0 ∈ B(y, sr) then it is the centre of a Λ-free +ball of radius (1 − s)r, and so (1 − s)r ≤ u(|γ|). I wish to use the ε-linear bound on u as I did before, +only this time u is only asymptotically smaller than εr. To circumvent this I just need to show that |γ| +is large enough. Indeed since B +� +γx0, (1 − s)r +� +is Λ-free it does not contain x0 ∈ Λ · x0 and in particular +(1 − s)r ≤ d(x0, γx0) = |γ|. For some R1(s) = R1(s, u) one therefore has for all r > R1(s) +(1 − s)r ≤ u(|γ|) ≤ ε|γ| +On the other hand |y| ≤ d(x, y) + d(x, x0) = r + |x|, and consequently |γ| ≤ (1 + s)r + |x|. For r > R1(s) +one has +17 + +(1 − s)r ≤ u(|γ|) ≤ ε|γ| ≤ ε +� +(1 + s)r + |x| +� +This means that s for which Γ · x0 ∩ B(y, sr) ̸= ∅ must admit, for all r > R1(s), +1 − s +1 + s + |x| +r += +(1 − s)r +(1 + s)r + |x| ≤ ε < 1 +(1) +The rest of the proof is just Calculus 1, and concerns with finding S = S(x, ε, u) ∈ (0, 1) so that for any +s ∈ (0, S) there is R(s) such that all r > R(s) satisfy +ε < +1 − S +1 + S + |x| +r +≤ +1 − s +1 + s + |x| +r +(2) +The lemma readily follows from inequalities 1,2. +Explicitly, fix ε′ > ε. As before, monotonic approach of +1−s +1+s to 1 allows to fix S ∈ (0, 1) for which +ε < ε′ < 1−s +1+s for all s ∈ (0, S). Next note that for any fixed s ∈ (0, S), limr→∞ +1−s +1+s+ |x| +r = 1−s +1+s, and that the +approach in monotonically increasing with r. Since ε < ε′, this limit implies that for some R2 > R1(S), all +r > R2 admit ε < +1−S +1+S+ |x| +r . Finally notice that for any fixed r the function +1−s +1+s+ |x| +r +is again monotonically +increasing as s tends to 0 from above. Therefore inequality 2 holds for every s ∈ (0, S) and all r > R2(S) +(capital S is intentional and important). +To conclude the proof, notice that if moreover r > R1(s) (again lowercase s is intentional and important) +then inequalities 1,2 both hold. This means that for any s ∈ (0, S) there is R(s) := max{R1(s), R2(S)} such +that r > R(s) ⇒ B(y, sr) is Γ-free. The constants R1(s), R2(S) have the desired dependencies, hence so +does R(s), proving the lemma. +Corollary 3.10 (Theorem 3.3). There is a uniform bound on {dγ}γ∈Γ, i.e., Γ ⊂ ND(Λ) for some D > 0. +In particular, Λ is a uniform lattice. +4 +Q-rank 1 Lattices +In this section I prove: +Theorem 4.1. Let G be a semisimple Lie group without compact factors and with finite centre, Γ ≤ G an +irreducible non-uniform Q-rank 1 lattice, Λ ≤ G a discrete irreducible subgroup. If Γ ⊂ Nu(Λ) for some +sublinear function u, then Λ is a lattice. Moreover, if Γ ̸⊂ ND(Λ) for any D > 0, then Λ is also of Q-rank 1. +If Γ ⊂ ND(Λ) for some D > 0, then Λ could be a uniform lattice. An obvious obstacle for that is if +Λ ⊂ Nu′(Γ) for some sublinear function u′. This condition turns out to be sufficient for commensurability. +Proposition 4.2. Let G be a semisimple Lie group without compact factors and with finite centre, Γ ≤ G an +irreducible non-uniform Q-rank 1 lattice, Λ ≤ G a discrete subgroup such that Γ ⊂ ND(Λ) for some D > 0, +and Λ ⊂ Nu(Γ) for some sublinear function u. Then Λ ⊂ ND′(Γ) for some D′. +As a result of Eskin’s and Schwartz’s arguments (see Section 4.3), I conclude: +Corollary 4.3. In the setting of Proposition 4.2 and unless G is locally isomorphic to SL2(R), Λ is com- +mensurable to Γ. +Theorem 1.8 is a result of Theorem 4.1 and Corollary 4.3 +18 + +4.1 +Strategy +Lattice Criteria. +I use three different lattice criteria, depending on the R-rank of G and on whether or +not Γ ⊂ ND(Λ) for some D > 0. My proof for Q-rank 1 lattices is motivated by a criterion of Benoist +and Miquel, resolving a conjecture of Margulis. It can be viewed as an algebraic converse to the geometric +structure of the compact core described in Theorem 2.20. +Theorem 4.4 (Theorem 2.16 in [5]). Let G be a semisimple real algebraic Lie group of real rank at least 2 +and U be a non-trivial horospherical subgroup of G. Let ∆ be a discrete Zariski dense subgroup of G that +contains an indecomposable lattice ∆U of U. Then ∆ is a non-uniform irreducible arithmetic lattice of G. +Remark 4.5. See Definition 4.47 for the precise meaning of an indecomposable horospherical lattice. +For R-rank 1 groups, one has the following theorem by Kapovich and Liu, stating that a group is +geometrically finite so long as ‘most’ of its limit points are conical. Recall L(∆) is the limit set of ∆ ≤ +Isom(X), and Lcon(∆) is the set of its conical limit points. +Theorem 4.6 (Theorem 1.5 in [30]). Let X be a R-rank 1 symmetric space. A discrete subgroup ∆ ≤ +Isom(X) is geometrically infinite if and only if the set L(∆) \ Lcon(∆) of non-conical limit points has the +cardinality of the continuum. +As a direct corollary I obtain the following criterion: +Corollary 4.7. Let X be a R-rank 1 symmetric space, Γ ≤ G = Isom(X) a non-uniform lattice and Λ ≤ G +a discrete subgroup. If L(Λ) = X(∞) and Lcon(Γ) ⊂ Lcon(Λ), then Λ is a lattice. +Proof. Since Γ is a lattice, L(Γ) = X(∞) and it is geometrically finite. Theorem 4.6 implies the cardinality +of X(∞) \ Lcon(Γ) is strictly smaller than the continuum. The assumption Lcon(Γ) ⊂ Lcon(Λ) implies the +same holds for Λ, and in particular that Λ is geometrically finite. The assumption that L(Λ) = X(∞) implies +that Λ is geometrically finite if and only if it is a lattice. +Corollary 4.8. Let X be a R-rank 1 symmetric space, Γ ≤ G = Isom(X) a non-uniform lattice and Λ ≤ G +a discrete subgroup. If Γ ⊂ ND(Λ) for some D > 0, then Λ is a lattice. +Proof. By definition of the limit set and of conical limit points, it is clear that every Γ-limit point is a Λ-limit +point, and every Γ-conical limit point is also Λ-conical limit point. I conclude from Corollary 4.7 that Λ is +a lattice. +Also in higher rank the inclusion Γ ⊂ ND(Λ) implies that Λ is a lattice. This result is due to Eskin. +Theorem 4.9 (Eskin, see Remark 4.11 below). Let G be a semisimple Lie group without compact factors and +of higher rank, Γ ≤ G an irreducible non-uniform lattice, Λ ≤ G a discrete subgroup such that Γ ⊂ ND(Λ) +for some D > 0. Then Λ is a lattice. +Theorem 4.9 was used in the proof of quasi-isometric rigidity for higher rank non-uniform lattices in [16], +[22]. In the (earlier) R-rank 1 case, Schwartz [52] used an analogous statement, which requires one extra +assumption. +Theorem 4.10 (Schwartz, see Section 10.4 in [52] and Remark 4.11 below). Let G be a real simple Lie group +of R-rank 1 and with finite centre, Γ ≤ G an irreducible non-uniform lattice, Λ ≤ G a discrete subgroup such +that both Γ ⊂ ND(Λ) and Λ ⊂ ND(Γ) for some D > 0. Then Λ is a lattice and commensurable to Γ. +Remark 4.11. Theorem 1.6 should be viewed as a generalization of the bounded case depicted in Theo- +rems 4.9 and 4.10, which were known to experts in the field in the late 1990’s. Complete proofs for these +statements were never given in print, and I take the opportunity to include them here. See Section 4.3, where +I also prove Proposition 4.2. I thank Rich Schwartz and Alex Eskin for supplying me with their arguments +and allowing me to include them in this paper. I also thank my thesis examiner Emmanuel Breuillard for +encouraging me to find and make these proofs public. +19 + +Outline of Proof and Use of Sublinearity. +Lattices of Q-rank 1 admit a concrete geometric structure +(see Section 2.2). This structure is manifested in the geometry of an orbit of such a lattice in the corresponding +symmetric space X = G/K. One important geometric property is the existence of a set of horoballs which +the orbit of the lattice intersects only in the bounding horospheres, and in each such horosphere the orbit +forms a (metric) cocompact lattice. +Corollary 4.8 and Theorem 4.9 reduce the proof to the case where Γ ̸⊂ ND(Λ) for any D > 0. In that +case, the essence lies in proving the existence of horospheres in X which a Λ-orbit intersects in a cocompact +lattice. +This is proved purely geometrically, using the geometric structure of Q-rank 1 lattices and the +sublinear constraint. Together with some control on the location of these horospheres, I prove two major +statements: +1. Λ · x0 intersects a horosphere H ⊂ X in a cocompact lattice (Proposition 4.12). +2. Every Γ-conical limit point is also a Λ-conical limit point (Corollary 4.27). +The R-rank 1 case of Theorem 4.1 follows directly from Corollary 4.7 using the second item above. The +higher rank case requires a bit more. namely it requires to deduce from the above items that Λ meets +the hypotheses of the Benoist-Miquel Theorem 4.4. To that end I use a well known geometric criterion +(Lemma 4.50) in order show that Λ is Zariski dense, and a lemma of Mostow (Lemma 4.36) to show that Λ +intersects a horospherical subgroup in a cocompact lattice. +Outline for Section 4. +Section 4.2 is the core of the original mathematics of this paper. It is devoted +to proving that Λ · x0 intersects some horospheres in a cocompact lattice. The proof is quite delicate and +somewhat involved, and I include a few figures and a detailed informal overview of the proof. The figures +are detailed and may take a few moments to comprehend, but I believe they are worth the effort. +Section 4.3 deals with the case where Γ ⊂ ND(Λ), and elaborates on Schwartz’s and Eskin’s proofs +of Theorem 4.9 and Theorem 4.10. Section 4.4 is devoted to the translation of the geometric results of +Section 4.2 to the algebraic language used in Theorem 4.4. +Though the work is indeed mainly one of +translation, some of it is non-trivial. Finally, in Section 4.5 I put everything together for a complete proof +of Theorem 4.1. +I highly recommend the reader to have a look at the uniform case in Section 3 before reading this one. +4.2 +A Λ-Cocompact Horosphere +Recall that dγ := d(γx0, λγx0). In this section I prove: +Proposition 4.12. If {dγ}γ∈Γ is unbounded, then there exists a horosphere H based at WQ(Γ) such that +� +Λ∩StabG(H) +� +·x0 intersects H in a cocompact metric lattice. Moreover, the bounded horoball HB is Λ-free. +Throughout Section 4.2 the standing assumptions are that {dγ}γ ∈ Γ is unbounded, and Γ is an irreducible +Q-rank 1 lattice. +The Argument. +The proof is by chasing down the geometric implications of unbounded dγ. +These +implications are delicate, but similar in spirit to the straight-forward proof for uniform lattices. The proof +consists of the following steps: +1. Unbounded dγ results in Λ-free horoballs HBΛ tangent to Λ-orbit points. Each such horoball is based +at WQ(Γ), giving rise to corresponding horoballs of Γ, denoted HBΓ. +2. If dγ is large, then γx0 must lie deep inside a unique Λ-free horoball tangent to λγx0. I use: +(a) A bound on the distance d(HΛ, HΓ). +(b) A bound on the angle ∠λγx0([λγx0, γx0], [λγx0, ξ)), where ξ ∈ X(∞) is the base point of a suitable +Λ-free horoball tangent to λγx0. +20 + +3. There exist horospheres of Γ, say HΓ, such that if γx0 ∈ HΓ then large dγ implies large Λ-free areas +along the bounding horosphere of some HBΛ. +4. If HBΛ is boundedly close to some Λ-orbit point, then HΛ is almost Λ-cocompact, that is HΛ ⊂ +ND(Λ·x0) for some universal D = D(Λ). Together with the previous step, this yields a uniform bound +on dγ along certain horospheres of Γ. +5. Finally I elevate the almost cocompactness to actual cocompactness and show HΛ ⊂ ND(Λ · x0 ∩ HΛ) +for some D > 0. This immediately elevates to HΛ ⊂ (Λ ∩ StabG(HΛ)) · x0, proving the proposition. +The Properties of Γ. +The geometric properties of Γ that are used in the proof are: +1. In higher rank, the characterization of WQ(Γ) using conical / non-horospherical limit points (Corol- +lary 2.30). In R-rank 1, the dichotomy of limit points being either non-horospherical or conical (The- +orem 2.29). +2. Γ-cocompactness along the horospheres of Γ. +3. For every point x ∈ X and C > 0 there is a bound K(C) on the number of horospheres of Γ that +intersect B(x, C) (Corollary 2.22). +4.2.1 +Λ-Free Horoballs +I retain the notations and objects defined in Section 3.1. +Lemma 4.13. There exists a Λ-free horoball tangent to x0. +Proof. Since {dγ}γ∈Γ is unbounded, there are γn ∈ Γ with dn = dγn = d(γn, λn) → ∞ monotonically, where +λn ∈ Λ is a Λ-orbit point closest to γn. Denote x′ +n = λ−1 +n γnx0, ηn := [x0, x′ +n], and vn ∈ Sx0X the initial +velocity vectors vn := +˙ηn(0). The tangent space Sx0X is compact, so up to a subsequence, vn converge +monotonically in angle to a direction v ∈ Sx0X. Let η be the unit speed geodesic ray emanating from x0 +with initial velocity ˙η(0) = v. Denote ξ := η(∞) the limit point of η in X(∞). +I claim that the horoball HB := ∪t>0B +� +η(t), t +� +, based at ξ and tangent to x0, is Λ-free. Let t > 0 and +consider η(t). For every ε > 0, there is some angle α = α(t, ε) such that any geodesic η′ with ∠x0(η, η′) < α +admits d +� +η(t), η′(t) +� +< ε/2. The convergence vn → v implies d +� +η(t), ηn(t) +� +< ε/2 for all but finitely many +n ∈ N. In particular, B +� +η(t), t +� +⊂ Nε +� +B +� +ηn(t), t +�� +for all such n ∈ N. +For a fixed t ≤ dn, it is clear from the definitions that B +� +ηn(t), t +� +⊂ B′ +n = B(x′ +n, dn). One has dn → ∞, +and so for a fixed t > 0 it holds that t < dn for all but finitely many n ∈ N. I conclude that for any fixed +t > 0 there is n large enough such that +B +� +η(t), t +� +⊂ Nε +� +B +� +ηn(t), t +�� +⊂ NεB′ +n +I conclude that for every ε > 0, HB ⊂ � +n Nε(B′ +n) = Nε +� � +n B′ +n +� +. This implies that any point in the +interior of HB is contained in the interior of one of the Λ-free balls B′ +n, proving HB is Λ-free. +Lemma 4.14. Suppose HB is a Λ-free horoball, based at some point ξ ∈ X(∞). Then ξ ∈ WQ(Γ). +Proof. For any geodesic η with limit ξ, the size d(x0, γx0) of the Γ-orbit points γx0 that lie boundedly close +to η grows linearly in the distance to any fixed horosphere based at ξ, and in particular to H = ∂HB. The +sublinear constraint d(γx0, λγx0) ≤ u(|γ|) together with the fact that HB is Λ-free imply that the size of +such γ is bounded. In R-rank 1 every limit point is either conical or in WQ(Γ), proving the lemma in this +case. For higher rank, the above argument actually shows more: it shows that a point ξ′ ∈ N π +2 (ξ) is not +conical, because every geodesic with limit ξ′ ∈ N π +2 (ξ) entres HB at a linear rate (Lemma 2.31). Hattori’s +characterization of WQ(Γ) (Corollary 2.30) implies ξ ∈ WQ(Γ). +21 + +Definition 4.15. Given a Λ-free horoball HBΛ, Lemma 4.14 gives rise to a horoball of Γ based at the same +point at X(∞). Call this the horoball corresponding to HBΛ, and denote it by HBΓ. The corresponding +horosphere is denoted HΓ. +Remark 4.16. In the course of my work I had had a few conversations with Omri Solan regarding the +penetration of geodesics into Λ-free horoballs. Assuming Λ ⊂ Nu(Γ) implies that Λ preserves WQ(Γ) (see +Lemma 4.32). This is the case in the motivating setting where Λ is an abstract finitely generated group that +is SBE to Γ, see Claim 6.4 in Chapter 6. In the case of SL2(R) Omri suggested to use the action of Λ on +the Bruhat-Tits tree of SL2(Qp) (for all primes p) and the classification of these elements into elliptic and +hyperbolic elements (separately for each p) in order to deduce that Λ actually lies in SL2(Z). We did not +pursue that path nor its possible generalization to the SLn case and general Bruhat-Tits buildings. +4.2.2 +A Γ-orbit point Lying Deep Inside a Λ-Free Horoball +I established the existence of Λ-free horoballs. It may seem odd that the first step in proving Λ · x0 is +‘almost everywhere’ is proving the existence of Λ-free regions. But this fits perfectly well with the algebraic +statement that non-uniform lattices must admit unipotent elements (see Proposition 5.3.1 in [39]). The goal +of this section is to obtain some control on the location of the Λ-free horoballs, in order to conclude that +some γx0 lies deep inside HBΛ. This results in yet more Λ-free regions, found on the bounding horosphere. +I need one property of sublinear functions. +I thank Panos Papazoglou for noticing a mistake in the +original formulation. +Lemma 4.17. Let u be a sublinear function, f, g : R≥0 −→ R>0 two positive monotone functions with +limx→∞ f(x) + g(x) = ∞. If for all large enough x it holds that f(x) ≤ u +� +f(x) + g(x) +� +, then for every 1 < s +and all large enough x it holds that f(x) ≤ u +� +s · g(x) +� +. In particular f(x) ≤ u′� +g(x) +� +for some sublinear +function u′. +Proof. Assume as one may that u is non-decreasing. By definition of sublinearity limx→∞ +u +� +f(x)+g(x) +� +f(x)+g(x) += 0, +so by hypothesis limx→∞ +f(x) +f(x)+g(x) = 0. This means that for every ε > 0 one has f(x) ≤ ε · g(x) for all large +enough x, resulting in +f(x) ≤ u +� +f(x) + g(x) +� +≤ u +� +(1 + ε) · g(x) +� +Notice that for a fixed s > 0, the function u′(x) = u(sx) is sublinear, as +lim +x→∞ +u(sx) +x += lim +x→∞ s · u(sx) +sx += 0 +Lemma 4.18. Let C > 0. There is L = L(C) such that if HBΛ is any Λ-free horoball tangent to a point +x ∈ B(x0, C) then d(HΛ, HΓ) ≤ L. Moreover, there is a sublinear function u′ such that: +L(C) ≤ +� +u′(C) +if HBΓ ⊂ HBΛ +C +if HBΛ ⊂ HBΓ +Proof. If HBΛ ⊂ HBΓ, then clearly d(HΛ, HΓ) ≤ C, simply because HBΓ is Γ-free and in particular cannot +contain x0. Therefore HΓ must separate HΛ from x0 and in particular d(HΛ, HΓ) ≤ C. +Assume that HBΓ ⊂ HBΛ, and denote l = d(HΛ, HΓ). The horoball HBΓ is a horoball of Γ, hence Γ · x0 +is DΓ-cocompact along HΓ and there is an element γ ∈ Γ with |γ| ≤ C + l + DΓ and γx0 ∈ HΓ. Since HBΛ +is Λ-free one has +l ≤ d(γx0, λγx0) ≤ u(|γ|) ≤ u(C + l + DΓ) +C, DΓ are fixed, so this inequality can only occur for boundedly small l, say l < L′(C) (DΓ is a universal +constant and may be ignored). Consult figure 2 for a geometric visualization of this situation. +22 + +Rad = C +x0 +Λ − free HBΛ +ξ +Γ − free HBΓ +≤ L(C) +γx0 +PHΓ(x0) +≤ L(C) + DΓ +Figure 2: Lemma 4.18. A Λ-free horoball HBΛ intersects a ball of radius C about x0. The associated +Γ-free horoball HBΓ is boundedly close, essentially due to the uniform cocompactness of Γ · x0 along the Γ +horospheres. +It remains to show that L′(C) is indeed sublinear in C. First define L(C) to be the minimal L that +bounds the distance d(HΛ, HΓ) for all possible HBΛ tangent to a point x ∈ B(x0, C). This is indeed a +minimum, since by Corollary 2.22 there are only finitely many horoballs of Γ intersecting B +� +x0, C + L′(C) +� +. +For every C there is thus a horoball HBΓ +C and an element γ ∈ Γ such that γx0 ∈ HΓ, d(HΛ +C, HΓ +C) = L(C) +and |γ| ≤ C + L(C) + DΓ. The fact that HBΛ +C is Λ-free implies +L(C) = d(HΛ +C, HΓ +C) ≤ u(|γ|) ≤ u +� +C + L(C) + DΓ +� +Lemma 4.17 implies that L(C) ≤ u′(C) for some sublinear function u′. +The following is an immediate corollary, apparent already in the above proof. +Corollary 4.19. For every C > 0 there is a bound K = K(C) and a fixed set ξ1, ξ2, . . . ξK ∈ WQ(Γ) ⊂ X(∞) +so that every Λ-free horoball HBΛ which is tangent to some point x ∈ B(x0, C) is based at ξi for some +i ∈ {1, . . ., K}. In particular, for any specific point x ∈ NC(Λ · x0) there are at most K Λ-free horoballs +tangent to x. +Proof. Let HBΛ be a horoball tangent to a point x ∈ B(x0, C). +Lemma 4.18 bounds d(HBΛ, HBΓ) by +L(C), hence HBΓ is tangent to a point x′ ∈ B +� +x0, C + L(C) +� +. By Corollary 2.22 there are only finitely +many possibilities for such HBΓ. In particular there are finitely many base-points for these horoballs, say +ξ1, ξ2, . . . , ξK(C) ∈ WQ(Γ). Finally, recall that a horoball is determined by a base point and a point x ∈ X +tangent to it, so the last statement of the corollary holds for any x ∈ B(x0, C). But the property in question +is Λ-invariant so the same holds for any point x ∈ Λ · B(x0, C) = NC(Λ · x0). +The bound on d(HBΛ, HBΓ) given by Lemma 4.18 further strengthen the relation between HBΛ and HBΓ. +The ultimate goal is to show that the HBΛ-s play the role of the Γ-horoballs in the geometric structure of +Q-rank 1 lattices, namely to show that Λ · x0 is cocompact on the HΛ-s. This requires to actually find +Λ-orbit points somewhere in X, and not just Λ-free regions as was done up to now. As one might suspect, +these points arise as λγx0 corresponding to points γx0 ∈ HΓ, which exist in abundance since Γ · x0 ∩ HΓ is +a cocompact lattice in HΓ. +The hope is that a Λ-free horoball HBΛ tangent to λγx0 would correspond to a horoball of Γ tangent to +γx0. This would have forced all the λγ to actually lie on the same bounding horosphere, and {λγx0 | γx0 ∈ +HΓ} would then be a cocompact lattice in HΛ. This hope turns out to be more or less true, but it requires +23 + +some work. The goal of the rest of this section is to establish a relation between a Λ-free horoball HBΛ +tangent to λγx0 and γx0. I start with some notations. +Definition 4.20. In light of Corollary 4.19, there is a finite number N of Λ-free horoballs tangent to x0. +Denote: +1. {HBΛ +i }N +1=1 are the Λ-free horoballs tangent to x0. +2. ξi ∈ WQ(Γ) is the base point of HBΛ +i . +3. vi ∈ Sx0X is the unit tangent vector in the direction ξi. +4. ηi := [x0, ξi) is the unit speed geodesic ray emanating from x0 with limit ξi. In particular vi = d +dtηi(0). +5. HBΓ +i is the horoball of Γ that corresponds to HBΛ +i , based at ξi. +6. HBΛ +λ,i, ξi +λ, ηi +λ are the respective λ-translates of the objects above. For example, HBΛ +λ,i := λ · HBΛ +i . +7. H decorated by the proper indices denotes the horosphere bounding HB, the horoball with respective +indices, e.g. HΛ +i := ∂HBΛ +i . +8. For an angle α > 0 and a tangent vector v0 ∈ SxX, define +(a) The α-sector of v in SxX is the set {v ∈ SxX | v ∈ Nα(v0)}. Recall that the metric on SxX is +the angular metric. +(b) The α-sector of v in X are all points y ∈ X for which the tangent vector at 0 of the unit speed +geodesic [x, y] lies in the α-sector of v in SxX. +Lemma 4.21. For every angle α ∈ (0, π +2 ) there exists D = D(α) such that if dγ > D then for some +i ∈ {1, . . . , N}, γx0 lies inside the α-sector of vi +λγ at λγx0. Furthermore whenever α is uniformly small +enough, there is a unique such i = i(γ), independent of α. +Proof. Translation by the isometry λ−1 +γ +preserves angles and distances, so it is enough to prove that there +is an i for which x′ +γ := λ−1γx0 lies inside the α-sector of vi, and that this i is unique if α is uniformly small. +Assume towards contradiction that there is α ∈ (0, π +2 ) and a sequence γn ∈ Γ, λn := λγn ∈ Λ with dγn +unbounded, and x′ +n := λ−1 +n γnx0 not lying in the union of the α-sectors of vi. By perhaps taking smaller α +I may assume all the α-sectors of the vi in Sx0X are pairwise disjoint. This can be done because there are +only finitely many vi. +Compactness of Sx0X allows me to take a converging subsequence v′ +n := +˙ +[x0, x′n], with limit direction +v′. Denote by η′ the geodesic ray emanating from x0 with initial velocity v′. The exact same argument of +Lemma 4.13 proves that η′(∞) is the base point of a Λ-free horoball tangent to x0. But this means v′ = vi +for some i ∈ {1, . . . , N}, contradicting the fact that all v′ +n lie outside the α-sectors of the vi. This proves +that there is a bound D = D(α) such that if dγ > D then x′ +γ lies within the α-sector of some vi. +The proof clearly shows that whenever α is small enough so that the α-sectors of the vi are disjoint, x′ +γ +lies in the α-sector of a unique vi as soon as dγ > D(α). +Remark 4.22. In the proof of Lemma 4.13 I used compactness of Sx0X to induce a converging subsequence +of directions. Lemma 4.21 actually shows that the fact there are finitely many Λ-free horoballs tangent to +x0 implies a posteriori that there was not much choice in the process - all directions [x0, x′ +γ] must fall into +one of the finitely many directions vi. +Next, I want to control the actual location of certain points with respect to the horoballs of interest, and +not just the angles. This turns out to be a more difficult of a task than one might suspect, since control on +angles does not immediately give control on distances. +Recall that large Λ-free balls near x0 imply large concentric Γ-free balls. The precise quantities and +bounds are given by Lemma 3.8 (one can use Lemma 3.9 to obtain a slightly cleaner statement). +24 + +Proposition 4.23. Let S ∈ (0, 1) be the constant given by Lemma 3.8, and let s ∈ (0, S). There is a bound +D = D(s) such that dγ > D implies that γx0 lies sdγ deep in HBΛ +λγ. +Proof. The proof is a bit delicate but very similar to that of Lemma 3.8. In essence, I use the Γ-free balls +near x0 to produce a Γ-free cylinder, which would force a certain geodesic not to cross a horosphere of Γ, +i.e. force it to stay inside a Γ-free horoball. +As in Lemma 4.21 it is only required to show that x′ = λ−1 +γ γx0 is sdγ deep inside HBΛ +i(γ). I start with +proving that x′ ∈ HBΓ +i(γ). I learned the hard way that even this is not a triviality. Recall the notation +Bγ = B(γx0, dγ). The ball B′ +γ = λ−1 +γ Bγ is a Λ-free ball of radius dγ about x′ = λ−1 +γ γx0. Denote by x′ +t +the point at time t along the unit speed geodesic η′ := [x0, x′]. It holds that |x′ +t| = t and, for t ≤ dγ, x′ +t +is the centre of a Λ-free ball of radius t tangent to x0. The constant s is fixed and by Lemma 3.8 there is +T ′ = T ′(s) such that if t > T ′, the ball sB +� +x′ +t, t +� +is Γ-free. +The next goal is to show that x′ +T ∈ HBΓ for some adequate T . For any time T > 0, let α = α(ε, T ) be the +angle for which d +� +η(T ), ηi(γ)(T ) +� +< ε for every η in the α-sector of vi(γ). By perhaps taking smaller α I may +assume that α is uniformly small as stated in Lemma 4.21. Let D(α) be the bound given by Lemma 4.21 +guaranteeing +D(α) < dγ ⇒ d +� +x′ +T , ηi(γ)(T ) +� +< ε +For my needs in this lemma ε may as well be chosen to be 1. I now choose a specific time T for which +I want x′ +T and ηi(γ)(T ) to be close. There are only finitely many Λ-free horoballs {HBΛ +i }i∈{1,...,N} tangent +to x0, giving rise to a uniform bound L = maxi∈{1,...,N}{d(HΛ +i , HΓ +i )} on the distance d(HΛ +i(γ), HΓ +i(γ)). Fix +T to be any time in the open interval (T ′ + L + ε, dγ). The fact that L + ε < T implies that ηi(γ)(T ) lies +at least ε-deep inside HBΓ +i(γ), and therefore η′(T ) ∈ HBΓ +i(γ). Recall that any point on HΓ is DΓ-close to a +point γx0 ∈ HΓ. By perhaps enlarging T and shrinking α if necessary, I may assume that DΓ < sT . Thus +for all T < t ≤ dγ, x′ +t is the centre of a Γ-free ball of radius st > sT > DΓ, hence {x′ +t}T ≤t≤dγ does not cross +a horosphere of Γ. Since x′ +T ∈ HBΓ +i(γ), this implies that x′ +t stays in HBΓ +i(γ) for all T < t ≤ dγ. In particular +x′ +dγ = x′ ∈ HBΓ +i(γ). +To get the result of the proposition, recall that sB′ +γ = B(x′, sdγ) is Γ-free, so x′ must be at distance at +least sdγ − DΓ from any horosphere of Γ, and in particular from HΓ +i(γ). In terms of Busemann functions, this +means that bηi(γ)(x′) ≤ −sdγ + DΓ whenever one can find such T ′ + L + ε < T < dγ. Since HBΛ +i(γ) is tangent +to x0, the corresponding horoball HBΓ +i(γ) lies inside it, and so x′ lies (sdγ − DΓ)-deep inside HBΛ +i(γ). A close +look at the argument yields the desired bound D = D(s) such that the above holds whenever dγ > D(s). +To help the reader take this closer look, I reiterate the choice of constants and their dependencies as they +appear in the proof: +1. Fix ε = 1. +2. Let T ′ = T ′(s) the constant from Lemma 3.8 and L = maxi∈{1,...,N}{d(HBΛ +i , HBΓ +i )}. +3. Fix T > T ′ + L + 1. +4. Fix α = α(1, T ). +5. Fix D(s) = max{D(α), T + 1}. +I remark, for the reader worried about the DΓ which appears in the final bound but not in the statement, +that (a) DΓ is a fixed universal constant and may as well be ignored, and (b) the discrepancy can be formally +corrected by taking a slightly larger s < s′ to begin with and as a result perhaps enlarging the bound D for +dγ). Also note that L = L(Λ) is a universal constant. +25 + +4.2.3 +Intersection of Λ-Free Regions and the Existence of a Λ-Cocompact Horosphere +In this section I find Λ-orbit points that lie close to the bounding horosphere of a Λ-free horoball HBΛ. In +order to find such points I need to make sure HBΛ is not contained inside a much larger Λ-free horoball. I +introduce the following definition. +Definition 4.24. A Λ-free horoball HBΛ is called maximal if it is tangent to a point x = λx0 ∈ Λ · x0. It +is called ε-almost maximal if d(Λ · x0, HΛ) < ε. +Remark 4.25. It may happen that a discrete group admits free but not no maximally free horoballs - see +discussion in section 4 of [55]. In any case it is clear that any Λ-free horoball can be ‘blown-up’ to an ε-almost +maximal Λ-free horoball, for every ε > 0. Moreover, every two ε-almost maximal horoballs based at the +same point ξ ∈ WQ(Γ) lie at distance at most ε of one another. For my needs any fixed ε would suffice, and +I fix ε = 1. +Lemma 4.26. There is DΛ > 0 such that if HBΛ is 1-almost maximal Λ-free horoball then HΛ ⊂ NDΛ(Λ·x0), +i.e. d(x, Λ · x0) ≤ DΛ for all x ∈ HΛ. +Notice that Lemma 4.26 does not state Λ · x0 even intersects HΛ. +Proof. I start with a short sketch of the proof. Consider a 1-maximal horoball and a point x on its bounding +horosphere with d(x, Λ · x0) = D. One may translate this situation to x0, which results in a Λ-free horoball +HBΛ intersecting the (closed) D-ball about x0 at a point w with B(w, D) Λ-free. +The proof differs depending on whether HBΓ ⊂ HBΛ or the other way round, since I use the bounds +from 4.18: +1. If HBΓ ⊂ HBΛ, there is a sublinear bound on d(HBΛ, HBΓ), which readily yields a bound on D. +2. if HBΛ ⊂ HBΓ there is a bound on d(x0, HBΓ) that is independent of D. So there are only finitely +many possibilities for HBΓ, independent of D. Hence there are only finitely many possible base points +for HBΓ. These in turn correspond to possible base points for such HBΛ, and this finiteness yields +a bound on the distance d(HBΓ, HBΛ) < L that is independent of D. The rest of the proof is quite +routine. +Let HBΛ be a 1-almost maximal Λ-free horoball. By definition there is λ ∈ Λ and z ∈ HΛ such that +d(λx0, z) < 1. Fix D > 0. I show that if there is some z′ ∈ HΛ for which d(z′, Λ · x0) ≥ D, then D must be +uniformly small. Exactly how small will be set in the course of the proof. +Fix D > 1 and assume that there is z′ ∈ HΛ with d(z, Λ · x0) ≥ D. Up to sliding z′ along HΛ, the +continuity of the function x �→ d(x, Λ · x0) together with Intermediate Value Theorem allows to assume that +d(z′, Λ · x0) = D. Let λ′ ∈ Λ be the element for which d(z′, λ′x0) = D. Translating by λ′−1 yields +1. A Λ-free horoball HBΛ +0 := λ′−1HBΛ. +2. A point w := λ′−1z′ ∈ HΛ +0 for which |w| = d(w, x0) = d(w, Λ · x0) = D. +Assume first that HBΓ +0 ⊂ HBΛ +0 . By Lemma 4.18 there is a sublinear function u′ such that d(HΓ +0 , HΛ +0 ) ≤ +u′(D). This yields a point γx0 ∈ HΓ +0 for which d(w, γx0) ≤ u′(D) + DΓ. Thus |γx0| ≤ D + u′(D) + DΓ and +the reverse triangle inequality gives +D − +� +u′(D) + DΓ +� +≤ d(w, λγx0) − d(w, γx0) < d(γx0, λγx0) +Together with the bound d(γx0, λγx0) ≤ u(|γx0|) and rearranging, one obtains +D ≤ u +� +D + u′(D) + DΓ +� ++ u′(D) + DΓ +The right hand side is clearly a sublinear function in D, hence this inequality may hold only for boundedly +small D, say D < D1. I conclude that HBΓ +0 ⊂ HBΛ +0 may occur only when D < D1. Notice that D1 depends +only on u and u′, and not on HBΛ. +26 + +w +x0 +Λ − free +B(w, D) +τ(t0) +τ +Λ − free HBΛ +ξ +Γ − free HBΓ +γx0 +Figure 3: Lemma 4.26, case HBΛ ⊂ HBΓ. The red horosphere of Γ is trapped between x0 and HΛ, and is +at distance t0 from x0. A Γ-orbit point on the red horosphere close to x0 allows to use sublinearity to get a +bound on t0. +Assume next that HBΛ +0 ⊂ HBΓ +0 , and that the containment is strict. Since x0 ∈ Γ · x0, the geodesic +τ := [x0, w] is of length D and intersects HΓ +0 . Denote by t0 ∈ [0, D) the time in which τ intersects HΓ +0 , and +let w′ := τ(t0) ∈ HΓ +0 be the intersection point. In particular |w′| = t0. It is clear that B(w′, t0) is Λ-free, +as a subset of the ball B(w, D). Again there is γx0 ∈ B(w′, DΓ) ∩ HΓ +0 and so |γx0| ≤ t0 + DΓ. By reverse +triangle inequality +t0 − DΓ ≤ d(w′, λγx0) − d(w′, γx0) ≤ d(γx0, λγx0) +and the sublinear constraint gives t0 − DΓ ≤ u(t0 + DΓ). This can only happen for boundedly small t0, +say t0 < T . I conclude that if HBΛ +0 ⊂ HBΓ +0 , then HBΓ +0 is a horoball of Γ tangent to some point y ∈ B(x0, T ). +By Corollary 2.22 there are finitely many horoballs of Γ tangent to points in B(x0, T ). In particular +there is a finite set {ξ′ +1, . . . , ξ′ +K} ∈ WQ(Γ) of possible base points for HBΓ +0. This set depends only on T , and +since the choice of T was completely independent of D, the set of possible base points is independent of D +as well. Let � +HBΓ +i be the horoball of Γ based at ξ′ +i. +I can now bound the distance d(HBΓ +0 , HBΛ +0 ). Let 1 ≤ i ≤ K be an index for which there is a Λ-free +horoball based at ξ′ +i that is contained in � +HBΓ +i . There is thus some 1-almost-maximal Λ-free horoball based at +ξ′ +i. Fix an arbitrary such 1-almost-maximal Λ-free horoball � +HBΛ +i for each such i, and let Li := d(� +HBΛ +i , � +HBΓ +i ). +Finally, define L := max{Li} + 1 among such i. As stated in Remark 4.25, d(HBΛ +0 , � +HBΛ +i ) ≤ 1 for some i, +therefore d(HBΓ +0, HBΛ +0 ) ≤ L. +Recall |w| = D and B(w, D) is Λ-free. It holds that d(w, HΓ +0 ) ≤ L, and so there is γx0 ∈ HΓ +0 for which +d(w, γx0) ≤ L + DΓ. In particular |γx0| ≤ D + L + DΓ (in fact it is clear that |γx0| ≤ T + DΓ, but this +won’t be necessary). Reverse triangle inequality gives +D − (L + DΓ) ≤ d(w, λγx0) − d(w, γx0) ≤ d(γx0, λγx0) +and from the sublinear constraint I conclude D − (L + DΓ) ≤ u(D + L + DΓ). Since L, DΓ are fixed +constants independent of D, this can only hold for boundedly small D, say D < D2. In particular, one gets +a uniform bound DΛ := max{D1, D2} such that x ∈ HΛ ⇒ d(x, Λ · x0) < DΛ. +Corollary 4.27. Every Γ-conical limit point is a Λ-conical limit point. +27 + +Proof. Let ξ ∈ X(∞) be a Γ-conical limit point. Let η : R≥0 → X be a geodesic with η(∞) = ξ. By +definition there is a bound D > 0 and sequences tn → ∞, γn ∈ Γ such that d +� +γnx0, η(tn) +� +< D. Consider the +corresponding λn := λγn and λnx0. If dn is uniformly bounded, then ξ is Λ-conical by definition. Otherwise, +assume dn is monotonically increasing to ∞. For some fixed s ∈ (0, 1) it holds that for all but finitely many +n ∈ N, γnx0 is sdn deep inside HBΛ +n := HBΛ +λn,i(γn). I assume dn is large enough so that sdn > D, and in +particular η(tn) ∈ HBΛ +n. Let ξn ∈ WQ(Γ) be the respective base points of HBΛ +n. The point ξ is Γ-conical, +and by Theorem 2.28 π +2 ≤ d(ξ, WQ(Γ)) ≤ d(ξ, ξn). +The proof differs depending on whether the above inequality is strict or not for any n ∈ N. Assume +first that for some m ∈ N, d(ξ, ξm) = π +2 . By item 2 of Lemma 2.31, d +� +HΛ +m, η(t) +� +is uniformly bounded, i.e., +there is C > 0 such that for every t > 0 there is xt ∈ HΛ +m for which d +� +xt, η(t) +� +< C. By Lemma 4.26, +d(xt, Λ · x0) < DΛ, hence d +� +η(tn), xtn +� +≤ C + DΛ. This means that ξ is Λ-conical. +Otherwise, for all n ∈ N it holds that π +2 < d(ξ, ξn). The fact that η(tn) ∈ HBΛ +n together with Lemma 2.31 +implies that at some later time the geodesic ray η leaves HBΛ +n. Thus there is sn > tn for which η(sn) ∈ HΛ +n. +Since HBΛ +n are maximal Λ-free horoballs, Lemma 4.26 gives rise to points λnx0 such that d +� +λnx0, η(sn) +� +≤ +DΛ. This renders ξ as a Λ-conical limit point, as wanted. +I now prove Proposition 4.12. +Proof of Proposition 4.12. The strategy is as follows. For HBΛ = HBΛ +λγ,i(γ), one uses Proposition 4.23 to +get that HBΓ ⊂ HBΛ and that the distance d(HΛ, HΓ) is large with dγ. The horosphere HΓ admits a +Γ-cocompact metric lattice, and so the projections of these metric lattice points onto HΛ form a cocompact +metric lattice in HΛ. It remains to show that for each γ′x0 ∈ Γ · x0 ∩ HΓ, the corresponding λ′ = λγ′ indeed +lies on the same HΛ and boundedly close to the projection PHΛ(γ′x0). This is done by putting together all +the geometric facts obtained up to this point, specifically Lemma 4.26. One delicate fact that will be of use +is that two maximal Λ-free horoballs that are based at the same point must be equal, because none of them +can contain a Λ-orbit point while on the other hand both bounding horospheres intersect Λ · x0. +Fix s > 0 for which Proposition 4.23 yields a corresponding bound D(s), and let γ ∈ Γ such that +sdγ > 2 · +� +DΛ + D(s) +� +. +Consider the (maximal) Λ-free horoball HBΛ +λγ,i(γ) based at ξi(γ) +λ +. +I show that +Λ · x0 ∩ HΛ +λγ,i(γ) is a cocompact metric lattice in HΛ +λγ,i(γ). I keep the subscript notation because the proof is +a game between HBΛ +λγ,i(γ) and another Λ-free horoball. +Let HBΓ +λγ,i(γ) be the Γ-horoball corresponding to HBΛ +λγ,i(γ). I can conclude that HBΓ +λγ,i(γ) ⊂ HBΛ +λγ,i(γ), +because the choice of dγ > D(s) guarantees γx0 is sdγ deep inside HBΛ +λγ,i(γ). In particular HBΛ +λγ,i(γ) is not +Γ-free. Moreover, it holds that L := d(HΛ +λγ,i(γ), HΓ +λγ,i(γ)) ≥ sdγ. Let γ′ ∈ Γ be any element in the cocompact +metric lattice Γ · x0 ∩ HΓ +λγ,i(γ), and consider two associated points: (a) λ′x0 = λγ′x0 and (b) the projection +of γ′x0 on HΛ +λγ,i(γ), denoted p′ +γ := PHΛ +λγ ,i(γ)(γ′x0) ∈ HΛ +λγ,i(γ). The horoball HBΛ +λγ,i(γ) is a maximal Λ-free +horoball so it is also 1-almost maximal, hence d(p′ +γ, Λ · x0) ≤ DΛ and the following holds: +sdγ ≤ L ≤ dγ′ ≤ d(γ′x0, p′ +γ) + d(p′ +γ, Λ · x0) ≤ L + DΛ +(3) +Consider ξi(γ′) +λ′ +, and assume towards contradiction that ξi(γ′) +λγ′ +̸= ξi(γ) +λγ . Both points lie in WQ(Γ) and +therefore must be at Tits distance π of each other. Therefore the fact that γ′x0 lies in HBΛ +λγ,i(γ) implies that +the geodesic [γ′x0, ξi(γ′)] leaves HBΛ +λγ,i(γ) at some point z ∈ HΛ +λγ,i(γ). +The fact that D(s) ≤ sdγ ≤ dγ′ implies that γ′x0 lies s2dγ deep inside HBΛ +λγ′ ,i(γ′). Therefore the point z +also lies at least s2dγ deep inside HBΛ +λγ′ ,i(γ′), and therefore z is the centre of a Λ-free horoball of radius at +least s2dγ. +By choice of dγ the point z therefore admits a 2DΛ neighbourhood that is Λ-free. But z lies on HΛ +λγ,i(γ), +a maximal horosphere of Λ, contradicting Lemma 4.26. I conclude that ξi(γ′) +λγ′ += ξi(γ) +λγ , so HBΛ +λγ,i(γ) and +28 + +γx0 +dγ +γ′x0 +λγ′x0 +HBΛ +λγ′ ,i(γ′) +z′ := πHBΛ +λγ′ ,i(γ′)(γ′x0) +D1 +���� +1 +ξi(γ′) +λγ′ +ξi(γ) +λγ +≥ 1 +2dγ +z +≥ 1 +2 dγ′ +Λ − free ball +B +� +z, 1 +2dγ +� +HBΛ +λγ,i(γ) +HBΓ +λγ,i(γ) +λγx0 +Figure 4: Proposition 4.12. Assuming towards contradiction that ξi(γ) +λγ +̸= ξi(γ′) +λγ′ +results in a point z ∈ HBΛ +λγ,i(γ) +(blue coloured and bold faced in the bottom part of the figure) admitting a large Λ-free neighbourhood, +contradicting almost cocompactness. +HBΛ +λγ′ ,i(γ′) are two Λ-free horoballs that are tangent to a Λ · x0 point and based at the same point at +∞. This implies HBΛ +λγ,i(γ) = HBΛ +λγ′ ,i(γ′), and in particular λγ′x0 ∈ HΛ. Finally, it is clearly seen from +Inequality 3 that +d(λ′x0, p′ +γ) ≤ d(λ′x0, γ′x0) + d(γ′x0, p′ +γ) ≤ dγ′ + L ≤ L + DΛ + L +The element γ′x0 ∈ Γ · x0 ∩ HΓ +λγ,i(γ) was as arbitrary element, and the above argument shows that the +corresponding Λ-orbit points satisfy: +1. λ′x0 all lie on HΛ +λγ,i(γ). +2. Each p′ +γ is 2L + DΛ close to the point λ′x0. +This shows that the cocompact metric lattice {p′ +γ | γ′x0 ∈ HΓ +λγ,i(γ)} lies in a bounded neighbourhood of +the set of points Λ · x0 ∩ HΛ +λγ,i(γ), proving that Λ · x0 ∩ HΛ +λγ,i(γ) is a cocompact metric lattice in HΛ +λγ,i(γ). +Lemma 2.24 elevates this to +� +Λ ∩ StabG(HΛ +λγ,i(γ)) +� +· x0 ∩ HΛ +λγ,i(γ) +being a cocompact metric lattice in HΛ +λγ,i(γ), completing the proof. +4.3 +The Bounded Case +Proposition 4.12 is enough in order to prove Theorem 4.1 in the case Γ ̸⊂ ND(Λ) for any D > 0, i.e. in case +Γ does not lie in a bounded neighbourhood of Λ. The case where Γ and Λ lie at bounded Hausdorff distance, +i.e. where Γ ⊂ ND(Λ) and Λ ⊂ ND(Γ), arose naturally in the context of the quasi-isometric classification +of non-uniform lattices in the works of Schwartz [52] (R-rank 1), Drut¸u [16] and Eskin [22] (higher rank). I +restate the theorems in the bounded setting. +29 + +Theorem 4.28 (Eskin [22], Theorem 4.9 above). Let G be a real centre-free semisimple Lie group without +compact factors and of higher rank, Γ ≤ G an irreducible non-uniform lattice, Λ ≤ G a discrete subgroup. If +Γ ⊂ ND(Λ) for some D > 0, then Λ is a lattice, and if moreover Λ ⊂ ND(Γ) then Λ and Γ are commensurable. +Theorem 4.29 (Schwartz [52], Theorem 4.10 above). Let G be a real simple Lie group of R-rank 1, Γ ≤ G +an irreducible non-uniform lattice, Λ ≤ G a discrete subgroup. If both Γ ⊂ ND(Λ) and Λ ⊂ ND(Γ) for some +D > 0, then Λ is a lattice, and if moreover G is not locally isomorphic to SL2(R), then Λ is commensurable +to Γ. +The notable difference between the two statements is that for higher rank groups, the inclusion Λ ⊂ ND(Γ) +is only required to prove commensurability. In view of Corollary 4.8, this allows me to omit that assumption +from Theorem 1.6. Notice also that for groups with property (T) the result easily follows from the (much +more recent) result by Leuzinger in Theorem 5.7. +In the context of commensurability in the sublinear setting, I can only prove a limited result, Namely +that Λ is commensurable to Γ if Γ is an irreducible Q-rank 1 lattice and both Γ ⊂ ND(Λ) and Λ ⊂ Nu(Γ) +for some constant D > 0 and a sublinear function u. This is done via a reduction to the bounded case. +Proposition 4.30. Let G be a real semisimple Lie group without compact factors and with finite centre, +Γ ≤ G an irreducible lattice of Q-rank 1, Λ ≤ G a discrete subgroup, and u a sublinear function. If Γ ⊂ ND(Λ) +for some D > 0 and Λ ⊂ Nu(Γ), then actually Λ ⊂ ND′(Γ) for some D′ > 0. Moreover, if G is of R-rank 1, +the conclusion holds under the relaxed assumption that u(r) ⪯∞ εr for some ε < 1. +Remark 4.31. While the setting of Proposition 4.2 is indeed rather limited, the situation that both Γ ⊂ +Nu(Λ) and Λ ⊂ Nu(Γ) arises naturally from the motivating example of SBE-rigidity in Theorem 6.9. Notice +however that Theorem 6.9 is not known for groups G that admit R-rank 1 factors, which is the only setting +for which I can prove Proposition 4.2. +4.3.1 +A Reduction +I start with the proof of Proposition 4.30. The first step is to establish the fact that Λ must preserve WQ(Γ). +Lemma 4.32. Let G be a real semisimple Lie group without compact factors and with finite centre, Γ ≤ G +an irreducible non-uniform lattice of Q-rank 1, Λ ≤ G a discrete subgroup. Assume that Γ ⊂ Nu(Λ) and +that Λ ⊂ Nu′(Γ) for sublinear functions u, u′. Then Λ · WQ(Γ) ⊂ WQ(Γ). Moreover, if G is of R-rank 1, the +conclusion holds under the relaxed assumption that u′(r) ⪯∞ εr for some ε < 1. +Proof. The proof is similar to the argument of Lemma 4.14, and uses the linear penetration rate of a geodesic +into a horoball. Let ξ ∈ WQ(Γ), and let HΓ be a horosphere bounding a Γ-free horoball HBΓ with HΓ ∩Γ·x0 +a metric lattice in HΓ. Assume first that u′ is sublinear. Since HBΓ is Γ-free and Λ ⊂ Nu′(Γ), I can conclude +that Λ · x0 ∩ HBΓ ⊂ Nu′(HΓ). Recall (Lemma 2.31) that every geodesic ray η with limit point ξ′ ∈ N π +2 (ξ) +penetrates HBΓ at linear rate. Therefore for every such geodesic ray η and every sublinear function v there +is R = R(η, v) > 0 for which Nv(η↾r>R) is Λ-free. +On the other hand, let λ ∈ Λ, and assume towards contradiction that λξ /∈ WQ(Γ). Then by Proposi- +tion 2.30 there is a Γ-conical limit point ξ′ ∈ N π +2 (λξ). The hypothesis that Γ ⊂ Nu(Λ) then implies that for +every R > 0, Nu(η↾r>R) ∩ Λ · x0 ̸= ∅. Translating by λ−1 yields a contradiction to the previous paragraph. +I conclude that λξ ∈ WQ(Γ). +I now modify the argument to include u′(r) ⪯∞ εr when G is of R-rank 1. In this case, the only point +ξ′ ∈ N π +2 (λξ) is λξ itself. Therefore by the same argument as above, the assumption that λξ /∈ WQ(Γ) +implies that the u-sublinear neighbourhood of every geodesic ray with limit point ξ intersects Λ · x0. I.e., for +every η with limit point ξ and every R > 0 it holds that Nu(η↾r>R) ∩ Λ · x0 ̸= ∅. On the other hand, every +such geodesic penetrates HBΓ at 1-linear rate. This amounts to the following fact: if v′(r) = εr for some +ε ∈ (0, 1), then for some R > 0, the set Nu(η↾r>R)∩Nv′(HΓ) = ∅. This is a contradiction to Λ ⊂ Nu′(Γ). +30 + +Proof of Proposition 4.30. Assume towards contradiction that there is a sequence λn such that d(λnx0, Γ · +x0) > n. Recall that Γ·x0 is a cocompact metric lattice in the compact core of Γ. This implies that there is a +number D′ > 0 such that any λ ∈ Λ for which λx0 /∈ ND′(Γ · x0) must lie at least 1 +2D′-deep inside a horoball +of Γ. I can assume that for all n ∈ N there are corresponding horoballs of Γ, which I denote HBΓ +n, for which +λn · x0 ∈ HBΓ +n. The fact that Γ ⊂ ND(Λ) then implies that ND(Λ · x0) covers a cocompact metric lattice in +HΓ +n, namely the metric lattice Γ · x0 ∩ HΓ +n. In the terminology of Section 4.2, HΓ +n is almost Λ-cocompact, or +D-almost Λ-cocompact. +I first prove that every horoball of Γ contains a Λ-free horoball (this is of course immediate if Λ ⊂ NC(Γ) +for some C > 0). Assume towards contradiction that there is a horoball HBΓ of Γ that does not contain a +Λ-free horoball. Denote HΓ := ∂HBΓ. In the notations of the previous paragraph, I can assume without loss +of generality that HBΓ = HBΓ +n for all n ∈ N. Denote by ξ the base point of HBΓ, fix some arbitrary x ∈ HΓ +and consider the geodesic ray η := [x, ξ). The constraint that Λ ⊂ Nu(Γ) implies that for every R > 0 there +is some L > 0 for which the ball B +� +η(L + t), R +� +is Λ-free, for all t ≥ 0. In particular, for all large enough +n ∈ N (depending on R), the horosphere H(ξ, λnx0) that is parallel to HΓ and that passes through λnx0 +contains a point that is the centre of Λ-free ball of radius R. This property is Λ-invariant, as well as the fact +that HBΓ is based at WQ(Γ). In particular, these two properties hold for the horoballs HBn := λ−1 +n +· HBΓ, +whose respective base points I denote ξn := λ−1 +n ξ ∈ WQ(Γ). +Fix R = D +2DΓ (recall that DΓ is such that every horosphere H of Γ admits H ⊂ NDΓ(Γ·x0 ∩H)). Let +L = L(D+2DΓ) be the corresponding bound from the previous paragraph. For every n > L the horoball HBn +has bounding horosphere Hn that admits a point zn ∈ Hn for which B(zn, D+2DΓ) is Λ-free. Moreover, the +same is true for every horosphere that is parallel to Hn which lies inside HBn. Since Γ ⊂ ND(Λ), this means +that every horosphere that lies inside HBn admits a point that is the centre of a Γ-free ball of radius 2DΓ. +I conclude that none of those horospheres could be the horosphere of Γ corresponding to the parabolic limit +point ξn ∈ WQ(Γ). Since x0 ∈ Hn it must therefore be that Hn is a horosphere of Γ. But this contradicts +the fact that zn ∈ Hn and B(zn, 2DΓ) is Γ-free. This shows that no horoball of Γ contains a sequence of +Λ-orbit points that lie deeper and deeper in that horoball. Put differently, it shows that every horoball of Γ +contains a Λ-free horoball. +I remark that the above argument shows something a bit stronger, which I will not use but which I find +illuminating. It proves that as soon as d(λx0, Γ · x0) is uniformly large enough, say more than M, then +λx0 must lie on a (D + 2DΓ)-almost Λ-cocompact horosphere parallel to HΓ, where HΓ is the bounding +horosphere of any horoball of Γ in which λx0 lies (recall that it must lie in at least one such horoball). On +the other hand if d(λx0, Γ · x0) < M, then since every point in the Γ-orbit lies on a horosphere of Γ one +concludes that λx0 lies on a horosphere H based at WQ(Γ) that is (M + D)-almost Λ-cocompact. +I can now assume that every HBΓ +n contains a Λ-free horoball. In particular it contains a 1-almost maximal +Λ-free horoball HBΛ +n (see Definition 4.24). By definition there is a point λ′ +nx0 that is at distance at most 1 +from HΛ +n = ∂HBΛ +n. Up to enlarging d(λnx0, Γ · x0) or decreasing it by at most 1, I can assume λn = λ′ +n to +begin with. Consider HBn := λ−1 +n +· HBΛ +n with Hn = ∂HBn. This is a sequence of horoballs, each of which +contains a Λ-free horoball at depth at most 1, based at corresponding parabolic limit points ξn ∈ WQ(Γ), +and tangent to points that are at distance at most 1 from x0, i.e., Hn ∩ B(x0, 1) ̸= ∅. +Since Γ ⊂ ND(Λ) I conclude that each of the HBn contain a horoball of depth at most D + 1 that +is Γ-free. Therefore the horoball of Γ that is based at ξn must have its bounding horosphere intersecting +B(x0, D + 2). By Corollary 2.22 there are only finitely such horoballs. I conclude that there are finitely +many points ξ′ +1, . . . , ξ′ +K ∈ WQ(Γ) such that for every n ∈ N there is i(n) ∈ {1, . . . , K} with ξn = ξ′ +i(n). From +the Pigeonhole Principle there is some ξ′ ∈ {ξ′ +1 . . . , ξ′ +K} for which ξn = ξ′ for infinitely many n ∈ N. Passing +to a subsequence I assume that this is the case for all n ∈ N. +To begin with the HBΓ +n are horoballs of Γ, and therefore as in the first case the bounding horospheres +HΓ +n are D + 2DΓ-almost Λ-cocompact. This is a Λ-invariant property and therefore the same holds for +the λ−1 +n +translate of it. These are the horospheres which are based at ξ′ and lie outside HBn at distance +d(λnx0, Γ · x0) > n − 1 from Hn. They form a sequence of outer and outer horospheres based at the same +point at WQ(Γ), all of which are D + 2DΓ-almost Λ-cocompact. This is a contradiction, since the union of +such horospheres intersect every horoball of X, contradicting the existence of Λ-free horoballs. Formally, +31 + +take some ζ ∈ WQ(Γ) different from ξ′. Since both ξ′ and ζ lie in WQ(Γ), they admit dT (ζ, ξ′) = π and +there is a geodesic η with η(−∞) = ξ′ and η(∞) = ζ. Let HBΓ +ζ be the horoball of Γ that is based at ζ. By +the first step of this proof, every such horoball must contain a Λ-free horoball HBΛ +ζ . Therefore there is some +T > 0 such that for all t > T the point η(t) lies 2(D + 2DΓ) deep in HBΛ +ζ . I conclude that for all t > T , +B +� +η(t), 2D +� +is Λ-free. On the other hand, for arbitrarily large t it holds that the horosphere based at ξ′ and +tangent to η(t) is D+2DΓ-almost Λ-cocompact, and in particular d +� +η(t), Λ·x0 +� +< D+2DΓ, a contradiction. +I conclude that Λ ⊂ ND′(Γ) for some D′ > 0, as claimed. +Corollary 4.33. In the setting of Proposition 4.30, Λ is a lattice commensurable to Γ. +4.3.2 +The Arguments of Schwartz and Eskin +The R-rank 1 case. +The statement of Theorem 4.10 is a slight modification of his original formulation. +His framework leads to a discrete subgroup ∆ ≤ G such that: +1. Every element of ∆ quasi-preserves the compact core of the lattice Γ. Namely, each element of ∆ is +an isometry of X that preserves WQ(Γ) and that maps every horosphere of Γ to within the D = D(∆) +neighbourhood of some other horosphere of Γ. +2. It holds that Γ ⊂ ND(∆). +From these two properties Schwartz is able to deduce that ∆ has finite covolume, i.e. that ∆ is a lattice +in G. Here is a sketch of his argument, which works whenever Γ is a Q-rank 1 lattice. +Theorem 4.34. In the setting described above, ∆ is a lattice in G. +Proof sketch. Consider X′ +0 := � +g∈∆ g · X0, where X0 is the compact core of Γ. +This space serves as a +‘compact core’ for ∆: the fact that ∆ quasi-preserves X0 implies that X′ +0 ⊂ ND(X0). It is a ∆-invariant +space, and therefore one gets an isometric action of ∆ on X′ +0. This action is cocompact: the reason is that +Γ acts cocompactly on X0, and Γ ⊂ ND(∆). Formally, every point in X′ +0 is D-close to a point in X0. Every +point in X0 is DΓ-close to a point in Γ · x0. Every point in Γ · x0 is D-close to a point in ∆ · x0. Therefore +the ball of radius 2D + DΓ contains a fundamental domain for the action of ∆ on X′ +0. +It remains to see that the action of ∆ on X \ X′ +0 is of finite covolume. As a result of the cocompact +action of ∆ on X′ +0, there is B := B(x0, R) so that X′ +0 ⊂ ∆·B. X′ +0 is the complement of a union of horoballs, +which one may call horoballs of ∆, with bounding horospheres of Λ. The fact that Γ is of Q-rank 1 means +that the horoballs of Γ are disjoint, and therefore those of Λ are almost disjoint: there is some C > 0 such +that for every horosphere H of Λ and every point x ∈ H, d(x, X′ +0) < C. Up to enlarging the radius of B by +C, I can assume that H ⊂ ∆ · B for every horosphere H of Λ. +Each horoball of Λ is based at WQ(Γ), and each lies uniformly boundedly close to the corresponding +horoballs of Γ. From Corollary 2.22 one therefore sees that there are finitely many horoballs of ∆ that inter- +sect B. Denote them by HB1, . . . , HBN, their bounding horospheres by Hi = ∂HBi, and their intersection +with B by Bi := B ∩ HBi. Let also ξi ∈ WQ(Γ) denote the base point of each HBi. Each Bi is pre-compact +and therefore the projection of each Bi on Hi is pre-compact as well (this is a consequence e.g. of the results +of Heintze-Im hof recalled in Remark 2.5). Let Di ⊂ Hi be a compact set that contains this projection, i.e. +PHi(Bi) ⊂ Di ⊂ Hi. In particular B ∩ Hi ⊂ Di. +Observe now that for every horoball HB of ∆, with bounding horosphere H = ∂HB, the ∆-orbit of every +point x ∈ H intersects some Di. First notice that for x ∈ H the choice of B implies that the ∆-orbit of x +must intersect B, say gx ∈ B. In particular gH ∩ B ̸= ∅, and since gHB is a horoball of ∆ then by definition +gH = Hi for some i ∈ {1, . . ., N}. One concludes that indeed gx ⊂ Di. +Moreover, let y ∈ X is any point that lies inside a horoball HB of ∆, and x = PH(y) its projection on +the bounding horosphere H = ∂HB. By the previous paragraph there is some g ∈ ∆ and i ∈ {1, . . . , N} for +which gx ∈ Di, and therefore it is clear that gy lies on a geodesic emanating from Di to ξi. +32 + +Finally, define Cone(Di) to be the set of all geodesic rays that emanate from Di and with limit point ξi. +The previous paragraph proves that �N +i=1 Cone(Di) contains a fundamental domain for the action of ∆ on +X \ X′ +0. Moreover, the fact that Di ⊂ Hi is compact readily implies that each Cone(Di) has finite volume, +and so this fundamental domain is of finite volume. +To conclude, B ∪ +� �N +i=1 Cone(Di) +� +is a set of finite volume and it contains a fundamental domain for the +∆-action on X, as claimed. The proof of commensurability of ∆ and Γ is given in full in [52]. +There is one essential difference between Theorem 4.10 and Theorem 4.34, namely the assumption that +Λ ⊂ ND(Γ) rather than quasi-preserving the compact core of Γ. In Schwartz’s work, the fact that ∆ ⊂ ND(Γ) +is not relevant (even though it easily follows from the construction of his embedding of ∆ in G). He only +uses the two properties described above, namely the quasi-preservation of X0 and Γ ⊂ ND(∆). +The assumption that Λ quasi-preserves the compact core of Γ does not feel appropriate in the context +of my thesis, while the metric condition Λ ⊂ ND(Γ) seems much more natural. It is a stronger condition +as I now show. By Lemma 4.32, Λ · WQ(Γ) ⊂ WQ(Γ). Let HΓ +1 be a horosphere of Γ, based at ξ ∈ WQ(Γ), +and let γx0 ∈ HΓ +1 be some point on the metric lattice of Γ · x0 on HΓ +1 . There is an element λ ∈ Λ such that +d(λx0, γx0) < D. Moreover, since Λ ⊂ ND(Γ) one knows that the parallel horoball that lies D-deep inside +HBΓ +1 is Λ-free. +Let λ′ ∈ Λ be an arbitrary element of Λ, and consider λ′ · HΓ +1 . +The last statement in the previous +paragraph is Λ-invariant, and so the horoball that lies D-deep inside λ′ · HBΓ +1 is Λ-free. +The fact that +Γ ⊂ ND(Λ) then implies that the parallel horoball that lies 2D deep inside λ′HBΓ +1 is Γ-free. Let HΓ +2 be the +horosphere of Γ that is based at λ′ξ. The last statement amounts to saying that HΓ +2 lies at most 2D-deep +inside λ′HBΓ +1 . On the other hand, one has d(λ′λx0, λ′HΓ +1 ) = d(λx0, HΓ +1 ) ≤ D, so there is a Λ-orbit point that +lies within D of λ′HΓ +1 . The parallel horoball that lies D-deep inside HBΓ +2 must also be Λ-free, so I conclude +that HΓ +2 must be contained in the parallel horoball to λ′HBΓ +1 which contains it and that is at distance D +from it. I conclude that d(λ′HΓ +1 , HΓ +2 ) ≤ 2D, and so that Λ quasi-preserves X0. +Remark 4.35. It is interesting to note that Schwartz’s arguments are similar in spirit to my arguments +in Section 4.2. In fact, one could also prove Theorem 4.10 using the same type of arguments that appear +repeatedly in section 4.2, namely by moving Λ-free horoballs around the space, specifically the proof of +Proposition 4.30. I do not present it here. +Higher rank. +Eskin’s proof is ergodic, and based on results of Mozes [42] and Shah [54]. I produce it here +without the necessary preliminaries, which are standard. +Proof of Theorem 4.9. To prove that Λ is a lattice amounts to finding a finite non-zero G-invariant measure +on Λ\G. By Theorem 2 in [42], if P ≤ G is a parabolic subgroup then every P-invariant measure on Λ\G is +automatically G-invariant. Fix a minimal parabolic subgroup P ≤ G and let µ0 be some fixed probability +measure on Λ\G. Since P is amenable it admits a tempered Følner sequence Fn ⊂ P, and one can average +µ0 along each Fn to get a sequence of probability measures µn. The weak* compactness of the unit ball +in the space of measures on Λ\G implies that there exists a weak* limit µ of the µn. The measure µ is +automatically a finite P-invariant measure. It remains to show that µ is not the zero measure. To see this +it is enough to show that for some compact set CΛ ⊂ Λ\G and some Λg = x ∈ Λ\G, one has +0 < lim inf +n +1 +|Fn| +� +Fn +1CΛ(xp−1)dp +(4) +Fix some compact neighbourhood CΓ ⊂ Γ\G of the trivial coset Γe. The hypothesis Γ ⊂ ND(Λ) implies +that there is a corresponding compact neighbourhood CΛ ⊂ Λ\G of the trivial coset Λe such that for any +p ∈ P, it holds that Γgp−1 ∈ CΓ ⇒ Λgp−1 ∈ CΛ (simply take CΛ to be the D + 1-blowup of CΓ). The action +of P on Γ\G is uniquely ergodic, therefore +0 < µΓ(CΓ) = lim +n +1 +|Fn| +� +Fn +1CΓ(Γp−1)dp +33 + +where µΓ denotes the natural G-invariant measure on Γ\G. The defining property of CΛ ensures that +Inequality (4) is satisfied, implying that µ is a non-zero P-invariant probability measure on Λ\G. I conclude +that µ is also G-invariant, and that Λ is a lattice. If moreover Λ ⊂ ND(Γ), one may use Shah’s Corollary +[54] to conclude that Λ is commensurable to Γ. +4.4 +Translating Geometry into Algebra +The goal of this section is to prove that the results of Section 4.2 imply that Λ satisfies the hypotheses +of the Benoist-Miquel criterion Theorem 4.4. +Namely, that Λ is Zariski dense, and that it intersects a +horospherical subgroup in a cocompact indecomposable lattice. +These are algebraic properties, and the +proof that Λ satisfies them is in essence just a translation of the geometric results of Section 4.2 to an +algebraic language. The geometric data given by Section 4.2 is that for some horosphere H bounding a +Λ-free horoball, Λ ∩ StabG(H) · x0 intersects H in a cocompact metric lattice (Proposition 4.12), and that +the set of Λ-conical limit points contains the set of Γ-conical limit points (Corollary 4.27). Note that since +K is compact the former implies that Λ ∩ StabG(H) is a uniform lattice in StabG(H). +4.4.1 +A Horospherical Lattice +I assume that Λ ∩ StabG(H) is a lattice in StabGH, and I want to show that Λ intersects a horospherical +subgroup U of G in a lattice. This step requires quite a bit of algebraic background, which I give below in full. +In short, the first goal is to show that StabG(H) admits a subgroup U ≤ StabG(H) that is a horospherical +subgroup of G. A lemma of Mostow (Lemma 4.36 below) allows to conclude that Λ intersects U in a lattice. +Lemma 4.36 (Lemma 3.9 in [40]). Let H be a Lie group having no compact connected normal semisimple +non-trivial Lie subgroups, and let N be the maximal connected nilpotent normal Lie subgroup of H. Let +Γ ≤ H be a lattice. Then N/N ∩ Γ is compact. +Remark 4.37. In the original statement Mostow uses the term ‘analytic group’, which I replaced here with +‘connected Lie subgroup’. This appears to be Mostow’s definition of an analytic group. See e.g. Section +10, Chapter 1 in [33]. In Chevalley’s Theory of Lie Groups, he defines a Lie group as a locally connected +topological group whose identity component is an analytic group (Definition 1, Section 8, Chapter 4 in [12]), +and proves (Theorem 1, Section 4, Chapter 4 therein) a 1-1 correspondence between analytic subgroups of +an analytic group and Lie subalgebras of the corresponding Lie algebra. +Lemma 4.36 lays the rationale for the rest of this section. Explicitly, I prove that StabG(H) admits +a subgroup that is a horospherical subgroup U of G (Corollary 4.39), and that U is maximal connected +nilpotent normal Lie subgroup of StabG(H) (Corollary 4.45). +In order to use Lemma 4.36, I show that the horospherical subgroup Nξ is a maximal normal nilpotent +connected Lie subgroup of StabG(H)◦, and that StabG(H)◦ admits no compact normal factors. This requires +to establish the structure of StabG(H)◦. +Definition 4.38. In the notation ht +ξ = exp(tX) and Aξ = exp +� +Z(X) ∩ p +� +of Proposition 2.7, define A⊥ +ξ to +be the codimension-1 submanifold of Aξ that is orthogonal to {hξ(t)}t∈R (with respect to the Killing form +in the Lie algebra). +Claim. Every element a ∈ A⊥ +ξ stabilizes H = H(x0, ξ). +Proof. An element in Aξ is an element that maps x0 to a point on a flat F ⊂ X that contains the geodesic +ray [x0, ξ). If a ∈ A⊥ +ξ , then the geodesic [x0, ax0] is orthogonal to [x0, ξ), and lies in F. From Euclidean +geometry and structure of horospheres in Euclidean spaces, it is clear that ax0 ∈ H(x,ξ). Since a ∈ Gξ, this +means aH = H(ax0, ξ) = H(x, ξ) = H. +Corollary 4.39. Let H be a horosphere based at ξ. Then StabG(H)◦ = (KξA⊥ +ξ )◦Nξ, and in particular it +contains a horospherical subgroup of G. Moreover, StabG(H)◦ is normal in StabG(ξ)◦ and acts transitively +on H. +34 + +Proof. Clearly (KξA⊥ +ξ )◦Nξ is a codimension-1 subgroup of StabG(ξ)◦. Since StabG(H) ̸= StabG(ξ) (e.g. +ht +ξ /∈ StabG(H) for t ̸= 0), it is enough to show that (KξA⊥ +ξ )◦Nξ ≤ StabG(H). Let kan ∈ (KξA⊥ +ξ )◦Nξ. It +fixes ξ, so it is enough to show that kanx0 ∈ H. Since k ∈ Kξ and kx0 = x0, it stabilizes H. From Claim 4.4.1 +a ∈ StabG(H). So it remains to check that Nξ stabilizes H, but this is more or less the definition: fixing +a base point x0, the horospheres based at ξ are parameterized by R. +Denote them by {Ht}t∈R, where +H = H0. In this parameterization, any element g ∈ Gξ acts on {Ht}t∈R by translation. I can thus define for +g ∈ StabG(ξ) the real number l(g) to be that number for which gHt = Ht+l(g). Clearly l +� +hξ(t) +� += t. The +element n fixes ξ, so one has +h−t +ξ nht +ξH0 = h−t +ξ Ht+l(n) = Ht+l(n)−t = Hl(n) +The fact that n ∈ Ker(Tξ), i.e. that limt→∞ h−t +ξ nht +ξ = eG readily implies that necessarily l(n) = 0. I +conclude that (KξA⊥ +ξ )◦Nξ = StabG(H)◦, as wanted. +Next recall that StabG(H)◦ acts transitively on X. Let x, y ∈ H, and consider g ∈ StabG(ξ)◦ with gx = y. +Writing an element g ∈ Gξ as kata⊥n ∈ Kξht +ξA⊥ +ξ Nξ, the argument above shows that kht +ξa⊥nH0 = H0 if +and only if t = 0, i.e., if and only if g ∈ StabG(H)◦. +Finally, let g ∈ StabG(ξ) and h ∈ StabG(H). By the discussion above h · Ht = Ht for all t ∈ R. Clearly +−l(g) = l(g−1), and therefore +ghg−1H0 = ghH−l(g) = g · H−l(g) = H0 +Therefore StabG(H) is normal in StabGξ, and the same is true for the respective identity components. +Corollary 4.40. StabG(H)◦ is a connected Lie group with no connected compact normal semisimple non- +trivial Lie subgroups. +Proof. Every compact subgroup of G fixes a point. Let H ≤ G be some closed subgroup. It is standard to +note that a normal N ≤ H that fixes a point x ∈ X must fix every point in the orbit H · x: hnh−1hx = hx. +Since H = StabG(H)◦ acts transitively on H, it shows that a normal compact subgroup of StabG(H)◦ fixes +every point in H. An isometry fixing a horosphere pointwise while fixing its base point is clearly the identity, +proving the claim. +The following fact is well known but I could not find it in the literature. +Corollary 4.41. A horosphere in X is not convex. +Proof. Let H′ be some horosphere in X, with base point ζ ∈ X(∞), and assume towards contradiction that +it is convex. Fix x ∈ H′ and a′ +t the one parameter subgroup with η′(∞) = a′ +tx, and denote H′ +t = H(a′ +tx, ζ). +Let eG ̸= n ∈ Nζ (Nζ defined with respect to a′ +t in a corresponding Langlands decomposition), and consider +the curve η′ +n(t) := a′ +tnx. I claim that this is a geodesic. On the one hand, the fact that H′ is convex implies +that the geodesic segment [x, nx] is contained in H′. Therefore a′ +t[x, nx] = [a′ +tx, a′ +tnx] ⊂ H′ +t. More generally +it is clear that because a′ +tH′ +s = H′ +s+t it holds that H′ +t is convex for every t as soon as it is convex for some t. +On the other hand, for every point y ∈ [x, nx], d(y, H′ +t) = t, and more generally for any y ∈ [a′ +snx, a′ +sx] +it holds that d(y, H′ +t) = |s − t|. In particular this is true for η′ +n(t) = yt := a′ +tnx. I get that d +� +η′ +n(t), η′ +n(s) +� += +|s − t|. Therefore η′ +n is a geodesic (to be pedantic one has to show that η′ +n is a continuous curve, which is a +result of the fact that a′ +t is a one parameter subgroup of isometries). Clearly +d +� +η′ +n(t), η′(t) +� += d(a′ +tnx, a′ +tx) = d(nx, x) +and therefore η′ +n is at uniformly bounded distance to η′. This bounds d(ηn, η′ +n) as bi-infinite geodesics, +i.e. for all t ∈ R, not just as infinite rays. The Flat Strip Theorem (Theorem 2.13, Chapter 2.2 in [11]), then +implies that the geodesics ηn, η′ +n bound a flat strip: an isometric copy of R × [0, l] (where l = d(x, nx)). +Up to now I did not use the fact that n ∈ Nξ, only that the point nx lies on a geodesic that is contained +in H′ = H′ +0. Therefore the entire bi-infinite geodesic that is determined by [x, nx] lies on a 2-dimensional +flat F that contains η′. The two elements n, a′ +t therefore admit nx, a′ +tx ∈ F. It is a fact that two such +elements must commute. I can conclude therefore that [n, a′ +t] = eG, which contradicts the fact that that +n ∈ Nζ = Ker(Tζ). +35 + +Lemma 4.42 (Theorem 11.13 in [51]). Let N be a connected real Lie group. Then Lie(N) is a nilpotent Lie +algebra if and only if N is a nilpotent Lie group. +Proposition 4.43 (Proposition 13, Section 4, Chapter 1 in [7]). In the notation of Proposition 2.7, nξ = +Lie(Nξ) is a maximal nilpotent ideal in gξ = Lie(Gξ). +Remark 4.44. +1. The presentation of nξ in [8] is given by means of the root space decomposition of +StabG(ξ), that appears in Proposition 2.17.13 in [20]. +2. There are two main objects in the literature that are referred to as the nilpotent radical or the nilradical +of a Lie algebra. These are: (a) the maximal nilpotent ideal of the Lie algebra, and (b) the intersection +of the kernels of all irreducible finite-dimensional representations. +Proposition 13 in Section 4 of +Chapter 9 in [7] shows that in the case of Lie algebras of parabolic Lie groups, these notions coincide. +Corollary 4.45. Nξ is a maximal connected nilpotent normal Lie subgroup of the identity connected com- +ponent StabG(H)◦. +Proof. Lemma 4.42 implies Nξ is nilpotent. Since StabGH ⊳ StabG(ξ), every normal subgroup of +StabG(H) containing Nξ is in fact a normal subgroup of StabG(ξ), still containing Nξ. It remains to +prove maximality of Nξ among all connected nilpotent normal Lie subgroups of StabG(ξ). Any such +subgroup N ′ ⊳ StabG(ξ) gives rise to an ideal n′ of gξ = Lie +� +StabG(ξ) +� +, and by Lemma 4.42 it is a +nilpotent ideal. Therefore by Proposition 4.43 it is contained in nξ = Lie(Nξ), implying that N ′ ≤ Nξ. +Corollary 4.46. A lattice in StabG(H) intersects the horospherical subgroup Nξ in a lattice. +Proof. Corollaries 4.40 and 4.45 imply that the pair Nξ ⊳ StabG(H) satisfy the hypotheses of Mostow’s +Lemma 4.36. +4.4.2 +Indecomposable Horospherical Lattices +The Benoist and Miquel criterion requires the horospherical lattice to be indecomposable. It is shown in [5] +that if this lattice is contained in a Zariski dense discrete subgroup, then the indecomposability condition is +equivalent to irreducibility of the ambient group. The precise definitions and statements are as follows. +Definition 4.47 (Definition 2.14 in [5]). For a semisimple real algebraic Lie group G and U a horospherical +subgroup of G, let ∆U be a lattice in U. +1. ∆U is irreducible if for any proper normal subgroup N of G◦, one has ∆U ∩ N = {e}. +2. ∆U is indecomposable if one cannot write G◦ as a product G◦ = N ′N ′′ of two proper normal subgroups +N ′, N ′′ ⊳ G with finite intersection such that the group +∆′ +U := (∆U ∩ N ′)(∆U ∩ N ′′) +has finite index in ∆U. +Definition 4.48 (See Section 2.4.1 in [5]). Let G be a semisimple real algebraic Lie group. A discrete +subgroup Λ ≤ G is said to be irreducible if, for all proper normal subgroups N ⊳ G, the intersection Λ ∩ N +is finite. +Lemma 4.49 (Lemma 4.3 in [5]). Let G be a semisimple real algebraic Lie group, U ⊂ G a non-trivial +horospherical subgroup, and ∆U ≤ U a lattice of U which is contained in a discrete Zariski dense subgroup +∆ of G. Then the following are equivalent: +1. ∆ is irreducible. +2. ∆U is irreducible. +3. ∆U is indecomposable. +36 + +4.4.3 +Zariski Density +The last requirement is for Λ to be Zariski dense. I use a geometric criterion which is well known to experts. +Lemma 4.50 (Proposition 2 in [31]). Let X be a symmetric space of noncompact type, G = Isom(X)◦. A +subgroup ∆ ≤ G is Zariski dense if and only if: +1. ∆ does not globally fix a point in X(∞), i.e. ∆ ̸≤ StabG(ζ) for any ζ ∈ X(∞). +2. The identity component of the Zariski closure of ∆ does not leave invariant any proper totally geodesic +submanifold in X. +In the proof I use several facts - mostly algebraic, and two geometric. I warmly thank Elyasheev Leibtag +for his help and erudition in algebraic groups. The first property I need is very basic. +Lemma 4.51. Let ∆ ≤ G be a discrete subgroup, and let H ≤ G be the Zariski closure of ∆. Then ∆ ∩ H◦ +is of finite index in ∆. +Proof. H◦ is normal and of finite index in H. +The following fact is probably known to experts. It appears in a recent work by Bader and Leibtag[2]. +Lemma 4.52 (Lemma 3.9 in [2]). Let k be a field, G a connected k algebraic group, P ≤ G = G(R) a +parabolic subgroup. Then the centre of G contains the centre of P. +Still on the algebraic side, I need a Theorem of Dani, generalizing the Borel Density Theorem. +Theorem 4.53 (See [15]). Let S be a real solvable algebraic group. If S = S(R) is R-split, then every lattice +ΓS ≤ S is Zariski dense. +Remark 4.54. It is a fact (see Theorem 15.4 and Section 18 in [6]) that: +1. Every unipotent group over R is R-split. +2. For a field k of characteristic 0, a solvable linear algebraic k-group is k-split if and only if its maximal +torus is k-split. +Finally I need two geometric facts. The first is a characterization determining when does a unipotent +element belongs to Nζ for some ζ ∈ X(∞). +Proposition 4.55 (Proposition 4.1.8 in [20]). Let X be a symmetric space of noncompact type and of higher +rank, n ∈ G = Isom(X)◦ a unipotent element, and ζ ∈ X(∞). The following are equivalent: +1. For Nζ as in Proposition 2.7, n ∈ Nζ. +2. For some geodesic ray η with η(∞) = ζ it holds that limt→∞ d +� +nη(t), η(t) +� += 0. +3. For every geodesic ray η with η(∞) = ζ it holds that limt→∞ d +� +nη(t), η(t) +� += 0. +The last property I need is a characterization of the displacement function for unipotent elements. +Proposition 4.56 (See proof of Proposition 3.4 in [4]). Let X be a symmetric space of noncompact type, +ζ ∈ X(∞) some point and n ∈ Nζ an element of the unipotent radical of StabG(ζ). +The displacement +function x �→ d(nx, x) is constant on horospheres based at ζ, and for every ε > 0 there is a horoball HBε +based at ζ such that d(nx, x) < ε for every x ∈ HBε. +Corollary 4.57. Assume that: +1. +� +Λ ∩ StabG(H) +� +· x0 is a cocompact metric lattice in a horosphere H ⊂ X bounding a Λ-free horoball. +37 + +2. Every Γ-conical limit point is a Λ-conical limit point. +Then Λ is Zariski dense. +Proof. I show the criteria of Lemma 4.50 are met, starting with Λ ̸≤ StabG(ζ) for any ζ ∈ X(∞). To this +end, I first prove that Λ · x0 is not contained in any bounded neighbourhood of any horosphere H′. Let +ξ′ be the base point of H′. By Hattori’s Lemma 2.31 (and Remark 2.32), it is enough to find a Λ-conical +limit point ζ′ with dT (ξ′, ζ′) ̸= π +2 . Take some ζ′′ ∈ X(∞) at Tits distance π of ξ′, i.e. take a flat F on +which ξ′ lies and let ζ′′ be the antipodal point to ξ′ in F. +Fix ε = +π +4 . +By Proposition 2.3, there are +neighbourhoods of the cone topology U, V ⊂ X(∞) of ξ′, ζ′′ (respectively) so that every point ζ′ ∈ V admits +dT (ξ′, ζ′) ≥ dT (ξ′, ζ′′)− π +4 = 3 +4π. Recall that the set of Γ-conical limit points is dense (in the cone topology), +so the second hypothesis implies there is indeed a Λ-conical limit point in V and therefore at Tits distance +different (in this case larger) than π +2 from ξ′. I conclude that Λ · x0 is not contained in any bounded metric +neighbourhood of any horosphere of X. +Assume towards contradiction that Λ ≤ StabG(ζ). +I show that this forces Λ ∩ Nζ ̸= ∅. By Propo- +sition 4.55 it is enough to find a unipotent element λ ∈ Λ and a geodesic η with η(∞) = ζ such that +limt→∞ d +� +λη(t), η(t) +� += 0. Let F be a maximal flat with ξ, ζ ∈ F(∞), x ∈ F some point and X, Y ∈ a ≤ p +two vectors such that exp(tY ) = η(t) for the unit speed geodesic η = [x, ζ), and exp(tX) = η′(t) for the unit +speed geodesic η′ = [x, ξ) (where a ≤ p a maximal abelian subalgebra in a suitable Cartan decomposition +g = p ⊕ k). Let StabG(ξ) = KξAξNξ be the decomposition described in Proposition 2.7 with respect to Y +(notice that Nξ does not depend on choice of Y , see item 3 of Proposition 2.17.7 in [20]). +The assumption that Λ ≤ StabG(ζ) implies that for any λ ∈ Λ the distance d +� +λη(t), η(t) +� +either tends +to 0 as t → ∞ or is uniformly bounded for t ∈ R. In the latter case there is some constant c > 0 for which +d +� +λη(t), η(t) +� += c for all t ∈ R. As in the proof of Corollary 4.41, the Flat Strip Theorem (Theorem 2.13, +Chapter 2.2 in [11]) implies that λ and at := exp(tY ) commute. +From the first hypothesis of the statement and Mostow’s result (Corollary 4.46) I know that Λ ∩ Nξ is a +cocompact lattice in Nξ (attention to subscripts). Therefore Λ ∩ Nξ is Zariski dense in Nξ (Theorem 4.53). +Moreover, since commuting with an element is an algebraic property, an element g ∈ G that commutes with +Λ∩Nξ must also commute with its Zariski closure, namely with Nξ. This means that if at commutes with all +Λ∩Nξ then it commutes with Nξ, i.e. atn = nat for all t ∈ R and all n ∈ Nξ. I know that at ∈ Aξ commutes +with both Kξ and Aξ (Proposition 2.7) therefore if at also commutes with Nξ then at lies in the centre +of StabG(ξ). This means that at is central in G (Lemma 4.52). For a group G with compact centre this +cannot happen, so there is indeed some unipotent element λ ∈ Λ ∩ Nξ for which limt→∞ d +� +λη(t), η(t) +� += 0. +I conclude from Proposition 4.55 that Λ ∩ Nζ ̸= ∅. +The first paragraph of the proof implies in particular that Λ·x0 does not lie in any bounded neighbourhood +of a horosphere H′ based at ζ. +The assumption that Λ ⊂ StabG(ζ) implies that every λ ∈ Λ acts by +translation on the filtration {H′ +t}t∈R by horospheres based at ζ. Therefore as soon as Λ · x0 ̸⊂ Ht for some +t ∈ R one concludes that ζ is a horospherical limit point of Λ, i.e. that every horoball based at ζ intersects +the orbit Λ · x0. +By Proposition 4.56 it holds that for a unipotent element g ∈ Nζ the displacement function x �→ d(gx, x) +depends only on the horosphere H′ +t in which x lies and that, for xt ∈ H′ +t it holds that limt→∞ d(gxt, xt) = 0 +(up to reorienting the filtration t ∈ R so that η(t) ∈ H′ +t). +For a non-trivial element λζ ∈ Λ ∩ Nζ the +previous paragraph therefore yields a sequence of elements λn ∈ Λ such that limn→∞ d(λζλnx0, λnx0) = 0, +contradicting the discreteness of Λ. I conclude that Λ ̸≤ StabG(ζ) for every ζ ∈ X(∞). +Assume that H := +� +Λ +Z�◦, the identity connected component of the Zariski closure of Λ, stabilizes a totally +geodesic submanifold Y ⊂ X. By Lemma 4.51, Λ0 := Λ ∩ H is of finite index in Λ, therefore Λ0 ∩ StabG(H) +is also a cocompact lattice in StabG(H). The fact that +� +Λ0 ∩ StabG(H) +� +· x0 is a cocompact metric lattice +in H readily implies that +� +Λ0 ∩ StabG(H) +� +· y is a cocompact metric lattice in Hy = H(y, ξ). This goes to +show that there is no loss of generality in assuming x0 ∈ H ∩ Y . It follows that Λ0 ∩ StabG(H) · x0 ⊂ Y ∩ H, +and therefore H ⊂ ND(Y ) for some D > 0. A horosphere is a codimension-1 submanifold, implying that Y +is either all of X or of codimension-1. The latter forces Y = H, which is impossible since H is not totally +geodesic (H is not convex, see Corollary 4.41). I conclude that H does not stabilize any totally geodesic +38 + +proper submanifold, and hence that Λ is Zariski dense. +4.5 +Proof of Theorem 4.1 +I now complete the proof of the main sublinear rigidity theorem for Q-rank 1 lattices. +Proof of Theorem 4.1. If {dγ}γ∈Γ is bounded, then Λ is a lattice by Corollary 4.7 or Theorem 4.9, depending +on the R-rank of G. +If {dγ}γ∈Γ is unbounded, then Proposition 4.12 and Corollary 4.27 both hold. In R-rank 1 the proof +again follows immediately from Corollary 4.7 using Lemma 2.16 and Corollary 4.27. +In higher rank, Section 4.4 allows one to conclude that Λ is an irreducible, discrete, Zariski dense subgroup +that contains a horospherical lattice. By Theorem 4.4, this renders Λ a lattice. It is a Q-rank 1 lattice as a +result of Theorem 2.20. +Remark 4.58. The sublinear nature of the hypothesis in Theorem 1.6 induces coarse metric constraints. +A horospherical lattice on the other hand is a very precise object. It is not clear how to produce unipotent +elements in Λ, or even general elements that preserve some horosphere. The proof sketched above produces +a whole lattice of unipotent elements in Λ (this is Corollary 4.46); it is also the only proof that I know which +produces even a single unipotent (or parabolic) element in Λ. +5 +Lattices with Property (T) +Recall that a lattice in a locally compact group G has property (T) if and only if G has property (T). In +this section I prove: +Theorem 5.1. Let G be a real centre-free semisimple Lie group without compact factors, Γ ≤ G a lattice, +Λ ≤ G a discrete subgroup such that Γ ⊂ Nu(Λ) for some sublinear function u. If Γ has property (T), then +Λ is a lattice. +As in the case of uniform lattices, lattices with property (T) admit the stronger version of ε-linear rigidity, +for suitable ε = ε(G): +Theorem 5.2. Let G be a real centre-free semisimple Lie group without compact factors, Γ ≤ G a lattice +and Λ ≤ G a discrete subgroup. If Γ has property (T) then there exists ε = ε(G) > 0 depending only on G +such that if Γ ⊂ Nu(Λ) for some function u(r) ⪯∞ εr, then Λ is a lattice. +Clearly Theorem 5.2 implies Theorem 5.1. I thank Emmanuel Breuillard for suggesting this generaliza- +tion. From now and until the end of this section, the standing assumptions are those of Theorem 5.2. +Lattice Criterion. +For groups with property (T) I use a criterion by Leuzinger, stating that being a +lattice is determined by the exponential growth rate. The formulation requires a definition. +Definition 5.3. Given a pointed metric space (X, dX, x0), denote: +1. bX(r) = |B(x0, r)| +2. bu +X(r) = supx∈X |B(x, r)| +When a group ∆ acts on a pointed metric space X, the orbit ∆ · x0 together with the metric induced +from X is a pointed metric space (∆ · x0, dX↾∆·x0, x0). In this setting b∆·x0(r) = |BX(xo, r) ∩ ∆ · x0|. When +the action is by isometries, i.e. ∆ ≤ Isom(X), it is straightforward to observe that this quantity does not +depend on the centre of the ball, and so bu +∆·x0(r) = b∆·x0(r). The pointed metric spaces of interest in this +section are the Γ and Λ orbits in the symmetric space X = G/K. +39 + +Definition 5.4. Let X be a symmetric space and ∆ ≤ G = Isom(X)◦ a subgroup of isometries. The critical +exponent of ∆ is defined to be +δ(∆) := lim sup +r→∞ +log +� +b∆·x0(r) +� +r +Remark 5.5. Throughout this section there is no risk of ambiguity, and I allow myself to ease notation and +let b∆(r) = b∆·x0(r), bu +∆(r) = bu +∆·x0(r). +To a semisimple Lie group G one can associate a quantity ∥ρ∥, where ρ = ρ(G) is the half sum of positive +roots in the root system of (g, a) (see Section 2 in [36]). +Theorem 5.6 (Theorem 2 in [36]). Let G be a real centre-free semisimple Lie group without compact factors. +Let ∆ be a discrete, torsion-free subgroup of G that is not a lattice. If G has Kazhdan’s property (T ), then +there is a constant c∗(G) (depending on G but not on ∆) such that δ(∆) ≤ 2∥ρ∥ − c∗(G). +It is known that the critical exponent of a discrete subgroup ∆ ≤ G is bounded above by 2∥ρ∥ (see +Section 2.2 in [36]). Moreover, every lattice Γ ≤ G admits δ(Γ) = 2∥ρ∥ (Example 2.3.5 in [36], Theorem C +in [1]). Combining these facts with Theorem 5.6 yield: +Theorem 5.7 (Theorem B in [36]). Let G be a real centre-free semisimple Lie group without compact +factors. Let ∆ be a discrete, torsion-free subgroup of G. If G has Kazhdan’s property (T ), then ∆ is a lattice +iff δ(∆) = 2∥ρ∥. +Line of Proof and the Use of ε-Linearity. +The proof of Theorem 5.1 goes by showing that ε-linear +distortion cannot decrease the exponential growth rate by much. This fact is essentially manifested in a +proposition by Cornulier [13], stated here in Proposition 5.8. This is the only use I make of ε-linearity, and +the computations involved are straightforward. Theorem 5.2 is then an immediate consequence of Leuzinger’s +criterion Theorem 5.6. +5.1 +Proof of Theorem 5.1 +In his study on SBE maps, Cornulier proves the following growth discrepancy result. +Proposition 5.8 (Proposition 3.6 in [13]). Let X, Y be two pointed metric spaces. Let u be a non-decreasing +sublinear function and p : X → Y a map such that for some L, R0 > 0: +1. |p(x)| ≤ max(|x|, R0), i.e. p(|x|) ≤ |x| for all large enough x ∈ X. +2. dY +� +p(x), p(x′) +� +≥ 1 +LdX(x, x′) − u(max{|x|, |x′|}) +Then for all r > R0, bY (r) ≥ bX(r)/bu +X +� +L · u(r) +� +. +I need a slightly modified version of Proposition 5.8: +Proposition 5.9. Let X, Y be two pointed metric spaces. Let u be a non-decreasing function that admits +u(r) ⪯∞ εr for some ε < 1, and p : X → Y a map such that for some L, R0 > 0: +1. |p(x)| ≤ max(|x| + u(|x|), R0), i.e. |p(x)| ≤ |x| + u(|x|) for all large enough x ∈ X. +2. dY +� +p(x), p(x′) +� +≥ 1 +LdX(x, x′) − u(max{|x|, |x′|}) +Then for all r > R0, +bY (r) ≥ bX +� +r − u(r) +� +/bu +X +� +L · u +� +r − u(r) +�� +Proof. Repeat verbatim the proof for Proposition 3.6. in [13]. +40 + +Corollary 5.10. Let u(r) ⪯∞ ε · r for some ε < 1 +2. Assume that Γ ⊂ Nu(Λ). Then δ(Λ) ≥ (1 − 4ε) · δ(Γ). +Remark 5.11. Restricting to ε < 1 +2 stems from a 2 factor that appears in the proof and could possibly be +dropped using a slightly more sophisticated approach. For my needs this is more than enough, since in any +case I eventually restrict attention to a small interval around 0. +Corollary 5.10 can be formulated in a slightly more general fashion. +Using the notation δ(W) := +lim supr→∞ +log +� +bW (r) +� +r +for a general subset W in a general metric space X, the following general version +holds: +Corollary 5.12. Let (X, x0) be a pointed metric space, Y, Z ⊂ X two subsets, and u(r) ⪯∞ εr for some +ε < 1 +2. Assume that bu +Z(r) = bZ(r). If Z ⊂ Nu(Y ), then δ(Y ) ≥ (1 − 4ε) · δ(Z). Moreover, if u is sublinear, +then δ(Z) = δ(Y ). +In particular, Corollary 5.10 holds even when the group G does not have property (T ). Corollary 5.10 +follows from Corollary 5.12 because the fact that Γ is a group of isometries implies bu +Γ(r) = bΓ(r). +Proof (Corollary 5.12). Observe that the closest point projection pY : Z → Y defined by z �→ zy for some +point in the closed ball zy ∈ B +� +z, u(|z|) +� +admits: +1. |yz| ≤ |z| + u(|z|) +2. d +� +yz, yz′� +≥ d(z, z′) − 2u(max{|z|, |z′|}) +The first item follows from triangle inequality: |yz| ≤ d(yz, z) + d(z, x0). The second item follows from +the quadrilateral inequality, i.e., using triangle inequality twice along the quadrilateral [z, z′, yz′, yz]. +The above properties allow me to use Proposition 5.9 with constant L = 1 and function u′ = 2u to get +bY (r) ≥ bZ +� +r − u′(r) +� +/bu +Z +� +u′� +r − u′(r) +�� +Since I assume bu +Z = bZ, I can omit the superscript u in the last expression. Recalling the definition +δ(W) = lim supr→∞ +bW (r) +r +, it remains to prove: +lim sup +r→∞ +1 +r · log +� +bZ +� +r − u′(r) +� +/bZ +� +u′� +r − u′(r) +��� +≥ (1 − 4ε) · δ(Z) +The proof of this inequality involves nothing more than log rules and arithmetic of limits: +lim sup +r→∞ +1 +r · log +� +bZ +� +r − u′(r) +� +/bZ +� +u′� +r − u′(r) +��� += lim sup +r→∞ +1 +r · +� +log +� +bZ +� +r − u′(r) +�� +− log +� +bZ +� +u′� +r − u′(r) +���� +≥ lim sup +r→∞ +� +1 +r · log +� +bZ +� +r − u′(r) +�� +− lim sup +s→∞ +�1 +s log +� +bZ +� +u′� +s − u′(s) +����� += lim sup +r→∞ +1 +r · log +� +bZ +� +r − u′(r) +�� +− lim sup +s→∞ +1 +s log +� +bZ +� +u′� +s − u′(s) +��� +≥ lim sup +r→∞ +1 +r · log +� +bZ +� +r − 2εr +�� +− lim sup +s→∞ +1 +s log +� +bZ +� +2εs +�� += (1 − 2ε)δ(Z) − 2εδ(Z) = (1 − 4ε)δ(Z) +(5) +Below I justify the steps in the above inequalities: +41 + +1. First equality is by rules of log. +2. Second and third inequalities are by arithmetic of limits: let (an)n, (bn)n be two sequences of positive +numbers, and A = lim supn an, B = lim supn bn. Then lim sup(an − bn) ≥ lim supn(an − B) = A − B. +3. Fourth inequality: u′(r) < 2ε(r) for all large enough r. +4. Fifth equality: definition of δ. +This completes the proof in the general case, which is what is needed for the proof of Theorem 5.2. +For the more refined statement in the case u is sublinear, one has to show a bit more. From inequality 5 +(specifically from the fourth line of the inequality) it is clearly enough to prove: +1. lim supr→∞ +1 +r · log +� +bZ +� +r − u′(r) +�� += δ(Z). +2. lim sups→∞ +1 +s log +� +bZ +� +u′� +s − u′(s) +��� += 0. +Starting from the second item, indeed it holds that +1 +s log +� +bZ +� +u′� +s − u′(s) +��� += u′� +s − u′(s) +� +s +· +log +� +bZ +� +u′� +s − u′(s) +��� +u′� +s − u′(s) +� +Clearly lim sup of the right factor in the above product is bounded by δ(Z), and in particular it is +uniformly bounded. On the other hand sublinearity of u′ implies that the left factor tends to 0. I conclude +that this product tends to 0 as s tends to ∞. +It remains to prove lim supr→∞ +1 +r · log +� +bZ +� +r − u′(r) +�� += δ(Z). In a similar fashion, +1 +r log +� +bZ +� +r − u′(r) +�� += r − u′(r) +r +· +log +� +bZ +� +r − u′(r) +�� +r − u′(r) +The left factor limits to 1 by sublinearity of u′. The right factor is nearly the expression in the definition +of δ(Z), and I want to prove that indeed taking lim sup of it equals δ(Z). A priori {r − u′(r)}r∈R>0 is just a +subset of R>0, so changing variable and writing t := r−u′(r) requires a justification. But there is no harm in +assuming that u′ is a non-decreasing continuous function, hence R≥R ⊂ {r − u′(r)}r∈R>0 for some R ∈ R>0. +Therefore for any sequence rn → ∞ there is a sequence r′ +n with rn = r′ +n − u′(r′ +n) for all large enough n (note +that in particular r′ +n → ∞). In the other direction, for every sequence r′ +n → ∞ there is clearly a sequence +rn → ∞ for which rn = r′ +n − u′(r′ +n). I conclude +lim sup +r→∞ +log +� +bZ +� +r − u′(r) +�� +r − u′(r) += lim sup +r→∞ +1 +r · log +� +bZ(r) +� += δ(Z) +This completes the proof. +Proof of Theorem 5.2. Define ε(G) = c∗(G) +4·2∥ρ∥, and assume u(r) ⪯∞ ε(G) · r. Notice that ε(G) < 1 +2, and since +δ(Γ) = 2∥ρ∥ Corollary 5.10 gives +δ(Λ) ≥ +� +1 − 4ε(G) +� +· 2∥ρ∥ = 2∥ρ∥ − 4ε(G) · 2∥ρ∥ ≥ 2∥ρ∥ − c∗(G) +By Theorem 5.6, Λ is a lattice. +42 + +Remark 5.13. The question of existence of interesting groups that coarsely cover a lattice is a key question +that arises naturally in the context of this paper. The first question that comes to mind is whether there +exist groups that are not commensurable to a lattice but that sublinearly, or even ε-linearly, cover one. +Perhaps the growth rate point of view could be used to rule out groups that cover a lattice ε-linearly but +not sublinearly. +6 +SBE Rigidity for Lattices of Higher Q-Rank +6.1 +Sublinear Distortion and SBE Maps +Denote a ∨ b := max(a, b) for a, b ∈ R. For a pointed metric space (X, x0, dX) and x, x1, x2 ∈ X, denote +|x|X := dX(x, x0) and |x1 −x2|X := dX(x1, x2) (or simply |x| and |x1 −x2| when there is no ambiguity about +the space X). +Following Cornulier [14], Pallier [44] makes the following definition: +Definition 6.1. A function u : R≥0 → R is admissible if it satisfies the following conditions: +• u is non-decreasing +• u grows sublinearly: lim sup +r�→∞ +u(r) +r += 0. +• u is doubling: +u(tr) +u(r) is bounded above for all t > 0. +• u ≥ 1. +The focus in this paper is condition 2, namely that the function u is strictly sublinear. I moreover require +it to be subadditive, resulting in the following terminology which I use from now on. +Definition 6.2. A function u : R≥0 → R is sublinear if it is admissible and subadditive, i.e. u(t + s) ≤ +u(t) + u(s) for all t, s > 0. +From now on by an SBE I mean an (L, u)-SBE where u is sublinear in the sense of Definition 6.2. +6.2 +SBE Rigidity +Two finitely generated groups Γ and Λ are said to be SBE if they are SBE when viewed as metric spaces with +some word metrics and base points eΓ, eΛ. Observe that every quasi-isometry is an SBE, and in particular +the word metric is an SBE-invariant. An SBE admits an SBE inverse, defined as quasi-inverse maps are +defined for quasi-isometries. +Definition 6.3. A class of groups A is said to be SBE complete if, for every finitely generated group Λ that +is SBE with some group Γ ∈ A, there is a short exact sequence +1 → F → Λ → Λ1 → 1 +for a finite group F ≤ Λ and some Λ1 ∈ A. +In this chapter I prove: +Theorem 6.4. Let G be a real centre-free semisimple Lie group without compact or R-rank 1 factors. +1. The class of uniform lattices of G is SBE complete. +2. The class of non-uniform lattices of G is SBE complete. +43 + +Remark 6.5. The proof I present is quite indifferent to whether the lattice Γ is uniform or not. In order +to have a unified proof and to ease notation, I fix the convention that for both uniform and non-uniform +lattices, X0 denotes the compact core of the lattice. This just means that X = X0 in case Γ is uniform. +My works heavily relies on that of Drut¸u in [16], where the theorems are stated for non-uniform lattices. +Nonetheless one readily sees that her proofs work perfectly well for uniform lattices. Indeed the arguments +of [16] are only much simpler in the uniform case. +6.2.1 +The Quasi-Isometry Case +The outline of the proof I present for Theorem 6.4 is identical to that of quasi-isometric rigidity, which I +now describe briefly. The main step is that for any quasi-isometry f : X0 → X0 of the compact core of Γ, +there exists an isometry g : X → X such that f, g are boundedly close, i.e. there is some D > 0 for which +d +� +f(x), g(x) +� +< D for all x ∈ X0. +Let Λ be an abstract group with a quasi-isometry q : Λ → Γ. +Using Lubotzki-Mozes-Raghunathan +([38], see Theorem 2.19 above), Γ is quasi-isometrically embedded in X as the compact core X0. One can +thus extend q to a quasi-isometry q0 : Λ → X0. A conjugation trick allows to associate to each λ ∈ Λ a +quasi-isometry fλ : X0 → X0 defined by fλ := q ◦ Lλ ◦ q−1 (Lλ : Λ → Λ is the left multiplication by λ in Λ). +By the first paragraph, there exists gλ ∈ Isom(X) that is boundedly close to fλ. Moreover, the proof also +shows that the bound D depends only on the quasi-isometry constants of fλ. These could be seen to depend +only on q and not on any specific λ. From this one concludes that the map λ �→ gλ is a group homomorphism +Φ : Λ → G. It is then straightforward to show that Φ has finite kernel and that Γ ⊂ ND +� +Im(Φ) +� +. One then +uses Theorem 4.3.2 (for higher rank groups, see Section 4.3) to deduce that Im(Φ) is a non-uniform lattice +in G that is commensurable to Γ. +6.2.2 +The SBE Case +Moving to SBE rigidity, one starts with an SBE q : Λ → Γ. The first step is to find an isometry of X that is +close to an SBE of X0. Drut¸u’s proof is preformed in the asymptotic cone of X, which allows for a smooth +transition to the SBE setting. Indeed, given an SBE f : X0 → X0, one can find an isometry g : X → X that +is close to it. The difference is that in the SBE setting, these maps are only sublinearly close. +Definition 6.6. Let (X, x0) be a pointed metric space, and f, g : X → X be two maps. Maps f, g are said +to be sublinearly close maps on X if there is a sublinear function u such that d +� +f(x), g(x) +� +≤ u(|x|). +Theorem 6.7 (Theorem 6.10). Let f : X0 → X0 be an SBE. Then there exists a unique isometry g ∈ +Isom(X) that is sublinearly close to f (in X0). +From this point on, one would like to continue as in the quasi-isometry case: define the map Φ : Λ → G +in a similar fashion and show that Γ ⊂ Nu +� +Im(Φ) +� +for some sublinear function u. That Im(Φ) is a lattice is +then a result of Theorem 1.6, proving Theorem 6.4. +There is however one additional obstacle that is unique to the SBE setting. Namely the SBE constants +of fλ do depend on λ, and the resulting sublinear bound on d +� +fλ(x), gλ(x) +� +in Theorem 6.7 is not enough +to define Φ properly. As far as I can see, one needs to get some uniform control on that bound in terms of +the SBE constants. The following statement is enough: +Lemma 6.8 (Lemma 6.11). Let {fr}r∈R>0 be a family of (L′, vr)-SBE maps fr : X0 → X0, where vr(s) = +L′·v(s)+v(r) for some sublinear function v ∈ O(u) and a constant L′. Let gr be the associated isometry given +by Theorem 6.10. Then for any x ∈ X0, there is a sublinear function ux ∈ O(u) such that d +� +fr(x), gr(x) +� +≤ +ux(r). +This type of uniform control is often needed when working with SBE maps, see e.g. Section I.3 in Pallier’s +thesis [43]. Using it, I am able to complete the argument as in the quasi-isometry case and prove: +Theorem 6.9. Let G be as in Theorem 6.4. In the notations described above, the map Φ : Λ → G is a group +homomorphism with Ker(Φ) finite, and there is a sublinear function u such that for Λ1 := Im(Φ) it holds +that Γ ⊂ Nu(Λ1) and Λ1 ⊂ Nu(Γ). +44 + +Theorem 6.4 is an immediate corollary of Theorem 6.9 and Theorem 1.6. +6.2.3 +Outline +This section is divided into two parts that correspond to the steps of the proof. Section 6.3 deals with the +task of finding an isometry that is sublinearly close to an SBE, and Section 6.4 establishes the properties of +the map Φ : Λ → G. I keep Section 6.3 slim and concise. The main reason for this choice is that the proof +of Theorem 6.10 is merely a mimic of Drut¸u’s argument in [16], or an adaptation of it to the SBE setting. +While these adaptations are somewhat delicate, giving a complete detailed proof would require reproducing +Drut¸u’s argument more or less in full. I felt that this is not desirable, and instead I only indicate the required +adaptations. I believe that a reader who is familiar with Drut¸u’s argument and with asymptotic cones could +easily produce a complete proof using these indications. In particular, there is no preliminary section. I do +not present buildings or dynamical results that go into Drut¸u’s argument. I only shortly present asymptotic +cones and some ideas from Drut¸u’s proof of the quasi-isometry version of Theorem 6.10. Section 6.4 is +elementary. +6.3 +SBE Maps are Close to Isometries +In this section I indicate how to adapt Drut¸u’s arguments in [16] in order to prove: +Theorem 6.10. There is a sublinear function v = v(L, u) such that for every +� +L, u +� +-SBE f : X0 → X0, +there exists a unique isometry g = g(f) ∈ Isom(X) such that d +� +f(x), g(x) +� +≤ v(|x|). +The proof of Theorem 6.4 requires some control on the sublinear distance between f and g, in terms of +the sublinear constants of f. This is the meaning of the following lemma. +Lemma 6.11. Let {fr}r∈R>0 be a family of (L′, vr)-SBE maps fr : X0 → X0, where vr = L′ · v + v(r) for +some sublinear function v ∈ O(u) and a constant L′. Let gr be the associated isometry given by Theorem 6.10. +Then for any x ∈ X0, there is a sublinear function ux ∈ O(u) such that d +� +fr(x), gr(x) +� +≤ ux(r). +Remark 6.12. Combined with Theorem 6.10, a different way to phrase the above statement is to say that +the function D : Λ× X0 → R≥0 defined by D(λ, x) = d +� +fλ(x), gλ(x) +� +is sublinear in each variable. I.e., there +is a function u : R≥0 × R≥0 → R≥1 such that u is sublinear in each variable and D(λ, x) ≤ u(|λ|, |x|). +Outline. +I begin with a short presentation of asymptotic cones. I then give an account of the original +proof of Theorem 6.10 when f : X0 → X0 is a quasi-isometry. I present the routines required to modify the +proof for the SBE setting. I exemplify the modification procedure in a specific representative example, and +finish with a road map for proving Theorem 6.10 and Lemma 6.11 using the aforementioned routines. +6.3.1 +Asymptotic Cones +Definition 6.13. Let (X, d) be a metric space. Fix an ultrafilter ω, a sequence of points xn ∈ X and a +sequence of scaling factors ın −→ +ω 0. The asymptotic cone of X w.r.t. xn, ın, denoted C(X), is the metric +ω-ultralimit of the sequence of pointed metric spaces (X, 1 +ın · d, xn). The metric on C(X) is denoted dω. +See Section 2.4 in [16] for an elaborate account, including the definitions of ultrafilters and ultralimits. +The strength of SBE maps is that they induce bi-Lipschitz maps between the respective asymptotic cones. +Lemma 6.14 (See e.g. Cornulier [13]). Let f : X → Y be an (L, u)-SBE. Then f induces an L-bi-Lipschitz +map C(f) : C(X) → C(Y ) between the corresponding asymptotic cones with the same scaling factors C(X) = +(X, 1 +ın dX, x0 +n) and C(Y ) = (Y, 1 +ın dY , y0 +n). +45 + +6.3.2 +The Argument +A High-Level Description. +The core of the argument lies in elevating an SBE f0 : X0 → X0 to an +isometry g0 ∈ G = Isom(X). There are two gaps to fill: first, Γ is non-uniform and so f0 is not even defined +on the whole space X. And obviously, f0 is just an SBE. +Assume for a moment that Γ is uniform and that f is defined on the whole space X. Elevating f : X → X +to an isometry is done by considering the map C(f) : C(X) → C(X) that f induces on an asymptotic cone +C(X). This map is bi-Lipschitz, and the work of Kleiner and Leeb [32] allows one to conclude that C(f) +is, up to a scalar, an isometry. In turn, this isometry induces an isometry ∂g on the spherical building +structure of X(∞). This is done by the relation between the Euclidean building structure of C(X) and the +spherical building structure of ∂∞X. A theorem of Tits [56] associates to ∂q a unique isometry g ∈ Isom(X) +that induces ∂f as its boundary map. By construction, it is then not too difficult to see that g and f are +‘close’. In case Γ is non-uniform, an SBE f : Γ → Γ does not readily yield a cone map on C(X), but only on +C(X0). Overcoming this difficulty requires substantial work and is the heart of Drut¸u’s proof. In short, she +uses dynamical results stating that the vast majority of flats in X are close enough to X0. As mentioned +in Section 2.2, the building structure on X(∞) is determined by the boundaries of flats F ⊂ X. The same +holds for the (Euclidean) building structure of C(X). Therefore the fact that the majority of flats in X are +‘close enough’ to X0 results in the fact that C(X0) composes the majority of C(X). This is a very rough +sketch of the logic behind Drut¸u’s argument. +The procedure described above results in an isometry g ∈ Isom(X) associated to f0. To complete the +argument one needs to verify that the map f0 �→ g is a group homomorphism between SBE(Γ) = SBE(X0) +and Isom(X). +Composing this map with a representation of Λ into SBE(Γ) yields a map Λ → G by +λ �→ fλ �→ gλ := gfλ. A computation then shows that this map has finite kernel and that Γ lies in a sublinear +neighbourhood of the image. +Flat Rigidity. +The adaptations that are required for the SBE setting lie mainly in the part of Drut¸u’s +work that concerns flat rigidity. That is, the proof that the quasi-isometry q0 maps a flat F ⊂ X0 to within +a uniformly bounded neighbourhood of another flat F ′ ⊂ X0. This is proved by passing to the cone map, +using an analogous result for bi-Lipschitz maps between Euclidean buildings, which translates back down to +the space X0. +Remark 6.15. Drut¸u’s argument requires many geometric and combinatorial definitions - some classical +and widely known (e.g., Weyl chambers of a symmetric space X) some less known (e.g. an asymptotic cone +with respect to an ultrafilter ω) and some new (e.g., the horizon of a set A ⊂ X). I use her definitions, +terminology and notations freely without giving the proper preliminaries or even the definitions. I assume +most readers who are interested in the question of SBE rigidity are familiar to some extent with most of these +objects. For the new definitions, I try to say as little as needed to allow the reader to follow the argument. +The proof consists of 6 steps: +1. The horizon of an image of a Weyl chamber is contained in the horizon of a finite union of Weyl +chambers, and the number of chambers in this union depend only on the Lipschitz constant. (Lemmas +3.3.5, 3.3.6 in [16], consult Remark 6.16 below for a sketchy definition of horizon). +2. The horizon of an image of a flat coincides with the horizon of a finite union of Weyl chambers, and +the number of chambers depends only on the Lipschitz constant of the quasi-isometry. +3. The union of Weyl chambers in the previous step limits to an apartment in the Tits building at X(∞). +Such a union is called a fan over an apartment. +4. For each Weyl chamber W there corresponds a unique chamber W ′ such that q0(W) and W ′ have the +same horizon. This amounts to an induced map on the Weyl chambers of the Tits building at X(∞) +(Lemma 4.2.1 in [16]). +46 + +5. Given a flat F through a point x, the unique flat F ′ asymptotic to the union of Weyl chambers obtained +in step 3 is at uniform bounded distance from f0(x). The bound depends only on the quasi-isometry +constants. The flat F ′ is called the flat associated to F. +6. If F1 and F2 are two flats through x which intersect along a hyperplane H, then the boundaries at +X(∞) of the associated flats F ′ +1 and F ′ +2 intersect along a hyperplane of the same codimension as H. +Remark 6.16. For a precise definition of horizon see [16], section 3. For now, it suffices to say the following. +The horizon of a set A ⊂ X is contained in the horizon of a set B ⊂ X if, looking far away at A from some +point x ∈ X, A appears to be contained in an ε-neighbourhood of B. This intuition is made precise by +considering the angle at x that a point a ∈ A makes with the set B. Two sets have the same horizon if each +set’s horizon is contained in the other. In the case A and B have the same horizon, an important aspect is +the distance R starting from which A and B seem to be ε-contained in one another. Call this distance the +horizon radius. It depends on x and ε. +The proofs for most of these steps have similar flavour: in any asymptotic cone C(X), f0 induces a +bi-Lipschitz map. Kleiner and Leeb [32] proved many results about such maps between cones of higher rank +symmetric spaces. One assumes towards contradiction that some assertion fails (say, in step 5, assume that +there is no bound on the distance between f0(xn) and the associated flat F ′ +n). This gives an unbounded +sequence of scalars (say, d +� +f0(xn), F ′ +n +� += ın → ∞). These scalars are used to define a cone in which one +obtains a contradiction to some fact about bi-Lipschitz maps (say, that the point [q0(xn)]ω is at dω-distance +1 from [F ′ +n]ω, while it should lie in [F ′ +n]ω). +Typically, the bounds obtained this way depend on the quasi-isometry constants. A priori, they also +depend on the specific point x ∈ X or flat F ⊂ X in which you work (e.g. the horizon radius for the chambers +in step 3 or the bound on d +� +q(x), F ′� +in step 5). However, it is easy to see that in the quasi-isometry setting, +the bounds are actually independent of the choice of point/flat/chamber. This independence stems from +the fact that one can pre-compose f0 with an isometry translating any given point/flat/chamber to a fixed +point/flat/chamber (resp.), without changing the quasi-isometry constants (see e.g. Remark 3.3.11 in [16]). +The fact that these bounds depend only on the quasi-isometry constants is essential for the proof that the +map Λ → Isom(X) has the desired properties. +Moving to the SBE setting, the essential difference is exactly that the bounds one obtains depend on +the specific point, Weyl chamber or flat. Indeed it is clear that these bounds should depend on the size |x|, +as they depend on the additive constant in the quasi-isometry case. It is sensible to guess though that the +bounds only grow sublinearly in |x|, which is enough in order to push the argument forward. In the next +section I show how to elevate a typical cone argument from the quasi-isometry setting to the SBE setting. I +focus on showing that the bound one obtains depend only on the SBE constants (L, u) and sublinearly |x|. +6.3.3 +Generalization to SBE: Adapting Cone Arguments +To adapt for the SBE setting, split each step into three sub-steps: the first two amount to proving Theo- +rem 6.10, and the third step amounts to proving Lemma 6.11. +Sub-Step 1. +Repeat the argument of the quasi-isometry setting verbatim, to obtain a bound c = c(x) +which depend on the point x. +Sub-Step 2. +Assume towards contradiction that there is a sequence of points xn for which limω +c(xn) +|xn| ̸= 0. +This means limω +|xn| +c(xn) ̸= ∞, and so the point (xn)n lies in the cone C(X) = Cone +� +X, x0, c(xn) +� +, and one +may proceed as in the corresponding quasi-isometry setting to obtain a contradiction. +Sub-step 3. +Fix x ∈ X and a sequence of SBE maps as in Lemma 6.11, i.e. {fn}n∈N with the same Lipschitz +constant and with sublinear constants vn(s) = v(s) + u(n), for some sublinear functions u, v. Denote by +cn(x) the constants that were achieved in the previous steps for x and the SBE map fn, and assume towards +47 + +contradiction that |cn(x)| is not bounded above by any function sublinear in n. This means in particular +that limω +u(n) +cn(x) = 0. One concludes that the cone map C(frn) is bi-Lipschitz, and gets a contradiction in the +same manner as in the first step. +Example 6.17. In order to give the reader a sense of what is actually required, I now demonstrate this +procedure in full in a specific claim. I chose to do this for proposition 4.2.7 of [16], which is complicated +enough to require some attention to details, but not too much. The statement is as follows: +Proposition 6.18 (SBE version of Proposition 4.2.7 in [16]). Let f : X → X be an (L, u)-SBE, and F ⊂ X +a flat through x to which f associates a fan over an apartment, ∪p +i=0Wi. If F ′ is the maximal flat asymptotic +to the fan, then d +� +f(x), F ′� +≤ c(x) where c(x) = c(|x|) is sublinear. +Moreover, let fn : X → X be a sequence of (L, vn) SBE maps for vn = v + u′(n) for v, u′ some sublinear +functions. The constant cn(x) associated to x and the SBE fn achieved in the first part of the proposition +admits cn(x) ≤ ux(n) for some sublinear function ux. +F ′ is then said to be the associated flat to F by f. +Proof. Proceed in two (sub-)steps. +Step 1. +I show that for a given x, there exists such a constant c(x) independent of the flat F. This is done +exactly as in [16], but I repeat the proof here because it contains the terminology and necessary preparation +for the second step. +Fix x ∈ X and assume towards contradiction that there exists a sequence Fn of flats through x and a +sequence fn : X → X of (L, u)-SBE maps such that cn := d +� +fn(x), F ′ +n +� +→ ∞. In Cone(X, x, c−1 +n ) one can +show that [∪p +i=0W n +i ], the union of Weyl chambers associated to fn(Fn), is a maximal flat (see Proposition +4.2.6 in [16]). Denote Fω := [∪p +i=0W n +i ]. Furthermore, since the bi-Lipschitz flat [fn(Fn)] ⊂ Cone(X, x, c−1 +n ) +is contained in it, it coincides with it: [fn(Fn)] = Fω. On the other hand, since the Hausdorff distance +between ∪p +i=0W n +i and F ′ +n is by assumption cn = d(x, F ′ +n), in the cone the maximal flats Fω and F ′ +ω := [F ′ +n] +are at Hausdorff distance 1. +This implies that Fω = F ′ +ω (see Corollary 4.6.4 in [32]). But since d +� +q(x), F ′ +n +� += cn the limit point +yω := Q(xω) = [q(x)] , which is contained in Fω, is at distance 1 from F ′ +ω - a contradiction. +Step 2. +Assume c(x) is taken to be the smallest possible for each x, and then modify the function c +so that c(x) = maxy:|y|=|x| c(y). +The function c : X → R now only depends on |x|. +I wish to show +that c(|x|) = O(u(|x|). +Assume towards contradiction that there exists a sequence xn with |xn| → ∞ +such that limω +c(xn) +u(|xn|) = ∞. Denote cn = c(xn) and consider the cone Cone(X, xn, c−1 +n ). The assumption +limω +c(xn) +u(|xn|) = ∞ implies (x0)ω = (xn)ω hence Cone(X, xn, cn) = Cone(X, x0, cn). By the definition of c(x) +this means that there is a sequence of flats Fn through xn such that d +� +q(xn), F ′ +n +� += cn, so one may proceed +as in step 1 for a cone with a fixed base point (x0)ω. In this cone the flat F ′ +ω := [F ′ +n] is at distance 1 +from the point [q(xn)], which lies on the maximal flat [∪p +i=0W n +i ]. The latter flat is, on the one hand, at +Hausdorff distance 1 from F ′ +ω (by the definition of the scaling factors cn), so they actually coincide. On +the other hand, [∪p +i=0W n +i ] coincides with Fω := Q(Fω) = [q(Fn)], so Fω = F ′ +ω, contradicting the fact that +dω([q(xn)], F ′ +ω) = 1. Thus c(|x|) = O(u(|x|)), as wanted. +Step 3. +For the moreover part (uniform control on the growth of c(x) as a function of the sublinear +constants), the proof is identical to Step 1. This time, consider a sequence fn as in the statement, and +denote by cn = cn(x) the constant obtained in step 1 w.r.t. the SBE constants (L, vn). Assume towards +contradiction that limω +u(n) +cn(x) = 0. The proof goes exactly as in step 1, with the sole difference that now one +might need convincing in the fact that in C(X), the cone with +1 +cn as scaling factors, the cone map C(fn) is +bi-Lipschitz. But indeed for any two cone points (xn), (yn) ∈ C(X) it holds: +dω +� +C(fn)(xn), C(fn)(yn) +� += lim +ω +1 +cn +d +� +fn(xn), fn(yn) +� +≤ lim +ω +1 +cn +L · d(xn, yn) + vn(|xn| ∨ |yn|) +48 + +By definition of the cone metric, limω +1 +cn L · d(xn, yn) = L · dω +� +(xn), (yn) +� +. +It thus remains to show +limω +1 +cn vn(|xn| ∨ |yn|) = 0. By definition of vn it amounts to proving limω +1 +cn v(|xn|) = 0 = limω +1 +cn v(|yn|) +and limω +1 +cn u(n) = 0. The former follows from the fact that (xn), (yn) ∈ C(X) and therefore both limω +1 +cn |xn| +and limω +1 +cn |xn| are finite. The latter follows from the assumption on the cn. One obtains a contradiction +identical to the one in Step 1. +Following the claims of Sections 3, 4, 5 in [16] carefully, and making the SBE adaptations as depicted +in the above example, one obtains flat rigidity in the SBE setting, that is Theorem 6.10 together with the +uniform control described in Lemma 6.11. Here is the complete list of claims involving cone arguments in +[16] that should be modified. +• Section 3. All claims starting from Lemma 3.3.5 through Corollary 3.3.10. All statements should +consider, instead of a quasi-isometry, a general (L, u)-SBE f and, when relevant, a general point x ∈ X +with f(x) = y (i.e., f(x) does not necessarily equal x). Also when relevant one should consider a family +fn of (L, vn) SBE maps as depicted in Lemma 6.11 above. +• Section 4. Propositions 4.2.6, 4.2.7, 4.2.9. When relevant, statements should be modified so that the +distance between f(x) and an associated flat of it should be uniformly sublinear in |x|. Also when +relevant one should consider a family fn of (L, vn) SBE maps as above. +• Section 5. Lemma 5.4.1 (D = D(x) should be uniformly linear in c = c(x)). Proposition 5.4.2 (the +constant D = D(x) should be replaced be a sublinear function D(|x|)). When relevant one should +consider a family fn of (L, vn) SBE maps as above. +This yields a proof for uniform flat rigidity in the SBE setting. The other major part of Drut¸u’s argument +concerns the fact that f0 is defined only on X0 and not on X. +The considerations for this aspect are +intertwined in the proof, but they all involve only Γ and the quasi-isometry between Γ and X0. For this +reason, the fact that f0 is an SBE to begin with does not effect any of these arguments. Moreover, this +argument is indifferent to whether or not Γ is uniform or not. If Γ is uniform all that changes is that that +part of Drut¸u’s argument dealing with extending the cone map from C(X0) to C(X) is not necessary since +X = X0. Her proof still works perfectly well also for uniform lattices. Therefore the argument above proves +Theorem 6.10 and Lemma 6.11. +6.3.4 +Some Remarks On R-rank 1 Factors +Quasi-isometric rigidity holds for groups of R-rank 1. It is worth mentioning that Schwartz’s proof also +relies on ‘flat rigidity’ - but in this case the flats are the horospheres of Γ. While these are not isometrically +embedded flats, the induced metric on horospheres is flat and Schwartz uses that in order to construct the +boundary map and find the associated isometry. +The same phenomena occurs in the SBE setting. Considering the compact core X0 ⊂ X of Γ, one can +use Proposition 5.6 in Drut¸u-Sapir [18], in order to show that horospheres are mapped boundedly close to +horospheres. In general, this work characterizes and explores a certain class of spaces they call asymptotically +tree graded, a class that is very suitable for the setting of the compact core of a non-uniform lattice in R- +rank 1. +A key ingredient in the proof is the fact that the boundary map ∂q induced by the quasi-isometry is +quasi-conformal. This in particular implies that it is almost everywhere differentiable. I spent some time +trying to generalize the proof of Schwartz to the SBE setting. One obstacle is that it is not clear that the +boundary map is going to be differentiable almost everywhere. Gabriel Pallier found that there are SBE +maps of the hyperbolic space whose boundary maps are not quasi-conformal (see Appendix A in [45]). For +this reason Pallier develops the notion of quasi-conformality [46]. While these maps may be differentiable, +he told me of examples he constructed where the differential is almost everywhere 0 - a property which also +nullifies Schwartz’s argument. +In the context of SBE rigidity, the maps I consider seem to indeed have ‘flat rigidity’, i.e. to map a +horosphere to within bounded distance of a unique horosphere. As in the higher rank flat rigidity, this +49 + +bound is not uniform but rather grows sublinearly with the distance of the horosphere to a fixed base point. +These are very specific maps, that coarsely preserve the compact core of that lattice X0 ⊂ X and basically +map horospheres to horospheres. This means there might still be hope for these specific maps to induce +boundary maps that admit the required analytic properties. +In a subsequent paper [53], Schwartz proves quasi-isometric rigidity for lattices in products of R-rank 1 +groups, i.e. in Hilbert modular groups. His proof there is different, but it also makes use of the fact that +horospheres are mapped to within uniformly bounded distance of horospheres. The fact that in the SBE +setting this bound is not uniform seems like a real obstruction to any attempt of generalizing his proof in +that case. +6.4 +From SBE to Sublinearly Close Groups +In this section I prove Theorem 6.9. I restate it here for convenience +Theorem 6.19. Let G be a real centre-free semisimple Lie group without compact or R-rank 1 factors. Let +Γ ≤ G be an irreducible lattice, and Λ an abstract finitely generated group that is SBE to Γ. Then there is +a group homomorphism Φ : Λ → G with finite kernel such that Γ ⊂ Nu +� +Φ(Λ) +� +and Φ(Λ) ⊂ Nu(Γ) for a +sublinear function u. +The proof is a sublinear adaptation of the classical arguments by Schwartz. The only difference is that +some calculations are in order, but there is no essential difference from Section 10.4 in [52]. +Before I start, I need one well known preliminary fact, namely that sublinearly close isometries are equal. +Lemma 6.20. Let X be a symmetric space of noncompact type and with no R-rank 1 factors. Let Γ ≤ +Isom(X) be a non-uniform lattice, X0 its compact core with respect to x0 ∈ X. Let g, h ∈ G = Isom(X) and +u a sublinear function such that for every x ∈ X0, d +� +g(x), h(x) +� +≤ u(|x|). Then g = h. +Proof. The proof is essentially just the fact that a sublinearly bounded convex function is uniformly bounded. +Up to multiplying by h−1, one may assume h = idX. First I show that the continuous map ∂g : X(∞) → +X(∞) is the identity map. Recall that the space X(∞) can be represented by all geodesics emanating from +the fixed point x0. Let η : [x0, ξ) be a Γ-periodic geodesic. By definition, there is some T > 0 and a sequence +γn ∈ G for which η(nT ) = γnx0. In particular, xn := γnx0 ∈ X0 hence d +� +g(xn), xn +� +≤ u(|xn|) = u(nT ). +On the other hand, the distance function d +� +η(t), g·η(t) +� +is convex. A convex sublinear function is bounded, +and so by definition in X(∞) one has [η] = [g · η], for all Γ-periodic geodesics η. The manifold X is of non- +positive curvature, hence ∂g is a homeomorphism of X(∞), and the density of Γ-periodic geodesics implies +∂g = idX(∞). This implies that g = idX (see Section 3.10 in [20] for a proof of this last implication). +Remark 6.21. The proof for the fact that ∂g = idX(∞) implies g = idX appears in [20] (section 3.10) as +part of the proof of the following important theorem of Tits: +Theorem 6.22 ([56], see Theorem 3.10.1 in [20]). Let X, X′ be symmetric spaces of noncompact type and +of higher R-rank. Assume X has no R-rank 1 factors, and let φ : X(∞) → X′(∞) be a bijection that is a +homeomorphism with respect to the cone topology and an isometry with respect to the Tits metric. Then, +after multiplying the metric of X by positive constants on de Rham factors, there exists a unique isometry +g : X → X′ such that φ = ∂g. +This theorem is actually a key ingredient in Drut¸u’s argument. Much of her work is directed towards +showing that the cone map C(q) will correspond to a map on X(∞) satisfying the above hypothesis. The +restriction to X with no R-rank 1 factors in Theorem 6.4 comes from this restriction in Tits’ Theorem 6.22 +Remark 6.23. The proof that ∂g = idX(∞) ⇒ g = idX only uses the fact that X has no Euclidean de +Rham factors (see pg. 251 in [20]). Since I only use it in the setting of no R-rank 1 factors, I added that +assumption to Lemma 6.20. +50 + +The Map Φ : Λ → G. +The orbit map q0 : Γ → X0 defined by γ �→ γx0 is a quasi-isometric embedding: +this is ˘Svarc-Milnor in case Γ is uniform and X0 = X, and Lubotzki-Mozes-Raghunathan (Theorem 2.19 +above) if Γ is non-uniform. An SBE f : Λ → Γ thus gives rise to an SBE Λ → X0, which I also denote by f. +For each λ ∈ Λ let fλ := f ◦ Lλ ◦ f −1 : X0 → X0, where Lλ is the left multiplication by λ. The left +translation Lλ is an isometry, hence fλ is a self SBE of X0. By Theorem 6.10, there exists a unique isometry +gλ ∈ Isom(X) that is sublinearly close to fλ. Define the map Φ : Λ → G by λ �→ gλ. The goal in this section +is to prove Φ is a homomorphism with finite kernel, and that Γ and Φ(Λ) are each contained in a sublinear +neighbourhood of the other. +I begin by controlling the SBE constants of the fλ. +Lemma 6.24. For each λ ∈ Λ, fλ is an (L2, vλ)-SBE, for +vλ(|x|) := (L + 1)u(|x|) + u(|λ|) +In particular vλ ∈ O(u). +Before the proof I state a corollary which follows immediately by combining Lemma 6.24 with Lemma 6.11. +Corollary 6.25. Assume Lemma 6.11 holds. Then for any x ∈ X there is a sublinear function ux such that +d +� +fλ(x), gλ(x) +� +≤ ux(|λ|) +. +Proof of Lemma 6.24. The proof is a straightforward computation. Up to an additive constant I may assume +f −1 is an (L, u)-SBE with f −1(eΓ) = eΛ. Let x1, x2 ∈ X0, and assume w.l.o.g |x2| ≤ |x1|. By the properties +of an SBE, this also means that for i ∈ {1, 2}: +|f −1(xi)| ≤ L|xi − x0| + u(|xi|) ≤ L|x1| + u(|x1|) +(6) +Notice that fλ(x) = f +� +λ · f −1(x) +� +, and f is an (L, u)-SBE. The following inequalities, justified below, +give the required upper bounds: +��fλ(x1) − fλ(x2) +�� ≤ L · +��λf −1(x1) − λf −1(x2) +�� + u +� +|λf −1(x1)| ∨ |λf −1(x2)| +� +≤ L2|x1 − x2| + Lu +� +|x1| +� ++ u +� +|λ|) + L|x1| + u(|x1|) +� +≤ L2|x1 − x2| + (L + 1)u +� +|x1| +� ++ u(|λ|). +(7) +From the first line to the second line I used: +1. For the first term: left multiplication in Λ is an isometry, and f −1 is an (L, u)-SBE. +2. For the second term: triangle inequality, left multiplication in Λ is an isometry, and Equation 6. +From the second to the third line I used the properties of u as an admissible function, namely that it is +sub-additive and doubling, so u +� +(L + 1)|x1| +� +≤ (L + 1)u(|x1|) for all large enough x1. +Remark 6.26. I remark that the proof of Lemma 6.24 is the only place where I use the properties of an +admissible function and not just the sublinearity of u. +Claim. Φ : Λ → G is a group homomorphism. +Proof. Let λ1, λ2 ∈ Λ. I begin with some notations: +1. f1 = fλ1, f2 = fλ2, f12 = fλ1λ2. By Lemma 6.24, these are all O(u) SBE maps with the same Lipschitz +constant L′ := L2 and sublinear constants v1, v2, v12 ∈ O(u). +51 + +2. g1 = Φ(λ1), g2 = Φ(λ2), g12 = Φ(λ1λ2) +3. u1, u2, u12 the sublinear functions that bound the respective distances between any g and f, e.g. +|g1(x) − f1(x)| ≤ u1(|x|). +One has to prove that gλ2 ◦ gλ1 = gλ1λ2. In view of Lemma 6.20, it is enough to find a sublinear function +v such that for all x ∈ X0 |g1g2(x) − g12(x)| ≤ v(|x|). By triangle inequality and the above definitions and +notation, it is enough to show that each of the following four terms are bounded by a function sublinear in +x: +1. |g1g2(x) − g1f2(x)| = |g2(x) − f2(x)| ≤ u2(|x|) (g1 is an isometry). +2. |g1f2(x) − f1f2(x)| ≤ u1 +� +|f2(x)| +� +≤ u1 +� +L2|x| + v2(|x|) +� +3. |f1f2(x) − f12(x)| +4. |f12(x) − g12(x)| ≤ u12(|x|) +Clearly items 1, 2, 4 are bounded by a sublinear function in |x|. It remains to bound |f1f2(x) − f12(x)|. +The map Λ → Aut(Λ) given by λ �→ Lλ is a group homomorphism, i.e. Lλ1λ2 = Lλ1Lλ2, so it remains to +bound: +|f1f2(x) − f12(x)| = |fLλ1f −1fLλ2f −1(x) − fLλ1Lλ2f −1(x)| +f ◦ Lλ is a composition of an isometry with an SBE, so it is still an SBE. Denote the SBE constants of +fLλ1 by L′, v (clearly one can take L′ = L and v ∈ O(u), but this is not needed). Writing y := Lλ2f −1(x), +this shows +|fLλ1f −1f(y) − fLλ1(y)| ≤ L|f −1fy − y| + v(|f −1fy| ∨ |y|) +By definition of an SBE inverse it holds that |f −1f(y) − y| ≤ u(|y|) and in particular also |f −1f(y)| ≤ +|y| + u(|y|). I conclude that +|f1f2(x) − f12(x)| ≤ L · u(|y|) + v +� +|y| + u(|y|) +� +The right-hand side is a sublinear function in |y|, hence it only remains to show that |y| is bounded by a +linear function in x. Indeed +|y| = |Lλ2f −1(x)| ≤ |λ2| + |f −1(x)| ≤ |λ2| + L|x| + u(|x|) +This completes the proof, rendering Φ a group homomorphism. +Claim. Φ has discrete image and finite kernel. +Proof. I show that for any radius R > 0, there are finitely many λ ∈ Λ for which gλx0 ∈ B(x0, R). I.e., that +there is a finite number of Φ(Λ)-orbit points, with multiplicities, inside an R ball in X. In particular the +set {λ ∈ Λ | gλx0 = x0} is finite, and clearly contains Ker(Φ). In addition, the actual number of Φ(Λ)-orbit +points inside that R ball is finite, so Φ(Λ) is discrete. +Let R > 0, and λ ∈ Λ. By the defining property of gλ and the definition of fλ, reverse triangle inequality +gives +d +� +x0, gλ(x0) +� +≥ d +� +x0, fλ(x0) +� +− d +� +gλ(x0), fλ(x0) +� +≥ |f(λ)| − uλ(|x0|) +Corollary 6.25 gives d +� +gλ(x0), fλ(x0) +� +≤ ux0(|λ|) for some sublinear function ux0 ∈ O(u). On the other +hand f is an SBE, and so |f(λ)| grows close to linearly in λ. Formally, +|f(λ)| = d +� +f(λ), x0 +� += d +� +f(λ), f(eΛ) +� +≥ 1 +Ld(λ, x0) − u(|λ| ∨ |eΛ|) ≥ 1 +L|λ| − u(|λ|) +52 + +To conclude, one has +d +� +x0, gλ(x0) +� +≥ 1 +L|λ| − u(|λ|) − ux0(|λ|) +and both u, ux0 are sublinear in |λ|. Therefore there is a bound S ∈ R>0 such that |λ| > S ⇒ 1 +L|λ| − +u(|λ|) − ux0(|λ|) > R. The group Λ is finitely generated and so only finitely many λ ∈ Λ admit |λ| ≤ S, +hence gλ(x0) ∈ B(x0, R) only for finitely many λ ∈ Λ. +Claim. There exists a sublinear function u′ : R≥0 → R≥1 such that +Γ · x0 ⊂ Nu′� +Φ(Λ) · x0 +� +Proof. I claim that there is a sublinear function u0, depending only on f and q0, such that for all γ ∈ G, +d +� +gλ(x0), γ(x0) +� +≤ u0(|γ|). +As before, I only have control on gλ via fλ, and so I use triangle inequality to get: +d +� +gλ(x0), γ(x0) +� +≤ d +� +gλ(x0), fλ(x0) +� ++ d +� +fλ(x0), γ(x0) +� +By Corollary 6.25, d +� +gλ(x0), fλ(x0) +� +≤ ux +0(|λ|) for a sublinear function ux0. +It is beneficial to distinguish between the SBE fΓ : Λ → Γ and the same SBE composed with the orbit +quasi-isometry q0 : Γ → X0. From now on I keep the notation fΓ : Λ → Γ for the SBE of the groups and f0 +for the same SBE composed with the orbit quasi-isometry so f0 = q0 ◦ fΓ. +Define λγ := f −1 +Γ (γ). I show that d +� +fλγ(x0), γ(x0) +� +is bounded by a function sublinear in γ. Indeed, +recall that I assumed without loss of generality q0 ◦f −1 +Γ (x0) = eΛ. Moreover, Γ is assumed to be torsion-free, +and so there is no ambiguity or trouble in defining the restriction of the map q−1 to the orbit Γ · x0 to be of +the form q−1(γx0) = γ. All together, this gives +d +� +fλγ(x0), γ(x0) +� += d +� +f0(λγ), γ(x0) +� += d +� +q0 ◦ fΓf −1 +Γ (γ), q0(γ) +� +Since fΓ is an SBE d +� +fΓf −1 +Γ (γ), γ +� +≤ u(γ). The fact that q0 is an (L′, C)-quasi-isometry implies that +d +� +q0 ◦ fΓf −1 +Γ (γ), q(γ) +� +≤ Ld +� +fΓf −1 +Γ (γ), γ +� ++ C ≤ L′u(|γ|) + C +Combining everything, one has +d +� +gλγ(x0), γ(x0) +� +≤ ux0(|λγ|) + L′u(|γ|) + C +As before, |λγ| = |f −1(γ)| ≤ |γ| + u(|γ|) ≤ 2|γ|, where the last inequality holds for all large enough γ. +What matters is that |λγ| is linear in |γ|. I conclude that indeed Γ · x0 ⊂ Nu′� +Φ(Λ) · x0 +� +for the sublinear +function u′ = ux0 + L′u + C ∈ O(u), as wanted. (To be pedantic, u′ = ux0 ◦ 2 + L′u + C ∈ O(u) where 2 is +the ‘multiplication by 2’ function, r �→ 2r). +Claim. 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Springer-Verlag, Berlin,New York, 1974. +56 + diff --git a/7tE4T4oBgHgl3EQfcwyw/content/tmp_files/load_file.txt b/7tE4T4oBgHgl3EQfcwyw/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e4ca8a0a26edcd047c0614ea228f9782ea8c9f9a --- /dev/null +++ b/7tE4T4oBgHgl3EQfcwyw/content/tmp_files/load_file.txt @@ -0,0 +1,3182 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf,len=3181 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='05086v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='GR] 12 Jan 2023 Sublinear Rigidity of Lattices in Semisimple Lie Groups Ido Grayevsky Abstract Let G be a real centre-free semisimple Lie group without compact factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I prove that irreducible lattices in G are rigid under two types of sublinear distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I show that if Λ ≤ G is a discrete subgroup that sublinearly covers a lattice, then Λ is itself a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I use this result to prove that the class of lattices in groups that do not admit R-rank 1 factors is SBE complete: if Λ is an abstract finitely generated group that is Sublinearly BiLipschitz Equivalent (SBE) to a lattice in G, then Λ can be homomorphically mapped into G with finite kernel and image a lattice in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This generalizes the well known quasi-isometric completeness of lattices in semisimple Lie groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 1 Introduction The quasi-isometric rigidity and classification of irreducible lattices in semisimple Lie groups was established in the 1990’s by the accumulated work of many authors - Pansu [48], Schwartz [52], Kleiner-Leeb [32], Eskin [22], Drut¸u [16], to name a few which are closely related to this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' See [23] for a concise survey of the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 (Quasi-Isometric Completeness, Theorem I in [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be a real finite-centre semisimple Lie group without compact factors, Γ ≤ G an irreducible lattice and Λ an abstract finitely generated group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If Γ and Λ are quasi-isometric, then there is a group homomorphism Φ : Λ → G with finite kernel whose image Λ′ := Φ(Λ) is a lattice in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Put differently, there is a lattice Λ′ ≤ G and a finite subgroup F ≤ Λ such that the sequence 1 → F → Λ → Λ′ → 1 is exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, Λ′ is uniform if and only if Γ is uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the standard terminology of metric rigidity, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 states that the classes of uniform and non- uniform lattices in a group G as in the statement are quasi-isometrically complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The main goal of the current work is to generalize this result to a sublinear setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Sublinear BiLipschitz Equivalences (SBE) are a sublinear generalization of quasi-isometries, brought forward by Cornulier [13] in the past decade or so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A function u : R≥0 → R≥1 is sublinear if limr→∞ u(r) r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For a, b ∈ R≥0 denote a ∨ b : max{a, b}, and for a pointed metric space (X, x0, dX) and x, x1, x2 ∈ X, let |x|X := dX(x, x0), |x1 − x2|X := dX(x1, x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let (X, dX, x0), (Y, dY , y0) be pointed metric spaces, L ∈ R>0 a constant, u : R≥0 → R≥1 a sublinear function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A map f : X → Y is an (L, u)-SBE if the following conditions are satisfied: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' f(x0) = y0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' ∀x1, x2 ∈ X, 1 L|x1 − x2|X − u � |x1|X ∨ |x2|X � ≤ |f(x1) − f(x2)|Y ≤ L|x1 − x2|X + u � |x1|X ∨ |x2|X � , 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' ∀y ∈ Y ∃x ∈ X such that |y − f(x)|Y ≤ u(|y|Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 1 A map is an SBE if it is an (L, u)-SBE for some L and u as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I prove SBE-completeness for irreducible lattices in groups without R-rank 1 factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3 (SBE-Completeness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be a real centre-free semisimple Lie group without compact or R-rank 1 factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let Γ ≤ G be an irreducible lattice and Λ an abstract finitely generated group, both considered as metric spaces with some word metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume there is an (L, u)-SBE f : Λ → Γ with u a subadditive sublinear function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then there is a group homomorphism Φ : Λ → G with finite kernel whose image Λ′ := Φ(Λ) is a lattice in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, Λ′ is uniform if and only if Γ is uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The main ingredients in the proof are of independent interest: the first is geometric rigidity for the corresponding symmetric space, stating that every self SBE of such a space is sublinearly close to an isom- etry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This generalizes Kleiner and Leeb’s result [32] on self quasi-isometries of symmetric spaces, as well as Eskin’s [22] and Drutu’s [16] results on self quasi-isometries of non-uniform lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For the definition of the ‘compact core’ of a lattice see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 (Sublinear Geometric Rigidity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let X be a symmetric space of noncompact type without R-rank 1 factors, Γ ≤ Isom(X) an irreducible lattice and X0 ⊂ X the compact core of Γ in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For any (L, u)-SBE map f : X0 → X0 there is a sublinear function v = v(L, u) and an isometry g : X → X such that d � q(x), g(x) � ≤ v(|x|) for all x ∈ X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3 actually requires a stronger version of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4, formulated in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Notice that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 both hold for uniform as well as non-uniform lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I remark that ‘generalized quasi-isometries’ already appeared in the context of geometric rigidity, as a technical tool in Eskin and Farb’s work [21] [22] on quasi-isometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Indeed much of their work is carried for maps which are even more general than SBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It seems however that their approach cannot yield a sublinear bound as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4, which is necessary for the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The second ingredient is a property I call sublinear rigidity, stating that a discrete subgroup Λ ≤ G which sublinearly covers a lattice is itself a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Sublinear rigidity holds for groups of any R-rank, and is the cornerstone of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Its proof contains the bulk of the original ideas that appear in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For a function u : R≥0 → R>0 and a subset Y ⊂ X, define the u-neighbourhood of Y to be Nu(Y ) := {x ∈ X | d(x, Y ) ≤ u(|x|)} A subset Y ⊂ X is said to sublinearly cover Z ⊂ X if Z ⊂ Nu(Y ) for some sublinear function u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For the definition of a Q-rank 1 lattice, see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6 (Sublinear Rigidity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be a real centre-free semisimple Lie group without compact factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let Γ ≤ G be an irreducible lattice, Λ ≤ G a discrete subgroup that sublinearly covers Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If Γ is of Q-rank 1, assume further that Λ is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then Λ is a lattice in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The notion of irreducibility in the non-standard context of a general (non-lattice) subgroup is explained Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 (see Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Sublinear neighbourhoods arise naturally in the presence of SBE maps: the essential difference between a quasi-isometry and an SBE is that ‘far away in the space’, the ‘additive’ error term of an SBE gets larger and larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' One is led to consider metric neighbourhoods that grow - sublinearly, yet unboundedly - with the distance to some (arbitrary) fixed base point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The hypothesis that u is sublinear is optimal in the sense that u could not be taken to be an arbitrary linear function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Indeed, the geometric meaning of Γ ⊂ Nu(Λ) is that for every element γ ∈ Γ, the ball BG � γ, u(|γ|) � ‘of sublinear radius about γ’ must intersect Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Observe that if f is the identity function f(r) = r, then by definition f(|g|) = f � d(g, eG) � = d(g, eG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular G lies in the f-neighbourhood of the trivial subgroup which is, after all, not a lattice in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For uniform lattices and for lattices with Kazhdan’s property (T) one can however relax the assumption of sublinearity: 2 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='7 (Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be a real centre-free semisimple Lie group without compact factors, Γ ≤ G an irreducible lattice, Λ ≤ G a discrete subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Fix ε > 0 and assume Γ ⊂ Nu(Λ) for the function u(r) = ε · r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If Γ is uniform and ε < 1, then Λ is a uniform lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If Γ has Kazhdan’s property (T) then there is a constant ε(G) such that if ε < ε(G) then Λ is a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6 generalizes the case where Γ lies in a bounded neighbourhood ND(Λ) for some D > 0, proved by Eskin and Schwartz in a slightly modified version (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the bounded case, the result is much stronger and states that Λ must be commensurable to Γ (except in groups locally isomorphic to SL2(R), see [52]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the sublinear setting I can only prove a limited commensurability result, which stems from a reduction to the bounded setting: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G, Γ and Λ be as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If Γ is uniform then so is Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If Γ is of Q-rank 1 then: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If Γ ̸⊂ ND(Λ) for any D > 0, then also Λ is of Q-rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If Γ ⊂ ND(Λ) for some D > 0 and in addition Γ sublinearly covers Λ, then Λ is commensurable to Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I stress that the case where Γ and Λ each sublinearly covers the other arises naturally in the context of SBE-completeness, see Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The most interesting case in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6 is when Γ is of Q-rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In that case, the proof is entirely geometric, relying on the following key proposition which might be of independent interest: Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the setting of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6, assume that Γ is a Q-rank 1 lattice which does not lie in any bounded neighbourhood of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then there exists a horosphere H based at the rational Tits building associated to Γ such that � Λ ∩ StabG(H) � x0 intersects H in a cocompact metric lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, the bounded horoball HB does not intersect the orbit Λ · x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 Outline of Proof My proofs rely and draw on the works on the quasi-isometric rigidity for non-uniform lattices, due to Schwartz [52] in R-rank 1, and to Drut¸u [16] and Eskin [22] independently in groups of R-rank greater than 1, often called higher rank groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The geometric proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6 for Q-rank 1 lattices is quite delicate and involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For this reason I give here a detailed sketch of the arguments and of the ideas one should have in mind when reading the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' What is written here is a good enough account if one wishes to understand the main ideas while avoiding the technical details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I end the section with a brief sketch of the proofs for SBE-completeness and for sublinear rigidity in the case of property (T) groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Strategy for Q-rank 1 Lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Denote dγ := d(γ, Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The novel case is when {dγ}γ∈Γ is unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The rationale for the proof comes from a conjecture of Margulis, recently proved in full generality by Benoist and Miquel ([5], see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Their result states that a discrete subgroup in a higher rank Lie group is a lattice as soon as it intersects a horospherical subgroup in a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This result could be seen as an algebraic converse to the geometric structure of a Q-rank 1 lattice, whose orbit in X intersects some parabolic horospheres in a cocompact (metric) lattice (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9 is the geometric analogue of the Benoist-Miquel criterion, and basically completes the proof in the higher R-rank case (some non-trivial translation work is needed, see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Also, it easily follows from Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9 that every Γ-conical limit point is also Λ-conical (Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proof for R-rank 1 groups is then a simple use of a criterion of Kapovich-Liu for geometrically finite groups ([30], see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I now give a detailed description of the proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 3 The ABC of Sublinear Constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Fix a point x0 ∈ X = G/K, identify Γ and Λ with Γ · x0 and Λ · x0 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Observe that by definition of dγ the interior of balls of the form B(γx0, dγ) does not intersect Λ · x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I call such balls (or general metric sets) Λ-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, these balls intersect Λ · x0 (only) in the bounding sphere: call such balls (sets) tangent to Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The Λ-free and, respectively, Γ-free regions in X are the main objects of interest in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6 is known in the case of bounded {dγ}γ∈Γ, it makes sense to think about large Λ-free regions as ‘problematic’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The state of mind of the proof relies on two easy observations that complete each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The sublinear constraint implies that dγn → ∞ forces |γn| → ∞, suggesting that ‘problematic’ Λ-free regions should appear only ‘far away’ in the space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' On the other hand Λ is a group, and being Λ-free is a Λ-invariant property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular any metric situation that can be described in terms of the Λ-orbit (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' B(γ, dγ) is a Λ-free ball tangent to Λ) can be translated back to x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This means that ‘problematic’ regions could actually be found near x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The moral of these observations can be formulated into a general principle that lies in the heart of the argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The sublinear constraint dγ ≤ u(|γ|) gives rise to many other constraints of ‘sublinear’ nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Each such constraint actually yields a uniform constraint inside any fixed bounded neighbourhood of x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since Λ and Γ are groups, many of these uniform bounds which are produced ‘near’ x0 turn out to be global bounds that depend only on the group and not on a specific orbit point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Put differently: the trick is to describe metric situations in terms of the Γ and Λ orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' One then uses the group invariance in order to move these metric situations around the space to a place where the sublinear constraint can be exploited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The Argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The above paragraph should more or less suffice the reader to produce a complete proof for uniform lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For non-uniform lattices, denote by λγ the closest Λ-orbit point to the point γ ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For H a cusp horosphere of Γ, let ΓH := {γ ∈ Γ | γx0 ∈ H}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The ultimate goal is to show that the metric lattice ΓH · x0 yields a metric lattice that is more or less {λγx0}γ∈ΓH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' One proceeds by the following steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Finding Λ-free horoballs (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1): The arbitrarily large Λ-free balls B(γn, u(|γn|)) are translated to x0, and the compactness of the unit tangent space at x0 yields a converging direction which is the base point at infinity of a Λ-free horoball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Translating by Λ, this yields Λ-free horoballs tangent to every Λ-orbit point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Controlling angles (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2): For every γ ∈ Γ with dγ uniformly large enough, one associates a point ξ at X(∞) such that ξ is the base point of a Λ-free horoball tangent to λγx0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The angle between the geodesics [λγx0, γx0] and [λγX0, ξ) is shown to be small as dγ grows large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This is used to show that arbitrarily large dγ give rise to arbitrarily deep Γ-orbit points inside Λ-free horoballs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A key step is Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='8, producing a Γ-free Euclidean cylinder between λγx0 and γx0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Λ-cocompact horospheres (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3): One uses uniform bounds near x0 to prove that every Λ-free horoball that is (almost) tangent to Λ must lie in a uniformly bounded neighbourhood of Λ · x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If dγ is large enough for some γ that lies on a horosphere HΓ of a cusp of Γ, then any γ′ ∈ ΓHΓ also admits large dγ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since the bounds from the previous steps only depend on dγ′, all λγ′ are forced to lie on the same horosphere HΛ parallel to HΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' One concludes that Λ · x0 intersects HΛ on the nose in a cocompact metric lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Property (T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Sublinear rigidity for groups with property (T) is established by the criterion that a discrete subgroup there is a lattice if and only if it has the same exponential growth rate as a lattice (Leuzinger [36]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It is quite straightforward that sublinear distortion cannot affect this growth rate (Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' SBE Rigidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The general scheme for SBE-completeness is parallel to the quasi-isometry case: each λ ∈ Λ naturally gives rise to an SBE X0 → X0 of the compact core of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Each self SBE is close to an isometry by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4, allowing to embed Λ as a discrete subgroup of isometries in G that sublinearly 4 covers Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proof for Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 heavily relies on Drut¸u’s argument for quasi-isometries [16], which uses the properties of the induced biLipschitz map on the asymptotic cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' As SBE also induce such a biLipschitz map - indeed that was a main motivation for Cornulier to study SBE [13] - it is possible to generally follow Drut¸u’s argument also in the SBE setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 Possible Improvements, Related and Future Work The proof suggests three natural improvements to the statement of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' One could probably relax the assumption of trivial centre and allow finite centre, if the same relaxation is applicable in Leuzinger’s work on property (T) groups [36] and in Prasad’s work [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular, the proofs for uniform lattices and for Q-rank 1 lattices hold also for groups G with finite centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The irreducibility of Λ may be derived directly from the irreducibility of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lastly, in view of the geometric characterization of Q-rank (see Corollary D in [35]), it is reasonable that the Q-rank of Λ should equal that of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There are problems that arise naturally from this work which seem to require new ideas: Question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be a real finite-centre semisimple Lie group without compact factors that admits a R- rank 1 factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Are the classes of uniform and non-uniform lattices of G SBE-complete?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular, is this true when G is of R-rank 1?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' See Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 below for a discussion on the case of R-rank 1 factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' SBE of R-rank 1 symmetric spaces is the main focus in Pallier’s work [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' He investigated the sublinearly large scale geometry of hyperbolic spaces, and proved that two R-rank 1 symmetric spaces that are SBE are homothetic, answering a question of Drut¸u (see Remarks 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='16 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='17 in [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Also in this context Pallier and Qing [47] recently showed that the sublinear Morse boundary is an SBE invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Another problem is to find a non-trivial example of the setting of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6: Question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be a real finite-centre semisimple Lie group without compact factors, Γ ≤ G a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Does there exist a finitely generated group that is SBE to Γ but not quasi-isometric to it?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Or, at least, not known to be quasi-isometric to one?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Does there exist Λ ≤ G discrete that ε-linearly covers Γ but which does not sublinearly cover it?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' On the side of the proof, it would be very interesting if the geometric ideas that prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6 for Q-rank 1 lattices could be applied to any Q-rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' While there are apparent places where the proof uses the unique geometry of Q-rank 1 lattices, most of the geometric arguments leading to Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9 seem to be susceptible to the higher Q-rank setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Such a generalization of the proof would definitely shed more light on the mysterious lattice arising from growth considerations in property (T) groups, and in particular on the question of commensurability of Λ and Γ in that case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It would also be interesting to see whether one can push the geometric argument forward in order to establish a complete geometric analogue of the Benoist-Miquel criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Namely, can one find a direct geometric proof that Λ admits finite co-volume (perhaps similarly to Schwartz’s argument in the bounded case, see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lastly, one could possibly relate this work to the work of Fraczyk and Gelander [24], who proved that a discrete subgroup (of a higher rank simple Lie group) is a lattice if and only if it has bounded injectivity radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' While their result seem very much related to the condition Γ ⊂ Nu(Λ), the nature of their work does not give explicit bounds on the injectivity radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Specifically, given r > 0 one cannot tell directly from their results how ‘far’ one must wander in X in order to find a point with injectivity radius r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Perhaps one could use the sublinear results of this work to say something about the relation of |x|X and InjRad(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3 Acknowledgments This paper is based on my DPhil thesis, supervised by Cornelia Drut¸u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I thank her for suggesting the question of SBE-completeness and for guiding me in my first steps in the theory of Lie groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I thank Uri Bader, Tsachik Gelander and the Midrasha on groups at the Weizmann institute, where I learned the basics 5 of symmetric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I thank my thesis examiners Emmanuel Breuillard and Yves Cornulier for their careful inspection and numerous remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I thank Elon Lindenstrauss for telling me about Leuzinger’s result [36] on property (T), and Or Landesberg, Omri Solan, Elyasheev Leibtag, Itamar Vigdorovich and Tal Cohen for many discussions on different aspects of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Finally I thank Gabriel Pallier for explaining his examples of some unusual SBE in R-rank 1, and for his interest in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2 Preliminaries For standard definitions and facts about fundamental domains, see Chapter 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The facts about fundamental domains for Q-rank 1 lattices appear in Raghunathan’s book [50] and in Prasad’s work on rigidity of Q-rank 1 lattices [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In notations and generalities I follow: Borel’s book on algebraic groups [6], Helgason’s books on Lie groups and symmetric spaces [27, 28], and Eberline’s book on the geometry of symmetric spaces of noncompact type [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 Generalities on Semisimple Lie Groups and their Lattices Let G be a real centre-free semisimple Lie group without compact factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A discrete subgroup Γ ≤ G is a lattice if Γ\\G carries a finite volume G-invariant measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Equivalently, Γ is a lattice if Γ\\X is a finite volume Riemannian manifold, where X = G/K is the symmetric space of noncompact type corresponding to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A lattice is irreducible if its projection to every simple factor of G is dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The group G can be viewed as an algebraic group via the adjoint representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If G is of R-rank greater than 1, then by the Margulis arithmeticity theorem every irreducible lattice of G is arithmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The Q-rank Γ is the Q-rank of the Q-structure associated to (G, Γ) be the arithmeticity theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A result of Prasad [49] states that if G admits a R-rank 1 factor, then a non-uniform irreducible lattice of G is of Q-rank 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The group G has Kazhdan’s property (T) if and only if it does not admit an SO(n, 1) or an SU(n, 1) factor, and an irreducible lattice Γ ≤ G has property (T) if and only if G has property (T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Together with Prasad’s result, I may conclude: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be a real centre-free semisimple Lie group without compact factors, and Γ ≤ G an irreducible lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then at least one of the following occurs: (a) G has property (T) (b) Γ is a non-uniform Q-rank 1 lattice (c) Γ is uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6 is therefore an immediate result of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 Cusps and the Rational Tits Building The facts about symmetric spaces of noncompact type can be found in Eberline’s book [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since the geometry of Q-rank 1 lattices resembles that of lattices in R-rank 1, the reader could for the most part simply have the image of the hyperbolic plane in mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If one wishes to see flats that are not geodesics, then a product of two hyperbolic planes is enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Even the product of the hyperbolic plane and R is helpful, albeit this space has a Euclidean factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 Basic Geometry of Symmetric Spaces of Noncompact Type Visual Boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The visual boundary X(∞) of X is the set of equivalence classes of geodesic rays, where two geodesic rays are equivalent if their Hausdorff distance is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For a ray η : [0, ∞) → X, η(∞) denotes the equivalence class of η in X(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There are two natural topologies on X(∞) that will be of use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The cone topology is the one given by viewing X(∞) as the set of all geodesic rays emanating from some fixed base point x0, with topology induced by the unit tangent space at x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There is a natural topology on X := X ∪ X(∞) such that X is the compactification of X and the induced topology on X(∞) is the cone topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A well known fact about geodesic rays in nonpositively curved spaces, stating that two ‘close’ geodesic rays fellow travel: 6 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Given time T and ε > 0, there is an angle α = α(T, ε) so that if η1, η2 are two geodesic rays with η1(0) = η2(0) = x for some x ∈ X and ∡x(η1, η2) ≤ α then dX � η1(t), η2(t) � < ε for all t ≤ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The Tits metric on X(∞) is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Given a totally geodesic submanifold Y ⊂ X, let Y (∞) ⊂ X(∞) be the subset of all points that admit a geodesic ray η lying inside Y (or, equivalently, those points that admit a ray lying at bounded Hausdorff distance to Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For any ξ1, ξ2 ∈ X(∞) there exists a flat F ⊂ X such that ξ1, ξ2 ∈ F(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Define dT (ξ1, ξ2) ∈ [0, π] to be the angle between two geodesic rays η1, η2 ⊂ F emanating from some point x ∈ F and with η1(∞) = ξ1, η2(∞) = ξ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This is a well defined metric on X(∞), called the Tits metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The pair � X(∞), dT � is a geodesic metric space, and isometries of X act on it by isometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I will use the following relation between the cone and the Tits topologies: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3 (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 in [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let X be a symmetric space of noncompact type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The Tits metric on X(∞) is semicontinuous with respect to the cone topology: for any ξ, ζ ∈ X(∞) and every ε > 0, there exists neighbourhoods of the cone topology U, V ⊂ X(∞) of ξ and ζ respectively such that for all ξ′ ∈ U, ζ′ ∈ V one has ∡(ξ′, ζ′) ≥ ∡(ξ, ζ) − ε Moreover, for any flat F ⊂ X, the cone topology and the Tits topology coincide on F(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Busemann Functions, Horoballs and Horospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Horoballs and horospheres play a crucial role in the proof, a role which stems from their role in the geometric description of the compact core of non-uniform lattices (see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='20 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A Busemann function on X is any function of the form fη(x) = lim t→∞ d � x, η(t) � − t for some geodesic ray η of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A horoball HB ⊂ X is an open sublevel set of a Busemann function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A horosphere H ⊂ X is a level set of a Busemann function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Two equivalent geodesic rays η1, η2 give rise to Busemann function which differ by a constant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' fη1 − fη2 = C for some C ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If HB is the sublevel set of fη, then η(∞) is called the base point of the horoball HB (and respectively of the horosphere H = ∂HB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The base point of a horoball is well defined, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' it depends only on η(∞) and not on η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For every choice of x ∈ X, ξ ∈ X(∞) there is a unique horosphere H based at ξ with x ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I denote this horosphere by H(x, ξ) and the bounded horoball HB(x, ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The following proposition collects some basic properties that will be of use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 (Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='5 in [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let x ∈ X, ξ ∈ X(∞), and let H = H(x, ξ), HB the horoball bounded by H and f the Busemann function based at ξ with f(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For any point y ∈ X, let η be the bi-infinite geodesic determined by the geodesic [y, ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then PH(y) = η ∩ H, where PH(y) is the unique point closest to y on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For any point y ∈ X, f(y) = ±d � y, PH(y) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, f(y) is negative if and only if y ∈ HB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If x′ ∈ X, then the horospheres H = H(x, ξ) and H′ = H(x′, ξ) are equidistant: if y ∈ H, y′ ∈ H′, then d(y, H′) = d(y′, H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Such horospheres are called parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A Busemann function fη thus naturally determines a filtration of X by the co-dimension 1 manifolds {Ht}t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By convention I usually assume that Ht := {x ∈ X | fη(x) = −t}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The stabilizer of a point ξ ∈ X(∞) acts transitively on the set of horospheres based at ξ, so every two such horospheres are isometric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, there is a close relation between the induced metrics on horospheres with the same base point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Briefly, if dH denotes the induced distance on a horosphere H ⊂ X, then dH � PH(x), PH(y) � for any two points x, y ∈ H′ can be bounded uniformly below and above as a function of the distance dH′(x, y) and the curvature bounds on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' See Heintze-Im hof [26] for precise statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 7 Using the above properties one can show that two horospheres are parallel if and only if they are based at the same point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular for every x ∈ X, ξ ∈ X(∞) it holds that StabG � H(x, ξ) � ⊂ StabG(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In addition, If H, H′ are two parallel horospheres based at the same ξ ∈ X(∞) and A ⊂ H is a cocompact metric lattice in H, then πH′(A) is a cocompact metric lattice in H′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For a point ξ ∈ X(∞) and a flat F with ξ ∈ F(∞), one readily observes that every horoball HB based at ξ intersects F in a Euclidean half space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular for every ζ ∈ X(∞) with dT (ξ, ζ) < π 2 , for every geodesic ray η with η(∞) = ζ and every horoball HB based at ξ there is some T for which η↾t>T ⊂ HB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Parabolic and Horospherical Subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The isometries of X are classified into elliptic, hyperbolic, and parabolic isometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Most significant for this paper are the parabolic isometries, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' those g ∈ G whose displacement function x �→ gx does not attain a minimum in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Every such isometry fixes (at least) one point in X(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A group P ≤ Isom(X) is called geometrically parabolic if it is of the form Gξ := StabG(ξ) for some ξ ∈ X(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Such groups act transitively on X, and in particular act transitively on the set of geodesic rays in the equivalence class of ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The same holds also for the identity component G◦ ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' An element g ∈ Gξ acts by permutation on the set of horoballs based at ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This permutation is a translation with respect to the filtration of the space X by horospheres based at ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Put differently, if {Ht}t∈R is a filtration of X by horospheres based at ξ, then for every g ∈ Gξ there is l(g) ∈ R such that gHt = Ht+l(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It is quite clear from all of the above that for every horosphere based on ξ, the group GH := StabG(H) acts transitively on H, and the same holds for G◦ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I now present a fundamental structure theorem for geometrically parabolic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Denote g := Lie(G), and let g = k ⊕ p be a Cartan decomposition defined using the maximal compact subgroup K ≤ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Recall that the Lie exponential map exp : g → G gives rise to a family of 1-parameter subgroups of the form exp(tX) for each X ∈ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6 (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3 in [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let x ∈ X(∞), and let X ∈ p be the tangent vector of the unit speed geodesic [x0, ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let ht ξ be the 1-parameter subgroup defined by t �→ exp(tX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then an element g ∈ G fixes ξ if and only if limt→∞ h−t ξ ght ξ exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='7 (Langlands Decomposition, Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='5 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='25 in [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let ξ ∈ X(∞) and ht ξ as in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let F be a flat containing [x0, ξ) and A ≤ G the maximal abelian subgroup such that Ax0 = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Denote Gξ := StabG(ξ), and define Tξ : Gξ → G by g �→ limn→∞ h−n ξ ghn ξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then Tξ is a homomorphism, and there are subgroups Nξ, Aξ, Kξ ≤ Gξ such that: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Aξ = exp � Z(X) ∩ p � , where Z(X) is the centralizer of X in g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, every element a ∈ Aξ lies in some conjugate Ag = gAg−1 with the property that [x0, ξ) ⊂ F g := Agx0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Kξ ≤ K = StabG(x0) is the compact subgroup fixing the bi-infinite geodesic determined by [x0, ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' KξAξ = AξKξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Nξ = Ker(Tξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It is a connected normal subgroup of Gξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Gξ = NξAξKξ, and the indicated decomposition of an element is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' G = NξAξK, and the indicated decomposition of an element is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In case ξ is a regular point at X(∞), this decomposition is the Iwasawa decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Gξ has finitely many connected components, and G◦ ξ = (KξAξ)◦Nξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Gξ is self normalizing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Viewing G as an algebraic group, the geometrically parabolic subgroups are exactly the (algebraically) non-trivial parabolic subgroups, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' proper subgroups of G that contain a normalizer of a maximal unipotent subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='7 is a geometric formulation of the algebraic Langlands decomposition of parabolic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Recall that a horospherical subgroup is the unipotent radical of a non-trivial parabolic group, or equivalently groups of the form Ug := {u ∈ G | limn→∞ g−nugn = idG}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The latter implies that Nξ a horospherical subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 8 Limit Set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' An important set associated to a discrete group ∆ ≤ G acting by isometries on X is the limit set L∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By definition L∆ := ∆ · x ∩ X(∞), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' it is the intersection with X(∞) of the closure, in the compactification X = X ∪ X(∞), of an orbit ∆ · x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It is clear that L∆ does not depend on the choice of x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The limit set of any lattice is always the entire X(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In fact, much more is true: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let ∆ ≤ G = Isom(X), and SX the unit tangent bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A vector v ∈ SX is ∆-periodic if there is δ ∈ ∆ and s > 0 such that δη(t) = η(t + s) for all t ∈ R, where η is the bi-infinite geodesic determined by the vector v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For a flat F ⊂ X (including geodesics), denote ∆F := {δ | δF = F}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The flat F is called a ∆-periodic flat if there exists a compact set C ⊂ F such that ∆F C = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9 (Propositions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=', 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='5, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='7 in [20], Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3′ in [41]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If Γ ≤ G = Isom(X) is a lattice, then: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The subset in SX of Γ-periodic vectors is dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let F ⊂ X be any flat, η any bi-infinite geodesic in F, and denote v = ˙η(0) ∈ SX the initial velocity vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There is a sequence vn ∈ SX of regular vectors such that (a) limn→∞ vn = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' (b) The bi-infinite geodesics ηn determined by vn are all Γ-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' (c) Denote by Fn the (unique) flat containing ηn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Each Fn is Γ-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Put differently, the set of Γ-periodic flats is dense in the set of flats of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' See Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='25 for the notion of regular tangent vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Points in the limit set of a group are classified according to how the orbit approaches them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let ∆ ≤ G be a discrete subgroup, ξ ∈ L∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The point ξ is called conical if for some (hence any) x and some (hence every) geodesic ray η with η(∞) = ξ there is a number D = D(x, η) such that for every T ∈ R>0 there is t > T for which B � η(t), D � ∩ ∆ · x ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since ∆ is discrete, this is equivalent to ∆ · x ∩ ND(η) being infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The point ξ is called horospherical if for every horoball HB based at ξ and every x ∈ X, ∆ · x ∩ HB is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular, a conical limit point is horospherical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The point ξ is non-horospherical if it is not horospherical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' As a corollary of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9, one has: Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If Γ ≤ Isom(X) is a lattice, then 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The set of Γ-conical limit points is dense in X(∞) with the cone topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' LΓ = X(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I finish this section with some results on geometrically finite subgroups of isometries in R-rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let X be a R-rank 1 symmetric space and ∆ ≤ Isom(X) a discrete subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Denote by Hull(∆) the closed convex hull in X = X ∪X(∞) of the limit set L∆, and Hull(∆) = X ∩Hull(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By virtue of negative curvature, Hull(∆) is the union of all geodesics η such that η(∞), η(−∞) ∈ L∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The convex core of ∆ is defined to be ∆\\Hull(∆) ⊂ ∆\\X, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=', the quotient of Hull(∆) by the ∆-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='14 (Bowditch [9], see Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 in [30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let X be a R-rank 1 symmetric space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' a symmetric space of pinched negative curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A discrete group ∆ ≤ G = Isom(X) is geometrically finite if for some δ > 0, the uniform δ-neighbourhood in ∆\\X of the convex core Nδ � ∆\\Hull(∆) � , has finite volume and there is a bound on the orders of finite subgroups of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A group is geometrically infinite if it is not geometrically finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 9 Immediately from the definition of geometrical finiteness, one gets a simple criterion for a subgroup to be a lattice: Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let X be a R-rank 1 symmetric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If ∆ ≤ Isom(X) is geometrically finite and admits L∆ = X(∞), then ∆ is a lattice in Isom(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Sublinear distortion does not effect the limit set, as the following lemma shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let Γ, Λ ≤ G be discrete subgroups and u : R≥0 → R>0 a sublinear function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If Γ ⊂ Nu(Λ), then LΓ ⊂ LΛ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' every Γ-limit point is a Λ-limit point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular, if Γ is a lattice then LΛ = X(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By definition, one has to show that given a point ξ ∈ X(∞) and a sequence γn ∈ Γ such that γnx0 → ξ (in the cone topology on X), there is a corresponding sequence λn ∈ Λ with λnx0 → ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Define λn := λγn to be the closest point to γn in Λ, and to ease notation denote xn := γnx0, x′ n = λnx0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let also ηn := [x0, xn] and η′ n := [x0, x′ n] be unit speed geodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Finally, let Tn denote the time in which ηn terminates, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' ηn(Tn) = xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Convergence in the cone topology xn → ξ is equivalent to the fact that the geodesics ηn converge to η := [x0, ξ) uniformly on compact sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This means in particular that Tn → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Non-positive curvature guarantees that the functions Fn(t) := d � η(t), ηn(t) � , F ′ n(t) := d � η(t), η′ n(t) � and Gn := d � ηn(t), η′ n(t) � are convex (F ′ n is just a notation, completely unrelated to the derivative of Fn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since Gn(0) = Fn(0) = F ′ n(0) = 0 any of these functions is either constant 0 or monotonically increasing, so proving uniform convergence of η′ n to η amounts to proving limn F ′ n(T ) = 0 for every T ∈ R≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Triangle inequality gives F ′ n(T ) ≤ Fn(T ) + Gn(T ), and by assumption limn Fn(T ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Notice that Gn(Tn) = d(xn, x′ n) ≤ u(|γn|) = u(Tn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Writing T = T Tn · Tn, convexity of Gn implies Gn(T ) ≤ (1 − T Tn )Gn(0) + T Tn Gn(Tn) ≤ 0 + T Tn u(Tn) = T · u(Tn) Tn As limn Tn = ∞ it follows from sublinearity that limn Gn(T ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I conclude that η′ n converge to η uniformly on compact sets, therefore ξ lies in the limit set of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 I prove that in the setting of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='12, the set of Λ-conical limit points contains the set of Γ-conical limit points (Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It holds that the conical limit points of Γ are dense in X(∞) (in the cone topology, see Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='12) and therefore LΛ = LΓ = X(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular, every Γ-limit point is a Λ-limit point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The strength of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='16 is that it does not assume anything on Γ other than that it is sublinearly covered by Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='16 does not require Γ to be a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 Cusps, Compact Core, and the Rational Tits Building In this section I present some of the structure theory of non-compact quotients of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The focus is on the structure of ‘cusps’ in noncompact finite volume quotients of symmetric spaces, and the ‘location’ of cusps on the visual boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Cusps and Compact Core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Consider V = Γ\\X, for Γ ≤ G a non-uniform lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This is a locally symmetric space of finite volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The term ‘cusps’ is an informal name given to those areas in a locally symmetric space through which one can ‘escape to infinity’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Another description is that cusps are the ends of the complement of a large enough compact set in V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In strictly negative curvature, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' in R-rank 1 locally symmetric spaces, these cusps have a precise description as submanifolds of the form C × R≥0 for a compact manifold C, and metrically (C, t) gets narrower as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There are finitely many cusps, each corresponding to a point at X(∞) called a ‘parabolic point’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Introduction in [3] or [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A fundamental feature of the cusps is that one can ‘chop’ them out of the quotient manifold V and get a compact manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This could be done in such a way so that: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The lifts of the chopped parts to the universal cover X are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Each cusp is covered in X by the Γ-orbit of a horoball, that is, the lift of a cusp is the Γ-orbit of a horoball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The respective base points are called parabolic points of Γ in X(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Γ acts on X \\ � � i∈I HBi � cocompactly, where {HBi}i∈I is the set of horoballs coming from the lifts of cusps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since there are only finitely many cusps and Γ discrete, there are exactly countably many such horoballs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' See for example Section 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6 in [17] where this is illustrated in the case of the real hyperbolic spaces Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Formally, one has;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='18 (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 in [19], see also Introduction therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume X is of R-rank 1, and Γ ≤ G a non-uniform lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The space V = Γ\\X has only finitely many (topological) ends and each end is parabolic and Riemannian collared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular, each cusp is a quotient of a horoball HB based at a parabolic limit point ξ such that Γ ∩ Gξ acts cocompactly on H = ∂HB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For symmetric spaces of higher rank, a similar construction is available (see [37]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By removing a countable family of horoballs from X, one obtains a subspace on which Γ acts cocompactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There are two main differences from the situation in R-rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' One is that an orbit map γ �→ γx is a quasi-isometric embedding of Γ (with the word metric) into X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='19 (Lubozki-Mozes-Raghunathan, Theorem A in [38]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be a semisimple Lie group of higher R-rank, dG a left invariant metric induced from some Riemannian metric on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let Γ an irreducible lattice, dΓ the corresponding word metric on Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then dG↾Γ×Γ and dΓ are Lipschitz equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This result plays a significant preliminary role in the proofs of quasi-isometric rigidity for non-uniform lattices in higher rank symmetric spaces in both [16] and [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The second difference in higher rank spaces is that the horoballs could not in general be taken to be disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' However, in the special case of Q-rank 1 lattices the horoballs can be taken to be disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Recall that the Q-structure of (G, Γ) is a Q-structure on G = G(R) in which Γ is an arithmetic lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The following theorem sums up the relevant properties for Q-rank 1 lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='20 (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 in [34], see also Remarks 3 and 4 in [37], Section 13 in [50], and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 in [49]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume X is of higher rank, and Γ ≤ G an irreducible torsion-free non-uniform lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' On the locally symmetric space V = Γ\\X there exists a continuous and piece-wise real analytic exhaustion function h : V → [0, ∞) such that, for any s > 0, the sublevel set V(s) := {h < s} is a compact submanifold with corners of V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover the boundary of V(s), which is a level set of h, consists of projections of subsets of horospheres in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The Γ-action on the above set of horospheres has finitely many orbits, and the following conditions are equivalent: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The corresponding horoballs bounded by these horospheres can be taken to be disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For each such horosphere H the action of Γ ∩ StabG(H) on H is cocompact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The Q-structure of (G, Γ) is of Q-rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The lattice Γ is of Q-rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the setting of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='18 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='20, the horoballs and horospheres that appear in the statement are called (global) horoballs (horospheres) of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Base points of these are called parabolic limit points of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The above geometric characterization of Q-rank 1 lattices is all that I use in order to prove the key Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The compact core of Γ is the complement in X of the horoballs of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The group Γ acts on it cocompactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The following corollary describes the orbit of Q-rank 1 lattices in X, and especially some finiteness properties which I will use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 11 Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let Γ ≤ G be a Q-rank 1 lattice, x ∈ X, and ξ a parabolic limit point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There is a unique horosphere H based at ξ such that both following conditions hold: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Γ · x ∩ H is a cocompact metric lattice in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Γ · x ∩ HB = ∅, where HB is the horoball bounded by H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Call H an x-horosphere of Γ at ξ, and the corresponding bounded horoball an x-horoball of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, one has: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For every C there exists a bound K = K(x, C) so that B(x, C) intersects at most K x-horospheres of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There is D = D(x) > 0 such that H ⊂ ND(Γ · x ∩ H) for any x-horosphere H of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The constant D is called the compactness number of (Γ, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There is a number N = N(Γ) such that every point x ∈ X admits exactly N x-horospheres of Γ that intersect x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' These are called the horospheres of (Γ, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since Γ is a Q-rank 1 lattice, the stabilizer in Γ of a parabolic limit point ξ ∈ X(∞) acts cocompactly on each horosphere based at ξ, and in particular on Hx := H(x, ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let D(x, H) be such that Hx ⊂ ND(Γ · x ∩ Hx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A priori D(x) depends on H, but the fact that Γ acts by isometries implies that for every horosphere of the form H′ := γH = H(γx, γξ) for some γ ∈ Γ, one has H′ ⊂ ND(Γ · x ∩ H′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' So D(x) depends only on the Γ-orbit of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since Γ acts on the set of parabolic limit points with finitely many orbits (that is to say Γ\\X has finitely many cusps) one may take D = D(x) to be the maximum of the respective bounds on each orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This gives the required compactness number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Thinking of horoballs of γ as lifts of ends of the complement of some compact subset of V = Γ\\X, one sees that there is some horoball HB based at ξ so that Γ · x /∈ HB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Denote H = ∂HB, and y = PHB(x) ∈ H be the projection on the closed convex set that is the closure of the horoball HB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Finally, Let η = [x, y] and denote l = d(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The existence of the required horosphere is equivalent to the fact that the following non-empty set admits a maximum: Hx,ξ := {t ∈ [0, l] | Γ · x ∩ H � η(t), ξ � ̸= ∅} Indeed 0 ∈ Hx,ξ and it is a bounded set, so it admits a supremum T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, this set is discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If t were an accumulation point, then for any small ε > 0 the geodesic segment η↾(t−ε,t+ε) would intersect D-cocompactly infinitely many horospheres of (Γ, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Orbit points on different horospheres are in particular different points, therefore the set B � η(t), D + ε � ∩ Γ · x would be infinite, contradicting discreteness of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I conclude that T is a maximum, and that H � η(T ), ξ � is the unique desired horosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The argument above generally shows that there cannot be an accumulation point in X of x-horospheres of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular, for every C, the ball B(x, C) intersects only finitely many x-horospheres of Γ, say K(C), proving item 1 in the ‘moreover’ statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For the last statement, simply note that the horoballs of Γ are the Γ-translates of finitely many horoballs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the terminology of the statement, infinitely many horospheres of (Γ, x) imply that infinitely many of them are in the same Γ-orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Suppose these are {Hn}n∈N, with base points ξn that are evidently pairwise different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Finally let γn ∈ Γ for which γnH1 = Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since Γ ∩ StabG(Hn) acts cocompactly on Hn, there is γ′ n ∈ Γ ∩ StabG(Hn) that maps γnx to B(x, D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Discreteness of Γ implies that the set γ′ nγnx is finite, hence infinitely many of these points are the same point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' However γ′ nγnξ1 = ξn and therefore γ′ nγn ̸= IdX, contradicting the fact that Γ is torsion-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For the most part, I am interested in a fixed base point x0 and the x0-horospheres and horoballs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By a slight abuse of terminology I omit x0 and call these objects ‘horospheres of Γ’ and ‘horoballs of Γ’, respectively, denoting the associated cocompactness number DΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='22 allows to upgrade a metric lattice of H to a metric lattice coming from StabG(H) only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 12 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume that a torsion-free discrete group ∆ ≤ G = Isom(X) admits the following: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There is a bound N such that at each point x ∈ ∆ · x0 there are at most N horoballs that are tangent to x and do not intersect ∆ · x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There is a horosphere H ⊂ X such that (a) The set ∆ · x0 ∩ H is a cocompact metric lattice in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' (b) The horoball HB bounded by H is ∆-free, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' ∆ · x0 ∩ HB ⊂ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then � ∆ ∩ StabG(H) � x0 is also a cocompact metric lattice in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This is the Pigeonhole Principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Fix x ∈ ∆ · x0 ∩ H, and let {HBi}i∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=',N} be the finite set of horoballs tangent to x that do not intersect ∆ · x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='g that H is the bounding horosphere of HB1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Fix a ∆-orbit point δ0x0 ∈ ∆ · x0 ∩ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Up to translating by some element of ∆, I may assume x = x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For any other ∆-orbit point δx0 ∈ ∆ · x0 ∩ H let i(δ) ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' , N} be the index of the horoball δ−1HB1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Notice that from hypothesis 1 it indeed follows that δ−1HB1 ∈ {HBi}i∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=',N}, because the action of ∆ is by isometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Define δi to be the element in ∆ for which: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' i(δi) = i, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' δ−1 i (HB1) = HBi 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' d(δix0, x0) is minimal among all such δ ∈ ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For some i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' , N} there is a δ ∈ ∆ with i = i(δ), while for others there might not be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I will only care about those i for which there is such δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='g that these are i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' M} for M ≤ N, and let L := max1≤i≤M{d(x0, δix0)} < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For such an index i0, the ∆-orbit points that share the same i(δ) = i0 are in the same ∆ ∩ StabG(H) orbit, namely {δx0 | i(δ) = i0} ⊂ � ∆ ∩ StabG(H) � δi0x0 Indeed, if δ−1HB1 = HBi0 then by definition δδ−1 i0 HB1 = δHBi0 = HB1, hence δδ−1 i0 ∈ StabG(H) is an element mapping δi0x0 to δx0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let now δx0 ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Its distance from the orbit � ∆ ∩ StabG(H) � x0 is � ∆ ∩ StabG(H) � invariant, therefore d � δx0, � ∆ ∩ StabG(H) � x0 � = d � δi(δ)x0, � ∆ ∩ StabG(H) � x0 � ≤ d(δi(δ), x0) The right-hand side is uniformly bounded by L, proving that (∆ · x0 ∩ H) ⊂ NL �� ∆ ∩ StabG(H) � x0 � The fact that ∆ · x0 ∩ H is a cocompact metric lattice in H renders (∆ ∩ StabG(H) � x0 a cocompact metric lattice in h as well, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Real and Rational Tits Buildings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The location of the parabolic points in X(∞) also plays an important role in the geometry of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In case Γ is an arithmetic lattice, the natural framework to consider these points is the so called rational Tits building.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This is a building structure on the subset of parabolic points at X(∞), sometimes referred to as ‘rational points’ in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' They are exactly those points in X(∞) whose stabilizers are Q-defined (algebraic) parabolic groups of G (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I present this object, denoted WQ(Γ), together with the more familiar real Tits building structure on X(∞) with the Tits metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The main goal is to present the results of Hattori [25], that give a good description of the rational Tits building in terms of conical and horospherical limit points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In case G is of R-rank 1, by WQ(Γ) I mean the (countable) set of parabolic limit points of Γ (so that X(∞) \\ WQ(Γ) is comprised of conical limit points only, see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 13 Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A geodesic η is said to be regular if it is contained in a unique maximal flat F ⊂ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The point η(∞) ∈ X(∞) is called a regular point of X(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A point ξ ∈ X(∞) is singular if it is not regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Regularity does not depend on the choice of representative geodesic ray η for ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A Weyl chamber of X(∞), or an open spherical chamber, is any connected component in the Tits topology of X(∞) \\ S, where S ⊂ X(∞) is the subset of singular points at X(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='26 (Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 in [29] and Section 8 in [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The Weyl chambers induce a simplicial complex structure on X(∞) that is a spherical Tits building.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The apartments of the building are exactly the sets of the form F(∞) ⊂ X(∞) for all flats F ⊂ X, and the chambers are exactly the Weyl chambers at X(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, the Tits metric completely determines the building structure, and vice versa, and � X(∞), dT � is a metric realization of the Tits building at X(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' None of the rich theory of buildings is used directly in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Given a non-uniform lattice of Γ ≤ G the rational Tits building WQ(Γ) is a building structure on the subset of parabolic points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It is not in general a sub-building of the real spherical building.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Flats of X correspond to real maximal split tori in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since G is an algebraic group defined over Q, one can consider the maximal Q-split tori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The rational flats of X are then the G(Q)-orbits of maximal Q-split tori of G, and the rational boundary are all points ξ ∈ X(∞) such that ξ ∈ FQ(∞) for some rational flat FQ(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' One defines regular rational directions and rational Weyl chambers in an analogous way to the real case, this time taking only rational flats into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For further details details see [29], and Section 2 in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='27 (Theorem A in [25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let X = G/K be a symmetric space of non-compact type and of higher rank, and let Γ ≤ Isom(X) be an irreducible non-uniform lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then WQ(Γ) does not include horospherical limit points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The π 2 -neighbourhood N π 2 � WQ(Γ) � := {ξ ∈ X(∞) | dT � ξ, WQ(Γ) � < π 2 } does not include conical limit points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In Q-rank 1 , the converse statement also holds: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='28 (Theorem B in [25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let X = G/K be a symmetric space of non-compact type of higher rank and Γ ≤ Isom(X) be an irreducible non-uniform lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let V = {ξ ∈ X(∞) | dT � ξ, WQ(Γ) � ≥ π 2 } Suppose that Γ is of Q-rank 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then V consists of conical limit points only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In groups of R-rank 1 one has the following well known fact: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='29 (Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 and Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 in [10], see also Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='29 in [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let X be a symmetric space of noncompact type and of R-rank 1, Γ ≤ Isom(X) a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then every ξ ∈ X(∞) is either conical or non-horospherical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' When Γ is of Q-rank 1 , the following holds: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' WQ(Γ) = {ξ ∈ X(∞) | N π 2 (ξ) does not contain conical limit points} 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Any two points ξ, ξ′ ∈ WQ(Γ) are at Tits distance = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In view of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='29, for R-rank 1 both statements hold trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In higher rank, both follow from the following observation: for any point ξ′ ∈ WQ(Γ) and any point ζ ∈ X(∞) with d(ζ, ξ′) = π 2 , ζ is conical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' To see this notice that ζ lies on the boundary of a horosphere based at ξ′: take a flat F ⊂ X with ξ′, ζ ∈ F(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Any geodesic with limit ζ is contained in (a Euclidean) horosphere based at ξ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The fact that Γ is cocompact on the horospheres based at WQ(Γ) implies that ζ is conical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 14 The second item of the corollary follows: let ξ, ξ′ ∈ WQ(Γ) and c : [0, α] a Tits geodesic joining them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There is a flat F ⊂ X containing both ξ, ξ′ as well as c ⊂ F(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Every point ζ that is at Tits distance π 2 from either ξ or ξ′ is conical, and no point inside the π 2 neighbourhood of either ξ or ξ′ is conical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In F(∞) the Tits metric is the same as the Tits metric on the Euclidean space of an equal rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Therefore one may prolong the geodesic c so that c(0) = ξ, c(α) = ξ′ and c(π) = ξ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If dT (ξ, ξ′) < π, then there is a point along this prolonged geodesic that is at Tits distance exactly π 2 from ξ (so it is conical by the first paragraph), but at Tits distance strictly less than π 2 from ξ′ (so it cannot be conical by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Therefore dT (ξ, ξ′) = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For the first item, one containment is just Hattori’s Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For the other containment, pretty much the same argument from above works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume for some ξ ∈ X(∞) that N π 2 (ξ) consists of non-conical limit points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular ξ itself is not conical, and by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='28 it holds that d(ξ, WQ(Γ)) < π 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let ξ′ ∈ WQ(Γ) be a point realizing this distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' As above, this gives rise to a flat F containing ξ, ξ′ and another point ζ that is at Tits distance π 2 from ξ′ but at Tits distance strictly less than π 2 from ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The first forces ζ to be conical, and the latter forces it to be non-conical, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Hattori’s characterization relies on a simple lemma which will also be of use in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It relates the (linear) penetration rate of a geodesic into a horoball to the Tits distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='31 (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 in [25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let X be a symmetric space of higher rank and of noncompact type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let η1, η2 : [0, ∞) → X be two geodesic rays, α := dT � η1(∞), η2(∞) � and b2 the Busemann function corresponding to η2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then there exists a positive constant C1, depending only on η1 and η2, such that: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If α > π 2 , then for all t ≥ 0 b2 � η1(t) � ≥ −t · cos α − C1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If α = π 2 , then b2 � η1(t) � is monotone non-increasing in t and −C1 ≤ b2 � η1(t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If α < π 2 , then for all t ≥ 0 b2 � η1(t) � ≤ −t · cos α − C1 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If X is a symmetric spaces of R-rank 1, maximal flats are geodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Therefore every two points ξ, ζ ∈ X(∞) admit dT (ξ, ζ) = π, and it is clear from the strict negative curvature that Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='31 is true also in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 3 Uniform Lattices In this section I prove: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be a semisimple Lie group without compact factors and with finite centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let Γ ≤ G be a lattice, Λ ≤ G a discrete subgroup such that Γ ⊂ Nu(Λ) for some sublinear function u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If Γ is uniform, then Λ is a uniform lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The focus of this paper is on sublinear distortion, however for uniform lattices (and also for lattices that have property (T), see Section 5), a slightly stronger result holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I call this ε-linear rigidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let f, g : R≥0 → R>0 be two monotonically increasing functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Call f asymptotically smaller than g if lim sup f g ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Denote this relation by f ⪯∞ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the setting of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1, the conclusion holds also under the relaxed assumption that u(r) ⪯∞ εr for any 0 < ε < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Clearly Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3 implies Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' From now and until the end of this section, the standing assumptions are those of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 15 Lattice Criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A discrete group is a uniform lattice if and only if it admits a relatively compact fundamental domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The criterion I use is the immediate consequence that if Γ is uniform and u is bounded (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Γ ⊂ ND(Λ) for some D > 0), then Λ is a uniform lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Outline of Proof and Use of ε-Linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The goal is to show that the ε-linearity of u forces Γ ⊂ ND(Λ) for some D > 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' that Γ actually lies inside a bounded neighbourhood of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proof is by way of contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If there is no such D > 0 then there are arbitrarily large balls that do not intersect Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proof goes by finding such large Λ-free balls that are all tangent to some fixed arbitrary point x ∈ X (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The ε-linearity then gives rise to concentric Γ-free balls that are arbitrarily large, contradicting the fact that Γ is a uniform lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The main difference from the non-uniform case is that for a non-uniform lattice Γ, the space X does admit arbitrarily large Γ-free balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This situation requires different lattice criteria and much extra work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Still the proof for the uniform case, though essentially no more than a few lines, lies the foundations for and presents the logic of the much more involved case of Q-rank 1 lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 Notations and Terminology Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let X be a metric space, Y, Z ⊂ X two closed subsets of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The closest point projection of Y to Z is the set theoretic map pZ : Y → Z defined by pZ(y) := zy, where zy ∈ Z is any point realizing the distance d(y, Z) = d(y, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If X is a proper metric space and Z discrete, then there are at most finitely many such points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In any case of multiple points, pZ chooses one arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The particular case of interest from now on is where the metric space is the pointed symmetric space (X, x0), the two subsets are the orbits Γ · x0 and Λ · x0, and the projection is pΛ·x0 : Γ · x0 → Λ · x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' To ease notation I often denote this projection by pΛ (there is no risk of ambiguity since the subgroups Λ and Γ are always considered in the context of their respective orbits in X and not in G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The following definitions will be used repeatedly in both this section and in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It mainly fixes terminology and notation of the geometric situation illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let H ≤ G = Isom(X)◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A set U ⊂ X is called H-free if H · x0 ∩ Int(U) = ∅, where Int(U) is the topological interior of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' That is, U is called H-free if its interior does not intersect the H-orbit H ·x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Denote PΛ(γx0) = PΛ(γ) = λγx0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' dγ := d(γx0, λγx0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Bγ := B(γx0, dγ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It is a Λ-free ball centred at γx0 and tangent to λγx0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' x′ γ := λ−1 γ γx0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Notice |x′ γ| = dγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' B′ γ := λ−1 γ Bγ = B(x′ γ, dγ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It is Λ-free as a Λ-translate of the Λ-free ball Bγ, and is tangent to x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For s ∈ R>0 and a ball B = B(x, r), denote sB := B(x, sr), the rescaled ball with same centre and radius sr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For a sequence γn, denote by λn, dn, Bn, B′ n, x′ n the respective λγn, dγn, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There exists S = S(x, u) ∈ (0, 1) such that for every s ∈ (0, S) there is R = R(s, S) such that if r > R and B = B(y, r) is a Λ-free ball tangent to x, then sB is Γ-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular, the existence of arbitrarily large Λ-free balls that are all tangent to a fixed point x ∈ X implies the existence of arbitrarily large Γ-free balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 16 x0 x′ γ = dγ = s · dγ Γ − free Λ − free γx0 = dγ Λ − free λγx0 Lλ−1 γ Figure 1: Basic Setting and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A Λ-free ball about γx0 of radius dγ, translated by λ−1 γ to a ball tangent to x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The linear ratio between |x′ γ| = dγ and the Λ-free radius forces the red ball to be Γ-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There is a slightly stronger version of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='8 if u is sublinear: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G, Γ, Λ and u be as in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 (in particular, u is a sublinear function and Γ ⊂ Nu(Γ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For every x ∈ X and every s ∈ (0, 1) there exists R = R(x, s) > 0 such that for every r > R, if B = B(y, r) is a Λ-free ball tangent to x then sB is Γ-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I omit the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9, which is a slightly simpler version of the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proof is more easily read if one assumes x = x0 and u(r) = εr so I begin with this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume B = B(y, R) is Λ-free for some y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The assumption x = x0 means that |y| = d(y, x0) = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume that for a given s ∈ (0, 1), the ball sB intersects Γ · x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This gives rise to an element γ ∈ Γ such that: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' |γ| = d(γx0, x0) ≤ d(y, x0) + d(γx0, y) = (1 + s)r (triangle inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular, d(γx0, Λ · x0) ≤ ε(1 + s)r 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' B � γx0, (1 − s)r � ⊂ B(y, r), so it is Λ-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I conclude that for s for which sB ∩ Γ · x0 ̸= ∅, one has the inequality (1 − s)r ≤ ε(1 + s)r, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 1−s 1+s ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The number ε is fixed and smaller than 1, while 1−s 1+s limit to 1 monotonically from below as s > 0 tend to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I conclude that there is a segment (0, S) ⊂ (0, 1) such that for all s ∈ (0, S), sB is Γ-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume now that x ̸= x0 and u(r) ⪯∞ εr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' As above, if γx0 ∈ B(y, sr) then it is the centre of a Λ-free ball of radius (1 − s)r, and so (1 − s)r ≤ u(|γ|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I wish to use the ε-linear bound on u as I did before, only this time u is only asymptotically smaller than εr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' To circumvent this I just need to show that |γ| is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Indeed since B � γx0, (1 − s)r � is Λ-free it does not contain x0 ∈ Λ · x0 and in particular (1 − s)r ≤ d(x0, γx0) = |γ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For some R1(s) = R1(s, u) one therefore has for all r > R1(s) (1 − s)r ≤ u(|γ|) ≤ ε|γ| On the other hand |y| ≤ d(x, y) + d(x, x0) = r + |x|, and consequently |γ| ≤ (1 + s)r + |x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For r > R1(s) one has 17 (1 − s)r ≤ u(|γ|) ≤ ε|γ| ≤ ε � (1 + s)r + |x| � This means that s for which Γ · x0 ∩ B(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' sr) ̸= ∅ must admit,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' for all r > R1(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 1 − s 1 + s + |x| r = (1 − s)r (1 + s)r + |x| ≤ ε < 1 (1) The rest of the proof is just Calculus 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' and concerns with finding S = S(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' u) ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 1) so that for any s ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' S) there is R(s) such that all r > R(s) satisfy ε < 1 − S 1 + S + |x| r ≤ 1 − s 1 + s + |x| r (2) The lemma readily follows from inequalities 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Explicitly, fix ε′ > ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' As before, monotonic approach of 1−s 1+s to 1 allows to fix S ∈ (0, 1) for which ε < ε′ < 1−s 1+s for all s ∈ (0, S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Next note that for any fixed s ∈ (0, S), limr→∞ 1−s 1+s+ |x| r = 1−s 1+s, and that the approach in monotonically increasing with r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since ε < ε′, this limit implies that for some R2 > R1(S), all r > R2 admit ε < 1−S 1+S+ |x| r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Finally notice that for any fixed r the function 1−s 1+s+ |x| r is again monotonically increasing as s tends to 0 from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Therefore inequality 2 holds for every s ∈ (0, S) and all r > R2(S) (capital S is intentional and important).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' To conclude the proof, notice that if moreover r > R1(s) (again lowercase s is intentional and important) then inequalities 1,2 both hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This means that for any s ∈ (0, S) there is R(s) := max{R1(s), R2(S)} such that r > R(s) ⇒ B(y, sr) is Γ-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The constants R1(s), R2(S) have the desired dependencies, hence so does R(s), proving the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10 (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There is a uniform bound on {dγ}γ∈Γ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=', Γ ⊂ ND(Λ) for some D > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular, Λ is a uniform lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 4 Q-rank 1 Lattices In this section I prove: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be a semisimple Lie group without compact factors and with finite centre, Γ ≤ G an irreducible non-uniform Q-rank 1 lattice, Λ ≤ G a discrete irreducible subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If Γ ⊂ Nu(Λ) for some sublinear function u, then Λ is a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, if Γ ̸⊂ ND(Λ) for any D > 0, then Λ is also of Q-rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If Γ ⊂ ND(Λ) for some D > 0, then Λ could be a uniform lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' An obvious obstacle for that is if Λ ⊂ Nu′(Γ) for some sublinear function u′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This condition turns out to be sufficient for commensurability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be a semisimple Lie group without compact factors and with finite centre, Γ ≤ G an irreducible non-uniform Q-rank 1 lattice, Λ ≤ G a discrete subgroup such that Γ ⊂ ND(Λ) for some D > 0, and Λ ⊂ Nu(Γ) for some sublinear function u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then Λ ⊂ ND′(Γ) for some D′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' As a result of Eskin’s and Schwartz’s arguments (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3), I conclude: Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the setting of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 and unless G is locally isomorphic to SL2(R), Λ is com- mensurable to Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='8 is a result of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3 18 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 Strategy Lattice Criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I use three different lattice criteria, depending on the R-rank of G and on whether or not Γ ⊂ ND(Λ) for some D > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' My proof for Q-rank 1 lattices is motivated by a criterion of Benoist and Miquel, resolving a conjecture of Margulis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It can be viewed as an algebraic converse to the geometric structure of the compact core described in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='16 in [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be a semisimple real algebraic Lie group of real rank at least 2 and U be a non-trivial horospherical subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let ∆ be a discrete Zariski dense subgroup of G that contains an indecomposable lattice ∆U of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then ∆ is a non-uniform irreducible arithmetic lattice of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' See Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='47 for the precise meaning of an indecomposable horospherical lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For R-rank 1 groups, one has the following theorem by Kapovich and Liu, stating that a group is geometrically finite so long as ‘most’ of its limit points are conical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Recall L(∆) is the limit set of ∆ ≤ Isom(X), and Lcon(∆) is the set of its conical limit points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6 (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='5 in [30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let X be a R-rank 1 symmetric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A discrete subgroup ∆ ≤ Isom(X) is geometrically infinite if and only if the set L(∆) \\ Lcon(∆) of non-conical limit points has the cardinality of the continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' As a direct corollary I obtain the following criterion: Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let X be a R-rank 1 symmetric space, Γ ≤ G = Isom(X) a non-uniform lattice and Λ ≤ G a discrete subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If L(Λ) = X(∞) and Lcon(Γ) ⊂ Lcon(Λ), then Λ is a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since Γ is a lattice, L(Γ) = X(∞) and it is geometrically finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6 implies the cardinality of X(∞) \\ Lcon(Γ) is strictly smaller than the continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The assumption Lcon(Γ) ⊂ Lcon(Λ) implies the same holds for Λ, and in particular that Λ is geometrically finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The assumption that L(Λ) = X(∞) implies that Λ is geometrically finite if and only if it is a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let X be a R-rank 1 symmetric space, Γ ≤ G = Isom(X) a non-uniform lattice and Λ ≤ G a discrete subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If Γ ⊂ ND(Λ) for some D > 0, then Λ is a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By definition of the limit set and of conical limit points, it is clear that every Γ-limit point is a Λ-limit point, and every Γ-conical limit point is also Λ-conical limit point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I conclude from Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='7 that Λ is a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Also in higher rank the inclusion Γ ⊂ ND(Λ) implies that Λ is a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This result is due to Eskin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9 (Eskin, see Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='11 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be a semisimple Lie group without compact factors and of higher rank, Γ ≤ G an irreducible non-uniform lattice, Λ ≤ G a discrete subgroup such that Γ ⊂ ND(Λ) for some D > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then Λ is a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9 was used in the proof of quasi-isometric rigidity for higher rank non-uniform lattices in [16], [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the (earlier) R-rank 1 case, Schwartz [52] used an analogous statement, which requires one extra assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10 (Schwartz, see Section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 in [52] and Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='11 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be a real simple Lie group of R-rank 1 and with finite centre, Γ ≤ G an irreducible non-uniform lattice, Λ ≤ G a discrete subgroup such that both Γ ⊂ ND(Λ) and Λ ⊂ ND(Γ) for some D > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then Λ is a lattice and commensurable to Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6 should be viewed as a generalization of the bounded case depicted in Theo- rems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10, which were known to experts in the field in the late 1990’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Complete proofs for these statements were never given in print, and I take the opportunity to include them here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' See Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3, where I also prove Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I thank Rich Schwartz and Alex Eskin for supplying me with their arguments and allowing me to include them in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I also thank my thesis examiner Emmanuel Breuillard for encouraging me to find and make these proofs public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 19 Outline of Proof and Use of Sublinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lattices of Q-rank 1 admit a concrete geometric structure (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This structure is manifested in the geometry of an orbit of such a lattice in the corresponding symmetric space X = G/K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' One important geometric property is the existence of a set of horoballs which the orbit of the lattice intersects only in the bounding horospheres, and in each such horosphere the orbit forms a (metric) cocompact lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='8 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9 reduce the proof to the case where Γ ̸⊂ ND(Λ) for any D > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In that case, the essence lies in proving the existence of horospheres in X which a Λ-orbit intersects in a cocompact lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This is proved purely geometrically, using the geometric structure of Q-rank 1 lattices and the sublinear constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Together with some control on the location of these horospheres, I prove two major statements: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Λ · x0 intersects a horosphere H ⊂ X in a cocompact lattice (Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Every Γ-conical limit point is also a Λ-conical limit point (Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The R-rank 1 case of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 follows directly from Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='7 using the second item above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The higher rank case requires a bit more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' namely it requires to deduce from the above items that Λ meets the hypotheses of the Benoist-Miquel Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' To that end I use a well known geometric criterion (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='50) in order show that Λ is Zariski dense, and a lemma of Mostow (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='36) to show that Λ intersects a horospherical subgroup in a cocompact lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Outline for Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 is the core of the original mathematics of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It is devoted to proving that Λ · x0 intersects some horospheres in a cocompact lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proof is quite delicate and somewhat involved, and I include a few figures and a detailed informal overview of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The figures are detailed and may take a few moments to comprehend, but I believe they are worth the effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3 deals with the case where Γ ⊂ ND(Λ), and elaborates on Schwartz’s and Eskin’s proofs of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 is devoted to the translation of the geometric results of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 to the algebraic language used in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Though the work is indeed mainly one of translation, some of it is non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Finally, in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='5 I put everything together for a complete proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I highly recommend the reader to have a look at the uniform case in Section 3 before reading this one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 A Λ-Cocompact Horosphere Recall that dγ := d(γx0, λγx0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In this section I prove: Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If {dγ}γ∈Γ is unbounded, then there exists a horosphere H based at WQ(Γ) such that � Λ∩StabG(H) � x0 intersects H in a cocompact metric lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, the bounded horoball HB is Λ-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Throughout Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 the standing assumptions are that {dγ}γ ∈ Γ is unbounded, and Γ is an irreducible Q-rank 1 lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The Argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proof is by chasing down the geometric implications of unbounded dγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' These implications are delicate, but similar in spirit to the straight-forward proof for uniform lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proof consists of the following steps: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Unbounded dγ results in Λ-free horoballs HBΛ tangent to Λ-orbit points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Each such horoball is based at WQ(Γ), giving rise to corresponding horoballs of Γ, denoted HBΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If dγ is large, then γx0 must lie deep inside a unique Λ-free horoball tangent to λγx0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I use: (a) A bound on the distance d(HΛ, HΓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' (b) A bound on the angle ∠λγx0([λγx0, γx0], [λγx0, ξ)), where ξ ∈ X(∞) is the base point of a suitable Λ-free horoball tangent to λγx0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 20 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There exist horospheres of Γ, say HΓ, such that if γx0 ∈ HΓ then large dγ implies large Λ-free areas along the bounding horosphere of some HBΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If HBΛ is boundedly close to some Λ-orbit point, then HΛ is almost Λ-cocompact, that is HΛ ⊂ ND(Λ·x0) for some universal D = D(Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Together with the previous step, this yields a uniform bound on dγ along certain horospheres of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Finally I elevate the almost cocompactness to actual cocompactness and show HΛ ⊂ ND(Λ · x0 ∩ HΛ) for some D > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This immediately elevates to HΛ ⊂ (Λ ∩ StabG(HΛ)) · x0, proving the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The Properties of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The geometric properties of Γ that are used in the proof are: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In higher rank, the characterization of WQ(Γ) using conical / non-horospherical limit points (Corol- lary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In R-rank 1, the dichotomy of limit points being either non-horospherical or conical (The- orem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Γ-cocompactness along the horospheres of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For every point x ∈ X and C > 0 there is a bound K(C) on the number of horospheres of Γ that intersect B(x, C) (Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 Λ-Free Horoballs I retain the notations and objects defined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There exists a Λ-free horoball tangent to x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since {dγ}γ∈Γ is unbounded, there are γn ∈ Γ with dn = dγn = d(γn, λn) → ∞ monotonically, where λn ∈ Λ is a Λ-orbit point closest to γn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Denote x′ n = λ−1 n γnx0, ηn := [x0, x′ n], and vn ∈ Sx0X the initial velocity vectors vn := ˙ηn(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The tangent space Sx0X is compact, so up to a subsequence, vn converge monotonically in angle to a direction v ∈ Sx0X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let η be the unit speed geodesic ray emanating from x0 with initial velocity ˙η(0) = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Denote ξ := η(∞) the limit point of η in X(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I claim that the horoball HB := ∪t>0B � η(t), t � , based at ξ and tangent to x0, is Λ-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let t > 0 and consider η(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For every ε > 0, there is some angle α = α(t, ε) such that any geodesic η′ with ∠x0(η, η′) < α admits d � η(t), η′(t) � < ε/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The convergence vn → v implies d � η(t), ηn(t) � < ε/2 for all but finitely many n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular, B � η(t), t � ⊂ Nε � B � ηn(t), t �� for all such n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For a fixed t ≤ dn, it is clear from the definitions that B � ηn(t), t � ⊂ B′ n = B(x′ n, dn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' One has dn → ∞, and so for a fixed t > 0 it holds that t < dn for all but finitely many n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I conclude that for any fixed t > 0 there is n large enough such that B � η(t), t � ⊂ Nε � B � ηn(t), t �� ⊂ NεB′ n I conclude that for every ε > 0, HB ⊂ � n Nε(B′ n) = Nε � � n B′ n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This implies that any point in the interior of HB is contained in the interior of one of the Λ-free balls B′ n, proving HB is Λ-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Suppose HB is a Λ-free horoball, based at some point ξ ∈ X(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then ξ ∈ WQ(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For any geodesic η with limit ξ, the size d(x0, γx0) of the Γ-orbit points γx0 that lie boundedly close to η grows linearly in the distance to any fixed horosphere based at ξ, and in particular to H = ∂HB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The sublinear constraint d(γx0, λγx0) ≤ u(|γ|) together with the fact that HB is Λ-free imply that the size of such γ is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In R-rank 1 every limit point is either conical or in WQ(Γ), proving the lemma in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For higher rank, the above argument actually shows more: it shows that a point ξ′ ∈ N π 2 (ξ) is not conical, because every geodesic with limit ξ′ ∈ N π 2 (ξ) entres HB at a linear rate (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Hattori’s characterization of WQ(Γ) (Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='30) implies ξ ∈ WQ(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 21 Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Given a Λ-free horoball HBΛ, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='14 gives rise to a horoball of Γ based at the same point at X(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Call this the horoball corresponding to HBΛ, and denote it by HBΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The corresponding horosphere is denoted HΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the course of my work I had had a few conversations with Omri Solan regarding the penetration of geodesics into Λ-free horoballs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assuming Λ ⊂ Nu(Γ) implies that Λ preserves WQ(Γ) (see Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This is the case in the motivating setting where Λ is an abstract finitely generated group that is SBE to Γ, see Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 in Chapter 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the case of SL2(R) Omri suggested to use the action of Λ on the Bruhat-Tits tree of SL2(Qp) (for all primes p) and the classification of these elements into elliptic and hyperbolic elements (separately for each p) in order to deduce that Λ actually lies in SL2(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' We did not pursue that path nor its possible generalization to the SLn case and general Bruhat-Tits buildings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 A Γ-orbit point Lying Deep Inside a Λ-Free Horoball I established the existence of Λ-free horoballs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It may seem odd that the first step in proving Λ · x0 is ‘almost everywhere’ is proving the existence of Λ-free regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' But this fits perfectly well with the algebraic statement that non-uniform lattices must admit unipotent elements (see Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 in [39]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The goal of this section is to obtain some control on the location of the Λ-free horoballs, in order to conclude that some γx0 lies deep inside HBΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This results in yet more Λ-free regions, found on the bounding horosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I need one property of sublinear functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I thank Panos Papazoglou for noticing a mistake in the original formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let u be a sublinear function, f, g : R≥0 −→ R>0 two positive monotone functions with limx→∞ f(x) + g(x) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If for all large enough x it holds that f(x) ≤ u � f(x) + g(x) � , then for every 1 < s and all large enough x it holds that f(x) ≤ u � s · g(x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular f(x) ≤ u′� g(x) � for some sublinear function u′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume as one may that u is non-decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By definition of sublinearity limx→∞ u � f(x)+g(x) � f(x)+g(x) = 0, so by hypothesis limx→∞ f(x) f(x)+g(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This means that for every ε > 0 one has f(x) ≤ ε · g(x) for all large enough x, resulting in f(x) ≤ u � f(x) + g(x) � ≤ u � (1 + ε) · g(x) � Notice that for a fixed s > 0, the function u′(x) = u(sx) is sublinear, as lim x→∞ u(sx) x = lim x→∞ s · u(sx) sx = 0 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There is L = L(C) such that if HBΛ is any Λ-free horoball tangent to a point x ∈ B(x0, C) then d(HΛ, HΓ) ≤ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, there is a sublinear function u′ such that: L(C) ≤ � u′(C) if HBΓ ⊂ HBΛ C if HBΛ ⊂ HBΓ Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If HBΛ ⊂ HBΓ, then clearly d(HΛ, HΓ) ≤ C, simply because HBΓ is Γ-free and in particular cannot contain x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Therefore HΓ must separate HΛ from x0 and in particular d(HΛ, HΓ) ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume that HBΓ ⊂ HBΛ, and denote l = d(HΛ, HΓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The horoball HBΓ is a horoball of Γ, hence Γ · x0 is DΓ-cocompact along HΓ and there is an element γ ∈ Γ with |γ| ≤ C + l + DΓ and γx0 ∈ HΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since HBΛ is Λ-free one has l ≤ d(γx0, λγx0) ≤ u(|γ|) ≤ u(C + l + DΓ) C, DΓ are fixed, so this inequality can only occur for boundedly small l, say l < L′(C) (DΓ is a universal constant and may be ignored).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Consult figure 2 for a geometric visualization of this situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 22 Rad = C x0 Λ − free HBΛ ξ Γ − free HBΓ ≤ L(C) γx0 PHΓ(x0) ≤ L(C) + DΓ Figure 2: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A Λ-free horoball HBΛ intersects a ball of radius C about x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The associated Γ-free horoball HBΓ is boundedly close, essentially due to the uniform cocompactness of Γ · x0 along the Γ horospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It remains to show that L′(C) is indeed sublinear in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' First define L(C) to be the minimal L that bounds the distance d(HΛ, HΓ) for all possible HBΛ tangent to a point x ∈ B(x0, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This is indeed a minimum, since by Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='22 there are only finitely many horoballs of Γ intersecting B � x0, C + L′(C) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For every C there is thus a horoball HBΓ C and an element γ ∈ Γ such that γx0 ∈ HΓ, d(HΛ C, HΓ C) = L(C) and |γ| ≤ C + L(C) + DΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The fact that HBΛ C is Λ-free implies L(C) = d(HΛ C, HΓ C) ≤ u(|γ|) ≤ u � C + L(C) + DΓ � Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='17 implies that L(C) ≤ u′(C) for some sublinear function u′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The following is an immediate corollary, apparent already in the above proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For every C > 0 there is a bound K = K(C) and a fixed set ξ1, ξ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' ξK ∈ WQ(Γ) ⊂ X(∞) so that every Λ-free horoball HBΛ which is tangent to some point x ∈ B(x0, C) is based at ξi for some i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=', K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular, for any specific point x ∈ NC(Λ · x0) there are at most K Λ-free horoballs tangent to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let HBΛ be a horoball tangent to a point x ∈ B(x0, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='18 bounds d(HBΛ, HBΓ) by L(C), hence HBΓ is tangent to a point x′ ∈ B � x0, C + L(C) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='22 there are only finitely many possibilities for such HBΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular there are finitely many base-points for these horoballs, say ξ1, ξ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' , ξK(C) ∈ WQ(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Finally, recall that a horoball is determined by a base point and a point x ∈ X tangent to it, so the last statement of the corollary holds for any x ∈ B(x0, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' But the property in question is Λ-invariant so the same holds for any point x ∈ Λ · B(x0, C) = NC(Λ · x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The bound on d(HBΛ, HBΓ) given by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='18 further strengthen the relation between HBΛ and HBΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The ultimate goal is to show that the HBΛ-s play the role of the Γ-horoballs in the geometric structure of Q-rank 1 lattices, namely to show that Λ · x0 is cocompact on the HΛ-s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This requires to actually find Λ-orbit points somewhere in X, and not just Λ-free regions as was done up to now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' As one might suspect, these points arise as λγx0 corresponding to points γx0 ∈ HΓ, which exist in abundance since Γ · x0 ∩ HΓ is a cocompact lattice in HΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The hope is that a Λ-free horoball HBΛ tangent to λγx0 would correspond to a horoball of Γ tangent to γx0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This would have forced all the λγ to actually lie on the same bounding horosphere, and {λγx0 | γx0 ∈ HΓ} would then be a cocompact lattice in HΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This hope turns out to be more or less true, but it requires 23 some work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The goal of the rest of this section is to establish a relation between a Λ-free horoball HBΛ tangent to λγx0 and γx0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I start with some notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In light of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='19, there is a finite number N of Λ-free horoballs tangent to x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Denote: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' {HBΛ i }N 1=1 are the Λ-free horoballs tangent to x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' ξi ∈ WQ(Γ) is the base point of HBΛ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' vi ∈ Sx0X is the unit tangent vector in the direction ξi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' ηi := [x0, ξi) is the unit speed geodesic ray emanating from x0 with limit ξi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular vi = d dtηi(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' HBΓ i is the horoball of Γ that corresponds to HBΛ i , based at ξi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' HBΛ λ,i, ξi λ, ηi λ are the respective λ-translates of the objects above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For example, HBΛ λ,i := λ · HBΛ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' H decorated by the proper indices denotes the horosphere bounding HB, the horoball with respective indices, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' HΛ i := ∂HBΛ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For an angle α > 0 and a tangent vector v0 ∈ SxX, define (a) The α-sector of v in SxX is the set {v ∈ SxX | v ∈ Nα(v0)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Recall that the metric on SxX is the angular metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' (b) The α-sector of v in X are all points y ∈ X for which the tangent vector at 0 of the unit speed geodesic [x, y] lies in the α-sector of v in SxX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For every angle α ∈ (0, π 2 ) there exists D = D(α) such that if dγ > D then for some i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' , N}, γx0 lies inside the α-sector of vi λγ at λγx0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Furthermore whenever α is uniformly small enough, there is a unique such i = i(γ), independent of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Translation by the isometry λ−1 γ preserves angles and distances, so it is enough to prove that there is an i for which x′ γ := λ−1γx0 lies inside the α-sector of vi, and that this i is unique if α is uniformly small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume towards contradiction that there is α ∈ (0, π 2 ) and a sequence γn ∈ Γ, λn := λγn ∈ Λ with dγn unbounded, and x′ n := λ−1 n γnx0 not lying in the union of the α-sectors of vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By perhaps taking smaller α I may assume all the α-sectors of the vi in Sx0X are pairwise disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This can be done because there are only finitely many vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Compactness of Sx0X allows me to take a converging subsequence v′ n := ˙ [x0, x′n], with limit direction v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Denote by η′ the geodesic ray emanating from x0 with initial velocity v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The exact same argument of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='13 proves that η′(∞) is the base point of a Λ-free horoball tangent to x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' But this means v′ = vi for some i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' , N}, contradicting the fact that all v′ n lie outside the α-sectors of the vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This proves that there is a bound D = D(α) such that if dγ > D then x′ γ lies within the α-sector of some vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proof clearly shows that whenever α is small enough so that the α-sectors of the vi are disjoint, x′ γ lies in the α-sector of a unique vi as soon as dγ > D(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='13 I used compactness of Sx0X to induce a converging subsequence of directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='21 actually shows that the fact there are finitely many Λ-free horoballs tangent to x0 implies a posteriori that there was not much choice in the process - all directions [x0, x′ γ] must fall into one of the finitely many directions vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Next, I want to control the actual location of certain points with respect to the horoballs of interest, and not just the angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This turns out to be a more difficult of a task than one might suspect, since control on angles does not immediately give control on distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Recall that large Λ-free balls near x0 imply large concentric Γ-free balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The precise quantities and bounds are given by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='8 (one can use Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9 to obtain a slightly cleaner statement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 24 Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let S ∈ (0, 1) be the constant given by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='8, and let s ∈ (0, S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There is a bound D = D(s) such that dγ > D implies that γx0 lies sdγ deep in HBΛ λγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proof is a bit delicate but very similar to that of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In essence, I use the Γ-free balls near x0 to produce a Γ-free cylinder, which would force a certain geodesic not to cross a horosphere of Γ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' force it to stay inside a Γ-free horoball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' As in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='21 it is only required to show that x′ = λ−1 γ γx0 is sdγ deep inside HBΛ i(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I start with proving that x′ ∈ HBΓ i(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I learned the hard way that even this is not a triviality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Recall the notation Bγ = B(γx0, dγ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The ball B′ γ = λ−1 γ Bγ is a Λ-free ball of radius dγ about x′ = λ−1 γ γx0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Denote by x′ t the point at time t along the unit speed geodesic η′ := [x0, x′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It holds that |x′ t| = t and, for t ≤ dγ, x′ t is the centre of a Λ-free ball of radius t tangent to x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The constant s is fixed and by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='8 there is T ′ = T ′(s) such that if t > T ′, the ball sB � x′ t, t � is Γ-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The next goal is to show that x′ T ∈ HBΓ for some adequate T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For any time T > 0, let α = α(ε, T ) be the angle for which d � η(T ), ηi(γ)(T ) � < ε for every η in the α-sector of vi(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By perhaps taking smaller α I may assume that α is uniformly small as stated in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let D(α) be the bound given by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='21 guaranteeing D(α) < dγ ⇒ d � x′ T , ηi(γ)(T ) � < ε For my needs in this lemma ε may as well be chosen to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I now choose a specific time T for which I want x′ T and ηi(γ)(T ) to be close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There are only finitely many Λ-free horoballs {HBΛ i }i∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=',N} tangent to x0, giving rise to a uniform bound L = maxi∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=',N}{d(HΛ i , HΓ i )} on the distance d(HΛ i(γ), HΓ i(γ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Fix T to be any time in the open interval (T ′ + L + ε, dγ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The fact that L + ε < T implies that ηi(γ)(T ) lies at least ε-deep inside HBΓ i(γ), and therefore η′(T ) ∈ HBΓ i(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Recall that any point on HΓ is DΓ-close to a point γx0 ∈ HΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By perhaps enlarging T and shrinking α if necessary, I may assume that DΓ < sT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Thus for all T < t ≤ dγ, x′ t is the centre of a Γ-free ball of radius st > sT > DΓ, hence {x′ t}T ≤t≤dγ does not cross a horosphere of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since x′ T ∈ HBΓ i(γ), this implies that x′ t stays in HBΓ i(γ) for all T < t ≤ dγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular x′ dγ = x′ ∈ HBΓ i(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' To get the result of the proposition, recall that sB′ γ = B(x′, sdγ) is Γ-free, so x′ must be at distance at least sdγ − DΓ from any horosphere of Γ, and in particular from HΓ i(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In terms of Busemann functions, this means that bηi(γ)(x′) ≤ −sdγ + DΓ whenever one can find such T ′ + L + ε < T < dγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since HBΛ i(γ) is tangent to x0, the corresponding horoball HBΓ i(γ) lies inside it, and so x′ lies (sdγ − DΓ)-deep inside HBΛ i(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A close look at the argument yields the desired bound D = D(s) such that the above holds whenever dγ > D(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' To help the reader take this closer look, I reiterate the choice of constants and their dependencies as they appear in the proof: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Fix ε = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let T ′ = T ′(s) the constant from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='8 and L = maxi∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=',N}{d(HBΛ i , HBΓ i )}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Fix T > T ′ + L + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Fix α = α(1, T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Fix D(s) = max{D(α), T + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I remark, for the reader worried about the DΓ which appears in the final bound but not in the statement, that (a) DΓ is a fixed universal constant and may as well be ignored, and (b) the discrepancy can be formally corrected by taking a slightly larger s < s′ to begin with and as a result perhaps enlarging the bound D for dγ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Also note that L = L(Λ) is a universal constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3 Intersection of Λ-Free Regions and the Existence of a Λ-Cocompact Horosphere In this section I find Λ-orbit points that lie close to the bounding horosphere of a Λ-free horoball HBΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In order to find such points I need to make sure HBΛ is not contained inside a much larger Λ-free horoball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I introduce the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A Λ-free horoball HBΛ is called maximal if it is tangent to a point x = λx0 ∈ Λ · x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It is called ε-almost maximal if d(Λ · x0, HΛ) < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It may happen that a discrete group admits free but not no maximally free horoballs - see discussion in section 4 of [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In any case it is clear that any Λ-free horoball can be ‘blown-up’ to an ε-almost maximal Λ-free horoball, for every ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, every two ε-almost maximal horoballs based at the same point ξ ∈ WQ(Γ) lie at distance at most ε of one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For my needs any fixed ε would suffice, and I fix ε = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There is DΛ > 0 such that if HBΛ is 1-almost maximal Λ-free horoball then HΛ ⊂ NDΛ(Λ·x0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' d(x, Λ · x0) ≤ DΛ for all x ∈ HΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Notice that Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='26 does not state Λ · x0 even intersects HΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I start with a short sketch of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Consider a 1-maximal horoball and a point x on its bounding horosphere with d(x, Λ · x0) = D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' One may translate this situation to x0, which results in a Λ-free horoball HBΛ intersecting the (closed) D-ball about x0 at a point w with B(w, D) Λ-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proof differs depending on whether HBΓ ⊂ HBΛ or the other way round, since I use the bounds from 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='18: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If HBΓ ⊂ HBΛ, there is a sublinear bound on d(HBΛ, HBΓ), which readily yields a bound on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' if HBΛ ⊂ HBΓ there is a bound on d(x0, HBΓ) that is independent of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' So there are only finitely many possibilities for HBΓ, independent of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Hence there are only finitely many possible base points for HBΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' These in turn correspond to possible base points for such HBΛ, and this finiteness yields a bound on the distance d(HBΓ, HBΛ) < L that is independent of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The rest of the proof is quite routine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let HBΛ be a 1-almost maximal Λ-free horoball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By definition there is λ ∈ Λ and z ∈ HΛ such that d(λx0, z) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Fix D > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I show that if there is some z′ ∈ HΛ for which d(z′, Λ · x0) ≥ D, then D must be uniformly small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Exactly how small will be set in the course of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Fix D > 1 and assume that there is z′ ∈ HΛ with d(z, Λ · x0) ≥ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Up to sliding z′ along HΛ, the continuity of the function x �→ d(x, Λ · x0) together with Intermediate Value Theorem allows to assume that d(z′, Λ · x0) = D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let λ′ ∈ Λ be the element for which d(z′, λ′x0) = D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Translating by λ′−1 yields 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A Λ-free horoball HBΛ 0 := λ′−1HBΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A point w := λ′−1z′ ∈ HΛ 0 for which |w| = d(w, x0) = d(w, Λ · x0) = D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume first that HBΓ 0 ⊂ HBΛ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='18 there is a sublinear function u′ such that d(HΓ 0 , HΛ 0 ) ≤ u′(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This yields a point γx0 ∈ HΓ 0 for which d(w, γx0) ≤ u′(D) + DΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Thus |γx0| ≤ D + u′(D) + DΓ and the reverse triangle inequality gives D − � u′(D) + DΓ � ≤ d(w, λγx0) − d(w, γx0) < d(γx0, λγx0) Together with the bound d(γx0, λγx0) ≤ u(|γx0|) and rearranging, one obtains D ≤ u � D + u′(D) + DΓ � + u′(D) + DΓ The right hand side is clearly a sublinear function in D, hence this inequality may hold only for boundedly small D, say D < D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I conclude that HBΓ 0 ⊂ HBΛ 0 may occur only when D < D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Notice that D1 depends only on u and u′, and not on HBΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 26 w x0 Λ − free B(w, D) τ(t0) τ Λ − free HBΛ ξ Γ − free HBΓ γx0 Figure 3: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='26, case HBΛ ⊂ HBΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The red horosphere of Γ is trapped between x0 and HΛ, and is at distance t0 from x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A Γ-orbit point on the red horosphere close to x0 allows to use sublinearity to get a bound on t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume next that HBΛ 0 ⊂ HBΓ 0 , and that the containment is strict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since x0 ∈ Γ · x0, the geodesic τ := [x0, w] is of length D and intersects HΓ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Denote by t0 ∈ [0, D) the time in which τ intersects HΓ 0 , and let w′ := τ(t0) ∈ HΓ 0 be the intersection point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular |w′| = t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It is clear that B(w′, t0) is Λ-free, as a subset of the ball B(w, D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Again there is γx0 ∈ B(w′, DΓ) ∩ HΓ 0 and so |γx0| ≤ t0 + DΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By reverse triangle inequality t0 − DΓ ≤ d(w′, λγx0) − d(w′, γx0) ≤ d(γx0, λγx0) and the sublinear constraint gives t0 − DΓ ≤ u(t0 + DΓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This can only happen for boundedly small t0, say t0 < T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I conclude that if HBΛ 0 ⊂ HBΓ 0 , then HBΓ 0 is a horoball of Γ tangent to some point y ∈ B(x0, T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='22 there are finitely many horoballs of Γ tangent to points in B(x0, T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular there is a finite set {ξ′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' , ξ′ K} ∈ WQ(Γ) of possible base points for HBΓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This set depends only on T , and since the choice of T was completely independent of D, the set of possible base points is independent of D as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let � HBΓ i be the horoball of Γ based at ξ′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I can now bound the distance d(HBΓ 0 , HBΛ 0 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let 1 ≤ i ≤ K be an index for which there is a Λ-free horoball based at ξ′ i that is contained in � HBΓ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There is thus some 1-almost-maximal Λ-free horoball based at ξ′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Fix an arbitrary such 1-almost-maximal Λ-free horoball � HBΛ i for each such i, and let Li := d(� HBΛ i , � HBΓ i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Finally, define L := max{Li} + 1 among such i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' As stated in Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='25, d(HBΛ 0 , � HBΛ i ) ≤ 1 for some i, therefore d(HBΓ 0, HBΛ 0 ) ≤ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Recall |w| = D and B(w, D) is Λ-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It holds that d(w, HΓ 0 ) ≤ L, and so there is γx0 ∈ HΓ 0 for which d(w, γx0) ≤ L + DΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular |γx0| ≤ D + L + DΓ (in fact it is clear that |γx0| ≤ T + DΓ, but this won’t be necessary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Reverse triangle inequality gives D − (L + DΓ) ≤ d(w, λγx0) − d(w, γx0) ≤ d(γx0, λγx0) and from the sublinear constraint I conclude D − (L + DΓ) ≤ u(D + L + DΓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since L, DΓ are fixed constants independent of D, this can only hold for boundedly small D, say D < D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular, one gets a uniform bound DΛ := max{D1, D2} such that x ∈ HΛ ⇒ d(x, Λ · x0) < DΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Every Γ-conical limit point is a Λ-conical limit point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 27 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let ξ ∈ X(∞) be a Γ-conical limit point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let η : R≥0 → X be a geodesic with η(∞) = ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By definition there is a bound D > 0 and sequences tn → ∞, γn ∈ Γ such that d � γnx0, η(tn) � < D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Consider the corresponding λn := λγn and λnx0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If dn is uniformly bounded, then ξ is Λ-conical by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Otherwise, assume dn is monotonically increasing to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For some fixed s ∈ (0, 1) it holds that for all but finitely many n ∈ N, γnx0 is sdn deep inside HBΛ n := HBΛ λn,i(γn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I assume dn is large enough so that sdn > D, and in particular η(tn) ∈ HBΛ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let ξn ∈ WQ(Γ) be the respective base points of HBΛ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The point ξ is Γ-conical, and by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='28 π 2 ≤ d(ξ, WQ(Γ)) ≤ d(ξ, ξn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proof differs depending on whether the above inequality is strict or not for any n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume first that for some m ∈ N, d(ξ, ξm) = π 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By item 2 of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='31, d � HΛ m, η(t) � is uniformly bounded, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=', there is C > 0 such that for every t > 0 there is xt ∈ HΛ m for which d � xt, η(t) � < C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='26, d(xt, Λ · x0) < DΛ, hence d � η(tn), xtn � ≤ C + DΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This means that ξ is Λ-conical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Otherwise, for all n ∈ N it holds that π 2 < d(ξ, ξn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The fact that η(tn) ∈ HBΛ n together with Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='31 implies that at some later time the geodesic ray η leaves HBΛ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Thus there is sn > tn for which η(sn) ∈ HΛ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since HBΛ n are maximal Λ-free horoballs, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='26 gives rise to points λnx0 such that d � λnx0, η(sn) � ≤ DΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This renders ξ as a Λ-conical limit point, as wanted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I now prove Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The strategy is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For HBΛ = HBΛ λγ,i(γ), one uses Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='23 to get that HBΓ ⊂ HBΛ and that the distance d(HΛ, HΓ) is large with dγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The horosphere HΓ admits a Γ-cocompact metric lattice, and so the projections of these metric lattice points onto HΛ form a cocompact metric lattice in HΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It remains to show that for each γ′x0 ∈ Γ · x0 ∩ HΓ, the corresponding λ′ = λγ′ indeed lies on the same HΛ and boundedly close to the projection PHΛ(γ′x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This is done by putting together all the geometric facts obtained up to this point, specifically Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' One delicate fact that will be of use is that two maximal Λ-free horoballs that are based at the same point must be equal, because none of them can contain a Λ-orbit point while on the other hand both bounding horospheres intersect Λ · x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Fix s > 0 for which Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='23 yields a corresponding bound D(s), and let γ ∈ Γ such that sdγ > 2 · � DΛ + D(s) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Consider the (maximal) Λ-free horoball HBΛ λγ,i(γ) based at ξi(γ) λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I show that Λ · x0 ∩ HΛ λγ,i(γ) is a cocompact metric lattice in HΛ λγ,i(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I keep the subscript notation because the proof is a game between HBΛ λγ,i(γ) and another Λ-free horoball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let HBΓ λγ,i(γ) be the Γ-horoball corresponding to HBΛ λγ,i(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I can conclude that HBΓ λγ,i(γ) ⊂ HBΛ λγ,i(γ), because the choice of dγ > D(s) guarantees γx0 is sdγ deep inside HBΛ λγ,i(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular HBΛ λγ,i(γ) is not Γ-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, it holds that L := d(HΛ λγ,i(γ), HΓ λγ,i(γ)) ≥ sdγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let γ′ ∈ Γ be any element in the cocompact metric lattice Γ · x0 ∩ HΓ λγ,i(γ), and consider two associated points: (a) λ′x0 = λγ′x0 and (b) the projection of γ′x0 on HΛ λγ,i(γ), denoted p′ γ := PHΛ λγ ,i(γ)(γ′x0) ∈ HΛ λγ,i(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The horoball HBΛ λγ,i(γ) is a maximal Λ-free horoball so it is also 1-almost maximal, hence d(p′ γ, Λ · x0) ≤ DΛ and the following holds: sdγ ≤ L ≤ dγ′ ≤ d(γ′x0, p′ γ) + d(p′ γ, Λ · x0) ≤ L + DΛ (3) Consider ξi(γ′) λ′ , and assume towards contradiction that ξi(γ′) λγ′ ̸= ξi(γ) λγ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Both points lie in WQ(Γ) and therefore must be at Tits distance π of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Therefore the fact that γ′x0 lies in HBΛ λγ,i(γ) implies that the geodesic [γ′x0, ξi(γ′)] leaves HBΛ λγ,i(γ) at some point z ∈ HΛ λγ,i(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The fact that D(s) ≤ sdγ ≤ dγ′ implies that γ′x0 lies s2dγ deep inside HBΛ λγ′ ,i(γ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Therefore the point z also lies at least s2dγ deep inside HBΛ λγ′ ,i(γ′), and therefore z is the centre of a Λ-free horoball of radius at least s2dγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By choice of dγ the point z therefore admits a 2DΛ neighbourhood that is Λ-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' But z lies on HΛ λγ,i(γ), a maximal horosphere of Λ, contradicting Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I conclude that ξi(γ′) λγ′ = ξi(γ) λγ , so HBΛ λγ,i(γ) and 28 γx0 dγ γ′x0 λγ′x0 HBΛ λγ′ ,i(γ′) z′ := πHBΛ λγ′ ,i(γ′)(γ′x0) D1 ���� 1 ξi(γ′) λγ′ ξi(γ) λγ ≥ 1 2dγ z ≥ 1 2 dγ′ Λ − free ball B � z, 1 2dγ � HBΛ λγ,i(γ) HBΓ λγ,i(γ) λγx0 Figure 4: Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assuming towards contradiction that ξi(γ) λγ ̸= ξi(γ′) λγ′ results in a point z ∈ HBΛ λγ,i(γ) (blue coloured and bold faced in the bottom part of the figure) admitting a large Λ-free neighbourhood, contradicting almost cocompactness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' HBΛ λγ′ ,i(γ′) are two Λ-free horoballs that are tangent to a Λ · x0 point and based at the same point at ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This implies HBΛ λγ,i(γ) = HBΛ λγ′ ,i(γ′), and in particular λγ′x0 ∈ HΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Finally, it is clearly seen from Inequality 3 that d(λ′x0, p′ γ) ≤ d(λ′x0, γ′x0) + d(γ′x0, p′ γ) ≤ dγ′ + L ≤ L + DΛ + L The element γ′x0 ∈ Γ · x0 ∩ HΓ λγ,i(γ) was as arbitrary element, and the above argument shows that the corresponding Λ-orbit points satisfy: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' λ′x0 all lie on HΛ λγ,i(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Each p′ γ is 2L + DΛ close to the point λ′x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This shows that the cocompact metric lattice {p′ γ | γ′x0 ∈ HΓ λγ,i(γ)} lies in a bounded neighbourhood of the set of points Λ · x0 ∩ HΛ λγ,i(γ), proving that Λ · x0 ∩ HΛ λγ,i(γ) is a cocompact metric lattice in HΛ λγ,i(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='24 elevates this to � Λ ∩ StabG(HΛ λγ,i(γ)) � x0 ∩ HΛ λγ,i(γ) being a cocompact metric lattice in HΛ λγ,i(γ), completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3 The Bounded Case Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='12 is enough in order to prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 in the case Γ ̸⊂ ND(Λ) for any D > 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' in case Γ does not lie in a bounded neighbourhood of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The case where Γ and Λ lie at bounded Hausdorff distance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' where Γ ⊂ ND(Λ) and Λ ⊂ ND(Γ), arose naturally in the context of the quasi-isometric classification of non-uniform lattices in the works of Schwartz [52] (R-rank 1), Drut¸u [16] and Eskin [22] (higher rank).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I restate the theorems in the bounded setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 29 Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='28 (Eskin [22], Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9 above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be a real centre-free semisimple Lie group without compact factors and of higher rank, Γ ≤ G an irreducible non-uniform lattice, Λ ≤ G a discrete subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If Γ ⊂ ND(Λ) for some D > 0, then Λ is a lattice, and if moreover Λ ⊂ ND(Γ) then Λ and Γ are commensurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='29 (Schwartz [52], Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10 above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be a real simple Lie group of R-rank 1, Γ ≤ G an irreducible non-uniform lattice, Λ ≤ G a discrete subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If both Γ ⊂ ND(Λ) and Λ ⊂ ND(Γ) for some D > 0, then Λ is a lattice, and if moreover G is not locally isomorphic to SL2(R), then Λ is commensurable to Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The notable difference between the two statements is that for higher rank groups, the inclusion Λ ⊂ ND(Γ) is only required to prove commensurability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In view of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='8, this allows me to omit that assumption from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Notice also that for groups with property (T) the result easily follows from the (much more recent) result by Leuzinger in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the context of commensurability in the sublinear setting, I can only prove a limited result, Namely that Λ is commensurable to Γ if Γ is an irreducible Q-rank 1 lattice and both Γ ⊂ ND(Λ) and Λ ⊂ Nu(Γ) for some constant D > 0 and a sublinear function u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This is done via a reduction to the bounded case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be a real semisimple Lie group without compact factors and with finite centre, Γ ≤ G an irreducible lattice of Q-rank 1, Λ ≤ G a discrete subgroup, and u a sublinear function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If Γ ⊂ ND(Λ) for some D > 0 and Λ ⊂ Nu(Γ), then actually Λ ⊂ ND′(Γ) for some D′ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, if G is of R-rank 1, the conclusion holds under the relaxed assumption that u(r) ⪯∞ εr for some ε < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' While the setting of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 is indeed rather limited, the situation that both Γ ⊂ Nu(Λ) and Λ ⊂ Nu(Γ) arises naturally from the motivating example of SBE-rigidity in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Notice however that Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9 is not known for groups G that admit R-rank 1 factors, which is the only setting for which I can prove Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 A Reduction I start with the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The first step is to establish the fact that Λ must preserve WQ(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be a real semisimple Lie group without compact factors and with finite centre, Γ ≤ G an irreducible non-uniform lattice of Q-rank 1, Λ ≤ G a discrete subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume that Γ ⊂ Nu(Λ) and that Λ ⊂ Nu′(Γ) for sublinear functions u, u′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then Λ · WQ(Γ) ⊂ WQ(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, if G is of R-rank 1, the conclusion holds under the relaxed assumption that u′(r) ⪯∞ εr for some ε < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proof is similar to the argument of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='14, and uses the linear penetration rate of a geodesic into a horoball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let ξ ∈ WQ(Γ), and let HΓ be a horosphere bounding a Γ-free horoball HBΓ with HΓ ∩Γ·x0 a metric lattice in HΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume first that u′ is sublinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since HBΓ is Γ-free and Λ ⊂ Nu′(Γ), I can conclude that Λ · x0 ∩ HBΓ ⊂ Nu′(HΓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Recall (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='31) that every geodesic ray η with limit point ξ′ ∈ N π 2 (ξ) penetrates HBΓ at linear rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Therefore for every such geodesic ray η and every sublinear function v there is R = R(η, v) > 0 for which Nv(η↾r>R) is Λ-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' On the other hand, let λ ∈ Λ, and assume towards contradiction that λξ /∈ WQ(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then by Proposi- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='30 there is a Γ-conical limit point ξ′ ∈ N π 2 (λξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The hypothesis that Γ ⊂ Nu(Λ) then implies that for every R > 0, Nu(η↾r>R) ∩ Λ · x0 ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Translating by λ−1 yields a contradiction to the previous paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I conclude that λξ ∈ WQ(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I now modify the argument to include u′(r) ⪯∞ εr when G is of R-rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In this case, the only point ξ′ ∈ N π 2 (λξ) is λξ itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Therefore by the same argument as above, the assumption that λξ /∈ WQ(Γ) implies that the u-sublinear neighbourhood of every geodesic ray with limit point ξ intersects Λ · x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=', for every η with limit point ξ and every R > 0 it holds that Nu(η↾r>R) ∩ Λ · x0 ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' On the other hand, every such geodesic penetrates HBΓ at 1-linear rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This amounts to the following fact: if v′(r) = εr for some ε ∈ (0, 1), then for some R > 0, the set Nu(η↾r>R)∩Nv′(HΓ) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This is a contradiction to Λ ⊂ Nu′(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 30 Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume towards contradiction that there is a sequence λn such that d(λnx0, Γ · x0) > n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Recall that Γ·x0 is a cocompact metric lattice in the compact core of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This implies that there is a number D′ > 0 such that any λ ∈ Λ for which λx0 /∈ ND′(Γ · x0) must lie at least 1 2D′-deep inside a horoball of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I can assume that for all n ∈ N there are corresponding horoballs of Γ, which I denote HBΓ n, for which λn · x0 ∈ HBΓ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The fact that Γ ⊂ ND(Λ) then implies that ND(Λ · x0) covers a cocompact metric lattice in HΓ n, namely the metric lattice Γ · x0 ∩ HΓ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the terminology of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2, HΓ n is almost Λ-cocompact, or D-almost Λ-cocompact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I first prove that every horoball of Γ contains a Λ-free horoball (this is of course immediate if Λ ⊂ NC(Γ) for some C > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume towards contradiction that there is a horoball HBΓ of Γ that does not contain a Λ-free horoball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Denote HΓ := ∂HBΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the notations of the previous paragraph, I can assume without loss of generality that HBΓ = HBΓ n for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Denote by ξ the base point of HBΓ, fix some arbitrary x ∈ HΓ and consider the geodesic ray η := [x, ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The constraint that Λ ⊂ Nu(Γ) implies that for every R > 0 there is some L > 0 for which the ball B � η(L + t), R � is Λ-free, for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular, for all large enough n ∈ N (depending on R), the horosphere H(ξ, λnx0) that is parallel to HΓ and that passes through λnx0 contains a point that is the centre of Λ-free ball of radius R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This property is Λ-invariant, as well as the fact that HBΓ is based at WQ(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular, these two properties hold for the horoballs HBn := λ−1 n HBΓ, whose respective base points I denote ξn := λ−1 n ξ ∈ WQ(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Fix R = D +2DΓ (recall that DΓ is such that every horosphere H of Γ admits H ⊂ NDΓ(Γ·x0 ∩H)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let L = L(D+2DΓ) be the corresponding bound from the previous paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For every n > L the horoball HBn has bounding horosphere Hn that admits a point zn ∈ Hn for which B(zn, D+2DΓ) is Λ-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, the same is true for every horosphere that is parallel to Hn which lies inside HBn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since Γ ⊂ ND(Λ), this means that every horosphere that lies inside HBn admits a point that is the centre of a Γ-free ball of radius 2DΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I conclude that none of those horospheres could be the horosphere of Γ corresponding to the parabolic limit point ξn ∈ WQ(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since x0 ∈ Hn it must therefore be that Hn is a horosphere of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' But this contradicts the fact that zn ∈ Hn and B(zn, 2DΓ) is Γ-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This shows that no horoball of Γ contains a sequence of Λ-orbit points that lie deeper and deeper in that horoball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Put differently, it shows that every horoball of Γ contains a Λ-free horoball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I remark that the above argument shows something a bit stronger, which I will not use but which I find illuminating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It proves that as soon as d(λx0, Γ · x0) is uniformly large enough, say more than M, then λx0 must lie on a (D + 2DΓ)-almost Λ-cocompact horosphere parallel to HΓ, where HΓ is the bounding horosphere of any horoball of Γ in which λx0 lies (recall that it must lie in at least one such horoball).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' On the other hand if d(λx0, Γ · x0) < M, then since every point in the Γ-orbit lies on a horosphere of Γ one concludes that λx0 lies on a horosphere H based at WQ(Γ) that is (M + D)-almost Λ-cocompact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I can now assume that every HBΓ n contains a Λ-free horoball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular it contains a 1-almost maximal Λ-free horoball HBΛ n (see Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By definition there is a point λ′ nx0 that is at distance at most 1 from HΛ n = ∂HBΛ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Up to enlarging d(λnx0, Γ · x0) or decreasing it by at most 1, I can assume λn = λ′ n to begin with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Consider HBn := λ−1 n HBΛ n with Hn = ∂HBn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This is a sequence of horoballs, each of which contains a Λ-free horoball at depth at most 1, based at corresponding parabolic limit points ξn ∈ WQ(Γ), and tangent to points that are at distance at most 1 from x0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=', Hn ∩ B(x0, 1) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since Γ ⊂ ND(Λ) I conclude that each of the HBn contain a horoball of depth at most D + 1 that is Γ-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Therefore the horoball of Γ that is based at ξn must have its bounding horosphere intersecting B(x0, D + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='22 there are only finitely such horoballs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I conclude that there are finitely many points ξ′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' , ξ′ K ∈ WQ(Γ) such that for every n ∈ N there is i(n) ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' , K} with ξn = ξ′ i(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' From the Pigeonhole Principle there is some ξ′ ∈ {ξ′ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' , ξ′ K} for which ξn = ξ′ for infinitely many n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Passing to a subsequence I assume that this is the case for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' To begin with the HBΓ n are horoballs of Γ, and therefore as in the first case the bounding horospheres HΓ n are D + 2DΓ-almost Λ-cocompact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This is a Λ-invariant property and therefore the same holds for the λ−1 n translate of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' These are the horospheres which are based at ξ′ and lie outside HBn at distance d(λnx0, Γ · x0) > n − 1 from Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' They form a sequence of outer and outer horospheres based at the same point at WQ(Γ), all of which are D + 2DΓ-almost Λ-cocompact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This is a contradiction, since the union of such horospheres intersect every horoball of X, contradicting the existence of Λ-free horoballs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Formally, 31 take some ζ ∈ WQ(Γ) different from ξ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since both ξ′ and ζ lie in WQ(Γ), they admit dT (ζ, ξ′) = π and there is a geodesic η with η(−∞) = ξ′ and η(∞) = ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let HBΓ ζ be the horoball of Γ that is based at ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By the first step of this proof, every such horoball must contain a Λ-free horoball HBΛ ζ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Therefore there is some T > 0 such that for all t > T the point η(t) lies 2(D + 2DΓ) deep in HBΛ ζ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I conclude that for all t > T , B � η(t), 2D � is Λ-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' On the other hand, for arbitrarily large t it holds that the horosphere based at ξ′ and tangent to η(t) is D+2DΓ-almost Λ-cocompact, and in particular d � η(t), Λ·x0 � < D+2DΓ, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I conclude that Λ ⊂ ND′(Γ) for some D′ > 0, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the setting of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='30, Λ is a lattice commensurable to Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 The Arguments of Schwartz and Eskin The R-rank 1 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The statement of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10 is a slight modification of his original formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' His framework leads to a discrete subgroup ∆ ≤ G such that: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Every element of ∆ quasi-preserves the compact core of the lattice Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Namely, each element of ∆ is an isometry of X that preserves WQ(Γ) and that maps every horosphere of Γ to within the D = D(∆) neighbourhood of some other horosphere of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It holds that Γ ⊂ ND(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' From these two properties Schwartz is able to deduce that ∆ has finite covolume, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' that ∆ is a lattice in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Here is a sketch of his argument, which works whenever Γ is a Q-rank 1 lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the setting described above, ∆ is a lattice in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Consider X′ 0 := � g∈∆ g · X0, where X0 is the compact core of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This space serves as a ‘compact core’ for ∆: the fact that ∆ quasi-preserves X0 implies that X′ 0 ⊂ ND(X0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It is a ∆-invariant space, and therefore one gets an isometric action of ∆ on X′ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This action is cocompact: the reason is that Γ acts cocompactly on X0, and Γ ⊂ ND(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Formally, every point in X′ 0 is D-close to a point in X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Every point in X0 is DΓ-close to a point in Γ · x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Every point in Γ · x0 is D-close to a point in ∆ · x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Therefore the ball of radius 2D + DΓ contains a fundamental domain for the action of ∆ on X′ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It remains to see that the action of ∆ on X \\ X′ 0 is of finite covolume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' As a result of the cocompact action of ∆ on X′ 0, there is B := B(x0, R) so that X′ 0 ⊂ ∆·B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' X′ 0 is the complement of a union of horoballs, which one may call horoballs of ∆, with bounding horospheres of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The fact that Γ is of Q-rank 1 means that the horoballs of Γ are disjoint, and therefore those of Λ are almost disjoint: there is some C > 0 such that for every horosphere H of Λ and every point x ∈ H, d(x, X′ 0) < C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Up to enlarging the radius of B by C, I can assume that H ⊂ ∆ · B for every horosphere H of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Each horoball of Λ is based at WQ(Γ), and each lies uniformly boundedly close to the corresponding horoballs of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' From Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='22 one therefore sees that there are finitely many horoballs of ∆ that inter- sect B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Denote them by HB1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' , HBN, their bounding horospheres by Hi = ∂HBi, and their intersection with B by Bi := B ∩ HBi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let also ξi ∈ WQ(Γ) denote the base point of each HBi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Each Bi is pre-compact and therefore the projection of each Bi on Hi is pre-compact as well (this is a consequence e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' of the results of Heintze-Im hof recalled in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let Di ⊂ Hi be a compact set that contains this projection, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' PHi(Bi) ⊂ Di ⊂ Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular B ∩ Hi ⊂ Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Observe now that for every horoball HB of ∆, with bounding horosphere H = ∂HB, the ∆-orbit of every point x ∈ H intersects some Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' First notice that for x ∈ H the choice of B implies that the ∆-orbit of x must intersect B, say gx ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular gH ∩ B ̸= ∅, and since gHB is a horoball of ∆ then by definition gH = Hi for some i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=', N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' One concludes that indeed gx ⊂ Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, let y ∈ X is any point that lies inside a horoball HB of ∆, and x = PH(y) its projection on the bounding horosphere H = ∂HB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By the previous paragraph there is some g ∈ ∆ and i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' , N} for which gx ∈ Di, and therefore it is clear that gy lies on a geodesic emanating from Di to ξi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 32 Finally, define Cone(Di) to be the set of all geodesic rays that emanate from Di and with limit point ξi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The previous paragraph proves that �N i=1 Cone(Di) contains a fundamental domain for the action of ∆ on X \\ X′ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, the fact that Di ⊂ Hi is compact readily implies that each Cone(Di) has finite volume, and so this fundamental domain is of finite volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' To conclude, B ∪ � �N i=1 Cone(Di) � is a set of finite volume and it contains a fundamental domain for the ∆-action on X, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proof of commensurability of ∆ and Γ is given in full in [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There is one essential difference between Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='34, namely the assumption that Λ ⊂ ND(Γ) rather than quasi-preserving the compact core of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In Schwartz’s work, the fact that ∆ ⊂ ND(Γ) is not relevant (even though it easily follows from the construction of his embedding of ∆ in G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' He only uses the two properties described above, namely the quasi-preservation of X0 and Γ ⊂ ND(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The assumption that Λ quasi-preserves the compact core of Γ does not feel appropriate in the context of my thesis, while the metric condition Λ ⊂ ND(Γ) seems much more natural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It is a stronger condition as I now show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='32, Λ · WQ(Γ) ⊂ WQ(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let HΓ 1 be a horosphere of Γ, based at ξ ∈ WQ(Γ), and let γx0 ∈ HΓ 1 be some point on the metric lattice of Γ · x0 on HΓ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There is an element λ ∈ Λ such that d(λx0, γx0) < D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, since Λ ⊂ ND(Γ) one knows that the parallel horoball that lies D-deep inside HBΓ 1 is Λ-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let λ′ ∈ Λ be an arbitrary element of Λ, and consider λ′ · HΓ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The last statement in the previous paragraph is Λ-invariant, and so the horoball that lies D-deep inside λ′ · HBΓ 1 is Λ-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The fact that Γ ⊂ ND(Λ) then implies that the parallel horoball that lies 2D deep inside λ′HBΓ 1 is Γ-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let HΓ 2 be the horosphere of Γ that is based at λ′ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The last statement amounts to saying that HΓ 2 lies at most 2D-deep inside λ′HBΓ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' On the other hand, one has d(λ′λx0, λ′HΓ 1 ) = d(λx0, HΓ 1 ) ≤ D, so there is a Λ-orbit point that lies within D of λ′HΓ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The parallel horoball that lies D-deep inside HBΓ 2 must also be Λ-free, so I conclude that HΓ 2 must be contained in the parallel horoball to λ′HBΓ 1 which contains it and that is at distance D from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I conclude that d(λ′HΓ 1 , HΓ 2 ) ≤ 2D, and so that Λ quasi-preserves X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It is interesting to note that Schwartz’s arguments are similar in spirit to my arguments in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In fact, one could also prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10 using the same type of arguments that appear repeatedly in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2, namely by moving Λ-free horoballs around the space, specifically the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I do not present it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Higher rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Eskin’s proof is ergodic, and based on results of Mozes [42] and Shah [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I produce it here without the necessary preliminaries, which are standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' To prove that Λ is a lattice amounts to finding a finite non-zero G-invariant measure on Λ\\G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By Theorem 2 in [42], if P ≤ G is a parabolic subgroup then every P-invariant measure on Λ\\G is automatically G-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Fix a minimal parabolic subgroup P ≤ G and let µ0 be some fixed probability measure on Λ\\G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since P is amenable it admits a tempered Følner sequence Fn ⊂ P, and one can average µ0 along each Fn to get a sequence of probability measures µn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The weak* compactness of the unit ball in the space of measures on Λ\\G implies that there exists a weak* limit µ of the µn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The measure µ is automatically a finite P-invariant measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It remains to show that µ is not the zero measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' To see this it is enough to show that for some compact set CΛ ⊂ Λ\\G and some Λg = x ∈ Λ\\G, one has 0 < lim inf n 1 |Fn| � Fn 1CΛ(xp−1)dp (4) Fix some compact neighbourhood CΓ ⊂ Γ\\G of the trivial coset Γe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The hypothesis Γ ⊂ ND(Λ) implies that there is a corresponding compact neighbourhood CΛ ⊂ Λ\\G of the trivial coset Λe such that for any p ∈ P, it holds that Γgp−1 ∈ CΓ ⇒ Λgp−1 ∈ CΛ (simply take CΛ to be the D + 1-blowup of CΓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The action of P on Γ\\G is uniquely ergodic, therefore 0 < µΓ(CΓ) = lim n 1 |Fn| � Fn 1CΓ(Γp−1)dp 33 where µΓ denotes the natural G-invariant measure on Γ\\G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The defining property of CΛ ensures that Inequality (4) is satisfied, implying that µ is a non-zero P-invariant probability measure on Λ\\G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I conclude that µ is also G-invariant, and that Λ is a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If moreover Λ ⊂ ND(Γ), one may use Shah’s Corollary [54] to conclude that Λ is commensurable to Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 Translating Geometry into Algebra The goal of this section is to prove that the results of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 imply that Λ satisfies the hypotheses of the Benoist-Miquel criterion Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Namely, that Λ is Zariski dense, and that it intersects a horospherical subgroup in a cocompact indecomposable lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' These are algebraic properties, and the proof that Λ satisfies them is in essence just a translation of the geometric results of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 to an algebraic language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The geometric data given by Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 is that for some horosphere H bounding a Λ-free horoball, Λ ∩ StabG(H) · x0 intersects H in a cocompact metric lattice (Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='12), and that the set of Λ-conical limit points contains the set of Γ-conical limit points (Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Note that since K is compact the former implies that Λ ∩ StabG(H) is a uniform lattice in StabG(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 A Horospherical Lattice I assume that Λ ∩ StabG(H) is a lattice in StabGH, and I want to show that Λ intersects a horospherical subgroup U of G in a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This step requires quite a bit of algebraic background, which I give below in full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In short, the first goal is to show that StabG(H) admits a subgroup U ≤ StabG(H) that is a horospherical subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A lemma of Mostow (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='36 below) allows to conclude that Λ intersects U in a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='36 (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9 in [40]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let H be a Lie group having no compact connected normal semisimple non-trivial Lie subgroups, and let N be the maximal connected nilpotent normal Lie subgroup of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let Γ ≤ H be a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then N/N ∩ Γ is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the original statement Mostow uses the term ‘analytic group’, which I replaced here with ‘connected Lie subgroup’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This appears to be Mostow’s definition of an analytic group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Section 10, Chapter 1 in [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In Chevalley’s Theory of Lie Groups, he defines a Lie group as a locally connected topological group whose identity component is an analytic group (Definition 1, Section 8, Chapter 4 in [12]), and proves (Theorem 1, Section 4, Chapter 4 therein) a 1-1 correspondence between analytic subgroups of an analytic group and Lie subalgebras of the corresponding Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='36 lays the rationale for the rest of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Explicitly, I prove that StabG(H) admits a subgroup that is a horospherical subgroup U of G (Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='39), and that U is maximal connected nilpotent normal Lie subgroup of StabG(H) (Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In order to use Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='36, I show that the horospherical subgroup Nξ is a maximal normal nilpotent connected Lie subgroup of StabG(H)◦, and that StabG(H)◦ admits no compact normal factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This requires to establish the structure of StabG(H)◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the notation ht ξ = exp(tX) and Aξ = exp � Z(X) ∩ p � of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='7, define A⊥ ξ to be the codimension-1 submanifold of Aξ that is orthogonal to {hξ(t)}t∈R (with respect to the Killing form in the Lie algebra).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Every element a ∈ A⊥ ξ stabilizes H = H(x0, ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' An element in Aξ is an element that maps x0 to a point on a flat F ⊂ X that contains the geodesic ray [x0, ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If a ∈ A⊥ ξ , then the geodesic [x0, ax0] is orthogonal to [x0, ξ), and lies in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' From Euclidean geometry and structure of horospheres in Euclidean spaces, it is clear that ax0 ∈ H(x,ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since a ∈ Gξ, this means aH = H(ax0, ξ) = H(x, ξ) = H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let H be a horosphere based at ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then StabG(H)◦ = (KξA⊥ ξ )◦Nξ, and in particular it contains a horospherical subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, StabG(H)◦ is normal in StabG(ξ)◦ and acts transitively on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 34 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Clearly (KξA⊥ ξ )◦Nξ is a codimension-1 subgroup of StabG(ξ)◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since StabG(H) ̸= StabG(ξ) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' ht ξ /∈ StabG(H) for t ̸= 0), it is enough to show that (KξA⊥ ξ )◦Nξ ≤ StabG(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let kan ∈ (KξA⊥ ξ )◦Nξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It fixes ξ, so it is enough to show that kanx0 ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since k ∈ Kξ and kx0 = x0, it stabilizes H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' From Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 a ∈ StabG(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' So it remains to check that Nξ stabilizes H, but this is more or less the definition: fixing a base point x0, the horospheres based at ξ are parameterized by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Denote them by {Ht}t∈R, where H = H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In this parameterization, any element g ∈ Gξ acts on {Ht}t∈R by translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I can thus define for g ∈ StabG(ξ) the real number l(g) to be that number for which gHt = Ht+l(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Clearly l � hξ(t) � = t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The element n fixes ξ, so one has h−t ξ nht ξH0 = h−t ξ Ht+l(n) = Ht+l(n)−t = Hl(n) The fact that n ∈ Ker(Tξ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' that limt→∞ h−t ξ nht ξ = eG readily implies that necessarily l(n) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I conclude that (KξA⊥ ξ )◦Nξ = StabG(H)◦, as wanted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Next recall that StabG(H)◦ acts transitively on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let x, y ∈ H, and consider g ∈ StabG(ξ)◦ with gx = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Writing an element g ∈ Gξ as kata⊥n ∈ Kξht ξA⊥ ξ Nξ, the argument above shows that kht ξa⊥nH0 = H0 if and only if t = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=', if and only if g ∈ StabG(H)◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Finally, let g ∈ StabG(ξ) and h ∈ StabG(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By the discussion above h · Ht = Ht for all t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Clearly −l(g) = l(g−1), and therefore ghg−1H0 = ghH−l(g) = g · H−l(g) = H0 Therefore StabG(H) is normal in StabGξ, and the same is true for the respective identity components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' StabG(H)◦ is a connected Lie group with no connected compact normal semisimple non- trivial Lie subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Every compact subgroup of G fixes a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let H ≤ G be some closed subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It is standard to note that a normal N ≤ H that fixes a point x ∈ X must fix every point in the orbit H · x: hnh−1hx = hx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since H = StabG(H)◦ acts transitively on H, it shows that a normal compact subgroup of StabG(H)◦ fixes every point in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' An isometry fixing a horosphere pointwise while fixing its base point is clearly the identity, proving the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The following fact is well known but I could not find it in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A horosphere in X is not convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let H′ be some horosphere in X, with base point ζ ∈ X(∞), and assume towards contradiction that it is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Fix x ∈ H′ and a′ t the one parameter subgroup with η′(∞) = a′ tx, and denote H′ t = H(a′ tx, ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let eG ̸= n ∈ Nζ (Nζ defined with respect to a′ t in a corresponding Langlands decomposition), and consider the curve η′ n(t) := a′ tnx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I claim that this is a geodesic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' On the one hand, the fact that H′ is convex implies that the geodesic segment [x, nx] is contained in H′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Therefore a′ t[x, nx] = [a′ tx, a′ tnx] ⊂ H′ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' More generally it is clear that because a′ tH′ s = H′ s+t it holds that H′ t is convex for every t as soon as it is convex for some t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' On the other hand, for every point y ∈ [x, nx], d(y, H′ t) = t, and more generally for any y ∈ [a′ snx, a′ sx] it holds that d(y, H′ t) = |s − t|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular this is true for η′ n(t) = yt := a′ tnx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I get that d � η′ n(t), η′ n(s) � = |s − t|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Therefore η′ n is a geodesic (to be pedantic one has to show that η′ n is a continuous curve, which is a result of the fact that a′ t is a one parameter subgroup of isometries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Clearly d � η′ n(t), η′(t) � = d(a′ tnx, a′ tx) = d(nx, x) and therefore η′ n is at uniformly bounded distance to η′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This bounds d(ηn, η′ n) as bi-infinite geodesics, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' for all t ∈ R, not just as infinite rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The Flat Strip Theorem (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='13, Chapter 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 in [11]), then implies that the geodesics ηn, η′ n bound a flat strip: an isometric copy of R × [0, l] (where l = d(x, nx)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Up to now I did not use the fact that n ∈ Nξ, only that the point nx lies on a geodesic that is contained in H′ = H′ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Therefore the entire bi-infinite geodesic that is determined by [x, nx] lies on a 2-dimensional flat F that contains η′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The two elements n, a′ t therefore admit nx, a′ tx ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It is a fact that two such elements must commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I can conclude therefore that [n, a′ t] = eG, which contradicts the fact that that n ∈ Nζ = Ker(Tζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 35 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='42 (Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='13 in [51]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let N be a connected real Lie group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then Lie(N) is a nilpotent Lie algebra if and only if N is a nilpotent Lie group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='43 (Proposition 13, Section 4, Chapter 1 in [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the notation of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='7, nξ = Lie(Nξ) is a maximal nilpotent ideal in gξ = Lie(Gξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The presentation of nξ in [8] is given by means of the root space decomposition of StabG(ξ), that appears in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='13 in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There are two main objects in the literature that are referred to as the nilpotent radical or the nilradical of a Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' These are: (a) the maximal nilpotent ideal of the Lie algebra, and (b) the intersection of the kernels of all irreducible finite-dimensional representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proposition 13 in Section 4 of Chapter 9 in [7] shows that in the case of Lie algebras of parabolic Lie groups, these notions coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Nξ is a maximal connected nilpotent normal Lie subgroup of the identity connected com- ponent StabG(H)◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='42 implies Nξ is nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since StabGH ⊳ StabG(ξ), every normal subgroup of StabG(H) containing Nξ is in fact a normal subgroup of StabG(ξ), still containing Nξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It remains to prove maximality of Nξ among all connected nilpotent normal Lie subgroups of StabG(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Any such subgroup N ′ ⊳ StabG(ξ) gives rise to an ideal n′ of gξ = Lie � StabG(ξ) � , and by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='42 it is a nilpotent ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Therefore by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='43 it is contained in nξ = Lie(Nξ), implying that N ′ ≤ Nξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A lattice in StabG(H) intersects the horospherical subgroup Nξ in a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Corollaries 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='40 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='45 imply that the pair Nξ ⊳ StabG(H) satisfy the hypotheses of Mostow’s Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 Indecomposable Horospherical Lattices The Benoist and Miquel criterion requires the horospherical lattice to be indecomposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It is shown in [5] that if this lattice is contained in a Zariski dense discrete subgroup, then the indecomposability condition is equivalent to irreducibility of the ambient group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The precise definitions and statements are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='47 (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='14 in [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For a semisimple real algebraic Lie group G and U a horospherical subgroup of G, let ∆U be a lattice in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' ∆U is irreducible if for any proper normal subgroup N of G◦, one has ∆U ∩ N = {e}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' ∆U is indecomposable if one cannot write G◦ as a product G◦ = N ′N ′′ of two proper normal subgroups N ′, N ′′ ⊳ G with finite intersection such that the group ∆′ U := (∆U ∩ N ′)(∆U ∩ N ′′) has finite index in ∆U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='48 (See Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 in [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be a semisimple real algebraic Lie group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A discrete subgroup Λ ≤ G is said to be irreducible if, for all proper normal subgroups N ⊳ G, the intersection Λ ∩ N is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='49 (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3 in [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be a semisimple real algebraic Lie group, U ⊂ G a non-trivial horospherical subgroup, and ∆U ≤ U a lattice of U which is contained in a discrete Zariski dense subgroup ∆ of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then the following are equivalent: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' ∆ is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' ∆U is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' ∆U is indecomposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 36 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3 Zariski Density The last requirement is for Λ to be Zariski dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I use a geometric criterion which is well known to experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='50 (Proposition 2 in [31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let X be a symmetric space of noncompact type, G = Isom(X)◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A subgroup ∆ ≤ G is Zariski dense if and only if: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' ∆ does not globally fix a point in X(∞), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' ∆ ̸≤ StabG(ζ) for any ζ ∈ X(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The identity component of the Zariski closure of ∆ does not leave invariant any proper totally geodesic submanifold in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the proof I use several facts - mostly algebraic, and two geometric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I warmly thank Elyasheev Leibtag for his help and erudition in algebraic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The first property I need is very basic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let ∆ ≤ G be a discrete subgroup, and let H ≤ G be the Zariski closure of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then ∆ ∩ H◦ is of finite index in ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' H◦ is normal and of finite index in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The following fact is probably known to experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It appears in a recent work by Bader and Leibtag[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='52 (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9 in [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let k be a field, G a connected k algebraic group, P ≤ G = G(R) a parabolic subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then the centre of G contains the centre of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Still on the algebraic side, I need a Theorem of Dani, generalizing the Borel Density Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='53 (See [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let S be a real solvable algebraic group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If S = S(R) is R-split, then every lattice ΓS ≤ S is Zariski dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It is a fact (see Theorem 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 and Section 18 in [6]) that: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Every unipotent group over R is R-split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For a field k of characteristic 0, a solvable linear algebraic k-group is k-split if and only if its maximal torus is k-split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Finally I need two geometric facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The first is a characterization determining when does a unipotent element belongs to Nζ for some ζ ∈ X(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='55 (Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='8 in [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let X be a symmetric space of noncompact type and of higher rank, n ∈ G = Isom(X)◦ a unipotent element, and ζ ∈ X(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The following are equivalent: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For Nζ as in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='7, n ∈ Nζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For some geodesic ray η with η(∞) = ζ it holds that limt→∞ d � nη(t), η(t) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For every geodesic ray η with η(∞) = ζ it holds that limt→∞ d � nη(t), η(t) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The last property I need is a characterization of the displacement function for unipotent elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='56 (See proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 in [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let X be a symmetric space of noncompact type, ζ ∈ X(∞) some point and n ∈ Nζ an element of the unipotent radical of StabG(ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The displacement function x �→ d(nx, x) is constant on horospheres based at ζ, and for every ε > 0 there is a horoball HBε based at ζ such that d(nx, x) < ε for every x ∈ HBε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume that: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' � Λ ∩ StabG(H) � x0 is a cocompact metric lattice in a horosphere H ⊂ X bounding a Λ-free horoball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 37 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Every Γ-conical limit point is a Λ-conical limit point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then Λ is Zariski dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I show the criteria of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='50 are met, starting with Λ ̸≤ StabG(ζ) for any ζ ∈ X(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' To this end, I first prove that Λ · x0 is not contained in any bounded neighbourhood of any horosphere H′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let ξ′ be the base point of H′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By Hattori’s Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='31 (and Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='32), it is enough to find a Λ-conical limit point ζ′ with dT (ξ′, ζ′) ̸= π 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Take some ζ′′ ∈ X(∞) at Tits distance π of ξ′, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' take a flat F on which ξ′ lies and let ζ′′ be the antipodal point to ξ′ in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Fix ε = π 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3, there are neighbourhoods of the cone topology U, V ⊂ X(∞) of ξ′, ζ′′ (respectively) so that every point ζ′ ∈ V admits dT (ξ′, ζ′) ≥ dT (ξ′, ζ′′)− π 4 = 3 4π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Recall that the set of Γ-conical limit points is dense (in the cone topology), so the second hypothesis implies there is indeed a Λ-conical limit point in V and therefore at Tits distance different (in this case larger) than π 2 from ξ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I conclude that Λ · x0 is not contained in any bounded metric neighbourhood of any horosphere of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume towards contradiction that Λ ≤ StabG(ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I show that this forces Λ ∩ Nζ ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By Propo- sition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='55 it is enough to find a unipotent element λ ∈ Λ and a geodesic η with η(∞) = ζ such that limt→∞ d � λη(t), η(t) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let F be a maximal flat with ξ, ζ ∈ F(∞), x ∈ F some point and X, Y ∈ a ≤ p two vectors such that exp(tY ) = η(t) for the unit speed geodesic η = [x, ζ), and exp(tX) = η′(t) for the unit speed geodesic η′ = [x, ξ) (where a ≤ p a maximal abelian subalgebra in a suitable Cartan decomposition g = p ⊕ k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let StabG(ξ) = KξAξNξ be the decomposition described in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='7 with respect to Y (notice that Nξ does not depend on choice of Y , see item 3 of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='7 in [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The assumption that Λ ≤ StabG(ζ) implies that for any λ ∈ Λ the distance d � λη(t), η(t) � either tends to 0 as t → ∞ or is uniformly bounded for t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the latter case there is some constant c > 0 for which d � λη(t), η(t) � = c for all t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' As in the proof of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='41, the Flat Strip Theorem (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='13, Chapter 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 in [11]) implies that λ and at := exp(tY ) commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' From the first hypothesis of the statement and Mostow’s result (Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='46) I know that Λ ∩ Nξ is a cocompact lattice in Nξ (attention to subscripts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Therefore Λ ∩ Nξ is Zariski dense in Nξ (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, since commuting with an element is an algebraic property, an element g ∈ G that commutes with Λ∩Nξ must also commute with its Zariski closure, namely with Nξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This means that if at commutes with all Λ∩Nξ then it commutes with Nξ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' atn = nat for all t ∈ R and all n ∈ Nξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I know that at ∈ Aξ commutes with both Kξ and Aξ (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='7) therefore if at also commutes with Nξ then at lies in the centre of StabG(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This means that at is central in G (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For a group G with compact centre this cannot happen, so there is indeed some unipotent element λ ∈ Λ ∩ Nξ for which limt→∞ d � λη(t), η(t) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I conclude from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='55 that Λ ∩ Nζ ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The first paragraph of the proof implies in particular that Λ·x0 does not lie in any bounded neighbourhood of a horosphere H′ based at ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The assumption that Λ ⊂ StabG(ζ) implies that every λ ∈ Λ acts by translation on the filtration {H′ t}t∈R by horospheres based at ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Therefore as soon as Λ · x0 ̸⊂ Ht for some t ∈ R one concludes that ζ is a horospherical limit point of Λ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' that every horoball based at ζ intersects the orbit Λ · x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='56 it holds that for a unipotent element g ∈ Nζ the displacement function x �→ d(gx, x) depends only on the horosphere H′ t in which x lies and that, for xt ∈ H′ t it holds that limt→∞ d(gxt, xt) = 0 (up to reorienting the filtration t ∈ R so that η(t) ∈ H′ t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For a non-trivial element λζ ∈ Λ ∩ Nζ the previous paragraph therefore yields a sequence of elements λn ∈ Λ such that limn→∞ d(λζλnx0, λnx0) = 0, contradicting the discreteness of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I conclude that Λ ̸≤ StabG(ζ) for every ζ ∈ X(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume that H := � Λ Z�◦, the identity connected component of the Zariski closure of Λ, stabilizes a totally geodesic submanifold Y ⊂ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='51, Λ0 := Λ ∩ H is of finite index in Λ, therefore Λ0 ∩ StabG(H) is also a cocompact lattice in StabG(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The fact that � Λ0 ∩ StabG(H) � x0 is a cocompact metric lattice in H readily implies that � Λ0 ∩ StabG(H) � y is a cocompact metric lattice in Hy = H(y, ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This goes to show that there is no loss of generality in assuming x0 ∈ H ∩ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It follows that Λ0 ∩ StabG(H) · x0 ⊂ Y ∩ H, and therefore H ⊂ ND(Y ) for some D > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A horosphere is a codimension-1 submanifold, implying that Y is either all of X or of codimension-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The latter forces Y = H, which is impossible since H is not totally geodesic (H is not convex, see Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I conclude that H does not stabilize any totally geodesic 38 proper submanifold, and hence that Λ is Zariski dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='5 Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 I now complete the proof of the main sublinear rigidity theorem for Q-rank 1 lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If {dγ}γ∈Γ is bounded, then Λ is a lattice by Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='7 or Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9, depending on the R-rank of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If {dγ}γ∈Γ is unbounded, then Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='12 and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='27 both hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In R-rank 1 the proof again follows immediately from Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='7 using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='16 and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In higher rank, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 allows one to conclude that Λ is an irreducible, discrete, Zariski dense subgroup that contains a horospherical lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4, this renders Λ a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It is a Q-rank 1 lattice as a result of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The sublinear nature of the hypothesis in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6 induces coarse metric constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A horospherical lattice on the other hand is a very precise object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It is not clear how to produce unipotent elements in Λ, or even general elements that preserve some horosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proof sketched above produces a whole lattice of unipotent elements in Λ (this is Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='46);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' it is also the only proof that I know which produces even a single unipotent (or parabolic) element in Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 5 Lattices with Property (T) Recall that a lattice in a locally compact group G has property (T) if and only if G has property (T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In this section I prove: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be a real centre-free semisimple Lie group without compact factors, Γ ≤ G a lattice, Λ ≤ G a discrete subgroup such that Γ ⊂ Nu(Λ) for some sublinear function u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If Γ has property (T), then Λ is a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' As in the case of uniform lattices, lattices with property (T) admit the stronger version of ε-linear rigidity, for suitable ε = ε(G): Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be a real centre-free semisimple Lie group without compact factors, Γ ≤ G a lattice and Λ ≤ G a discrete subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If Γ has property (T) then there exists ε = ε(G) > 0 depending only on G such that if Γ ⊂ Nu(Λ) for some function u(r) ⪯∞ εr, then Λ is a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Clearly Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 implies Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I thank Emmanuel Breuillard for suggesting this generaliza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' From now and until the end of this section, the standing assumptions are those of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lattice Criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For groups with property (T) I use a criterion by Leuzinger, stating that being a lattice is determined by the exponential growth rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The formulation requires a definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Given a pointed metric space (X, dX, x0), denote: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' bX(r) = |B(x0, r)| 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' bu X(r) = supx∈X |B(x, r)| When a group ∆ acts on a pointed metric space X, the orbit ∆ · x0 together with the metric induced from X is a pointed metric space (∆ · x0, dX↾∆·x0, x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In this setting b∆·x0(r) = |BX(xo, r) ∩ ∆ · x0|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' When the action is by isometries, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' ∆ ≤ Isom(X), it is straightforward to observe that this quantity does not depend on the centre of the ball, and so bu ∆·x0(r) = b∆·x0(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The pointed metric spaces of interest in this section are the Γ and Λ orbits in the symmetric space X = G/K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 39 Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let X be a symmetric space and ∆ ≤ G = Isom(X)◦ a subgroup of isometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The critical exponent of ∆ is defined to be δ(∆) := lim sup r→∞ log � b∆·x0(r) � r Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Throughout this section there is no risk of ambiguity, and I allow myself to ease notation and let b∆(r) = b∆·x0(r), bu ∆(r) = bu ∆·x0(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' To a semisimple Lie group G one can associate a quantity ∥ρ∥, where ρ = ρ(G) is the half sum of positive roots in the root system of (g, a) (see Section 2 in [36]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6 (Theorem 2 in [36]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be a real centre-free semisimple Lie group without compact factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let ∆ be a discrete, torsion-free subgroup of G that is not a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If G has Kazhdan’s property (T ), then there is a constant c∗(G) (depending on G but not on ∆) such that δ(∆) ≤ 2∥ρ∥ − c∗(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It is known that the critical exponent of a discrete subgroup ∆ ≤ G is bounded above by 2∥ρ∥ (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 in [36]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, every lattice Γ ≤ G admits δ(Γ) = 2∥ρ∥ (Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='5 in [36], Theorem C in [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Combining these facts with Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6 yield: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='7 (Theorem B in [36]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be a real centre-free semisimple Lie group without compact factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let ∆ be a discrete, torsion-free subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If G has Kazhdan’s property (T ), then ∆ is a lattice iff δ(∆) = 2∥ρ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Line of Proof and the Use of ε-Linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 goes by showing that ε-linear distortion cannot decrease the exponential growth rate by much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This fact is essentially manifested in a proposition by Cornulier [13], stated here in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This is the only use I make of ε-linearity, and the computations involved are straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 is then an immediate consequence of Leuzinger’s criterion Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 In his study on SBE maps, Cornulier proves the following growth discrepancy result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='8 (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6 in [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let X, Y be two pointed metric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let u be a non-decreasing sublinear function and p : X → Y a map such that for some L, R0 > 0: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' |p(x)| ≤ max(|x|, R0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' p(|x|) ≤ |x| for all large enough x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' dY � p(x), p(x′) � ≥ 1 LdX(x, x′) − u(max{|x|, |x′|}) Then for all r > R0, bY (r) ≥ bX(r)/bu X � L · u(r) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I need a slightly modified version of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='8: Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let X, Y be two pointed metric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let u be a non-decreasing function that admits u(r) ⪯∞ εr for some ε < 1, and p : X → Y a map such that for some L, R0 > 0: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' |p(x)| ≤ max(|x| + u(|x|), R0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' |p(x)| ≤ |x| + u(|x|) for all large enough x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' dY � p(x), p(x′) � ≥ 1 LdX(x, x′) − u(max{|x|, |x′|}) Then for all r > R0, bY (r) ≥ bX � r − u(r) � /bu X � L · u � r − u(r) �� Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Repeat verbatim the proof for Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 40 Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let u(r) ⪯∞ ε · r for some ε < 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume that Γ ⊂ Nu(Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then δ(Λ) ≥ (1 − 4ε) · δ(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Restricting to ε < 1 2 stems from a 2 factor that appears in the proof and could possibly be dropped using a slightly more sophisticated approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For my needs this is more than enough, since in any case I eventually restrict attention to a small interval around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10 can be formulated in a slightly more general fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Using the notation δ(W) := lim supr→∞ log � bW (r) � r for a general subset W in a general metric space X, the following general version holds: Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let (X, x0) be a pointed metric space, Y, Z ⊂ X two subsets, and u(r) ⪯∞ εr for some ε < 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume that bu Z(r) = bZ(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If Z ⊂ Nu(Y ), then δ(Y ) ≥ (1 − 4ε) · δ(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, if u is sublinear, then δ(Z) = δ(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10 holds even when the group G does not have property (T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10 follows from Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='12 because the fact that Γ is a group of isometries implies bu Γ(r) = bΓ(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof (Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Observe that the closest point projection pY : Z → Y defined by z �→ zy for some point in the closed ball zy ∈ B � z, u(|z|) � admits: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' |yz| ≤ |z| + u(|z|) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' d � yz, yz′� ≥ d(z, z′) − 2u(max{|z|, |z′|}) The first item follows from triangle inequality: |yz| ≤ d(yz, z) + d(z, x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The second item follows from the quadrilateral inequality, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=', using triangle inequality twice along the quadrilateral [z, z′, yz′, yz].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The above properties allow me to use Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9 with constant L = 1 and function u′ = 2u to get bY (r) ≥ bZ � r − u′(r) � /bu Z � u′� r − u′(r) �� Since I assume bu Z = bZ, I can omit the superscript u in the last expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Recalling the definition δ(W) = lim supr→∞ bW (r) r ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' it remains to prove: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='lim sup ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='r→∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='r · log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='bZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='r − u′(r) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='/bZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='u′� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='r − u′(r) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='≥ (1 − 4ε) · δ(Z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='The proof of this inequality involves nothing more than log rules and arithmetic of limits: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='lim sup ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='r→∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='r · log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='bZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='r − u′(r) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='/bZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='u′� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='r − u′(r) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='= lim sup ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='r→∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='r · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='bZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='r − u′(r) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='− log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='bZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='u′� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='r − u′(r) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='≥ lim sup ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='r→∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='r · log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='bZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='r − u′(r) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='− lim sup ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='s→∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='�1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='s log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='bZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='u′� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='s − u′(s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='= lim sup ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='r→∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='r · log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='bZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='r − u′(r) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='− lim sup ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='s→∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='s log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='bZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='u′� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='s − u′(s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='≥ lim sup ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='r→∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='r · log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='bZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='r − 2εr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='− lim sup ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='s→∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='s log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='bZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2εs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='= (1 − 2ε)δ(Z) − 2εδ(Z) = (1 − 4ε)δ(Z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='(5) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='Below I justify the steps in the above inequalities: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='41 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' First equality is by rules of log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Second and third inequalities are by arithmetic of limits: let (an)n, (bn)n be two sequences of positive numbers, and A = lim supn an, B = lim supn bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then lim sup(an − bn) ≥ lim supn(an − B) = A − B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Fourth inequality: u′(r) < 2ε(r) for all large enough r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Fifth equality: definition of δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This completes the proof in the general case, which is what is needed for the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For the more refined statement in the case u is sublinear, one has to show a bit more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' From inequality 5 (specifically from the fourth line of the inequality) it is clearly enough to prove: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' lim supr→∞ 1 r · log � bZ � r − u′(r) �� = δ(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' lim sups→∞ 1 s log � bZ � u′� s − u′(s) ��� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Starting from the second item, indeed it holds that 1 s log � bZ � u′� s − u′(s) ��� = u′� s − u′(s) � s log � bZ � u′� s − u′(s) ��� u′� s − u′(s) � Clearly lim sup of the right factor in the above product is bounded by δ(Z), and in particular it is uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' On the other hand sublinearity of u′ implies that the left factor tends to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I conclude that this product tends to 0 as s tends to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It remains to prove lim supr→∞ 1 r · log � bZ � r − u′(r) �� = δ(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In a similar fashion, 1 r log � bZ � r − u′(r) �� = r − u′(r) r log � bZ � r − u′(r) �� r − u′(r) The left factor limits to 1 by sublinearity of u′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The right factor is nearly the expression in the definition of δ(Z), and I want to prove that indeed taking lim sup of it equals δ(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A priori {r − u′(r)}r∈R>0 is just a subset of R>0, so changing variable and writing t := r−u′(r) requires a justification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' But there is no harm in assuming that u′ is a non-decreasing continuous function, hence R≥R ⊂ {r − u′(r)}r∈R>0 for some R ∈ R>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Therefore for any sequence rn → ∞ there is a sequence r′ n with rn = r′ n − u′(r′ n) for all large enough n (note that in particular r′ n → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the other direction, for every sequence r′ n → ∞ there is clearly a sequence rn → ∞ for which rn = r′ n − u′(r′ n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I conclude lim sup r→∞ log � bZ � r − u′(r) �� r − u′(r) = lim sup r→∞ 1 r · log � bZ(r) � = δ(Z) This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Define ε(G) = c∗(G) 4·2∥ρ∥, and assume u(r) ⪯∞ ε(G) · r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Notice that ε(G) < 1 2, and since δ(Γ) = 2∥ρ∥ Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10 gives δ(Λ) ≥ � 1 − 4ε(G) � 2∥ρ∥ = 2∥ρ∥ − 4ε(G) · 2∥ρ∥ ≥ 2∥ρ∥ − c∗(G) By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6, Λ is a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 42 Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The question of existence of interesting groups that coarsely cover a lattice is a key question that arises naturally in the context of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The first question that comes to mind is whether there exist groups that are not commensurable to a lattice but that sublinearly, or even ε-linearly, cover one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Perhaps the growth rate point of view could be used to rule out groups that cover a lattice ε-linearly but not sublinearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 6 SBE Rigidity for Lattices of Higher Q-Rank 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 Sublinear Distortion and SBE Maps Denote a ∨ b := max(a, b) for a, b ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For a pointed metric space (X, x0, dX) and x, x1, x2 ∈ X, denote |x|X := dX(x, x0) and |x1 −x2|X := dX(x1, x2) (or simply |x| and |x1 −x2| when there is no ambiguity about the space X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Following Cornulier [14], Pallier [44] makes the following definition: Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A function u : R≥0 → R is admissible if it satisfies the following conditions: u is non-decreasing u grows sublinearly: lim sup r�→∞ u(r) r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' u is doubling: u(tr) u(r) is bounded above for all t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' u ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The focus in this paper is condition 2, namely that the function u is strictly sublinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I moreover require it to be subadditive, resulting in the following terminology which I use from now on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A function u : R≥0 → R is sublinear if it is admissible and subadditive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' u(t + s) ≤ u(t) + u(s) for all t, s > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' From now on by an SBE I mean an (L, u)-SBE where u is sublinear in the sense of Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 SBE Rigidity Two finitely generated groups Γ and Λ are said to be SBE if they are SBE when viewed as metric spaces with some word metrics and base points eΓ, eΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Observe that every quasi-isometry is an SBE, and in particular the word metric is an SBE-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' An SBE admits an SBE inverse, defined as quasi-inverse maps are defined for quasi-isometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A class of groups A is said to be SBE complete if, for every finitely generated group Λ that is SBE with some group Γ ∈ A, there is a short exact sequence 1 → F → Λ → Λ1 → 1 for a finite group F ≤ Λ and some Λ1 ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In this chapter I prove: Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be a real centre-free semisimple Lie group without compact or R-rank 1 factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The class of uniform lattices of G is SBE complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The class of non-uniform lattices of G is SBE complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 43 Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proof I present is quite indifferent to whether the lattice Γ is uniform or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In order to have a unified proof and to ease notation, I fix the convention that for both uniform and non-uniform lattices, X0 denotes the compact core of the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This just means that X = X0 in case Γ is uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' My works heavily relies on that of Drut¸u in [16], where the theorems are stated for non-uniform lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Nonetheless one readily sees that her proofs work perfectly well for uniform lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Indeed the arguments of [16] are only much simpler in the uniform case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 The Quasi-Isometry Case The outline of the proof I present for Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 is identical to that of quasi-isometric rigidity, which I now describe briefly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The main step is that for any quasi-isometry f : X0 → X0 of the compact core of Γ, there exists an isometry g : X → X such that f, g are boundedly close, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' there is some D > 0 for which d � f(x), g(x) � < D for all x ∈ X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let Λ be an abstract group with a quasi-isometry q : Λ → Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Using Lubotzki-Mozes-Raghunathan ([38], see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='19 above), Γ is quasi-isometrically embedded in X as the compact core X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' One can thus extend q to a quasi-isometry q0 : Λ → X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A conjugation trick allows to associate to each λ ∈ Λ a quasi-isometry fλ : X0 → X0 defined by fλ := q ◦ Lλ ◦ q−1 (Lλ : Λ → Λ is the left multiplication by λ in Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By the first paragraph, there exists gλ ∈ Isom(X) that is boundedly close to fλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, the proof also shows that the bound D depends only on the quasi-isometry constants of fλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' These could be seen to depend only on q and not on any specific λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' From this one concludes that the map λ �→ gλ is a group homomorphism Φ : Λ → G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It is then straightforward to show that Φ has finite kernel and that Γ ⊂ ND � Im(Φ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' One then uses Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 (for higher rank groups, see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3) to deduce that Im(Φ) is a non-uniform lattice in G that is commensurable to Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 The SBE Case Moving to SBE rigidity, one starts with an SBE q : Λ → Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The first step is to find an isometry of X that is close to an SBE of X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Drut¸u’s proof is preformed in the asymptotic cone of X, which allows for a smooth transition to the SBE setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Indeed, given an SBE f : X0 → X0, one can find an isometry g : X → X that is close to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The difference is that in the SBE setting, these maps are only sublinearly close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let (X, x0) be a pointed metric space, and f, g : X → X be two maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Maps f, g are said to be sublinearly close maps on X if there is a sublinear function u such that d � f(x), g(x) � ≤ u(|x|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='7 (Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let f : X0 → X0 be an SBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then there exists a unique isometry g ∈ Isom(X) that is sublinearly close to f (in X0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' From this point on, one would like to continue as in the quasi-isometry case: define the map Φ : Λ → G in a similar fashion and show that Γ ⊂ Nu � Im(Φ) � for some sublinear function u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' That Im(Φ) is a lattice is then a result of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6, proving Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There is however one additional obstacle that is unique to the SBE setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Namely the SBE constants of fλ do depend on λ, and the resulting sublinear bound on d � fλ(x), gλ(x) � in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='7 is not enough to define Φ properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' As far as I can see, one needs to get some uniform control on that bound in terms of the SBE constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The following statement is enough: Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='8 (Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let {fr}r∈R>0 be a family of (L′, vr)-SBE maps fr : X0 → X0, where vr(s) = L′·v(s)+v(r) for some sublinear function v ∈ O(u) and a constant L′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let gr be the associated isometry given by Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then for any x ∈ X0, there is a sublinear function ux ∈ O(u) such that d � fr(x), gr(x) � ≤ ux(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This type of uniform control is often needed when working with SBE maps, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Section I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3 in Pallier’s thesis [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Using it, I am able to complete the argument as in the quasi-isometry case and prove: Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be as in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the notations described above, the map Φ : Λ → G is a group homomorphism with Ker(Φ) finite, and there is a sublinear function u such that for Λ1 := Im(Φ) it holds that Γ ⊂ Nu(Λ1) and Λ1 ⊂ Nu(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 44 Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 is an immediate corollary of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3 Outline This section is divided into two parts that correspond to the steps of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3 deals with the task of finding an isometry that is sublinearly close to an SBE, and Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 establishes the properties of the map Φ : Λ → G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I keep Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3 slim and concise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The main reason for this choice is that the proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10 is merely a mimic of Drut¸u’s argument in [16], or an adaptation of it to the SBE setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' While these adaptations are somewhat delicate, giving a complete detailed proof would require reproducing Drut¸u’s argument more or less in full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I felt that this is not desirable, and instead I only indicate the required adaptations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I believe that a reader who is familiar with Drut¸u’s argument and with asymptotic cones could easily produce a complete proof using these indications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular, there is no preliminary section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I do not present buildings or dynamical results that go into Drut¸u’s argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I only shortly present asymptotic cones and some ideas from Drut¸u’s proof of the quasi-isometry version of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 is elementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3 SBE Maps are Close to Isometries In this section I indicate how to adapt Drut¸u’s arguments in [16] in order to prove: Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There is a sublinear function v = v(L, u) such that for every � L, u � SBE f : X0 → X0, there exists a unique isometry g = g(f) ∈ Isom(X) such that d � f(x), g(x) � ≤ v(|x|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 requires some control on the sublinear distance between f and g, in terms of the sublinear constants of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This is the meaning of the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let {fr}r∈R>0 be a family of (L′, vr)-SBE maps fr : X0 → X0, where vr = L′ · v + v(r) for some sublinear function v ∈ O(u) and a constant L′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let gr be the associated isometry given by Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then for any x ∈ X0, there is a sublinear function ux ∈ O(u) such that d � fr(x), gr(x) � ≤ ux(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Combined with Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10, a different way to phrase the above statement is to say that the function D : Λ× X0 → R≥0 defined by D(λ, x) = d � fλ(x), gλ(x) � is sublinear in each variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=', there is a function u : R≥0 × R≥0 → R≥1 such that u is sublinear in each variable and D(λ, x) ≤ u(|λ|, |x|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I begin with a short presentation of asymptotic cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I then give an account of the original proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10 when f : X0 → X0 is a quasi-isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I present the routines required to modify the proof for the SBE setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I exemplify the modification procedure in a specific representative example, and finish with a road map for proving Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10 and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='11 using the aforementioned routines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 Asymptotic Cones Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let (X, d) be a metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Fix an ultrafilter ω, a sequence of points xn ∈ X and a sequence of scaling factors ın −→ ω 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The asymptotic cone of X w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' xn, ın, denoted C(X), is the metric ω-ultralimit of the sequence of pointed metric spaces (X, 1 ın · d, xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The metric on C(X) is denoted dω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' See Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 in [16] for an elaborate account, including the definitions of ultrafilters and ultralimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The strength of SBE maps is that they induce bi-Lipschitz maps between the respective asymptotic cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='14 (See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Cornulier [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let f : X → Y be an (L, u)-SBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then f induces an L-bi-Lipschitz map C(f) : C(X) → C(Y ) between the corresponding asymptotic cones with the same scaling factors C(X) = (X, 1 ın dX, x0 n) and C(Y ) = (Y, 1 ın dY , y0 n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 45 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 The Argument A High-Level Description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The core of the argument lies in elevating an SBE f0 : X0 → X0 to an isometry g0 ∈ G = Isom(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There are two gaps to fill: first, Γ is non-uniform and so f0 is not even defined on the whole space X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' And obviously, f0 is just an SBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume for a moment that Γ is uniform and that f is defined on the whole space X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Elevating f : X → X to an isometry is done by considering the map C(f) : C(X) → C(X) that f induces on an asymptotic cone C(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This map is bi-Lipschitz, and the work of Kleiner and Leeb [32] allows one to conclude that C(f) is, up to a scalar, an isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In turn, this isometry induces an isometry ∂g on the spherical building structure of X(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This is done by the relation between the Euclidean building structure of C(X) and the spherical building structure of ∂∞X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A theorem of Tits [56] associates to ∂q a unique isometry g ∈ Isom(X) that induces ∂f as its boundary map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By construction, it is then not too difficult to see that g and f are ‘close’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In case Γ is non-uniform, an SBE f : Γ → Γ does not readily yield a cone map on C(X), but only on C(X0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Overcoming this difficulty requires substantial work and is the heart of Drut¸u’s proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In short, she uses dynamical results stating that the vast majority of flats in X are close enough to X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' As mentioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2, the building structure on X(∞) is determined by the boundaries of flats F ⊂ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The same holds for the (Euclidean) building structure of C(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Therefore the fact that the majority of flats in X are ‘close enough’ to X0 results in the fact that C(X0) composes the majority of C(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This is a very rough sketch of the logic behind Drut¸u’s argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The procedure described above results in an isometry g ∈ Isom(X) associated to f0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' To complete the argument one needs to verify that the map f0 �→ g is a group homomorphism between SBE(Γ) = SBE(X0) and Isom(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Composing this map with a representation of Λ into SBE(Γ) yields a map Λ → G by λ �→ fλ �→ gλ := gfλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A computation then shows that this map has finite kernel and that Γ lies in a sublinear neighbourhood of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Flat Rigidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The adaptations that are required for the SBE setting lie mainly in the part of Drut¸u’s work that concerns flat rigidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' That is, the proof that the quasi-isometry q0 maps a flat F ⊂ X0 to within a uniformly bounded neighbourhood of another flat F ′ ⊂ X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This is proved by passing to the cone map, using an analogous result for bi-Lipschitz maps between Euclidean buildings, which translates back down to the space X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Drut¸u’s argument requires many geometric and combinatorial definitions - some classical and widely known (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=', Weyl chambers of a symmetric space X) some less known (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' an asymptotic cone with respect to an ultrafilter ω) and some new (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=', the horizon of a set A ⊂ X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I use her definitions, terminology and notations freely without giving the proper preliminaries or even the definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I assume most readers who are interested in the question of SBE rigidity are familiar to some extent with most of these objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For the new definitions, I try to say as little as needed to allow the reader to follow the argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proof consists of 6 steps: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The horizon of an image of a Weyl chamber is contained in the horizon of a finite union of Weyl chambers, and the number of chambers in this union depend only on the Lipschitz constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' (Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='5, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6 in [16], consult Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='16 below for a sketchy definition of horizon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The horizon of an image of a flat coincides with the horizon of a finite union of Weyl chambers, and the number of chambers depends only on the Lipschitz constant of the quasi-isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The union of Weyl chambers in the previous step limits to an apartment in the Tits building at X(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Such a union is called a fan over an apartment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For each Weyl chamber W there corresponds a unique chamber W ′ such that q0(W) and W ′ have the same horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This amounts to an induced map on the Weyl chambers of the Tits building at X(∞) (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 in [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 46 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Given a flat F through a point x, the unique flat F ′ asymptotic to the union of Weyl chambers obtained in step 3 is at uniform bounded distance from f0(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The bound depends only on the quasi-isometry constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The flat F ′ is called the flat associated to F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If F1 and F2 are two flats through x which intersect along a hyperplane H, then the boundaries at X(∞) of the associated flats F ′ 1 and F ′ 2 intersect along a hyperplane of the same codimension as H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For a precise definition of horizon see [16], section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For now, it suffices to say the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The horizon of a set A ⊂ X is contained in the horizon of a set B ⊂ X if, looking far away at A from some point x ∈ X, A appears to be contained in an ε-neighbourhood of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This intuition is made precise by considering the angle at x that a point a ∈ A makes with the set B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Two sets have the same horizon if each set’s horizon is contained in the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the case A and B have the same horizon, an important aspect is the distance R starting from which A and B seem to be ε-contained in one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Call this distance the horizon radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It depends on x and ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proofs for most of these steps have similar flavour: in any asymptotic cone C(X), f0 induces a bi-Lipschitz map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Kleiner and Leeb [32] proved many results about such maps between cones of higher rank symmetric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' One assumes towards contradiction that some assertion fails (say, in step 5, assume that there is no bound on the distance between f0(xn) and the associated flat F ′ n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This gives an unbounded sequence of scalars (say, d � f0(xn), F ′ n � = ın → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' These scalars are used to define a cone in which one obtains a contradiction to some fact about bi-Lipschitz maps (say, that the point [q0(xn)]ω is at dω-distance 1 from [F ′ n]ω, while it should lie in [F ′ n]ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Typically, the bounds obtained this way depend on the quasi-isometry constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A priori, they also depend on the specific point x ∈ X or flat F ⊂ X in which you work (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' the horizon radius for the chambers in step 3 or the bound on d � q(x), F ′� in step 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' However, it is easy to see that in the quasi-isometry setting, the bounds are actually independent of the choice of point/flat/chamber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This independence stems from the fact that one can pre-compose f0 with an isometry translating any given point/flat/chamber to a fixed point/flat/chamber (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' ), without changing the quasi-isometry constants (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='11 in [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The fact that these bounds depend only on the quasi-isometry constants is essential for the proof that the map Λ → Isom(X) has the desired properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moving to the SBE setting, the essential difference is exactly that the bounds one obtains depend on the specific point, Weyl chamber or flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Indeed it is clear that these bounds should depend on the size |x|, as they depend on the additive constant in the quasi-isometry case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It is sensible to guess though that the bounds only grow sublinearly in |x|, which is enough in order to push the argument forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the next section I show how to elevate a typical cone argument from the quasi-isometry setting to the SBE setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I focus on showing that the bound one obtains depend only on the SBE constants (L, u) and sublinearly |x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3 Generalization to SBE: Adapting Cone Arguments To adapt for the SBE setting, split each step into three sub-steps: the first two amount to proving Theo- rem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10, and the third step amounts to proving Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Sub-Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Repeat the argument of the quasi-isometry setting verbatim, to obtain a bound c = c(x) which depend on the point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Sub-Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume towards contradiction that there is a sequence of points xn for which limω c(xn) |xn| ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This means limω |xn| c(xn) ̸= ∞, and so the point (xn)n lies in the cone C(X) = Cone � X, x0, c(xn) � , and one may proceed as in the corresponding quasi-isometry setting to obtain a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Sub-step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Fix x ∈ X and a sequence of SBE maps as in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='11, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' {fn}n∈N with the same Lipschitz constant and with sublinear constants vn(s) = v(s) + u(n), for some sublinear functions u, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Denote by cn(x) the constants that were achieved in the previous steps for x and the SBE map fn, and assume towards 47 contradiction that |cn(x)| is not bounded above by any function sublinear in n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This means in particular that limω u(n) cn(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' One concludes that the cone map C(frn) is bi-Lipschitz, and gets a contradiction in the same manner as in the first step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In order to give the reader a sense of what is actually required, I now demonstrate this procedure in full in a specific claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I chose to do this for proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='7 of [16], which is complicated enough to require some attention to details, but not too much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The statement is as follows: Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='18 (SBE version of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='7 in [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let f : X → X be an (L, u)-SBE, and F ⊂ X a flat through x to which f associates a fan over an apartment, ∪p i=0Wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If F ′ is the maximal flat asymptotic to the fan, then d � f(x), F ′� ≤ c(x) where c(x) = c(|x|) is sublinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, let fn : X → X be a sequence of (L, vn) SBE maps for vn = v + u′(n) for v, u′ some sublinear functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The constant cn(x) associated to x and the SBE fn achieved in the first part of the proposition admits cn(x) ≤ ux(n) for some sublinear function ux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' F ′ is then said to be the associated flat to F by f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proceed in two (sub-)steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I show that for a given x, there exists such a constant c(x) independent of the flat F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This is done exactly as in [16], but I repeat the proof here because it contains the terminology and necessary preparation for the second step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Fix x ∈ X and assume towards contradiction that there exists a sequence Fn of flats through x and a sequence fn : X → X of (L, u)-SBE maps such that cn := d � fn(x), F ′ n � → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In Cone(X, x, c−1 n ) one can show that [∪p i=0W n i ], the union of Weyl chambers associated to fn(Fn), is a maximal flat (see Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6 in [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Denote Fω := [∪p i=0W n i ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Furthermore, since the bi-Lipschitz flat [fn(Fn)] ⊂ Cone(X, x, c−1 n ) is contained in it, it coincides with it: [fn(Fn)] = Fω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' On the other hand, since the Hausdorff distance between ∪p i=0W n i and F ′ n is by assumption cn = d(x, F ′ n), in the cone the maximal flats Fω and F ′ ω := [F ′ n] are at Hausdorff distance 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This implies that Fω = F ′ ω (see Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 in [32]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' But since d � q(x), F ′ n � = cn the limit point yω := Q(xω) = [q(x)] , which is contained in Fω, is at distance 1 from F ′ ω - a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume c(x) is taken to be the smallest possible for each x, and then modify the function c so that c(x) = maxy:|y|=|x| c(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The function c : X → R now only depends on |x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I wish to show that c(|x|) = O(u(|x|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume towards contradiction that there exists a sequence xn with |xn| → ∞ such that limω c(xn) u(|xn|) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Denote cn = c(xn) and consider the cone Cone(X, xn, c−1 n ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The assumption limω c(xn) u(|xn|) = ∞ implies (x0)ω = (xn)ω hence Cone(X, xn, cn) = Cone(X, x0, cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By the definition of c(x) this means that there is a sequence of flats Fn through xn such that d � q(xn), F ′ n � = cn, so one may proceed as in step 1 for a cone with a fixed base point (x0)ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In this cone the flat F ′ ω := [F ′ n] is at distance 1 from the point [q(xn)], which lies on the maximal flat [∪p i=0W n i ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The latter flat is, on the one hand, at Hausdorff distance 1 from F ′ ω (by the definition of the scaling factors cn), so they actually coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' On the other hand, [∪p i=0W n i ] coincides with Fω := Q(Fω) = [q(Fn)], so Fω = F ′ ω, contradicting the fact that dω([q(xn)], F ′ ω) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Thus c(|x|) = O(u(|x|)), as wanted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For the moreover part (uniform control on the growth of c(x) as a function of the sublinear constants), the proof is identical to Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This time, consider a sequence fn as in the statement, and denote by cn = cn(x) the constant obtained in step 1 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' the SBE constants (L, vn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume towards contradiction that limω u(n) cn(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proof goes exactly as in step 1, with the sole difference that now one might need convincing in the fact that in C(X), the cone with 1 cn as scaling factors, the cone map C(fn) is bi-Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' But indeed for any two cone points (xn), (yn) ∈ C(X) it holds: dω � C(fn)(xn), C(fn)(yn) � = lim ω 1 cn d � fn(xn), fn(yn) � ≤ lim ω 1 cn L · d(xn, yn) + vn(|xn| ∨ |yn|) 48 By definition of the cone metric, limω 1 cn L · d(xn, yn) = L · dω � (xn), (yn) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It thus remains to show limω 1 cn vn(|xn| ∨ |yn|) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By definition of vn it amounts to proving limω 1 cn v(|xn|) = 0 = limω 1 cn v(|yn|) and limω 1 cn u(n) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The former follows from the fact that (xn), (yn) ∈ C(X) and therefore both limω 1 cn |xn| and limω 1 cn |xn| are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The latter follows from the assumption on the cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' One obtains a contradiction identical to the one in Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Following the claims of Sections 3, 4, 5 in [16] carefully, and making the SBE adaptations as depicted in the above example, one obtains flat rigidity in the SBE setting, that is Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10 together with the uniform control described in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Here is the complete list of claims involving cone arguments in [16] that should be modified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' All claims starting from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='5 through Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' All statements should consider, instead of a quasi-isometry, a general (L, u)-SBE f and, when relevant, a general point x ∈ X with f(x) = y (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=', f(x) does not necessarily equal x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Also when relevant one should consider a family fn of (L, vn) SBE maps as depicted in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='11 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Propositions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='7, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' When relevant, statements should be modified so that the distance between f(x) and an associated flat of it should be uniformly sublinear in |x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Also when relevant one should consider a family fn of (L, vn) SBE maps as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 (D = D(x) should be uniformly linear in c = c(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 (the constant D = D(x) should be replaced be a sublinear function D(|x|)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' When relevant one should consider a family fn of (L, vn) SBE maps as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This yields a proof for uniform flat rigidity in the SBE setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The other major part of Drut¸u’s argument concerns the fact that f0 is defined only on X0 and not on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The considerations for this aspect are intertwined in the proof, but they all involve only Γ and the quasi-isometry between Γ and X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For this reason, the fact that f0 is an SBE to begin with does not effect any of these arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, this argument is indifferent to whether or not Γ is uniform or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If Γ is uniform all that changes is that that part of Drut¸u’s argument dealing with extending the cone map from C(X0) to C(X) is not necessary since X = X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Her proof still works perfectly well also for uniform lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Therefore the argument above proves Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10 and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 Some Remarks On R-rank 1 Factors Quasi-isometric rigidity holds for groups of R-rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It is worth mentioning that Schwartz’s proof also relies on ‘flat rigidity’ - but in this case the flats are the horospheres of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' While these are not isometrically embedded flats, the induced metric on horospheres is flat and Schwartz uses that in order to construct the boundary map and find the associated isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The same phenomena occurs in the SBE setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Considering the compact core X0 ⊂ X of Γ, one can use Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6 in Drut¸u-Sapir [18], in order to show that horospheres are mapped boundedly close to horospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In general, this work characterizes and explores a certain class of spaces they call asymptotically tree graded, a class that is very suitable for the setting of the compact core of a non-uniform lattice in R- rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A key ingredient in the proof is the fact that the boundary map ∂q induced by the quasi-isometry is quasi-conformal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This in particular implies that it is almost everywhere differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I spent some time trying to generalize the proof of Schwartz to the SBE setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' One obstacle is that it is not clear that the boundary map is going to be differentiable almost everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Gabriel Pallier found that there are SBE maps of the hyperbolic space whose boundary maps are not quasi-conformal (see Appendix A in [45]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For this reason Pallier develops the notion of quasi-conformality [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' While these maps may be differentiable, he told me of examples he constructed where the differential is almost everywhere 0 - a property which also nullifies Schwartz’s argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the context of SBE rigidity, the maps I consider seem to indeed have ‘flat rigidity’, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' to map a horosphere to within bounded distance of a unique horosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' As in the higher rank flat rigidity, this 49 bound is not uniform but rather grows sublinearly with the distance of the horosphere to a fixed base point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' These are very specific maps, that coarsely preserve the compact core of that lattice X0 ⊂ X and basically map horospheres to horospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This means there might still be hope for these specific maps to induce boundary maps that admit the required analytic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In a subsequent paper [53], Schwartz proves quasi-isometric rigidity for lattices in products of R-rank 1 groups, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' in Hilbert modular groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' His proof there is different, but it also makes use of the fact that horospheres are mapped to within uniformly bounded distance of horospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The fact that in the SBE setting this bound is not uniform seems like a real obstruction to any attempt of generalizing his proof in that case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 From SBE to Sublinearly Close Groups In this section I prove Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I restate it here for convenience Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be a real centre-free semisimple Lie group without compact or R-rank 1 factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let Γ ≤ G be an irreducible lattice, and Λ an abstract finitely generated group that is SBE to Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then there is a group homomorphism Φ : Λ → G with finite kernel such that Γ ⊂ Nu � Φ(Λ) � and Φ(Λ) ⊂ Nu(Γ) for a sublinear function u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proof is a sublinear adaptation of the classical arguments by Schwartz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The only difference is that some calculations are in order, but there is no essential difference from Section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 in [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Before I start, I need one well known preliminary fact, namely that sublinearly close isometries are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let X be a symmetric space of noncompact type and with no R-rank 1 factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let Γ ≤ Isom(X) be a non-uniform lattice, X0 its compact core with respect to x0 ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let g, h ∈ G = Isom(X) and u a sublinear function such that for every x ∈ X0, d � g(x), h(x) � ≤ u(|x|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then g = h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proof is essentially just the fact that a sublinearly bounded convex function is uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Up to multiplying by h−1, one may assume h = idX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' First I show that the continuous map ∂g : X(∞) → X(∞) is the identity map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Recall that the space X(∞) can be represented by all geodesics emanating from the fixed point x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let η : [x0, ξ) be a Γ-periodic geodesic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By definition, there is some T > 0 and a sequence γn ∈ G for which η(nT ) = γnx0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular, xn := γnx0 ∈ X0 hence d � g(xn), xn � ≤ u(|xn|) = u(nT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' On the other hand, the distance function d � η(t), g·η(t) � is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A convex sublinear function is bounded, and so by definition in X(∞) one has [η] = [g · η], for all Γ-periodic geodesics η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The manifold X is of non- positive curvature, hence ∂g is a homeomorphism of X(∞), and the density of Γ-periodic geodesics implies ∂g = idX(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This implies that g = idX (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10 in [20] for a proof of this last implication).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proof for the fact that ∂g = idX(∞) implies g = idX appears in [20] (section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10) as part of the proof of the following important theorem of Tits: Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='22 ([56], see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 in [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let X, X′ be symmetric spaces of noncompact type and of higher R-rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume X has no R-rank 1 factors, and let φ : X(∞) → X′(∞) be a bijection that is a homeomorphism with respect to the cone topology and an isometry with respect to the Tits metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then, after multiplying the metric of X by positive constants on de Rham factors, there exists a unique isometry g : X → X′ such that φ = ∂g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This theorem is actually a key ingredient in Drut¸u’s argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Much of her work is directed towards showing that the cone map C(q) will correspond to a map on X(∞) satisfying the above hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The restriction to X with no R-rank 1 factors in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 comes from this restriction in Tits’ Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='22 Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proof that ∂g = idX(∞) ⇒ g = idX only uses the fact that X has no Euclidean de Rham factors (see pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 251 in [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since I only use it in the setting of no R-rank 1 factors, I added that assumption to Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 50 The Map Φ : Λ → G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The orbit map q0 : Γ → X0 defined by γ �→ γx0 is a quasi-isometric embedding: this is ˘Svarc-Milnor in case Γ is uniform and X0 = X, and Lubotzki-Mozes-Raghunathan (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='19 above) if Γ is non-uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' An SBE f : Λ → Γ thus gives rise to an SBE Λ → X0, which I also denote by f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For each λ ∈ Λ let fλ := f ◦ Lλ ◦ f −1 : X0 → X0, where Lλ is the left multiplication by λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The left translation Lλ is an isometry, hence fλ is a self SBE of X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10, there exists a unique isometry gλ ∈ Isom(X) that is sublinearly close to fλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Define the map Φ : Λ → G by λ �→ gλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The goal in this section is to prove Φ is a homomorphism with finite kernel, and that Γ and Φ(Λ) are each contained in a sublinear neighbourhood of the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I begin by controlling the SBE constants of the fλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For each λ ∈ Λ, fλ is an (L2, vλ)-SBE, for vλ(|x|) := (L + 1)u(|x|) + u(|λ|) In particular vλ ∈ O(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Before the proof I state a corollary which follows immediately by combining Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='24 with Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='11 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then for any x ∈ X there is a sublinear function ux such that d � fλ(x), gλ(x) � ≤ ux(|λ|) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proof is a straightforward computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Up to an additive constant I may assume f −1 is an (L, u)-SBE with f −1(eΓ) = eΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let x1, x2 ∈ X0, and assume w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='g |x2| ≤ |x1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By the properties of an SBE, this also means that for i ∈ {1, 2}: |f −1(xi)| ≤ L|xi − x0| + u(|xi|) ≤ L|x1| + u(|x1|) (6) Notice that fλ(x) = f � λ · f −1(x) � , and f is an (L, u)-SBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The following inequalities, justified below, give the required upper bounds: ��fλ(x1) − fλ(x2) �� ≤ L · ��λf −1(x1) − λf −1(x2) �� + u � |λf −1(x1)| ∨ |λf −1(x2)| � ≤ L2|x1 − x2| + Lu � |x1| � + u � |λ|) + L|x1| + u(|x1|) � ≤ L2|x1 − x2| + (L + 1)u � |x1| � + u(|λ|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' (7) From the first line to the second line I used: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For the first term: left multiplication in Λ is an isometry, and f −1 is an (L, u)-SBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For the second term: triangle inequality, left multiplication in Λ is an isometry, and Equation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' From the second to the third line I used the properties of u as an admissible function, namely that it is sub-additive and doubling, so u � (L + 1)|x1| � ≤ (L + 1)u(|x1|) for all large enough x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I remark that the proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='24 is the only place where I use the properties of an admissible function and not just the sublinearity of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Φ : Λ → G is a group homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let λ1, λ2 ∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I begin with some notations: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' f1 = fλ1, f2 = fλ2, f12 = fλ1λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='24, these are all O(u) SBE maps with the same Lipschitz constant L′ := L2 and sublinear constants v1, v2, v12 ∈ O(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 51 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' g1 = Φ(λ1), g2 = Φ(λ2), g12 = Φ(λ1λ2) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' u1, u2, u12 the sublinear functions that bound the respective distances between any g and f, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' |g1(x) − f1(x)| ≤ u1(|x|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' One has to prove that gλ2 ◦ gλ1 = gλ1λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In view of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='20, it is enough to find a sublinear function v such that for all x ∈ X0 |g1g2(x) − g12(x)| ≤ v(|x|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By triangle inequality and the above definitions and notation, it is enough to show that each of the following four terms are bounded by a function sublinear in x: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' |g1g2(x) − g1f2(x)| = |g2(x) − f2(x)| ≤ u2(|x|) (g1 is an isometry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' |g1f2(x) − f1f2(x)| ≤ u1 � |f2(x)| � ≤ u1 � L2|x| + v2(|x|) � 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' |f1f2(x) − f12(x)| 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' |f12(x) − g12(x)| ≤ u12(|x|) Clearly items 1, 2, 4 are bounded by a sublinear function in |x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It remains to bound |f1f2(x) − f12(x)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The map Λ → Aut(Λ) given by λ �→ Lλ is a group homomorphism, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lλ1λ2 = Lλ1Lλ2, so it remains to bound: |f1f2(x) − f12(x)| = |fLλ1f −1fLλ2f −1(x) − fLλ1Lλ2f −1(x)| f ◦ Lλ is a composition of an isometry with an SBE, so it is still an SBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Denote the SBE constants of fLλ1 by L′, v (clearly one can take L′ = L and v ∈ O(u), but this is not needed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Writing y := Lλ2f −1(x), this shows |fLλ1f −1f(y) − fLλ1(y)| ≤ L|f −1fy − y| + v(|f −1fy| ∨ |y|) By definition of an SBE inverse it holds that |f −1f(y) − y| ≤ u(|y|) and in particular also |f −1f(y)| ≤ |y| + u(|y|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I conclude that |f1f2(x) − f12(x)| ≤ L · u(|y|) + v � |y| + u(|y|) � The right-hand side is a sublinear function in |y|, hence it only remains to show that |y| is bounded by a linear function in x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Indeed |y| = |Lλ2f −1(x)| ≤ |λ2| + |f −1(x)| ≤ |λ2| + L|x| + u(|x|) This completes the proof, rendering Φ a group homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Φ has discrete image and finite kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I show that for any radius R > 0, there are finitely many λ ∈ Λ for which gλx0 ∈ B(x0, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=', that there is a finite number of Φ(Λ)-orbit points, with multiplicities, inside an R ball in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular the set {λ ∈ Λ | gλx0 = x0} is finite, and clearly contains Ker(Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In addition, the actual number of Φ(Λ)-orbit points inside that R ball is finite, so Φ(Λ) is discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let R > 0, and λ ∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By the defining property of gλ and the definition of fλ, reverse triangle inequality gives d � x0, gλ(x0) � ≥ d � x0, fλ(x0) � − d � gλ(x0), fλ(x0) � ≥ |f(λ)| − uλ(|x0|) Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='25 gives d � gλ(x0), fλ(x0) � ≤ ux0(|λ|) for some sublinear function ux0 ∈ O(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' On the other hand f is an SBE, and so |f(λ)| grows close to linearly in λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Formally, |f(λ)| = d � f(λ), x0 � = d � f(λ), f(eΛ) � ≥ 1 Ld(λ, x0) − u(|λ| ∨ |eΛ|) ≥ 1 L|λ| − u(|λ|) 52 To conclude, one has d � x0, gλ(x0) � ≥ 1 L|λ| − u(|λ|) − ux0(|λ|) and both u, ux0 are sublinear in |λ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Therefore there is a bound S ∈ R>0 such that |λ| > S ⇒ 1 L|λ| − u(|λ|) − ux0(|λ|) > R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The group Λ is finitely generated and so only finitely many λ ∈ Λ admit |λ| ≤ S, hence gλ(x0) ∈ B(x0, R) only for finitely many λ ∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There exists a sublinear function u′ : R≥0 → R≥1 such that Γ · x0 ⊂ Nu′� Φ(Λ) · x0 � Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I claim that there is a sublinear function u0, depending only on f and q0, such that for all γ ∈ G, d � gλ(x0), γ(x0) � ≤ u0(|γ|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' As before, I only have control on gλ via fλ, and so I use triangle inequality to get: d � gλ(x0), γ(x0) � ≤ d � gλ(x0), fλ(x0) � + d � fλ(x0), γ(x0) � By Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='25, d � gλ(x0), fλ(x0) � ≤ ux 0(|λ|) for a sublinear function ux0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It is beneficial to distinguish between the SBE fΓ : Λ → Γ and the same SBE composed with the orbit quasi-isometry q0 : Γ → X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' From now on I keep the notation fΓ : Λ → Γ for the SBE of the groups and f0 for the same SBE composed with the orbit quasi-isometry so f0 = q0 ◦ fΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Define λγ := f −1 Γ (γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I show that d � fλγ(x0), γ(x0) � is bounded by a function sublinear in γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Indeed, recall that I assumed without loss of generality q0 ◦f −1 Γ (x0) = eΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, Γ is assumed to be torsion-free, and so there is no ambiguity or trouble in defining the restriction of the map q−1 to the orbit Γ · x0 to be of the form q−1(γx0) = γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' All together, this gives d � fλγ(x0), γ(x0) � = d � f0(λγ), γ(x0) � = d � q0 ◦ fΓf −1 Γ (γ), q0(γ) � Since fΓ is an SBE d � fΓf −1 Γ (γ), γ � ≤ u(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The fact that q0 is an (L′, C)-quasi-isometry implies that d � q0 ◦ fΓf −1 Γ (γ), q(γ) � ≤ Ld � fΓf −1 Γ (γ), γ � + C ≤ L′u(|γ|) + C Combining everything, one has d � gλγ(x0), γ(x0) � ≤ ux0(|λγ|) + L′u(|γ|) + C As before, |λγ| = |f −1(γ)| ≤ |γ| + u(|γ|) ≤ 2|γ|, where the last inequality holds for all large enough γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' What matters is that |λγ| is linear in |γ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I conclude that indeed Γ · x0 ⊂ Nu′� Φ(Λ) · x0 � for the sublinear function u′ = ux0 + L′u + C ∈ O(u), as wanted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' (To be pedantic, u′ = ux0 ◦ 2 + L′u + C ∈ O(u) where 2 is the ‘multiplication by 2’ function, r �→ 2r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There exists a sublinear function u′ : R≥0 → R≥1 such that Φ(Λ) · x0 ⊂ Nu′(Γ · x0) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let λ ∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let γ = f(λ) and consider the distance d(gλx0, γx0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' From triangle inequality and Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='25 one has d(gλx0, γx0) ≤ d(gλx0, fλx0) + d(fλx0, γx0) ≤ u′(|λ|) + d(fλx0, γx0) By definition of fλ it holds that fλ(x0) = f(λ) · x0 = γ · x0 and so d � fλ(x0), γx0 � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Claims 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 result in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 then follows from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 53 References [1] Paul Albuquerque.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Patterson-Sullivan theory in higher rank symmetric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' GAFA Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Anal.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Tata Institute of Fundamental Research, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' [55] Dennis Sullivan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' On The Ergodic Theory at Infinity of an Arbitrary Discrete Group of Hyperbolic Motions, pages 465–496.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Princeton University Press, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' [56] Jacques Tits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Buildings of spherical type and finite BN-pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Springer-Verlag, Berlin,New York, 1974.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 56' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} diff --git a/89E1T4oBgHgl3EQfCALf/content/tmp_files/2301.02860v1.pdf.txt b/89E1T4oBgHgl3EQfCALf/content/tmp_files/2301.02860v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d9369c3967088001a8fea9cbd10752b71136681b --- /dev/null +++ b/89E1T4oBgHgl3EQfCALf/content/tmp_files/2301.02860v1.pdf.txt @@ -0,0 +1,2141 @@ +THERMODYNAMICAL MODELING OF MULTIPHASE FLOW +SYSTEM WITH SURFACE TENSION AND FLOW +HAJIME KOBA +Abstract. We consider the governing equations for the motion of the viscous +fluids in two moving domains and an evolving surface from both energetic +and thermodynamic points of view. We make mathematical models for multi- +phase flow with surface flow by our energetic variational and thermodynamic +approaches. +More precisely, we apply our energy densities, the first law of +thermodynamics, and the law of conservation of total energy to derive our +multiphase flow system with surface tension and flow. We study the conserva- +tive forms and conservation laws of our system by using the surface transport +theorem and integration by parts. Moreover, we investigate the enthalpy, the +entropy, the Helmholtz free energy, and the Gibbs free energy of our model by +applying the thermodynamic identity. The key idea of deriving surface tension +and viscosities is to make use of both the first law of thermodynamics and our +energy densities. +1. Introduction +Figure 1. Moving Domains, Surfaces and Notations +We are interested in a mathematical modeling of a soap bubble floating in the +air. When we focus on a soap bubble, we can see the fluid flow in the bubble. We +2020 Mathematics Subject Classification. 80M30, 35Q79, 76-10, 80-10, 35A15. +Key words and phrases. Multiphase flow, Surface tension, Surface flow, Mathematical model- +ing, First law of thermodynamics, Energetic variational approach. +This work was partly supported by the Japan Society for the Promotion of Science (JSPS) +KAKENHI Grant Number JP21K03326. +1 +arXiv:2301.02860v1 [math-ph] 7 Jan 2023 + +PA,PB, Ps: Density +n2 +UA, UB, Us: Velocity +μA, μB, μs, 入A,入B, 入s: Viscosity +nr +TA,TB, s: Pressure +I(t) +A,OB,Os: Temperature +A(t) +B(t) +eA, eB, es: Internal energy +KA, KB, Ks: Thermal conductivity +hA,hB,hs: Enthalpy +SA, SB, Ss: Entropy +2 +AB += A(t) UI(t) U 2B(t2 +HAJIME KOBA +call the fluid flow in the bubble a surface flow. We can consider a surface flow +as a fluid-flow on an evolving surface. To make a mathematical model for a soap +bubble floating in the air, we have to study the dependencies among fluid-flows in +two moving domains and surface flow. We consider the governing equations for +the motion of the viscous fluids in the two moving domains and surface from both +energetic and thermodynamic points of view. More precisely, we apply the first law +of thermodynamics and our energy densities to derive our multiphase flow system +with surface tension and flow. +Let us first introduce fundamental notations. Let t ≥ 0 be the time variable, +and x(= t(x1, x2, x3)) ∈ R3 the spatial variable. +Fix T > 0. +Let Ω ⊂ R3 be +a bounded domain with a smooth boundary ∂Ω. +The symbol nΩ = nΩ(x) = +t(nΩ +1 , nΩ +2 , nΩ +3 ) denotes the unit outer normal vector at x ∈ ∂Ω. +Let ΩA(t)(= +{ΩA(t)}0≤t0.01 +Yes +Finding parameters +For satellites +For HAPS +sAT,dtrop,dion +OHAPS +Applying elevation mask +For satellites +For HAPS +sAT,dtrop,dion,PRN,dT,PsAT +HAPS,PHAPS +Pseudorangecorrection +PSAT,PHAPS +Combining the +corrected +pseudoranges +P'=[PSAT,PHAPS] +Correcting for the Sagnac effect +(i.e., Earth rotation) +PSAT,PHAPS +Combiningthecorrected +ranging source positions +P° = [PSAT, PHAPS] +Finding parameters +V,P,b,H,Q +Output +Computingtheposition solution +using the Least Square method +x,dt +x,dt𝑹 = [ +−sin 𝜆 +cos 𝜆 +0 +− cos 𝜆 sin 𝜑 +− sin 𝜆 sin 𝜑 +cos 𝜑 +cos 𝜆 cos 𝜑 +sin 𝜆 cos 𝜑 +sin 𝜑 +] (15) +where 𝜆 and 𝜑 represent the longitude and latitude of the +receiver, respectively. Then the HDOP is described by +𝐻𝐷𝑂𝑃 = √𝜎𝑛2 + 𝜎𝑒2 (16) +where 𝜎𝑛, 𝜎𝑒, and 𝜎𝑑 represent the receiver position errors in +the local north, east and down directions, respectively. +III. SIMULATION OF THE HAPS-AIDED GPS SYSTEM +A. Simulation Setup +The system model is established using the default Earth +orientation parameters of the Skydel GNSS simulation +software [13] which considers all GPS satellites orbiting +around the Earth and transmitting the L1 C/A code. The +Saastamoinen model is chosen to emulate the tropospheric +effect and the Klobuchar model is chosen to emulate the +ionospheric effect along with the software default Klobuchar +parameters (i.e., alpha and beta). The output from Skydel +contains the ECEF coordinates of satellites at the signal +emission time, the ionospheric corrections, the tropospheric +corrections, the satellite clock offsets, the ECEF coordinates +of the receiver, the signal emission time, and so forth, at each +time stamp from the start of the simulation. The receiver clock +offset in the simulation is zero by default. The correction terms +in the pseudorange equation of satellite including the satellite +orbit error, the multipath and the receiver noise are not +separately considered in the simulation, instead a pseudorange +error is introduced to reflect the presence of those effect. The +pseudorange error of satellite is featured using the built-in first +order Gauss-Markov process with the default time constant of +10 s and the standard deviation of 6 m. The continuous model +for the first order Gauss-Markov process is described by [16] +𝑥̇ = − +1 +𝑇𝑐 𝑥 + 𝑤 (17) +where 𝑥 represents a random process with zero mean, +correlation time 𝑇𝑐, and noise 𝑤. The autocorrelation of the +first order Gauss-Markov process is described by [17] +𝑅(∆𝑡) = 𝜎2𝑒−|∆𝑡| +𝜏 (18) +where ∆𝑡 represents the sampling interval, 𝜎 and 𝜏 denote the +standard deviation and the time constant of the first order +Gauss-Markov process, respectively. The pseudorange of +HAPS is simulated by adding Gaussian noise to the geometric +range between HAPS and receiver, where the Gaussian noise +represents the sum of all kinds of estimation residuals +including the HAPS position, the HAPS clock offset, the +tropospheric delay, the multipath and the receiver noise. The +pseudorange error for HAPS is modelled using the Gaussian +noise with standard deviations of 2 m and 5 m representing the +suburban and the dense urban scenario, respectively. The +characteristics of the pseudorange errors for the suburban +scenario and the dense urban scenario are set to be the same +for satellites. Note that by doing this, the positioning +performance of the GPS-only system stays the same in both +suburban scenario and dense urban scenario. The standard +deviation for the HAPS pseudorange error is enforced to be +smaller than that for the satellite pseudorange error in both +suburban scenario and dense urban scenario. All the available +satellites (i.e., satellites with elevation angles greater than the +predefined elevation mask) are simultaneously utilized for +positioning as if all satellites above the elevation mask are in +line of sight (LOS) with the receiver. Under this setting, we +examine the 3D positioning performance for the GPS-only +system, the one-HAPS with GPS system, the four-HAPS with +GPS system and the four-HAPS-only system. For the one- +HAPS with GPS system, we use the HAPS on top of the +downtown Ottawa area which elevation is above 80°. +B. Simulation Results +The cumulative distribution functions of the 3D +positioning accuracy for different systems with the standard +deviations of the HAPS pseudorange error being 2 m and 5 m +are shown in Fig. 4 and Fig. 5, respectively. From Fig. 4, we +can observe that with much less pseudorange error for HAPS, +the four-HAPS with GPS system achieves the best +positioning performance, the one-HAPS with GPS system +achieves almost the same positioning performance as the +GPS-only system, and the four-HAPS-only system achieves +slightly worse performance than the four-HAPS with GPS +system. The reasons why the four-HAPS-only system does +not achieve the best positioning performance is potentially +due to the following reasons 1) it has much fewer ranging +sources in receiver position computation; 2) the ranging +source geometry is poor as the elevation angles for all four +HAPS at any given time are above 40° with one even above +80°. From Fig. 5, we see that with the HAPS pseudorange +error similar but slightly smaller than the satellites’ +pseudorange error, the four-HAPS-only system achieves the +worst positioning performance but the four- HAPS with GPS + +Fig. 4: CDF for 3D position accuracy (suburban scenario). + +Fig. 5: CDF for 3D position accuracy (dense urban scenario). + +Denseurbanscenario(HAPSprstd=5m) +0.9 +0.8 +0.7 +0.6 +DF +0.5 +0.4 +0.3 +0.2 +GPS-onlysystem +One-HAPSwithGPSsystem +0.1 +Four-HAPSwithGPSsystem +Four-HAPS-only system +0 +0 +5 +10 +15 +20 +25 +30 +35 +3Dpositionalaccuracy(m)inlocalNEDframeSuburbanscenario(HAPSprstd=2m) +0.9 +0.8 +0.7 +0.6 +DF +0.5 +C +0.4 +0.3 +0.2 +GPS-onlysystem +One-HAPSwithGPSsystem +0.1 +Four-HAPS with GPS system +Four-HAPS-only system +0 +0 +5 +10 +15 +20 +25 +30 +35 +3Dpositionalaccuracy(m)inlocalNEDframesystem still outperforms the other systems considered. +IV. FIELD EXPERIMENTS +A. Experiment Setup +To verify and support the simulation results, we also +process the raw GNSS data collected along the vehicle +trajectory which is similar to the one shown in Fig. 2 with a +slight difference due to partial road closure on the day of data +collection. The raw GNSS data are collected using the Ublox +EVK-M8T GNSS unit and processed using the single point +positioning package developed by Napat Tongkasem [18] +with proper modification so that HAPS can be incorporated +in the single point positioning algorithm. Table I gives the +specifications of the EVK-M8T GNSS unit. To reflect +realistic LOS conditions for HAPS, the LOS probability with +respect to the HAPS elevation angle in the urban area is +implemented based on [19] and [20]. Note that the LOS +probability for HAPS provided by [19] is generated based on +the city of Chicago and enforcing the LOS probability on +HAPS in the dense urban area in Ottawa might be too harsh +considering the incompatible city scale. The pseudorange of +HAPS in the experiment is modeled as the addition of the +geometric range between the satellite and receiver, the +receiver clock offset multiplied by the speed of light and the +pseudorange error representing the sum of all kinds of +estimation residuals. The pseudorange errors for HAPS in the +suburban area and in the dense urban area are simulated as +Gaussian noise with standard deviations of 2 m and 5 m, +respectively. Since the vehicle trajectory involves both +suburban area and dense urban area, the entire route is +divided into two parts where the first part is considered as the +suburban scenario and the second part is considered as the +dense urban scenario (see Fig. 2). By observing the +positioning performance of the GPS-only system using the +real GPS data, the LOS probability for the suburban area is +applied to HAPS for epochs less than 380 s, and the LOS +probability for the dense urban area is applied to HAPS for +epochs greater than or equal to 380 s (refer to Fig. 6). Since +the GNSS receiver does not provide accurate receiver clock +offset with respect to the GPS time, the receiver clock offset +in each epoch is estimated by making use of the ground truth +receiver position. The ground truth data is provided by Ublox +EVK-M8U +GNSS +unit, +which +is +equipped +with +accelerometer and gyroscope, hence it can perform sensor +fusion to get better positioning performance and dead +reckoning when the signal quality degrades. +TABLE I. +EVK-M8T GNSS UNIT SPECIFICATIONS [21] +Parameter +Specification +Serial Interfaces +1 USB V2.0 +1 RS232, max.baud rate 921,6 kBd +DB9 +/- 12 V level +14 pin – 3.3 V logic +1 DDC (I2C compatible) max. 400 kHz +1 SPI-clock signal max. 5,5 MHz – SPI DATA +max. 1 Mbit/s +Timing Interfaces +2 Time-pulse outputs +1 Time-mark input +Dimensions +105 × 64 × 26 mm +Power Supply +5 V via USB or external powered via extra power +supply pin 14 (V5_IN) 13 (GND) +Normal Operating +Temperature +−40℃ to +65℃ + +Fig. 6: HDOP (top) and 3D position accuracy (bottom). +B. Experiment Results +Fig. 6 shows the HDOP, and the 3D positioning accuracy +overlapped with the number of visible HAPS at each epoch. +As we can see from Fig. 6, the HDOP and 3D positioning +accuracy of the HAPS-aided GPS system are better than that +of the GPS-only system in both suburban area and dense +urban area. Moreover, we can observe that the positioning +performance of the HAPS-aided GPS system is more stable +than the GPS-only system as there are less spikes for the +HAPS-aided GPS system. Note that, the pseudorange of +HAPS in the experiment is modeled as a function of the +receiver clock offset, which is estimated with the best effort, +additional error should be expected in the pseudorange of +HAPS with the magnitude depending on the quality of all +visible satellite signals and the ground truth receiver position. +As we would expect the quality of the satellite signals in the +suburban area is better compared to that in the dense urban +area, the receiver clock offset would also be expected to be + +Fig. 7: CDF of 3D position accuracy in the suburban area. + +Fig. 8: CDF of 3D position accuracy in the dense urban area. + +Suburbanarea +1 +0.9 +0.8 +0.7 +0.6 +CDF +0.5 +C +0.4 +0.3 +0.2 +0.1 +GPS-onlysystem +HAPS-aided GPS system +0 +0 +5 +10 +15 +20 +25 +30 +35 +3Dpositioningerror(m)Denseurbanarea +1 +0.9 +0.8 +0.7 +0.6 +CDF +0.5 +0.4 +0.3 +0.2 +0.1 +GPS-only system +HAPS-aidedGPSsystem +0 +0 +50 +100 +150 +200 +250 +3Dpositioningerror(m)GPS-only system +HAPS-aided GPS system +number of HAPS +0 +100 +200 +300 +400 +500 +600 +700 +epoch (s)GPS-cnly system +4 +HAPS-aided GPS system +300 +Hoe +number of HAPS +3 +DOSI +100 +0 +100 +200 +300 +400 +500 +600 +700 +epoch (s)estimated with higher accuracy in the suburban area than in +the dense urban area, hence the HDOP of the HAPS-aided +GPS system in the suburban area is better. The cumulative +distribution functions of the 3D positioning accuracy in the +suburban and dense urban areas are shown in Fig. 7 and Fig. +8, respectively. From Fig. 7 and Fig. 8, we can observe that +the HAPS-aided GPS system outperforms the GPS-only +system, especially in the suburban area. +V. CONCLUSION +As we are passing 5G and soon entering 6G and beyond, +HAPS can be of invisible treasure as it can be used for +computation offloading [22], edge computing [23], even base +station [24] to meet human needs. HAPS can be another type +of ranging source which is quasi-stationary and much closer +to the ground of the Earth. Compared to satellite, HAPS +exhibits the advantages of lower latency, lower pathloss, +lower pseudorange error, and it can provide continuous +coverage to reduce the number of handovers for the users in +a certain region. Since urban area is the region where GNSS +positioning performance degrades severely and where most +people live in, deploying several HAPS acting as another type +of ranging source on top of a metro city would improve the +GNSS positioning performance and maximize the value of +the extra payload on HAPS. The HAPS-aided GNSS can also +be deployed in the regions with extreme environment such as +the Arctic region where the satellite availability is low, and +the ionospheric disturbances is severe [25]. From both the +simulation and physical experiment results, we observe that +HAPS can indeed improve the 3D positioning accuracy, +especially in the suburban area. To improve the results of +HAPS-aided GPS system in the dense urban area, the receiver +clock offset should be estimated with higher accuracy. In +future work, the received signal powers of HAPS and satellite +will jointly be considered, a satellite selection algorithm will +be applied to better emulate the way a modern GNSS receiver +processes the raw GNSS data. +ACKNOWLEDGMENT +This paper is supported in part by Huawei Canada. The +Skydel software is a formal donation from Orolia. +REFERENCES +[1] X. Li et al., “Precise positioning with current multi-constellation +Global Navigation Satellite Systems: GPS, GLONASS, Galileo and +BeiDou,” Sci Rep 5, 8328 (2015). +[2] “Global Positioning System standard positioning service performance +standard,” GPS.GOV, USA, Apr. 2020. 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Li, “Joint optimization +of transmission and computation resources for satellite and high +altitude platform assisted edge computing,” IEEE Transactions on +Wireless Communications, vol. 21, no. 2, pp. 1362-1377, Feb. 2022. +[24] M. S. Alam, G. K. Kurt, H. Yanikomeroglu, P. Zhu, and N. D. Dao, +“High altitude platform station based super macro base station +constellations”, IEEE Communications Magazine, vol. 59, no. 1, pp. +103-109, Jan. 2021. +[25] A. Yastrebova, M. Höyhtyä, S. Boumard, E. S. Lohan, and A. Ometov, +“Positioning in the Arctic region: State-of-the-art and future +perspectives,” IEEE Access, vol. 9, pp. 53964-53978, Mar. 2021. + + diff --git a/8dAyT4oBgHgl3EQf2_nF/content/tmp_files/load_file.txt b/8dAyT4oBgHgl3EQf2_nF/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3ee6b3ff942327722b8eda5aeafcc9f1c3727dcf --- /dev/null +++ b/8dAyT4oBgHgl3EQf2_nF/content/tmp_files/load_file.txt @@ -0,0 +1,425 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf,len=424 +page_content='High Altitude Platform Station (HAPS)-Aided GNSS for Urban Areas Hongzhao Zheng, Mohamed Atia, Halim Yanikomeroglu Department of Systems and Computer Engineering, Carleton University, Ottawa, Canada hongzhaozheng@cmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='carleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='ca Abstract—Today the global averaged civilian positioning accuracy is still at meter level for all existing Global Navigation Satellite Systems (GNSSs), and the civilian positioning performance is even worse in regions such as the Arctic region and the urban areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' In this work, we examine the positioning performance of the High Altitude Platform Station (HAPS)- aided GPS system in an urban area via both simulation and physical experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' HAPS can support GNSS in many ways, herein we treat the HAPS as an additional ranging source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' From both simulation and experiment results, we can observe that HAPS can improve the horizontal dilution of precision (HDOP) and the 3D positioning accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' The simulated positioning performance of the HAPS-aided GPS system is subject to the estimation accuracy of the receiver clock offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' This work also presents the future work and challenges in modelling the pseudorange of HAPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' Keywords—High Altitude Platform Station (HAPS), Global Navigation Satellite System (GNSS), pseudorange, horizontal dilution of precision (HDOP) I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' INTRODUCTION The global navigation satellite system (GNSS) has been around for decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' Since the first launch of a legacy GNSS in 1978, the global positioning system (GPS) owned by the US, the positioning accuracy brought by satellites has been improving thanks to the ongoing research in the associated scientific fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' Depending on the application, centimeter level accuracy can be obtained by techniques such as differential GPS (DGPS), real-time kinematic (RTK), multi- constellation GNSS and so forth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' For example, the multi- constellation GNSS (BeiDou + Galileo + GLONASS + GPS) has been shown to not only shorten the convergence time, but also to provide centimeter-level positioning accuracy even with 40° cut-off elevation using the precise positioning algorithm [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' Although numerous techniques have been developed to achieve centimeter-level positioning accuracy, many of which are not suitable for civilian applications such as smartphones, smartwatches, bikes, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' Most civilian applications use single frequency and low cost receivers for navigation and positioning, hence precise positioning is not applicable due to reasons such as the incomplete elimination of the ionospheric delay which appears to be one of the largest error sources in the pseudorange measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' For similar reasons, the most common algorithm used in the civilian applications is therefore the single point positioning algorithm which only requires a single frequency in localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' However the global averaged positioning accuracy of using the single point positioning algorithm is still at meter level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' For example, the 95% global averaged horizontal error is less than or equal to 8 m and the 95% global averaged vertical error is less than or equal to 13 m for the GPS system [2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' the 95% global averaged horizontal error is less than or equal to 9 m and the 95 % global averaged vertical error is less than or equal to 10 m for the BeiDou Navigation Satellite System (BDS) [3];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' the 95% global averaged horizontal error is less than or equal to 5 m and the 95% global averaged vertical error is less than or equal to 9 m for GLONASS [4];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' the 95% global averaged positioning error is less than or equal to 7 m for Galileo [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' The positioning performance of the GNSS is even worse in the urban area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' Today there are many low Earth orbit (LEO) satellites launched into space, people are also interested in utilizing LEO satellites to aid positioning service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' For instance, Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' prove that the LEO enhanced GNSS can provide centimeter level Signal-In-Space Ranging Error (SISRE) in real-time precise point positioning (PPP) application [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' Furthermore, researchers have also been investigating the navigation performance which relies exclusively on the LEO satellite signals in case the GNSS signals are unavailable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' Khalife et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' have shown that a position root mean squared error (RMSE) of 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='8 m for an unmanned aerial vehicle (UAV) can be achieved with only two Orbcomm LEO satellites using the carrier phase differential algorithm [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' Compare LEO satellites with the typical satellites used in the traditional GNSS which are the medium Earth orbit (MEO) satellites, LEO exhibits the advantages including but not limited to shorter propagation delay and lower pathloss due to shorter distance to the ground user, wider coverage and higher availability due to the enormous number of satellites simultaneously visible/available for positioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' To further enhance high bandwidth networking coverage in challenging areas, High Altitude Platform Stations (HAPS), which resides in the stratosphere with a typical altitude of about 20 km, can be introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' As urban area is the region where the GNSS positioning performance degrades most severely, we could utilize HAPS as another ranging source by equipping it with a satellite-grade atomic clock on top of a metro city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' Since HAPS is only 20 km above the ground, the pathloss of the HAPS signal is expected to be much less than that for any satellites, making the received signal power of HAPS stronger than that of satellite, which likely renders less estimation error in the multipath mitigation of the HAPS signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' HAPS is quasi-stationary as it does not orbit around the globe, this can reduce the number of handovers during the course of positioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' Moreover the HAPS signal does not suffer from the ionospheric effect since it is transmitted from below the ionosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' Therefore the pseudorange from HAPS can likely be estimated with less error compared with that from satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' Similar to the pseudorange from satellites which incorporates satellite position error, we should also consider the position error in the pseudorange measurement for HAPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' Fortunately, there are ongoing research in the literature investigating the positioning of HAPS and showing HAPS positioning error can be comparable or even better than the satellite orbit error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' For example, a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='5 m positioning accuracy (circular error probable [CEP] 68 percent) for HAPS has been shown achievable using the modified RTK method [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' In fact, there are a handful of papers in the literature investigating the HAPS-aided GNSS positioning performance [9]-[12], but none of which considers utilizing HAPS for the sole mission of improving the GNSS positioning performance in the urban area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' In this work we examine the HAPS-aided GNSS positioning performance in the urban area via both simulation and physical experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' For simplicity, the GNSS signal only involves the GPS C/A L1 signal, and the single point positioning algorithm is used to compute the position solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' SYSTEM MODEL The general system model is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' The HAPS is situated at 20 km above ground in the stratosphere which is below the ionosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' There are four satellites shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' 1, this is just a reminder that at least four satellites are required to perform precise 3D localization using GNSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' Although only a selection of visible satellites is used in position solution calculation in reality, in this work all available satellites are used in position solution calculation for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' The elevation masks for both satellite and HAPS are chosen to be 15 degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' The pseudorange equation for satellite is given by 𝑝𝑆𝐴𝑇 = 𝜌𝑆𝐴𝑇 + 𝑑𝑆𝐴𝑇 + 𝑐(𝑑𝑡 − 𝑑𝑇𝑆𝐴𝑇) + 𝑑𝑖𝑜𝑛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='𝑆𝐴𝑇 + 𝑑𝑡𝑟𝑜𝑝,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='𝑆𝐴𝑇 + 𝜖𝑚𝑝,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='𝑆𝐴𝑇 + 𝜖𝑝 (1) where 𝑝𝑆𝐴𝑇 denotes the satellite pseudorange measurement,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' 𝜌𝑆𝐴𝑇 is the geometric range between the satellite and receiver,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' 𝑑𝑆𝐴𝑇 represents the satellite orbit error,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' 𝑐 is the speed of light,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' 𝑑𝑡 is the receiver clock offset from GPS time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' 𝑑𝑇𝑆𝐴𝑇 is the satellite clock offset from GPS time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' 𝑑𝑖𝑜𝑛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='𝑆𝐴𝑇 denotes the ionospheric delay for satellite signals,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' 𝑑𝑡𝑟𝑜𝑝,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='𝑆𝐴𝑇 denotes the tropospheric delay for satellite signals,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' 𝜖𝑚𝑝,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='𝑆𝐴𝑇 is the delay caused by the multipath for satellite signals and 𝜖𝑝 is the delay caused by the receiver noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' The pseudorange equation for HAPS is described by 𝑝𝐻𝐴𝑃𝑆 = 𝜌𝐻𝐴𝑃𝑆 + 𝑑𝐻𝐴𝑃𝑆 + 𝑐(𝑑𝑡 − 𝑑𝑇𝐻𝐴𝑃𝑆) + 𝑑𝑡𝑟𝑜𝑝,𝐻𝐴𝑃𝑆 + 𝜖𝑚𝑝,𝐻𝐴𝑃𝑆 + 𝜖𝑝 (2) where 𝑝𝐻𝐴𝑃𝑆 denotes the HAPS pseudorange measurement, 𝜌𝐻𝐴𝑃𝑆 represents the geometric range between the HAPS and the receiver, 𝑑𝐻𝐴𝑃𝑆 represents the HAPS position error, 𝑑𝑇𝐻𝐴𝑃𝑆 is the HAPS clock offset from GPS time, 𝑑𝑡𝑟𝑜𝑝,𝐻𝐴𝑃𝑆 denotes the tropospheric delay for HAPS signals, 𝜖𝑚𝑝,𝐻𝐴𝑃𝑆 is the delay caused by the multipath for HAPS signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' In this work, the satellite orbit error, the HAPS position error, and the HAPS clock offset are assumed to be zero for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' The simulated vehicle trajectory originates from Carleton University in the suburban area and ends at the Rideau Street of Ottawa in the dense urban area (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' There are four simulated HAPS where one HAPS is following a circular trajectory on top of the downtown Ottawa area, and the other three HAPS are following similar circular trajectories on top of three populated regions near Ottawa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' Note that HAPS is quasi-stationary due to factors such as wind, it can move within a confined space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' 3 shows the flowchart of the single point positioning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' Since the HAPS clock offset in this work is assumed zero, we simply use 𝑑𝑇 to denote the satellite clock offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' From the data collected by the GNSS receiver, we shall obtain both the receiver independent exchange (RINEX) format observation file and the RINEX navigation file, which contains the satellite information such as the pseudorange, the ionospheric parameters, 𝜶, the Keplerian parameters, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' With that information, we know the pseudo-random noise (𝑷𝑹𝑵) code which represents the unique number of each satellite, the day of year (𝐷𝑂𝑌) which represents the day of year at the time of measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' Note that 𝑷𝑹𝑵 is in bold to represent a vector Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' 1: System model of the HAPS-aided GPS system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' 2: Vehicle trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' containing the pseudo-random noise code of all visible satellites at the current epoch and the current iteration of estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' We can compute the satellite positions, 𝑷𝑺𝑨𝑻, and satellite clock offset, 𝒅𝑻, using the Keplerian parameters contained in the navigation file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' 𝑷𝑯𝑨𝑷𝑺 denotes a vector containing the positions of all HAPS which are generated using the Skydel GNSS simulator [13], and 𝒑𝑯𝑨𝑷𝑺 denotes a vector containing the HAPS pseudorange which will be explained in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' To compute the position solution, 𝒙, we firstly initialize the receiver position to the center of the Earth, and the receiver clock offset is initialized to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' The change in estimate, 𝒅𝒙, is initialized to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' For each epoch of measurement, we will first check if the number of available ranging sources is more than three as at least four ranging sources are required to perform precise 3D localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' Since the receiver position is iteratively estimated, we calculate the elevation angles for both satellites and HAPS with respect to the recently estimated receiver position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' Since both the tropospheric delay and the ionospheric delay are functions of the receiver position, these two atmospheric delays are estimated iteratively as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' The Ionosphere HAPS HAPS 20km HAPSfootprint HAPSfootprint 15° cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' cellOSM+ relief shading V Tracks: nelByDrivi Dense urban Areas OldOrtowa Huram tonbu neGleb Suburban Are eas Ottawa enEza e45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='40751.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='-75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='60857 Googe Map created at GpSVisualiz Madutn con Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' 3: Flow chart of the single point positioning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' elevation angle, satellite pseudorange, HAPS pseudorange, satellite position, satellite clock offset, the tropospheric delay, 𝒅𝒕𝒓𝒐𝒑, the ionospheric delay, 𝒅𝒊𝒐𝒏, and the pseudo- random noise (𝑷𝑹𝑵) code are modified iteratively based on the re-computed elevation angles for both satellites and HAPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' To prepare the parameters needed for the least square methods, the corrected pseudorange needs to be computed as follows: 𝒑𝑺𝑨𝑻 𝒄 = 𝒑𝑺𝑨𝑻 + 𝑐 ∙ 𝒅𝑻 − 𝒅𝒕𝒓𝒐𝒑,𝑺𝑨𝑻 − 𝒅𝒊𝒐𝒏,𝑺𝑨𝑻 (3) where 𝒑𝑺𝑨𝑻 𝒄 represents the corrected pseudorange for satellite, 𝒑𝑺𝑨𝑻 represents the uncorrected pseudorange for satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' Since the pseudorange error of HAPS is modeled as Gaussian noise representing the estimation residual, the HAPS pseudorange does not need to be corrected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' Due to the Earth rotation, the positions of satellites and HAPS at the signal emission time are different from that at the signal reception time, this is known as the Sagnac effect [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' The coordinates of satellite/HAPS can be transformed from the signal emission time to the signal reception time by [14] ∆𝑡𝑅𝑂𝑇 = 𝑡𝑟𝑥 − 𝑡𝑡𝑥 (4) 𝑃𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='𝑟𝑥 = 𝑀𝑅𝑂𝑇(𝜔𝐸 × ∆𝑡𝑅𝑂𝑇)𝑃𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='𝑡𝑥 (5) where ∆𝑡𝑅𝑂𝑇 denotes the signal propagation time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' 𝑡𝑟𝑥 represents the signal reception time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' 𝑡𝑡𝑥 represents the signal emission time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' 𝑃𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='𝑟𝑥 is the 𝑖𝑡ℎ satellite/HAPS coordinates at the signal reception time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' 𝑃𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='𝑡𝑥 is the 𝑖𝑡ℎ satellite/HAPS coordinates at the signal emission time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' 𝜔𝐸 denotes the Earth’s rotation rate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' and 𝑀𝑅𝑂𝑇(𝜔𝐸 × ∆𝑡𝑅𝑂𝑇) is known as the rotation matrix which is described by 𝑀𝑅𝑂𝑇(𝜔𝐸 × ∆𝑡𝑅𝑂𝑇) = [ cos(𝜔𝐸 × ∆𝑡𝑅𝑂𝑇) sin(𝜔𝐸 × ∆𝑡𝑅𝑂𝑇) 0 − sin(𝜔𝐸 × ∆𝑡𝑅𝑂𝑇) 0 cos(𝜔𝐸 × ∆𝑡𝑅𝑂𝑇) 0 0 1 ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' (6) The line-of-sight vector, 𝒗, and the geometric range between ranging sources and receiver, 𝝆 , are then calculated to compute the a-priori range residual vector 𝒃 and the design matrix 𝑯, where 𝒃 = 𝒑𝒄 − 𝝆 (7) 𝑯 = [𝒗, 𝟏𝑙𝑒𝑛𝑔𝑡ℎ(𝒑𝐜)×1)] (8) where 𝒑𝒄 is the corrected satellite pseudorange combined with the corrected HAPS pseudorange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' At last, the least square solution is computed as 𝑸 = (𝑯′𝑯)−1 (9) 𝒅𝒙 = 𝑸𝑯′𝒃 (10) 𝑑𝑡 = 𝒅𝒙(4)/𝑐 (11) where Q is known as the covariance matrix, 𝒅𝒙(4) means the fourth element in the vector 𝒅𝒙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' The covariance matrix, 𝑸, is described by 𝑸 = [ 𝜎𝑥 2 𝜎𝑥𝑦 𝜎𝑥𝑧 𝜎𝑥𝑡 𝜎𝑥𝑦 𝜎𝑦 2 𝜎𝑦𝑧 𝜎𝑦𝑡 𝜎𝑥𝑧 𝜎𝑦𝑧 𝜎𝑧 2 𝜎𝑧𝑡 𝜎𝑥𝑡 𝜎𝑦𝑡 𝜎𝑧𝑡 𝜎𝑡 2 ] (12) where 𝜎𝑥, 𝜎𝑦, 𝜎𝑧 and 𝜎𝑡 represent the standard deviations of the receiver coordinates x, y, z in the Earth-centered Earth- fixed (ECEF) coordinate frame and the receiver clock offset, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' The least square solution shall be found when the norm of the change in receiver position, 𝒅𝒙(1: 3), is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' In this work, this threshold is chosen to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='01 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' We use the horizontal dilution of precision (HDOP) and the 3D positioning accuracy as the metrics to examine the positioning performance of the proposed HAPS-aided GPS system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' To compute the HDOP, we must convert the covariance matrix into the local north-east-down (NED) coordinate frame, which can be done with the following equation [15]: 𝑸𝑵𝑬𝑫 = 𝑹′𝑸̃𝑹 (13) where 𝑸̃ and 𝑹 are defined as 𝑸̃ = [ 𝜎𝑥 2 𝜎𝑥𝑦 𝜎𝑥𝑧 𝜎𝑥𝑦 𝜎𝑦 2 𝜎𝑦𝑧 𝜎𝑥𝑧 𝜎𝑦𝑧 𝜎𝑧 2 ] (14) Initialization PsAT,PHAPS,PRN,DOY x = 04x1 Input dt = x(4)/c PHAPs,PsAT,dT,α dx =x+Inf stop = 0 No Exit 4 NsAT + NHAPs ≥ 4 Yes No Idx(1:3)I>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content="01 Yes Finding parameters For satellites For HAPS sAT,dtrop,dion OHAPS Applying elevation mask For satellites For HAPS sAT,dtrop,dion,PRN,dT,PsAT HAPS,PHAPS Pseudorangecorrection PSAT,PHAPS Combining the corrected pseudoranges P'=[PSAT,PHAPS] Correcting for the Sagnac effect (i." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=', Earth rotation) PSAT,PHAPS Combiningthecorrected ranging source positions P° = [PSAT, PHAPS] Finding parameters V,P,b,H,Q Output Computingtheposition solution using the Least Square method x,dt x,dt𝑹 = [ −sin 𝜆 cos 𝜆 0 − cos 𝜆 sin 𝜑 − sin 𝜆 sin 𝜑 cos 𝜑 cos 𝜆 cos 𝜑 sin 𝜆 cos 𝜑 sin 𝜑 ] (15) where 𝜆 and 𝜑 represent the longitude and latitude of the receiver, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' Then the HDOP is described by 𝐻𝐷𝑂𝑃 = √𝜎𝑛2 + 𝜎𝑒2 (16) where 𝜎𝑛, 𝜎𝑒, and 𝜎𝑑 represent the receiver position errors in the local north, east and down directions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' SIMULATION OF THE HAPS-AIDED GPS SYSTEM A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' Simulation Setup The system model is established using the default Earth orientation parameters of the Skydel GNSS simulation software [13] which considers all GPS satellites orbiting around the Earth and transmitting the L1 C/A code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' The Saastamoinen model is chosen to emulate the tropospheric effect and the Klobuchar model is chosen to emulate the ionospheric effect along with the software default Klobuchar parameters (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=', alpha and beta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' The output from Skydel contains the ECEF coordinates of satellites at the signal emission time, the ionospheric corrections, the tropospheric corrections, the satellite clock offsets, the ECEF coordinates of the receiver, the signal emission time, and so forth, at each time stamp from the start of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' The receiver clock offset in the simulation is zero by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' The correction terms in the pseudorange equation of satellite including the satellite orbit error, the multipath and the receiver noise are not separately considered in the simulation, instead a pseudorange error is introduced to reflect the presence of those effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' The pseudorange error of satellite is featured using the built-in first order Gauss-Markov process with the default time constant of 10 s and the standard deviation of 6 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' The continuous model for the first order Gauss-Markov process is described by [16] 𝑥̇ = − 1 𝑇𝑐 𝑥 + 𝑤 (17) where 𝑥 represents a random process with zero mean, correlation time 𝑇𝑐, and noise 𝑤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' The autocorrelation of the first order Gauss-Markov process is described by [17] 𝑅(∆𝑡) = 𝜎2𝑒−|∆𝑡| 𝜏 (18) where ∆𝑡 represents the sampling interval, 𝜎 and 𝜏 denote the standard deviation and the time constant of the first order Gauss-Markov process, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' The pseudorange of HAPS is simulated by adding Gaussian noise to the geometric range between HAPS and receiver, where the Gaussian noise represents the sum of all kinds of estimation residuals including the HAPS position, the HAPS clock offset, the tropospheric delay, the multipath and the receiver noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' The pseudorange error for HAPS is modelled using the Gaussian noise with standard deviations of 2 m and 5 m representing the suburban and the dense urban scenario, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' The characteristics of the pseudorange errors for the suburban scenario and the dense urban scenario are set to be the same for satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' Note that by doing this, the positioning performance of the GPS-only system stays the same in both suburban scenario and dense urban scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' The standard deviation for the HAPS pseudorange error is enforced to be smaller than that for the satellite pseudorange error in both suburban scenario and dense urban scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' All the available satellites (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=', satellites with elevation angles greater than the predefined elevation mask) are simultaneously utilized for positioning as if all satellites above the elevation mask are in line of sight (LOS) with the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' Under this setting, we examine the 3D positioning performance for the GPS-only system, the one-HAPS with GPS system, the four-HAPS with GPS system and the four-HAPS-only system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' For the one- HAPS with GPS system, we use the HAPS on top of the downtown Ottawa area which elevation is above 80°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' Simulation Results The cumulative distribution functions of the 3D positioning accuracy for different systems with the standard deviations of the HAPS pseudorange error being 2 m and 5 m are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' 4 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' 4, we can observe that with much less pseudorange error for HAPS, the four-HAPS with GPS system achieves the best positioning performance, the one-HAPS with GPS system achieves almost the same positioning performance as the GPS-only system, and the four-HAPS-only system achieves slightly worse performance than the four-HAPS with GPS system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' The reasons why the four-HAPS-only system does not achieve the best positioning performance is potentially due to the following reasons 1) it has much fewer ranging sources in receiver position computation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' 2) the ranging source geometry is poor as the elevation angles for all four HAPS at any given time are above 40° with one even above 80°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' 5, we see that with the HAPS pseudorange error similar but slightly smaller than the satellites’ pseudorange error, the four-HAPS-only system achieves the worst positioning performance but the four- HAPS with GPS Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' 4: CDF for 3D position accuracy (suburban scenario).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' 5: CDF for 3D position accuracy (dense urban scenario).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' Denseurbanscenario(HAPSprstd=5m) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='6 DF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='2 GPS-onlysystem One-HAPSwithGPSsystem 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='1 Four-HAPSwithGPSsystem Four-HAPS-only system 0 0 5 10 15 20 25 30 35 3Dpositionalaccuracy(m)inlocalNEDframeSuburbanscenario(HAPSprstd=2m) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='6 DF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='5 C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='2 GPS-onlysystem One-HAPSwithGPSsystem 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='1 Four-HAPS with GPS system Four-HAPS-only system 0 0 5 10 15 20 25 30 35 3Dpositionalaccuracy(m)inlocalNEDframesystem still outperforms the other systems considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' FIELD EXPERIMENTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' Experiment Setup To verify and support the simulation results, we also process the raw GNSS data collected along the vehicle trajectory which is similar to the one shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' 2 with a slight difference due to partial road closure on the day of data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' The raw GNSS data are collected using the Ublox EVK-M8T GNSS unit and processed using the single point positioning package developed by Napat Tongkasem [18] with proper modification so that HAPS can be incorporated in the single point positioning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' Table I gives the specifications of the EVK-M8T GNSS unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' To reflect realistic LOS conditions for HAPS, the LOS probability with respect to the HAPS elevation angle in the urban area is implemented based on [19] and [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' Note that the LOS probability for HAPS provided by [19] is generated based on the city of Chicago and enforcing the LOS probability on HAPS in the dense urban area in Ottawa might be too harsh considering the incompatible city scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' The pseudorange of HAPS in the experiment is modeled as the addition of the geometric range between the satellite and receiver, the receiver clock offset multiplied by the speed of light and the pseudorange error representing the sum of all kinds of estimation residuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' The pseudorange errors for HAPS in the suburban area and in the dense urban area are simulated as Gaussian noise with standard deviations of 2 m and 5 m, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' Since the vehicle trajectory involves both suburban area and dense urban area, the entire route is divided into two parts where the first part is considered as the suburban scenario and the second part is considered as the dense urban scenario (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' By observing the positioning performance of the GPS-only system using the real GPS data, the LOS probability for the suburban area is applied to HAPS for epochs less than 380 s, and the LOS probability for the dense urban area is applied to HAPS for epochs greater than or equal to 380 s (refer to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' Since the GNSS receiver does not provide accurate receiver clock offset with respect to the GPS time, the receiver clock offset in each epoch is estimated by making use of the ground truth receiver position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' The ground truth data is provided by Ublox EVK-M8U GNSS unit, which is equipped with accelerometer and gyroscope, hence it can perform sensor fusion to get better positioning performance and dead reckoning when the signal quality degrades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' EVK-M8T GNSS UNIT SPECIFICATIONS [21] Parameter Specification Serial Interfaces 1 USB V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='0 1 RS232, max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='baud rate 921,6 kBd DB9 +/- 12 V level 14 pin – 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='3 V logic 1 DDC (I2C compatible) max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' 400 kHz 1 SPI-clock signal max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' 5,5 MHz – SPI DATA max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' 1 Mbit/s Timing Interfaces 2 Time-pulse outputs 1 Time-mark input Dimensions 105 × 64 × 26 mm Power Supply 5 V via USB or external powered via extra power supply pin 14 (V5_IN) 13 (GND) Normal Operating Temperature −40℃ to +65℃ Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' 6: HDOP (top) and 3D position accuracy (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' Experiment Results Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' 6 shows the HDOP, and the 3D positioning accuracy overlapped with the number of visible HAPS at each epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' As we can see from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' 6, the HDOP and 3D positioning accuracy of the HAPS-aided GPS system are better than that of the GPS-only system in both suburban area and dense urban area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' Moreover, we can observe that the positioning performance of the HAPS-aided GPS system is more stable than the GPS-only system as there are less spikes for the HAPS-aided GPS system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' Note that, the pseudorange of HAPS in the experiment is modeled as a function of the receiver clock offset, which is estimated with the best effort, additional error should be expected in the pseudorange of HAPS with the magnitude depending on the quality of all visible satellite signals and the ground truth receiver position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' As we would expect the quality of the satellite signals in the suburban area is better compared to that in the dense urban area, the receiver clock offset would also be expected to be Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' 7: CDF of 3D position accuracy in the suburban area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' 8: CDF of 3D position accuracy in the dense urban area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' Suburbanarea 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='6 CDF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='5 C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='1 GPS-onlysystem HAPS-aided GPS system 0 0 5 10 15 20 25 30 35 3Dpositioningerror(m)Denseurbanarea 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='6 CDF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content='1 GPS-only system HAPS-aidedGPSsystem 0 0 50 100 150 200 250 3Dpositioningerror(m)GPS-only system HAPS-aided GPS system number of HAPS 0 100 200 300 400 500 600 700 epoch (s)GPS-cnly system 4 HAPS-aided GPS system 300 Hoe number of HAPS 3 DOSI 100 0 100 200 300 400 500 600 700 epoch (s)estimated with higher accuracy in the suburban area than in the dense urban area, hence the HDOP of the HAPS-aided GPS system in the suburban area is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' The cumulative distribution functions of the 3D positioning accuracy in the suburban and dense urban areas are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' 7 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' 8, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' 7 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' 8, we can observe that the HAPS-aided GPS system outperforms the GPS-only system, especially in the suburban area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' CONCLUSION As we are passing 5G and soon entering 6G and beyond, HAPS can be of invisible treasure as it can be used for computation offloading [22], edge computing [23], even base station [24] to meet human needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' HAPS can be another type of ranging source which is quasi-stationary and much closer to the ground of the Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' Compared to satellite, HAPS exhibits the advantages of lower latency, lower pathloss, lower pseudorange error, and it can provide continuous coverage to reduce the number of handovers for the users in a certain region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' Since urban area is the region where GNSS positioning performance degrades severely and where most people live in, deploying several HAPS acting as another type of ranging source on top of a metro city would improve the GNSS positioning performance and maximize the value of the extra payload on HAPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' The HAPS-aided GNSS can also be deployed in the regions with extreme environment such as the Arctic region where the satellite availability is low, and the ionospheric disturbances is severe [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' From both the simulation and physical experiment results, we observe that HAPS can indeed improve the 3D positioning accuracy, especially in the suburban area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' To improve the results of HAPS-aided GPS system in the dense urban area, the receiver clock offset should be estimated with higher accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' In future work, the received signal powers of HAPS and satellite will jointly be considered, a satellite selection algorithm will be applied to better emulate the way a modern GNSS receiver processes the raw GNSS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' ACKNOWLEDGMENT This paper is supported in part by Huawei Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' The Skydel software is a formal donation from Orolia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' REFERENCES [1] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQf2_nF/content/2301.00762v1.pdf'} +page_content=' Li et al.' metadata={'source': 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Dias,1, † Adolfo Guevara,1, ‡ +Feng-Kun Guo,1, 2, 3, § and Bing-Song Zou1, 2, 4, ¶ +1CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, +Chinese Academy of Sciences, Beijing 100190, China +2School of Physical Sciences, University of Chinese Academy of Sciences, +Beijing 100049, China +3Peng Huanwu Collaborative Center for Research and Education, Beihang University, Beijing 100191, China +4Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China +In this work, we interpret the newly observed η1(1855) resonance with exotic JP C = +1−+ quantum numbers in the I = 0 sector, reported by the BESIII Collaboration, as a +dynamically generated state from the interaction between the lightest pseudoscalar mesons +and axial-vector mesons. The interaction is derived from the lowest order chiral Lagrangian +from which the Weinberg-Tomozawa term is obtained, describing the transition amplitudes +among the relevant channels, which are then unitarized using the Bethe-Salpeter equation, +according to the chiral unitary approach. We evaluate the η1(1855) decays into the ηη′ and +K ¯K∗π channels and find that the latter has a larger branching fraction. We also investigate +its SU(3) partners, and according to our findings, the π1(1400) and π1(1600) structures +may correspond to dynamically generated states, with the former one coupled mostly to +the b1π component and the latter one coupled to the K1(1270) ¯K channel. In particular, +our result for the ratio Γ(π1(1600) → f1(1285)π)/Γ(π1(1600) → η′π) is consistent with the +measured value, which supports our interpretation for the higher π1 state. We also report +two poles with a mass about 1.7 GeV in the I = 1/2 sector, which may be responsible for +the K∗(1680). We suggest searching for two additional η1 exotic mesons with masses around +1.4 and 1.7 GeV. In particular, the predicted η1(1700) is expected to have a width around +0.1 GeV and can decay easily into K ¯Kππ. +∗ yanmaojun@itp.ac.cn +† jorgivan.mdias@itp.ac.cn +‡ aguevara@itp.ac.cn +§ fkguo@itp.ac.cn +¶ zoubs@itp.ac.cn +arXiv:2301.04432v1 [hep-ph] 11 Jan 2023 + +2 +I. +INTRODUCTION +Over the last two decades, the experimental observation of many new hadronic states is chal- +lenging our current understanding of hadrons as conventional mesons and baryons with valence +contents of quark-antiquark and three quarks, respectively, since most of them do not fit in the +well-known quark model. This difficulty brought back a long-standing discussion on the exotic +hadronic structures, i.e., multiquark configurations that might have quantum numbers beyond +those assigned to the conventional mesons and baryons [1, 2]. +Exotic quark configurations such as tetraquarks [3, 4], hadron-hadron molecules [5], glueballs, +and hybrids [6, 7], among others, have been suggested to describe suitably most of the properties of +these new states, such as the JPC quantum numbers, mass, and decay width, especially for those +lying in the charmonium and bottomonium spectra. +On the other hand, distinguishing the exotic states from the conventional hadrons is a more +complicated task in the light quark sector. Many states have their masses close to each other, and +the possibility of mixing brings additional difficulty to the problem. The situation improves as +the quantum numbers do not fall into those allowed by the conventional quark model. It seems +to be the case of the newly discovered state, dubbed η1(1855), by the BESIII Collaboration [8, 9], +observed in the invariant mass distribution of the η η′ meson pair in the J/ψ → γ η η′ decay +channel with a significance of 19σ. Its mass and width reported by BESIII are 1855 ± 9+6 +−1 MeV +and 188 ± 18+3 +−8 MeV, respectively, with likely JPC = 1−+ quantum numbers, which cannot be +formed by a pair of quark and antiquark. The η1(1855) is not the only state experimentally found +with that set of quantum numbers. As of today, three other hadronic structures, called π1(1400), +π1(1600) and π1(2015), with JPC = 1−+, were observed by several collaborations [7, 10]. +From the theoretical point of view, the hybrid model has been used to investigate these exotic +meson states, in particular the 1−+ ones. Lattice quantum chromodynamics (QCD) calculations +have pointed out hybrid supermultiplets with exotic JPC quantum numbers, including the 1−+ +one [11–16]. In this picture, however, the mass of the lightest 1−+ state and decay modes are in- +consistent with the corresponding experimental results, while the π1(1600) and π1(2015) structures +can fit into the nonets predicted by lattice QCD [7]. +The newly observed η1(1855) state has also been the focus of some studies. In particular, the +authors in Ref. [17] proposed two hybrid nonet schemes in which the η1(1855) resonance can be +either the lower or higher mass state with isospin I = 0. In Ref. [18], an effective Lagrangian +respecting flavor, parity, and charge conjugation symmetries is used to study the hybrid nonet + +3 +decays into two-body meson states. The authors have fixed the couplings to those two-body meson +states by performing a combined fit to the experimental and lattice results available. As a result, +the decay width value estimated for the isoscalar member of the hybrid nonet agrees with the +one observed for η1(1855) state. Also addressing the same picture, Ref. [19] applied the approach +of QCD sum rules to describe the η1(1855) mass. By contrast, within the same approach, the +η1(1855) resonance is described as a tetraquark state in Ref. [20]. +The η1(1855) resonance also supports a meson-meson molecule interpretation due to its prox- +imity to the K ¯K1(1400) threshold, as put forward by Refs. [21, 22]. In particular, the authors +in Ref. [21] have investigated the K ¯K1(1400) interaction through the one-boson exchange model. +According to their findings, the K ¯K1(1400) system binds for cutoff values above 2 GeV with a +monopole form factor. In addition, the comparison between their result for the branching fraction +B(η1 → η η′) to the experimental one led them to conclude that the K ¯K1(1400) molecule can +explain the η1(1855) structure. +An important point to be addressed is the meson-meson interaction around the K1(1400) ¯K +threshold for the JPC = 1−+ quantum numbers. In this sector, many meson-meson pairs may +contribute to that interaction, so a coupled-channel treatment seems appropriate to take these +contributions into account. In particular, hadron-hadron interactions in coupled channels have +been studied in many works to describe the properties of the new hadronic systems experimentally +observed. In those cases, these hadronic structures are called dynamically generated states. +Following this approach, in this work, we aim to explore the η1(1855), π1(1400), and π1(1600) +hadronic systems as dynamically generated states from pseudoscalar-axial vector meson interactions +in coupled channels. Specifically, the low-energy interactions are given by the Weinberg-Tomozawa +(WT) term from chiral Lagrangians at the leading order of the chiral expansion by treating the +axial vector mesons as matter fields and the pseudoscalar mesons as the pseudo-Nambu-Goldstone +bosons of the spontaneous breaking of chiral symmetry. +Such Lagrangians have been used to +study many hadron structures stemming from meson-meson and meson-baryon interactions in +coupled channels in light and heavy sectors, see, e.g., Refs. [23–27]. In our case, the amplitudes +obtained from the WT term are unitarized via the Bethe-Salpeter equation from which bound +states/resonances manifest as poles in the physical/unphysical Riemann sheets of the scattering +matrices. The existence of a whole family of kaonic bound states has been pointed out in Ref. [28] +based on unitarizing the WT term for the scattering of the kaon off isospin-1/2 matter fields +taking heavy mesons and doubly-charmed baryons as examples. As we shall show in this work, +the newly observed η1(1855) structure may correspond to a dynamically generated state from the + +4 +pseudoscalar-axial vector interaction in the isospin I = 0 sector coupling strongly to the K1(1400) ¯K +channel. Moreover, the π1(1400) and π1(1600), may be assigned as the η1(1855) SU(3) partners +which are also dynamically generated from the pseudoscalar-axial vector meson interactions in the +I = 1 sector. The former resonance couples mainly to the b1π channel, and the latter has the +K1(1270) ¯K as its main coupled channel. +In addition, we have also found two poles around 1.7 GeV in the I = 1/2 sector. These poles +are particularly interesting as they could be the origin of the K∗(1680) structure observed experi- +mentally [10], which is the main component of the 1− contribution to the φK mass distribution in +the B → J/ψφK decays recently measured by LHCb [29]. +This paper is organized as follows. In Section II, we discuss the relevant channels contributing +to the pseudoscalar-axial vector meson interactions and the use of the chiral unitary approach +(ChUA) for the evaluation of the transition amplitudes among those channels. In Sections III +and IV, we investigate the dynamical generation of poles stemming from those interactions in the +I = 0 and I = 1 sectors and discuss their possible decay channels. Finally, in Section V, we also +explore the dynamical generation of poles for I = 1/2 and their connection to the vector K∗(1680) +structure observed experimentally. Section VI gives a summary. +II. +COUPLED CHANNEL SCATTERING IN CHIRAL UNITARY APPROACH +We investigate the interactions between axial and pseudoscalar mesons in coupled channels in +the 1300 ∼ 2000 MeV energy range. First, we need to determine the space of states contributing +to the interaction in this energy range. +In Tables I, II, III, and IV, we list all the relevant channels for the problem under consideration +along with their corresponding mass thresholds. The channels are organized from the lower to +higher mass values and by the isospin, 0, 1 and 1/2, respectively. +TABLE I. JP C = 1−+ meson-meson channels with I = 0. The threshold masses are in the units of MeV. +Channel +a1π +K1(1270) ¯K +f1(1285)η +K1(1400) ¯K +f1(1420)η +Threshold +1368 +1748 +1829 +1898 +1973 +TABLE II. JP C = 1−+ meson-meson channels with I = 1. The threshold masses are in the units of MeV. +Channel +b1π +f1(1285)π +f1(1420)π +K1(1270) ¯K +a1η +K1(1400) ¯K +Threshold +1367 +1419 +1564 +1748 +1777 +1895 + +5 +TABLE III. JP = 1− meson-meson channels with I = 1/2. The threshold masses are in the units of MeV. +Here the flavor-neutral axial vector mesons have JP C = 1++. +Channel +a1K +f1(1285)K+ +K1(1270)η +f1(1420)K +K1(1400)η +Threshold +1725 +1777 +1800 +1921 +1947 +TABLE IV. JP = 1− meson-meson channels with I = 1/2. The threshold masses are in the units of MeV. +Here the flavor-neutral axial vector mesons have JP C = 1+−. +Channel +h1(1170)K +b1K +K1(1270)η +h1(1415)K +K1(1400)η +Threshold +1661 +1725 +1800 +1911 +1947 +In what follows, we shall discuss the relevant scattering amplitudes among all those channels +above for each isospin sector. These transitions can be written in the form of the WT term which +then is unitarized. Notice that the channels displayed in Tables III and IV, in principle, should be +grouped in the same space of states since they share identical isospin and JP quantum numbers. +However, the relevant transitions among them arise only at the next-to-leading order in the chiral +expansion; see the discussion around Eq. (17) below. Thus, such transitions are of higher order +than that of the WT term and will be neglected here. +A. +The Weinberg-Tomozawa term +In order to study the interactions among all the channels listed in the previous tables, we have +to evaluate the interactions between the pseudoscalar and axial-vector mesons. +The latter are +organized in two SU(3) octets according to their JPC quantum numbers. +A1 = +� +� +� +� +� +a0 +1 +√ +2 + f8 +1 +√ +6 +a+ +1 +K+ +1A +a− +1 +− a0 +1 +√ +2 + f8 +1 +√ +6 +K0 +1A +K− +1A +¯K0 +1A +− 2f8 +1 +√ +6 +� +� +� +� +� +(1) +is the octet of resonances of axial-vector states with JPC = 1++ for the flavor-neutral mesons, and +B1 = +� +� +� +� +� +b0 +1 +√ +2 + h8 +1 +√ +6 +b+ +1 +K+ +1B +b− +1 +− b0 +1 +√ +2 + h8 +1 +√ +6 +K0 +1B +K− +1B +K0 +1B +− 2 +√ +6h8 +1 +� +� +� +� +� +(2) +describes the octet of axial-vector resonances with JPC = 1+−. +The singlet and I = 0 octet +flavor eigenstates are not mass eigenstates; that is, the pairs of f1(1420), h1(1415) (also known as + +6 +TABLE V. Two sets of values of the axial-vector meson mixing angles taken from Ref. [30]. Set B is preferred +in Ref. [30]. The η-η′ mixing angle θP is taken from Ref. [31]. For more discussions about these mixing +angles, we refer to the review of Quark Model in the Review of Particle Physics [10]. +Angles +θK1 +θ3P1 +θ1P1 +θP +Set A +57◦ +52◦ +−17.5◦ −17◦ +Set B +34◦ +23.1◦ +28.0◦ +−17◦ +h1(1380)) and f1(1285), h1(1170) mesons are mixtures of the singlet (1) and octet (8) mesons such +that +� +� |f1(1285)⟩ +|f1(1420)⟩ +� +� = +� +� cos θ3P1 +sin θ3P1 +− sin θ3P1 cos θ3P1 +� +� +� +� +��f1 +1 +� +��f8 +1 +� +� +� , +(3) +and +� +� |h1(1170)⟩ +|h1(1415)⟩ +� +� = +� +� cos θ1P1 +sin θ1P1 +− sin θ1P1 cos θ1P1 +� +� +� +� +��h1 +1 +� +��h8 +1 +� +� +� . +(4) +Furthermore, the K1A and K1B members of the multiplets in Eqs. (1) and (2) are the strange +partners of the a1(1260) and b1(1235), and their mixture contributes to the physical K1(1270) and +K1(1400) mesons, that is +� +� |K1(1270)⟩ +|K1(1400)⟩ +� +� = +� +� sin θK1 +cos θK1 +cos θK1 − sin θK1 +� +� +� +� |K1A⟩ +|K1B⟩ +� +� . +(5) +The corresponding values for the mixing angles in Eqs. (3), (4), and (5) are listed in Table V, where +they are grouped into two sets, denoted by A and B. Although set B is preferred in Ref. [30], we +will use both sets to have an estimate of the uncertainties caused by such an angle. +In order to determine the WT term we start with the Lagrangian (see, e.g., Ref. [32]) +L0 = −1 +4 +� +VµνV µν − 2M2 +V VµV µ� +, +(6) +where ⟨, ⟩ takes trace in the SU(3) flavor space, +Vµν = DµVν − DνVµ , +(7) +while Dµ is the chirally covariant derivative, which when acting on SU(3) octet matter fields reads +as +Dµ = ∂µ + [Γµ, ] , +(8) + +7 +with [ , ] the usual commutator. In addition, Γµ stands for the chiral connection, given by +Γµ = 1 +2 +� +u†∂µu + u∂µu†� +, +(9) +with +u = exp +� +i +√ +2Fπ +φ8 +� +, +(10) +where Fπ = 92.1 MeV is the pion decay constant [10], and φ8 is the pseudoscalar SU(3) octet, that +is +φ8 = +� +� +� +� +� +π0 +√ +2 + +1 +√ +6η8 +π+ +K+ +π− +− 1 +√ +2π0 + +1 +√ +6η8 +K0 +K− +¯K0 +− 2 +√ +6η8 +� +� +� +� +� . +(11) +In addition, the physical η and η′ mesons are the mixtures of η8 and η1 +� +� |η⟩ +|η′⟩ +� +� = +� +� − sin θP cos θP +cos θP +sin θP +� +� +� +� +��η1� +��η8� +� +� , +(12) +where η1 becomes the ninth pseudo-Goldstone boson in large Nc QCD [33–36]. The Goldstone +boson nonet is written as +φ9 = φ8 + 1 +√ +3η1, +(13) +which leads to a relation in the commutator +� +φ9, ∂µφ9� += +� +φ8, ∂µφ8� +. +(14) +Therefore, only the scattering of the octet Goldstone bosons off the axial-vector mesons in +Weinberg-Tomozawa term contributes to JP(C) = 1−(+) spectrum. +The covariant derivative Dµ by means of the connection Γµ encodes the leading order interaction +between the pseudoscalar mesons and the vector field Vµ [32, 37, 38]. Therefore, by replacing the +Vµ field to the axial-vector field Aµ corresponding to either the A1 or B1 multiplet, the chiral tran- +sition between φ8 (pseudoscalar) and A (1+) (axial-vector) is described by the following interaction +Lagrangian +LI = − 1 +4F 2π +� +[Aµ, ∂νAµ] +� +φ8, ∂νφ8�� +, +(15) + +8 +which accounts for the WT interaction term for the PA → PA process, with P and A corresponding +to the pseudoscalar and axial-vector mesons, respectively. From this Lagrangian we obtain the S- +wave transition amplitude among the channels listed in Tables I, II, III and IV, that is +Vij(s) = −ϵ · ϵ′ +8F 2π +Cij +� +3s − +� +M2 + m2 + M′2 + m′2� +− 1 +s +� +M2 − m2� � +M′2 − m′2�� +, +(16) +where ϵ (ϵ′) stands for the polarization four-vector of the incoming (outgoing) axial-vector me- +son [25, 39]. The masses M (M′) , m (m′) correspond to the initial (final) axial-vector mesons and +initial (final) pseudoscalar mesons, respectively. The indices i and j represent the initial and final +PA states, respectively. The coefficients Cij are given in Tables VI, VII, VIII, and IX. +TABLE VI. Cij coefficients in Eq. (16) for axial and pseudoscalar pairs coupled to JP C = 1−+ in S-wave +and I = 0. +Cij +a1π +K1(1270) ¯K +f1(1285)η +K1(1400) ¯K +f1(1420)η +a1π +−4 +� +3 +2 sin θK1 +0 +� +3 +2 cos θK1 +0 +K1(1270) ¯K +−3 +− 3 +√ +2 sin θ3P1 sin θK1 +0 +− 3 +√ +2 cos θ3P1 sin θK1 +f1(1285)η +0 +− 3 +√ +2 cos θK1 sin θ3P1 +0 +K1(1400) ¯K +−3 +− 3 +√ +2 cos θ3P1 cos θK1 +f1(1420)η +0 +TABLE VII. Cij coefficients in Eq. (16) for axial and pseudoscalar pairs coupled to JP C = 1−+ in S-wave +and I = 1. +Cij +b1π +f1(1285)π +f1(1420)π +K1(1270) ¯K +a1η +K1(1400) ¯K +b1π +−2 +0 +0 +cos θK1 +0 +− sin θK1 +f1(1285)π +0 +0 +� +3 +2 sin θK1 sin θ3P1 +0 +� +3 +2 cos θK1 sin θ3P1 +f1(1420)π +0 +� +3 +2 cos θ3P1 sin θK1 +0 +� +3 +2 cos θK1 cos θ3P1 +K1(1270) ¯K +−1 +− +� +3 +2 sin θK1 +0 +a1η +0 +− +� +3 +2 cos θK1 +K1(1400) ¯K +−1 +Before proceeding, let us discuss the A1φ8 → B1φ8 transitions, with A1 and B1 the two SU(3) +octets of axial-vector mesons and φ8 the octet of the pseudo-Nambu-Goldstone bosons. Let A1µ +and B1µ denote the fields for the 1++ and 1+− axial-vector meson multiplets, respectively. Under +parity transformation, we have A1µ → −Aµ +1 and B1µ → −Bµ +1 ; under charge conjugation, we have +A1µ → AT +1µ and B1µ → −BT +1µ. Then the A1φ8 → B1φ8 transitions can only arise at O +� +p2� +with p + +9 +TABLE VIII. Cij coefficients in Eq. (16) for axial and pseudoscalar pairs coupled to JP = 1− in S-wave +and I = 1/2. Here the flavor-neutral axial mesons have JP C = 1++. +Cij +a1K +f1(1285)K +K1(1270)η +f1(1420)K +K1(1400)η +a1K +−2 +0 +− 3 +2 sin θK1 +0 +− 3 +2 cos θK1 +f1(1285)K +0 +3 +2 sin θK1 sin θ3P1 +0 +3 +2 sin θK1 cos θK1 +K1(1270)η +0 +3 +2 cos θ3P1 sin θK1 +0 +f1(1420)K +0 +3 +2 cos θ3P1 cos θK1 +K1(1400)η +0 +TABLE IX. Cij coefficients in Eq. (16) for axial and pseudoscalar pairs coupled to JP = 1− in S-wave and +I = 1/2. Here the flavor-neutral axial mesons have JP C = 1+−. +Cij +h1(1170)K +b1K +K1(1270)η +h1(1415)K +K1(1400)η +h1(1170)K +0 +0 +3 +2 cos θK1 sin θ1P1 +0 +3 +2 sin θK1 sin θ1P1 +b1K +−2 +− 3 +2 cos θK1 +0 +− 3 +2 sin θK1 +K1(1270)η +0 +3 +2 cos θK1 cos θ1P1 +0 +h1(1415)K +0 +3 +2 sin θK1 cos θ1P1 +K1(1400)η +0 +the momentum scale in the chiral power counting. They are given by operators such as +⟨A1µ[B1ν, [uµ, uν]]⟩ , +(17) +with +uµ = i +� +u†∂µu − u∂µu†� +. +(18) +Such terms are one order higher in the chiral power counting than the WT terms describing the +A1φ8 → A1φ8 and B1φ8 → B1φ8 transitions, and thus will be neglected. +B. +Unitarization procedure +The unitarization procedure we adopt follows ChUA in which the scattering amplitudes in +Eq. (16) are the elements of a matrix, denoted by V , such that it enters as an input to solve the +Bethe-Salpeter equation, which in its on-shell factorization form, reads [23] +T = (1 − V G)−1 V . +(19) + +10 +The V matrix describes the transition between the channels listed in Tables I, II, III, and IV. In +addition, G is the diagonal loop function matrix whose diagonal matrix elements are given by +Gl = i +� +d4q +(2π)4 +1 +q2 − m2 +l + iϵ +1 +(q − P)2 − M2 +l + iϵ , +(20) +with ml and Ml the masses of the pseudoscalar and axial-vector mesons, respectively, involved in +the loop in the channel l, and P the total four-momentum of those mesons (P 2 = s). After the +integration over the temporal component q0, Eq. (20) becomes +Gl(s) = +� +d3q +(2π)3 +ω1 + ω2 +2ω1ω2 +1 +(P 0)2 − (ω1 + ω2)2 + iϵ +, +(21) +with ω1 = +� +Ml2 + |⃗q|2 and ω2 = +� +ml2 + |⃗q|2, and can be regularized by means of a cutoff in +the three-momentum qmax. On the other hand, the function Gl can also be regularized using a +subtraction constant as [40] +GDR +l +(s) = +1 +16π2 +� +αl(µ) + log M2 +l +µ2 + m2 +l − M2 +l + s +2s +log m2 +l +M2 +l ++ pl +√s +� +log s − m2 +l + M2 +l + 2pl +√s +−s + m2 +l − M2 +l + 2pl +√s ++ log s + m2 +l − M2 +l + 2pl +√s +−s − m2 +l + M2 +l + 2pl +√s +�� +, +(22) +where pl is the three-momentum of the mesons in the center-of-mass (c.m.) frame +pl = +�� +s − (Ml + ml)2� � +s − (Ml − ml)2� +2√s +, +(23) +while µ is an arbitrary scale of the regularization. Any changes in the µ scale can be absorbed by the +subtraction constant αl(µ) such that the result is independent of the scale. We may determine the +subtraction constant for each intermediate state of the scattering problem by comparing Eqs. (21), +regularized using qmax, and (22) at the threshold. The equivalence between the two prescriptions +for the loop-function is discussed in, e.g., Refs. [41–43]. In this work, we follow Ref. [44] and set +µ = 1 GeV and α = −1.35, which is obtained by matching to hard cutoff regularization with +qmax ≃ 0.7 GeV in the f1(1285)η channel. This set of parameters are used for all channels, and +a variation of the cutoff within qmax = (0.7 ± 0.1) GeV, and correspondingly α(µ = 1 GeV) = +−1.35 ± 0.17, will be used to show the dependence of the results on this parameter. +C. +Searching for poles +We move on to the complex energy plane to search for poles in the T-matrix. Specifically, for a +single-channel problem, there are two Riemann sheets for the complex energy plane. Bound states + +11 +show up as poles, below the threshold, in the transition matrix on the real energy axis on the first +Riemann sheet, while virtual states manifest themselves below the threshold on the real axis on the +second Riemann sheet, and resonances correspond to poles off the real axis on the second Riemann +sheet. The Riemann sheets come about because the G loop function has a cut extending from +the threshold to infinity which is usually chosen to be along the positive real axis. For n coupled +channels, there are n cuts and thus 2n Riemann sheets. From unitarity and the Schwarz reflection +principle, the discontinuity of the Gl function can be read off from its imaginary part, +Im Gl(s) = − +pl +8π√s , +(24) +which we can use to perform an analytic continuation to the entire complex plane. In this case, +the Gl loop function on the “second” Riemann sheet with respect to the lth channel reads +GII +l (s) = GI +l(s) + i +pl +4π√s ; +(25) +the lower half plane of this Riemann sheet is directly connected to the physical region when the lth +channel is open, i.e., Re(√s) ≥ m + M. We will label the Riemann sheets according to the sign of +the imaginary part of the corresponding c.m. momentum for each channel (see the next section). +Furthermore, it is also possible to determine the pole couplings to the lth channel. Note that +close to the pole singularity the T-matrix elements Tij(s) admit a Laurent expansion, +Tij(s) = gi gj +s − zp ++ regular terms, +(26) +where zp = (Mp −iΓ/2)2 is the pole location on the complex energy plane, with Mp and Γ standing +for the pole mass and width, respectively. Therefore, the product of couplings gigj is the residue +at the pole in Tij(s) which takes values on the Riemann sheet where the pole is located. In this +way, the couplings can be evaluated straightforwardly. For instance, for a diagonal transition it is +given by +g2 +i = r +2π +� 2π +0 +Tii(z(θ))eiθdθ += lim +s→zp(s − zp)Tii(s) = +� d +ds +1 +Tii(s) +�−1 +s=zp +, +(27) +where z(θ) = zp + i reiθ with r the radius of contour for the integral, and the two lines give two +equivalent ways of computing residues. + +12 +TABLE X. The poles (in GeV) and their corresponding couplings (in GeV) to the channels contributing to the +PA interaction with I = 0 and exotic quantum numbers JP C = 1−+. The corresponding Riemann sheet for +each pole is listed below the pole position. The dominantly coupled channel is emphasized in boldface for each +pole. The errors of the poles are from varying the subtraction constant within α(µ = 1 GeV) = −1.35±0.17, +and only the central values of the couplings are given. +Poles (Set A) +Channels +1.39 ± 0.01 − i(0.04 ± 0.01) +a1π +K1(1270) ¯ +K +f1(1285)η +K1(1400) ¯ +K +f1(1420)η +(− + + + +) +gl +5.21 + i3.01 +1.22 + i0.78 +0.01 + i0.02 +0.36 + i0.35 +0.00 +1.69 ± 0.03 +a1π +K1(1270) ¯ +K +f1(1285)η +K1(1400) ¯ +K +f1(1420)η +(− + + + +) +gl +0.36 + i0.98 +8.16 − i0.17 +3.64 + i0.01 +0.09 − i0.15 +2.46 + i0.01 +1.84 ± 0.03 +a1π +K1(1270) ¯ +K +f1(1285)η +K1(1400) ¯ +K +f1(1420)η +(− − − + +) +gl +0.07 + i0.28 +0.69 + i0.55 +1.68 + i0.08 9.33 + i0.15 1.16 + i0.06 +Poles (Set B) +Channels +1.39 ± 0.01 − i(0.04 ± 0.01) +a1π +K1(1270) ¯ +K +f1(1285)η +K1(1400) ¯ +K +f1(1420)η +(− + + + +) +gl +5.21 + i3.03 +0.81 + i0.53 +0.00 +0.55 + i0.54 +0.00 +1.70 ± 0.02 +a1π +K1(1270) ¯ +K +f1(1285)η +K1(1400) ¯ +K +f1(1420)η +(− + + + +) +gl +0.25 + i0.67 +8.34 − i0.08 +1.27 − i0.01 +0.37 + i0.17 +2.58 − i0.01 +1.84 ± 0.03 +a1π +K1(1270) ¯ +K +f1(1285)η +K1(1400) ¯ +K +f1(1420)η +(− − − + +) +gl +0.15 + i0.62 +0.33 − i0.27 +1.83 + i0.09 9.05 + i0.17 3.81 − i0.20 +III. +η1(1855) AND ITS DECAYS +A. +Dynamical generation of the η1(1855) +Following the unitarization procedure described previously, we seek dynamically generated +states stemming from the S-wave interactions between pseudoscalar and axial-vector mesons. For +the I = 0 case, the transition amplitudes among the channels listed in Table I are determined using +Eq. (16) with the Cij coefficients given in Table VI. In Table X, we show the isoscalar poles with +exotic quantum numbers JPC = 1−+ obtained by solving Eq. (19) using those coefficients as well +as each set of mixing angles listed in Table V. We also show the couplings of these poles to the +channels spanning the space of states in Table I. +Furthermore, in Table X we also highlight the Riemann sheets, the first and the second one +for each channel, denoted by the + and − signs, respectively. We get three poles such that their + +13 +locations are barely affected by the change of the mixing angles from set A to set B listed in +Table V. The lower pole is at 1.39 GeV with a width of about 0.04 GeV, which is above the a1π +threshold. In particular, this channel is open for decay, and the fact that it is this channel the +one for which the pole couples mostly, as pointed out in Table X, explains why that pole has such +a value for its width. By contrast, although the a1π channel is also open for decay, the pole at +1.69 GeV has a much smaller width because its coupling to this channel is small compared to +the one for K1(1400) ¯K, which is the dominant channel for that pole. Similarly, the highest pole, +located at 1.84 GeV, couples mostly to the K1(1400) ¯K channel, and has a small imaginary part. +In addition, we can also understand why the highest pole couples more to the K1(1400) ¯K than +to the f1(1285)η. The latter channel is closer to the pole than the former, but from Table VI, +the diagonal f1(1285)η transition is not allowed since its WT term is zero. Nevertheless, the pole +couples to f1(1285)η through the nondiagonal K1(1400) ¯K–f1(1285)η transition, which leads to a +small coupling. +B. +Effects of the widths of the axial-vector mesons +So far we have neglected the nonzero widths of the axial-vector mesons. In order to investigate +their effects on the results, we use complex masses for the intermediate resonances, that is, Mi → +Mi − iΓi/2. However, by doing that, the analytic properties are lost such that the poles of the +T matrix do not correspond to the masses and widths of the obtained resonances any more. On +the other hand, we can see the impact of such nonzero widths on the lineshapes of the transition +matrix elements. +In Fig. 1 we show a comparison between the lineshape for the T-matrix element corresponding +to the elastic transition TK1(1400) ¯ +K→K1(1400) ¯ +K with and without including the widths for the inter- +mediate particles. This channel has the strongest coupling to the pole at 1.84 GeV; therefore, we +expect that any nontrivial structure should manifest most in its associated T-matrix element. The +dashed and solid lines are the TK1(1400) ¯ +K→K1(1400) ¯ +K with zero and nonzero width, respectively, for +both sets A and B of mixing angles in Table 1. Notice that, for the case of zero width approxima- +tion, the TK1(1400) ¯ +K→K1(1400) ¯ +K lineshape has narrow peaks around 1845 MeV, right at the range +of energy where we expect the η1(1855) manifests in our model. The inclusion of finite widths for +the axial-vector mesons changes the sharp peak to a broad bump with a width of about 0.2 GeV, +which is around the width of the K1(1400) [10]. Notice that the width matches nicely that of the +η1(1855) measured by BESIII, +� +188 ± 18+3 +−8 +� +MeV [8]. In the following, we will continue to present + +14 +w/o Γ +w/o Γ +w/ Γ +w/ Γ +1600 +1700 +1800 +1900 +2000 +0 +10 +20 +30 +40 +50 +s [MeV] +|T44 +2 +Set A +Set B +FIG. 1. The blue dashed and solid lines are, respectively, the modulus squared of the T-matrix element, cor- +responding to the diagonal K1(1400) ¯K → K1(1400) ¯K transition, evaluated with and without the inclusion +of the widths associated with the axial-vector mesons taking part in the loop function Gl (Eq. (20)). +predictions neglecting the width effects of the axial-vector mesons. +Let us briefly discuss the other two predicted isoscalar exotic η1 mesons in Table X. The one +with a mass of about 1.39 GeV, denoted as η1(1400), is expected to be rather broad due to the +large width of the a1(1260) as it couples most strongly to the a1π channel. It can be searched for +in final states such as ρππ and K ¯Kππ. The one with a mass around 1.7 GeV, denoted as η1(1700), +couples most strongly to the K1(1270) ¯K and is expected to have a width similar to that of the +K1(1270), i.e., around 0.1 GeV. It can also be searched for in final states of K ¯Kππ. +C. +The η1(1855) → η′η and K∗ ¯Kπ decays +Let us first discuss the η1 → ηη′ decay, whose Feynman diagram is shown in Fig. 2. Within +our approach the η1(1855) structure decays via its K1(1400) ¯K component, with the corresponding +coupling constant listed in Table X. We also need to evaluate the K1(1400) ¯K → ηη′ transition, for +which we use the resonance chiral theory (RχT) operators given in Ref. [45]. +The RχT operators can be divided regarding the intrinsic-parity sector to which they contribute. +Due to its nature, the odd-intrinsic parity sector will contain a Levi-Civita tensor [46–48]; for the +η1 → ηη′ decay one cannot saturate the Lorentz indices in such tensor to get a nonzero contribution. +Thus, only the even-intrinsic parity operators must give a nonvanishing contribution. Since the +chiral O(p2) Lagrangian does not contribute to such processes [49], we will use the O(p4) Lagrangian +given in Ref. [45]. From these operators, only three will contribute to this decay. To get the largest +possible contribution from such operators, we use the upper bounds imposed from chiral counting + +15 +as done in Ref. [50]. This amounts to making equal the three coupling constants and setting them +to λA +1 = λA +2 = λA +3 = g = 0.025 GeV−1, which gives a Lagrangian +L = g +� +⟨Aµν (uµuαhνα + hναuαuµ)⟩ + ⟨Aµν (uαuµhνα + hναuµuα)⟩ + ⟨Aµν (uµhναuα + uαhναuµ)⟩ +� +, +(28) +where uµ has been given in Eq. (18), hµν = D{µuν} is the symmetrized covariant derivative of uµ +and the spin-1 resonance field is given in the antisymmetric tensor formalism [37]. However, since +the η1 → K1 ¯K transition is given in terms of Proca fields, we need to express the K1 as a Proca +field. Following Ref. [49], the antisymmetric tensor field can be expressed in terms of the Proca +one as follows, +Rµ = +1 +MR +∂νRνµ, +(29) +where MR is the mass of the resonance. Using the Lagrangian of Eq.(28) and expressing the axial +resonance in the Proca representation, we get the η1 → ηη′ decay amplitude +Mη1→ ηη′ = − +4m2 +η1 +3F 3πmK1 +ggK1(1400) ¯ +KGK1 ¯ +K +�� +αp2 +η′ + 1 +√ +2βp2 +η +� +εη1 · pη + +� +pη ↔ pη′�� +, +(30) +where Fπ is the pion decay constant, gK1 ¯ +K is the coupling constant of the pole to the K1(1400) ¯K +channel, GK1 ¯ +K is the loop function for the K1 and ¯K mesons , εη1 is the η1 vector polarization, +and pη(′) is the momentum of the η(′). Here, α and β are given in terms of the η-η′ mixing angle +α = cos 2θP + 2 +√ +2 sin 2θP , +(31a) +β = 2 +√ +2 cos 2θP − sin 2θP . +(31b) +K1 +¯K +η +η′ +η1 (1855) +FIG. 2. Diagram corresponding to the η1 → ηη′ decay through the K1 ¯K loop. +Although one might try to rely in a much simpler way to describe the direct coupling of one axial- +vector and three pseudoscalar fields by means of the Hidden Local Symmetry (HLS) Lagrangian + +16 +[51–53], it is worth to notice that nonetheless, the total amplitude for this process given by the +HLS Lagrangian vanishes, which coincides with Eq.(30) in the chiral limit. +The decay of η1 state into ηη′ is given by +Γ2B = +1 +2J + 1 +1 +8πM2η1 +|Mη1→ ηη′|2 q , +(32) +with the amplitude Mη1→ ηη′ in Eq. (30), while J stands for the η1 spin. Besides that, q reads +q = +1 +2Mη1 +λ1/2 � +M2 +η1, m2 +η′, m2 +η +� +, +(33) +with Mη1, mη′, and mη the masses for the η1(1855), η′, and η mesons, respectively, where +λ (x, y, z) = x2 + y2 + z2 − 2xy − 2yz − 2zx is the K¨all´en triangle function. +Therefore, we +get the following results for the decay width in this channel +Γ2B = +� +� +� +(19 ± 4) MeV (set A) , +(7 ± 2) MeV (set B) , +(34) +where the error is from choosing subtraction constant to be in the range α(µ = 1GeV) = −1.35 ± +0.17, corresponding to the hard cutoff qmax = (0.7±0.1) GeV as discussed at the end of Section II B. +For set A, our result agrees with that of Ref. [21], where the η1(1855) was assumed to be a K1 ¯K +molecule and the same θK1 mixing angle was used for accounting for the K1A and K1B mixture +contributing to the physical K1(1270) and K1(1400) states. +AB5HicbZDLSsNA +FIZP6q3W9Slm8EiIuSFGXRTeCm4r2Am0sk+lJO3RyYWYilNA30JWoO5/IF/BtnNYstPVfXP+f+D8x08EV9pxvqzC0vLK6lpxvbSxubW9Y+/uNVWcSoYNFotYtn2qUPAIG5 +prge1EIg19gS1/dDX1W48oFY+jez1O0AvpIOIBZ1Sb0d3Nw0nPLjsVZyayCG4OZchV79mf3X7M0hAjzQRVquM6ifYyKjVnAielbqowoWxEB9gxGNEQlZfNVp2QoyCWRA+RzN6/s +xkNlRqHvsmEVA/VvDcd/ud1Uh1ceBmPklRjxEzEeEqiI7JtDHpc4lMi7EByiQ3WxI2pJIybe5SMvXd+bKL0DytuGeV6m21XLvMD1GEAziEY3DhHGpwDXVoAIMBPMbvFuB9WS +9WK8/0YKV/9mHP7I+vgGwsosJK⇤ +AB5HicbZDLSgMxFIZP6q3 +W9Wlm2ARXJUZKeqy6EZwU9FeoB1KJj3ThmYuJBmhDH0DXYm684l8Ad/GtM5CW/Vl/P/gfMfP5FCG8f5IoWV1bX1jeJmaWt7Z3evH/Q0nGqODZ5LGPV8ZlGKSJsGmEkdhKFLPQltv3x9cxvP6LS +Io4ezCRBL2TDSASCM2NH97d9t1+uOFVnLroMbg4VyNXolz97g5inIUaGS6Z13US42VMGcElTku9VGPC+JgNsWsxYiFqL5uvOqUnQayoGSGdv39nMxZqPQl9mwmZGelFbzb8z+umJrj0MhElqcGI2 +4j1glRSE9NZYzoQCrmREwuMK2G3pHzEFOPG3qVk67uLZehdVZ1z6u1u1qlfpUfoghHcAyn4MIF1OEGtAEDkN4hjd4JwF5Ii/k9SdaIPmfQ/gj8vENvKOLEQ=K1 +AB6HicbZDLSgMxFIZP6q3 +W9Wlm2ARXJUZKeqy6EZwU8FeoB1KJj3TxmYuJBmhDH0HXYm683l8Ad/GtM5CW/Vl/P/gfMfP5FCG8f5IoWV1bX1jeJmaWt7Z3evH/Q0nGqODZ5LGPV8ZlGKSJsGmEkdhKFLPQltv3x9cxvP6LS +Io7uzSRBL2TDSASCM2NH7Z7PVHY7ZcrTtWZiy6Dm0MFcjX65c/eIOZpiJHhkmndZ3EeBlTRnCJ01Iv1ZgwPmZD7FqMWIjay+brTulJECtqRkjn79/ZjIVaT0LfZkJmRnrRmw3/87qpCS69TERJa +jDiNmK9IJXUxHTWmg6EQm7kxALjStgtKR8xbixtynZ+u5i2WVonVXd82rtrlapX+WHKMIRHMpuHABdbiBjSBwxie4Q3eyQN5Ii/k9SdaIPmfQ/gj8vENSTmNMg= ¯K +AB53icbZDLTgJBEVr8IX4Ql26UhMXJEZQ9Q +l0Y1LTARJYEJ6mhpo6Xmku8aEL5BV0bd+T/+gH9jg7NQ8K5O172d1K0gVdK +Q6345hZXVtfWN4mZpa3tnd6+8f9AySaYFNkWiEt0OuElY2ySJIXtVCOPAoX +3weh65t8/ojYyie9onKIf8UEsQyk42VGri8R7Xq9cavuXGwZvBwqkKvRK39 +2+4nIoxJKG5Mx3NT8idckxQKp6VuZjDlYsQH2LEY8wiNP5lvO2UnYaIZDZH +N37+zEx4ZM4Cm4k4Dc2iNxv+53UyCi/9iYzTjDAWNmK9MFOMEjYrzfpSoyA +1tsCFlnZLJoZc0H2NCVb31suwyts6p3Xq3d1ir1q/wQRTiCYzgFDy6gDjf +QgCYIeIBneIN3RzpPzovz+hMtOPmfQ/gj5+MbHk6Meg=⌘1 +AB5HicbZDLTgJBEVr8IX4Ql26UhMXJEZY9Ql0Y1LjPJIYEJ6mhro0PNId40JIfyBroy684v8Af/GBmeh4F2drns7q +VtBqQh1/1yCiura+sbxc3S1vbO7l5/6BpkwLbIhEJbodcINKxtgSQrbqUYeBQpbwehm5rceURuZxA80TtGP+CWoRSc7 +Oi+m8peueJW3bnYMng5VCBXvVf+7PYTkUYk1DcmI7npuRPuCYpFE5L3cxgysWID7BjMeYRGn8yX3XKTsJEMxoim79/Zyc8M +mYcBTYTcRqaRW82/M/rZBRe+RMZpxlhLGzEemGmGCVs1pj1pUZBamyBCy3tlkwMueaC7F1Ktr63WHYZmdV76J6fndeqV3nh +yjCERzDKXhwCTW4hTo0QMAnuEN3p3QeXJenNefaMHJ/xzCHzkf30Mxi2s=⇡ +FIG. 3. Feynman Diagram associated with the three-body decay of the pole through its main component +K1 ¯K. +As for the η1 → ¯KK∗π three-body decay, Fig. 3 shows the Feynman diagrams contributing +to this process. In particular, the η1(1855) decays through its molecular components, that in our +approach are the K1(1270) ¯K and K1(1400) ¯K. In this case, the contribution from the K1(1270) ¯K +component can be ignored for the following reasons: 1) from Table X, we see that the relative +coupling strength for the K1(1270) ¯K channel is much smaller than that for the K1(1400) ¯K one; + +117 +2) the branching ratio B[K1(1270) → K∗π] is only 16%, while 96% of the K1(1400) decays is +dominated by the K∗π. Therefore, from Fig. 3 the η1(1855) → ¯KK∗π amplitude is written as +M3B = gK1(1400) ¯ +K +� +−gµν + pµpν +M2 +K1 +� +1 +p2 − M2 +K1 + i MK1ΓK1 +gK∗π εµ +η1εν +K∗ , +(35) +where gK1(1400) ¯ +K is the coupling of the pole associated with the η1 state to the K1(1400) ¯K channel, +gK∗π is the K1(1400)K∗π coupling extracted from the K1(1400) → K∗π reaction in the Review of +Particle Physics (RPP) [10], and εµ +η1 and εν +K∗ are the polarization vectors of the η1 and K∗ mesons, +respectively. +The differential decay width for the η1 → ¯KK∗π process is given by +dΓ +dMK1 ¯ +K += +1 +(2π)3 +pK ˜pπ +4M2η1 +|M3B|2 +1 +2J + 1 , +(36) +where +˜pπ = +1 +2MK1 +λ1/2 � +M2 +K1, m2 +K∗, m2 +π +� +, +(37) +and +pK = +1 +2Mη1 +λ1/2 � +M2 +η1, m2 +K, M2 +K1 +� +, +(38) +with MK1, mK∗, mπ being the masses of the K1(1400), K∗ and π mesons. +From Eq. (36) we obtain the following results for the η1 → ¯KK∗π decay width +Γ3B = +� +81+11 +−24 MeV +�A , +Γ3B = +� +74+12 +−24 MeV +�B , +(39) +where the uncertainties come from the subtraction constant (cutoff) used to regularize the loops +in Eq. (22) (Eq. (21)). As can be seen from Eq. (39), we obtain similar results whether we use the +sets A or B. For the sake of comparison to other works, we evaluate the ratio Γ2B/Γ3B, and get +Γ2B +Γ3B += +� +0.23−0.08 ++0.16 +�A or +� +0.10−0.03 ++0.08 +�B , +(40) +which is consistent to the results in Ref. [21], where the η1 is also assumed to be a K1(1400) ¯K +molecular state. On the other hand, adopting the same multiquark configuration than the present +work and Ref. [21], the authors of Ref. [22] have found a different result for the ratio, Γ2B/Γ3B ≈ +0.03. Nevertheless, in all the cases the results point out that the ¯KK∗π three-body channel is more +likely than the ηη′ one. + +18 +IV. +THE π1(1400/1600) DYNAMICAL GENERATION +The WT amplitudes for the pseudoscalar-axial vector meson interactions with I = 1 are given +by Eq. (16), with the corresponding Cij coefficients listed in Table VII. In this case, from Eq. (19), +we get two π1 poles shown in Table XI. +TABLE XI. Poles and their corresponding couplings to the channels contributing to the PA interaction +with JP C = 1−+ and I = 1. The errors of the poles are from varying the subtraction constant within +α(µ = 1 GeV) = −1.35 ± 0.17, and only the central values of the couplings are given. +Poles (Set A) +Channels +1.47 ± 0.01 − i(0.12 ± 0.02) +b1π +f1(1285)π +f1(1420)π +K1(1270) ¯ +K +a1η +K1(1400) ¯ +K +(− − + + ++) +gl +5.22 + i4.40 0.02 − i0.09 +0.03 − i0.05 +1.25 + i1.27 +0.02 − i0.12 1.33 + i1.63 +1.75 ± 0.02 − i(0.02 ± 0.01) +b1π +f1(1285)π +f1(1420)π +K1(1270) ¯ +K +a1η +K1(1400) ¯ +K +(− − − + ++) +gl +0.10 + i0.95 +2.73 − i0.02 +1.89 +5.84 − i1.85 3.49 − i0.03 2.65 − i0.53 +Poles (Set B) +Channels +1.47 ± 0.01 − i(0.12 ± 0.02) +b1π +f1(1285)π +f1(1420)π +K1(1270) ¯ +K +a1η +K1(1400) ¯ +K +(− − + + ++) +gl +5.27 + i4.31 0.01 − i0.03 +0.03 − i0.06 +1.97 − i1.81 +0.02 − i0.08 0.91 + i1.07 +1.77 ± 0.01 − i(0.01 ± 0.01) +b1π +f1(1285)π +f1(1420)π +K1(1270) ¯ +K +a1η +K1(1400) ¯ +K +(− − − + ++) +gl +0.13 + i1.44 +1.37 − i0.25 +2.86 − i0.50 +4.80 − i2.29 3.53 − i0.64 4.54 − i1.77 +Similar to the previous section, we also provide the couplings of these dynamically generated +states to the channels listed in Table II. Table XI shows a broad π1 pole at 1.47 GeV, and a width of +about 0.12 GeV.1 This state is above the b1π and f1(1285)π thresholds. Its large width stems from +the large coupling to the b1π and the fact that this channel is open for decaying. The f1(1285)π +channel is also open. However, according to Table VII, the corresponding WT term in Eq. (16) is +zero for the diagonal f1(1285)π transition. On the other hand, the next π1 pole in Table XI has a +sizeable dependence on the mixing angles. Using set A, we find that pole at 1.75 GeV. It couples +most strongly to the K1(1270) ¯K channel, which is closed for decaying. Nonetheless, the state +can decay into b1π and f1(1285)π, albeit their corresponding couplings are small compared to the +K1(1270) ¯K one, but still large enough to provide a sizeable width for the pole. In contrast, when +set B is adopted, the higher π1 pole is now located at 1.77 GeV, above the f1(1420)π threshold, +1 As discussed in Section III B, the widths of the dynamically generated poles will be significantly increased once +the width effects of the axial-vector mesons are taken into account; see also the discussions below. + +19 +which is now open. One might think that the width should increase since now three channels are +open for decaying. However, although the coupling to the f1(1420)π has increased in this case, at +the same time the couplings to the other open channels have decreased. Hence, the overall effect +leads to a smaller width compared to the previous case. +π +f1(1285) +π1 (1600) +K1/a1 +¯K/η +π +η′ +π1 (1600) +FIG. 4. a) Diagram corresponding to the π1(1600) → f1(1285)π reaction, and b) the π1(1600) → η′π decay +also via the AP loop. The filled circles represent the effective couplings of the π1 to the AP meson pairs +calculated from the residues. The rectangles are the AP → η′π transition amplitudes at tree level. +The lower pole mass is slightly higher than the mass of the π1(1400) state listed in RPP, +(1354 ± 25) MeV [10]. +Notice that we use the same subtraction constant for all channels. +In +principle, it can take different values and lead to a shift of the poles. In addition, we did not +include in the loops the b1 width, that is relatively large and whose effects could influence the pole +position. However, it is expected to affect more the imaginary part of the pole than the real one +(see Fig. 5(a) below). We can get a rough estimate of this change by adding the b1 width to the +previous result for Im(z1), with z1 the lower π1 pole, i.e., +Γb1 + 2Im(z1) ≈ 0.4 GeV , +(41) +which is close to the π1(1400) width reported in RPP, (330 ± 35) MeV [10]. From these results, +we are led to claim that the lower π1 pole may explain the π1(1400) resonance; in other words, the +π1(1400) is suitably described in our approach as a dynamically generated state with the b1π as +its main component. +Alternatively, following the prescription used in Section III, we can also study the changes in +the results caused by the inclusion of the finite widths for the axial-vector mesons by looking at +the line shape for the relevant T-matrix elements. In Fig. 5(a) we show the line shapes for the +T-matrix element corresponding to the elastic b1π → b1π transition, which is the one we would +expect the lower pole in Table XI manifests most due to its large coupling to the b1π channel. It +becomes clear that the bumps become broader when the widths of axial-vector mesons are taken +into account. A similar behavior can be seen in Fig. 5(b) for the T-matrix element associated + +20 +with the scattering of K1 (1270) ¯K, which is the channel to which the higher π1 pole couples most +strongly. +w/o Γ +w/o Γ +w/ Γ +w/ Γ +1300 +1400 +1500 +1600 +1700 +0.0 +0.5 +1.0 +1.5 +2.0 +s [MeV] +|T11 +2 +Set A +Set B +(a) Modulus square of elastic b1π scattering +w/o Γ +w/o Γ +w/ Γ +w/Γ +1400 +1500 +1600 +1700 +1800 +1900 +0 +1 +2 +3 +4 +s [MeV] +|T44 +2 +Set A +Set B +(b) Modulus square of elastic K1 (1270) ¯K +scattering +FIG. 5. The dashed and solid lines correspond to zero and full widths of the axial-vector mesons in G. +The higher π1 pole, denoted now by z2, has a mass consistent with that of the π1(1600), whose +pole mass has been reported to be +� +1623 ± 47+24 +−75 +� +MeV in Ref. [54] and (1564 ± 24 ± 86) MeV in +Ref. [55]. It can decay into the η′π and f1(1285)π channels. The corresponding diagrams for both +amplitudes are illustrated in Fig. 4, from which we have +Mf1(1285)π = gf1(1285)πεη1 · εf1 , +(42) +and +Mη′π = gK1 ¯ +KGK1 ¯ +KVK1 ¯ +K,η′π · εη1 + ga1ηGa1ηVa1η,η′π · εη1 , +(43) +with εη1 and εf1 the polarization vectors of the η1 and f1 (1285) mesons. Here gf1(1285)π, gK1 ¯ +K and +ga1η are the effective coupling of the z2 pole to the corresponding couplings, and GK1 ¯ +K and Ga1η +are the loops involving the K1 ¯K and a1η mesons, respectively. Notice that the effective couplings +are computed from the residues of the T matrix elements; thus they contain contributions from all +coupled channels. +In order to compare our findings with the experimental information, we evaluate the ratio +R1 = |Mf1(1285)π|2 q +|Mη′π|2 ˜q +, +(44) +where q and ˜q are the momentum in the c.m. frame of the f1(1285)π and η′π pairs, respectively. +Numerically, Eq. (44) gives +R1 = +� +� +� +� +2.4+0.8 +−0.6 +�A , +� +2.1+0.4 +−0.3 +�B . +(45) + +21 +The ratio is slightly bigger for the mixing angles in the set A. Nevertheless, the result in Eq. (45) +is consistent to the corresponding ratio 3.80 ± 0.78 reported by the E852 Collaboration [56]. This +good agreement with the experimental data supports the molecular picture for the π1(1600) state. +V. +DYNAMICAL GENERATION IN I = 1/2 SECTOR +In the I = 1/2 sector, the corresponding WT amplitudes are given by Eq. (16) with the Cij +coefficients given in Tables VIII and IX. For each case, we have found two poles for parameter sets +A and B, as shown in Table XII and XIII. +TABLE XII. Poles and their corresponding couplings to the channels contributing to the PA interaction +with JP = 1−. Here the flavor-neutral axial mesons have JP C = 1++. The errors of the poles are from +varying the subtraction constant within α(µ = 1 GeV) = −1.35 ± 0.17, and only the central values of the +couplings are given. +Poles (Set A) +Channels +1.69 ± 0.02 +a1K +f1(1285)K +K1(1270)η f1(1420)K +K1(1400)η +(+ + + + +) +gl +6.89 +0.89 +3.75 +0.54 +2.10 +Poles (Set B) +Channels +1.70 ± 0.02 +a1K +f1(1285)K +K1(1270)η f1(1420)K +K1(1400)η +(+ + + + +) +gl +6.58 +0.25 +2.45 +0.27 +3.15 +TABLE XIII. Poles and their corresponding couplings to the channels contributing to the PA interaction +with JP = 1−. Here the flavor-neutral axial mesons have JP C = 1+−. The errors of the poles are from +varying the subtraction constant within α(µ = 1 GeV) = −1.35 ± 0.17, and only the central values of the +couplings are given. +Poles (Set A) +Channels +1.70 ± 0.02 +h1(1170)K +b1K +K1(1270)η +h1(1415)K +K1(1400)η +(− + + + +) +gl +0.20 +6.46 +2.38 − i0.01 +0.50 +3.21 − i0.02 +Poles (Set B) +Channels +1.69 ± 0.02 +h1(1170)K +b1K +K1(1270)η +h1(1415)K +K1(1400)η +(− + + + +) +gl +0.55 − i0.01 6.78 + i0.02 3.69 − i0.06 0.83 − i0.01 2.17 − i0.04 +Similarly to the previous cases, the poles are located on the same Riemann sheets in both sets +of mixing angles. The interactions in the a1K and b1K channels are strong to generate a bound + +22 +state in each of them. The existence of a lower h1 (1170) K channel below the b1K threshold moves +the pole in Table XIII to Riemann sheet (− + + + +). It has a nonzero imaginary part of a few +MeV, which is not shown in the table due to precision. +As discussed before, the I = 1/2 poles in Tables +XII and XIII will receive sizeable widths +once the width effects of the axial-vector mesons are taken into account, and it is expected that +the widths are of the order of a few hundred MeV, like those of the b1 and a1 mesons. Although +we neglected the transitions between the A1P and B1P sectors as discussed around Eq. (17) in +Section II, strange mesons are not C-parity eigenstates and the two dynamically generated I = 1/2 +1− states will inevitably mix. The two mixed states together could correspond to the 1− K∗ (1680) +structure [10]. +VI. +CONCLUSIONS +We have studied the interactions between the pseudoscalar and axial-vector mesons in coupled +channels with JPC = 1−(+) quantum numbers for the isospin 0, 1, and 1/2 sectors. Using the +chiral unitary approach, we describe the interaction with the Weinberg-Tomozawa term derived +from chiral Lagrangians. The transition amplitudes among all the relevant channels are unitarized +using the Bethe-Salpeter equation from which resonances (bound states) manifest themselves as +poles on the (un)physical Riemann sheets of the complex energy plane. +We consider the physical isoscalar axial-vector states as mixtures of the corresponding SU(3) +singlets and octets. In addition, the K1(1270) and K1(1400) physical states are also mixtures of +the K1A and K1B mesons, which are the strange partners of the a1 and b1 resonances, respectively. +We group into two sets, called A and B, the mixing angles accounting for such mechanisms and +investigate their influence on the pole positions. +According to our findings, we obtain poles with JP(C) = 1−(+) quantum numbers in the energy +range from 1.30 to 2.00 GeV, in each isospin sector studied (I = 0, 1, 1/2). The 1−+ quantum +numbers are exotic in the sense that they cannot be formed from a pair of quark and antiquark. +In particular, we have found an isoscalar state that may correspond to the η1(1855) state, newly +observed by the BESIII Collaboration [8]. In addition, we have also found two dynamically gener- +ated isovector states that we assign to be the π1(1400) and π1(1600) resonances. Hence, within our +formalism, they are dynamically generated through the pseudoscalar-axial vector meson interac- +tions, with the η1(1855) state coupling mostly to K1(1400) ¯K channel, while the π1(1400) couples +strongly to the b1π, and π1(1600) structure couples most strongly to the K1(1270) ¯K. We also + +23 +find two I = 1/2 JP = 1− states with a mass around 1.7 GeV. They combined together could be +responsible to the observed K∗(1680) structure. +In addition, we also evaluate the decays of the η1(1855) and the π1(1600). We find that the +three-body decay channel ¯KK∗π has a significantly larger branching fraction than the η′η, which +is the channel where the observation of the η1(1855) was made. The obtained ratio between the +π1(1600) → f1(1285)π and π1(1600) → η′π decays, given by Eq. (45), is consistent with the +corresponding experimental value. +We suggest searching for two additional η1 exotic mesons with masses of about 1.4 and 1.7 GeV, +respectively. In particular, the latter should be relatively narrow with a width around 0.1 GeV +and one of its main decay channels is K ¯Kππ. +ACKNOWLEDGMENTS +M. J. Y is grateful to Shuang-Shi Fang and M. P. Valderrama for valuable discussions. This +project is supported in part by the National Natural Science Foundation of China (NSFC) under +Grants No. 12125507, No. 11835015, and No. 12047503; by the China Postdoctoral Science Foun- +dation under Grant No. 2022M713229; by the NSFC and the Deutsche Forschungsgemeinschaft +(DFG) through the funds provided to the Sino-German Collaborative Research Center TRR110 +“Symmetries and the Emergence of Structure in QCD” (NSFC Grant No. 12070131001, DFG +Project-ID 196253076); and by the Chinese Academy of Sciences under Grant No. XDB34030000. +[1] A. Chodos, R. L. Jaffe, K. Johnson, C. B. Thorn, and V. F. Weisskopf, A New Extended Model of +Hadrons, Phys. Rev. D 9, 3471 (1974). +[2] R. L. Jaffe, Multi-Quark Hadrons. 1. The Phenomenology of (2 Quark 2 anti-Quark) Mesons, Phys. +Rev. D 15, 267 (1977). +[3] L. Maiani, F. Piccinini, A. D. Polosa, and V. Riquer, Diquark-antidiquarks with hidden or open charm +and the nature of X(3872), Phys. Rev. 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B 595, 109 (2004), arXiv:hep-ex/0401004. + diff --git a/99E3T4oBgHgl3EQfSQn_/content/tmp_files/load_file.txt b/99E3T4oBgHgl3EQfSQn_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..abd05c172fac4710231b349ee1b0b1f978224504 --- /dev/null +++ b/99E3T4oBgHgl3EQfSQn_/content/tmp_files/load_file.txt @@ -0,0 +1,1152 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf,len=1151 +page_content='On the η1(1855), π1(1400) and π1(1600) as dynamically generated states and their SU(3) partners Mao-Jun Yan,1, ∗ Jorgivan M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Dias,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' † Adolfo Guevara,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' ‡ Feng-Kun Guo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' § and Bing-Song Zou1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' ¶ 1CAS Key Laboratory of Theoretical Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Institute of Theoretical Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Beijing 100190,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' China 2School of Physical Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' University of Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Beijing 100049,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' China 3Peng Huanwu Collaborative Center for Research and Education,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Beihang University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Beijing 100191,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' China 4Institute of Modern Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Lanzhou 730000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' China In this work,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' we interpret the newly observed η1(1855) resonance with exotic JP C = 1−+ quantum numbers in the I = 0 sector,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' reported by the BESIII Collaboration,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' as a dynamically generated state from the interaction between the lightest pseudoscalar mesons and axial-vector mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The interaction is derived from the lowest order chiral Lagrangian from which the Weinberg-Tomozawa term is obtained, describing the transition amplitudes among the relevant channels, which are then unitarized using the Bethe-Salpeter equation, according to the chiral unitary approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' We evaluate the η1(1855) decays into the ηη′ and K ¯K∗π channels and find that the latter has a larger branching fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' We also investigate its SU(3) partners, and according to our findings, the π1(1400) and π1(1600) structures may correspond to dynamically generated states, with the former one coupled mostly to the b1π component and the latter one coupled to the K1(1270) ¯K channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In particular, our result for the ratio Γ(π1(1600) → f1(1285)π)/Γ(π1(1600) → η′π) is consistent with the measured value, which supports our interpretation for the higher π1 state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' We also report two poles with a mass about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='7 GeV in the I = 1/2 sector, which may be responsible for the K∗(1680).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' We suggest searching for two additional η1 exotic mesons with masses around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='4 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='7 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In particular, the predicted η1(1700) is expected to have a width around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='1 GeV and can decay easily into K ¯Kππ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' ∗ yanmaojun@itp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='cn † jorgivan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='mdias@itp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='cn ‡ aguevara@itp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='cn § fkguo@itp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='cn ¶ zoubs@itp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='cn arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='04432v1 [hep-ph] 11 Jan 2023 2 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' INTRODUCTION Over the last two decades, the experimental observation of many new hadronic states is chal- lenging our current understanding of hadrons as conventional mesons and baryons with valence contents of quark-antiquark and three quarks, respectively, since most of them do not fit in the well-known quark model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' This difficulty brought back a long-standing discussion on the exotic hadronic structures, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=', multiquark configurations that might have quantum numbers beyond those assigned to the conventional mesons and baryons [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Exotic quark configurations such as tetraquarks [3, 4], hadron-hadron molecules [5], glueballs, and hybrids [6, 7], among others, have been suggested to describe suitably most of the properties of these new states, such as the JPC quantum numbers, mass, and decay width, especially for those lying in the charmonium and bottomonium spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' On the other hand, distinguishing the exotic states from the conventional hadrons is a more complicated task in the light quark sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Many states have their masses close to each other, and the possibility of mixing brings additional difficulty to the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The situation improves as the quantum numbers do not fall into those allowed by the conventional quark model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' It seems to be the case of the newly discovered state, dubbed η1(1855), by the BESIII Collaboration [8, 9], observed in the invariant mass distribution of the η η′ meson pair in the J/ψ → γ η η′ decay channel with a significance of 19σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Its mass and width reported by BESIII are 1855 ± 9+6 −1 MeV and 188 ± 18+3 −8 MeV, respectively, with likely JPC = 1−+ quantum numbers, which cannot be formed by a pair of quark and antiquark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The η1(1855) is not the only state experimentally found with that set of quantum numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' As of today, three other hadronic structures, called π1(1400), π1(1600) and π1(2015), with JPC = 1−+, were observed by several collaborations [7, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' From the theoretical point of view, the hybrid model has been used to investigate these exotic meson states, in particular the 1−+ ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Lattice quantum chromodynamics (QCD) calculations have pointed out hybrid supermultiplets with exotic JPC quantum numbers, including the 1−+ one [11–16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In this picture, however, the mass of the lightest 1−+ state and decay modes are in- consistent with the corresponding experimental results, while the π1(1600) and π1(2015) structures can fit into the nonets predicted by lattice QCD [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The newly observed η1(1855) state has also been the focus of some studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In particular, the authors in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' [17] proposed two hybrid nonet schemes in which the η1(1855) resonance can be either the lower or higher mass state with isospin I = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' [18], an effective Lagrangian respecting flavor, parity, and charge conjugation symmetries is used to study the hybrid nonet 3 decays into two-body meson states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The authors have fixed the couplings to those two-body meson states by performing a combined fit to the experimental and lattice results available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' As a result, the decay width value estimated for the isoscalar member of the hybrid nonet agrees with the one observed for η1(1855) state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Also addressing the same picture, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' [19] applied the approach of QCD sum rules to describe the η1(1855) mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' By contrast, within the same approach, the η1(1855) resonance is described as a tetraquark state in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The η1(1855) resonance also supports a meson-meson molecule interpretation due to its prox- imity to the K ¯K1(1400) threshold, as put forward by Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In particular, the authors in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' [21] have investigated the K ¯K1(1400) interaction through the one-boson exchange model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' According to their findings, the K ¯K1(1400) system binds for cutoff values above 2 GeV with a monopole form factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In addition, the comparison between their result for the branching fraction B(η1 → η η′) to the experimental one led them to conclude that the K ¯K1(1400) molecule can explain the η1(1855) structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' An important point to be addressed is the meson-meson interaction around the K1(1400) ¯K threshold for the JPC = 1−+ quantum numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In this sector, many meson-meson pairs may contribute to that interaction, so a coupled-channel treatment seems appropriate to take these contributions into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In particular, hadron-hadron interactions in coupled channels have been studied in many works to describe the properties of the new hadronic systems experimentally observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In those cases, these hadronic structures are called dynamically generated states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Following this approach, in this work, we aim to explore the η1(1855), π1(1400), and π1(1600) hadronic systems as dynamically generated states from pseudoscalar-axial vector meson interactions in coupled channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Specifically, the low-energy interactions are given by the Weinberg-Tomozawa (WT) term from chiral Lagrangians at the leading order of the chiral expansion by treating the axial vector mesons as matter fields and the pseudoscalar mesons as the pseudo-Nambu-Goldstone bosons of the spontaneous breaking of chiral symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Such Lagrangians have been used to study many hadron structures stemming from meson-meson and meson-baryon interactions in coupled channels in light and heavy sectors, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' [23–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In our case, the amplitudes obtained from the WT term are unitarized via the Bethe-Salpeter equation from which bound states/resonances manifest as poles in the physical/unphysical Riemann sheets of the scattering matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The existence of a whole family of kaonic bound states has been pointed out in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' [28] based on unitarizing the WT term for the scattering of the kaon off isospin-1/2 matter fields taking heavy mesons and doubly-charmed baryons as examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' As we shall show in this work, the newly observed η1(1855) structure may correspond to a dynamically generated state from the 4 pseudoscalar-axial vector interaction in the isospin I = 0 sector coupling strongly to the K1(1400) ¯K channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Moreover, the π1(1400) and π1(1600), may be assigned as the η1(1855) SU(3) partners which are also dynamically generated from the pseudoscalar-axial vector meson interactions in the I = 1 sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The former resonance couples mainly to the b1π channel, and the latter has the K1(1270) ¯K as its main coupled channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In addition, we have also found two poles around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='7 GeV in the I = 1/2 sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' These poles are particularly interesting as they could be the origin of the K∗(1680) structure observed experi- mentally [10], which is the main component of the 1− contribution to the φK mass distribution in the B → J/ψφK decays recently measured by LHCb [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In Section II, we discuss the relevant channels contributing to the pseudoscalar-axial vector meson interactions and the use of the chiral unitary approach (ChUA) for the evaluation of the transition amplitudes among those channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In Sections III and IV, we investigate the dynamical generation of poles stemming from those interactions in the I = 0 and I = 1 sectors and discuss their possible decay channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Finally, in Section V, we also explore the dynamical generation of poles for I = 1/2 and their connection to the vector K∗(1680) structure observed experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Section VI gives a summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' COUPLED CHANNEL SCATTERING IN CHIRAL UNITARY APPROACH We investigate the interactions between axial and pseudoscalar mesons in coupled channels in the 1300 ∼ 2000 MeV energy range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' First, we need to determine the space of states contributing to the interaction in this energy range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In Tables I, II, III, and IV, we list all the relevant channels for the problem under consideration along with their corresponding mass thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The channels are organized from the lower to higher mass values and by the isospin, 0, 1 and 1/2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' JP C = 1−+ meson-meson channels with I = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The threshold masses are in the units of MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Channel a1π K1(1270) ¯K f1(1285)η K1(1400) ¯K f1(1420)η Threshold 1368 1748 1829 1898 1973 TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' JP C = 1−+ meson-meson channels with I = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The threshold masses are in the units of MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Channel b1π f1(1285)π f1(1420)π K1(1270) ¯K a1η K1(1400) ¯K Threshold 1367 1419 1564 1748 1777 1895 5 TABLE III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' JP = 1− meson-meson channels with I = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The threshold masses are in the units of MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Here the flavor-neutral axial vector mesons have JP C = 1++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Channel a1K f1(1285)K+ K1(1270)η f1(1420)K K1(1400)η Threshold 1725 1777 1800 1921 1947 TABLE IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' JP = 1− meson-meson channels with I = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The threshold masses are in the units of MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Here the flavor-neutral axial vector mesons have JP C = 1+−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Channel h1(1170)K b1K K1(1270)η h1(1415)K K1(1400)η Threshold 1661 1725 1800 1911 1947 In what follows, we shall discuss the relevant scattering amplitudes among all those channels above for each isospin sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' These transitions can be written in the form of the WT term which then is unitarized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Notice that the channels displayed in Tables III and IV, in principle, should be grouped in the same space of states since they share identical isospin and JP quantum numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' However, the relevant transitions among them arise only at the next-to-leading order in the chiral expansion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' see the discussion around Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (17) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Thus, such transitions are of higher order than that of the WT term and will be neglected here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The Weinberg-Tomozawa term In order to study the interactions among all the channels listed in the previous tables, we have to evaluate the interactions between the pseudoscalar and axial-vector mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The latter are organized in two SU(3) octets according to their JPC quantum numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' A1 = � � � � � a0 1 √ 2 + f8 1 √ 6 a+ 1 K+ 1A a− 1 − a0 1 √ 2 + f8 1 √ 6 K0 1A K− 1A ¯K0 1A − 2f8 1 √ 6 � � � � � (1) is the octet of resonances of axial-vector states with JPC = 1++ for the flavor-neutral mesons, and B1 = � � � � � b0 1 √ 2 + h8 1 √ 6 b+ 1 K+ 1B b− 1 − b0 1 √ 2 + h8 1 √ 6 K0 1B K− 1B K0 1B − 2 √ 6h8 1 � � � � � (2) describes the octet of axial-vector resonances with JPC = 1+−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The singlet and I = 0 octet flavor eigenstates are not mass eigenstates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' that is, the pairs of f1(1420), h1(1415) (also known as 6 TABLE V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Two sets of values of the axial-vector meson mixing angles taken from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Set B is preferred in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The η-η′ mixing angle θP is taken from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' For more discussions about these mixing angles, we refer to the review of Quark Model in the Review of Particle Physics [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Angles θK1 θ3P1 θ1P1 θP Set A 57◦ 52◦ −17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='5◦ −17◦ Set B 34◦ 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='1◦ 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='0◦ −17◦ h1(1380)) and f1(1285), h1(1170) mesons are mixtures of the singlet (1) and octet (8) mesons such that � � |f1(1285)⟩ |f1(1420)⟩ � � = � � cos θ3P1 sin θ3P1 − sin θ3P1 cos θ3P1 � � � � ��f1 1 � ��f8 1 � � � , (3) and � � |h1(1170)⟩ |h1(1415)⟩ � � = � � cos θ1P1 sin θ1P1 − sin θ1P1 cos θ1P1 � � � � ��h1 1 � ��h8 1 � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (4) Furthermore, the K1A and K1B members of the multiplets in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (1) and (2) are the strange partners of the a1(1260) and b1(1235), and their mixture contributes to the physical K1(1270) and K1(1400) mesons, that is � � |K1(1270)⟩ |K1(1400)⟩ � � = � � sin θK1 cos θK1 cos θK1 − sin θK1 � � � � |K1A⟩ |K1B⟩ � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (5) The corresponding values for the mixing angles in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (3), (4), and (5) are listed in Table V, where they are grouped into two sets, denoted by A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Although set B is preferred in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' [30], we will use both sets to have an estimate of the uncertainties caused by such an angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In order to determine the WT term we start with the Lagrangian (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=', Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' [32]) L0 = −1 4 � VµνV µν − 2M2 V VµV µ� , (6) where ⟨, ⟩ takes trace in the SU(3) flavor space, Vµν = DµVν − DνVµ , (7) while Dµ is the chirally covariant derivative, which when acting on SU(3) octet matter fields reads as Dµ = ∂µ + [Γµ, ] , (8) 7 with [ , ] the usual commutator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In addition, Γµ stands for the chiral connection, given by Γµ = 1 2 � u†∂µu + u∂µu†� , (9) with u = exp � i √ 2Fπ φ8 � , (10) where Fπ = 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='1 MeV is the pion decay constant [10], and φ8 is the pseudoscalar SU(3) octet, that is φ8 = � � � � � π0 √ 2 + 1 √ 6η8 π+ K+ π− − 1 √ 2π0 + 1 √ 6η8 K0 K− ¯K0 − 2 √ 6η8 � � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (11) In addition, the physical η and η′ mesons are the mixtures of η8 and η1 � � |η⟩ |η′⟩ � � = � � − sin θP cos θP cos θP sin θP � � � � ��η1� ��η8� � � , (12) where η1 becomes the ninth pseudo-Goldstone boson in large Nc QCD [33–36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The Goldstone boson nonet is written as φ9 = φ8 + 1 √ 3η1, (13) which leads to a relation in the commutator � φ9, ∂µφ9� = � φ8, ∂µφ8� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (14) Therefore, only the scattering of the octet Goldstone bosons off the axial-vector mesons in Weinberg-Tomozawa term contributes to JP(C) = 1−(+) spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The covariant derivative Dµ by means of the connection Γµ encodes the leading order interaction between the pseudoscalar mesons and the vector field Vµ [32, 37, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Therefore, by replacing the Vµ field to the axial-vector field Aµ corresponding to either the A1 or B1 multiplet, the chiral tran- sition between φ8 (pseudoscalar) and A (1+) (axial-vector) is described by the following interaction Lagrangian LI = − 1 4F 2π � [Aµ, ∂νAµ] � φ8, ∂νφ8�� , (15) 8 which accounts for the WT interaction term for the PA → PA process, with P and A corresponding to the pseudoscalar and axial-vector mesons, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' From this Lagrangian we obtain the S- wave transition amplitude among the channels listed in Tables I, II, III and IV, that is Vij(s) = −ϵ · ϵ′ 8F 2π Cij � 3s − � M2 + m2 + M′2 + m′2� − 1 s � M2 − m2� � M′2 − m′2�� , (16) where ϵ (ϵ′) stands for the polarization four-vector of the incoming (outgoing) axial-vector me- son [25, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The masses M (M′) , m (m′) correspond to the initial (final) axial-vector mesons and initial (final) pseudoscalar mesons, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The indices i and j represent the initial and final PA states, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The coefficients Cij are given in Tables VI, VII, VIII, and IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' TABLE VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Cij coefficients in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (16) for axial and pseudoscalar pairs coupled to JP C = 1−+ in S-wave and I = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Cij a1π K1(1270) ¯K f1(1285)η K1(1400) ¯K f1(1420)η a1π −4 � 3 2 sin θK1 0 � 3 2 cos θK1 0 K1(1270) ¯K −3 − 3 √ 2 sin θ3P1 sin θK1 0 − 3 √ 2 cos θ3P1 sin θK1 f1(1285)η 0 − 3 √ 2 cos θK1 sin θ3P1 0 K1(1400) ¯K −3 − 3 √ 2 cos θ3P1 cos θK1 f1(1420)η 0 TABLE VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Cij coefficients in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (16) for axial and pseudoscalar pairs coupled to JP C = 1−+ in S-wave and I = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Cij b1π f1(1285)π f1(1420)π K1(1270) ¯K a1η K1(1400) ¯K b1π −2 0 0 cos θK1 0 − sin θK1 f1(1285)π 0 0 � 3 2 sin θK1 sin θ3P1 0 � 3 2 cos θK1 sin θ3P1 f1(1420)π 0 � 3 2 cos θ3P1 sin θK1 0 � 3 2 cos θK1 cos θ3P1 K1(1270) ¯K −1 − � 3 2 sin θK1 0 a1η 0 − � 3 2 cos θK1 K1(1400) ¯K −1 Before proceeding,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' let us discuss the A1φ8 → B1φ8 transitions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' with A1 and B1 the two SU(3) octets of axial-vector mesons and φ8 the octet of the pseudo-Nambu-Goldstone bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Let A1µ and B1µ denote the fields for the 1++ and 1+− axial-vector meson multiplets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Under parity transformation, we have A1µ → −Aµ 1 and B1µ → −Bµ 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' under charge conjugation, we have A1µ → AT 1µ and B1µ → −BT 1µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Then the A1φ8 → B1φ8 transitions can only arise at O � p2� with p 9 TABLE VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Cij coefficients in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (16) for axial and pseudoscalar pairs coupled to JP = 1− in S-wave and I = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Here the flavor-neutral axial mesons have JP C = 1++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Cij a1K f1(1285)K K1(1270)η f1(1420)K K1(1400)η a1K −2 0 − 3 2 sin θK1 0 − 3 2 cos θK1 f1(1285)K 0 3 2 sin θK1 sin θ3P1 0 3 2 sin θK1 cos θK1 K1(1270)η 0 3 2 cos θ3P1 sin θK1 0 f1(1420)K 0 3 2 cos θ3P1 cos θK1 K1(1400)η 0 TABLE IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Cij coefficients in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (16) for axial and pseudoscalar pairs coupled to JP = 1− in S-wave and I = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Here the flavor-neutral axial mesons have JP C = 1+−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Cij h1(1170)K b1K K1(1270)η h1(1415)K K1(1400)η h1(1170)K 0 0 3 2 cos θK1 sin θ1P1 0 3 2 sin θK1 sin θ1P1 b1K −2 − 3 2 cos θK1 0 − 3 2 sin θK1 K1(1270)η 0 3 2 cos θK1 cos θ1P1 0 h1(1415)K 0 3 2 sin θK1 cos θ1P1 K1(1400)η 0 the momentum scale in the chiral power counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' They are given by operators such as ⟨A1µ[B1ν, [uµ, uν]]⟩ , (17) with uµ = i � u†∂µu − u∂µu†� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (18) Such terms are one order higher in the chiral power counting than the WT terms describing the A1φ8 → A1φ8 and B1φ8 → B1φ8 transitions, and thus will be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Unitarization procedure The unitarization procedure we adopt follows ChUA in which the scattering amplitudes in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (16) are the elements of a matrix, denoted by V , such that it enters as an input to solve the Bethe-Salpeter equation, which in its on-shell factorization form, reads [23] T = (1 − V G)−1 V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (19) 10 The V matrix describes the transition between the channels listed in Tables I, II, III, and IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In addition, G is the diagonal loop function matrix whose diagonal matrix elements are given by Gl = i � d4q (2π)4 1 q2 − m2 l + iϵ 1 (q − P)2 − M2 l + iϵ , (20) with ml and Ml the masses of the pseudoscalar and axial-vector mesons, respectively, involved in the loop in the channel l, and P the total four-momentum of those mesons (P 2 = s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' After the integration over the temporal component q0, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (20) becomes Gl(s) = � d3q (2π)3 ω1 + ω2 2ω1ω2 1 (P 0)2 − (ω1 + ω2)2 + iϵ , (21) with ω1 = � Ml2 + |⃗q|2 and ω2 = � ml2 + |⃗q|2, and can be regularized by means of a cutoff in the three-momentum qmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' On the other hand, the function Gl can also be regularized using a subtraction constant as [40] GDR l (s) = 1 16π2 � αl(µ) + log M2 l µ2 + m2 l − M2 l + s 2s log m2 l M2 l + pl √s � log s − m2 l + M2 l + 2pl √s −s + m2 l − M2 l + 2pl √s + log s + m2 l − M2 l + 2pl √s −s − m2 l + M2 l + 2pl √s �� , (22) where pl is the three-momentum of the mesons in the center-of-mass (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=') frame pl = �� s − (Ml + ml)2� � s − (Ml − ml)2� 2√s , (23) while µ is an arbitrary scale of the regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Any changes in the µ scale can be absorbed by the subtraction constant αl(µ) such that the result is independent of the scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' We may determine the subtraction constant for each intermediate state of the scattering problem by comparing Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (21), regularized using qmax, and (22) at the threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The equivalence between the two prescriptions for the loop-function is discussed in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' [41–43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In this work, we follow Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' [44] and set µ = 1 GeV and α = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='35, which is obtained by matching to hard cutoff regularization with qmax ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='7 GeV in the f1(1285)η channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' This set of parameters are used for all channels, and a variation of the cutoff within qmax = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='1) GeV, and correspondingly α(µ = 1 GeV) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='17, will be used to show the dependence of the results on this parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Searching for poles We move on to the complex energy plane to search for poles in the T-matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Specifically, for a single-channel problem, there are two Riemann sheets for the complex energy plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Bound states 11 show up as poles, below the threshold, in the transition matrix on the real energy axis on the first Riemann sheet, while virtual states manifest themselves below the threshold on the real axis on the second Riemann sheet, and resonances correspond to poles off the real axis on the second Riemann sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The Riemann sheets come about because the G loop function has a cut extending from the threshold to infinity which is usually chosen to be along the positive real axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' For n coupled channels, there are n cuts and thus 2n Riemann sheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' From unitarity and the Schwarz reflection principle, the discontinuity of the Gl function can be read off from its imaginary part, Im Gl(s) = − pl 8π√s , (24) which we can use to perform an analytic continuation to the entire complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In this case, the Gl loop function on the “second” Riemann sheet with respect to the lth channel reads GII l (s) = GI l(s) + i pl 4π√s ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (25) the lower half plane of this Riemann sheet is directly connected to the physical region when the lth channel is open, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=', Re(√s) ≥ m + M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' We will label the Riemann sheets according to the sign of the imaginary part of the corresponding c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' momentum for each channel (see the next section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Furthermore, it is also possible to determine the pole couplings to the lth channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Note that close to the pole singularity the T-matrix elements Tij(s) admit a Laurent expansion, Tij(s) = gi gj s − zp + regular terms, (26) where zp = (Mp −iΓ/2)2 is the pole location on the complex energy plane, with Mp and Γ standing for the pole mass and width, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Therefore, the product of couplings gigj is the residue at the pole in Tij(s) which takes values on the Riemann sheet where the pole is located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In this way, the couplings can be evaluated straightforwardly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' For instance, for a diagonal transition it is given by g2 i = r 2π � 2π 0 Tii(z(θ))eiθdθ = lim s→zp(s − zp)Tii(s) = � d ds 1 Tii(s) �−1 s=zp , (27) where z(θ) = zp + i reiθ with r the radius of contour for the integral, and the two lines give two equivalent ways of computing residues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' 12 TABLE X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The poles (in GeV) and their corresponding couplings (in GeV) to the channels contributing to the PA interaction with I = 0 and exotic quantum numbers JP C = 1−+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The corresponding Riemann sheet for each pole is listed below the pole position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The dominantly coupled channel is emphasized in boldface for each pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The errors of the poles are from varying the subtraction constant within α(µ = 1 GeV) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='35±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='17, and only the central values of the couplings are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Poles (Set A) Channels 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='01 − i(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='01) a1π K1(1270) ¯ K f1(1285)η K1(1400) ¯ K f1(1420)η (− + + + +) gl 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='21 + i3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='22 + i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='01 + i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='36 + i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='69 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='03 a1π K1(1270) ¯ K f1(1285)η K1(1400) ¯ K f1(1420)η (− + + + +) gl 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='36 + i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='98 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='16 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='17 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='64 + i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='09 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='46 + i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='84 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='03 a1π K1(1270) ¯ K f1(1285)η K1(1400) ¯ K f1(1420)η (− − − + +) gl 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='07 + i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='69 + i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='55 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='68 + i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='08 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='33 + i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='16 + i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='06 Poles (Set B) Channels 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='01 − i(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='01) a1π K1(1270) ¯ K f1(1285)η K1(1400) ¯ K f1(1420)η (− + + + +) gl 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='21 + i3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='81 + i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='55 + i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='70 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='02 a1π K1(1270) ¯ K f1(1285)η K1(1400) ¯ K f1(1420)η (− + + + +) gl 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='25 + i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='67 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='34 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='27 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='37 + i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='17 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='58 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='84 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='03 a1π K1(1270) ¯ K f1(1285)η K1(1400) ¯ K f1(1420)η (− − − + +) gl 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='15 + i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='33 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='27 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='83 + i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='09 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='05 + i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='17 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='81 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='20 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' η1(1855) AND ITS DECAYS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Dynamical generation of the η1(1855) Following the unitarization procedure described previously, we seek dynamically generated states stemming from the S-wave interactions between pseudoscalar and axial-vector mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' For the I = 0 case, the transition amplitudes among the channels listed in Table I are determined using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (16) with the Cij coefficients given in Table VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In Table X, we show the isoscalar poles with exotic quantum numbers JPC = 1−+ obtained by solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (19) using those coefficients as well as each set of mixing angles listed in Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' We also show the couplings of these poles to the channels spanning the space of states in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Furthermore, in Table X we also highlight the Riemann sheets, the first and the second one for each channel, denoted by the + and − signs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' We get three poles such that their 13 locations are barely affected by the change of the mixing angles from set A to set B listed in Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The lower pole is at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='39 GeV with a width of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='04 GeV, which is above the a1π threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In particular, this channel is open for decay, and the fact that it is this channel the one for which the pole couples mostly, as pointed out in Table X, explains why that pole has such a value for its width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' By contrast, although the a1π channel is also open for decay, the pole at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='69 GeV has a much smaller width because its coupling to this channel is small compared to the one for K1(1400) ¯K, which is the dominant channel for that pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Similarly, the highest pole, located at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='84 GeV, couples mostly to the K1(1400) ¯K channel, and has a small imaginary part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In addition, we can also understand why the highest pole couples more to the K1(1400) ¯K than to the f1(1285)η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The latter channel is closer to the pole than the former, but from Table VI, the diagonal f1(1285)η transition is not allowed since its WT term is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Nevertheless, the pole couples to f1(1285)η through the nondiagonal K1(1400) ¯K–f1(1285)η transition, which leads to a small coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Effects of the widths of the axial-vector mesons So far we have neglected the nonzero widths of the axial-vector mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In order to investigate their effects on the results, we use complex masses for the intermediate resonances, that is, Mi → Mi − iΓi/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' However, by doing that, the analytic properties are lost such that the poles of the T matrix do not correspond to the masses and widths of the obtained resonances any more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' On the other hand, we can see the impact of such nonzero widths on the lineshapes of the transition matrix elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' 1 we show a comparison between the lineshape for the T-matrix element corresponding to the elastic transition TK1(1400) ¯ K→K1(1400) ¯ K with and without including the widths for the inter- mediate particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' This channel has the strongest coupling to the pole at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='84 GeV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' therefore, we expect that any nontrivial structure should manifest most in its associated T-matrix element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The dashed and solid lines are the TK1(1400) ¯ K→K1(1400) ¯ K with zero and nonzero width, respectively, for both sets A and B of mixing angles in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Notice that, for the case of zero width approxima- tion, the TK1(1400) ¯ K→K1(1400) ¯ K lineshape has narrow peaks around 1845 MeV, right at the range of energy where we expect the η1(1855) manifests in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The inclusion of finite widths for the axial-vector mesons changes the sharp peak to a broad bump with a width of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='2 GeV, which is around the width of the K1(1400) [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Notice that the width matches nicely that of the η1(1855) measured by BESIII, � 188 ± 18+3 −8 � MeV [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In the following, we will continue to present 14 w/o Γ w/o Γ w/ Γ w/ Γ 1600 1700 1800 1900 2000 0 10 20 30 40 50 s [MeV] |T44 2 Set A Set B FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The blue dashed and solid lines are, respectively, the modulus squared of the T-matrix element, cor- responding to the diagonal K1(1400) ¯K → K1(1400) ¯K transition, evaluated with and without the inclusion of the widths associated with the axial-vector mesons taking part in the loop function Gl (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (20)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' predictions neglecting the width effects of the axial-vector mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Let us briefly discuss the other two predicted isoscalar exotic η1 mesons in Table X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The one with a mass of about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='39 GeV, denoted as η1(1400), is expected to be rather broad due to the large width of the a1(1260) as it couples most strongly to the a1π channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' It can be searched for in final states such as ρππ and K ¯Kππ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The one with a mass around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='7 GeV, denoted as η1(1700), couples most strongly to the K1(1270) ¯K and is expected to have a width similar to that of the K1(1270), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=', around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='1 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' It can also be searched for in final states of K ¯Kππ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The η1(1855) → η′η and K∗ ¯Kπ decays Let us first discuss the η1 → ηη′ decay, whose Feynman diagram is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Within our approach the η1(1855) structure decays via its K1(1400) ¯K component, with the corresponding coupling constant listed in Table X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' We also need to evaluate the K1(1400) ¯K → ηη′ transition, for which we use the resonance chiral theory (RχT) operators given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The RχT operators can be divided regarding the intrinsic-parity sector to which they contribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Due to its nature, the odd-intrinsic parity sector will contain a Levi-Civita tensor [46–48];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' for the η1 → ηη′ decay one cannot saturate the Lorentz indices in such tensor to get a nonzero contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Thus, only the even-intrinsic parity operators must give a nonvanishing contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Since the chiral O(p2) Lagrangian does not contribute to such processes [49], we will use the O(p4) Lagrangian given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' From these operators, only three will contribute to this decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' To get the largest possible contribution from such operators, we use the upper bounds imposed from chiral counting 15 as done in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' This amounts to making equal the three coupling constants and setting them to λA 1 = λA 2 = λA 3 = g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='025 GeV−1, which gives a Lagrangian L = g � ⟨Aµν (uµuαhνα + hναuαuµ)⟩ + ⟨Aµν (uαuµhνα + hναuµuα)⟩ + ⟨Aµν (uµhναuα + uαhναuµ)⟩ � , (28) where uµ has been given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (18), hµν = D{µuν} is the symmetrized covariant derivative of uµ and the spin-1 resonance field is given in the antisymmetric tensor formalism [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' However, since the η1 → K1 ¯K transition is given in terms of Proca fields, we need to express the K1 as a Proca field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Following Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' [49], the antisymmetric tensor field can be expressed in terms of the Proca one as follows, Rµ = 1 MR ∂νRνµ, (29) where MR is the mass of the resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Using the Lagrangian of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (28) and expressing the axial resonance in the Proca representation, we get the η1 → ηη′ decay amplitude Mη1→ ηη′ = − 4m2 η1 3F 3πmK1 ggK1(1400) ¯ KGK1 ¯ K �� αp2 η′ + 1 √ 2βp2 η � εη1 · pη + � pη ↔ pη′�� , (30) where Fπ is the pion decay constant, gK1 ¯ K is the coupling constant of the pole to the K1(1400) ¯K channel, GK1 ¯ K is the loop function for the K1 and ¯K mesons , εη1 is the η1 vector polarization, and pη(′) is the momentum of the η(′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Here, α and β are given in terms of the η-η′ mixing angle α = cos 2θP + 2 √ 2 sin 2θP , (31a) β = 2 √ 2 cos 2θP − sin 2θP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (31b) K1 ¯K η η′ η1 (1855) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Diagram corresponding to the η1 → ηη′ decay through the K1 ¯K loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Although one might try to rely in a much simpler way to describe the direct coupling of one axial- vector and three pseudoscalar fields by means of the Hidden Local Symmetry (HLS) Lagrangian 16 [51–53], it is worth to notice that nonetheless, the total amplitude for this process given by the HLS Lagrangian vanishes, which coincides with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (30) in the chiral limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The decay of η1 state into ηη′ is given by Γ2B = 1 2J + 1 1 8πM2η1 |Mη1→ ηη′|2 q , (32) with the amplitude Mη1→ ηη′ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (30), while J stands for the η1 spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Besides that, q reads q = 1 2Mη1 λ1/2 � M2 η1, m2 η′, m2 η � , (33) with Mη1, mη′, and mη the masses for the η1(1855), η′, and η mesons, respectively, where λ (x, y, z) = x2 + y2 + z2 − 2xy − 2yz − 2zx is the K¨all´en triangle function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Therefore, we get the following results for the decay width in this channel Γ2B = � � � (19 ± 4) MeV (set A) , (7 ± 2) MeV (set B) , (34) where the error is from choosing subtraction constant to be in the range α(µ = 1GeV) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='17, corresponding to the hard cutoff qmax = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='1) GeV as discussed at the end of Section II B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' For set A, our result agrees with that of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' [21], where the η1(1855) was assumed to be a K1 ¯K molecule and the same θK1 mixing angle was used for accounting for the K1A and K1B mixture contributing to the physical K1(1270) and K1(1400) states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Feynman Diagram associated with the three-body decay of the pole through its main component K1 ¯K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' As for the η1 → ¯KK∗π three-body decay, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' 3 shows the Feynman diagrams contributing to this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In particular, the η1(1855) decays through its molecular components, that in our approach are the K1(1270) ¯K and K1(1400) ¯K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In this case, the contribution from the K1(1270) ¯K component can be ignored for the following reasons: 1) from Table X, we see that the relative coupling strength for the K1(1270) ¯K channel is much smaller than that for the K1(1400) ¯K one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' 117 2) the branching ratio B[K1(1270) → K∗π] is only 16%, while 96% of the K1(1400) decays is dominated by the K∗π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Therefore, from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' 3 the η1(1855) → ¯KK∗π amplitude is written as M3B = gK1(1400) ¯ K � −gµν + pµpν M2 K1 � 1 p2 − M2 K1 + i MK1ΓK1 gK∗π εµ η1εν K∗ , (35) where gK1(1400) ¯ K is the coupling of the pole associated with the η1 state to the K1(1400) ¯K channel, gK∗π is the K1(1400)K∗π coupling extracted from the K1(1400) → K∗π reaction in the Review of Particle Physics (RPP) [10], and εµ η1 and εν K∗ are the polarization vectors of the η1 and K∗ mesons, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The differential decay width for the η1 → ¯KK∗π process is given by dΓ dMK1 ¯ K = 1 (2π)3 pK ˜pπ 4M2η1 |M3B|2 1 2J + 1 , (36) where ˜pπ = 1 2MK1 λ1/2 � M2 K1, m2 K∗, m2 π � , (37) and pK = 1 2Mη1 λ1/2 � M2 η1, m2 K, M2 K1 � , (38) with MK1, mK∗, mπ being the masses of the K1(1400), K∗ and π mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (36) we obtain the following results for the η1 → ¯KK∗π decay width Γ3B = � 81+11 −24 MeV �A , Γ3B = � 74+12 −24 MeV �B , (39) where the uncertainties come from the subtraction constant (cutoff) used to regularize the loops in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (22) (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (21)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' As can be seen from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (39), we obtain similar results whether we use the sets A or B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' For the sake of comparison to other works, we evaluate the ratio Γ2B/Γ3B, and get Γ2B Γ3B = � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='23−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='08 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='16 �A or � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='10−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='03 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='08 �B , (40) which is consistent to the results in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' [21], where the η1 is also assumed to be a K1(1400) ¯K molecular state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' On the other hand, adopting the same multiquark configuration than the present work and Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' [21], the authors of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' [22] have found a different result for the ratio, Γ2B/Γ3B ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Nevertheless, in all the cases the results point out that the ¯KK∗π three-body channel is more likely than the ηη′ one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' 18 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' THE π1(1400/1600) DYNAMICAL GENERATION The WT amplitudes for the pseudoscalar-axial vector meson interactions with I = 1 are given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (16), with the corresponding Cij coefficients listed in Table VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In this case, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (19), we get two π1 poles shown in Table XI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' TABLE XI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Poles and their corresponding couplings to the channels contributing to the PA interaction with JP C = 1−+ and I = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The errors of the poles are from varying the subtraction constant within α(µ = 1 GeV) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='17, and only the central values of the couplings are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Poles (Set A) Channels 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='47 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='01 − i(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='02) b1π f1(1285)π f1(1420)π K1(1270) ¯ K a1η K1(1400) ¯ K (− − + + ++) gl 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='22 + i4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='02 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='03 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='25 + i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='02 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='33 + i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='63 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='02 − i(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='01) b1π f1(1285)π f1(1420)π K1(1270) ¯ K a1η K1(1400) ¯ K (− − − + ++) gl 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='10 + i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='95 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='73 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='89 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='84 − i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='85 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='49 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='03 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='65 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='53 Poles (Set B) Channels 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='47 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='01 − i(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='02) b1π f1(1285)π f1(1420)π K1(1270) ¯ K a1η K1(1400) ¯ K (− − + + ++) gl 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='27 + i4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='01 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='03 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='97 − i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='02 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='91 + i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='77 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='01 − i(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='01) b1π f1(1285)π f1(1420)π K1(1270) ¯ K a1η K1(1400) ¯ K (− − − + ++) gl 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='13 + i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='44 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='37 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='86 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='50 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='80 − i2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='29 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='53 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='64 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='54 − i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='77 Similar to the previous section, we also provide the couplings of these dynamically generated states to the channels listed in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Table XI shows a broad π1 pole at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='47 GeV, and a width of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='12 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='1 This state is above the b1π and f1(1285)π thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Its large width stems from the large coupling to the b1π and the fact that this channel is open for decaying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The f1(1285)π channel is also open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' However, according to Table VII, the corresponding WT term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (16) is zero for the diagonal f1(1285)π transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' On the other hand, the next π1 pole in Table XI has a sizeable dependence on the mixing angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Using set A, we find that pole at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='75 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' It couples most strongly to the K1(1270) ¯K channel, which is closed for decaying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Nonetheless, the state can decay into b1π and f1(1285)π, albeit their corresponding couplings are small compared to the K1(1270) ¯K one, but still large enough to provide a sizeable width for the pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In contrast, when set B is adopted, the higher π1 pole is now located at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='77 GeV, above the f1(1420)π threshold, 1 As discussed in Section III B, the widths of the dynamically generated poles will be significantly increased once the width effects of the axial-vector mesons are taken into account;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' see also the discussions below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' 19 which is now open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' One might think that the width should increase since now three channels are open for decaying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' However, although the coupling to the f1(1420)π has increased in this case, at the same time the couplings to the other open channels have decreased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Hence, the overall effect leads to a smaller width compared to the previous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' π f1(1285) π1 (1600) K1/a1 ¯K/η π η′ π1 (1600) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' a) Diagram corresponding to the π1(1600) → f1(1285)π reaction, and b) the π1(1600) → η′π decay also via the AP loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The filled circles represent the effective couplings of the π1 to the AP meson pairs calculated from the residues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The rectangles are the AP → η′π transition amplitudes at tree level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The lower pole mass is slightly higher than the mass of the π1(1400) state listed in RPP, (1354 ± 25) MeV [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Notice that we use the same subtraction constant for all channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In principle, it can take different values and lead to a shift of the poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In addition, we did not include in the loops the b1 width, that is relatively large and whose effects could influence the pole position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' However, it is expected to affect more the imaginary part of the pole than the real one (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' 5(a) below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' We can get a rough estimate of this change by adding the b1 width to the previous result for Im(z1), with z1 the lower π1 pole, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=', Γb1 + 2Im(z1) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='4 GeV , (41) which is close to the π1(1400) width reported in RPP, (330 ± 35) MeV [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' From these results, we are led to claim that the lower π1 pole may explain the π1(1400) resonance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' in other words, the π1(1400) is suitably described in our approach as a dynamically generated state with the b1π as its main component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Alternatively, following the prescription used in Section III, we can also study the changes in the results caused by the inclusion of the finite widths for the axial-vector mesons by looking at the line shape for the relevant T-matrix elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' 5(a) we show the line shapes for the T-matrix element corresponding to the elastic b1π → b1π transition, which is the one we would expect the lower pole in Table XI manifests most due to its large coupling to the b1π channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' It becomes clear that the bumps become broader when the widths of axial-vector mesons are taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' A similar behavior can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' 5(b) for the T-matrix element associated 20 with the scattering of K1 (1270) ¯K, which is the channel to which the higher π1 pole couples most strongly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' w/o Γ w/o Γ w/ Γ w/ Γ 1300 1400 1500 1600 1700 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='0 s [MeV] |T11 2 Set A Set B (a) Modulus square of elastic b1π scattering w/o Γ w/o Γ w/ Γ w/Γ 1400 1500 1600 1700 1800 1900 0 1 2 3 4 s [MeV] |T44 2 Set A Set B (b) Modulus square of elastic K1 (1270) ¯K scattering FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The dashed and solid lines correspond to zero and full widths of the axial-vector mesons in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The higher π1 pole, denoted now by z2, has a mass consistent with that of the π1(1600), whose pole mass has been reported to be � 1623 ± 47+24 −75 � MeV in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' [54] and (1564 ± 24 ± 86) MeV in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' It can decay into the η′π and f1(1285)π channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The corresponding diagrams for both amplitudes are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' 4, from which we have Mf1(1285)π = gf1(1285)πεη1 · εf1 , (42) and Mη′π = gK1 ¯ KGK1 ¯ KVK1 ¯ K,η′π · εη1 + ga1ηGa1ηVa1η,η′π · εη1 , (43) with εη1 and εf1 the polarization vectors of the η1 and f1 (1285) mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Here gf1(1285)π, gK1 ¯ K and ga1η are the effective coupling of the z2 pole to the corresponding couplings, and GK1 ¯ K and Ga1η are the loops involving the K1 ¯K and a1η mesons, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Notice that the effective couplings are computed from the residues of the T matrix elements;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' thus they contain contributions from all coupled channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In order to compare our findings with the experimental information, we evaluate the ratio R1 = |Mf1(1285)π|2 q |Mη′π|2 ˜q , (44) where q and ˜q are the momentum in the c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' frame of the f1(1285)π and η′π pairs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Numerically, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (44) gives R1 = � � � � 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='4+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='6 �A , � 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='3 �B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (45) 21 The ratio is slightly bigger for the mixing angles in the set A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Nevertheless, the result in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (45) is consistent to the corresponding ratio 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='78 reported by the E852 Collaboration [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' This good agreement with the experimental data supports the molecular picture for the π1(1600) state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' DYNAMICAL GENERATION IN I = 1/2 SECTOR In the I = 1/2 sector, the corresponding WT amplitudes are given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (16) with the Cij coefficients given in Tables VIII and IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' For each case, we have found two poles for parameter sets A and B, as shown in Table XII and XIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' TABLE XII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Poles and their corresponding couplings to the channels contributing to the PA interaction with JP = 1−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Here the flavor-neutral axial mesons have JP C = 1++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The errors of the poles are from varying the subtraction constant within α(µ = 1 GeV) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='17, and only the central values of the couplings are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Poles (Set A) Channels 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='69 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='02 a1K f1(1285)K K1(1270)η f1(1420)K K1(1400)η (+ + + + +) gl 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='89 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='54 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='10 Poles (Set B) Channels 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='70 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='02 a1K f1(1285)K K1(1270)η f1(1420)K K1(1400)η (+ + + + +) gl 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='27 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='15 TABLE XIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Poles and their corresponding couplings to the channels contributing to the PA interaction with JP = 1−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Here the flavor-neutral axial mesons have JP C = 1+−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The errors of the poles are from varying the subtraction constant within α(µ = 1 GeV) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='17, and only the central values of the couplings are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Poles (Set A) Channels 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='70 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='02 h1(1170)K b1K K1(1270)η h1(1415)K K1(1400)η (− + + + +) gl 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='20 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='46 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='38 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='50 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='21 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='02 Poles (Set B) Channels 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='69 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='02 h1(1170)K b1K K1(1270)η h1(1415)K K1(1400)η (− + + + +) gl 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='55 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='01 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='78 + i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='02 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='69 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='83 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='17 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='04 Similarly to the previous cases, the poles are located on the same Riemann sheets in both sets of mixing angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The interactions in the a1K and b1K channels are strong to generate a bound 22 state in each of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The existence of a lower h1 (1170) K channel below the b1K threshold moves the pole in Table XIII to Riemann sheet (− + + + +).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' It has a nonzero imaginary part of a few MeV, which is not shown in the table due to precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' As discussed before, the I = 1/2 poles in Tables XII and XIII will receive sizeable widths once the width effects of the axial-vector mesons are taken into account, and it is expected that the widths are of the order of a few hundred MeV, like those of the b1 and a1 mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Although we neglected the transitions between the A1P and B1P sectors as discussed around Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (17) in Section II, strange mesons are not C-parity eigenstates and the two dynamically generated I = 1/2 1− states will inevitably mix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The two mixed states together could correspond to the 1− K∗ (1680) structure [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' CONCLUSIONS We have studied the interactions between the pseudoscalar and axial-vector mesons in coupled channels with JPC = 1−(+) quantum numbers for the isospin 0, 1, and 1/2 sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Using the chiral unitary approach, we describe the interaction with the Weinberg-Tomozawa term derived from chiral Lagrangians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The transition amplitudes among all the relevant channels are unitarized using the Bethe-Salpeter equation from which resonances (bound states) manifest themselves as poles on the (un)physical Riemann sheets of the complex energy plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' We consider the physical isoscalar axial-vector states as mixtures of the corresponding SU(3) singlets and octets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In addition, the K1(1270) and K1(1400) physical states are also mixtures of the K1A and K1B mesons, which are the strange partners of the a1 and b1 resonances, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' We group into two sets, called A and B, the mixing angles accounting for such mechanisms and investigate their influence on the pole positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' According to our findings, we obtain poles with JP(C) = 1−(+) quantum numbers in the energy range from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='30 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='00 GeV, in each isospin sector studied (I = 0, 1, 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The 1−+ quantum numbers are exotic in the sense that they cannot be formed from a pair of quark and antiquark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In particular, we have found an isoscalar state that may correspond to the η1(1855) state, newly observed by the BESIII Collaboration [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In addition, we have also found two dynamically gener- ated isovector states that we assign to be the π1(1400) and π1(1600) resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Hence, within our formalism, they are dynamically generated through the pseudoscalar-axial vector meson interac- tions, with the η1(1855) state coupling mostly to K1(1400) ¯K channel, while the π1(1400) couples strongly to the b1π, and π1(1600) structure couples most strongly to the K1(1270) ¯K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' We also 23 find two I = 1/2 JP = 1− states with a mass around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='7 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' They combined together could be responsible to the observed K∗(1680) structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In addition, we also evaluate the decays of the η1(1855) and the π1(1600).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' We find that the three-body decay channel ¯KK∗π has a significantly larger branching fraction than the η′η, which is the channel where the observation of the η1(1855) was made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The obtained ratio between the π1(1600) → f1(1285)π and π1(1600) → η′π decays, given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (45), is consistent with the corresponding experimental value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' We suggest searching for two additional η1 exotic mesons with masses of about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='4 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='7 GeV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In particular, the latter should be relatively narrow with a width around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='1 GeV and one of its main decay channels is K ¯Kππ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' ACKNOWLEDGMENTS M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Y is grateful to Shuang-Shi Fang and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Valderrama for valuable discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' This project is supported in part by the National Natural Science Foundation of China (NSFC) under Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' 12125507, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' 11835015, and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' 12047503;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' by the China Postdoctoral Science Foun- dation under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' 2022M713229;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' by the NSFC and the Deutsche Forschungsgemeinschaft (DFG) through the funds provided to the Sino-German Collaborative Research Center TRR110 “Symmetries and the Emergence of Structure in QCD” (NSFC Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' 12070131001, DFG Project-ID 196253076);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' and by the Chinese Academy of Sciences under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' XDB34030000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Chodos, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Jaffe, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Johnson, C.' metadata={'source': 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Recent research put efforts on the tokeniza- +tion, i.e. the conversion of data into sequences +of integers intelligible to such models. This can +be achieved by many ways as music can be com- +posed of simultaneous tracks, of simultaneous +notes with several attributes. Until now, the pro- +posed tokenizations are based on small vocabular- +ies describing the note attributes and time events, +resulting in fairly long token sequences. In this +paper, we show how Byte Pair Encoding (BPE) +can improve the results of deep learning models +while improving its performances. We experiment +on music generation and composer classification, +and study the impact of BPE on how models learn +the embeddings, and show that it can help to in- +crease their isotropy, i.e., the uniformity of the +variance of their positions in the space. +1. Introduction +Deep learning tasks on symbolic music are nowadays mostly +tackled by sequential models1, such as the Transformers +(Vaswani et al., 2017). These models receive sequences of +tokens as input, and convert them to learned embedding +vectors. A token is an integer associated to a high level +element, such as a word or sub-word in natural language, +and both are linked in a vocabulary that acts as a look-up +table. An embedding represents the semantic information of +a token as a vector of fixed-size, and is learning contextually +by the model. To use such models for symbolic music, one +needs to tokenize the data, i.e., convert it to sequences of +tokens that can be decoded back. This can be achieved by +several ways, as music can be composed of simultaneous +tracks, of simultaneous notes with several attributes such as +1LIP6, Sorbonne University - CNRS, Paris, France 2Aubay, +Boulogne-Billancourt, France 3 ESEO-TECH / ERIS, Angers, +France 4University of Angers, Angers, France. Correspondence to: +Nathan Fradet . +1Commonly referred as Language Models (LM) +their pitch and duration. +Recently, the token representation of symbolic music has +been extensively studied, with the goal to improve 1) the +results, e.g. the quality of generated results or the accuracy +of a certain Music Information Retrieval (MIR) task, and; +2) the efficiency of the models. The former is tackled with +more expressive representations (Huang & Yang, 2020; Ker- +marec et al., 2022), and the latter by representations based +on either token combinations (Payne, 2019; Donahue et al., +2019), or embedding pooling (Hsiao et al., 2021; Zeng et al., +2021; Ren et al., 2020), which reduce the overall sequence +length. Still, current tokenizations only use tokens represent- +ing the values of time and note attributes, such as Pitch or +Duration. This comes with a big limitation: these tokens do +not carry much information by themselves, and neither their +associated embeddings. By analogy to natural language, +these tokens are closer to the characters than words. Yet, the +expressive information carried by music is deduced by the +combinations of its notes and their attributes. Considering +the infinite possible arrangements, deep learning models +may struggle to implicitly learn their common features. +In this paper, we study the application of Byte Pair En- +coding (BPE, described in Section 3) for symbolic music +generation, aiming to improve the two objectives mentioned +above, while making the models learn more isotropic em- +bedding representations in some cases. To the best of our +knowledge, BPE has yet not been studied for the symbolic +music modality, although it can be applied on top of any +music tokenization that do not perform embedding pooling. +This work aims at closing this gap by shedding light on the +results and performance gains of using BPE: +• We experiment on two public datasets (Wang et al., +2020b; Kong et al., 2021), with two base tokenizations, +on which BPE is learned with several vocabulary sizes, +on the generation and composer classification tasks, +and show that it improves the results; +• We compare BPE with other sequence reduction tech- +niques introduced in recent research; +• We study the geometry of the learned embeddings, and +show that BPE can improve their isotropy; +• We show some limits of BPE, such as on the proportion +arXiv:2301.11975v1 [cs.LG] 27 Jan 2023 + +Byte Pair Encoding for Symbolic Music +of sampled tokens, and that the vocabulary size has to +be carefully chosen. +The source code is provided for reproducibility: https: +//github.com/Natooz/BPE-Symbolic-Music +The paper is organised as follows: Section 2 reviews the +related work while Section 3 sheds light on the BPE tech- +nique. Section 4 describes our experimental settings and +Section 5 describes the evaluation metrics that we use for +the experimental evaluation. Section 6 presents the results +and analysis. Furthermore, Section 7 provides an additional +study on the impact of BPE on how the models learn the +embeddings. Finally, Section 8 presents our conclusion and +perspectives. +2. Related work +In this section we start by reminding research of specific +music representation of symbolic music generation. Then, +we present how recent works put efforts on different strate- +gies to reduce the sequence length. Finally, we explain their +limitations which conduce us to propose our novel approach +that is to apply Byte Pair Encoding in the field of symbolic +music for reducing sequence length. +2.1. Representation of symbolic music +Most works on symbolic music generation from deep learn- +ing use a specific music representation. Early research in- +troduced representations specifically tied to the training +data being used, such as DeepBach (Hadjeres et al., 2017), +FolkRNN (Sturm et al., 2015) or BachBot (Liang et al., +2017). Non-sequential models such as MuseGAN (Dong +et al., 2018) often represent music as pianoroll matrices. +Since, more universal representations have been studied, al- +lowing to convert any sequence of (simultaneous) notes into +tokens (Oore et al., 2018; Huang & Yang, 2020; Hadjeres +& Crestel, 2021; Fradet et al., 2021). Some of them are +depicted in Figure 1. +2.2. Sequence reduction strategies +In more recent works, efforts have been put towards the +efficiency. Indeed, most recent models are based on the +Transformer architecture (Vaswani et al., 2017). The atten- +tion mechanism, at the heart of Transformers, has however +a time and space complexity that grows quadratically with +the input sequence length. This is a well known bottleneck, +that led researchers to work on more efficient attention esti- +mations (Tay et al., 2021), down to linear complexity. In the +field of symbolic music specifically, researchers worked on +strategies to reduce the sequence length in order to increase +1) the efficiency of the models; 2) the scope of the attention +mechanism; 3) the quality of the generated results. These +Bar Pos. 0 +Pitch D3 +Vel. 22Dur. 7Pos. 7 +Pitch A3 +Vel. 24Dur. 7Pos. 15 +Pitch E4 +Vel. 24Dur. 7Pos. 27 +Pitch G3 +Vel. 16Dur. 3 Bar Pos. 0 +Pitch A3 +Vel. 20Dur. 31 +Ti.-Sh. 0 +Pitch D3 +Vel. 22Dur. 7 +Pitch A3 +Vel. 24Dur. 7 +Pitch E4 +Vel. 24Dur. 7 +Pitch G3 +Vel. 16Dur. 3 +Pitch A3 +Vel. 20Dur. 31 +Ti.-Sh. 8 +Ti.-Sh. 8 +Ti.-Sh. 12 +Ti.-Sh. 4 +N.-On D3 +Vel. 22 +N.-On A3 +Vel. 24 +N.-Off A3 +N.-On E4 +Vel. 24 +N.-Off E4 N.-On G3 +Vel. 16 +N.-On A3 +Vel. 20 +Ti.-Sh. 7 +Ti.-Sh. 7 +Ti.-Sh. 7 +N.-Off D3 +Ti.-Sh. 3 +Ti.-Sh. 3 +N.-Off G3 +Ti.-Sh. 31 +N.-Off A3 +Music score +MIDI-Like +REMI +Structured +Figure 1. A sheet music and several token representations. +strategies can be split in two categories: 1) embedding pool- +ing strategies such as Compound Word (Hsiao et al., 2021) +(CPWord), Octuple (Zeng et al., 2021) or PopMag (Ren +et al., 2020); 2) token combination strategies such as in +MuseNet (Payne, 2019) or LakhNES (Donahue et al., 2019). +Embedding pooling consists in merging the embeddings of +several distinct tokens with a pooling operation. This is +often done by concatenating the embeddings and projecting +the sequence, resulting in an aggregated embedding of fixed +size. Token combinations is simply the use of a vocabu- +lary containing tokens that represent several values, e.g., +Pitch-x Duration-y that represent both the pitch and +velocity information. +2.3. Limitations +However, these strategies show the following limitations. +Embedding pooling: 1) requires a more complex training +procedure; 2) for generation, inferring from such model +can be seen as sampling from a multivariate distribution, +which can be a delicate operation; 3) the results can easily +degenerate if the pooling does not yield semantically rich +embeddings that represent the underlying tokens. On the +other hand, token combinations of entire types of tokens can +lead to large vocabularies with unused tokens and potentially +non-optimized or unbalanced token distributions. +To the best of our knowledge, no work has been conducted +on applying BPE, introduced in Section 3, to symbolic mu- +sic generation. A similar technique is used with Sympho- +nyNet (Liu et al., 2022), which does not rely on token adja- +cency but rather on the concurrence of multiple notes, and +they only experimented with a vocabulary size of 1k tokens. +The following section describes the Byte Pair Encoding +technique, its algorithm and depicts how it can be relevant +to use in the field of symbolic music. + +1 ++Byte Pair Encoding for Symbolic Music +3. Byte Pair Encoding +Byte Pair Encoding (BPE) (Gage, 1994) is a data com- +pression technique. It converts the most recurrent succes- +sive bytes (or in our case tokens) in a corpus into newly +created ones. +For instance, in the character sequence +aabaabaacaa, the sub-sequence aa occurs three times +and is the most recurrent. Learning and applying BPE on +this sequence would replace aa with a new symbol, e.g., d, +resulting in a reduced sequence dbdbdcd. The latter can +be reduced again by replacing the db subsequence, giving +eedcd. In the context of deep learning, BPE naturally in- +creases the size of the vocabulary, while reducing the overall +sequence lengths. In practice BPE is learned on a corpus +until the vocabulary reaches a target size. BPE learning is +described by the pseudo-code of Algorithm 1. +Algorithm 1 Learning of BPE pseudo-code +Require: Base vocabulary V, target vocabulary size N, +dataset X +1: while |V|< N do +2: +Find s = {t1, t2} ∈ V2, from X, the most recurrent +token succession +3: +Add a new token t in V, mapping to s +4: +Substitute every occurrence of s in X with t +5: end while +6: return V +BPE is nowadays largely used in the NLP field as it allows +to encode rare words and segmenting unknown or com- +posed words as sequences of sub-word units (Sennrich et al., +2016). +In symbolic music, notes are represented by successions +of tokens that represent the values of their attributes. In +this context, BPE can allow to represent a note, or even a +succession of notes, that is very recurrent in the dataset, as +a single token. For instance, a note that would be coded +as the succession of tokens Pitch D3, Velocity 60, +Duration 2.0 could be replaced by a single new one. +Rare note (and attributes) would still be encoded as non- +BPE tokens. The same logic applies to time tokens, that can +also be associated to note tokens. +4. Experimental settings +This section details the experimental protocol by describing +the models, the training and the datasets used along with the +specific tokenization processes. +4.1. Model and training +As we specifically focus on sequential models, we exper- +iment with the state of the art deep learning architecture +for most NLP tasks at the time of writing, the Transformer +(Vaswani et al., 2017) architecture. The generator uses a +causal attention mask and is trained with teacher forcing, +while the classifier does not use attention mask and is first +pre-trained to retrieve randomized tokens then finetuned to +classify the input sequences. They are respectively similar +to GPT2 (Radford et al., 2019) and BERT (Devlin et al., +2019). The details of implementation, such as their sizes +and training, can be found in Appendix A +All models receive sequences between 384 and 460 tokens, +beginning with special BOS (Beginning of Sequence) and +ending EOS (End of Sequence) tokens. We split datasets +in two subsets: one only used for training and updating +the models, one for validation to monitor trainings, that is +also used to test the models after training. These subsets +represent respectively 65% and 35% of the original datasets. +4.2. Datasets +We experiment with two datasets: POP909 (Wang et al., +2020b) and GiantMIDI (Kong et al., 2021). +The POP909 dataset (Wang et al., 2020b) is composed of +909 piano tracks of Pop musics, with aligned MIDI and +audio versions. Each MIDI file contains three tracks: the +first is the lead melody, the second is secondary melodies +and bridges, the third is the arrangements with chords and +arpeggios. For our experiments we merge all three tracks +into a single one. +The GiantMIDI dataset (Kong et al., 2021) is composed +of 10k piano MIDI files, transcribed from audio to MIDI +without downbeat and tempo estimation. Each file contains +a single track of non-interrupted piano music, often with +complex melodies and harmonies. Considering the com- +plexity of its content, we make the assumption that it is a +difficult dataset for a model to learn from. +We perform data augmentation on the pitch dimension on +both datasets. Each MIDI file is augmented up and down to +two octaves. +4.3. Music tokenization +We experiment with Remi (Huang & Yang, 2020) and +TSD (for Time Shift Duration) as base tokenizations, on +which BPE will be applied on top. Both tokenizations de- +scribe notes as a succession of the Pitch, Velocity and +Duration tokens. Remi represents time with Bar and +Position tokens, which respectively indicates when a +new bar is beginning and at which position within the time +is. TSD represents time with TimeShift tokens, indicat- +ing explicitly time movements. +When tokenizing symbolic music, it is common to down- +sample continuous features to discrete sets of values. For +instance, velocities can be downsampled from 128 to 32 + +Byte Pair Encoding for Symbolic Music +values. These sets should be sufficiently precise so that +the global information remains coherent (Huang & Yang, +2020; Oore et al., 2018; Hadjeres & Crestel, 2021). Down- +sampling features helps models to learn more easily, as it +allows to reduce the perplexity of the predictions, especially +for values which are less commons in the training set. The +details of our downsamplings can be found in Appendix B. +BPE is learned from tokenized corpuses, up to a maximum +of 1500 randomly picked files, to reduce the learning time. +We choose to experiment with six vocabulary sizes. One +without BPE, and five where the original vocabulary size is +multiplied by 4, 10, 20, 50 and 100. +To extend our analysis, we also experiment with a version +of TSD and Remi where Pitch and Velocity tokens +are merged (PVm), and one where Pitch, Velocity and +Duration are merged (PVDm). PVm is similar to the +strategy used with MuseNet (Payne, 2019). We finally ex- +periment with the CPWord (Hsiao et al., 2021) and Octuple +(Zeng et al., 2021) embedding pooling strategies, that we +group with Remi in our experiments as they represent time +similarly. We use the same pooling strategy, and sample +independently from the logits of each output modules. For +implementation simplicity reasons, all embeddings have the +same size than the model dimension. +5. Evaluation metrics +Generative models are often evaluated with automatic met- +rics on the generated results. Image and audio models are +assessed with the Fr´echet Inception Distance (FID) (Heusel +et al., 2017) and Fr´echet Audio Distance (FAD) (Kilgour +et al., 2019), both comparing the distribution of original data +and generated results. Language models are often assessed +with BLEU (Papineni et al., 2002), ROUGE (Lin, 2004) or +other metrics that compare generated results with reference +sentences. +Automatic evaluation of symbolic music remains however +an open issue. It exists no reference-free metric measuring +its quality or fidelity. Metrics with reference such as BLEU +may be suited for machine translation tasks, but remains +irrelevant for open-ended generation, such as in our case. +We then perform both human and automatic evaluations, as +commonly done for symbolic music (Huang & Yang, 2020; +Huang et al., 2018; Hsiao et al., 2021). Our automatic met- +rics aim to measure the errors of prediction of the models, +and the similarity of some features. +5.1. Tokenization syntax error +Every tokenization has an underlying syntax of token type +and value successions, that can normally be made. For +instance, if the last token of an input sequence is of type +Pitch, a tokenization could require that the next token to +predict must be of type Velocity. We could also expect +a model to not predict more than once the same note at a +same moment, or to not go back in time. +Successions of incorrect token types can be interpreted as +errors of prediction. These errors can help us to measure +if a model has efficiently learned the music representation +and if it can yield coherent results. With this motivation, +we introduce a new metric we called Tokenization Syntax +Errors (TSE). +Velocity +Pitch +Duration +Position +Bar +(a) REMI. +Velocity +Note-On +Note-Off +Time-Shift +(b) MIDI-Like +Figure 2. Directed graphs of the token types succession (without +additional tokens) for a) REMI (Huang & Yang, 2020) and b) +MIDI-Like (Oore et al., 2018). +We distinguish five categories of errors: +• TSEtype: the predicted token does not have a type +that should follow the previous one. For any tokeniza- +tion, we can draw a directed graph representing the +possible token types successions, such as in Figure 2. +• TSEtime: when using Position tokens, the pre- +dicted Position value is inferior or equal to the +current one, making the time goes backward. +• TSEdupn (duplicated note): when the model predicts +a note that has already been played at the current mo- +ment (by the same instrument). +• TSEnnof (no NoteOff): when using NoteOn and +NoteOff, and that a NoteOn token has been pre- +dicted with no NoteOff later to end it, or too distant +in time. +• TSEnnon (no NoteOn): when a NoteOff token is +predicted but the corresponding note has not been +played. +For a given sequence of tokens, TSE measures the ratio, +scaled between 0 and 1, of errors for these five categories. +A TSE of 0 means that there is no error in the sequence, +while a ratio of 1 means only errors were predicted. Our +experiments are not concerned by the last two categories as +we do not use NoteOff tokens. +Finally, we should mention that most of these errors can +be avoided by a ruled-based sampling. When predicting a +token, one can easily keep track of the time, notes played +and token types to automatically exclude invalid predictions. + +Byte Pair Encoding for Symbolic Music +In practice, this can be achieved by setting the invalid indices +of the predicted logits to −∞ before applying softmax. +5.2. Feature similarity +We expect models to generate continuations that keep the +features of the input prompt consistent. For instance, it +should predict first notes within the same scale and with +the same velocity range. We measure this similarity by +calculating the overlapping area of distributions of features, +for the prompt and the first 16 generated beats. +Previous works (Yang & Lerch, 2020; Choi et al., 2020; +Mittal et al., 2021; von R¨utte et al., 2022) use the proba- +bility density function of the distributions, estimated with +kernel density estimations, and emphasizes that it smooths +and transforms the distributions into more general repre- +sentations. While this method can be suited for continuous +modalities, it can lead to inaccuracies with categorical ones. +Here, pitch and duration features can be considered as dis- +crete. Their distributions are both sparse, containing for +instance many white keys and fewer black keys, yet adja- +cent and corresponding to close integer values in the MIDI +format. In order to be more accurate, we measure this simi- +larity with the histogram intersection of these features, as +described in Equation (1). +Similarity (D1, D2) = HI (Hist (D1) , Hist (D2)) +HI(x, y) = � +i min(xi, yi), xi ≥ 0, yi ≥ 0 +(1) +Hist : R|D| �→ Ne returns the normalized histogram of +a distribution of a feature with e elements, HI stands for +Histogram Intersection. +5.3. Human evaluations +For each experiment, we select 40 prompts of 8 beats. For +each prompts, we generate continuations of 1k tokens with +the benchmarked models. Three musicians open the con- +tinuations as a MIDI file, allowing them to listen the tracks +and also visualize them as piano rolls. Among the tracks, +they are asked to select the one: 1) with the highest fidelity +on pitch scale, velocity, note density and rhythm, with the +prompt; 2) they subjectively prefer overall, considering its +correctness, structure and richness. +6. Results and analysis +We focus on how BPE is learned on the corpuses, then on its +benefits for music generation and composer classification. +6.1. BPE learning +Figure 3 shows the distribution of token types combina- +tions of the learned BPE tokens. +We observe that the +majority of the combinations learned on the Remi tok- +enization represent notes, by their Pitch, Velocity and +4 +10 +20 +50 +100 +BPE Factor +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +Proportion +Pch-Vel-Dur +Pch-Vel-Dur-TimeShift +Vel-Dur-TimeShift +Vel-Dur +Pch-Vel-Dur-Pch-Vel-Dur +TimeShift-Pch +Other +(a) TSD +4 +10 +20 +50 +100 +BPE Factor +0.0 +0.2 +0.4 +0.6 +0.8 +Proportion +Pch-Vel-Dur +Pch-Vel-Dur-Pos +Vel-Dur +Pos-Pch-Vel-Dur +Pch-Vel-Dur-Pch-Vel-Dur +Pos-Pch +Other +(b) Remi +Figure 3. Normalized distributions of the token types of the BPE +tokens, per BPE factor for the POP909 dataset. +0 +2k +4k +6k +8k +10k +12k +14k +Vocabulary size +2.0 +2.5 +3.0 +3.5 +4.0 +Avg. token combinations +POP909 TSD +POP909 REMI +GiantMIDI TSD +GiantMIDI REMI +0 +2k +4k +6k +8k +10k +12k +14k +Vocabulary size +5 +10 +15 +20 +25 +30 +35 +Max. token combinations +Figure 4. Average (left) and maximum (right) number of token +combinations represented by BPE tokens in function of the vocab- +ulary size. +Duration attributes. For TSD, the combinations also in- +clude TimeShift tokens early in the learning. This dif- +ference mostly comes from common TimeShift tokens +following notes, whereas for Remi the notes are distributed +at different Position(s). As the vocabulary grows, the +combinations tend to be more diverse. The distribution for +the GiantMIDI dataset are showned in Appendix C. +Figure 4 shows the evolution of the average number of non- +BPE token combinations represented by the BPE tokens. +At the beginning of the learning, the mean number of com- +binations grows more quickly as the most recurrent token +successions are often made of more than two tokens. The +POP909 dataset being smaller than GiantMIDI, it naturally +leads to a higher maximum number of combinations as the +latter is more diverse. When the vocabulary begins to con- + +Byte Pair Encoding for Symbolic Music +Table 1. Metrics of generated results. TSE numbers are all scaled at e-3 for better readability. Sim stands for similarity, the best results are +the closest to the datasets. Hum. Fidelity and Overall are the human evaluations. +Data / Strategy +TSEtype (↓) +TSEdupn (↓) +TSEtime (↓) +Sim. pit. +Sim. vel. +Sim. dur. +Hum. Fidelity (↑) +Hum. Overall (↑) +POP909 TSD +0.66 ± 0.13 +0.84 ± 0.12 +0.69 ± 0.14 +No BPE +1.0 ± 1.8 +13.6 ± 8.0 +- +0.59 ± 0.08 +0.82 ± 0.10 +0.64 ± 0.09 +0.00 +0.00 +BPE×4 +0.2 ± 0.9 +21.9 ± 19.9 +- +0.65 ± 0.07 +0.82 ± 0.10 +0.74 ± 0.08 +0.24 +0.19 +BPE×10 +0.5 ± 2.2 +13.4 ± 14.6 +- +0.64 ± 0.07 +0.78 ± 0.12 +0.74 ± 0.07 +0.53 +0.42 +BPE×20 +0.8 ± 2.1 +12.8 ± 11.0 +- +0.62 ± 0.07 +0.79 ± 0.11 +0.70 ± 0.09 +0.20 +0.31 +BPE×50 +22.4 ± 24.0 +4.4 ± 5.3 +- +0.56 ± 0.07 +0.70 ± 0.12 +0.62 ± 0.11 +0.02 +0.02 +BPE×100 +21.5 ± 40.2 +35.6 ± 56.0 +- +0.54 ± 0.08 +0.66 ± 0.14 +0.63 ± 0.10 +0.00 +0.00 +PVm +6.1 ± 6.6 +6.9 ± 9.3 +- +0.59 ± 0.08 +0.78 ± 0.12 +0.73 ± 0.08 +0.01 +0.06 +PVDm +23.6 ± 19.3 +0.2 ± 0.7 +- +0.43 ± 0.09 +0.57 ± 0.19 +0.54 ± 0.12 +0.00 +0.00 +POP909 REMI +0.66 ± 0.13 +0.84 ± 0.12 +0.69 ± 0.14 +No BPE +0.0 ± 0.1 +115.4 ± 33.8 +74.7 ± 26.7 +0.61 ± 0.08 +0.85 ± 0.09 +0.72 ± 0.07 +0.02 +0.03 +BPE×4 +0.1 ± 0.4 +65.7 ± 21.7 +154.9 ± 27.6 +0.55 ± 0.09 +0.77 ± 0.12 +0.70 ± 0.09 +0.27 +0.34 +BPE×10 +0.3 ± 1.1 +52.3 ± 18.3 +167.1 ± 30.5 +0.49 ± 0.08 +0.77 ± 0.10 +0.63 ± 0.09 +0.52 +0.44 +BPE×20 +0.8 ± 2.2 +81.8 ± 37.3 +242.6 ± 46.5 +0.46 ± 0.08 +0.71 ± 0.13 +0.61 ± 0.10 +0.12 +0.12 +BPE×50 +37.8 ± 35.5 +128.2 ± 22.2 +324.1 ± 21.5 +0.30 ± 0.12 +0.56 ± 0.20 +0.55 ± 0.12 +0.00 +0.00 +BPE×100 +83.9 ± 78.0 +136.3 ± 32.4 +324.6 ± 28.8 +0.28 ± 0.11 +0.54 ± 0.22 +0.55 ± 0.12 +0.00 +0.00 +PVm +2.3 ± 7.1 +160.0 ± 75.3 +102.7 ± 48.2 +0.60 ± 0.08 +0.77 ± 0.12 +0.69 ± 0.09 +0.05 +0.04 +PVDm +49.3 ± 46.2 +99.8 ± 25.1 +301.9 ± 26.5 +0.32 ± 0.13 +0.50 ± 0.24 +0.45 ± 0.12 +0.02 +0.02 +CPWord +331.9 ± 33.8 +144.5 ± 46.8 +99.3 ± 16.6 +0.57 ± 0.08 +0.85 ± 0.07 +0.73 ± 0.09 +0.00 +0.00 +Octuple +- +789.3 ± 111.1 +891.9 ± 76.1 +0.05 ± 0.15 +0.07 ± 0.21 +0.06 ± 0.17 +0.00 +0.00 +GiantMIDI TSD +0.49 ± 0.17 +0.74 ± 0.18 +0.52 ± 0.23 +No BPE +0.2 ± 1.1 +3.9 ± 4.6 +- +0.50 ± 0.10 +0.77 ± 0.12 +0.63 ± 0.13 +0.24 +0.19 +BPE×4 +0.5 ± 1.4 +15.2 ± 18.1 +- +0.51 ± 0.10 +0.75 ± 0.13 +0.62 ± 0.14 +0.33 +0.27 +BPE×10 +1.5 ± 3.3 +35.2 ± 45.6 +- +0.51 ± 0.11 +0.68 ± 0.17 +0.65 ± 0.13 +0.29 +0.37 +BPE×20 +0.0 ± 0.0 +17.5 ± 29.3 +- +0.52 ± 0.09 +0.73 ± 0.15 +0.65 ± 0.12 +0.11 +0.08 +BPE×50 +0.0 ± 0.3 +6.8 ± 8.5 +- +0.50 ± 0.09 +0.70 ± 0.13 +0.64 ± 0.11 +0.00 +0.00 +BPE×100 +1.5 ± 3.7 +1.1 ± 1.5 +- +0.46 ± 0.09 +0.63 ± 0.17 +0.53 ± 0.13 +0.01 +0.01 +PVm +3.0 ± 3.7 +0.7 ± 1.3 +- +0.46 ± 0.11 +0.69 ± 0.15 +0.67 ± 0.11 +0.02 +0.09 +PVDm +35.6 ± 56.1 +0.5 ± 1.2 +- +0.39 ± 0.13 +0.61 ± 0.18 +0.25 ± 0.18 +0.00 +0.00 +GiantMIDI REMI +0.49 ± 0.17 +0.74 ± 0.18 +0.52 ± 0.23 +No BPE +0.2 ± 0.9 +57.8 ± 40.2 +95.1 ± 42.8 +0.53 ± 0.10 +0.75 ± 0.14 +0.63 ± 0.13 +0.00 +0.01 +BPE×4 +0.2 ± 0.8 +44.3 ± 23.5 +82.3 ± 36.4 +0.46 ± 0.11 +0.71 ± 0.15 +0.62 ± 0.12 +0.41 +0.43 +BPE×10 +2.5 ± 3.5 +31.7 ± 20.2 +175.6 ± 60.3 +0.43 ± 0.10 +0.63 ± 0.21 +0.54 ± 0.15 +0.53 +0.52 +BPE×20 +0.7 ± 2.4 +36.6 ± 29.3 +221.9 ± 66.4 +0.33 ± 0.12 +0.65 ± 0.16 +0.46 ± 0.15 +0.02 +0.01 +BPE×50 +34.8 ± 11.1 +80.5 ± 53.1 +316.4 ± 54.1 +0.36 ± 0.11 +0.58 ± 0.18 +0.30 ± 0.23 +0.00 +0.00 +BPE×100 +476.1 ± 148.3 +159.8 ± 60.1 +285.3 ± 31.5 +0.19 ± 0.10 +0.59 ± 0.20 +0.20 ± 0.19 +0.00 +0.00 +PVm +0.7 ± 2.4 +53.8 ± 47.4 +181.5 ± 56.9 +0.46 ± 0.11 +0.70 ± 0.15 +0.60 ± 0.14 +0.00 +0.01 +PVDm +31.9 ± 63.9 +65.6 ± 28.8 +285.6 ± 32.6 +0.33 ± 0.14 +0.58 ± 0.19 +0.29 ± 0.17 +0.02 +0.02 +CPWord +408.9 ± 28.3 +160.1 ± 54.4 +69.3 ± 16.7 +0.51 ± 0.11 +0.81 ± 0.09 +0.69 ± 0.12 +0.00 +0.00 +Octuple +- +763.8 ± 134.4 +894.3 ± 62.1 +0.03 ± 0.11 +0.06 ± 0.19 +0.04 ± 0.15 +0.00 +0.00 +tain BPE tokens with a large number of combinations, it +starts to specialize on very specific note successions that +may appear in few data samples. In particular, big jumps +of maximum number of combinations, e.g. from 14 to 27 +for POP909 Remi, indicate that two already big BPE tokens +represent the most recurrent succession. These numbers, +correlated with the model, dataset sizes and overall token +distribution of the dataset, might help to choose an optimal +vocabulary size. +Further analysis in Appendix C shows that BPE consider- +ably reduces the sequence length, and so the training and +generation time, at the cost of an increased tokenization +time. Tokenization of data is however often performed once, +and the training time gain is very likely to be larger than the +tokenization time loss. +6.2. Generated results +For the generation task, we generate continuations of input +prompt from the validation subset. The continuations are +autoregressively generated with 1024 steps, with nucleus +sampling (Holtzman et al., 2020), with p = 0.9. +The results of all metrics are reported in Table 1. For TSD, +BPE allows to reduce both the token type and note dupli- +cation errors in most cases, while the time errors slightly +increase for Remi baselines. These results show that models +can easily scale to bigger vocabularies, up to a certain limit. +Here, starting from a BPE factor of 50, the TSE seems to +increase, as do the other results. BPE tends to however +produce results with features slightly less similar, especially +with big vocabulary sizes. +We gathered a total of 400 human evaluations. They show +that BPE with factors of 4 and 10 significantly outperform +other baselines, in all experiments. BPE helps models to + +Byte Pair Encoding for Symbolic Music +Table 2. Number of tokens sampled and not sampled by generative +models, respectively right and left separated by |. +Strategy +POP909 TSD +POP909 Remi +GiantMIDI TSD +GiantMIDI Remi +No BPE +116 | 23 (16%) +141 | 11 (7%) +136 | 3 (2%) +151 | 1 (0%) +BPE×4 +454 | 102 (18%) +487 | 121 (19%) +456 | 100 (17%) +386 | 222 (36%) +BPE×10 +479 | 911 (65%) +514 | 1006 (66%) +456 | 934 (67%) +618 | 902 (59%) +BPE×20 +592 | 2188 (78%) +552 | 2488 (81%) +478 | 2302 (82%) +504 | 2536 (83%) +BPE×50 +521 | 6429 (92%) +249 | 7351 (96%) +401 | 6549 (94%) +155 | 7445 (97%) +BPE×100 +521 | 13379 (96%) +244 | 14956 (98%) +281 | 13619 (97%) +89 | 15111 (99%) +PVm +321 | 426 (57%) +338 | 422 (55%) +342 | 405 (54%) +369 | 391 (51%) +PVDm +391 | 13712 (97%) +144 | 13972 (98%) +252 | 13851 (98%) +166 | 13950 (98%) +Table 3. Average accuracy of classification models. +Strategy +TSD (↑) +Remi (↑) +TSD Large (↑) +Remi Large (↑) +No BPE +0.196 ± 0.031 +0.169 ± 0.021 +0.208 ± 0.033 +0.175 ± 0.022 +BPE×4 +0.218 ± 0.033 +0.168 ± 0.021 +0.226 ± 0.034 +0.171 ± 0.022 +BPE×10 +0.226 ± 0.038 +0.190 ± 0.030 +0.228 ± 0.037 +0.201 ± 0.034 +BPE×20 +0.236 ± 0.038 +0.195 ± 0.026 +0.240 ± 0.039 +0.210 ± 0.029 +BPE×50 +0.199 ± 0.027 +0.207 ± 0.032 +0.247 ± 0.041 +0.216 ± 0.035 +BPE×100 +0.122 ± 0.009 +0.119 ± 0.008 +0.243 ± 0.037 +0.126 ± 0.010 +PVm +0.199 ± 0.027 +0.150 ± 0.016 +0.213 ± 0.029 +0.188 ± 0.025 +PVDm +0.226 ± 0.035 +0.192 ± 0.028 +0.228 ± 0.036 +0.194 ± 0.029 +CPWord +- +0.204 ± 0.28 +- +0.214 ± 0.024 +Octuple +- +0.274 ± 0.041 +- +0.283 ± 0.043 +generate more correct and pleasant music. We make the +assumption that having a larger set of learned embeddings +help the model to capture more easily the global melody, har- +mony and music structure, and in turn improve the generated +results. These embedding, when well learned contextually, +may represent richer and more explicit information. +Table 2 shows that while models give high probabilities to +more unique tokens with BPE in absolute number, the pro- +portion of sampled tokens decreases. Models tend to focus +on the sets of more recurrent tokens and omitting more rare +ones. Beyond a BPE factor of 20 (or vocabulary size be- +tween 2k and 2.5k tokens), the models are even focusing on +a more restricting sets of tokens. These numbers highlight +the limitations of using a too large vocabulary size, as the +extra effort is unlikely to result in better results. +6.3. Composer classification +Composer classification is performed with the top-10 most +present composers of the GiantMIDI dataset. The results, +reported in Table 3, show that BPE outperforms other base- +lines. Here, the model seems to benefit from larger vocabu- +lary sizes. We also remark that the model size plays in its +capacity to handle large vocabularies. While the results of +BPE100 for the small model indicate it was unable to learn +anything, the larger one performed almost as good as the top +baseline. A second observation is the good performances +of embedding pooling strategies (CPWord and OCtuple). +While they performed poorly for generative tasks, they are +among the best for this classification task. They seem to be +better for MIR tasks than generation. As stated in Section 1, +generation implies sampling, and sampling from several +distributions is delicate, as for training a model with an +autoregressive objective on several output distributions. +Table 4. IsoScore results. +Generator +POP909 TSD +POP909 Remi +GiantMIDI TSD +GiantMIDI Remi +No BPE +0.09 +0.14 +0.08 +0.09 +BPE×4 +0.02 +0.04 +0.02 +0.02 +BPE×10 +0.12 +0.11 +0.02 +0.07 +BPE×20 +0.13 +0.05 +0.02 +0.02 +BPE×50 +0.02 +0.01 +0.01 +0.01 +BPE×100 +0.01 +0.01 +0.01 +0.00 +PVm +0.02 +0.02 +0.01 +0.02 +PVDm +0.00 +0.00 +0.00 +0.00 +CPWord +- +0.04 +- +0.08 +Octuple +- +0.04 +- +0.02 +Classifier +TSD (↑) +Remi (↑) +TSD Large (↑) +Remi Large (↑) +No BPE +0.74 +0.71 +0.80 +0.77 +BPE×4 +0.35 +0.33 +0.54 +0.37 +BPE×10 +0.36 +0.31 +0.48 +0.50 +BPE×20 +0.54 +0.57 +0.64 +0.53 +BPE×50 +0.77 +0.80 +0.75 +0.82 +BPE×100 +0.82 +0.90 +0.87 +0.89 +PVm +0.27 +0.27 +0.32 +0.32 +PVDm +0.69 +0.88 +0.88 +0.88 +CPWord +- +0.08 +- +0.05 +Octuple +- +0.08 +- +0.06 +7. Learned embedding spaces +Results presented in this section rely on Table 4, and Fig- +ures 5 and 6. Isotropy is a measure of the uniformity of the +space occupied by a distribution, across all dimensions. In +our case, the distribution is a manifold X ∈ RN×d where +N = |V | and d is the model/embedding dimension. It +has been associated with improved performances with lan- +guage models (Bi´s et al., 2021; Liang et al., 2021), mostly +because embeddings are more discriminative and enable +models to capture and distinguish more easily subtle seman- +tic information. It has been observed that representations +from Transformers often exhibit anisotropy, i.e., they tend +to occupy only a small subspace of the embedding space, +and often not uniformly (Gao et al., 2019; Ethayarajh, 2019; +Wang et al., 2020a; Gong et al., 2018; Reif et al., 2019), +especially causal generative models (Ethayarajh, 2019). +Isotropy is often estimated by different ways: singular value +decomposition (Bi´s et al., 2021; Gao et al., 2019; Liang +et al., 2021; Wang et al., 2020a), intrinsic dimension (Cai +et al., 2021), partition function (Arora et al., 2016; Mu & +Viswanath, 2018), average cosine similarity (Ethayarajh, +2019). Although these methods are correlated with isotropy, +recent research shed light on some of their limits (Rudman +et al., 2022). We choose to estimate it with intrinsic value, +IsoScore (Rudman et al., 2022), singular value and cosine +similarity, to have results that corroborate and complement +themselves. The results of the two latter can be found in +Appendix D. For tokenizations with embedding pooling, we +used 50k randomly sampled embeddings of combinations +of tokens representing notes, as using all the embedding +combinations would be intractable and would not reflect the +ones actually learned by the models. Results for tokeniza- +tions where N ≲ d (no BPE) have to be interpreted loosely. +Isotropy cannot be reliably measured with less samples than +the number of dimensions they occupy. The estimations are +more accurate when N ≫ d, as more samples populate all + +Byte Pair Encoding for Symbolic Music +Gen. POP909 TSD +noBPE +bpe4 +bpe10 +bpe20 +bpe50 +bpe100 +PVm +PVDm +0 +20 +40 +60 +80 +Dimension +lPCA +MLE +MOM +TLE +TwoNN +FisherS +Gen. POP909 Remi +noBPE +bpe4 +bpe10 +bpe20 +bpe50 +bpe100 +PVm +PVDm +CPWord +Octuple +0 +20 +40 +60 +lPCA +MLE +MOM +TLE +TwoNN +FisherS +Gen. GiantMIDI TSD +noBPE +bpe4 +bpe10 +bpe20 +bpe50 +bpe100 +PVm +PVDm +0 +5 +10 +15 +20 +25 +lPCA +MLE +MOM +TLE +TwoNN +FisherS +Gen. GiantMIDI Remi +noBPE +bpe4 +bpe10 +bpe20 +bpe50 +bpe100 +PVm +PVDm +CPWord +Octuple +0 +20 +40 +60 +80 +lPCA +MLE +MOM +TLE +TwoNN +FisherS +Clasmall TSD +noBPE +bpe4 +bpe10 +bpe20 +bpe50 +bpe100 +PVm +PVDm +0 +20 +40 +60 +80 +Dimension +200 +300 +400 +500 +600 +700 +lPCA +MLE +MOM +TLE +TwoNN +FisherS +Clasmall Remi +noBPE +bpe4 +bpe10 +bpe20 +bpe50 +bpe100 +PVm +PVDm +CPWord +Octuple +0 +20 +40 +60 +80 +100 +200 +300 +400 +500 +600 +700 +800 +lPCA +MLE +MOM +TLE +TwoNN +FisherS +Clalarge TSD +noBPE +bpe4 +bpe10 +bpe20 +bpe50 +bpe100 +PVm +PVDm +0 +20 +40 +60 +80 +100 +200 +400 +600 +800 +1000 +lPCA +MLE +MOM +TLE +TwoNN +FisherS +Clalarge Remi +noBPE +bpe4 +bpe10 +bpe20 +bpe50 +bpe100 +PVm +PVDm +CPWord +Octuple +0 +20 +40 +60 +80 +100 +200 +400 +600 +800 +1000 +lPCA +MLE +MOM +TLE +TwoNN +FisherS +Figure 5. Intrinsic dimension estimations. A second x axis has been added on the right for lPCA on classifier plots for better readability. +Gen. POP909 no BPE +Gen. POP909 BPE×20 +Clasmall GiantMIDI no BPE +Clasmall GiantMIDI BPE×20 +Figure 6. 3d UMAP representations of learning embedding spaces, +with TSD tokenization. Abbreviations in legend stand for: Pi: +Pitch; V: Velocity; D: Duration; Po: Position: TS: TimeShift. +dimensions of Rd. +The IsoScore results (See Table 4), show that BPE does +not increase the score for generative models. It seems that +big vocabularies with BPE yield lower IsoScore results, +that corroborate with the intrinsic dimension results (Fig- +ure 5). Causal generative models have been shown to learn +anisotropic embedding representations (Cai et al., 2021; +Ethayarajh, 2019). Embeddings form cones and clusters, +that can be observed in Figure 6. As we estimated isotropy +on all embeddings altogether, the presence of clusters nat- +urally correlate with anisotropy, as the variance is mostly +pronounced on their distances. The cluster themselves might +be more isotropic (Cai et al., 2021). +On the other hand, BPE can help bi-directional models +to learn more isotropic embedding representations. The +IsoScore grows with the vocabulary size, as do the intrinsic +dimension. In Figure 6 we observe that the embeddings +have no preferred direction in space, forming a sphere (See +more figures in Appendix D). +8. Conclusion +We showed that BPE can increase the quality of results +for symbolic music generation, and composer classifica- +tion, while improving the performances, with a well chosen +vocabulary size. BPE can be applied on top of any tokeniza- +tion, and we advice the reader to do so for projects involving +symbolic music. The drawbacks are a time-consuming vo- +cabulary learning, and a slower tokenization of data. BPE +can also helps models to learn more isotropic embedding +representations. Future work will explore more in depth +the isotropy of clusters of embeddings of generative models. +We also plan to experiment with larger model, dataset and +vocabulary sizes, hoping to find guidelines for choosing an +optimum vocabulary size. + +Special +Pitch +Velocity +Duration +TimeShift +8 +7 +6 +5 +4 +3 +8 +7 +6 +-3 +5 +-2 +-1 +4 +0Special +Pitch +Velocity +Duration +Time-Shift +Pi-V-D +7 +Pi-V-D-TS +V-D-TS +6 +V-D +TS-Pi +5 +Other BPE +4 +5 +4 +3 +5 +6 +7 +2 +8 +9 +10 +1 +11 +12Special +Pitch +Velocity +Duration +TimeShift +4.5 +4.0 +3.5 +3.0 +2.5 +2.0 +3.5 +3.0 +2.5 +-4.0 +2.0 +-3.5 +-3.0 +1.5 +-2.5 +-2.0 +1.0 +-1.5Special +Pitch +Velocity +Duration +Time-shift +8.0 +Pi-V-D +7.5 +V-D-Pi +7.0 +V-D-TS +6.5 +Pi-V-D-TS +6.0 +V-D +5.5 +Other BPE +5.0 +8.5 +8.0 +7.5 +7.0 +3.0 +3.5 +6.5 +4.0 +6.0 +4.5 +5.0 +5.5 +5.5 +6.0 +5.0Byte Pair Encoding for Symbolic Music +References +Arora, S., Li, Y., Liang, Y., Ma, T., and Risteski, A. 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Association for Compu- +tational Linguistics. doi: 10.18653/v1/2021.findings-acl. +70. +URL https://aclanthology.org/2021. +findings-acl.70. + +Byte Pair Encoding for Symbolic Music +A. Model and training +Table 5. Model configurations. The number of parameters is based on the baseline with no BPE, and may vary depending on the baseline +with the size of the first and last layers. Gen stands for generator and Cla for classifier. +Gen +Clasmall +Clalarge +Dimension +512 +768 +1024 +Nb attention heads +8 +12 +16 +Nb layers +10 +10 +18 +Feedforward size +2048 +2048 +3078 +Parameters +32.6M +58.0M +193.3M +The sizes of the models are reported in Table 5. The generator is trained with a teacher forcing objective on 100k steps. The +classifier pre-trained on 60k steps to retrieve the value of randomized positions. Between 1 to 15% of each input sequences +is randomized during pre-training. It is then fine-tuned on 100k steps to predict the composer of the input sequence, from +the first output hidden state, i.e., the BOS position, which is projected through an output classification layer. The input +embedding and output pre-training module weights are tied to improve the performances (Press & Wolf, 2017). +The batch size is set to 16 for the generator, and 24 for the classifier. All trainings are done with automatic mixed-precision +(Micikevicius et al., 2018), the Adam optimizer (Kingma & Ba, 2015) with β1 = 0.9, β2 = 0.999 and ϵ = 10−8, and +dropout, weight decay and a gradient clip norm of respectively 10−1, 10−2 and 3. We use a one cycle learning rate scheduler: +the initial learning rate is close to 0 and gradually grows for the 30% first steps to 5e−6, 1e−6 and 5e−7 for the generators, +classifier pre-training and classifier fine-tuning respectively, then slowly decreases down to 0. We perform 5 validations steps +every 30 training steps, and compute their average accuracy and loss. The model parameters are saved when the validation +loss is the lowest ever observed, and after training the last version saved is used for testing. The training is stopped early if +the validation losses did not decrease for 15k steps and 25k steps for respectively the generator and classifier. +B. Data downsampling +0 +1 +2 +3 +4 +5 +6 +7 +duration +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +density +Dataset +POP909 +GiantMIDI +0 +20 +40 +60 +80 +100 +120 +velocity +0.000 +0.005 +0.010 +0.015 +0.020 +0.025 +0.030 +density +Dataset +POP909 +GiantMIDI +Figure 7. Distributions of the note durations and velocities of the POP909 and GiantMIDI datasets. The duration axis is limited to 7 beats. +Figure 7 shows the distributions of velocity and duration values of the notes from the two datasets we use. As there is a +larger proportion of low note durations (below two beats), we decided to downsample the Duration and TimeShift +tokens with different resolutions: those up to one beat are downsampled to 8 samples per beat (spb), those from one to +two beats to 4 spb, those from two to four beats to 2 spb, and those from four to eight beats to 1 spb. This way, short +notes are represented more precisely than longer ones, reducing the vocabulary size. For Remi, Position tokens are +downsampled to 8 spb, resulting in 32 different tokens as we only consider the 4/* time signature. This allows to represent +the 16th note. We only consider pitches within the recommended range for piano (program 0) specified in the General MIDI +2 specifications2: 21 to 108. We then deduplicate all duplicated notes. Velocities are downsampled to 8 distinct values. No +additional token (e.g., Chord, Tempo) is used. +2Available on the MIDI Manufacturers Association website + +Byte Pair Encoding for Symbolic Music +C. BPE Learning +Table 6. Vocabulary sizes, mean tokens per beat (tpb), and variation of tpb from without BPE, average tokenizing time and detokenizing +time. A maximum of 1000k randomly sampled MIDI files were used for each row. Vocabulary sizes for CPWord and Octuple are the +product of the sizes of their ”sub-vocabularies”, or in other words the number of possible token combinations, and are rounded for better +readability. Tokenizing and detokenizing times were run on an Intel Xeon Gold 5128 CPU. +Data +Vocab. size +tpb +tpb variation (%) +Tok. time (sec) +Detok. time (sec) +POP909 TSD +No BPE +139 +17.81 ± 4.12 +- +0.04 ± 0.02 +0.01 ± 0.02 +BPE×4 +556 +9.71 ± 2.12 +-45.50 +0.20 ± 0.05 +0.02 ± 0.02 +BPE×10 +1390 +8.05 ± 1.75 +-54.80 +0.44 ± 0.10 +0.02 ± 0.02 +BPE×20 +2780 +6.95 ± 1.53 +-60.99 +0.77 ± 0.18 +0.02 ± 0.02 +BPE×50 +6950 +5.84 ± 1.28 +-67.20 +1.59 ± 0.37 +0.02 ± 0.02 +BPE×100 +13.9k +5.33 ± 1.16 +-70.10 +2.72 ± 0.63 +0.02 ± 0.02 +PVm +747 +12.72 ± 2.92 +-28.59 +0.03 ± 0.01 +0.01 ± 0.01 +PVDm +14.1k +7.63 ± 1.73 +-57.17 +0.02 ± 0.01 +0.01 ± 0.01 +POP909 Remi +No BPE +152 +18.06 ± 4.12 +- +0.03 ± 0.02 +0.01 ± 0.01 +BPE×4 +608 +10.55 ± 2.26 +-41.61 +0.21 ± 0.05 +0.02 ± 0.02 +BPE×10 +1520 +8.85 ± 1.90 +-51.00 +0.47 ± 0.11 +0.02 ± 0.02 +BPE×20 +3040 +8.01 ± 1.74 +-55.64 +0.86 ± 0.19 +0.02 ± 0.02 +BPE×50 +7600 +7.32 ± 1.58 +-59.46 +1.97 ± 0.43 +0.02 ± 0.02 +BPE×100 +15.2k +6.70 ± 1.43 +-62.92 +3.64 ± 0.79 +0.02 ± 0.02 +PVm +760 +12.97 ± 2.92 +-28.19 +0.02 ± 0.01 +0.01 ± 0.01 +PVDm +14k +7.88 ± 1.73 +-56.38 +0.02 ± 0.01 +0.01 ± 0.01 +CPWord +49k +7.88 ± 1.73 +-56.38 +0.03 ± 0.01 +0.03 ± 0.02 +Octuple +161k +5.09 ± 1.21 +-71.81 +0.02 ± 0.01 +0.02 ± 0.02 +GiantMIDI TSD +No BPE +139 +15.64 ± 6.29 +- +0.08 ± 0.10 +0.03 ± 0.05 +BPE×4 +556 +8.87 ± 3.30 +-43.26 +0.45 ± 0.57 +0.08 ± 0.16 +BPE×10 +1390 +7.88 ± 2.86 +-49.64 +1.04 ± 1.29 +0.07 ± 0.16 +BPE×20 +2780 +7.04 ± 2.40 +-54.98 +1.90 ± 2.34 +0.07 ± 0.15 +BPE×50 +6950 +5.94 ± 2.20 +-62.03 +4.11 ± 5.03 +0.07 ± 0.14 +BPE×100 +13.9k +5.45 ± 2.04 +-65.15 +7.49 ± 9.16 +0.07 ± 0.14 +PVm +747 +11.26 ± 4.46 +-28.03 +0.06 ± 0.08 +0.03 ± 0.04 +PVDm +14.1k +6.57 ± 2.59 +-57.98 +0.06 ± 0.07 +0.02 ± 0.03 +GiantMIDI Remi +no BPE +152 +15.89 ± 6.42 +- +0.08 ± 0.10 +0.04 ± 0.05 +BPE×4 +608 +9.58 ± 3.39 +-39.70 +0.53 ± 0.67 +0.08 ± 0.18 +BPE×10 +1520 +8.18 ± 2.96 +-48.51 +1.22 ± 1.51 +0.08 ± 0.17 +BPE×20 +3040 +7.22 ± 2.78 +-54.56 +2.18 ± 2.70 +0.08 ± 0.17 +BPE×50 +7600 +6.41 ± 2.42 +-59.67 +4.87 ± 5.98 +0.08 ± 0.16 +BPE×100 +15.2k +5.96 ± 2.20 +-62.51 +9.08 ± 11.12 +0.07 ± 0.15 +PVm +760 +11.30 ± 4.66 +-28.90 +0.06 ± 0.08 +0.03 ± 0.04 +PVDm +14k +6.94 ± 2.63 +-56.29 +0.05 ± 0.06 +0.02 ± 0.03 +CPWord +49k +6.90 ± 2.55 +-56.60 +0.07 ± 0.09 +0.07 ± 0.10 +Octuple +161k +4.37 ± 1.88 +-72.51 +0.05 ± 0.06 +0.05 ± 0.07 +4 +10 +20 +50 +100 +BPE Factor +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +Proportion +Vel-Dur-TimeShift +Pch-Vel-Dur +Pch-Vel-Dur-TimeShift +Vel-Dur-Pch +Vel-Dur +Pch-Vel-Dur-Pch +Other +(a) TSD +4 +10 +20 +50 +100 +BPE Factor +0.0 +0.2 +0.4 +0.6 +0.8 +Proportion +Pch-Vel-Dur +Pch-Vel-Dur-Pos +Vel-Dur +Pch-Vel-Dur-Pch-Vel-Dur +Pos-Pch-Vel-Dur +Pos-Pch +Other +(b) Remi +Figure 8. Normalized distributions of token types per BPE factor for the GiantMIDI dataset. +Table 6 shows the vocabulary sizes, sequence length variation and tokenization times of all baselines. When learning BPE, +the average number of tokens per beat (tpb) quickly decreases, so the sequence length. As the vocabulary grows, the tpb +decreases more slowly, as the most recurrent token successions have already be learned and replaced. A lower tpb allows to +generate faster. + +Byte Pair Encoding for Symbolic Music +The tradeoff of BPE, besides the vocabulary learning time, is the tokenization time, as a MIDI file is first tokenized without +BPE, then BPE is applied by finding the token subsequences to be replaced by the BPE tokens. The decoding step time, i.e., +the time of the conversion of tokens to a MIDI file, is almost not impacted by BPE. The tokenization and detokenization +times have been gotten with MidiTok (Fradet et al., 2021) which is implemented in Python. The tokenization time could be +decreased if performed by a faster compiled language such as Rust or C. The Figure 8 complements the Figure 3, with the +GiantMIDI dataset. +D. Learned embedding space +Figure 9 shows the singular values for the generative and classification models. As the different tokenizations features +vocabularies with very different sizes, the values are normalized for better readability. Note that the NoBPE tokenizations +feature vocabularies with a size inferior to the embedding dimension. NoBPE adj. corresponds to the NoBPE results adjusted +to cover the x-axis on the whole embedding size. +Figure 10 shows the pairwise cosine similarity of the learned embedding vectors, for the TSD and Remi representation on +the POP909 dataset. The first tokens up to 90 are Pitches, followed by Velocities up to 125, Durations up to 160 +and then Time-Shift or Position. Without BPE, we can clearly distinguish patterns in the cosine similarity matrices. +These high similarities shows that embeddings are close to each other. With BPE and larger vocabulary sizes, the average +cosine similarity tend to decrease, especially between BPE tokens. Embeddings are less similar and more discriminative. +UMAP (McInnes et al., 2018) representations shown in Figure 6, Figure 12 and Figure 11 have been calculated with the +default parameters of the official Python package. We clearly see that generative models learn clusters of embeddings of the +same type, distant from each other. The embeddings do not occupy the space uniformly. On the other hand, pre-trained +bi-directional models learn more isotropic embedding representations. The embeddings are spread uniformly across all +directions, for all token types. + +Byte Pair Encoding for Symbolic Music +Gen POP909 +100 +101 +102 +Dimension +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Singular value +noBPE +bpe4 +bpe10 +bpe20 +bpe50 +bpe100 +PVm +PVDm +noBPE adj. +100 +101 +102 +Dimension +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Singular value +noBPE +bpe4 +bpe10 +bpe20 +bpe50 +bpe100 +PVm +PVDm +CPWord +Octuple +noBPE adj. +Gen GiantMIDI +100 +101 +102 +Dimension +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Singular value +noBPE +bpe4 +bpe10 +bpe20 +bpe50 +bpe100 +PVm +PVDm +noBPE adj. +100 +101 +102 +Dimension +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Singular value +noBPE +bpe4 +bpe10 +bpe20 +bpe50 +bpe100 +PVm +PVDm +CPWord +Octuple +noBPE adj. +Clasmall +100 +101 +102 +Dimension +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Singular value +noBPE +bpe4 +bpe10 +bpe20 +bpe50 +bpe100 +PVm +PVDm +noBPE adj. +100 +101 +102 +Dimension +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Singular value +noBPE +bpe4 +bpe10 +bpe20 +bpe50 +bpe100 +PVm +PVDm +CPWord +Octuple +noBPE adj. +Clalarge +100 +101 +102 +103 +Dimension +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Singular value +noBPE +bpe4 +bpe10 +bpe20 +bpe50 +bpe100 +PVm +PVDm +noBPE adj. +TSD +100 +101 +102 +103 +Dimension +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Singular value +noBPE +bpe4 +bpe10 +bpe20 +bpe50 +bpe100 +PVm +PVDm +CPWord +Octuple +noBPE adj. +Remi +Figure 9. Normalized singular values of embedding matrices of classifier models. + +Byte Pair Encoding for Symbolic Music +TSD +0 +20 +40 +60 +80 +100 +120 +0 +20 +40 +60 +80 +100 +120 +0.6 +0.4 +0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0 +100 +200 +300 +400 +500 +0 +100 +200 +300 +400 +500 +0.4 +0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0 +200 +400 +600 +800 +1000 +1200 +0 +200 +400 +600 +800 +1000 +1200 +0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0 +500 +1000 +1500 +2000 +2500 +0 +500 +1000 +1500 +2000 +2500 +0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Remi +0 +20 +40 +60 +80 +100 +120 +140 +0 +20 +40 +60 +80 +100 +120 +140 +0.4 +0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +No BPE +0 +100 +200 +300 +400 +500 +600 +0 +100 +200 +300 +400 +500 +600 +0.6 +0.4 +0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +BPE x4 +0 +200 +400 +600 +800 +1000 +1200 +1400 +0 +200 +400 +600 +800 +1000 +1200 +1400 +0.4 +0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +BPE x10 +0 +500 +1000 +1500 +2000 +2500 +3000 +0 +500 +1000 +1500 +2000 +2500 +3000 +0.6 +0.4 +0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +BPE x20 +Figure 10. Pairwise cosine similarity matrix of learned embedding of the generative models, on the POP909 dataset. +No BPE +BPE x4 +BPE x10 +BPE x20 +BPE x50 +BPE x100 +PVm +PVDm +Figure 11. UMAP 2d representations of the embeddings of classifier models pre-trained with the GiantMIDI dataset and TSD tokenization. +Abbreviations in legend stand for: Pi: Pitch; V: Velocity; D: Duration; Po: Position; TS: TimeShift. + +4.0 +Special +Pitch +3.5 +Velocity +Duration +3.0 +TimeShift +2.5 +2.0 +1.5 +1.0 +0.5 +0.0 +6.5 +7.0 +7.5 +8.0 +8.5 +9.0 +9.5 +10.06 +5 +Special +Pitch +Velocity +！ +C +4 +Duration +Time-Shift +V-D-TS +V-D +3 +Pi-V-D +V-D-Pi +Other BPE +5 +6 +7 +8 +9 +10Special +Pitch +Velocity +8 +Duration +Time-Shift +V-D-TS +7 +Pi-V-D +V-D-Pi +Pi-V-D-TS +V-D +6 +Other BPE +5 +4 +3 +4 +5 +6 +7Special +9 +Pitch +Velocity +Duration +Time-Shift +8 +Pi-V-D +V-D-Pi +V-D-TS +7 +Pi-V-D-TS +V-D +Other BPE +6 +5 +2 +3 +4 +5 +6Special +Pitch +Velocity +9 +Duration +Time-Shift +Pi-V-D-TS +8 +Pi-V-D +V-D-Pi +V-D-TS +7 +Pi-V-D-Pi +Other BPE +6 +5 +4 +5 +6 +7 +8 +97 +6 +Special +Pitch +Velocity +5 +Duration +Time-Shift +Pi-V-D-TS +4 +Pi-V-D +V-D-Pi +Pi-V-D-Pi +3 +V-D-TS +Other BPE +4 +5 +6 +7 +8 +97.5 +7.0 +6.5 +6.0 +5.5 +5.0 +4.5 +4.0 +Special +PitchVel +3.5 +Duration +Time-Shift +3 +4 +5 +63 +2 +0 +-1 +Special +PitchVelDur +-2 +Time-Shift +6 +7 +8 +9 +10 +11 +1284009 0400344Byte Pair Encoding for Symbolic Music +TSD No BPE +TSD BPE×4 +TSD BPE×10 +TSD BPE×20 +TSD BPE×50 +TSD BPE×100 +TSD PVm +TSD PVDm +Remi No BPE +Remi BPE×4 +Remi BPE×10 +Remi BPE×20 +Remi BPE×50 +Remi BPE×100 +Remi PVm +Remi PVDm +Figure 12. UMAP 3d representations of the embeddings of generative models with the POP909 dataset. Abbreviations in legend stand for: +Pi: Pitch; V: Velocity; D: Duration; Po: Position: TS: TimeShift. + +Special +Pitch +Velocity +Duration +TimeShift +8 +7 +6 +5 +4 +3 +8 +7 +6 +-3 +5 +-2 +-1 +4 +0Special +Pitch +Velocity +Duration +Time-Shift +Pi-V-D +7 +Pi-V-D-TS +V-D-TS +6 +V-D +TS-Pi +5 +Other BPE +4 +5 +4 +3 +5 +6 +7 +2 +8 +9 +10 +1 +11 +128 +7 +6 +5 +Special +4 +Pitch +3 +Velocity +2 +Duration +Time-Shift +Pi-V-D +6 +V-D-TS +4 +V-D +O Pi-V-D-TS +2 +5 +D-TS: +5.0 +0 +7.5 +Other BPE +10.0 +12.5 +-2Special +Pitch +Velocity +Duration +Time-Shift +14 +Pi-V-D +13 +Pi-V-D-TS +12 +V-D-TS +11 +V-D +10 +D-TS +9 +Other BPE +8 +7 +1 +0 +-1 +-9 +-2 +-8 +-3 +-7 +-6 +-4 +-58 +6 +4 +Special +2 +Pitch +Velocity +0 +Duration +Time-Shift +Pi-V-D-TS +12.5 +Pi-V-D +10.0 +V-D-TS +7.5 +5.0 +5 Pi-V-D-Pi-V-D +0.0 +2.5 +V-D +5.0 +0.0 +Other BPE +7.5 +10.0 +-2.58 +6 +4 +2 +Special +Pitch +0 +Velocity +-2 +Duration +-4 +Time-Shift +Pi-V-D-TS +12.5 +Pi-V-D +10.0 +7.5 +Pi-V-D-Pi-V-D +5.0 +V-D-TS +2.5 +Pi-V-D-TS-Pi-V-D-TS +0.0 +8 +Other BPE +10 +-2.5 +12Special +PitchVel +Duration +Time-Shift +3 +2 +1 +0 +-1 +-2 +8 +6 +7 +8 +9 +4 +10 +11 +2Special +PitchVeiDur +Time-Shift +8 +7 +6 +5 +4 +3 +2 +10 +8 +6 +0.0 +4 +2 +0 +-2Special +Bar +Pitch +Velocity +Duration +2 +Position +1 +0 +-1 +-2 +8.5 +8.0 +7.5 +7.0 +0 +6.5 +1 +6.0 +2 +5.5 +3 +5.010 +8 +6 +Special +Bar +4 +Pitch +Velocity +Duration +Position +Pi-V-D +V-D +0 +5 +Bar-Po +10 +Other BPE +15Special +Bar +Pitch +Velocity +Duration +Position +-3 +Pi-V-D +-4 +V-D +V-D-Po +-5 +Bar-Po +-6 +V-D-Bar-Po +Other BPE +-7 +-7 +-8 +-7 +-6 +-9 +-5 +-4 +-3 +-10 +-28 +6 +Special +4 +Bar +Pitch +2 +Velocity +Duration +0 +Position +19 +Pi-V-D +V-D +18 +Po-Pi +4 +V-D-Po +6 +Pi-V-D-Pi-V-D +8 +Other BPE +10 +15 +1212 +10 +8 +Special +6 +Bar +4 +Pitch +2 +Velocity +0 +Duration +Position +Pi-V-D +7.5 +Pi-V-D-Po +5.0 +Po-Pi-V-D +2.5 +2.Pi-V-D-Pi-V-D +0.0 +2.5 +-2.5 +Po-Pi +5 +0 +-5.0 +Other BPE +7.5 +10.0 +-7.5 +12.5Special +Bar +Pitch +Velocity +Duration 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symbolic music modality is nowadays mostly represented as discrete and used with sequential models such as Transformers, for deep learning tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfCS7y/content/2301.11975v1.pdf'} +page_content=' Recent research put efforts on the tokeniza- tion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfCS7y/content/2301.11975v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfCS7y/content/2301.11975v1.pdf'} +page_content=' the conversion of data into sequences of integers intelligible to such models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfCS7y/content/2301.11975v1.pdf'} +page_content=' This can be achieved by many ways as music can be com- posed of simultaneous tracks, of simultaneous notes with several attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfCS7y/content/2301.11975v1.pdf'} +page_content=' Until now, the pro- posed tokenizations are based on small vocabular- ies describing the note attributes and time events, resulting in fairly long token sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfCS7y/content/2301.11975v1.pdf'} +page_content=' In this paper, we show how Byte Pair Encoding (BPE) can improve the results of deep learning models while improving its performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfCS7y/content/2301.11975v1.pdf'} +page_content=' We experiment on music generation and composer classification, and study the impact of BPE on how models learn the embeddings, and show that it can help to in- crease their isotropy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfCS7y/content/2301.11975v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfCS7y/content/2301.11975v1.pdf'} +page_content=', the uniformity of the variance of their positions in the space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfCS7y/content/2301.11975v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfCS7y/content/2301.11975v1.pdf'} +page_content=' Introduction Deep learning tasks on symbolic music are nowadays mostly tackled by sequential models1, such as the Transformers (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfCS7y/content/2301.11975v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfCS7y/content/2301.11975v1.pdf'} +page_content=' These models receive sequences of tokens as input, and convert them to learned embedding vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfCS7y/content/2301.11975v1.pdf'} +page_content=' A token is an integer associated to a high level element, such as a word or sub-word in natural language, and both are linked in a vocabulary that acts as a look-up table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfCS7y/content/2301.11975v1.pdf'} +page_content=' An embedding represents the semantic information of a token as a vector of fixed-size, and is learning contextually by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfCS7y/content/2301.11975v1.pdf'} +page_content=' To use such models for symbolic music, one needs to tokenize the data, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfCS7y/content/2301.11975v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfCS7y/content/2301.11975v1.pdf'} +page_content=', convert it to sequences of tokens that can be decoded back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfCS7y/content/2301.11975v1.pdf'} +page_content=' This can be achieved by several ways, as music can be composed of simultaneous tracks, of simultaneous notes with several attributes such as 1LIP6, Sorbonne University - CNRS, Paris, France 2Aubay, Boulogne-Billancourt, France 3 ESEO-TECH / ERIS, Angers, France 4University of Angers, Angers, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfCS7y/content/2301.11975v1.pdf'} +page_content=' Correspondence to: Nathan Fradet 1K +40.81% +62.61% +11.87% +8.61% +All +19.01% +26.30% +6.22% +3.94% +MDAM +POMO +BQ (ours) +Set (size) +bs50 +augx8 +greedy +bs16 +A (32-80) +6.17% +4.86% +5.85% +1.96% +B (30-77) +8.77% +5.13% +7.04% +3.50% +F (44-134) +16.96% +15.49% +7.20% +3.04% +M (100-200) +5.92% +4.99% +6.69% +1.85% +P (15-100) +8.44% +14.69% +4.71% +1.32% +X (100-1K) +34.17% +21.62% +10.74% +8.35% +All (15-1K) +22.36% +15.58% +8.58% +5.60% +(c) Experimental results on TSPLib (left) and CVRPLib (right). +10 +−4 +10 +−3 +10 +−2 +10 +−1 +10 +0 +Inference time (per instance, in seconds) +0 +5 +10 +15 +20 +25 +30 +35 +40 +Optimality gap +AM bs1024 +MDAM bs50 +POMO augx8 +Att- GCN+MCTS +BQ (ours) greedy +BQ (ours) bs16 +TSP100 +TSP200 +TSP500 +TSP1000 +10 +−4 +10 +−3 +10 +−2 +10 +−1 +10 +0 +Inference time (per instance, in seconds) +0 +20 +40 +60 +80 +100 +120 +140 +Optimality gap +AM bs1024 +MDAM bs50 +POMO augx8 +NeuralRewriter +BQ (ours) greedy +BQ (ours) bs16 +CVRP100 +CVRP200 +CVRP500 +CVRP1000 +Figure 2: Generalization results on different graph sizes for TSP (left) and CVRP (right). Lower +and further left is better. +9 + +Published as a conference paper at ICLR 2023 +REPRODUCIBILITY STATEMENT +In order to ensure the reproducibility of our approach, we have: +• described precisely our generic theoretical framework (Section ??) and provided a detailed +proof of Proposition 1 in Appendix F. This should in particular serve to adapt the frame- +work to other CO problems; +• explained in detail our proposed model (Section 4 for TSP and Appendix A for CVRP), +described precisely the training procedure and listed the hyperparameters (Section 6); +• used public datasets referenced in Section 6. +Furthermore, we plan to make our code public upon acceptance. +REFERENCES +David Applegate, Robert Bixby, Vasek Chvatal, and William Cook. Concorde TSP solver. Univer- +sity of Waterloo, 2015. +Jimmy Lei Ba, Jamie Ryan Kiros, and Geoffrey E. Hinton. Layer Normalization, July 2016. +Thomas Bachlechner, Bodhisattwa Prasad Majumder, Henry Mao, Gary Cottrell, and Julian +McAuley. 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IEEE Transactions on Industrial Informatics, 17(7):4861–4871, July 2021b. +ISSN 1941-0050. doi: 10.1109/TII.2020.3031409. +Cong Zhang, Wen Song, Zhiguang Cao, Jie Zhang, Puay Siew Tan, and Chi Xu. +Learning to +Dispatch for Job Shop Scheduling via Deep Reinforcement Learning. arXiv:2010.12367 [cs, +stat], October 2020. +12 + +Published as a conference paper at ICLR 2023 +A +APPLICATION TO THE CVRP +Problem definition and specification +The Capacitated Vehicle Routing Problem (CVRP) is a +vehicle routing problem in which a vehicle (here, a single one) with limited capacity must deliver +items from a depot location to various customer locations. Each customer has an associated demand, +which represents an amount of items, and the problem is for the vehicle to provide all the customers +in the least travel distance, returning as many times as needed to the depot to refill, but without ever +exceeding the vehicle capacity. +Formally, we assume given a set of customer nodes, each with a demand (positive scalar), plus a +depot node. A CVRP solution (in X) is a finite sequence of nodes starting at the depot, which are +pairwise distinct except for the depot, and respecting the capacity constraint: the total demand of +any contiguous sub-sequence of customer nodes is below the vehicle capacity. A CVRP instance (in +F) is given by a finite set D of nodes, including the depot, their coordinates in the Euclidian space +V , and maps any solution to the length of the corresponding path using the distances in V , if the +path visits exactly all the nodes of D, or ∞ otherwise (unfeasible solutions). +A possible specification ⟨T , SOL, VAL⟩ for the CVRP is defined as follows. The step space T is the +set of pairs of a non depot node and a binary flag indicating whether that node is to be reached via +the depot or directly. The extension ¯t of a step t is either the singleton of its node component if its +flag is 0 or the pair of the depot node and its node component if its flag is 1. For a given problem +instance f and sequence t1:n of steps, SOL(f, t1:n) is either the sequence ¯t1:n if it forms a d-path +which visits exactly all the nodes of f , or ⊥ otherwise. VAL(f, t1:n) is either the total length of ¯t1:n +(closed at its end) if it forms a d-path which visits only nodes of f (maybe not all), or ∞ otherwise. +It is easy to show that ⟨T , SOL, VAL⟩ forms a specification for the CVRP (i.e. satisfies the axioms of +specifications introduced in Section ??). The naive MDP obtained from it is denoted CVRP-MDP. +Bisimulation quotienting +Just as with TSP, we can define a mapping Φ from CVRP-MDP states +to a new “path-CVRP” state space, informally described by the following diagram. +C=10 +1 +1 +4 +3 +2 +4 +1 +1 +4 +3 +CVRP state +C=3 +1 +4 +3 +path-CVRP state +Φ +Here, the capacity of the vehicle is C=10, shown next to the (colourless) depot node, and the demand +of each node is shown next to it, in orange. The black dotted line indicates that the action which +introduced the node with demand 2 was via the depot: its flag was set to 1 (all the other actions had +their flag set to 0 in this simple example). The green dotted line indicates how the path is closed +to measure its length. After the node with demand 2, the path of visited nodes (in red) continues +with nodes with demand 4 and 1, respectively, so that the remaining capacity at the end of the path +is C−(2+4+1)=3. Compared to TSP, this is the additional piece of information in the summary +of the “past” (path of visited nodes) which is preserved in the path-CVRP state, together with the +origin and destination of the path. Mapping Φ thus defined satisfies Equation 3, hence induces a +bisimulation on CVRP-MDP states, and by quotienting, one obtains an MDP which can be defined +directly on path-CVRP states. +Model architecture for CVRP +The model architecture for CVRP is almost the same as for TSP, +with a slight difference in the input sequence and in the output layer. In the TSP model, the input to +the node embedding layer for a N-node state is a 2×N matrix of coordinates. For CVRP, we use two +additional channels: one for node demands, and one for the current vehicle capacity, repeated across +all nodes. The demand is set to zero for the origin and destination nodes. We obtain a 4×N matrix of +features, which is passed through a learned embedding layer. As for TSP, a learned origin-signalling +(resp. destination-signalling) vector is added to the corresponding embeddings. The rest of the +architecture, in the form of attention layers, is identical to TSP, until after the action scores projection +layer. In the case of TSP, the projection layer returns a vector of N scores, where each score, after +13 + +Published as a conference paper at ICLR 2023 +a softmax, represents the probability of choosing the node as the next step in the construction. In +the case of CVRP, the model returns a matrix of scores of dimension N×2, corresponding to each +possible actions (node-flag pair) and the softmax scopes over this whole matrix. As usual, a mask is +always applied to unfeasible actions before the softmax operator: those which have higher demand +than the remaining vehicle capacity, as well as the origin and destination nodes. +B +APPLICATION TO THE KNAPSACK PROBLEM +Problem definition and specification +The Knapsack Problem (KP) is classical combinatorial op- +timization problem in which we need to pack items, with given values and weights, into a knapsack +with a given capacity. The objective is to maximize the total value of packed items. Formally, we +assume given a set of items, each with a value and weight. A KP solution (in X) is a subset of the +items which respects a capacity constraint (“c-subset”): total weight of the items of the subset must +not exceed the knapsack capacity. A KP instance (in F) is given by a finite set of D items and maps +any c-subset to the sum of values of its items. +A simple problem specification ⟨T , SOL, VAL⟩ can be defined as follows. The step space T is equal +to the set of items, . For a partial solution (f, t1:n), if the selected items satisfy the capacity con- +straints and adding any of the remaining items results in an infeasible solution, then SOL(f, t1:n) +returns the subset of selected items; otherwise it returns ⊥. Finally, VAL(f, t1:n) is either the sum +of the values of the items in t1:n if they satisfy the capacity constraint and ∞ otherwise. Similarly +to the TSP and CVRP cases, it is easy to show that ⟨T , SOL, VAL⟩ forms a specification for the KP. +The naive MDP obtained from it is denoted MDP-KP. +Bisimulation quotienting +As it was the case for TSP and CVRP, we can define a mapping Φ +from KP-MDP state to a new “BQ-KP” state space, informally described by the following diagram. +3 +7 +9 +1 +1 +2 +4 +5 +8 +8 +6 +weights +values +1 +9 +2 +8 +3 +7 +1 +6 +7 +3 +9 +C = 20 +3 +9 +1 +4 +5 +8 +8 +6 +1 +2 +3 +1 +6 +7 +3 +9 +C = 10 +KP-state +BQ-KP-state +Φ +Here, capacity of the knapsack is C = 20 and each item is defined by its weight (bottom cell) and +value (top cell). Mapping Φ for KP is straightforward - simply saying, it removes all picked items +and update the remaining capacity by subtracting total weight of removed items from the previous +capacity. +Model architecture for KP +The model architecture for KP is again very similar to previously +described models for TSP and CVRP. The input to the model is a 3 × N tensor composed of items +properties (values, weights) and the additional channel for the remaining knapsack’s capacity. By +definition, the solution has no order (the result is a set of items), so there is no need to add tokens for +origin and destination. A part from excluding these tokens and different input dimensions, the rest of +the model is identical to the TSP model. The output is a vector of N probabilities over all items with +a mask over the unfeasible ones (with weights larger than remaining knapsack’s capacity). In the +training, at each construction step, any item of the ground-truth solution is a valid choice. Therefore +we use a multi-class cross-entropy loss. +Experimental results for KP +We generate the training dataset as described in Kwon et al. (2021). +We train our model on 1M KP instances of size 200 and capacity 25, with values and weights ran- +domly sampled from the unit interval. We use the dynamic programming algorithm from ORTools +to compute the ground-truth optinal solutions. As hyperparameters, we use the same as for the TSP. +Then, we evaluate our model on test datasets with the number of items equal 200, 500 and 1000 +and capacity of 25 and 50, for each problem size. Table B shows the performance of our model +compared to POMO, one of the best performing NCO models on KP. Although our model does not +outperform it in every dataset, it achieves better overall performance. It should be noted again that +POMO builds N solutions per instance and choose the best one, while our model generate a single +solution per instance but still achieves better results. +14 + +Published as a conference paper at ICLR 2023 +Optimal +POMO (single traj.) +POMO (all traj.) +BQ (greedy) +value +value +opt gap +value +opt gap +value +opt gap +N=200 +C=25 +58.023 +57.740 +0.476% +58.007 +0.017% +57.970 +0.081% +C=50 +80.756 +79.483 +1.544% +79.787 +1.170% +80.710 +0.056% +N=500 +C=25 +90.986 +85.309 +6.217% +86.516 +4.897% +90.150 +0.904% +C=50 +129.326 +128.950 +0.291% +129.272 +0.042% +128.369 +0.739% +N=1000 +C=25 +128.692 +120.757 +5.386% +123.572 +3.973% +121.217 +5.808% +C=50 +182.898 +170.920 +6.545% +172.427 +5.724% +175.093 +4.267% +All +- +3.552% +2.648% +1.980% +Table 2: Experimental results on KP. +Greedy +Beam size 16 +Beam size 64 +Full graph +0.79% +5s +0.17% +1m +0.08% +5m +TSP200 +100KNNs +1.31% +3s +0.23% +33s +0.10% +3m +Full graph +1.71% +1m +0.68% +15m +0.54% +1h +TSP500 +100KNNs +2.58% +18s +0.92% +3m +0.69% +12m +250KNNs +1.56% +32s +0.67% +9m +0.53% +30m +Full graph +2.34% +7m +1.31% +1.8h +1.19% +7.3h +TSP1000 +100KNNs +3.34% +25s +1.69% +6m +1.45% +24m +250KNNs +2.53% +1m +1.43% +23m +1.19% +1.4h +Full graph +4.80% +5s +2.42% +1m +1.82% +5m +CVRP200 +100KNNs +5.18% +3s +2.12% +33s +1.68% +3m +Full graph +4.74% +1m +2.10% +15m +1.59% +1h +CVRP500 +100KNNs +5.14% +18s +2.02% +3m +1.74% +12m +250KNNs +4.58% +32s +1.86% +9m +1.14% +30m +Full graph +8.00% +7m +3.19% +1.8h +2.39% +7.3h +CVRP1000 +100KNNs +8.25% +25s +4.76% +6m +3.58% +24m +250KNNs +7.51% +1m +3.08% +23m +2.28% +1.4h +Table 3: Improving the model performance using a k-nearest-neighbor heuristic. +C +IMPROVING THE MODEL PERFORMANCE WITH A k-NEAREST-NEIGHBOR +HEURISTIC +Our decoding strategy could be further improved by using a k-nearest-neighbor heuristic to restrict +the search space and reduce the inference time. For both greedy and beam search strategies, at +every step, it is possible to reduce the remaining graph by considering only a certain number of +neighbouring nodes. Table 3 presents the experiments on TSP and CVRP where we apply the model +just on a certain number on nearest neighbours of the origin. This approach clearly reduces the +execution time, but also in some cases even improves the performance in terms of optimality gap. +The same heuristic can be applied on Knapsack problem, where model could be applied just on a +certain number of items with highest values. +D +ABLATION STUDY +D.1 +TRANSFORMER VS HYPERMIXER AS MODEL +In Section 6 we have shown that our model has an excellent generalization ability to graphs of +larger size. In Section ??, we hypothesize that this has to do with the fact that a subproblem of +size t spends O(t2) computation operations due to the quadratic complexity of the Transformer +encoder’s self-attention component, which is responsible for mixing node representations. To test +this hypothesis, we experiment with replacing self-attention with an efficient mixing component (see +Tay et al. (2022) for an overview), namely the recent linear-time HyperMixer (Mai et al., 2022). We +chose this model because it does not assume that the input is ordered, unlike e.g. sparse attention +alternatives. +15 + +Published as a conference paper at ICLR 2023 +Seed +TSP100 +TSP200 +TSP500 +TSP1000 +1 +2.10% +8.38% +34.91% +71.30% +2 +1.38% +3.54% +98.59% +628.71% +3 +1.93% +4.14% +120.18% +216.77% +4 +1.37% +4.54% +46.23% +104.85% +5 +1.25% +3.66% +61.99% +524.43% +Table 4: Experimental results on TSP with HyperMixer for five different seeds. +Experimental Details +For comparability, we set the model and training parameters to the same as +for Transformers, so the experiments only differ in token mixing component that is used. The only +other difference is that we used Layer Normalization Ba et al. (2016) instead of ReZero Bachlechner +et al. (2021), which leads to more stable training for HyperMixer. Since we observed relatively large +sensitivity to model initialization, we are reporting the results for 5 different seeds. +Results +Table 4 shows the result for HyperMixer with greedy decoding. While the model reaches +lower but acceptable performance than Transformers on TSP100, it generalizes poorly to instances +of larger size. Moreover, performance is very sensitive to the seed. These results suggest that the +computation spent by self-attention is indeed necessary to reach the generalization ability of our +model, which increases the compute with the size of the (sub)problem. +D.2 +APPROXIMATED MODEL +As mentioned in Section 5, existing works have also noted the importance of accounting for the +change of the state after each action: Xin et al. (2021b; 2020) claimed that models should recompute +the embeddings after each action. However because of the additional training cost, they proposed +the following approximation: fixing lower encoder levels and recomputing just the top level with a +mask of already visited nodes. They hypothesis a kind of hierarchical feature extraction property +that may make the last layers more important for the fine-grained next decision. In contrast, we call +our entire model after each construction step; effectively recomputing the embeddings of each state. +We hypothesis that this property may explain the superior performance (Table 1) w.r.t MDAM model +Xin et al. (2020). In order to support this hypothesis, we have implemented an approximated version +of our model as follows. We fixed the bottom layers of our model and recomputed just the top layer, +by masking already visited nodes and adding the updated information (origin and destination tokens +for TSP). As expected, inference time is 1.6 times shorter, but performance is severely degraded: we +obtained optimality gap of 9.833% (vs 0.540% with original model) on TSP100. +D.3 +REZERO VS BATCHNORM AS NORMALIZATION +Most NCO works that use transformer networks (Kool et al., 2019)(Kwon et al., 2021)(Xin et al., +2020) use batch normalization(Ioffe & Szegedy, 2015) rather than layer normalization (Ba et al., +2016) in attention layers. We find ReZero normalization (Bachlechner et al., 2021) to work even +better. Figure 3 shows the effect of using ReZero compared to batch normalization in our Trans- +former network. Using it leads to more stable training, better performance, and drastically lower +variance between seeds. +E +ON THE IMPACT OF EXPERT SOLUTIONS +Our datasets consist of pairs of a problem instance and a solution (tour). On the other hand, in this +paper, we use imitation learning, which requires instead pairs of a problem instance and (expert) +trajectory in the MDP. Now, a solution may be obtained from multiple trajectories in the MDP. For +example, with TSP, a solution is a loop in a graph, and one has to decide at which node its construc- +tion started and in which direction it proceeded. With CVRP, the order in which the subtours are +constructed needs also to be decided. Hence, all our datasets are pre-processed to transform solu- +tions into corresponding construction trajectories (a choice for each or even all possible ones). We +experimentally observed that this transformation has an impact on the performance. For example, +with CVRP, choosing, for each solution, the construction in the order in which LKH3 displays it, +16 + +Published as a conference paper at ICLR 2023 +0 +200 +400 +600 +800 +1000 +Epochs +0 +5 +10 +15 +20 +Optimality gap +BatchNorm, seed 0 +BatchNorm, seed 1 +ReZero, seed 0 +ReZero, seed 1 +Figure 3: Training curves showing the effect of the choice of normalization layer on validation +performance +which does not seem arbitrary, yields to 1.3 point better opt-gap performance compared to following +a random ordering of the sub-tours. We hypothesize that if there is any bias in the display of the +optimal solution - for example, shorter tour first, or closest node first - it requires slightly less model +capacity to learn action imitation for this display rather than for all possible displays. +F +PROOF OF PROPOSITION 1 (SOUNDNESS OF THE NAIVE MDP) +We show here that procedure SOLVE satisfies SOLVE(f)= arg minx∈X f(x). We first show the +following general lemma: +Let Y +ψ→X +f→R∪{∞} be arbitrary mappings, if ψ is surjective then +arg min +x∈X f(x) = ψ(arg min +y∈Y f(ψ(y))) +Simple application of the definition of arg min (as a set). The subscript ∗ denotes the steps where +the assumption that ψ is a surjection is used: +x′ ∈ ψ(arg min +y +f(ψ(y))) +iff +∃y′ ∈ arg min +y +f(ψ(y)) x′ = ψ(y′) +iff +∃y′ x′ = ψ(y′) ∀y f(ψ(y′)) ≤ f(ψ(y)) +iff +∃y′ x′ = ψ(y′) ∀y f(x′) ≤ f(ψ(y)) +iff∗ +∀y f(x′) ≤ f(ψ(y)) +iff∗ +∀x f(x′) ≤ f(x) +iff +x′ ∈ arg min +x f(x) +Let (F, X) be a CO problem with specification ⟨T , SOL, VAL⟩ and M the naive MDP obtained from +it. For each f∈F, let vf=VAL(f, ϵ), Xf={x∈X|f(x)<∞} and let Yf be the set of M-trajectories +which start at (f, ϵ) and end at a stop state. +17 + +Published as a conference paper at ICLR 2023 +... +Transformer encoder +activation = ReLU +normalization = ReZero +input embedding layer +Linear +softmax +dest. +emb. +origin +emb. ++ ++ +... +... +Figure 4: Computation flow at the t-th time step, when a partial solution of length t − 1 already +exists. The input state consist of the destination node (i.e. the first and last node in the TSP tour), +the origin node (i.e., the most recent node in the tour), and the set of remaining nodes. After passing +all input nodes through an embedding layer, we add special, learnable vector embeddings to the +origin and current node to signal their special meaning. Finally, a Transformer encoder followed by +a linear classifier head selects the next node at step t. +• For any M-trajectory τ=s0t1r1s1 · · · tnrnsn in Yf, define ψ(τ) =def SOL(sn). Since +τ∈Yf, we have s0=(f, ϵ) and sn is a stop state, i.e. SOL(sn)=ψ(τ)∈X, and by Equa- +tion 2a, f(ψ(τ))<∞. Hence ψ:Yf �→ Xf. +• By construction, sm=(f, t1:m) for all m∈1:n and each transition in τ has a finite reward +VAL(sm−1)−VAL(sm) (condition for it to be valid). Hence the cumulated reward is given +by R(τ)=VAL(s0)−VAL(sn). Now, VAL(s0)=vf which is independent of τ and by Equa- +tion 2c, VAL(sn)=f(ψ(τ)). Hence f(ψ(τ))=vf−R(τ). +• Let’s show that ψ is surjective. Let x∈Xf. Equation 2a ensures that x=SOL(f, t1:n) +for some t1:n∈T ∗. +For each m∈{0:n}, let sm=(f, t1:m) and consider the sequence +τ=s0t1r1s1 · · · tnrnsn. Now, SOL(sn)=x̸=⊥ hence τ ends in a stop state and starts at +(f, ϵ). By Equation 2c we have VAL(sn)=f(x), hence VAL(sn)<∞, and VAL(sm)<∞ for +all m∈{0:n−1}. And by Equation 2b SOL(sm)=⊥, hence all the transitions in τ are valid +in M. Hence τ∈Yf and by definition, ψ(τ)=x. +Therefore we can apply the lemma proved above: +arg min +x∈Xf f(x) = ψ(arg min +τ∈Yf f(ψ(τ))) = ψ(arg min +τ∈Yf vf−R(τ)) += ψ(arg max +τ∈Yf R(τ)) = ψ(SOLVEMDP +M (f, ϵ)) = SOLVE(f) +Now, obviously, arg minx∈X f(x) = arg minx∈Xf f(x), since by definition f is infinite on X\Xf. +18 + +Published as a conference paper at ICLR 2023 +G +PLOTS OF SOME TSPLIB AND CVRPLIB SOLUTIONS +(a) Optimal solution +(b) Our model (BS16), opt_gap 0.549% +(c) MDAM (BS50), opt_gap 11.501% +(d) POMO (x8), opt_gap 18.614% +Instance pcb442 +(a) Optimal solution +(b) Our model (BS16), opt_gap 4.253% +(c) MDAM (BS50), opt_gap 20.916% +(d) POMO (x8), opt_gap 44.664% +Instance pr1002 +19 + +Published as a conference paper at ICLR 2023 +(a) Optimal solution +(b) Our model (BS16), opt_gap 3.464% +(c) MDAM (BS50), opt_gap 45.669% +(d) POMO (x8), opt_gap 11.416% +Instance X-n284-k15 +(a) Best known solution +(b) Our model (BS16), opt_gap 2.667% +(c) MDAM (BS50), opt_gap 19.739% +(d) POMO (x8), opt_gap 46.603% +Instance X-n513-k21 +20 + +Published as a conference paper at ICLR 2023 +H +BACKGROUND ON BISIMULATION-BISIMILARITY +H.1 +BISIMULATION IN LABELLED TRANSITION SYSTEMS +Bisimulation is a very broad concept which applies to arbitrary Labelled Transition Systems (LTS). It +has been declined in various flavours of LTS, such as Process Calculi, Finite State Automata, Game +theory, and of course MDP (initially deterministic MDP such as those used here, later extended +to stochastic MDP which we are not concerned with here). A bisimulation is a binary relation R +among states which “commutes” with the transitions of the LTS in the following diagram, which +should informally be read as follows: if the pair of arrows connected to p (resp. q) exists then so +does the “opposite” pair (w.r.t. the centre of the diagram). +p +q +p′ +q′ +ℓ +ℓ +R +R +The notation p +ℓ +−−→ p′ means the transition from p to p′ with label ℓ is valid. Thus, formally, +Definition 1. A binary relation R on states is a bisimulation if for all label ℓ and states p, q such +that pRq +∀p′ s.t. p +ℓ +−−→ p′ ∃q′ s.t. q +ℓ +−−→ q′ , p′Rq′ +∀q′ s.t. q +ℓ +−−→ q′ ∃p′ s.t. p +ℓ +−−→ p′ , p′Rq′ +Note that this definition is extended to the “heterogeneous” case where R is bi-partite, relating +the state spaces of two LTS L1, L2 sharing the same label space. One just forms a new LTS L +whose state space is the disjoint union of the state spaces of L1, L2 and the transitions are those of +L1, L2 in their respective (disjoint) component. An heterogeneous bisimulation on L1, L2 is then a +(homogeneous) bisimulation on L. Most results below also have a heterogeneous version. +Proposition 2. The set of bisimulations (subset of the set of binary relations on states) is stable by +union, composition, and inversion, hence also by reflexive-symmetric-transitive closure. +In particular, the union of all bisimulations, called the bisimilarity of the LTS, is itself a bisimulation, +and it is also an equivalence relation. +Proof. (outline) Let’s detail stability by composition, the other cases are similarly obvious. +If +R1, R2 are the two bisimulations being composed, apply the commutation property to each cell +of the following diagram (from top to bottom). +p +r +q +p′ +r′ +q′ +ℓ +ℓ +ℓ +R1 +R2 +R1 +R2 +Definition 2. Given an LTS L, its transitive closure is another LTS denoted L∗ on the same state +space, where the labels are the sequences of labels of L and the transitions are defined by +p +ℓ1:n +−−−−→ +(L∗) +p′ +if +∃p0:n such that p = p0 +ℓ1 +−−−→ +(L) +p1 · · · +ℓn−1 +−−−−→ +(L) +pn−1 +ℓn +−−−→ +(L) +pn = p′ +Proposition 3. If R is a bisimulation on L, then it is also a bisimulation on L∗. +Proof. (outline) This is essentially shown by successively applying the commutation property to +each cell of the following diagram (from left to right): +21 + +Published as a conference paper at ICLR 2023 +p0 +q0 +p1 +q1 +pn−1 +qn−1 +pn +qn +ℓ1 +ℓn +ℓ1 +ℓn +R +R +R +R +Definition 3. Given an LTS L and an equivalence relation R on its state space, we can define the +quotient LTS L/R with the same label space, where the states are the R-equivalence classes and +the transitions are defined, for any classes ˙p, ˙p′, by +˙p +ℓ +−−−−→ +L/R +˙p′ +if +∀p ∈ ˙p ∃p′ ∈ ˙p′ +p +ℓ +−−→ +L +p′ +Proposition 4. Let R be an equivalence on the state space of L. R is a bisimulation on L if and +only if ∈ is a (heterogeneous) bisimulation on L, L/R. +Proof. We show both implications: +• Assume R is a bisimulation on L. +– Let p ∈ ˙q and p +ℓ−→ p′. Let q ∈ ˙q. Hence pRq and p +ℓ−→ p′. Since R is a bisimulation, +there exists q′ such that q +ℓ−→ q′ and p′Rq′. Hence for all q ∈ ˙q, there exists q′ ∈ ¯p′ +such that q +ℓ−→ q′. Hence by definition ˙q +ℓ−→ ¯p′ while p′ ∈ ¯p′. +– Let p ∈ ˙q and ˙q +ℓ−→ ˙q′. Hence by definition, there exists p′ ∈ ˙q′ such that p +ℓ−→ p′. +• Assume ∈ is a (heterogeneous) bisimulation on L, L/R. +– Let pRq and p +ℓ−→ p′. Hence p ∈ ¯q and p +ℓ−→ p′. Since ∈ is a bisimulation, there exists +˙q′ such that p′ ∈ ˙q′ and ¯q +ℓ−→ ˙q′. Now q ∈ ¯q, hence, by definition, there exists q′ ∈ ˙q′ +such that q +ℓ−→ q′. And p′Rq′ since p′, q′ ∈ ˙q′. +– Let pRq and q +ℓ−→ q′. Hence qRp and q +ℓ−→ q′, and we are in the previous case up to a +permutation of variables. +Proposition 5. Let R be an equivalence relation on the state space of L. If R is a bisimulation on +L, then for any L-state p, L/R-state ˙p and L∗-label ℓ +¯p +ℓ +−−−−−−→ +(L/R)∗ +˙p′ +if and only if +∃p′ ∈ ˙p′ +p +ℓ +−−→ +L∗ +p′ +Proof. Simple combination of Propositions 4 and 3. R is a bisimulation on L, hence ∈ is a het- +erogeneous bisimulation on L, L/R (Proposition 4), hence also a heterogeneous bisimulation on +L∗, (L/R)∗ (Proposition 3, heterogeneous version). +• If ¯p +ℓ +−−−−−−→ +(L/R)∗ +˙p′, since p∈¯p and ∈ is a bisimulation, we have p +ℓ +−−→ +L∗ +p′ for some p′∈ ˙p′. +• Conversely, if p +ℓ +−−→ +L∗ +p′ for some p′∈ ˙p′, since p∈¯p and ∈ is a bisimulation, we have +¯p +ℓ +−−−−−−→ +(L/R)∗ +˙q′ and p′∈ ˙q′ for some ˙q′. Now p′∈ ˙p′∩ ˙q′ hence ˙p′= ˙q′ and ¯p +ℓ +−−−−−−→ +(L/R)∗ +˙p′. +H.2 +BISIMULATION IN DETERMINISTIC MDP +Definition 4. An MDP is a pair (L, ⊤) where L is a LTS with label space A × R for some action +space A (action-reward pairs denoted a|r) and ⊤ is a subset of states (the stop states). It is said to +be deterministic if +if s +a|r1 +−−→ s′ +1 and s +a|r2 +−−→ s′ +2 then r1 = r2 and s′ +1 = s′ +2 +22 + +Published as a conference paper at ICLR 2023 +Given an L-trajectory τ, i.e. a sequence s0a1r1s1 · · · anrnsn where si−1 +ai|ri +−−−→ si for all i∈{1:n}, +its cumulated reward is defined by R(τ)= �n +i=1 ri. The generic problem statement of the MDP +solution framework is, given an MDP (L, ⊤) and one of its states so, to solve the following optimi- +sation: +SOLVEMDP((L, ⊤), so) = arg max +τ +R(τ) | τ is a L-trajectory starting at so and ending in ⊤ +This definition of MDP and the standard textbook one coincide only in the deterministic case (in +the standard definition, an MDP is deterministic if the distribution of output state-reward pairs for a +given input state and allowed action is “one-hot”). The non deterministic case in the definition above +does not match the standard definition: it would be wrong to interpret two distinct transitions for the +same input state s and action a as meaning that the outcome of applying a to state s is distributed +between the two output reward-state pairs according to a specific distribution (e.g. uniform). Also, +in the problem statement, the objective R(τ) has no expectation, which, with the standard definition, +only makes sense in the case of a deterministic MDP. Similarly, the standard problem statement is +expressed in terms of policies rather than trajectories directly, but in the deterministic case, the two +are equivalent. Observe that there is a one-to-one correspondence between trajectories in L and +transitions in the LTS L∗, so the problem statement can be formulated equivalently as +SOLVEMDP((L, ⊤), so) = arg max +ℓ +R(ℓ) | ∃s ∈ ⊤, so +ℓ +−−→ +L∗ +s +(4) +Proposition 6. Let (L, ⊤) be an MDP and R an equivalence relation on its state space. +1. (L/R, ¯⊤) is also an MDP, where ¯⊤={¯s|s∈⊤}, and if L is deterministic, so is L/R. +2. If R is a bisimulation on L preserving ⊤ (i.e. � +s∈⊤ ¯s = ⊤), then for any state so and label +ℓ in L∗ we have +∃s ∈ ⊤, so +ℓ +−−→ +L∗ +s +if and only if +∃ ˙s ∈ ¯⊤, ¯so +ℓ +−−−−−−→ +(L/R)∗ +˙s +Proof. The second property is a direct consequence of Proposition 5 and the assumption that ⊤ is +preserved by R. For the first, assume that L is deterministic. Let ˙s, ˙s1, ˙s2 be L/R states, such that +˙s +a|r1 +−−→ ˙s1 and ˙s +a|r2 +−−→ ˙s2. Choose s ∈ ˙s. Hence, by definition, there exist s1∈ ˙s1 and s2∈ ˙s2 such +that s +a|r1 +−−→ s1 and s +a|r2 +−−→ s2. Since L is deterministic, we have r1=r2 and s1=s2∈ ˙s1∩ ˙s2, hence +˙s1 = ˙s2. Hence L/R is also deterministic. +Therefore, when R is a bisimulation equivalence on L preserving ⊤, the generic MDP problem +statement of Eq. equation 4 can be reformulated as +SOLVEMDP((L, ⊤), so) = SOLVEMDP((L/R, ¯⊤), ¯so) = arg max +ℓ +R(ℓ) | ∃ ˙s ∈ ¯⊤, ¯so +ℓ +−−−−−−→ +(L/R)∗ +˙s +(5) +Note that a bisimulation on L preserving ⊤ is simply a bisimulation on the LTS ˙L defined as follows: +˙L has the same state space as L and an additional transition s +·−→ s for each s∈⊤, where “·” is a +distinguished label not present in L. +A bisimulation R on ˙L captures some symmetries of the state space of ˙L. If R is taken to be the +bisimilarity of ˙L, i.e. the union of all the bisimulations on ˙L, i.e. the union of all the bisimulations +on L preserving ⊤, then it captures all the possible symmetries of the state space. This should be +seen as an asymptotic result, since constructing and working with the full bisimilarity of ˙L is not +feasible. But Proposition 6 remains valuable as it applies to all bisimulation, not just the maximal +bisimulation of ˙L (its bisimilarity). +23 + diff --git a/9NE1T4oBgHgl3EQfnwSt/content/tmp_files/load_file.txt b/9NE1T4oBgHgl3EQfnwSt/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..29f962262227f1bf0682603e90e669967fa1815c --- /dev/null +++ b/9NE1T4oBgHgl3EQfnwSt/content/tmp_files/load_file.txt @@ -0,0 +1,1065 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf,len=1064 +page_content='Published as a conference paper at ICLR 2023 BQ-NCO: BISIMULATION QUOTIENTING FOR GENER- ALIZABLE NEURAL COMBINATORIAL OPTIMIZATION Darko Drakulic1 Sofia Michel1 Florian Mai2,* Arnaud Sors1 Jean-Marc Andreoli1 1 NAVER Labs Europe {firstname.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='lastname}@naverlabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='com 2 Idiap Research Institute and EPFL florian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='mai@idiap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='ch Work was done as part of an internship at NAVER Labs Europe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' ABSTRACT Despite the success of Neural Combinatorial Optimization methods for end-to- end heuristic learning, out-of-distribution generalization remains a challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' In this paper, we present a novel formulation of combinatorial optimization (CO) problems as Markov Decision Processes (MDPs) that effectively leverages sym- metries of the CO problems to improve out-of-distribution robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Starting from the standard MDP formulation of constructive heuristics, we introduce a generic transformation based on bisimulation quotienting (BQ) in MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' This transformation allows to reduce the state space by accounting for the intrinsic symmetries of the CO problem and facilitates the MDP solving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' We illustrate our approach on the Traveling Salesman, Capacitated Vehicle Routing and Knapsack Problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' We present a BQ reformulation of these problems and introduce a sim- ple attention-based policy network that we train by imitation of (near) optimal solutions for small instances from a single distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' We obtain new state-of- the-art generalization results for instances with up to 1000 nodes from synthetic and realistic benchmarks that vary both in size and node distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' 1 INTRODUCTION Combinatorial Optimization problems are crucial in many application domains such as transporta- tion, energy, logistics, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Because they are generally NP-hard (Cook et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=', 1997), their resolution at real-life scales is mainly done by heuristics, which are efficient algorithms that generally produce good quality solutions (Boussa¨ıd et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' However, strong heuristics are generally problem- specific and designed by domain experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Neural Combinatorial Optimization (NCO) is a relatively recent line of research that focuses on using deep neural networks to learn such heuristics from data, possibly exploiting information on the specific distribution of problem instances of interest (Bengio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Cappart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Despite the impressive progress in this field over the last few years, their out-of-distribution generalization, especially to larger instances, remains a major hurdle (Joshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Manchanda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' In this paper, we are interested in constructive NCO methods, which build a solution incrementally, by applying a sequence of elementary steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' These methods are often quite generic, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' the seminal papers by Khalil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Kool et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Most CO problems can indeed be rep- resented in this way, although the representation is not unique as the nature of the steps is, to a large extent, a matter of choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Given a choice of step space, solving the CO problem amounts to computing an optimal policy for sequentially selecting the steps in the construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' This task can typically be performed in the framework of Markov Decision Processes (MDP), through imitation or reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' The exponential size of the state space, inherent to the NP-hardness of combinatorial problems, usually precludes other methods such as (tabular) dynamic programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Whatever the learning method used to solve the MDP, its efficiency, and in particular its out-of- distribution generalization capabilities, greatly depends on the state representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' The state space is often characterized by deep symmetries, which, if they are not adequately identified and lever- aged, hinders the training process by forcing it to independently learn the policy at states which in fact are essentially the same (modulo some symmetry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='03313v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='LG] 9 Jan 2023 Published as a conference paper at ICLR 2023 In this work, we investigate a type of symmetries which often occurs in MDP formulations of con- structive CO heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' We first introduce a generic framework to systematically derive a naive CO problem-specific MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' We formally demonstrate the equivalence between solving the MDP and solving the CO problem and highlight the flexibility of the MDP formulation, by defining a mini- mal set of conditions for the equivalence to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Our framework is general and easy to specialize to encompass previously proposed learning-based construction heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' We then show that the state space of this naive MDP is inefficient because it fails to capture deep symmetries of the CO problem, even though such symmetries are easy to identify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Therefore, we propose a method to transform the naive MDP, based on the concept of bisimulation quotienting (BQ), in order to get a reduced state space, which is easier to process by the usual (approximate) MDP solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' We il- lustrate our approach on three well-known CO problems, the Traveling Salesman Problem (TSP), the Capacitated Vehicle Routing Problem (CVRP) and Knapsack Problem (KP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Furthermore, we propose a simple transformer-based architecture for these problems, that we train by imitation of expert trajectories derived from (near) optimal solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' In particular, we show that our model is well-suited for our BQ formulation: it spends a monotonically increasing amount of computation as a function of the subproblem size (and therefore complexity), in contrast to most previous models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Finally, extensive experiments confirm the validity of our approach, and in particular its state-of-the- art out-of-distribution generalization capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' In summary, our contributions are as follows: 1) We present a generic and flexible framework to define a construction heuristic MDP for arbitrary CO problems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' 2) We propose a method to simplify commonly used “naive” MDPs for constructive NCO via symmetry-focused bisimulation quotienting;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' 3) We design an adequate transformer-based archi- tecture for the new MDP, for the TSP, CVRP and KP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' 4) We achieve state-of-the-art generalization performance on these three problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' 2 COMBINATORIAL OPTIMIZATION AS A MARKOV DECISION PROBLEM In this section, we present a generic formalization of constructive heuristics which underlies their MDP formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' A deterministic CO problem is denoted by a pair (F, X), where F is its ob- jective function space and X its (discrete) solution space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' A problem instance f∈F is a mapping f:X→R∪{∞}, with the convention that f(x)=∞ if x is infeasible for instance f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' A solver for problem (F, X) is a functional: SOLVE : F → X satisfying SOLVE(f) = arg min x∈X f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' (1) Incremental solution construction Constructive heuristics for CO problems build a solution se- quentially, starting at an empty partial solution and expanding it at each step until a finalized solution is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Many NCO approaches are based on a formalization of that process as an MDP, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Khalil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Kool et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Such an MDP can be obtained, for an arbitrary CO problem (F, X), using the following ingredients: Steps: T is a set of available steps to construct solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' A partial solution is a pair (f, t1:n) of a problem instance f∈F and a sequence of steps t1:n∈T ∗ (the set of sequences of elements of T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Observe that a partial solution (in F×T ∗) is not a solution (in X), but may represent one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Representation: SOL:F×T ∗→X∪{⊥} is a mapping that takes a partial solution and returns ei- ther a feasible solution (in which case the partial solution is said to be finalized), or ⊥ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Evaluation: VAL:F×T ∗→R∪{∞} is a mapping that takes a partial solution and returns an esti- mate of the minimum value of its expansions into finalized solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' If the returned value is finite, the partial solution is said to be admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' In order to define an MDP using these ingredients, we assume they satisfy the following axioms: ∀f∈F, x∈X, f(x) < ∞ ⇔ ∃t1:n ∈ T ∗ such that SOL(f, t1:n) = x, (2a) ∀f∈F, t1:n∈T ∗, SOL(f, t1:n) ̸= ⊥ ⇒ ∀m∈{1:n−1}, SOL(f, t1:m) = ⊥, (2b) ∀f∈F, t1:n∈T ∗, x∈X, SOL(f, t1:n) = x ⇒ � VAL(f, t1:n) = f(x), ∀m ∈ {1:n−1}, VAL(f, t1:m) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' (2c) Equation 2a states that the feasible solutions are exactly those represented by a finalized partial solution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Equation 2b states that if a partial solution is finalized then none of its preceding partial solutions in the construction can also be finalized;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Equation 2c states that the evaluation of a finalized 2 Published as a conference paper at ICLR 2023 partial solution is the value of the solution it represents, and all its preceding partial solutions are admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' We call a triplet ⟨T , SOL, VAL⟩ satisfying the above axioms a specification of problem (F, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Note that a specification is not intrinsic to the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' The step space T results from a choice of how to construct a solution sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Once T is chosen, SOL is determined, and must satisfy Equation 2a and 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Then VAL is only loosely constrained by Equation 2c, and can be chosen among a wide range of alternatives, including the following straightforward, uninformed one and the ideal, but intractable one (and, more likely, somewhere in between these two extremes): VALuninformed(f, t1:n) =def f(x) if [SOL(f, t1:n) = x ̸= ⊥] else 0, VALideal(f, t1:n) =def min{f(x)|x ∈ X, ∃u1:m ∈ T ∗ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' SOL(f, t1:nu1:m) = x}, with the convention min ∅=∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Value 0 in the uninformed case can be replaced by any constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Solution construction as an MDP Using a specification ⟨T , SOL, VAL⟩ of problem (F, X), one can derive a “naive” MDP as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' States are partial solutions (in F×T ∗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' actions are steps (in T );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' a state is terminal if it is a finalized partial solution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' transitions: action u∈T applied to a non-terminal state (f, t1:n) leads to state (f, t1:nu) where u is appended to the sequence so far, with reward VAL(f, t1:n)−VAL(f, t1:nu), conditioned on VAL(f, t1:nu) being finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Note that VAL has the double role of providing a reward and specifying the set of allowed actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' The number of these is expected to be linear, or at worst polynomial, in the size of the instance, since picking a step should not be as complex as solving the whole problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Now, assume we have access to a generic solver SOLVEMDP, which, given an MDP M and one of its states so, returns an optimal trajectory starting at that state, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' arg maxτ R(τ) where τ ranges over the M-trajectories starting at so and ending in a terminal state, and R(τ) denotes its cumulated reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Note that because we are dealing with deterministic MDPs, looking for an optimal policy is the same as looking for an optimal trajectory for a given set of initial states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' That is why SOLVEMDP is defined here directly in terms of trajectories rather than policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' SOLVEMDP can then be specialized into a solver for the specific CO problem (F, X): Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Let Mo be the naive MDP obtained from specification ⟨T , SOL, VAL⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' The proce- dure defined as follows (where ϵ denotes the empty sequence) satisfies the requirement of equation 1: SOLVE(f ∈ F) =def {SOL(s)|s is the last state of the trajectory SOLVEMDP(Mo, (f, ϵ))}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' In other words, solving the naive MDP is equivalent to solving the CO problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' The detailed proof of Proposition 1 is in Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Of course, procedure SOLVEMDP may be approximate, in which case so is procedure SOLVE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Moreover, its performance depends on that of SOLVEMDP, esp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' its out- of-distribution generalization capacity, but also on the choice of specification, esp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' of action space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' It is a distinguishing feature of CO from an MDP perspective that the action space is not prescribed by the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' The impact of the choice of the VAL mapping depends on the type of learning used by SOLVEMDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' When SOLVEMDP learns by reinforcement, VAL is essential, as it provides the rewards which guide the resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' For example, VALuninformed leads to the notoriously hard case of sparse rewards, while VALideal (were it tractable) would lead to the trivial case where a myopic policy (greedy in its immediate reward) is optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Although we do not provide a generic method to design VAL, we argue that there are natural candidates, typically based on extending the objective function to partial solutions (not just finalized ones).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' When SOLVEMDP learns by imitation instead, the choice of VAL has a much more limited impact: it only serves to define the allowed actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' The critical factor in that case is the construction of the training dataset of expert trajectories to imitate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Example on TSP Consider the widespread CO problem known as the Traveling Salesman Prob- lem (TSP) in a Euclidian space V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' A TSP solution (in X) is a path, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' a finite sequence of pairwise distinct nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' A TSP instance (in F) is given by a finite set D of nodes as points in V , and maps any solution (path) to the length of that path (closed at its ends) if it visits exactly all the nodes of D, and ∞ otherwise (infeasible solutions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' A simple specification ⟨T , SOL, VAL⟩ for the TSP is given by: the step space T is the set of nodes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' for a given instance f and sequence t1:n of steps, SOL(f, t1:n) is either the sequence t1:n if it forms 3 Published as a conference paper at ICLR 2023 u a v u a v u a v step|u−(a+v) step|u−(a+v) ≡Φ ≡Φ step|u−(a+v) Figure 1: An example of bisimulation commutation in TSP-MDP, and the corresponding path-TSP- MDP transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' The step is the same in all three transitions: it is the end node of the dashed arrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' And the reward is also the same: it depends only on the distances a, u, v, and not on any of the previously visited nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' a path which visits exactly all the nodes of f, or ⊥ otherwise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' and VAL(f, t1:n) is either the length of path t1:n (closed at its ends) if it forms a path which visits only nodes of f (maybe not all), or ∞ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' It is easy to show that we thus obtain a specification (as defined by the axioms above) of the TSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' In TSP-MDP, the naive MDP obtained from it, the reward of taking action u (a node) at state (f, t1:n) is δf(tn, t1)−(δf(tn, u)+δf(u, t1)) where δf is the node distance measured on the corresponding points of V in f, conditioned on t1:nu being pairwise distinct nodes of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Observe that when allowed, the reward depends only on the start and end nodes t1, tn of the step sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' 3 BISIMULATION QUOTIENTING FOR COMBINATORIAL OPTIMIZATION In our context of deterministic CO problems and therefore deterministic MDPs, the general notion of bisimilarity is simplified (Givan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=', 2003): two states are said to be bisimilar if they spawn exactly the same action-reward sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Likewise, the notion of a binary relation R on states being a bisimulation reduces to a commutation between the (deterministic) transitions of the MDP and that relation: if s1Rs2 and action a applied to state s1 leads to state s′ 1 with reward r, then action a applied to state s2 leads to a state s′ 2 with the same reward r, and s′ 1Rs′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' An illustration is given in Fig 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Bisimilarity is equivalently defined as the largest bisimulation (see Appendix H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Bisimilarity-induced symmetries In the naive MDP obtained from a specification of a given CO problem, a state is a partial solution and carries the whole information about the “past” decisions (steps) leading to it, which may not all be useful for the “future” decisions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' the completion of that partial solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Consider for example the following two states in TSP-MDP, in which the sequence of steps of the partial solution is represented as a directed path in red among some of the problem instance nodes: s1 s2 Observe that s1, s2 differ only in the inner nodes of the red path (black diamond-shaped nodes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Now, it is easy to see that the successful completions of these two partial solutions are identical: they each consist of a path visiting the (same) unvisited (blue) nodes, starting at the end node of the red path and ending at its start node, with the same rewards defined by VAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Consequently, in the MDP, the two states s1, s2 spawn exactly the same action-reward sequences and form a bisimilar pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' This is the kind of deep symmetries of the problem which we want the MDP to leverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Of course, there exist other kinds of symmetries, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' rotational symmetries: if s2 is obtained from s1 by applying an isometric transformation to all the points in the problem instance, then s1, s2 also form a bisimilar pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' However, the latter symmetry is specific to the Euclidian version of the TSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' We focus here on the former kind of symmetry as it is more general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Although it has previously been noted for routing problems (Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Xin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=', 2021b), we show here that it is an inherent characteristic of constructive CO approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' 4 Published as a conference paper at ICLR 2023 Bisimulation quotienting A classical result on MDPs (Givan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=', 2003) states that all such symmetries in any MDP can be leveraged by quotienting it by its bisimilarity relation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' the set of all bisimilar pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Of course, there is no free lunch: constructing the bisimilarity of an MDP is in general intractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Still, the result remains valuable because it holds for any bisimulation, not just the bisimilarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Therefore one can control the amount of symmetries captured in the quotienting by carefully choosing the bisimulation, trading off its closeness to full bisimilarity for tractability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' We now assume that, for a given CO problem (F, X) we have access not only to a specification ⟨T , SOL, VAL⟩ with its associated naive MDP, but also to a mapping Φ:F×T ∗→ ˆS from partial solutions to some new space ˆS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Typically, Φ(f, t1:n) should capture, within the partial solution (f, t1:n), a piece of information as small as possible but sufficient to determine the set of action- reward sequences it spawns in the MDP – in other words, a summary of its “past” which is sufficient to determine its “future”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' We can then define an equivalence relation ≡Φ where two partial solutions are equivalent if they have the same image by Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' For it to be a bisimulation, Φ must satisfy: ∀s1, s2∈F×T ∗, Φ(s1)=Φ(s2) ⇒ � ∀u∈T , Φ(s1u)=Φ(s2u) and VAL(s1)−VAL(s1u)=VAL(s2)−VAL(s2u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' (3) Under that assumption, we can construct a new MDP (the quotient of the original one by the bisim- ulation) which is equivalent, as far as policy optimization is concerned, to the original one, but captures more symmetries of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' This allows to reduce the state space and should lead to a better performance, whatever the generic MDP solver used afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Furthermore, by construc- tion, the equivalence classes are in one-to-one correspondence with the states in ˆS, so that the new MDP can be formulated on that space directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Application to the TSP, CVRP and KP Let Φ be the mappings from TSP-MDP states (TSP states for short) into new objects called “path-TSP” states, informally described by the following diagram: TSP state path-TSP state Φ The inner nodes (black diamonds) on the red path of visited nodes in the TSP state are removed, leaving only the two ends of the red path which constitute two distinguished nodes in the path-TSP state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Mapping Φ has been designed to satisfy equation 3, so it induces a bisimulation on TSP-MDP (see Figure 1), and TSP-MDP can be turned into an equivalent “path-TSP-MDP” on path-TSP states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' This path-TSP-MDP can be viewed as solving a variant of the TSP known as path-TSP, hence its name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' However it is not the naive MDP for that variant since it forgets as it progresses, while naive MDPs always accumulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' With the CVRP, we define a step as the pair of a node and a binary flag specifying whether that node is reached via the depot or directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' We can define a mapping Φ similarly to the TSP case, except it is not sufficient to summarize the “past” (the visited nodes) by just the two ends of their path: to guarantee equation 3 and the bisimulation property, an additional piece of information must be preserved from the past, namely the remaining capacity at the end of the current path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' For the KP, intuitively, the summary of the “past” is captured by the remaining items and the remaining knapsack capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' This idea can be leveraged to design a bisimulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Formal descriptions of the specifications and bisimulation quotienting for the CVRP and KP are provided in Appendices A and B, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' 4 NEURAL ARCHITECTURE FOR PATH-TSP We now describe our proposed policy network for the path-TSP-MDP above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Figure 4 (Appendix) provides a quick overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' The models for path-CVRP and BQ-KP differ only slightly and are presented in Appendix A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Most neural models for TSP utilize an encoder-decoder architecture, in which the encoder computes a representation of the entire graph once, and the decoder constructs a solution by taking into consideration the representation of the whole graph 5 Published as a conference paper at ICLR 2023 and the partial solution, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Attention Model (Kool et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=', 2019), or PointerNetworks (Vinyals et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' In our case, the path-TSP formulation allows us to forget the nodes in the graph that have already been visited, except the distinguished origin and destination nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' As a corollary, it also requires re-encoding the remaining nodes at each prediction step – hence removing the need for a separate auto-regressive decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' To encode a path-TSP state, we use a Transformer model (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Each node is represented by its (x, y) coordinates, so that the input feature matrix for an N-node state is an N×2 matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' We embed these features via a linear layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' The remainder of the encoder is based on Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' (2017) with the following differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' First, we do not use positional encoding since the input nodes have no order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Instead, we learn an origin (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' destination) embedding that is added to the feature embedding of the origin (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' destination) node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Second, we use ReZero (Bachlechner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=', 2021) normalization, which leads to more stable training and better performance in our experiments (see ablation study in Appendix D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Finally, a last linear layer projects the encoder’s output into a vector of size N, from which unfea- sible actions, corresponding to the origin and destination nodes, are masked out, before applying a softmax operator so as to interpret the scalar node values for all allowed nodes as action probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Training We train our model by imitation of expert trajectories, using a plain cross-entropy loss (behaviour cloning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Such trajectories are extracted from pre-computed optimal (or near optimal) solutions for instances of a (relatively small) fixed size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Note that (optimal) solutions are not directly in the form of trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Equation 2a guarantees that a trajectory exists for any solution, but it is usually far from unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Besides, optimal solutions are costly, so we seek to make the most out of each of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' In the TSP case, we observe that given an optimal tour, any sub-path of that tour is also an optimal solution to the associated path-TSP sub-problem, hence amenable to our path-TSP model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' We therefore form minibatches by first sampling a number n between 4 and N (path-TSP problems with less than 4 nodes are trivial), then sampling sub-paths of length n – the same n for all the minibatch entries so as to simplify batching – from the initial solution set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' For the CVRP, the procedure is similar, except that, first, the extracted sub-paths must end at the depot, and, second, they can follow the sub-tours of the full solution in any order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' We observed experimentally that the way that order is sampled has an impact on the performance (see Appendix E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Complexity Because of the quadratic complexity of self-attention, and the fact that we call our model at each construction step, the total complexity is O(N 3)1 where N is the instance size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Note that closely related Transformer-based models such as the TransformerTSP (Bresson & Laurent, 2021) and the Attention Model (Kool et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=', 2019) have a total complexity of O(N 2)2 At each decision step, for t remaining nodes, our model has a budget of O(t2) compute whereas previous models only spend O(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' We believe that this is a useful inductive bias, which enables better generalization in particular for larger problem sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' This hypothesis is supported by the fact that replacing the self-attention component with a linear time alternative (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=', spending O(t) operations per step) drastically degrades the generalization ability to larger instances, as we show in Appendix D, Summary By reformulating TSP-MDP into path-TSP-MDP, the state is made to contain only a very concise summary of the “past” of a partial solution (how it was formed) as two distinguished nodes, but sufficient to determine its “future” (how it can be completed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Furthermore, at train time, we sample optimal solutions and associated path-TSP states amongst all the possible infixes of solutions of full problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' These proposed modifications go hand-in-hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Thanks to the transformation to path-TSP-MDP, our model enables better generalization in two important ways: (i) Due to re- encoding at each step, the encoder produces a graph representation that is specific to the current path-TSP-MDP state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Graphs in these states vary in size and distribution, implicitly encouraging the model to work well across sizes and node distributions, and generalize better than if such variations were not seen during the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' In this regard, our model is similar to the SW-AM model (Xin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=', 2021b), except that they only approximate the re-embedding process in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' (ii) By sampling subsequences from our training instances, we automatically get an augmented dataset, which some previous models had to explicitly design their model for (Kwon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' 1More precisely, the complexity is proportional to �N t=1 t2 = N(N + 1)(2N + 1)/6 hence the O(N 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' 2After an encoder of complexity O(N 2), the decoder has linear complexity O(N − t) at step t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' 6 Published as a conference paper at ICLR 2023 5 RELATED WORK NCO approaches Many NCO approaches construct solutions sequentially, via auto-regressive models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Starting with the seminal work by Vinyals et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' (2015), which proposed the Pointer network that was based on RNNs and trained in a supervised way, Bello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' (2017) trained the same model by RL for the TSP and Nazari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' (2018) adapted it for the CVRP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Kool et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' (2019) introduced an attention-based encoder-decoder architecture (AM) trained with RL to solve several variants of routing problems – which is reused by Kwon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' (2021) along with a few extensions (POMO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' TransformerTSP Bresson & Laurent (2021) use a similar architecture with a different decoder on TSP problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Another line of works is concerned with directly producing a heat-map of solution segments: Nowak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' (2018) trained a Graph Neural Network in a supervised manner to output an adjacency matrix, which is converted into a feasible solution using beam search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Joshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' (2019) followed a similar framework and trained a deep Graph Convolutional Network instead, that was used by (Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=', 2020) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Step-wise methods Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' (2020) first pointed out the limitation of the original AM (Kool et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=', 2019) approach in representing the dynamic nature of routing problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' They proposed to update the encoding after each subtour completion for the CVRP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Xin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' (2021b) proposed a similar step-wise strategy but the encodings recomputed after each decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' In practice, their architecture is the most similar to ours for the TSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' However, thanks to our principled MDP transformations based on bisimulation quotienting, we obtain a superior representation for CVRP: In contrast to our approach, their CVRP architecture only provides censored information by omitting the remaining vehicle capacity and simply restricting the state to the nodes whose demand is below the remaining capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Xin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' (2020) extended on this idea by proposing the Multi-Decoder Attention Model (MDAM) that in particular contains a special layer to efficiently approximate the re-embedding process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' As MDAM constitutes the most advanced version, we employ it as a baseline in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Generalizable NCO Generalization to different instances distributions, and esp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' larger instances, is regarded as one of the major limitations of current NCO approaches (Joshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Mazyavkina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' (2020) trained a Graph Convolution model in a supervised manner on small graphs and used it to solve large TSP instances, by applying the model on sampled subgraphs and us- ing an expensive MCTS search to improve the final solution (Att-GCN+MCTS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' While this method achieves excellent generalization on TSP instances, MCTS requires a lot of computing resources and is essentially a post-learning search strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Geisler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' (2022) investigate the robustness of NCO solvers through adversarial attacks and find that existing neural solvers are highly non-robust to out-of-distribution examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' They conclude that one way to address this issue is through adver- sarial training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' In particular, Xin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' (2021a) trains a GAN to generate instances that are difficult to solve for the current model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Manchanda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' (2022) take a different approach and leverage meta- learning to learn a model in such a way that it is easily adaptable to new distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Accounting for symmetries in a given CO problem is a powerful idea to boost the generalization performance of neural solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Both Kwon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' (2021) and Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' (2022) make use of solution symmetricity as part of their loss function during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Problem instance symmetry can be used at training time to augment the dataset (Kwon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=', 2021) or enforce robust representations (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=', 2022), or it can be used at inference time to augment the set of solutions (Kwon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Please note that all of the above are orthogonal to our approach: rather than augmenting data or changing the training paradigm, our approach simplifies the state space by transforming the MDP, which has beneficial effects irrespective of the method of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' 6 EXPERIMENTS To verify the effectiveness of our method, we test it on TSP, CVRP and KP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' This section presents experimental results for TSP and CVRP, while results for KP are presented in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' We train our model and all baselines on synthetic TSP and CVRP instances of size 100, generated as in Kool et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' We choose graphs of size 100 because it is the largest size for which (near) optimal solutions are still reasonably fast to obtain, and such training datasets are commonly used in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Then, we evaluate trained models on synthetic instances of size 100, 200, 500 and 1K generated from the same distribution, as well as the standard TSPLib and CVRPLib datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' 7 Published as a conference paper at ICLR 2023 Hyperparameters and training procedure We use the same hyperparameters for all problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' The model has 9 layers, each built with 8 attention heads with embedding size of 128 and dimension of feed-forward layer of 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Our model is trained on 1 million instances with 100 nodes split into batches of size 1024, for 1000 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Solutions of these problems are obtained by using the Concorde solver (Applegate et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=', 2015) for TSP and LKH heuristic (Helsgaun, 2017) for CVRP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' We use Adam (Kingma & Ba, 2017) as optimizer with an initial learning rate of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='5e−4 and decay of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='98 every 20 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Evaluation We compare our model with existing state-of-the-art methods: OR-Tools (Perron & Furnon, 2022), LKH (Helsgaun, 2017), and Hybrid Genetic Search (HGS) for the CVRP (Vidal, 2022) as traditional non-neural methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Att-GCN+MCTS and NeuralRewriter (Chen & Tian, 2019) as hybrid methods for TSP and CVRP respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' and deep learning-based constructive methods: AM, TransformerTSP, MDAM and POMO, which were discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' For all deep learning baselines we use the model trained on graphs of size 100 and the best decoding strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Following the same procedure as in Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' (2020), we generate four test datasets with graphs of sizes 100, 200, 500 and 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' For CVRP, we use capacities of 50, 80, 100 and 250, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' In addition, we report the results on TSPLib instances with up to 4461 nodes and all CVRPLib instances with node coordinates in the Euclidian space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' For all models, we report the optimality gap and the inference time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' The optimality gap for TSP is based on the optimal solutions obtained with Concorde.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' For CVRP, although HGS gives better results than LKH, we use the LKH solution as a reference to compute the ”optimality” gap, in order to be consistent (and easily comparable) with previous works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' While the optimality gap is easy to compute and compare, measurements of running times are much harder: they may vary due to the implementation platforms (Python, C++), hardware (GPU, CPU), parallelization, batch size, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Therefore, we also report the number of solutions generated by each of the constructive deep learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' In our experiments, we run all deep learning models on a single Nvidia Tesla V100-S GPU with 24GB memory, and other solvers on Intel(R) Xeon(R) CPU E5-2670 with 256GB memory, in one thread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Results Tables 1a and 1b summarize our results on TSP and CVRP, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' For both problems, our model shows superior generalization on larger graphs, even with the greedy decoding strategy, which generates only a single solution while all others generate several hundreds (and select the best among them).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' In terms of running time with greedy decoding, our model is competitive with the POMO baseline, and significantly faster than other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Beam search decoding with beam size 16 further improves the quality of solutions, but as expected, it takes approximately 16 times longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Figure 2 shows optimality gap versus running time for our model and other baseline models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Our model clearly outperforms other models in terms of generalization to larger instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' The only model that is competitive with ours is Att-GCN+MCTS, but it is 2-15 times slower and is designed for TSP only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' In addition to synthetic datasets, we test our model on TSPLib and VRPLib instances, which are of varying graph sizes, node distributions, demand distributions and vehicle capacities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Table 1c shows our model’s strength over MDAM and POMO, even with the greedy decoding strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' The effectiveness of our MDP transformation method and the resulting neural architecture is confirmed by the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Thanks to our more principled approach that leads to better state representations and a simpler architecture without a decoder, by generating a single solution, it is able to outperform MDAM (with 250 solutions), which is closest to our model conceptually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Moreover, an ablation study in Appendix D suggests that spending appropriate amounts of compute for each subproblem is a crucial factor in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' 7 CONCLUSION We have presented a flexible framework to derive MDPs that sequentially construct solutions to CO problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Starting from a naive MDP, we introduced a generic transformation using bisim- ulation quotienting, which reduces the state space by leveraging its symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' We applied this transformation on the TSP and CVRP, for which we also designed a simple attention-based model, well-suited to the transformed state representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' We show experimentally that this combination of state representation, simple model, and training procedure yields state-of-the-art generalization re- sults on diverse benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' While training on relatively small instances allowed us to use imitation learning, our approach and model could be similarly used with reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Finally, we have focused on deterministic CO problems, leaving the adaptation of our framework to stochastic problems as future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' 8 Published as a conference paper at ICLR 2023 Table 1: Summary of the experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' The bold values represent the best optimality gap (lower is better) and fastest inference time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' The underlined cells represent the best ratio between the quality of the solution and the inference time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' #s refers to number of generated solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' #s TSP100 TSP200 TSP500 TSP1000 Concorde 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='000% 38m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='000% 2m 0.' metadata={'source': 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+page_content='142% 9m 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='215% 37m MDAM bs50 250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='395% 45m 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='044% 3m 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='878% 13m 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='965% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='1h POMO augx8 8N 0.' metadata={'source': 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10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='74% 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='35% All (15-1K) 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='36% 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='58% 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='58% 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='60% (c) Experimental results on TSPLib (left) and CVRPLib (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' 10 −4 10 −3 10 −2 10 −1 10 0 Inference time (per instance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' in seconds) 0 5 10 15 20 25 30 35 40 Optimality gap AM bs1024 MDAM bs50 POMO augx8 Att- GCN+MCTS BQ (ours) greedy BQ (ours) bs16 TSP100 TSP200 TSP500 TSP1000 10 −4 10 −3 10 −2 10 −1 10 0 Inference time (per instance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' in seconds) 0 20 40 60 80 100 120 140 Optimality gap AM bs1024 MDAM bs50 POMO augx8 NeuralRewriter BQ (ours) greedy BQ (ours) bs16 CVRP100 CVRP200 CVRP500 CVRP1000 Figure 2: Generalization results on different graph sizes for TSP (left) and CVRP (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Lower and further left is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' 9 Published as a conference paper at ICLR 2023 REPRODUCIBILITY STATEMENT In order to ensure the reproducibility of our approach, we have: described precisely our generic theoretical framework (Section ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=') and provided a detailed proof of Proposition 1 in Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' This should in particular serve to adapt the frame- work to other CO problems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' explained in detail our proposed model (Section 4 for TSP and Appendix A for CVRP), described precisely the training procedure and listed the hyperparameters (Section 6);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' used public datasets referenced in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Furthermore, we plan to make our code public upon acceptance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' REFERENCES David Applegate, Robert Bixby, Vasek Chvatal, and William Cook.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Attention Is All You Need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' arXiv:1706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='03762 [cs], June 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' 11 Published as a conference paper at ICLR 2023 Thibaut Vidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Hybrid genetic search for the CVRP: Open-source implementation and SWAP* neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Computers & Operations Research, 140:105643, April 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' ISSN 0305-0548.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='105643.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Lee, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Sugiyama, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Garnett (eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' ), Advances in Neural Information Pro- cessing Systems 28, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' 2692–2700.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Curran Associates, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=', 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Liang Xin, Wen Song, Zhiguang Cao, and Jie Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Multi-Decoder Attention Model with Embed- ding Glimpse for Solving Vehicle Routing Problems, December 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Liang Xin, Wen Song, Zhiguang Cao, and Jie Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Generative Adversarial Training for Neural Combinatorial Optimization Models, September 2021a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Liang Xin, Wen Song, Zhiguang Cao, and Jie Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Step-Wise Deep Learning Models for Solving Routing Problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' IEEE Transactions on Industrial Informatics, 17(7):4861–4871, July 2021b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' ISSN 1941-0050.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='1109/TII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='3031409.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Cong Zhang, Wen Song, Zhiguang Cao, Jie Zhang, Puay Siew Tan, and Chi Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Learning to Dispatch for Job Shop Scheduling via Deep Reinforcement Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='12367 [cs, stat], October 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' 12 Published as a conference paper at ICLR 2023 A APPLICATION TO THE CVRP Problem definition and specification The Capacitated Vehicle Routing Problem (CVRP) is a vehicle routing problem in which a vehicle (here, a single one) with limited capacity must deliver items from a depot location to various customer locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Each customer has an associated demand, which represents an amount of items, and the problem is for the vehicle to provide all the customers in the least travel distance, returning as many times as needed to the depot to refill, but without ever exceeding the vehicle capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Formally, we assume given a set of customer nodes, each with a demand (positive scalar), plus a depot node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' A CVRP solution (in X) is a finite sequence of nodes starting at the depot, which are pairwise distinct except for the depot, and respecting the capacity constraint: the total demand of any contiguous sub-sequence of customer nodes is below the vehicle capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' A CVRP instance (in F) is given by a finite set D of nodes, including the depot, their coordinates in the Euclidian space V , and maps any solution to the length of the corresponding path using the distances in V , if the path visits exactly all the nodes of D, or ∞ otherwise (unfeasible solutions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' A possible specification ⟨T , SOL, VAL⟩ for the CVRP is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' The step space T is the set of pairs of a non depot node and a binary flag indicating whether that node is to be reached via the depot or directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' The extension ¯t of a step t is either the singleton of its node component if its flag is 0 or the pair of the depot node and its node component if its flag is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' For a given problem instance f and sequence t1:n of steps, SOL(f, t1:n) is either the sequence ¯t1:n if it forms a d-path which visits exactly all the nodes of f , or ⊥ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' VAL(f, t1:n) is either the total length of ¯t1:n (closed at its end) if it forms a d-path which visits only nodes of f (maybe not all), or ∞ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' It is easy to show that ⟨T , SOL, VAL⟩ forms a specification for the CVRP (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' satisfies the axioms of specifications introduced in Section ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' The naive MDP obtained from it is denoted CVRP-MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Bisimulation quotienting Just as with TSP, we can define a mapping Φ from CVRP-MDP states to a new “path-CVRP” state space, informally described by the following diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' C=10 1 1 4 3 2 4 1 1 4 3 CVRP state C=3 1 4 3 path-CVRP state Φ Here, the capacity of the vehicle is C=10, shown next to the (colourless) depot node, and the demand of each node is shown next to it, in orange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' The black dotted line indicates that the action which introduced the node with demand 2 was via the depot: its flag was set to 1 (all the other actions had their flag set to 0 in this simple example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' The green dotted line indicates how the path is closed to measure its length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' After the node with demand 2, the path of visited nodes (in red) continues with nodes with demand 4 and 1, respectively, so that the remaining capacity at the end of the path is C−(2+4+1)=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Compared to TSP, this is the additional piece of information in the summary of the “past” (path of visited nodes) which is preserved in the path-CVRP state, together with the origin and destination of the path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Mapping Φ thus defined satisfies Equation 3, hence induces a bisimulation on CVRP-MDP states, and by quotienting, one obtains an MDP which can be defined directly on path-CVRP states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Model architecture for CVRP The model architecture for CVRP is almost the same as for TSP, with a slight difference in the input sequence and in the output layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' In the TSP model, the input to the node embedding layer for a N-node state is a 2×N matrix of coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' For CVRP, we use two additional channels: one for node demands, and one for the current vehicle capacity, repeated across all nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' The demand is set to zero for the origin and destination nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' We obtain a 4×N matrix of features, which is passed through a learned embedding layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' As for TSP, a learned origin-signalling (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' destination-signalling) vector is added to the corresponding embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' The rest of the architecture, in the form of attention layers, is identical to TSP, until after the action scores projection layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' In the case of TSP, the projection layer returns a vector of N scores, where each score, after 13 Published as a conference paper at ICLR 2023 a softmax, represents the probability of choosing the node as the next step in the construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' In the case of CVRP, the model returns a matrix of scores of dimension N×2, corresponding to each possible actions (node-flag pair) and the softmax scopes over this whole matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' As usual, a mask is always applied to unfeasible actions before the softmax operator: those which have higher demand than the remaining vehicle capacity, as well as the origin and destination nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' B APPLICATION TO THE KNAPSACK PROBLEM Problem definition and specification The Knapsack Problem (KP) is classical combinatorial op- timization problem in which we need to pack items, with given values and weights, into a knapsack with a given capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' The objective is to maximize the total value of packed items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Formally, we assume given a set of items, each with a value and weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' A KP solution (in X) is a subset of the items which respects a capacity constraint (“c-subset”): total weight of the items of the subset must not exceed the knapsack capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' A KP instance (in F) is given by a finite set of D items and maps any c-subset to the sum of values of its items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' A simple problem specification ⟨T , SOL, VAL⟩ can be defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' The step space T is equal to the set of items, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' For a partial solution (f, t1:n), if the selected items satisfy the capacity con- straints and adding any of the remaining items results in an infeasible solution, then SOL(f, t1:n) returns the subset of selected items;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' otherwise it returns ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Finally, VAL(f, t1:n) is either the sum of the values of the items in t1:n if they satisfy the capacity constraint and ∞ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Similarly to the TSP and CVRP cases, it is easy to show that ⟨T , SOL, VAL⟩ forms a specification for the KP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' The naive MDP obtained from it is denoted MDP-KP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Bisimulation quotienting As it was the case for TSP and CVRP, we can define a mapping Φ from KP-MDP state to a new “BQ-KP” state space, informally described by the following diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' 3 7 9 1 1 2 4 5 8 8 6 weights values 1 9 2 8 3 7 1 6 7 3 9 C = 20 3 9 1 4 5 8 8 6 1 2 3 1 6 7 3 9 C = 10 KP-state BQ-KP-state Φ Here, capacity of the knapsack is C = 20 and each item is defined by its weight (bottom cell) and value (top cell).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Mapping Φ for KP is straightforward - simply saying, it removes all picked items and update the remaining capacity by subtracting total weight of removed items from the previous capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Model architecture for KP The model architecture for KP is again very similar to previously described models for TSP and CVRP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' The input to the model is a 3 × N tensor composed of items properties (values, weights) and the additional channel for the remaining knapsack’s capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' By definition, the solution has no order (the result is a set of items), so there is no need to add tokens for origin and destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' A part from excluding these tokens and different input dimensions, the rest of the model is identical to the TSP model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' The output is a vector of N probabilities over all items with a mask over the unfeasible ones (with weights larger than remaining knapsack’s capacity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' In the training, at each construction step, any item of the ground-truth solution is a valid choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Therefore we use a multi-class cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Experimental results for KP We generate the training dataset as described in Kwon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' We train our model on 1M KP instances of size 200 and capacity 25, with values and weights ran- domly sampled from the unit interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' We use the dynamic programming algorithm from ORTools to compute the ground-truth optinal solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' As hyperparameters, we use the same as for the TSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Then, we evaluate our model on test datasets with the number of items equal 200, 500 and 1000 and capacity of 25 and 50, for each problem size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Table B shows the performance of our model compared to POMO, one of the best performing NCO models on KP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Although our model does not outperform it in every dataset, it achieves better overall performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' It should be noted again that POMO builds N solutions per instance and choose the best one, while our model generate a single solution per instance but still achieves better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' 14 Published as a conference paper at ICLR 2023 Optimal POMO (single traj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=') POMO (all traj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=') BQ (greedy) value value opt gap value opt gap value opt gap N=200 C=25 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='023 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='740 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='476% 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} 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KP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Greedy Beam size 16 Beam size 64 Full graph 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='79% 5s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='17% 1m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='08% 5m TSP200 100KNNs 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='31% 3s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='23% 33s 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='19% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='4h Full graph 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='80% 5s 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='42% 1m 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='82% 5m CVRP200 100KNNs 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='18% 3s 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='74% 12m 250KNNs 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='58% 32s 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='86% 9m 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='14% 30m Full graph 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='00% 7m 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='19% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='8h 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='39% 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='3h CVRP1000 100KNNs 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='25% 25s 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='76% 6m 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='58% 24m 250KNNs 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='51% 1m 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='08% 23m 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='28% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='4h Table 3: Improving the model performance using a k-nearest-neighbor heuristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' C IMPROVING THE MODEL PERFORMANCE WITH A k-NEAREST-NEIGHBOR HEURISTIC Our decoding strategy could be further improved by using a k-nearest-neighbor heuristic to restrict the search space and reduce the inference time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' For both greedy and beam search strategies, at every step, it is possible to reduce the remaining graph by considering only a certain number of neighbouring nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Table 3 presents the experiments on TSP and CVRP where we apply the model just on a certain number on nearest neighbours of the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' This approach clearly reduces the execution time, but also in some cases even improves the performance in terms of optimality gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' The same heuristic can be applied on Knapsack problem, where model could be applied just on a certain number of items with highest values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' D ABLATION STUDY D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='1 TRANSFORMER VS HYPERMIXER AS MODEL In Section 6 we have shown that our model has an excellent generalization ability to graphs of larger size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' In Section ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=', we hypothesize that this has to do with the fact that a subproblem of size t spends O(t2) computation operations due to the quadratic complexity of the Transformer encoder’s self-attention component, which is responsible for mixing node representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' To test this hypothesis, we experiment with replacing self-attention with an efficient mixing component (see Tay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' (2022) for an overview), namely the recent linear-time HyperMixer (Mai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' We chose this model because it does not assume that the input is ordered, unlike e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' sparse attention alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' 15 Published as a conference paper at ICLR 2023 Seed TSP100 TSP200 TSP500 TSP1000 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='10% 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='38% 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='91% 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='30% 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='38% 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='54% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='59% 628.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='71% 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='93% 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='14% 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='18% 216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='77% 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='37% 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='54% 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='23% 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='85% 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='25% 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='66% 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='99% 524.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='43% Table 4: Experimental results on TSP with HyperMixer for five different seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Experimental Details For comparability, we set the model and training parameters to the same as for Transformers, so the experiments only differ in token mixing component that is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' The only other difference is that we used Layer Normalization Ba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' (2016) instead of ReZero Bachlechner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' (2021), which leads to more stable training for HyperMixer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Since we observed relatively large sensitivity to model initialization, we are reporting the results for 5 different seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Results Table 4 shows the result for HyperMixer with greedy decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' While the model reaches lower but acceptable performance than Transformers on TSP100, it generalizes poorly to instances of larger size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Moreover, performance is very sensitive to the seed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' These results suggest that the computation spent by self-attention is indeed necessary to reach the generalization ability of our model, which increases the compute with the size of the (sub)problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='2 APPROXIMATED MODEL As mentioned in Section 5, existing works have also noted the importance of accounting for the change of the state after each action: Xin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' (2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' 2020) claimed that models should recompute the embeddings after each action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' However because of the additional training cost, they proposed the following approximation: fixing lower encoder levels and recomputing just the top level with a mask of already visited nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' They hypothesis a kind of hierarchical feature extraction property that may make the last layers more important for the fine-grained next decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' In contrast, we call our entire model after each construction step;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' effectively recomputing the embeddings of each state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' We hypothesis that this property may explain the superior performance (Table 1) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='t MDAM model Xin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' In order to support this hypothesis, we have implemented an approximated version of our model as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' We fixed the bottom layers of our model and recomputed just the top layer, by masking already visited nodes and adding the updated information (origin and destination tokens for TSP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' As expected, inference time is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='6 times shorter, but performance is severely degraded: we obtained optimality gap of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='833% (vs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='540% with original model) on TSP100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='3 REZERO VS BATCHNORM AS NORMALIZATION Most NCO works that use transformer networks (Kool et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=', 2019)(Kwon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=', 2021)(Xin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=', 2020) use batch normalization(Ioffe & Szegedy, 2015) rather than layer normalization (Ba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=', 2016) in attention layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' We find ReZero normalization (Bachlechner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=', 2021) to work even better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Figure 3 shows the effect of using ReZero compared to batch normalization in our Trans- former network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Using it leads to more stable training, better performance, and drastically lower variance between seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' E ON THE IMPACT OF EXPERT SOLUTIONS Our datasets consist of pairs of a problem instance and a solution (tour).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' On the other hand, in this paper, we use imitation learning, which requires instead pairs of a problem instance and (expert) trajectory in the MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Now, a solution may be obtained from multiple trajectories in the MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' For example, with TSP, a solution is a loop in a graph, and one has to decide at which node its construc- tion started and in which direction it proceeded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' With CVRP, the order in which the subtours are constructed needs also to be decided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Hence, all our datasets are pre-processed to transform solu- tions into corresponding construction trajectories (a choice for each or even all possible ones).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' We experimentally observed that this transformation has an impact on the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' For example, with CVRP, choosing, for each solution, the construction in the order in which LKH3 displays it, 16 Published as a conference paper at ICLR 2023 0 200 400 600 800 1000 Epochs 0 5 10 15 20 Optimality gap BatchNorm, seed 0 BatchNorm, seed 1 ReZero, seed 0 ReZero, seed 1 Figure 3: Training curves showing the effect of the choice of normalization layer on validation performance which does not seem arbitrary, yields to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='3 point better opt-gap performance compared to following a random ordering of the sub-tours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' We hypothesize that if there is any bias in the display of the optimal solution - for example, shorter tour first, or closest node first - it requires slightly less model capacity to learn action imitation for this display rather than for all possible displays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' F PROOF OF PROPOSITION 1 (SOUNDNESS OF THE NAIVE MDP) We show here that procedure SOLVE satisfies SOLVE(f)= arg minx∈X f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' We first show the following general lemma: Let Y ψ→X f→R∪{∞} be arbitrary mappings, if ψ is surjective then arg min x∈X f(x) = ψ(arg min y∈Y f(ψ(y))) Simple application of the definition of arg min (as a set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' The subscript ∗ denotes the steps where the assumption that ψ is a surjection is used: x′ ∈ ψ(arg min y f(ψ(y))) iff ∃y′ ∈ arg min y f(ψ(y)) x′ = ψ(y′) iff ∃y′ x′ = ψ(y′) ∀y f(ψ(y′)) ≤ f(ψ(y)) iff ∃y′ x′ = ψ(y′) ∀y f(x′) ≤ f(ψ(y)) iff∗ ∀y f(x′) ≤ f(ψ(y)) iff∗ ∀x f(x′) ≤ f(x) iff x′ ∈ arg min x f(x) Let (F, X) be a CO problem with specification ⟨T , SOL, VAL⟩ and M the naive MDP obtained from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' For each f∈F, let vf=VAL(f, ϵ), Xf={x∈X|f(x)<∞} and let Yf be the set of M-trajectories which start at (f, ϵ) and end at a stop state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' 17 Published as a conference paper at ICLR 2023 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Transformer encoder activation = ReLU normalization = ReZero input embedding layer Linear softmax dest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' emb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' origin emb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' + + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Figure 4: Computation flow at the t-th time step, when a partial solution of length t − 1 already exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' The input state consist of the destination node (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' the first and last node in the TSP tour), the origin node (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=', the most recent node in the tour), and the set of remaining nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' After passing all input nodes through an embedding layer, we add special, learnable vector embeddings to the origin and current node to signal their special meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Finally, a Transformer encoder followed by a linear classifier head selects the next node at step t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' For any M-trajectory τ=s0t1r1s1 · · · tnrnsn in Yf, define ψ(τ) =def SOL(sn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Since τ∈Yf, we have s0=(f, ϵ) and sn is a stop state, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' SOL(sn)=ψ(τ)∈X, and by Equa- tion 2a, f(ψ(τ))<∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Hence ψ:Yf �→ Xf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' By construction, sm=(f, t1:m) for all m∈1:n and each transition in τ has a finite reward VAL(sm−1)−VAL(sm) (condition for it to be valid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Hence the cumulated reward is given by R(τ)=VAL(s0)−VAL(sn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Now, VAL(s0)=vf which is independent of τ and by Equa- tion 2c, VAL(sn)=f(ψ(τ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Hence f(ψ(τ))=vf−R(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Let’s show that ψ is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Let x∈Xf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Equation 2a ensures that x=SOL(f, t1:n) for some t1:n∈T ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' For each m∈{0:n}, let sm=(f, t1:m) and consider the sequence τ=s0t1r1s1 · · · tnrnsn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Now, SOL(sn)=x̸=⊥ hence τ ends in a stop state and starts at (f, ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' By Equation 2c we have VAL(sn)=f(x), hence VAL(sn)<∞, and VAL(sm)<∞ for all m∈{0:n−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' And by Equation 2b SOL(sm)=⊥, hence all the transitions in τ are valid in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Hence τ∈Yf and by definition, ψ(τ)=x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Therefore we can apply the lemma proved above: arg min x∈Xf f(x) = ψ(arg min τ∈Yf f(ψ(τ))) = ψ(arg min τ∈Yf vf−R(τ)) = ψ(arg max τ∈Yf R(τ)) = ψ(SOLVEMDP M (f, ϵ)) = SOLVE(f) Now, obviously, arg minx∈X f(x) = arg minx∈Xf f(x), since by definition f is infinite on X\\Xf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' 18 Published as a conference paper at ICLR 2023 G PLOTS OF SOME TSPLIB AND CVRPLIB SOLUTIONS (a) Optimal solution (b) Our model (BS16), opt_gap 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='549% (c) MDAM (BS50), opt_gap 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='501% (d) POMO (x8), opt_gap 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='614% Instance pcb442 (a) Optimal solution (b) Our model (BS16), opt_gap 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='253% (c) MDAM (BS50), opt_gap 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='916% (d) POMO (x8), opt_gap 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='664% Instance pr1002 19 Published as a conference paper at ICLR 2023 (a) Optimal solution (b) Our model (BS16), opt_gap 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='464% (c) MDAM (BS50), opt_gap 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='669% (d) POMO (x8), opt_gap 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='416% Instance X-n284-k15 (a) Best known solution (b) Our model (BS16), opt_gap 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='667% (c) MDAM (BS50), opt_gap 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='739% (d) POMO (x8), opt_gap 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='603% Instance X-n513-k21 20 Published as a conference paper at ICLR 2023 H BACKGROUND ON BISIMULATION-BISIMILARITY H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='1 BISIMULATION IN LABELLED TRANSITION SYSTEMS Bisimulation is a very broad concept which applies to arbitrary Labelled Transition Systems (LTS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' It has been declined in various flavours of LTS, such as Process Calculi, Finite State Automata, Game theory, and of course MDP (initially deterministic MDP such as those used here, later extended to stochastic MDP which we are not concerned with here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' A bisimulation is a binary relation R among states which “commutes” with the transitions of the LTS in the following diagram, which should informally be read as follows: if the pair of arrows connected to p (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' q) exists then so does the “opposite” pair (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' the centre of the diagram).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' p q p′ q′ ℓ ℓ R R The notation p ℓ −−→ p′ means the transition from p to p′ with label ℓ is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Thus, formally, Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' A binary relation R on states is a bisimulation if for all label ℓ and states p, q such that pRq ∀p′ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' p ℓ −−→ p′ ∃q′ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' q ℓ −−→ q′ , p′Rq′ ∀q′ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' q ℓ −−→ q′ ∃p′ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' p ℓ −−→ p′ , p′Rq′ Note that this definition is extended to the “heterogeneous” case where R is bi-partite, relating the state spaces of two LTS L1, L2 sharing the same label space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' One just forms a new LTS L whose state space is the disjoint union of the state spaces of L1, L2 and the transitions are those of L1, L2 in their respective (disjoint) component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' An heterogeneous bisimulation on L1, L2 is then a (homogeneous) bisimulation on L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Most results below also have a heterogeneous version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' The set of bisimulations (subset of the set of binary relations on states) is stable by union, composition, and inversion, hence also by reflexive-symmetric-transitive closure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' In particular, the union of all bisimulations, called the bisimilarity of the LTS, is itself a bisimulation, and it is also an equivalence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' (outline) Let’s detail stability by composition, the other cases are similarly obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' If R1, R2 are the two bisimulations being composed, apply the commutation property to each cell of the following diagram (from top to bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' p r q p′ r′ q′ ℓ ℓ ℓ R1 R2 R1 R2 Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Given an LTS L, its transitive closure is another LTS denoted L∗ on the same state space, where the labels are the sequences of labels of L and the transitions are defined by p ℓ1:n −−−−→ (L∗) p′ if ∃p0:n such that p = p0 ℓ1 −−−→ (L) p1 · · · ℓn−1 −−−−→ (L) pn−1 ℓn −−−→ (L) pn = p′ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' If R is a bisimulation on L, then it is also a bisimulation on L∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' (outline) This is essentially shown by successively applying the commutation property to each cell of the following diagram (from left to right): 21 Published as a conference paper at ICLR 2023 p0 q0 p1 q1 pn−1 qn−1 pn qn ℓ1 ℓn ℓ1 ℓn R R R R Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Given an LTS L and an equivalence relation R on its state space, we can define the quotient LTS L/R with the same label space, where the states are the R-equivalence classes and the transitions are defined, for any classes ˙p, ˙p′, by ˙p ℓ −−−−→ L/R ˙p′ if ∀p ∈ ˙p ∃p′ ∈ ˙p′ p ℓ −−→ L p′ Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Let R be an equivalence on the state space of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' R is a bisimulation on L if and only if ∈ is a (heterogeneous) bisimulation on L, L/R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' We show both implications: Assume R is a bisimulation on L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' – Let p ∈ ˙q and p ℓ−→ p′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Let q ∈ ˙q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Hence pRq and p ℓ−→ p′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Since R is a bisimulation, there exists q′ such that q ℓ−→ q′ and p′Rq′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Hence for all q ∈ ˙q, there exists q′ ∈ ¯p′ such that q ℓ−→ q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Hence by definition ˙q ℓ−→ ¯p′ while p′ ∈ ¯p′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' – Let p ∈ ˙q and ˙q ℓ−→ ˙q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Hence by definition, there exists p′ ∈ ˙q′ such that p ℓ−→ p′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Assume ∈ is a (heterogeneous) bisimulation on L, L/R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' – Let pRq and p ℓ−→ p′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Hence p ∈ ¯q and p ℓ−→ p′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Since ∈ is a bisimulation, there exists ˙q′ such that p′ ∈ ˙q′ and ¯q ℓ−→ ˙q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Now q ∈ ¯q, hence, by definition, there exists q′ ∈ ˙q′ such that q ℓ−→ q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' And p′Rq′ since p′, q′ ∈ ˙q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' – Let pRq and q ℓ−→ q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Hence qRp and q ℓ−→ q′, and we are in the previous case up to a permutation of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Let R be an equivalence relation on the state space of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' If R is a bisimulation on L, then for any L-state p, L/R-state ˙p and L∗-label ℓ ¯p ℓ −−−−−−→ (L/R)∗ ˙p′ if and only if ∃p′ ∈ ˙p′ p ℓ −−→ L∗ p′ Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Simple combination of Propositions 4 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' R is a bisimulation on L, hence ∈ is a het- erogeneous bisimulation on L, L/R (Proposition 4), hence also a heterogeneous bisimulation on L∗, (L/R)∗ (Proposition 3, heterogeneous version).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' If ¯p ℓ −−−−−−→ (L/R)∗ ˙p′, since p∈¯p and ∈ is a bisimulation, we have p ℓ −−→ L∗ p′ for some p′∈ ˙p′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Conversely, if p ℓ −−→ L∗ p′ for some p′∈ ˙p′, since p∈¯p and ∈ is a bisimulation, we have ¯p ℓ −−−−−−→ (L/R)∗ ˙q′ and p′∈ ˙q′ for some ˙q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Now p′∈ ˙p′∩ ˙q′ hence ˙p′= ˙q′ and ¯p ℓ −−−−−−→ (L/R)∗ ˙p′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='2 BISIMULATION IN DETERMINISTIC MDP Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' An MDP is a pair (L, ⊤) where L is a LTS with label space A × R for some action space A (action-reward pairs denoted a|r) and ⊤ is a subset of states (the stop states).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' It is said to be deterministic if if s a|r1 −−→ s′ 1 and s a|r2 −−→ s′ 2 then r1 = r2 and s′ 1 = s′ 2 22 Published as a conference paper at ICLR 2023 Given an L-trajectory τ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' a sequence s0a1r1s1 · · · anrnsn where si−1 ai|ri −−−→ si for all i∈{1:n}, its cumulated reward is defined by R(τ)= �n i=1 ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' The generic problem statement of the MDP solution framework is, given an MDP (L, ⊤) and one of its states so, to solve the following optimi- sation: SOLVEMDP((L, ⊤), so) = arg max τ R(τ) | τ is a L-trajectory starting at so and ending in ⊤ This definition of MDP and the standard textbook one coincide only in the deterministic case (in the standard definition, an MDP is deterministic if the distribution of output state-reward pairs for a given input state and allowed action is “one-hot”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' The non deterministic case in the definition above does not match the standard definition: it would be wrong to interpret two distinct transitions for the same input state s and action a as meaning that the outcome of applying a to state s is distributed between the two output reward-state pairs according to a specific distribution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' uniform).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Also, in the problem statement, the objective R(τ) has no expectation, which, with the standard definition, only makes sense in the case of a deterministic MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Similarly, the standard problem statement is expressed in terms of policies rather than trajectories directly, but in the deterministic case, the two are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Observe that there is a one-to-one correspondence between trajectories in L and transitions in the LTS L∗, so the problem statement can be formulated equivalently as SOLVEMDP((L, ⊤), so) = arg max ℓ R(ℓ) | ∃s ∈ ⊤, so ℓ −−→ L∗ s (4) Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Let (L, ⊤) be an MDP and R an equivalence relation on its state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' (L/R, ¯⊤) is also an MDP, where ¯⊤={¯s|s∈⊤}, and if L is deterministic, so is L/R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' If R is a bisimulation on L preserving ⊤ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' � s∈⊤ ¯s = ⊤), then for any state so and label ℓ in L∗ we have ∃s ∈ ⊤, so ℓ −−→ L∗ s if and only if ∃ ˙s ∈ ¯⊤, ¯so ℓ −−−−−−→ (L/R)∗ ˙s Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' The second property is a direct consequence of Proposition 5 and the assumption that ⊤ is preserved by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' For the first, assume that L is deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Let ˙s, ˙s1, ˙s2 be L/R states, such that ˙s a|r1 −−→ ˙s1 and ˙s a|r2 −−→ ˙s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Choose s ∈ ˙s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Hence, by definition, there exist s1∈ ˙s1 and s2∈ ˙s2 such that s a|r1 −−→ s1 and s a|r2 −−→ s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Since L is deterministic, we have r1=r2 and s1=s2∈ ˙s1∩ ˙s2, hence ˙s1 = ˙s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Hence L/R is also deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' Therefore, when R is a bisimulation equivalence on L preserving ⊤, the generic MDP problem statement of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' equation 4 can be reformulated as SOLVEMDP((L, ⊤), so) = SOLVEMDP((L/R, ¯⊤), ¯so) = arg max ℓ R(ℓ) | ∃ ˙s ∈ ¯⊤, ¯so ℓ −−−−−−→ (L/R)∗ ˙s (5) Note that a bisimulation on L preserving ⊤ is simply a bisimulation on the LTS ˙L defined as follows: ˙L has the same state space as L and an additional transition s −→ s for each s∈⊤, where “·” is a distinguished label not present in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' A bisimulation R on ˙L captures some symmetries of the state space of ˙L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' If R is taken to be the bisimilarity of ˙L, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' the union of all the bisimulations on ˙L, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' the union of all the bisimulations on L preserving ⊤, then it captures all the possible symmetries of the state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' This should be seen as an asymptotic result, since constructing and working with the full bisimilarity of ˙L is not feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' But Proposition 6 remains valuable as it applies to all bisimulation, not just the maximal bisimulation of ˙L (its bisimilarity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} +page_content=' 23' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfnwSt/content/2301.03313v1.pdf'} diff --git a/BNE2T4oBgHgl3EQfRgdU/content/tmp_files/2301.03781v1.pdf.txt b/BNE2T4oBgHgl3EQfRgdU/content/tmp_files/2301.03781v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ab3ebed262f1617a3e692c01ecfb96250dfba93f --- /dev/null +++ b/BNE2T4oBgHgl3EQfRgdU/content/tmp_files/2301.03781v1.pdf.txt @@ -0,0 +1,807 @@ +REDUCED CLIQUE GRAPHS: A CORRECTION TO +“CHORDAL GRAPHS AND THEIR CLIQUE GRAPHS” +DILLON MAYHEW AND ANDREW PROBERT +Abstract. Galinier, Habib, and Paul introduced the reduced clique +graph of a chordal graph G. +The nodes of the reduced clique graph +are the maximal cliques of G, and two nodes are joined by an edge if +and only if they form a non-disjoint separating pair of cliques in G. In +this case the weight of the edge is the size of the intersection of the two +cliques. A clique tree of G is a tree with the maximal cliques of G as +its nodes, where for any v ∈ V (G), the subgraph induced by the nodes +containing v is connected. Galinier et al. prove that a spanning tree of +the reduced clique graph is a clique tree if and only if it has maximum +weight, but their proof contains an error. We explain and correct this +error. +In addition, we initiate a study of the structure of reduced clique +graphs by proving that they cannot contain any induced cycle of length +five (although they may contain induced cycles of length three, four, or +six). We show that no cycle of length four or more is isomorphic to a +reduced clique graph. We prove that the class of clique graphs of chordal +graphs is not comparable to the class of reduced clique graphs of chordal +graphs by providing examples that are in each of these classes without +being in the other. +1. Introduction +We consider only simple graphs. A chord of a cycle is an edge that joins +two vertices of the cycle without being in the cycle itself. A graph is chordal +if any cycle with at least four vertices has a chord. A clique is a set of +pairwise adjacent vertices. If S is a set of vertices and P is a path, then P +is S-avoiding if no internal vertex of P is in S. Assuming that a and b are +distinct vertices, an ab-separator is a set S of vertices not containing either +a or b such that there is no S-avoiding path from a to b. If, in addition, S +does not properly contain an ab-separator then it is a minimal ab-separator. +If G is a chordal graph, then C(G) is the corresponding clique graph +(also known as the clique intersection graph). +The vertices of C(G) are +the maximal cliques of G, and two maximal cliques are adjacent in C(G) if +and only if they have a non-empty intersection. The vertices of the reduced +clique graph, CR(G), are again the maximal cliques of G, but C and C′ are +adjacent in CR(G) if and only if C ∩ C′ ̸= ∅ and C and C′ form a separating +pair: that is, there is no (C ∩ C′)-avoiding path from a vertex in C − C′ +1 +arXiv:2301.03781v1 [math.CO] 10 Jan 2023 + +2 +MAYHEW AND PROBERT +to a vertex in C′ − C. Note that the vertices of CR(G) are identical to the +vertices of C(G), and every edge of CR(G) is an edge of C(G). +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +234589 +234689 +235789 +123 +8910 +G +C(G) +CR(G) +234589 +234689 +235789 +123 +8910 +Figure 1. A chordal graph, its clique graph, and its reduced +clique graph. +The reduced clique graph was introduced in [3] (where it is called a clique +graph) and studied further in [5–8]. +Let G be a graph, and let T be a tree whose vertices are the maximal +cliques of G. If, for every v ∈ V (G), the maximal cliques of G that contain +v induce a subtree of T, then T is a clique tree. Clique trees were introduced +by Gavril [4], who proved that a graph has a clique tree exactly when it is +chordal. +We weight each edge of CR(G) as follows: the edge joining cliques C and +C′ is weighted with |C ∩ C′|. The following result is [3, Theorem 6]. +Theorem 1.1. Let G be a connected chordal graph. Let T be a spanning +tree of CR(G). Then T is a clique tree if and only if it is a maximum-weight +spanning tree. +Although the statement of Theorem 1.1 is correct, it is not proved in +[3, Theorem 6] because of a flaw in the argument. The issue arises in the +proof that a maximum-weight spanning tree must be a clique tree. +We +illustrate the error by using the same argument to prove a false statement. +Non-theorem 1.2. Let G be a chordal graph. Let C0, C1, . . . , Cn be the +sequence of maximal cliques in a path of CR(G) where n > 1. Assume that +there is a vertex v of G such that v is in C0 ∩ Cn, but in none of the cliques +C1, . . . , Cn−1. Then C0 and Cn are adjacent in CR(G). +Non-proof. Consider the subgraph G′ of G induced by C0 ∪ C1 ∪ · · · ∪ Cn. +Thus G′ is chordal. From [10, Corollary 2] we see that either v is a simplicial +vertex (meaning that the neighbours of v in G′ form a clique), or there is a +pair, a, b, of vertices such that v belongs to a minimal ab-separator of G′. +In the former case v is in a unique maximal clique of G′ ([1, Theorem 3.1]). +But C0 and Cn are distinct maximal cliques of G′ that contain v. Therefore + +REDUCED CLIQUE GRAPHS +3 +we can let S be a minimal ab-separator of G′, where v is in S. The proof +of [2, Lemma 2.3] shows that there are two distinct maximal cliques, Da +and Db, of G′ such that Da and Db properly contain S, and Da − S is in +the same connected component of G′ − S as a, while Db − S is in the same +component as b. Thus Da and Db are maximal cliques of G′ that contain v. +But the only maximal cliques of G′ that contain v are C0 and Cn. Therefore +we can assume without loss of generality that Da = C0 and Db = Cn. Any +path from a vertex of C0 − Cn to a vertex of Cn − C0 must contain a vertex +in S = C0 ∩ Cn. Therefore C0 and Cn form a non-disjoint separating pair, +so C0 and Cn are adjacent in CR(G), as claimed. +□ +We can see that this non-theorem is, indeed, not a theorem by examining +Figure 1. Set C0, C1, and C2 to be the maximal cliques {2, 3, 4, 6, 8, 9}, +{1, 2, 3}, and {2, 3, 5, 7, 8, 9}, respectively. +Thus C0, C1, C2 is the vertex +sequence of a path in CR(G). The vertex 8 is in C0 ∩ C2, but not in C1. +However C0 and C2 are not adjacent in CR(G). The error in the “proof” +lies in the claim that “the only maximal cliques of G′ that contain v are C0 +and Cn”. This need not be true. Indeed, {2, 3, 4, 5, 8, 9} is a maximal clique +in the subgraph induced by C0 ∪ C1 ∪ C2, and it contains 8, but it is not +equal to either C0 or C2. Exactly the same error appears in the proof of +[3, Theorem 6]. Nonetheless, Theorem 1.1 is true, and we prove it in the +next section. +2. Reduced clique graphs and clique trees +In [9] we will apply our main theorem to some matroid problems. For +these purposes we would like to extend its scope somewhat. +Instead of +weighting the edges of CR(G) with sizes of intersections, we consider more +general weightings. +Definition 2.1. Let G be a chordal graph. We consider a function σ which +takes +{∅} ∪ {C ∩ C′ : C, C are distinct maximal cliques of G} +to non-negative integers. We insist that σ(∅) = 0 and if X and X′ are in the +domain of σ and X ⊂ X′, then σ(X) < σ(X′). In such a case the function +σ is a legitimate weighting of G. +Theorem 2.2. Let G be a connected chordal graph and let σ be a legitimate +weighting of G. Every clique tree is a spanning tree of CR(G) and every edge +of CR(G) is contained in a clique tree. Moreover, a spanning tree of CR(G) +is a clique tree if and only if it has maximum weight amongst all spanning +trees. +Note that the function that takes each intersection C ∩ C′ to |C ∩ C′| is +a legitimate weighting, so Theorem 2.2 does indeed imply Theorem 1.1. We +now start proving the intermediate results required for the proof of Theorem +2.2. + +4 +MAYHEW AND PROBERT +Proposition 2.3. Let G be a chordal graph, and let C and C′ be maximal +cliques of G. Let S be a set of vertices that contains C∩C′. Let v0, v1, . . . , vk +be the vertex sequence of P, a shortest-possible S-avoiding path from a vertex +in C − C′ to a vertex in C′ − C. Then (C ∩ C′) ∪ {vi, vi+1} is a clique for +each i = 0, 1, . . . , k − 1. +Proof. If C ∩ C′ = ∅ then the result holds trivially, so we assume C ∩ C′ +is non-empty. Note that every vertex in C ∩ C′ is adjacent to v0, and also +to vk, since these vertices are in C − C′ and C′ − C. Now the result can +only fail if there is a vertex x ∈ C ∩ C′ that is not adjacent to vi for some +i ∈ {1, . . . , k − 1}. Let p be the largest integer such that p < i and x is +adjacent to vp. Similarly, let q be the smallest integer such that q > i and +x is adjacent to vq. Consider the cycle obtained by adding the edges vpx +and vqx to vp, vp+1, . . . , vq. This cycle contains the distinct vertices vp, vi, +vq, and x, so it must contain a chord. No chord can join two vertices in the +path P, since P is as short as possible. Thus any chord is incident with x. +But x is not adjacent to any of the vertices in vp+1, . . . , vq−1 by the choice +of p and q, so we have a contradiction. +□ +Proposition 2.4. Let G be a chordal graph, and let C and C′ be maximal +cliques of G where C ∩ C′ ̸= ∅. If C and C′ are not adjacent in CR(G), +then they are joined by a path of CR(G) with vertex sequence C0, C1, . . . , Cs, +where each Ci ∩ Ci+1 properly contains C ∩ C′. +Proof. Assume this fails for C and C′, and they have been chosen so that +C ∩ C′ is as large as possible. Let S be C ∩ C′. Because C and C′ are not +adjacent in CR(G), but S ̸= ∅, it follows that there is an S-avoiding path +from a vertex in C − C′ to a vertex in C′ − C. Let v0, v1, . . . , vk be the +vertex sequence of such a path, where k is as small as possible. We assume +v0 is in C − C′ while vk is in C′ − C. We apply Proposition 2.3 and for each +i = 1, . . . , k, we let Di be a maximal clique of G that contains S ∪{vi−1, vi}. +Set D0 to be C and set Dk+1 to be C′. Note that Di ̸= Dj when i < j, +because vi−1 is not adjacent to vj. For each i = 0, 1, . . . , k, the intersection +of Di and Di+1 contains S as well as vi. If Di and Di+1 are adjacent in +CR(G) then we let Pi be the path of CR(G) consisting of Di, Di+1, and +the edge between them. Otherwise Di and Di+1 are not adjacent in CR(G) +and the assumption on the cardinality of S means that there is a path Pi of +CR(G) from Di to Di+1 such that every intersection of consecutive cliques +in Pi properly contains S ∪ vi. +We concatenate the paths P0, P1, . . . , Pk +and obtain a walk of CR(G) from C to C′. The intersection of any two +consecutive cliques in this walk properly contains S. It follows that there is +a path of CR(G) from C to C′ with exactly the same property, and now C +and C′ fail to provide a counterexample after all. +□ +Figure 2 illustrates Proposition 2.4. +The intersection of cliques C = +{1, 2, 3} and C′ = {3, 5, 7, 8} is {3} ̸= ∅, but C and C′ are not adjacent + +REDUCED CLIQUE GRAPHS +5 +in CR(G). However, there is a path between C and C′ in CR(G), and the +intersection of any consecutive two cliques in the path properly contains {3}. +1 +6 +4 +7 +5 +2 +3 +8 +123 +2345 +3567 +3456 +3578 +G +CR(G) +Figure 2. +Proposition 2.5. Let G be a connected chordal graph. Let T be a clique +tree of G. Assume that C and C′ are maximal cliques of G that are adjacent +in T. Then C and C′ are adjacent in CR(G). +Proof. Assume C and C′ are adjacent in T, but not in CR(G). We partition +the maximal cliques of G as follows. Let U be the set of maximal cliques of +G such that D is in U if and only if the path of T from D to C does not +contain C′. Similarly, define U′ so that D′ is in U′ if and only if the path +of T from D′ to C′ does not contain C. Note that every maximal clique +of G is in exactly one of U or U′, since T is a tree. Furthermore C is in U +and C′ is in U′. Let U be the union of the cliques in U, and let U ′ be the +union of the cliques in U′. Every vertex is in at least one maximal clique +so U ∪ U ′ = V (G). Note that C ⊆ U and C′ ⊆ U ′, so neither U nor U ′ is +empty. +If U ∩ U ′ = ∅, then we choose u ∈ U and u′ ∈ U ′ so that u and u′ are +adjacent in G. (We are able to do so because G is connected.) The edge +between u and u′ is contained in a maximal clique. If this maximal clique +is in U then u′ is in U ∩ U ′, and if it is in U′ then u is in U ∩ U ′. In either +case we have a contradiction, so U ∩ U ′ ̸= ∅. +Choose an arbitrary vertex v in U ∩ U ′. Choose D ∈ U and D′ ∈ U′ such +that v is in D ∩ D′. Because T is a clique tree, it follows that v is contained +in all the cliques belonging to the path of T from D to D′. In particular, v +is contained in C and C′. Thus U ∩ U ′ ⊆ C ∩ C′ and C ∩ C′ is non-empty. +Let S be C ∩ C′. Since C and C′ are not adjacent in CR(G), we can +apply Proposition 2.4 and find a path P of CR(G) from C to C′, where the +intersection of each pair of consecutive cliques in this path properly contains +S. Since C is in U and C′ is in U′, there is an edge of P that joins a clique +D ∈ U to a clique D′ ∈ U′. Then D∩D′ properly contains S, so we choose v +in (D ∩D′)−S. Again using the fact that T is a clique tree, we see that the +path of T from D to D′ consists of cliques that contain v. In particular, v +is in C ∩ C′ = S, and we have a contradiction that completes the proof. +□ + +6 +MAYHEW AND PROBERT +It follows from Proposition 2.5 that every clique tree of G is a spanning +tree of CR(G). +Proposition 2.6. Let G be a connected chordal graph and let σ be a legiti- +mate weighting of G. Let T be a clique tree of G. Let C and C′ be maximal +cliques of G that are adjacent in C(G) and let P be the path of T between C +and C′. The weight of any edge in P is at least σ(C ∩ C′). Moreover, if C +and C′ are adjacent in CR(G), then at least one edge in P has weight equal +to σ(C ∩ C′). +Proof. Let S be C ∩ C′. Let P be the path of T from C to C′, and let the +cliques in this path be C0, C1, . . . , Cn, where C0 = C and Cn = C′. Note that +P is a path of CR(G) by Proposition 2.5. Thus any two consecutive cliques +in the path have a non-empty intersection. Assume σ(Ci ∩ Ci+1) < σ(S) +for some i. If S were a subset of Ci ∩ Ci+1, then we would have σ(S) ≤ +σ(Ci ∩Ci+1) by the definition of a legitimate weighting, but this is not true. +Therefore we can choose v to be a vertex in S − (Ci ∩ Ci+1). Now v is a +vertex of both C and C′, but the path of T between C and C′ contains at +least one maximal clique (either Ci or Ci+1) that does not contain v. This +contradicts the fact that T is a clique tree. Therefore the weight of any edge +in P is at least equal to σ(S). +Now assume that C and C′ are adjacent in CR(G), so that they form a +separating pair. That is, there are distinct connected components of G − S +that contain, respectively, C −S and C′−S. There must be maximal cliques +D and D′ that are adjacent in P, where D − S is in the same connected +component of G−S as C−S, and D′−S is not in this connected component. +This means that D ∩ D′ is contained in S. Hence σ(D ∩ D′) ≤ σ(S). The +previous paragraph shows that σ(D ∩ D′) ≥ σ(S), so the result follows. +□ +The proof of the next result is a straightforward adaptation of a proof +given by Blair and Peyton [1, Theorem 3.6]. +Lemma 2.7. Let G be a connected chordal graph. Let σ be a legitimate +weighting of G and let T be a spanning tree of C(G). Then T is a clique +tree of G if and only if it is a maximum-weight spanning tree of C(G). +Proof. If T is a clique tree, then for any pair of maximal cliques, C and C′, +such that C and C′ are adjacent in C(G), the weight of the edge between C +and C′ is no greater than the weight of any edge in the path of T between +C and C′ (Proposition 2.6). It immediately follows that T has maximum +weight. +For the other direction, we assume that T is a maximum-weight spanning +tree. Because every chordal graph has a clique tree, and any clique tree is a +spanning tree of CR(G) (and hence of C(G)), we can choose a clique tree T ′ +so that T and T ′ have as many edges in common as possible. We can choose +an edge in T that is not in T ′, because otherwise there is nothing left for us +to prove. So let e be such an edge, and assume that e joins maximal cliques +C and C′. There are two connected components of T\e, one containing C + +REDUCED CLIQUE GRAPHS +7 +and the other containing C′. Let P be the path of T ′ from C to C′. We let f +be an edge of P which joins two cliques that are not in the same component +of T\e. Note that f is an edge of T ′, and hence an edge of C(G). +If (T − e) ∪ f is not a spanning tree of C(G), then there is a path of T +between the end-vertices of f that does not use e. But the end-vertices of +f are in different connected components of T\e, so (T − e) ∪ f is indeed a +spanning tree. Similarly, if (T ′ − f) ∪ e is not a spanning tree, then there +is a path of T ′ between C and C′ that does not contain f. But P is the +unique path of T ′ between C and C′, and f is an edge of P. So (T − e) ∪ f +and (T ′ − f) ∪ e are both spanning trees of C(G). +Applying Proposition 2.6 to the clique tree T ′ shows that the weight of f +is at least the weight of e. Since T is a maximum-weight spanning tree, and +(T − e) ∪ f is a spanning tree it follows that the weights on e and f must +be equal. Let D and D′ be the maximal cliques joined by f. Any element +that is in both C and C′ must be in all the cliques in P, since T ′ is a clique +tree. This shows that C ∩ C′ ⊆ D ∩ D′. If C ∩ C′ were a proper subset of +D ∩ D′, then the definition of a legitimate weighting would mean that the +weight of e is strictly less than the weight of f, which is not true. Therefore +C ∩ C′ = D ∩ D′. +We note that (T ′−f)∪e cannot be a clique tree, since it has one more edge +in common with T than T ′ does. Therefore we choose a vertex v ∈ V (G) so +that the maximal cliques containing v do not induce a subtree of (T ′−f)∪e. +Let T ′′ be the subtree of T ′ induced by the maximal cliques containing v. +Then f is in T ′′, or else T ′′ would be a subtree of (T ′ − f) ∪ e. This means +that v is in D ∩ D′ = C ∩ C′. +So both C and C′ are in T ′′, but they +are not in the same component of T ′′\f, because in that case (T ′ − f) ∪ e +would contain a cycle. So e joins two vertices of T ′′ that are in different +components of T ′′\f. Thus (T ′′ − f) ∪ e is a subtree of (T ′ − f) ∪ e, and we +have a contradiction that completes the proof. +□ +Proof of Theorem 2.2. We have already noted that every clique tree is a +spanning tree of CR(G). Let T be a clique tree of G. Then T is a maximum- +weight spanning tree of C(G) by Lemma 2.7. But every edge of T is an edge +of CR(G), by Proposition 2.5. Since CR(G) is a subgraph of C(G) it follows +that T is a maximum-weight spanning tree of CR(G). +For the other direction, we let T be a maximum-weight spanning tree +of CR(G). +We claim that T is also a maximum-weight spanning tree of +C(G). To prove this claim, let e be an arbitrary edge of C(G) that is not +in T, let C and C′ be the maximal cliques of G joined by e, and let P +be the path of T that joins C and C′. If e is an edge of CR(G), then the +weight of e is no greater than the weight of any edge in P, since T is a +maximum-weight spanning tree of CR(G). Therefore we assume that e is +not an edge of CR(G). Now it follows from Proposition 2.4 and the definition +of a legitimate weighting that the edges in P all have weight strictly greater +than the weight of e. In either case, the weight of e does not exceed the + +8 +MAYHEW AND PROBERT +weight of any edge in P. This implies that T is indeed a maximum-weight +spanning tree of C(G), and thus T is a clique tree of G by Lemma 2.7. +To complete the proof, we let e be an arbitrary edge of CR(G). We will +prove that e is in a maximum-weight spanning tree of CR(G). We let C and +C′ be the maximal cliques joined by e. Let T be an arbitrary maximum- +weight spanning tree of CR(G), so that T is a clique tree by the previous +paragraph. If e is in T then we have nothing left to prove, so assume that +P is the path of T joining C to C′, where P contains more than one edge. +Proposition 2.6 shows that P contains an edge, f, with weight equal to the +weight of e. Now (T − f) ∪ e is a maximum-weight spanning tree of CR(G) +that contains e, and we are done. +□ +From the previous arguments we can deduce further additional facts, both +noted in [3]: any edge that is in C(G) but not CR(G) cannot be in any +maximum-weight spanning tree of C(G). Secondly, CR(G) is in fact the +union of all clique trees of G. +Although the next fact is incidental to our main results here, we note it +for a future application in [9]. +Proposition 2.8. Let G be a connected chordal graph, and let T be a clique +tree of G. Let C and C′ be adjacent in T and let S be C ∩ C′. Assume +that D and D′ are maximal cliques of G and the path of T from D to D′ +contains both C and C′. Then D − S and D′ − S are in different connected +components of G − S. +Proof. Let U be the family of maximal cliques of G such that D is in U if +and only if the path of T from D to C does not contain C′. Similarly, we let +U′ be the family of maximal cliques where D′ is in U′ if and only if the path +of T from D′ to C′ does not contain C. Note that every maximal clique of +G belongs to exactly one of U and U′. We are asserting that if D ∈ U and +D′ ∈ U′, then D − S and D′ − S are in different connected components of +G − S. Assume that this fails for D and D′, where D ∩ D′ is as large as +possible. Let H be the connected component of G − S that contains both +D − S and D′ − S. +Let P be the path of T from D to D′. Therefore P contains both C and +C′. Let v be an arbitrary vertex of D ∩ D′. Then v is in every maximal +clique that appears in P, since T is a clique tree. In particular, v is in C +and C′. Thus v is in S, and this shows that D ∩ D′ is contained in S. +Let v0, v1, . . . , vk be the vertex sequence of a shortest-possible path of H +from a vertex v0 ∈ D − S to a vertex vk ∈ D′ − S. This is an S-avoiding +path, where S contains D ∩ D′. Thus we can apply Proposition 2.3. For +i = 1, 2, . . . , k we let Di be a maximal clique of G that contains (D ∩ D′) ∪ +{vi−1, vi}. Let D0 be D and let Dk+1 be D′. Note that each Di − S is +contained in H. This is true for D0 and Dk+1 by definition, and every other +Di contains the edge vi−1vi, which is in the path of H from v0 to vk. Since +D0 is in U and Dk+1 is in U′, we can choose i so that Di is in U and Di+1 +is in U′. The intersection of Di and Di+1 is larger than D ∩ D′, since it + +REDUCED CLIQUE GRAPHS +9 +contains (D ∩ D′) ∪ vi. As Di − S and Di+1 − S are both contained in H +we have a contradiction to the choice of D and D′. +□ +3. The structure of reduced clique graphs +Habib and Stacho comment on the possibility of investigating the struc- +ture of graphs that are isomorphic to reduced clique graphs [6, p. 714]. +In this section we make a contribution to this investigation. We start by +answering an obvious question that requires a non-trivial proof. +Corollary 3.1. Let G be a chordal graph. Then CR(G) is connected if and +only if G is connected. +Proof. Assume that H and H′ are distinct connected components of G. No +maximal clique of H can share a vertex with a maximal clique of H′. It +follows that there be no path of CR(G) that joins two such cliques. Thus +CR(G) is not connected. +The other direction is stated without proof in [6, p. 716]. Assume that +G is connected. Since G is chordal it has a clique tree [4, Theorem 2], and +Proposition 2.5 shows that every edge of the clique tree is an edge of CR(G). +Thus CR(G) has a spanning tree, so it is connected. +□ +Next we note a characterisation of clique graphs due to Szwarcfiter and +Bornstein. +Theorem 3.2 ([11, Theorem 2.1]). The graph H is isomorphic to C(G) for +some connected chordal graph G if and only if H has a spanning tree T such +that whenever u and v are adjacent in H, the path of T from u to v induces +a clique of H. +3.1. Induced cycles. Next we observe that clique graphs can have induced +cycles of any length. We will later show that this is not true for reduced +clique graphs. For an integer n ≥ 3 the wheel graph with n spokes is obtained +from a cycle of n vertices by adding a new vertex and making it adjacent to +all vertices of the cycle. Thus the wheel graph with n spokes has an induced +cycle of n vertices. +Proposition 3.3. For each integer n ≥ 3 the wheel graph with n spokes is +isomorphic to the clique graph of a chordal graph. +Proof. This is easy to prove using Theorem 3.2, but we will give a direct +construction. Start with a clique on the n + 1 vertices u0, u1, . . . , un−1, x. +For each i ∈ Z/nZ, add a new vertex vi and make it adjacent to ui and ui+1. +Call the resulting graph G. It is easy to verify that G is chordal, and its +maximal cliques are {u0, u1, . . . , un−1, x} along with {vi, ui, ui+1} for each +i ∈ Z/nZ. The result follows. +□ +Definition 3.4. Let G be a chordal graph. Let C0, C1, . . . , Cn−1 be a cyclic +ordering of the maximal cliques in an induced cycle of CR(G). We take the +indices to be from Z/nZ, so Ci and Cj are adjacent in CR(G) if and only if + +10 +MAYHEW AND PROBERT +j ∈ {i − 1, i + 1}. If |Ci ∩ Ci+1| ≤ |Cj ∩ Cj+1| for every j ∈ Z/nZ, then we +say that the edge between Ci and Ci+1 is a minimal edge of the cycle. +Lemma 3.5. Let G be a chordal graph. Let C0, C1, . . . , Cn−1 be a cyclic +ordering of the maximal cliques in an induced cycle of CR(G), where n ≥ 4 +and the indices are from Z/nZ. Assume that the edge between C0 and C1 is +a minimal edge of the induced cycle. Let S be C0 ∩ C1 and for i = 0, 1 let +Hi be the connected component of G−S that contains Ci −S. Then H0 and +H1 are distinct connected components and Ci − S is contained in H0 or H1 +for every i ∈ Z/nZ. Furthermore, either: +(i) H0 contains all of C0 − S, C2 − S, . . . , Cn−1 − S, +(ii) H1 contains all of C1 − S, C2 − S, . . . , Cn−1 − S, or +(iii) n = 4, and H0 contains C0 −S and C2 −S while H1 contains C1 −S +and C3 − S. +Proof. Note that because C0, C1, . . . , Cn−1 are distinct maximal cliques of +G, none of them is contained in S. Thus Ci − S is non-empty for all i. We +consider the connected components of G − S. Any set Ci − S is contained +in such a component. Because C0 and C1 form a separating pair, C0 − S +and C1 − S are contained in different connected components of G − S, so +H0 and H1 are distinct components. +Claim 3.5.1. Assume that i and j are distinct indices in Z/nZ such that +there are distinct connected components of G−S, call them Hi and Hj, that +contain Ci − S and Cj − S respectively. Assume also that Ci is adjacent in +CR(G) to Cp, where Cp − S is not contained in Hi and that Cj is adjacent +to Cq, where Cq − S is not contained in Hj. Then Ci and Cj are adjacent +in CR(G). +Proof. Note that because the cycle of CR(G) is induced, p is in {i − 1, i + 1} +and q is in {j − 1, j + 1}. Note also that Ci ∩ Cp is contained in S. If this +containment is proper then |Ci ∩ Cp| < |S| = |C0 ∩ C1| and we have violated +our assumption that the edge between C0 and C1 is minimal. Therefore Ci +and Cp both contain S. The same argument shows S ⊆ Cj ∩Cq. Now Ci∩Cj +is equal to S. Moreover Ci − S and Cj − S are in different components of +G − S, so Ci and Cj form a separating pair of maximal cliques. Hence they +are adjacent in CR(G). +□ +We colour the cliques of C0, C1, . . . , Cn−1 in the following way. For each +i ∈ Z/nZ, if Ci − S is contained in H0 we colour Ci red, and if Cj − S is in +H1 we colour Ci blue. Thus C0 is red and C1 is blue. +Claim 3.5.2. Any maximal clique Ci is either red or blue. +Proof. If the claim fails then there is some i ∈ Z/nZ−{0, 1} such that Ci−S +is contained in neither H0 nor H1. Let Hi be the connected component of +G − S that contains Ci − S. We colour any clique Cj in C0, C1, . . . , Cn−1 +green if Cj −S is contained in Hi. We know that the collections of red, blue, + +REDUCED CLIQUE GRAPHS +11 +and green cliques are all non-empty. So therefore we can find a red clique, +Cred, adjacent to a clique that is not red. We can similarly find Cblue, a blue +clique that is adjacent to a non-blue clique, and Cgreen, a green clique that +is adjacent to a clique that is not green. Now Claim 3.5.1 implies that Cred, +Cblue, and Cgreen are adjacent to each other in CR(G). As they are three +distinct vertices in an induced cycle of CR(G) with at least four vertices, +this is an immediate contradiction. +□ +If C1 is the only blue clique, then statement (i) holds and we have nothing +left to prove. Similarly, if C0 is the only red clique, then (ii) holds and we +are done. So we assume there are at least two red cliques and at least two +blue cliques. We can choose Cred and C′ +red to be distinct red cliques that are +adjacent to blue cliques, and we can choose Cblue and C′ +blue to be two distinct +blue cliques that are adjacent to red cliques. Now Claim 3.5.1 implies that +Cred and C′ +red are adjacent to both Cblue and C′ +blue. Thus the four cliques +induce a cycle in CR(G). This is impossible if n ≥ 5, so we conclude that +n = 4. Now C0 is a red clique and it is adjacent to two blue cliques. Thus +C1 and C3 are blue, C2 is red, and we are finished. +□ +The example in Figure 1 shows that a reduced clique graph may contain +an induced cycle with four vertices. +We will next show that there is no +example with an induced cycle of five vertices. +Lemma 3.6. There is no chordal graph G such that CR(G) has an induced +cycle with exactly five vertices. +Proof. Assume otherwise and let G be a chordal graph such that CR(G) +contains an induced cycle with five vertices. Let C0, C1, C2, C3, C4 be the +maximal cliques in this cycle, where the indices are from Z/5Z and Ci is +adjacent to Cj if and only if j ∈ {i−1, i+1}. By adding a constant to these +indices as necessary, we may assume that +|C0 ∩ C1| ≤ |Ci ∩ Ci+1| +for all i ∈ Z/5Z, so that the edge between C0 and C1 is a minimal edge of +the cycle. Let S be C0 ∩ C1. Note that S is non-empty. +Now we apply Lemma 3.5. By applying the permutation ρ: i �→ 1 − i +as necessary, we may assume that statement (ii) in Lemma 3.5 applies. +Therefore we let H0 and H1 be connected components of G − S such that +H0 contains C0 − S and H1 contains C1 − S, C2 − S, C3 − S, and C4 − S. +Claim 3.6.1. C0 ∩ C4 = S = C0 ∩ C1. +Proof. Because C0 − S and C4 − S are contained in different components of +G − S, it follows that C0 ∩ C4 ⊆ S. All we have left to prove is that this +containment is not proper. If it were proper, then we would contradict the +assumption that the edge between C0 and C1 is minimal. +□ +Claim 3.6.2. Neither C2 nor C3 contains S. + +12 +MAYHEW AND PROBERT +Proof. Note that C0 ∩ C2 ⊆ S because C0 − S and C2 − S are contained +in different components of G − S. +Certainly any path from a vertex of +C0 − C2 to a vertex of C2 − C0 must use a vertex of S. If C0 ∩ C2 = S, +then C0 and C2 form a separating pair, so C0 and C2 are adjacent in CR(G). +This contradicts the fact that C0 and C2 are non-consecutive vertices in an +induced cycle. The same argument shows that C3 does not contain S. +□ +Claim 3.6.3. C2 ∩ C4 ⊆ C1 and C3 ∩ C1 ⊆ C4. +Proof. Assume that x is a vertex of C2 ∩ C4 that is not in C1. By Claim +3.6.2 we can let y be a vertex in S − C2. Thus y is in C1 − C2. So x is in +C2 −C1 and y is in C1 −C2. Claim 3.6.1 implies that y is in C4. As x is also +in C4 we see that x and y are adjacent. Because C1 and C2 are adjacent in +CR(G) they have a non-empty intersection, but now the edge xy shows that +C1 and C2 do not form a separating pair and we have a contradiction. A +symmetric argument shows C3 ∩ C1 ⊆ C4. +□ +Claim 3.6.4. C2 contains a vertex of C1 − C4 and C3 contains a vertex of +C4 − C1. +Proof. By symmetry it suffices to prove the first statement. Assume that C2 +contains no vertex of C1 − C4. Because C1 and C2 are adjacent in CR(G), +they have at least one vertex in common. By our assumption, no vertex of +C1 ∩ C2 is in C1 − C4, so any such vertex must be in C1 ∩ C4. Therefore C2 +and C4 are not disjoint. Since C2 and C4 are not adjacent in CR(G), we can +let P be a (C2 ∩C4)-avoiding path from a vertex x ∈ C2 −C4 to y ∈ C4 −C2. +Our assumption means that x is not in C1 − C4, so it is in C2 − C1. Our +assumption and Claim 3.6.3 imply that C2 ∩ C4 = C2 ∩ C1. Therefore P +is a (C2 ∩ C1)-avoiding path. But Claim 3.6.2 shows that we can choose a +vertex z in S − C2. Thus z is in C1 − C2 and Claim 3.6.1 shows that z is +in C4. Assuming that z and y are not equal, they are adjacent, as both are +in C4. By appending (if necessary) the edge yz to the end of P we obtain +a (C1 ∩ C2)-avoiding path from a vertex in C2 − C1 to a vertex in C1 − C2. +Hence C1 and C2 do not form a separating pair and this contradicts the fact +that they are adjacent in CR(G). +□ +Claim 3.6.5. Either C2 ∩ (C1 ∩ C4) ⊆ C3 or C3 ∩ (C1 ∩ C4) ⊆ C2. +Proof. Note that C2 ∩ C3 is non-empty, since C2 and C3 are adjacent in +CR(G). If the claim fails, then we choose x ∈ (C2 ∩ C1 ∩ C4) − C3 and +y ∈ (C3 ∩ C1 ∩ C4) − C2. Now x and y are both in C1 ∩ C4, so they are +adjacent. Moreover x is in C2 − C3 and y is in C3 − C2. Thus C2 and C3 do +not form a separating pair and we have a contradiction. +□ +By using Claim 3.6.5, we will assume that C2 ∩ (C1 ∩ C4) is a subset of +C3. The other outcome from Claim 3.6.5 yields to a symmetric argument. +Using Claim 3.6.2 we choose a vertex x ∈ S that is not in C3. Note that +Claim 3.6.1 implies that S is contained in C1 ∩C4. So x is in (C1 ∩C4)−C3. +Our assumption therefore implies that x is not in C2. + +REDUCED CLIQUE GRAPHS +13 +By Claim 3.6.4 we can also choose y in C2 ∩(C1 −C4) and z in C3 ∩(C4 − +C1). Claim 3.6.3 implies that y is in C2 −C3 and z is in C3 −C2. Now x and +y are adjacent as they are both in C1, and x and z are adjacent as they are +both in C4. Note that C2 ∩ C3 is non-empty as C2 and C3 are adjacent in +CR(G). But the path with vertex sequence y, x, z is (C2 ∩ C3)-avoiding, so +C2 and C3 do not form a separating pair. This final contradiction completes +the proof. +□ +Lemma 3.6 shows that the class of reduced clique graphs is contained in +the class of graphs with no length-five induced cycle. We next show that +this containment is proper. +Proposition 3.7. Let n ≥ 4 be an integer. There is no chordal graph G +such that either C(G) or CR(G) is a cycle with n vertices. +Proof. Szwarcfiter and Bornstein characterise the clique graphs of chordal +graphs [11]. In particular H is isomorphic to C(G) for some chordal graph +G if and only if H has a spanning tree T such that whenever u and v are +adjacent in H, the path of T from u to v induces a clique of H. +Now +assume H is a cycle with at least four vertices. Any spanning tree of H is a +Hamiltonian path. The end vertices of this path are adjacent in H, but the +path of the spanning tree between these vertices does not induce a clique. +Therefore H is not isomorphic to C(G) for any chordal graph G. +We turn to reduced chordal graphs. Assume for a contradiction that G is +a chordal graph with C0, C1, . . . , Cn−1 as its list of maximal cliques, where +the indices are from Z/nZ, and Ci is adjacent to Cj in CR(G) if and only if +j ∈ {i − 1, i + 1}. We can assume without loss of generality that the edge +between C0 and C1 is a minimal edge of CR(G). Let S be C0 ∩ C1. Assume +that statement (iii) in Lemma 3.5 holds. Thus n = 4 and there are distinct +connected components, H0 and H1, of G − S such that H0 contains C0 − S +and C2 − S while H1 contains C1 − S and C3 − S. Note that C0 ∩ C3 ⊆ S, +and in fact C0 ∩ C3 is equal to S, or else the minimality of the C0-C1 edge +is contradicted. +Either C0 ∩ C2 is empty, or it is not. In the latter case, we can apply +Proposition 2.4 to C0 and C2. +We see that either C0 ∩ C1 or C0 ∩ C3 +properly contains C0 ∩C2. By symmetry, we can assume C0 ∩C2 is a proper +subset of C0 ∩ C1 = S. Thus C0 − S and C2 − S are disjoint sets. They are +contained in the same connected component of G − S, so we can let P be a +shortest-possible path of H0 from a vertex of C0 − S to a vertex of C2 − S. +On the other hand, if C0 ∩ C2 is empty, then C0 − S and C2 − S are again +disjoint subsets in H0, so we again let P be a shortest-possible path of H0 +from C0 − S to C2 − S. In either case, P contains exactly one vertex of C0 +and exactly one vertex of C2. Then P must contain at least one edge, and +this edge is in a maximal clique that is equal to neither C0 nor C2. Nor can +this maximal clique be C1 or C3, because any edge of P is contained in H0. +So we have a contradiction in the case that (iii) in Lemma 3.5 holds. + +14 +MAYHEW AND PROBERT +Now we assume that either (i) or (ii) holds. By applying the permutation +ρ: i �→ 1 − i as necessary, we will assume that H0 and H1 are distinct +connected components of G − S, and that H0 contains C0 − S while H1 +contains Ci − S for i = {1, 2, . . . , n − 1}. By the same argument as earlier, +we can see that Cn−1 contains S, or else the choice of the C0 − C1 edge is +contradicted. +Now C1 ∩ Cn−1 contains S, and C1 and Cn−1 are non-adjacent in CR(G). +We apply Proposition 2.4 and see that there is a path of CR(G) from C1 to +Cn−1 such that every intersection of consecutive cliques in the path properly +contains C1∩Cn−1. This path is either C1, C0, Cn−1, or it is C1, C2, . . . , Cn−1. +Assume the former. Then C1 ∩ C0 = S properly contains C1 ∩ Cn−1 ⊇ S +and we have a contradiction. Hence any intersection of consecutive cliques +in C1, C2, . . . , Cn−1 properly contains C1 ∩ Cn−1, and hence contains S. It +follows that C2 contains S and thus C0 ∩ C2 is non-empty. +Since C0 − S and C2 − S are contained in different components of G − S, +any path of a vertex from C0 − C2 to a vertex of C2 − C0 must contain a +vertex of S = C0 ∩ C2. Thus C0 and C2 form a separating pair in G, and +hence they are adjacent in CR(G), which is a contradiction. +□ +3.2. Clique graphs vs. reduced clique graphs. Consider the classes +{C(G)} and {CR(G)}, where G ranges over all chordal graphs. Proposition +3.3 and Lemma 3.6 show that the wheel with five spokes is isomorphic to +a graph in the former class but not the latter. +Is there a graph that is +isomorphic to a graph in the latter class but not the former? We will show +that the answer is, once again, yes. Recall that if G and G′ are disjoint +graphs, then G ⊠ G′ is obtained from the union of G and G′ by making +every vertex of G adjacent to every vertex of G′. We use Pn to denote the +path of length n. +Lemma 3.8. Let m, n ≥ 1 be integers. Then Pm ⊠ Pn is isomorphic to +the reduced clique graph of a chordal graph. If n ≥ 22, then Pn ⊠ Pn is not +isomorphic to the clique graph of a chordal graph. +Proof. Let G be the graph obtained from the disjoint union of Pm and Pn +and adding a new vertex that is adjacent to every vertex of the disjoint +union. It is easy to confirm that G is chordal, and that CR(G) is isomorphic +to Pm ⊠ Pn. +For the second statement, we let H be a graph with disjoint induced paths +Pu = u0, u1, . . . , un−1 and Pv = v0, v1, . . . , vn−1, where n ≥ 22 and every ui +is adjacent to every vj. Thus H is isomorphic to Pn ⊠ Pn. We will assume +for a contradiction that H is isomorphic to C(G) for some chordal graph G. +Because C(G) is connected it follows easily that G is connected, so we can +apply Theorem 3.2 and deduce that H has a spanning tree T, where the +path of T from u to v induces a clique of H whenever u and v are adjacent +in H. + +REDUCED CLIQUE GRAPHS +15 +Claim 3.8.1. Let i and j be integers satisfying 0 < i, j < n−1. The path of +T from ui to vj is contained in one of: {ui, ui+1, vj, vj+1}, {ui, ui+1, vj−1, vj}, +{ui−1, ui, vj, vj+1}, {ui−1, ui, vj−1, vj}. +Proof. Let P be the path of T from ui to vj. Since ui is adjacent to vj it +follows that P induces a clique of H. As ui is not adjacent to any of the +vertices in u0, . . . , ui−2, ui+2, . . . , un−1, it follows that the vertices of P that +are in Pu belong to {ui−1, ui, ui+1}. Similarly, the vertices of P that are in +Pv belong to {vj−1, vj, vj+1}. But ui−1 is not adjacent to ui+1, so P does +not contain both. The claim follows by symmetry. +□ +Claim 3.8.1 implies that the path of T between ui and vj has at most +three edges. +Let P be a longest-possible path of T and let p0, p1, . . . , pk−1 be the ver- +tices of P. For i = 0, 1, . . . , k − 1, let Ui be the set of vertices in Pu such +that u is in Ui if and only if the shortest path of T from u to a vertex in P +contains pi. We define Vi to be the analogous set of vertices in Pv. Note that +(U0, U1, . . . , Uk−1) is a partition of the vertices of Pu, and (V0, V1, . . . , Vk−1) +is a partition of the vertices of Pv. +Claim 3.8.2. Either +max{|Ui|: 0 ≤ i ≤ k − 1} ≤ 3 +or +max{|Vi|: 0 ≤ i ≤ k − 1} ≤ 3. +Proof. Assume for a contradiction that |Ui| ≥ 4 and |Vj| ≥ 4. Let p and q, +respectively, be the smallest (largest) integers such that up, uq ∈ Ui. Then +q − 1 > p + 1 because |Ui| ≥ 4. In the same way, let s and t be the smallest +(largest) integers such that vs, vt ∈ Vj. Then t−1 > s+1. It is simple to see +from Claim 3.8.1 that the path of T from up to vs has no vertex in common +with the path of T from uq to vt. But this contradicts the fact that both +paths contain pi and pj. +□ +By using Claim 3.8.2, we will assume without loss of generality that |Vi| ≤ +3 for each i = 0, 1, . . . , k − 1. Since (U0, U1, . . . , Uk−1) is a partition of the +vertices in Pu we can choose i so that Ui contains a vertex x. We claim that +if j ≤ i − 4 or j ≥ i + 4, then Vj = ∅. If this fails, then the path of T from a +vertex in Vj to x contains at least four edges of P. But this contradicts our +earlier conclusion that any path of T from a vertex of Pu to a vertex of Pv +contains at most three edges. So now the vertices of Pv belong to +Vi−3 ∪ Vi−2 ∪ · · · ∪ Vi+2 ∪ Vi+3 +and this union has cardinality at most 7 × 3. Thus Pv contains at most 21 +vertices and this contradicts n ≥ 22. +□ +4. Conclusions and open problems +Given Lemma 3.6 it might be natural to believe that reduced clique graphs +cannot have any induced cycles with five or more vertices. But Figure 3 +shows a chordal graph G where CR(G) has an induced cycle with six vertices. + +16 +MAYHEW AND PROBERT +5 +10 +235 +346 +1234 +G +C(G) +CR(G) +2 +4 +3 +6 +1 +8 +9 +3479 +2378 +710 +2347 +2347 +235 +346 +1234 +3479 +2378 +710 +2347 +2347 +7 +Figure 3. +Nonetheless we believe the following to be true. +Conjecture 4.1. There is no chordal graph G such that CR(G) contains +an induced cycle with seven or more vertices. +So far as we have been able to tell, every chordal graph is isomorphic +to both a clique graph, and to a reduced clique graph. We conjecture this +holds generally. +Conjecture 4.2. Let H be a chordal graph. There are chordal graphs G +and G′ such that H is isomorphic to both C(G) and CR(G′). +Szwarcfiter and Bornstein present a polynomial-time algorithm for decid- +ing whether a given graph is isomorphic to C(G) for some chordal graph G +[11]. Their techniques do not obviously extend to recognising reduced clique +graphs. Nonetheless, we will make the following conjecture. +Conjecture 4.3. There is a polynomial-time algorithm for deciding whether +a given graph is isomorphic to CR(G) for some chordal graph G. +More informally, we ask if there is a structural description for reduced +clique graphs that is analogous to Theorem 3.2. +References +[1] Jean R. S. Blair and Barry Peyton, An introduction to chordal graphs and clique +trees, Graph theory and sparse matrix computation, IMA Vol. Math. Appl., vol. 56, +Springer, New York, 1993, pp. 1–29. +[2] Peter Buneman, A characterisation of rigid circuit graphs, Discrete Math. 9 (1974), +205–212. +[3] Philippe Galinier, Michel Habib, and Christophe Paul, Chordal graphs and their clique +graphs, Graph-theoretic concepts in computer science (Aachen, 1995), Lecture Notes +in Comput. Sci., vol. 1017, Springer, Berlin, 1995, pp. 358–371. +[4] F˘anic˘a Gavril, The intersection graphs of subtrees in trees are exactly the chordal +graphs, J. Combinatorial Theory Ser. B 16 (1974), 47–56. + +REDUCED CLIQUE GRAPHS +17 +[5] Michel Habib and Vincent Limouzy, On some simplicial elimination schemes for +chordal graphs, DIMAP Workshop on Algorithmic Graph Theory, Electron. Notes +Discrete Math., vol. 32, Elsevier Sci. B. V., Amsterdam, 2009, pp. 125–132. +[6] Michel Habib and Juraj Stacho, Reduced clique graphs of chordal graphs, European +J. Combin. 33 (2012), no. 5, 712–735. +[7] Terry A. McKee, Minimal weak separators of chordal graphs, Ars Combin. 101 (2011), +321–331. +[8] Yasuko Matsui, Ryuhei Uehara, and Takeaki Uno, Enumeration of the perfect se- +quences of a chordal graph, Theoret. Comput. Sci. 411 (2010), no. 40-42, 3635–3641. +[9] Dillon Mayhew and Andrew Probert, Supersolvable saturated matroids and chordal +graphs. In preparation. +[10] Donald J. Rose, Triangulated graphs and the elimination process, J. Math. Anal. Appl. +32 (1970), 597–609. +[11] Jayme L. Szwarcfiter and Claudson F. Bornstein, Clique graphs of chordal and path +graphs, SIAM J. Discrete Math. 7 (1994), no. 2, 331–336. + diff --git a/BNE2T4oBgHgl3EQfRgdU/content/tmp_files/load_file.txt b/BNE2T4oBgHgl3EQfRgdU/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9fc842d337f1c12e1de3589e9fdf3120170bc547 --- /dev/null +++ b/BNE2T4oBgHgl3EQfRgdU/content/tmp_files/load_file.txt @@ -0,0 +1,818 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf,len=817 +page_content='REDUCED CLIQUE GRAPHS: A CORRECTION TO “CHORDAL GRAPHS AND THEIR CLIQUE GRAPHS” DILLON MAYHEW AND ANDREW PROBERT Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Galinier, Habib, and Paul introduced the reduced clique graph of a chordal graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' The nodes of the reduced clique graph are the maximal cliques of G, and two nodes are joined by an edge if and only if they form a non-disjoint separating pair of cliques in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' In this case the weight of the edge is the size of the intersection of the two cliques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' A clique tree of G is a tree with the maximal cliques of G as its nodes, where for any v ∈ V (G), the subgraph induced by the nodes containing v is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Galinier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' prove that a spanning tree of the reduced clique graph is a clique tree if and only if it has maximum weight, but their proof contains an error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We explain and correct this error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' In addition, we initiate a study of the structure of reduced clique graphs by proving that they cannot contain any induced cycle of length five (although they may contain induced cycles of length three, four, or six).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We show that no cycle of length four or more is isomorphic to a reduced clique graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We prove that the class of clique graphs of chordal graphs is not comparable to the class of reduced clique graphs of chordal graphs by providing examples that are in each of these classes without being in the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Introduction We consider only simple graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' A chord of a cycle is an edge that joins two vertices of the cycle without being in the cycle itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' A graph is chordal if any cycle with at least four vertices has a chord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' A clique is a set of pairwise adjacent vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' If S is a set of vertices and P is a path, then P is S-avoiding if no internal vertex of P is in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Assuming that a and b are distinct vertices, an ab-separator is a set S of vertices not containing either a or b such that there is no S-avoiding path from a to b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' If, in addition, S does not properly contain an ab-separator then it is a minimal ab-separator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' If G is a chordal graph, then C(G) is the corresponding clique graph (also known as the clique intersection graph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' The vertices of C(G) are the maximal cliques of G, and two maximal cliques are adjacent in C(G) if and only if they have a non-empty intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' The vertices of the reduced clique graph, CR(G), are again the maximal cliques of G, but C and C′ are adjacent in CR(G) if and only if C ∩ C′ ̸= ∅ and C and C′ form a separating pair: that is, there is no (C ∩ C′)-avoiding path from a vertex in C − C′ 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='03781v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='CO] 10 Jan 2023 2 MAYHEW AND PROBERT to a vertex in C′ − C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Note that the vertices of CR(G) are identical to the vertices of C(G), and every edge of CR(G) is an edge of C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' 1 2 3 4 5 6 7 8 9 10 234589 234689 235789 123 8910 G C(G) CR(G) 234589 234689 235789 123 8910 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' A chordal graph, its clique graph, and its reduced clique graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' The reduced clique graph was introduced in [3] (where it is called a clique graph) and studied further in [5–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let G be a graph, and let T be a tree whose vertices are the maximal cliques of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' If, for every v ∈ V (G), the maximal cliques of G that contain v induce a subtree of T, then T is a clique tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Clique trees were introduced by Gavril [4], who proved that a graph has a clique tree exactly when it is chordal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We weight each edge of CR(G) as follows: the edge joining cliques C and C′ is weighted with |C ∩ C′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' The following result is [3, Theorem 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let G be a connected chordal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let T be a spanning tree of CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Then T is a clique tree if and only if it is a maximum-weight spanning tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Although the statement of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='1 is correct, it is not proved in [3, Theorem 6] because of a flaw in the argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' The issue arises in the proof that a maximum-weight spanning tree must be a clique tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We illustrate the error by using the same argument to prove a false statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Non-theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let G be a chordal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , Cn be the sequence of maximal cliques in a path of CR(G) where n > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Assume that there is a vertex v of G such that v is in C0 ∩ Cn, but in none of the cliques C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , Cn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Then C0 and Cn are adjacent in CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Non-proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Consider the subgraph G′ of G induced by C0 ∪ C1 ∪ · · · ∪ Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Thus G′ is chordal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' From [10, Corollary 2] we see that either v is a simplicial vertex (meaning that the neighbours of v in G′ form a clique), or there is a pair, a, b, of vertices such that v belongs to a minimal ab-separator of G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' In the former case v is in a unique maximal clique of G′ ([1, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' But C0 and Cn are distinct maximal cliques of G′ that contain v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Therefore REDUCED CLIQUE GRAPHS 3 we can let S be a minimal ab-separator of G′, where v is in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' The proof of [2, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='3] shows that there are two distinct maximal cliques, Da and Db, of G′ such that Da and Db properly contain S, and Da − S is in the same connected component of G′ − S as a, while Db − S is in the same component as b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Thus Da and Db are maximal cliques of G′ that contain v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' But the only maximal cliques of G′ that contain v are C0 and Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Therefore we can assume without loss of generality that Da = C0 and Db = Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Any path from a vertex of C0 − Cn to a vertex of Cn − C0 must contain a vertex in S = C0 ∩ Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Therefore C0 and Cn form a non-disjoint separating pair, so C0 and Cn are adjacent in CR(G), as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' □ We can see that this non-theorem is, indeed, not a theorem by examining Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Set C0, C1, and C2 to be the maximal cliques {2, 3, 4, 6, 8, 9}, {1, 2, 3}, and {2, 3, 5, 7, 8, 9}, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Thus C0, C1, C2 is the vertex sequence of a path in CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' The vertex 8 is in C0 ∩ C2, but not in C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' However C0 and C2 are not adjacent in CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' The error in the “proof” lies in the claim that “the only maximal cliques of G′ that contain v are C0 and Cn”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' This need not be true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Indeed, {2, 3, 4, 5, 8, 9} is a maximal clique in the subgraph induced by C0 ∪ C1 ∪ C2, and it contains 8, but it is not equal to either C0 or C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Exactly the same error appears in the proof of [3, Theorem 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Nonetheless, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='1 is true, and we prove it in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Reduced clique graphs and clique trees In [9] we will apply our main theorem to some matroid problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' For these purposes we would like to extend its scope somewhat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Instead of weighting the edges of CR(G) with sizes of intersections, we consider more general weightings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let G be a chordal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We consider a function σ which takes {∅} ∪ {C ∩ C′ : C, C are distinct maximal cliques of G} to non-negative integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We insist that σ(∅) = 0 and if X and X′ are in the domain of σ and X ⊂ X′, then σ(X) < σ(X′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' In such a case the function σ is a legitimate weighting of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let G be a connected chordal graph and let σ be a legitimate weighting of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Every clique tree is a spanning tree of CR(G) and every edge of CR(G) is contained in a clique tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Moreover, a spanning tree of CR(G) is a clique tree if and only if it has maximum weight amongst all spanning trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Note that the function that takes each intersection C ∩ C′ to |C ∩ C′| is a legitimate weighting, so Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='2 does indeed imply Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We now start proving the intermediate results required for the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' 4 MAYHEW AND PROBERT Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let G be a chordal graph, and let C and C′ be maximal cliques of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let S be a set of vertices that contains C∩C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let v0, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , vk be the vertex sequence of P, a shortest-possible S-avoiding path from a vertex in C − C′ to a vertex in C′ − C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Then (C ∩ C′) ∪ {vi, vi+1} is a clique for each i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' If C ∩ C′ = ∅ then the result holds trivially, so we assume C ∩ C′ is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Note that every vertex in C ∩ C′ is adjacent to v0, and also to vk, since these vertices are in C − C′ and C′ − C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Now the result can only fail if there is a vertex x ∈ C ∩ C′ that is not adjacent to vi for some i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , k − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let p be the largest integer such that p < i and x is adjacent to vp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Similarly, let q be the smallest integer such that q > i and x is adjacent to vq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Consider the cycle obtained by adding the edges vpx and vqx to vp, vp+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , vq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' This cycle contains the distinct vertices vp, vi, vq, and x, so it must contain a chord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' No chord can join two vertices in the path P, since P is as short as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Thus any chord is incident with x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' But x is not adjacent to any of the vertices in vp+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , vq−1 by the choice of p and q, so we have a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' □ Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let G be a chordal graph, and let C and C′ be maximal cliques of G where C ∩ C′ ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' If C and C′ are not adjacent in CR(G), then they are joined by a path of CR(G) with vertex sequence C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , Cs, where each Ci ∩ Ci+1 properly contains C ∩ C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Assume this fails for C and C′, and they have been chosen so that C ∩ C′ is as large as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let S be C ∩ C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Because C and C′ are not adjacent in CR(G), but S ̸= ∅, it follows that there is an S-avoiding path from a vertex in C − C′ to a vertex in C′ − C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let v0, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , vk be the vertex sequence of such a path, where k is as small as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We assume v0 is in C − C′ while vk is in C′ − C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We apply Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='3 and for each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , k, we let Di be a maximal clique of G that contains S ∪{vi−1, vi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Set D0 to be C and set Dk+1 to be C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Note that Di ̸= Dj when i < j, because vi−1 is not adjacent to vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' For each i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , k, the intersection of Di and Di+1 contains S as well as vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' If Di and Di+1 are adjacent in CR(G) then we let Pi be the path of CR(G) consisting of Di, Di+1, and the edge between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Otherwise Di and Di+1 are not adjacent in CR(G) and the assumption on the cardinality of S means that there is a path Pi of CR(G) from Di to Di+1 such that every intersection of consecutive cliques in Pi properly contains S ∪ vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We concatenate the paths P0, P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , Pk and obtain a walk of CR(G) from C to C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' The intersection of any two consecutive cliques in this walk properly contains S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' It follows that there is a path of CR(G) from C to C′ with exactly the same property, and now C and C′ fail to provide a counterexample after all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' □ Figure 2 illustrates Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' The intersection of cliques C = {1, 2, 3} and C′ = {3, 5, 7, 8} is {3} ̸= ∅, but C and C′ are not adjacent REDUCED CLIQUE GRAPHS 5 in CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' However, there is a path between C and C′ in CR(G), and the intersection of any consecutive two cliques in the path properly contains {3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' 1 6 4 7 5 2 3 8 123 2345 3567 3456 3578 G CR(G) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let G be a connected chordal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let T be a clique tree of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Assume that C and C′ are maximal cliques of G that are adjacent in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Then C and C′ are adjacent in CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Assume C and C′ are adjacent in T, but not in CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We partition the maximal cliques of G as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let U be the set of maximal cliques of G such that D is in U if and only if the path of T from D to C does not contain C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Similarly, define U′ so that D′ is in U′ if and only if the path of T from D′ to C′ does not contain C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Note that every maximal clique of G is in exactly one of U or U′, since T is a tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Furthermore C is in U and C′ is in U′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let U be the union of the cliques in U, and let U ′ be the union of the cliques in U′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Every vertex is in at least one maximal clique so U ∪ U ′ = V (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Note that C ⊆ U and C′ ⊆ U ′, so neither U nor U ′ is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' If U ∩ U ′ = ∅, then we choose u ∈ U and u′ ∈ U ′ so that u and u′ are adjacent in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' (We are able to do so because G is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=') The edge between u and u′ is contained in a maximal clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' If this maximal clique is in U then u′ is in U ∩ U ′, and if it is in U′ then u is in U ∩ U ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' In either case we have a contradiction, so U ∩ U ′ ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Choose an arbitrary vertex v in U ∩ U ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Choose D ∈ U and D′ ∈ U′ such that v is in D ∩ D′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Because T is a clique tree, it follows that v is contained in all the cliques belonging to the path of T from D to D′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' In particular, v is contained in C and C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Thus U ∩ U ′ ⊆ C ∩ C′ and C ∩ C′ is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let S be C ∩ C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Since C and C′ are not adjacent in CR(G), we can apply Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='4 and find a path P of CR(G) from C to C′, where the intersection of each pair of consecutive cliques in this path properly contains S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Since C is in U and C′ is in U′, there is an edge of P that joins a clique D ∈ U to a clique D′ ∈ U′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Then D∩D′ properly contains S, so we choose v in (D ∩D′)−S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Again using the fact that T is a clique tree, we see that the path of T from D to D′ consists of cliques that contain v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' In particular, v is in C ∩ C′ = S, and we have a contradiction that completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' □ 6 MAYHEW AND PROBERT It follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='5 that every clique tree of G is a spanning tree of CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let G be a connected chordal graph and let σ be a legiti- mate weighting of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let T be a clique tree of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let C and C′ be maximal cliques of G that are adjacent in C(G) and let P be the path of T between C and C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' The weight of any edge in P is at least σ(C ∩ C′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Moreover, if C and C′ are adjacent in CR(G), then at least one edge in P has weight equal to σ(C ∩ C′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let S be C ∩ C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let P be the path of T from C to C′, and let the cliques in this path be C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , Cn, where C0 = C and Cn = C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Note that P is a path of CR(G) by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Thus any two consecutive cliques in the path have a non-empty intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Assume σ(Ci ∩ Ci+1) < σ(S) for some i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' If S were a subset of Ci ∩ Ci+1, then we would have σ(S) ≤ σ(Ci ∩Ci+1) by the definition of a legitimate weighting, but this is not true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Therefore we can choose v to be a vertex in S − (Ci ∩ Ci+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Now v is a vertex of both C and C′, but the path of T between C and C′ contains at least one maximal clique (either Ci or Ci+1) that does not contain v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' This contradicts the fact that T is a clique tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Therefore the weight of any edge in P is at least equal to σ(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Now assume that C and C′ are adjacent in CR(G), so that they form a separating pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' That is, there are distinct connected components of G − S that contain, respectively, C −S and C′−S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' There must be maximal cliques D and D′ that are adjacent in P, where D − S is in the same connected component of G−S as C−S, and D′−S is not in this connected component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' This means that D ∩ D′ is contained in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Hence σ(D ∩ D′) ≤ σ(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' The previous paragraph shows that σ(D ∩ D′) ≥ σ(S), so the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' □ The proof of the next result is a straightforward adaptation of a proof given by Blair and Peyton [1, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let G be a connected chordal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let σ be a legitimate weighting of G and let T be a spanning tree of C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Then T is a clique tree of G if and only if it is a maximum-weight spanning tree of C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' If T is a clique tree, then for any pair of maximal cliques, C and C′, such that C and C′ are adjacent in C(G), the weight of the edge between C and C′ is no greater than the weight of any edge in the path of T between C and C′ (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' It immediately follows that T has maximum weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' For the other direction, we assume that T is a maximum-weight spanning tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Because every chordal graph has a clique tree, and any clique tree is a spanning tree of CR(G) (and hence of C(G)), we can choose a clique tree T ′ so that T and T ′ have as many edges in common as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We can choose an edge in T that is not in T ′, because otherwise there is nothing left for us to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' So let e be such an edge, and assume that e joins maximal cliques C and C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' There are two connected components of T\\e, one containing C REDUCED CLIQUE GRAPHS 7 and the other containing C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let P be the path of T ′ from C to C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We let f be an edge of P which joins two cliques that are not in the same component of T\\e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Note that f is an edge of T ′, and hence an edge of C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' If (T − e) ∪ f is not a spanning tree of C(G), then there is a path of T between the end-vertices of f that does not use e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' But the end-vertices of f are in different connected components of T\\e, so (T − e) ∪ f is indeed a spanning tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Similarly, if (T ′ − f) ∪ e is not a spanning tree, then there is a path of T ′ between C and C′ that does not contain f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' But P is the unique path of T ′ between C and C′, and f is an edge of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' So (T − e) ∪ f and (T ′ − f) ∪ e are both spanning trees of C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Applying Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='6 to the clique tree T ′ shows that the weight of f is at least the weight of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Since T is a maximum-weight spanning tree, and (T − e) ∪ f is a spanning tree it follows that the weights on e and f must be equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let D and D′ be the maximal cliques joined by f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Any element that is in both C and C′ must be in all the cliques in P, since T ′ is a clique tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' This shows that C ∩ C′ ⊆ D ∩ D′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' If C ∩ C′ were a proper subset of D ∩ D′, then the definition of a legitimate weighting would mean that the weight of e is strictly less than the weight of f, which is not true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Therefore C ∩ C′ = D ∩ D′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We note that (T ′−f)∪e cannot be a clique tree, since it has one more edge in common with T than T ′ does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Therefore we choose a vertex v ∈ V (G) so that the maximal cliques containing v do not induce a subtree of (T ′−f)∪e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let T ′′ be the subtree of T ′ induced by the maximal cliques containing v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Then f is in T ′′, or else T ′′ would be a subtree of (T ′ − f) ∪ e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' This means that v is in D ∩ D′ = C ∩ C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' So both C and C′ are in T ′′, but they are not in the same component of T ′′\\f, because in that case (T ′ − f) ∪ e would contain a cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' So e joins two vertices of T ′′ that are in different components of T ′′\\f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Thus (T ′′ − f) ∪ e is a subtree of (T ′ − f) ∪ e, and we have a contradiction that completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' □ Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We have already noted that every clique tree is a spanning tree of CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let T be a clique tree of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Then T is a maximum- weight spanning tree of C(G) by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' But every edge of T is an edge of CR(G), by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Since CR(G) is a subgraph of C(G) it follows that T is a maximum-weight spanning tree of CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' For the other direction, we let T be a maximum-weight spanning tree of CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We claim that T is also a maximum-weight spanning tree of C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' To prove this claim, let e be an arbitrary edge of C(G) that is not in T, let C and C′ be the maximal cliques of G joined by e, and let P be the path of T that joins C and C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' If e is an edge of CR(G), then the weight of e is no greater than the weight of any edge in P, since T is a maximum-weight spanning tree of CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Therefore we assume that e is not an edge of CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Now it follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='4 and the definition of a legitimate weighting that the edges in P all have weight strictly greater than the weight of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' In either case, the weight of e does not exceed the 8 MAYHEW AND PROBERT weight of any edge in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' This implies that T is indeed a maximum-weight spanning tree of C(G), and thus T is a clique tree of G by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' To complete the proof, we let e be an arbitrary edge of CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We will prove that e is in a maximum-weight spanning tree of CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We let C and C′ be the maximal cliques joined by e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let T be an arbitrary maximum- weight spanning tree of CR(G), so that T is a clique tree by the previous paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' If e is in T then we have nothing left to prove, so assume that P is the path of T joining C to C′, where P contains more than one edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='6 shows that P contains an edge, f, with weight equal to the weight of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Now (T − f) ∪ e is a maximum-weight spanning tree of CR(G) that contains e, and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' □ From the previous arguments we can deduce further additional facts, both noted in [3]: any edge that is in C(G) but not CR(G) cannot be in any maximum-weight spanning tree of C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Secondly, CR(G) is in fact the union of all clique trees of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Although the next fact is incidental to our main results here, we note it for a future application in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let G be a connected chordal graph, and let T be a clique tree of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let C and C′ be adjacent in T and let S be C ∩ C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Assume that D and D′ are maximal cliques of G and the path of T from D to D′ contains both C and C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Then D − S and D′ − S are in different connected components of G − S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let U be the family of maximal cliques of G such that D is in U if and only if the path of T from D to C does not contain C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Similarly, we let U′ be the family of maximal cliques where D′ is in U′ if and only if the path of T from D′ to C′ does not contain C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Note that every maximal clique of G belongs to exactly one of U and U′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We are asserting that if D ∈ U and D′ ∈ U′, then D − S and D′ − S are in different connected components of G − S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Assume that this fails for D and D′, where D ∩ D′ is as large as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let H be the connected component of G − S that contains both D − S and D′ − S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let P be the path of T from D to D′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Therefore P contains both C and C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let v be an arbitrary vertex of D ∩ D′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Then v is in every maximal clique that appears in P, since T is a clique tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' In particular, v is in C and C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Thus v is in S, and this shows that D ∩ D′ is contained in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let v0, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , vk be the vertex sequence of a shortest-possible path of H from a vertex v0 ∈ D − S to a vertex vk ∈ D′ − S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' This is an S-avoiding path, where S contains D ∩ D′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Thus we can apply Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' For i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , k we let Di be a maximal clique of G that contains (D ∩ D′) ∪ {vi−1, vi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let D0 be D and let Dk+1 be D′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Note that each Di − S is contained in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' This is true for D0 and Dk+1 by definition, and every other Di contains the edge vi−1vi, which is in the path of H from v0 to vk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Since D0 is in U and Dk+1 is in U′, we can choose i so that Di is in U and Di+1 is in U′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' The intersection of Di and Di+1 is larger than D ∩ D′, since it REDUCED CLIQUE GRAPHS 9 contains (D ∩ D′) ∪ vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' As Di − S and Di+1 − S are both contained in H we have a contradiction to the choice of D and D′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' The structure of reduced clique graphs Habib and Stacho comment on the possibility of investigating the struc- ture of graphs that are isomorphic to reduced clique graphs [6, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' 714].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' In this section we make a contribution to this investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We start by answering an obvious question that requires a non-trivial proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let G be a chordal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Then CR(G) is connected if and only if G is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Assume that H and H′ are distinct connected components of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' No maximal clique of H can share a vertex with a maximal clique of H′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' It follows that there be no path of CR(G) that joins two such cliques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Thus CR(G) is not connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' The other direction is stated without proof in [6, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' 716].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Assume that G is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Since G is chordal it has a clique tree [4, Theorem 2], and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='5 shows that every edge of the clique tree is an edge of CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Thus CR(G) has a spanning tree, so it is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' □ Next we note a characterisation of clique graphs due to Szwarcfiter and Bornstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='2 ([11, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' The graph H is isomorphic to C(G) for some connected chordal graph G if and only if H has a spanning tree T such that whenever u and v are adjacent in H, the path of T from u to v induces a clique of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Induced cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Next we observe that clique graphs can have induced cycles of any length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We will later show that this is not true for reduced clique graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' For an integer n ≥ 3 the wheel graph with n spokes is obtained from a cycle of n vertices by adding a new vertex and making it adjacent to all vertices of the cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Thus the wheel graph with n spokes has an induced cycle of n vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' For each integer n ≥ 3 the wheel graph with n spokes is isomorphic to the clique graph of a chordal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' This is easy to prove using Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='2, but we will give a direct construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Start with a clique on the n + 1 vertices u0, u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , un−1, x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' For each i ∈ Z/nZ, add a new vertex vi and make it adjacent to ui and ui+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Call the resulting graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' It is easy to verify that G is chordal, and its maximal cliques are {u0, u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , un−1, x} along with {vi, ui, ui+1} for each i ∈ Z/nZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' The result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' □ Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let G be a chordal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , Cn−1 be a cyclic ordering of the maximal cliques in an induced cycle of CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We take the indices to be from Z/nZ, so Ci and Cj are adjacent in CR(G) if and only if 10 MAYHEW AND PROBERT j ∈ {i − 1, i + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' If |Ci ∩ Ci+1| ≤ |Cj ∩ Cj+1| for every j ∈ Z/nZ, then we say that the edge between Ci and Ci+1 is a minimal edge of the cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let G be a chordal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , Cn−1 be a cyclic ordering of the maximal cliques in an induced cycle of CR(G), where n ≥ 4 and the indices are from Z/nZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Assume that the edge between C0 and C1 is a minimal edge of the induced cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let S be C0 ∩ C1 and for i = 0, 1 let Hi be the connected component of G−S that contains Ci −S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Then H0 and H1 are distinct connected components and Ci − S is contained in H0 or H1 for every i ∈ Z/nZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Furthermore, either: (i) H0 contains all of C0 − S, C2 − S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , Cn−1 − S, (ii) H1 contains all of C1 − S, C2 − S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , Cn−1 − S, or (iii) n = 4, and H0 contains C0 −S and C2 −S while H1 contains C1 −S and C3 − S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Note that because C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , Cn−1 are distinct maximal cliques of G, none of them is contained in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Thus Ci − S is non-empty for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We consider the connected components of G − S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Any set Ci − S is contained in such a component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Because C0 and C1 form a separating pair, C0 − S and C1 − S are contained in different connected components of G − S, so H0 and H1 are distinct components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Assume that i and j are distinct indices in Z/nZ such that there are distinct connected components of G−S, call them Hi and Hj, that contain Ci − S and Cj − S respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Assume also that Ci is adjacent in CR(G) to Cp, where Cp − S is not contained in Hi and that Cj is adjacent to Cq, where Cq − S is not contained in Hj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Then Ci and Cj are adjacent in CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Note that because the cycle of CR(G) is induced, p is in {i − 1, i + 1} and q is in {j − 1, j + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Note also that Ci ∩ Cp is contained in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' If this containment is proper then |Ci ∩ Cp| < |S| = |C0 ∩ C1| and we have violated our assumption that the edge between C0 and C1 is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Therefore Ci and Cp both contain S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' The same argument shows S ⊆ Cj ∩Cq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Now Ci∩Cj is equal to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Moreover Ci − S and Cj − S are in different components of G − S, so Ci and Cj form a separating pair of maximal cliques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Hence they are adjacent in CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' □ We colour the cliques of C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , Cn−1 in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' For each i ∈ Z/nZ, if Ci − S is contained in H0 we colour Ci red, and if Cj − S is in H1 we colour Ci blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Thus C0 is red and C1 is blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Any maximal clique Ci is either red or blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' If the claim fails then there is some i ∈ Z/nZ−{0, 1} such that Ci−S is contained in neither H0 nor H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let Hi be the connected component of G − S that contains Ci − S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We colour any clique Cj in C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , Cn−1 green if Cj −S is contained in Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We know that the collections of red, blue, REDUCED CLIQUE GRAPHS 11 and green cliques are all non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' So therefore we can find a red clique, Cred, adjacent to a clique that is not red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We can similarly find Cblue, a blue clique that is adjacent to a non-blue clique, and Cgreen, a green clique that is adjacent to a clique that is not green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Now Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='1 implies that Cred, Cblue, and Cgreen are adjacent to each other in CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' As they are three distinct vertices in an induced cycle of CR(G) with at least four vertices, this is an immediate contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' □ If C1 is the only blue clique, then statement (i) holds and we have nothing left to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Similarly, if C0 is the only red clique, then (ii) holds and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' So we assume there are at least two red cliques and at least two blue cliques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We can choose Cred and C′ red to be distinct red cliques that are adjacent to blue cliques, and we can choose Cblue and C′ blue to be two distinct blue cliques that are adjacent to red cliques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Now Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='1 implies that Cred and C′ red are adjacent to both Cblue and C′ blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Thus the four cliques induce a cycle in CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' This is impossible if n ≥ 5, so we conclude that n = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Now C0 is a red clique and it is adjacent to two blue cliques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Thus C1 and C3 are blue, C2 is red, and we are finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' □ The example in Figure 1 shows that a reduced clique graph may contain an induced cycle with four vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We will next show that there is no example with an induced cycle of five vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' There is no chordal graph G such that CR(G) has an induced cycle with exactly five vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Assume otherwise and let G be a chordal graph such that CR(G) contains an induced cycle with five vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let C0, C1, C2, C3, C4 be the maximal cliques in this cycle, where the indices are from Z/5Z and Ci is adjacent to Cj if and only if j ∈ {i−1, i+1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' By adding a constant to these indices as necessary, we may assume that |C0 ∩ C1| ≤ |Ci ∩ Ci+1| for all i ∈ Z/5Z, so that the edge between C0 and C1 is a minimal edge of the cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let S be C0 ∩ C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Note that S is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Now we apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' By applying the permutation ρ: i �→ 1 − i as necessary, we may assume that statement (ii) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='5 applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Therefore we let H0 and H1 be connected components of G − S such that H0 contains C0 − S and H1 contains C1 − S, C2 − S, C3 − S, and C4 − S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' C0 ∩ C4 = S = C0 ∩ C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Because C0 − S and C4 − S are contained in different components of G − S, it follows that C0 ∩ C4 ⊆ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' All we have left to prove is that this containment is not proper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' If it were proper, then we would contradict the assumption that the edge between C0 and C1 is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' □ Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Neither C2 nor C3 contains S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' 12 MAYHEW AND PROBERT Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Note that C0 ∩ C2 ⊆ S because C0 − S and C2 − S are contained in different components of G − S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Certainly any path from a vertex of C0 − C2 to a vertex of C2 − C0 must use a vertex of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' If C0 ∩ C2 = S, then C0 and C2 form a separating pair, so C0 and C2 are adjacent in CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' This contradicts the fact that C0 and C2 are non-consecutive vertices in an induced cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' The same argument shows that C3 does not contain S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' □ Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' C2 ∩ C4 ⊆ C1 and C3 ∩ C1 ⊆ C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Assume that x is a vertex of C2 ∩ C4 that is not in C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' By Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='2 we can let y be a vertex in S − C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Thus y is in C1 − C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' So x is in C2 −C1 and y is in C1 −C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='1 implies that y is in C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' As x is also in C4 we see that x and y are adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Because C1 and C2 are adjacent in CR(G) they have a non-empty intersection, but now the edge xy shows that C1 and C2 do not form a separating pair and we have a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' A symmetric argument shows C3 ∩ C1 ⊆ C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' □ Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' C2 contains a vertex of C1 − C4 and C3 contains a vertex of C4 − C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' By symmetry it suffices to prove the first statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Assume that C2 contains no vertex of C1 − C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Because C1 and C2 are adjacent in CR(G), they have at least one vertex in common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' By our assumption, no vertex of C1 ∩ C2 is in C1 − C4, so any such vertex must be in C1 ∩ C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Therefore C2 and C4 are not disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Since C2 and C4 are not adjacent in CR(G), we can let P be a (C2 ∩C4)-avoiding path from a vertex x ∈ C2 −C4 to y ∈ C4 −C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Our assumption means that x is not in C1 − C4, so it is in C2 − C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Our assumption and Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='3 imply that C2 ∩ C4 = C2 ∩ C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Therefore P is a (C2 ∩ C1)-avoiding path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' But Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='2 shows that we can choose a vertex z in S − C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Thus z is in C1 − C2 and Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='1 shows that z is in C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Assuming that z and y are not equal, they are adjacent, as both are in C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' By appending (if necessary) the edge yz to the end of P we obtain a (C1 ∩ C2)-avoiding path from a vertex in C2 − C1 to a vertex in C1 − C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Hence C1 and C2 do not form a separating pair and this contradicts the fact that they are adjacent in CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' □ Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Either C2 ∩ (C1 ∩ C4) ⊆ C3 or C3 ∩ (C1 ∩ C4) ⊆ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Note that C2 ∩ C3 is non-empty, since C2 and C3 are adjacent in CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' If the claim fails, then we choose x ∈ (C2 ∩ C1 ∩ C4) − C3 and y ∈ (C3 ∩ C1 ∩ C4) − C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Now x and y are both in C1 ∩ C4, so they are adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Moreover x is in C2 − C3 and y is in C3 − C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Thus C2 and C3 do not form a separating pair and we have a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' □ By using Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='5, we will assume that C2 ∩ (C1 ∩ C4) is a subset of C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' The other outcome from Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='5 yields to a symmetric argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Using Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='2 we choose a vertex x ∈ S that is not in C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Note that Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='1 implies that S is contained in C1 ∩C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' So x is in (C1 ∩C4)−C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Our assumption therefore implies that x is not in C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' REDUCED CLIQUE GRAPHS 13 By Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='4 we can also choose y in C2 ∩(C1 −C4) and z in C3 ∩(C4 − C1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='3 implies that y is in C2 −C3 and z is in C3 −C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Now x and y are adjacent as they are both in C1, and x and z are adjacent as they are both in C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Note that C2 ∩ C3 is non-empty as C2 and C3 are adjacent in CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' But the path with vertex sequence y, x, z is (C2 ∩ C3)-avoiding, so C2 and C3 do not form a separating pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' This final contradiction completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='6 shows that the class of reduced clique graphs is contained in the class of graphs with no length-five induced cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We next show that this containment is proper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let n ≥ 4 be an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' There is no chordal graph G such that either C(G) or CR(G) is a cycle with n vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Szwarcfiter and Bornstein characterise the clique graphs of chordal graphs [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' In particular H is isomorphic to C(G) for some chordal graph G if and only if H has a spanning tree T such that whenever u and v are adjacent in H, the path of T from u to v induces a clique of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Now assume H is a cycle with at least four vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Any spanning tree of H is a Hamiltonian path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' The end vertices of this path are adjacent in H, but the path of the spanning tree between these vertices does not induce a clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Therefore H is not isomorphic to C(G) for any chordal graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We turn to reduced chordal graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Assume for a contradiction that G is a chordal graph with C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , Cn−1 as its list of maximal cliques, where the indices are from Z/nZ, and Ci is adjacent to Cj in CR(G) if and only if j ∈ {i − 1, i + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We can assume without loss of generality that the edge between C0 and C1 is a minimal edge of CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let S be C0 ∩ C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Assume that statement (iii) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='5 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Thus n = 4 and there are distinct connected components, H0 and H1, of G − S such that H0 contains C0 − S and C2 − S while H1 contains C1 − S and C3 − S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Note that C0 ∩ C3 ⊆ S, and in fact C0 ∩ C3 is equal to S, or else the minimality of the C0-C1 edge is contradicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Either C0 ∩ C2 is empty, or it is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' In the latter case, we can apply Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='4 to C0 and C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We see that either C0 ∩ C1 or C0 ∩ C3 properly contains C0 ∩C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' By symmetry, we can assume C0 ∩C2 is a proper subset of C0 ∩ C1 = S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Thus C0 − S and C2 − S are disjoint sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' They are contained in the same connected component of G − S, so we can let P be a shortest-possible path of H0 from a vertex of C0 − S to a vertex of C2 − S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' On the other hand, if C0 ∩ C2 is empty, then C0 − S and C2 − S are again disjoint subsets in H0, so we again let P be a shortest-possible path of H0 from C0 − S to C2 − S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' In either case, P contains exactly one vertex of C0 and exactly one vertex of C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Then P must contain at least one edge, and this edge is in a maximal clique that is equal to neither C0 nor C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Nor can this maximal clique be C1 or C3, because any edge of P is contained in H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' So we have a contradiction in the case that (iii) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='5 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' 14 MAYHEW AND PROBERT Now we assume that either (i) or (ii) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' By applying the permutation ρ: i �→ 1 − i as necessary, we will assume that H0 and H1 are distinct connected components of G − S, and that H0 contains C0 − S while H1 contains Ci − S for i = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , n − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' By the same argument as earlier, we can see that Cn−1 contains S, or else the choice of the C0 − C1 edge is contradicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Now C1 ∩ Cn−1 contains S, and C1 and Cn−1 are non-adjacent in CR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We apply Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='4 and see that there is a path of CR(G) from C1 to Cn−1 such that every intersection of consecutive cliques in the path properly contains C1∩Cn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' This path is either C1, C0, Cn−1, or it is C1, C2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , Cn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Assume the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Then C1 ∩ C0 = S properly contains C1 ∩ Cn−1 ⊇ S and we have a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Hence any intersection of consecutive cliques in C1, C2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , Cn−1 properly contains C1 ∩ Cn−1, and hence contains S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' It follows that C2 contains S and thus C0 ∩ C2 is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Since C0 − S and C2 − S are contained in different components of G − S, any path of a vertex from C0 − C2 to a vertex of C2 − C0 must contain a vertex of S = C0 ∩ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Thus C0 and C2 form a separating pair in G, and hence they are adjacent in CR(G), which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Clique graphs vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' reduced clique graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Consider the classes {C(G)} and {CR(G)}, where G ranges over all chordal graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='3 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='6 show that the wheel with five spokes is isomorphic to a graph in the former class but not the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Is there a graph that is isomorphic to a graph in the latter class but not the former?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We will show that the answer is, once again, yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Recall that if G and G′ are disjoint graphs, then G ⊠ G′ is obtained from the union of G and G′ by making every vertex of G adjacent to every vertex of G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We use Pn to denote the path of length n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let m, n ≥ 1 be integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Then Pm ⊠ Pn is isomorphic to the reduced clique graph of a chordal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' If n ≥ 22, then Pn ⊠ Pn is not isomorphic to the clique graph of a chordal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let G be the graph obtained from the disjoint union of Pm and Pn and adding a new vertex that is adjacent to every vertex of the disjoint union.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' It is easy to confirm that G is chordal, and that CR(G) is isomorphic to Pm ⊠ Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' For the second statement, we let H be a graph with disjoint induced paths Pu = u0, u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , un−1 and Pv = v0, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , vn−1, where n ≥ 22 and every ui is adjacent to every vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Thus H is isomorphic to Pn ⊠ Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We will assume for a contradiction that H is isomorphic to C(G) for some chordal graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Because C(G) is connected it follows easily that G is connected, so we can apply Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='2 and deduce that H has a spanning tree T, where the path of T from u to v induces a clique of H whenever u and v are adjacent in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' REDUCED CLIQUE GRAPHS 15 Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let i and j be integers satisfying 0 < i, j < n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' The path of T from ui to vj is contained in one of: {ui, ui+1, vj, vj+1}, {ui, ui+1, vj−1, vj}, {ui−1, ui, vj, vj+1}, {ui−1, ui, vj−1, vj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let P be the path of T from ui to vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Since ui is adjacent to vj it follows that P induces a clique of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' As ui is not adjacent to any of the vertices in u0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , ui−2, ui+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , un−1, it follows that the vertices of P that are in Pu belong to {ui−1, ui, ui+1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Similarly, the vertices of P that are in Pv belong to {vj−1, vj, vj+1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' But ui−1 is not adjacent to ui+1, so P does not contain both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' The claim follows by symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' □ Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='1 implies that the path of T between ui and vj has at most three edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let P be a longest-possible path of T and let p0, p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , pk−1 be the ver- tices of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' For i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , k − 1, let Ui be the set of vertices in Pu such that u is in Ui if and only if the shortest path of T from u to a vertex in P contains pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We define Vi to be the analogous set of vertices in Pv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Note that (U0, U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , Uk−1) is a partition of the vertices of Pu, and (V0, V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , Vk−1) is a partition of the vertices of Pv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Either max{|Ui|: 0 ≤ i ≤ k − 1} ≤ 3 or max{|Vi|: 0 ≤ i ≤ k − 1} ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Assume for a contradiction that |Ui| ≥ 4 and |Vj| ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let p and q, respectively, be the smallest (largest) integers such that up, uq ∈ Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Then q − 1 > p + 1 because |Ui| ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' In the same way, let s and t be the smallest (largest) integers such that vs, vt ∈ Vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Then t−1 > s+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' It is simple to see from Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='1 that the path of T from up to vs has no vertex in common with the path of T from uq to vt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' But this contradicts the fact that both paths contain pi and pj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' □ By using Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='2, we will assume without loss of generality that |Vi| ≤ 3 for each i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Since (U0, U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' , Uk−1) is a partition of the vertices in Pu we can choose i so that Ui contains a vertex x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We claim that if j ≤ i − 4 or j ≥ i + 4, then Vj = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' If this fails, then the path of T from a vertex in Vj to x contains at least four edges of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' But this contradicts our earlier conclusion that any path of T from a vertex of Pu to a vertex of Pv contains at most three edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' So now the vertices of Pv belong to Vi−3 ∪ Vi−2 ∪ · · · ∪ Vi+2 ∪ Vi+3 and this union has cardinality at most 7 × 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Thus Pv contains at most 21 vertices and this contradicts n ≥ 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Conclusions and open problems Given Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='6 it might be natural to believe that reduced clique graphs cannot have any induced cycles with five or more vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' But Figure 3 shows a chordal graph G where CR(G) has an induced cycle with six vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' 16 MAYHEW AND PROBERT 5 10 235 346 1234 G C(G) CR(G) 2 4 3 6 1 8 9 3479 2378 710 2347 2347 235 346 1234 3479 2378 710 2347 2347 7 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Nonetheless we believe the following to be true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Conjecture 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' There is no chordal graph G such that CR(G) contains an induced cycle with seven or more vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' So far as we have been able to tell, every chordal graph is isomorphic to both a clique graph, and to a reduced clique graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' We conjecture this holds generally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Conjecture 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Let H be a chordal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' There are chordal graphs G and G′ such that H is isomorphic to both C(G) and CR(G′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Szwarcfiter and Bornstein present a polynomial-time algorithm for decid- ing whether a given graph is isomorphic to C(G) for some chordal graph G [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Their techniques do not obviously extend to recognising reduced clique graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Nonetheless, we will make the following conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Conjecture 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' There is a polynomial-time algorithm for deciding whether a given graph is isomorphic to CR(G) for some chordal graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' More informally, we ask if there is a structural description for reduced clique graphs that is analogous to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' References [1] Jean R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Blair and Barry Peyton, An introduction to chordal graphs and clique trees, Graph theory and sparse matrix computation, IMA Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' 56, Springer, New York, 1993, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' 1–29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' [2] Peter Buneman, A characterisation of rigid circuit graphs, Discrete Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' 9 (1974), 205–212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' [3] Philippe Galinier, Michel Habib, and Christophe Paul, Chordal graphs and their clique graphs, Graph-theoretic concepts in computer science (Aachen, 1995), Lecture Notes in Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' 1017, Springer, Berlin, 1995, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' 358–371.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' [4] F˘anic˘a Gavril, The intersection graphs of subtrees in trees are exactly the chordal graphs, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Combinatorial Theory Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' B 16 (1974), 47–56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' REDUCED CLIQUE GRAPHS 17 [5] Michel Habib and Vincent Limouzy, On some simplicial elimination schemes for chordal graphs, DIMAP Workshop on Algorithmic Graph Theory, Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Notes Discrete Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' 32, Elsevier Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=', Amsterdam, 2009, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' 125–132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' [6] Michel Habib and Juraj Stacho, Reduced clique graphs of chordal graphs, European J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Combin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' 33 (2012), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' 5, 712–735.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' [7] Terry A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' McKee, Minimal weak separators of chordal graphs, Ars Combin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' 101 (2011), 321–331.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' [8] Yasuko Matsui, Ryuhei Uehara, and Takeaki Uno, Enumeration of the perfect se- quences of a chordal graph, Theoret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' 411 (2010), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' 40-42, 3635–3641.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' [9] Dillon Mayhew and Andrew Probert, Supersolvable saturated matroids and chordal graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' In preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' [10] Donald J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Rose, Triangulated graphs and the elimination process, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' 32 (1970), 597–609.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' [11] Jayme L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Szwarcfiter and Claudson F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Bornstein, Clique graphs of chordal and path graphs, SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' Discrete Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' 7 (1994), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} +page_content=' 2, 331–336.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf'} diff --git a/C9E5T4oBgHgl3EQfUA9Q/content/tmp_files/2301.05540v1.pdf.txt b/C9E5T4oBgHgl3EQfUA9Q/content/tmp_files/2301.05540v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f29cc2b3b84aa0bd7c5a5e175c2cc2a6fb07b9ed --- /dev/null +++ b/C9E5T4oBgHgl3EQfUA9Q/content/tmp_files/2301.05540v1.pdf.txt @@ -0,0 +1,1715 @@ +arXiv:2301.05540v1 [math.NA] 13 Jan 2023 +SOLVING PDES WITH INCOMPLETE INFORMATION +PETER BINEV, ANDREA BONITO, ALBERT COHEN, WOLFGANG DAHMEN +RONALD DEVORE, AND GUERGANA PETROVA +Abstract. We consider the problem of numerically approximating the solutions to a partial differential +equation (PDE) when there is insufficient information to determine a unique solution. Our main example +is the Poisson boundary value problem, when the boundary data is unknown and instead one observes +finitely many linear measurements of the solution. We view this setting as an optimal recovery problem +and develop theory and numerical algorithms for its solution. The main vehicle employed is the derivation +and approximation of the Riesz representers of these functionals with respect to relevant Hilbert spaces of +harmonic functions. +1. Introduction +The questions we investigate sit in the broad research area of using measurements to enhance the numer- +ical recovery of the solution u to a PDE. The particular setting addressed in this paper is to numerically +approximate the solution to an elliptic boundary value problem when there is insufficient information on the +boundary value to determine a unique solution to the PDE. In place of complete boundary information, we +have a finite number of data observations of the solution u. This data serves to narrow the set of possible +solutions. We ask what is the optimal accuracy to which we can recover u and what is a near optimal +numerical algorithm to approximate u. Problems of this particular type arise in several fields of science and +engineering (see e.g. [28, 3, 7] for examples in fluid dynamics), where a lack of full information on boundary +conditions arises for various reasons. For example, the correct physics might not be fully understood [22, 24], +or the boundary values are not accessible [11], or they must be appropriately modified in numerical schemes +[8, 23]. Other examples of application domains for the results of the present paper can be found in the +introduction of [9]. +1.1. A model for PDEs with incomplete data. In this paper, we consider the model elliptic problem +(1.1) +− ∆u = f +in Ω, +u = g +on Γ := ∂Ω, +where Ω ⊂ Rd is a bounded Lipschitz domain with d = 2 or 3. The Lax-Milgram theorem [29] implies the +existence and uniqueness of a solution u from the Sobolev space H1(Ω) to (1.1), once f and g are prescribed +in H−1(Ω) (the dual of H1 +0(Ω)) and in H1/2(Γ) (the image of H1(Ω) by the trace operator), respectively. +Recall that the trace operator T is defined on a function w ∈ C(¯Ω) as the restriction of w to Γ and this +definition is then generalized to functions in Sobolev spaces by a denseness argument. In particular, the +trace operator is well defined on H1(Ω). For any function v in H1(Ω) we denote by vΓ its trace, +(1.2) +vΓ := T (v) = v|Γ, +v ∈ H1(Ω). +The Lax-Milgram analysis also yields the inequalities +(1.3) +c0∥v∥H1(Ω) ≤ ∥∆v∥H−1(Ω) + ∥vΓ∥H1/2(Γ) ≤ c1∥v∥H1(Ω), +v ∈ H1(Ω). +Here the constants c0, c1 depend on Ω and on the particular choice of norms employed on H1(Ω) and H1/2(Γ). +Our interest centers on the question of how well we can numerically recover u in the H1 norm when we +do not have sufficient knowledge to guarantee a unique solution to (1.1). There are many possible settings +to which our techniques apply, but we shall focus on the following scenario: +Date: January 16, 2023. +This research was supported by the NSF Grants DMS 2110811 (AB), DMS 2038080 (PB and WD), DMS-2012469 (WD), +DMS 21340077 (RD and GP), the MURI ONR Grant N00014-20-1-278 (RD and GP), the ARO Grant W911NF2010318 (PB), +and the SFB 1481, funded by the German Research Foundation (WD). +1 + +(i) We have a complete knowledge of f but we do not know g. +(ii) The function g belongs to a known compact subset KB of H +1 +2 (Γ). +Thus, membership in KB +describes our knowledge of the boundary data. The function u we wish to recover comes from the +set +(1.4) +K := {u : u solves (1.1) for some g ∈ KB}, +which is easily seen from (1.3) to be a compact subset of H1(Ω). +(iii) We have access to finitely many data observations of the unknown solution u, in terms of a vector +(1.5) +λ(u) := (λ1(u), . . . , λm(u)) ∈ Rm, +where the λj are fixed and known linear functionals defined on the functions from K. +Natural candidates for the compact set KB are balls of Sobolev spaces that are compactly embedded in +H +1 +2 (Γ). We thus restrict our attention for the remainder of this paper to the case +(1.6) +KB := U(Hs(Γ)), +for some s > 1 +2,where the precise definition of Hs(Γ) and its norm ∥ · ∥Hs(Γ) is described later. Note that +U(Y ) denotes the unit ball of a Banach space Y with respect to the norm ∥ · ∥Y . +1.2. The optimal recovery benchmark. Let wj := λj(u), j = 1, . . . , m, and +(1.7) +w := (w1, . . . , wm) = λ(u) ∈ Rm, +be the vector of data observations. Therefore, the totality of information we have about u is that it lies in +the compact set +(1.8) +Kw := {u ∈ K : λ(u) = w}. +Our problem is to numerically find a function ˆu ∈ H1(Ω) which approximates simultaneously all the +u ∈ Kw. This is a special case of the problem of optimal recovery from data (see [15, 27, 19]). The optimal +recovery, i.e. the best choice for ˆu, has the following well known theoretical description. Let B(Kw) be a +smallest ball in H1(Ω) which contains Kw and let R(Kw) := R(Kw)H1(Ω) be its radius. Then, R(Kw) is the +optimal recovery error, that is, the smallest error we can have for recovering u in the norm of H1(Ω), and +the center of B(Kw) is an optimal recovery of u. +We are interested in understanding how small R(Kw) is and what are the numerical algorithms which are +near optimal in recovering u from the given data w. We say that an algorithm w �→ ˆu = ˆu(w) delivers near +optimal recovery with constant C if +(1.9) +∥u − ˆu(w)∥H1(Ω) ≤ CR(Kw), +w ∈ Rm. +Of course, we want C to be a reasonable constant independent of m. Our results actually deliver a recovery +estimate of the form +(1.10) +∥u − ˆu(w)∥H1(Ω) ≤ R(Kw) + ε, +w ∈ Rm, +where ε > 0 can made arbitrarily small at the price of higher computational cost. In this sense, the recovery +is near optimal with constant C > 1 in (1.9) that can be made arbitrarily close to 1. +1.3. A connection with the recovery of harmonic functions. There is a natural restatement of our +recovery problem in terms of harmonic functions. Let f be the right side of (1.1), where f is a known fixed +element of H−1(Ω). Let u0 be the function in H1(Ω) which is the solution to (1.1) with g = 0. Then, we +can write any function u ∈ K as +(1.11) +u = u0 + uH, +where uH is a harmonic function in H1(Ω) which has boundary value g = T (uH) with g ∈ KB. Recall our +assumption that KB is the unit ball of Hs(Γ) with s > 1 +2. +2 + +Let Hs(Ω) denote the set of harmonic functions v defined on Ω for which vΓ ∈ Hs(Γ). We refer the reader +to [2], where a detailed study of spaces like Hs(Ω) is presented. We define the norm on Hs(Ω) to be the one +induced by the norm on Hs(Γ), namely, +(1.12) +∥v∥Hs(Ω) := ∥vΓ∥Hs(Γ), +v ∈ Hs(Ω). +There exist several equivalent definitions of norms on Hs(Γ), as discussed later. For the moment, observe +that from (1.3) it follows the existence of a constant Cs such that +(1.13) +∥v∥H1(Ω) ≤ Cs∥v∥Hs(Ω), +v ∈ Hs(Ω). +Indeed, the space Hs(Ω) is a Hilbert space that is compactly embedded in H1(Ω), as a consequence of the +compact embedding of Hs(Γ) in H1/2(Γ). We denote by KH the unit ball of Hs(Ω), +(1.14) +KH := U(Hs(Ω)). +Since the function u0 in (1.11) is fixed, it follows from (1.6) that +(1.15) +R(Kw) = R(KH +w′)H1(Ω), +w′ := λ(uH) = w − λ(u0). +There are two conclusions that can be garnered from this reformulation. The first is that the optimal +error in recovering u ∈ Kw is the same as that in recovering the harmonic function uH ∈ KH +w′ in the H1(Ω) +norm. The harmonic recovery problem does not involve f except in determining w′. The second point is +that one possible numerical algorithm for our original problem is to first construct a sufficiently accurate +approximation ˆu0 to u0 and then to numerically implement an optimal recovery of a harmonic function in +KH from data observations. This numerical approach requires the computation of w′. In theory, u0 is known +to us since we have a complete knowledge of f. However, u0 must be computed and any approximation ˆu0 +will induce an error. Although this error can be made arbitrarily small, it means that we only know w′ up to +a certain numerical accuracy. One can thus view the harmonic reformulation as an optimal recovery problem +with perturbed observations of w′. The numerical algorithm presented here follows this approach. Its central +constituent, namely the recovery of harmonic functions from a finite number of noisy observations, can be +readily employed as well in a number of different application scenarios described e.g. in [9]. +1.4. Objectives and outline. Our main goal is to create numerical algorithms which are guaranteed to +produce a function ˆu which is near optimal and to discuss their practical implementation. We begin in §2 +with some remarks on the definition of the space Hs(Γ) and its norm, which are of importance both in the +accuracy analysis and the practical implementation of recovery algorithms. +The general approach for optimal recovery that was introduced in [15, 14] is recalled in §3. We describe +a solution algorithm which takes into consideration the effect of numerical perturbations. We first consider +the case when the linear functionals λj are defined on all of H1(Ω) and then adapt this algorithm to the +case when the linear functionals are point evaluations +(1.16) +λj(u) := u(xj), +xj ∈ Ω, +j = 1, . . . , m. +Point evaluations are not defined on all of H1(Ω) when d > 1, however, they are defined on K when the +smoothness order s is large enough. +The critical ingredient in our proposed algorithm is the numerical computation of the Riesz representers +φj of the restrictions of λj to the Hilbert space Hs(Ω). Each of these Riesz representers is characterized +as a solution to an elliptic problem and can be computed offline since it does not involve the measurement +vector w. Our suggested numerical method for approximating φj is based on finite element discretizations +and is discussed in §4. We establish quantitative error bounds for the numerical approximation in terms of +the mesh size. Numerical illustrations of the optimal recovery algorithm are given in §5. +Note that the optimal recovery error over the class K strongly depends on the choice of the linear +functionals λj. For example, in the case of point evaluation, this error can be very large if the data sites +{xj}m +j=1 are poorly positioned, or small if they are optimally positioned. This points to the importance of +the Gelfand widths and sampling numbers. They describe the optimal recovery error over K with optimal +choice of functionals in the general case and the point evaluation case, respectively. The numerical behaviour +of these quantities in our specific setting is discussed in §6. +3 + +2. The spaces Hs(Γ) and Hs(Ω) +In this section, we discuss the definition and basic properties of the spaces Hs(Γ) and Hs(Ω). We refer +to [1] for a general treatment of Sobolev spaces on domains D ⊂ Rd. Recall that for fractional orders r > 0, +the norm of Hr(D) is defined as +∥v∥2 +Hr(D) := ∥v∥2 +Hk(D) + +� +|α|=k +� +D×D +|∂αv(x) − ∂αv(y)|2 +|x − y|d+2(r−k) +dxdy, +where k is the integer such that k < r < k+1, and ∥v∥2 +Hk(D) := � +|α|≤k ∥∂αv∥2 +L2(D) is the standard Hk-norm. +2.1. Equivalent definitions of Hs(Γ). Let Ω be any bounded Lipschitz domain in Rd. We recall the trace +operator T introduced in §1.1. One first possible definition of the space Hs(Γ), for any s ≥ 1 +2, is as the +restriction of Hs+ 1 +2 (Ω) to Γ, that is, +Hs(Γ) = T (Hs+ 1 +2 (Ω)), +with norm +(2.1) +∥g∥Hs(Γ) := min +� +∥v∥Hs+ 1 +2 (Ω) : vΓ = g +� +. +The resulting norm is referred to as the trace norm definition for Hs(Γ). +There is a second, more intrinsic way to define Hs(Γ), by properly adapting the notion of Sobolev +smoothness to the boundary. This can be done by locally mapping the boundary onto domains of Rd−1 +and requiring that the pullback of g by such transformation have Hs smoothness on such domains. We refer +the reader to [10] and [17] for the complete intrinsic definition, where it is proved to be equivalent to the +trace definition for a range of s that depends on the smoothness of the boundary Γ. +For small values of s, Sobolev norms for Hs(Γ) may also be equivalently defined without the help of local +parameterizations, as contour integrals. For example, if 0 < s < 1 and Ω is a Lipschitz domain, we define +∥g∥2 +Hs(Γ) := ∥g∥2 +L2(Γ) + +� +Γ×Γ +|g(x) − g(y)|2 +|x − y|d−1+2s dxdy, +and if s = 1 and Ω is a polygonal domain, we define +(2.2) +∥g∥2 +H1(Γ) := ∥g∥2 +L2(Γ) + ∥∇Γg∥2 +L2(Γ), +where ∇Γ is the tangential gradient, and likewise +∥g∥2 +Hs(Γ) := ∥g∥2 +H1(Γ) + +� +Γ×Γ +|∇Γg(x) − ∇Γg(y)|2 +|x − y|d−1+2(s−1) dxdy, +for 1 < s < 2. In the numerical illustration given in §5, we will specifically take the value s = 1 and a square +domain, using the definition (2.2). +When Ω has smooth boundary, it is known that the trace definition and intrinsic definition of the Hs(Γ) +norms are equivalent for all s ≥ 1/2. On the other hand, when Ω does not have a smooth boundary, it is +easily seen that the two definition are not equivalent unless restrictions are made on s. Consider for example +the case of polygonal domains of R2: it is easily seen that the trace vΓ of a smooth function v ∈ C∞(Ω) +has a tangential gradient ∇ΓvΓ that generally has jump discontinuities at the corner points and thus does +not belong to H1/2(Γ). In turn, the equivalence between the trace and intrinsic norms only holds for s < 3 +2 +and in such case we limit the value of s to this range. The same restriction s < 3/2 applies to a polyhedral +domain in the case d = 3. +2.2. The regularity of functions in Hs(Ω). We next give some remarks on the Sobolev smoothness of +functions from the space Hs(Ω) when s > 1/2. Clearly such harmonic functions are infinitely smooth inside +Ω and also belong to H1(Ω), but one would like to know for which value of r they belong to Hr(Ω). To +answer this question, we consider v ∈ Hs(Ω). By the definition of Hs(Ω), v is harmonic in Ω and vΓ ∈ Hs(Γ). +4 + +Having assumed that s in the admissible range where all above definitions of the Hs(Γ) norms are equivalent, +and using the first one, we know that there exists a function ˜v ∈ Hs+ 1 +2 (Ω) such that ˜vΓ = vΓ +∥˜v∥Hs+ 1 +2 (Ω) = ∥vΓ∥Hs(Γ) = ∥v∥Hs(Ω). +We define v := v − ˜v so that v = ˜v +v. We are interested in the regularity of v since it will give the regularity +of v. Notice that vΓ = 0 and +−∆v = f := ∆˜v. +The function f belongs to the Sobolev space Hs− 3 +2 (Ω) and we are left with the classical question of the +regularizing effect in Sobolev scales when solving the Laplace equation with Dirichlet boundary conditions. +Obviously, when Ω is smooth, we find that v ∈ Hs+ 1 +2 (Ω) and so we have obtained the continuous embedding +Hs(Ω) ⊂ Hr(Ω), +r = s + 1 +2. +For less smooth domains, the smoothing effect is limited (in particular by the presence of singularities on +the boundary of Ω), i.e., v is only guaranteed to be in Hr(Ω) where r may be less than s + 1/2, see [10]. +More precisely +Hs(Ω) ⊂ Hr(Ω), +where +(2.3) +r := min +� +s + 1 +2, r∗� +, +Here, r∗ = r∗(Ω) is the limiting bound for the smoothing effect: +(i) For smooth domains r∗ = ∞. +(ii) For convex domains r∗ = 2. +(iii) For non-convex polygonal domains in R2, or a polyhedron in R3, one has 3/2 < r∗ < 2 where the +value of r∗ depends on the reentrant angles. +(iv) In particular for polygons, we can take r∗ = 1 + π +ω − ε, for any ε > 0 where ω is the largest inner +angle. +Note that r∗ could be strictly smaller than s + 1 +2. +In summary, for an admissible range of r > 1 that depends on s and Ω one has the continuous embedding +Hs(Ω) ⊂ Hr(Ω), and so there exists a constant C1 that depends on (r, s) and Ω, such that +(2.4) +∥v∥Hr(Ω) ≤ C1∥v∥Hs(Ω) = C1∥vΓ∥Hs(Γ), +v ∈ Hs(Ω). +3. A near optimal recovery algorithm +In this section, we present a numerical algorithm for solving (1.1) when the information about the bound- +ary value g is incomplete. We first work under the assumption that the λj’s are continuous over H1(Ω), and +assumed to be linearly independent (linear independence can be guaranteed by throwing away dependent +functionals when necessary). We prove that the proposed numerical recovery algorithm is near optimal. +We then adapt our approach to the case where the λj’s are point evaluations, see (1.16), and therefore not +continuous over H1(Ω) when d ≥ 2. +3.1. Minimum norm data fitting. As noted in §1.3, the problem of recovering u ∈ Kw is directly related +to the problem of recovering the harmonic component uH ∈ KH from the given data observations w′. Note +that KH is the unit ball of the Hilbert space Hs(Ω). There is a general approach for optimal recovery +from data observations in this Hilbert space setting, as discussed e.g. in [15]. We first describe the general +principles of this technique and then apply them to our specific setting. +Let H be any Hilbert space and suppose that λ1, . . . , λm ∈ H∗ are linearly independent functionals from +H∗. Let X be a Banach space such that H is continuously embedded in X. We are interested in optimal +recovery of a function v in the norm ∥ · ∥X, knowing that v ∈ K := U(H), the unit ball of H. If w ∈ Rm is +the vector of observations, we define the minimal norm interpolant as +v∗(w) = argmin{∥v∥H : v ∈ H and λ(v) = w}. +5 + +It is easily checked that when Kw is non-empty, the function v∗(w) coincides with the Chebyshev center of +Kw in X. To see this, first note that any v ∈ Kw may be written as v = v∗(w) + η where η belongs to the +null space N of λ. Because v∗(w) has minimal norm, v − v∗(w) = η is orthogonal to v∗(w) and hence from +the Pythagorean theorem +∥v − v∗∥2 +H = ∥v∥2 +H − ∥v∗(w)∥2 +H ≤ 1 − ∥v∗(w)∥2 +H =: r2, +because ∥v∥H ≤ 1. Notice that v∗(w) − η is also in Kw. It follows that Kw is precisely the ball in the affine +space v∗(w) + N centered at v∗(w) and of radius r. In particular, Kw is centrally symmetric around v∗(w). +Therefore, v∗(w) is the Chebyshev center for Kw for any norm, in particular for the ∥ · ∥X norm. Therefore, +∥v − v∗(w)∥X ≤ R(Kw)X, +v ∈ Kw, +that is, the minimal norm interpolant gives optimal recovery with constant C = 1. +Standard Hilbert space analysis shows that the mapping w �→ v∗(w) is a linear operator. More importantly, +it has a natural expression that is useful for numerical computation. Namely, from the Riesz representation +theorem each λj can be described as +λj(v) = ⟨v, φj⟩H, +v ∈ H, +where φj ∈ H is called the Riesz representer of λj. The minimal norm interpolant has the representation +(3.1) +v∗ = +m +� +j=1 +a∗ +jφj, +where a∗ = (a∗ +1, . . . , a∗ +m) solves the system of equations +Ga∗ = w, +G := (⟨φi, φj⟩H)i,j=1,...,m, +with G being the Gramian matrix associated to φ1, . . . , φm. +Remark 3.1. In the case where H is a more general Banach space, we are still ensured that the minimal +norm interpolation is a near-optimal recovery with constant C = 2. However, its dependence on the data w +is no longer linear and the above observation regarding its computation does not apply. +Let us now apply this general principle to our particular setting in which the Hilbert space H is Hs(Ω) +and X = H1(Ω). Let φj ∈ Hs(Ω) be the Riesz representer of the functional λj when viewed as a functional +on Hs(Ω). In other words +λj(v) = ⟨v, φj⟩Hs(Ω), +v ∈ Hs(Ω). +We assume that the λj are linearly independent on Hs(Ω) and thus the Gramian matrix +G = +� +gi,j +� +i,j=1,...,m, +gi,j := ⟨φi, φj⟩Hs = λj(φi), +is invertible. +Now, let u = u0 + uH, with uH ∈ KH = U(Hs(Ω)) be the function in K that gave rise to our data +observation w. So, we have +w′ = w − λ(u0) = λ(uH). +If a∗ is the vector in Rm which satisfies Ga∗ = w′, then u∗ +H := �m +j=1 a∗ +jφj is the function of minimum Hs(Ω) +norm which satisfies the data w′, i.e., λ(u∗ +H) = w′. We have seen that +∥uH − u∗ +H∥H1(Ω) ≤ R(KH +w′)H1(Ω), +namely, u∗ +H is the optimal recovery of the functions in KH +w′. Note that the recovery error is measured in H1 +not in Hs(Ω). In turn, see (1.15), the function u∗ := u∗ +H + u0 is the optimal recovery for functions in Kw: +∥u − u∗∥H1(Ω) ≤ R(Kw)H1(Ω). +The idea behind our proposed numerical method is to numerically construct a function ˆu ∈ H1 that +approximates u∗ well. If, for example, we have for ε > 0 the bound +∥u∗ − ˆu∥H1(Ω) ≤ ε, +then for any u ∈ K, we have by the triangle inequality +∥u − ˆu∥H1(Ω) ≤ R(Kw)H1(Ω) + ε. +6 + +Given any C > 1, by taking ε small enough, we have that ˆu is a near best recovery of the functions in Kw +with constant C. +3.2. The numerical recovery algorithm for H1-continuous functionals. Motivated by the above +analysis, we propose the following numerical algorithm for solving our recovery problem. The algorithm +involves approximations of the function u0 and the Riesz representers φj, typically computed by finite +element discretizations, and the application of the linear functionals λj to these approximations. In order to +avoid extra technicalities, we make here the assumption that the applications of the functionals to a known +finite element function can be exactly computed. +We first work under the additional assumption that the linear functionals λj are not only defined on K +but that they are continuous over H1(Ω). We define Λ as the maximum of the norms of the λj on H1(Ω). +In this case +(3.2) +|λj(v)| ≤ Λ∥v∥H1(Ω), +v ∈ H1(Ω). +In what follows, throughout this paper, we use the following weighted ℓ2 norm on Rm, +∥z∥ := + + 1 +m +m +� +j=1 +|zj|2 + + +1/2 += m−1/2∥z∥ℓ2, +z = (z1, . . . , zm) ∈ Rm. +In particular, we have +∥λ(v)∥ ≤ Λ∥v∥H1(Ω), +v ∈ H1(Ω). +Given a user prescribed accuracy ε > 0, our algorithm does the following four steps involving intermediate +tolerances (ε1, ε2). +Step 1: We numerically find an approximation ˆu0 to u0 which satisfies +(3.3) +∥u0 − ˆu0∥H1(Ω) ≤ ε1. +To find such a ˆu0, we use standard or adaptive FEM methods. Given that ˆu0 has been constructed, we +define ˆw := w − λ(ˆu0). Then, for w′ := w − λ(u0) we have, see (3.3), +(3.4) +∥w′ − ˆw∥ ≤ Λε1. +On the other hand, since |λj(v)| ≤ Λ∥v∥H1(Ω) ≤ Λs∥v∥Hs(Ω) ≤ Λs, where +Λs := CsΛ, +see (3.2), (1.13) and (1.14), we derive that +(3.5) +∥w′∥ ≤ Λs. +Thus by triangle inequality, we also find that +(3.6) +∥ ˆw∥ ≤ Λs + Λε1. +Step 2: For each j = 1, . . . , m, we numerically compute an approximation ˆφj ∈ H1(Ω) to φj which satisfies +(3.7) +∥φj − ˆφj∥H1(Ω) ≤ ε2, +j = 1, . . . , m. +This numerical computation is crucial and is performed during the offline phase of the algorithm. We detail +it in §4. Note that the ˆφj’s are not assumed to be in Hs(Ω), and in particular not assumed to be harmonic +functions. +Step 3: We define and compute the matrix +ˆG = (ˆgi,j)i,j=1,...,m, +ˆgi,j := λj(ˆφi), +and thus |ˆgi,j − gi,j| ≤ Λε2 for all i, j. +7 + +It follows that for the matrix R := G − ˆG we have +∥R∥1 ≤ mΛε2, +where we use the shorthand notation ∥ · ∥1 := ∥ · ∥ℓ1→ℓ1 for matrices. Since G is invertible, we are ensured +that ˆG is also invertible for ε2 small enough. We define +M := ∥G−1∥1, +ˆ +M := ∥ ˆG−1∥1. +While these two norms are finite, their size will depend on the nature and the positioning of the linear +functionals λj, j = 1, . . . , m, as it will be seen in the section on numerical experiments. These two numbers +are close to one another when ε2 is small since ˆ +M converges towards the unknown quantity M as ε2 → 0. +In particular, we have +|M − ˆ +M| = |∥G−1∥1 − ∥ ˆG−1∥1| ≤ ∥G−1 − ˆG−1∥1 = ∥ ˆG−1RG−1∥1 ≤ M ˆ +MmΛε2, +from which we obtain that +(3.8) +M ≤ +ˆ +M +1 − m ˆ +MΛε2 +and +ˆ +M ≤ +M +1 − mMΛε2 +, +provided that mMΛε2 < 1 and m ˆ +MΛε2 < 1. We also have the bound +(3.9) +∥ ˆG−1 − G−1∥1 ≤ +M 2 +1 − mMΛε2 +mΛε2 =: δ. +It is important to observe that δ can be made arbitrarily small by diminishing ε2. +Step 4: We numerically solve the m×m algebraic system ˆGˆa = ˆw, thereby finding a vector ˆa = (ˆa1, . . . , ˆam). +We then define ˆuH := �m +j=1 ˆaj ˆφj and our recovery of u is ˆu := ˆu0 + ˆuH. +This step can be implemented by standard linear algebra solvers. +One major advantage of the above algorithm is that Steps 1-2-3 can be performed offline since they do not +involve the data w. That is, we can compute ˆu0, the approximate Riesz representers ˆφj and the approximate +Gramian ˆG and its inverse without knowing w. In this way, the computation of ˆu from given data w can be +done fast online by Step 4 which only involves solving an m × m linear system. This may be a significant +advantage, for example, when having to process a large number of measurements for the same set of sensors. +3.3. A near optimal recovery bound. The following theorem shows that a near optimal recovery of u +can be reached provided that the tolerances in the above described algorithm are chosen small enough. +Theorem 3.2. For any prescribed ε > 0, if the tolerances (ε1, ε2), are small enough such that mMΛε2 < 1 +and +(3.10) +ε1 + mMΛsε2 + (C0 + ε2)(mMΛε1 + m(Λs + Λε1)δ) ≤ ε, +where C0 := maxj=1,...,m ∥φj∥H1(Ω) and δ := +M2 +1−mMΛε2 mΛε2, then the function ˆu generated by the above +algorithm satisfies +∥u − ˆu∥H1(Ω) ≤ R(Kw)H1(Ω) + ε, +for every +u ∈ Kw. +Thus, for any C > 1 it is a near optimal recovery of u with constant C provided ε is taken sufficiently small. +Proof. Let u = u0 + v be our target function in Kw. We define w′ = w − λ(u0) and v∗ := v∗(w′) which is +the Chebyshev center of KH +w′. We recall the algebraic system Ga∗ = w′ associated to the characterization of +v∗ (see (3.1)). We write +(3.11) ∥u∗ +H− ˆuH∥H1(Ω) ≤ +��� +m +� +j=1 +a∗ +j(φj − ˆφj) +��� +H1(Ω) + +��� +m +� +j=1 +(a∗ +j −ˆaj)ˆφj +��� +H1(Ω) ≤ ∥a∗∥ℓ1ε2+∥a∗−ˆa∥ℓ1(C0 +ε2), +where we have used (3.7) and the fact that +∥ˆφj∥H1(Ω) ≤ ∥φj∥H1(Ω) + ∥φj − ˆφj∥H1(Ω) ≤ C0 + ε2. +8 + +Note that +(3.12) +∥a∗∥ℓ1 = ∥G−1w′∥ℓ1 ≤ M∥w′∥ℓ1 ≤ Mm∥w′∥ ≤ mMΛs, +where we have used that ∥w′∥ℓ1 ≤ m∥w′∥ and inequality (3.5). Therefore it follows from (3.11) and (3.12) +that +(3.13) +∥u∗ +H − ˆuH∥H1 ≤ mMΛsε2 + ∥a∗ − ˆa∥ℓ1(C0 + ε2). +For the estimation of ∥a∗ − ˆa∥ℓ1, we introduce the intermediate vector ˜a ∈ Rm, which is the solution to the +system G˜a = ˆw. Clearly, +∥˜a − a∗∥ℓ1 = ∥G−1( ˆw − w′)∥ℓ1 ≤ M∥ ˆw − w′∥ℓ1 ≤ Mm∥ ˆw − w′∥ ≤ mMΛε1, +where we invoked (3.4). On the other hand, in view of (3.9) and (3.6), we have +∥˜a − ˆa∥ℓ1 = ∥(G−1 − ˆG−1) ˆw∥ℓ1 ≤ δ∥ ˆw∥ℓ1 ≤ mδ∥ ˆw∥ ≤ m(Λs + Λε1)δ. +Combining these two estimates, we find that +∥a∗ − ˆa∥ℓ1 ≤ mMΛε1 + m(Λs + Λε1)δ. +We now insert this bound into (3.13) to obtain +∥u∗ +H − ˆuH∥H1(Ω) ≤ mMΛsε2 + (C0 + ε2)(mMΛε1 + m(Λs + Λε1)δ). +Thus, for u∗ := u0 + u∗ +H and using (3.3), we have +∥u∗ − ˆu∥H1(Ω) +≤ +∥u0 − ˆu0∥H1(Ω) + ∥u∗ +H − ˆuH∥H1(Ω) +≤ +ε1 + mMΛsε2 + (C0 + ε2)(mMΛε1 + m(Λs + Λε1)δ) ≤ ε, +(3.14) +Since u = u0 + uH, we have +∥u − u∗∥H1(Ω) = ∥uH − u∗ +H∥H1(Ω) ≤ R(KH +w′)H1(Ω) = R(Kw)H1(Ω), +and the statement of the theorem follows from this inequality and (3.14). +Remark 3.3. Note that in numerical computations the quantity ˆ +M is available while M is unknown. Thus +in practice, in order to achieve the prescribed accuracy ε, we can first impose that ε2 < (2m ˆ +MΛ)−1 and +derive the inequalities, see (3.8), +M ≤ +ˆ +M +1 − m ˆ +MΛε2 +≤ 2 ˆ +M, +∥G−1 − ˆG−1∥1 ≤ +ˆ +M 2 +1 − m ˆ +MΛε2 +mΛε2 ≤ 2 ˆ +M 2mΛε2 =: ˆδ, +where the last inequality is proven in a similar fashion to (3.9). If we then follow the proof of Theorem 3.2, +the requirement in (3.14) can be substituted by +ε1 + 2m ˆ +MΛsε2 + (C0 + ε2)(2m ˆ +MΛε1 + m(Λs + Λε1)ˆδ) ≤ ε, +and thus all participating quantities are computable. +Remark 3.4. The result in Theorem 3.2 can easily be extended to the case of noisy data, that is, to the case +when the observations +˜w = w + η, +where η is a noise vector of norm ∥η∥ ≤ κ. Indeed, the application of the algorithm to this noisy data leads +to finding in Step 1 the vector ˆw := w + η − λ(ˆu0) that satisfies +∥w′ − ˆw∥ ≤ Λε1 + κ, +and +∥ ˆw∥ ≤ Λs + ε1Λ + κ, +where w′ = w − λ(u0). Inspection of the above proof shows that under the same assumption as in Theorem +3.2, one has the recovery bound +∥u − ˆu∥H1(Ω) ≤ R(Kw)H1(Ω) + ε + Cκ, +for every +u ∈ Kw, +where C := (M + δ)m(C0 + ε2). +9 + +Remark 3.5. For simplicity, we did not introduce in the above analysis the possible errors in the application +of the λi to the approximations ˆu0 and ˆφj, and in the numerical solution to the system ˆGˆa = ˆw, which would +simply result in similar conditions involving the extra tolerance parameters. +3.4. Point evaluation data. We now want to extend the numerical algorithm and its analysis to the case +when the data functionals λj, j = 1, . . . , m, are point evaluations +λj(h) := h(xj), +xj ∈ Ω, +j = 1, . . . , m. +Of course these functionals are not defined for general functions h from H1(Ω). However, we can formulate +the recovery problem whenever the functionals λj are well defined on K. We now discuss settings when this +is possible. +Recall that any u ∈ K can be written as u = u0 + uH, where u0 is the solution to (1.1) with right side +f and g = 0 and uH ∈ Hs(Ω). Point evaluation is well defined for the harmonic functions uH ∈ Hs(Ω), +provided the points are in Ω. In addition, they are well defined for points on the boundary Γ if the space +Hs(Ω) continuously embeds into C(Ω). For d = 2, this is the case when s > 1/2 and when d = 3, this is the +case when s > 1. +Concerning u0, we will need some additional assumption to guarantee that point evaluation of u0 makes +sense at the data sites xj, j = 1, . . . , m. For example, it is enough to assume that u0 is globally continuous +or at least in a neighborhood of each of these points. This can be guaranteed by assuming an appropriate +regularity of f. In this section, we assume that one of these settings holds. We then write +w′ +j := uH(xj) = wj − u0(xj), +j = 1, . . . , m, +and follow the algorithm of the previous section with the following simple modifications: +Modified Step 1: We numerically find an approximation ˆu0 to u0, which in addition to +∥u0 − ˆu0∥H1(Ω) ≤ ε1, +satisfies the requirement +(3.15) +max +i=1,...,m |u0(xi) − ˆu0(xi)| ≤ ε1. +To find such a ˆu0 we use standard or adaptive FEM methods. Given that ˆu0 has been constructed, we define +ˆwj := wj − ˆu0(xj), j = 1, . . . , m, and thus, using (3.15), we have ∥w′ − ˆw∥ ≤ ε1. +Modified Step 2: For each j = 1, . . . , m, we numerically compute an approximation ˆφj to φj, which in +addition to +∥φj − ˆφj∥H1(Ω) ≤ ε2, +j = 1, . . . , m, +satisfies the condition +(3.16) +max +i=1,...,m |φj(xi) − ˆφj(xi)| ≤ ε2, +i, j = 1, . . . , m. +Condition (3.16) ensures that in Step 3 we can choose the entries ˆgi,j of the matrix ˆG as +ˆgi,j = ˆφj(xi), +i, j = 1, . . . , m. +The Steps 3 and 4 of our algorithm remain the same as in the previous section. +Theorem 3.6. With the above modifications, Theorem 3.2 holds with the exact same statement in this point +evaluation setting. +Proof. The proof is the same as that of Theorem 3.2. +10 + +4. Finite element approximations of the Riesz representers +The computation of an approximation ˆu0 to u0, required in Step 1 of the algorithm, can be carried out by +standard finite element Galerkin schemes. Depending on our knowledge on f one can resort to known a priori +estimates for ε1, or may employ standard a posteriori estimates to ensure that the underlying discretization +provides a desired target accuracy. Therefore, in the remainder of this section, we focus on a numerical +implementation of Step 2 of the proposed algorithm. +Our proposed numerical algorithm for Step 2 is to use finite element methods to generate the approx- +imations ˆφj of the Riesz representers φj. Note that each of the functions φj is harmonic on Ω but we do +not require that the sought after numerical approximation ˆφj is itself harmonic but only that it provides +an accurate H1(Ω) approximation to φj. This allows us to use finite element approximations which are +themselves not harmonic. However, the ˆφj will necessarily have to be close to being harmonic since they +approximate a harmonic function in the H1(Ω) norm. +Our numerical approach to constructing a ˆφj, discussed in §4.1, is to use discretely harmonic finite +elements. Here, ˆφj is a discrete harmonic extension of a finite element approximation to the trace ψj = T (φj) +computed by solving a Galerkin problem. +In order to reduce computational cost (see Remark 4.2), we +incorporate discrete harmonicity as constraints and introduce in §4.2 an equivalent saddle point formulation +that has the same solution ˆφj, and which is the one that we practically employ in the numerical experiments +given in §5. We give in §4.4 an a priori analysis with error bounds for ∥φj − ˆφj∥H1 in terms of the finite +element mesh size, in the case where the measurement functionals are continuous on H1(Ω). These error +bounds can in turn be used to ensure the prescribed accuracy ε2 in Step 2. We finally discuss in §4.5 the +extensions to the point value case where pointwise error bounds on |ˆφj(xi) − ˆφj(xi)| are also needed. +In order to simplify notation, we describe these procedures for finding an approximation ˆφ to the Riesz +representer φ ∈ Hs = Hs(Ω) of a given linear functional ν on Hs. This numerical procedure is then applied +with ν = λj, to find the numerical approximations ˆφj to the Riesz representer φj. +For simplicity, throughout this section, we work under the assumption that Ω is a polygonal domain of R2 +or polyhedral domain of R3. This allows us to define finite element spaces based on triangular or simplicial +partitions of Ω that in turn induce similar partitions on the boundary. We assume that 1 +2 < s < 3 +2, which +is the relevant range for such domains, as explained in §2. Our analysis can be extended to more general +domains with smooth or piecewise smooth boundaries, for example by using isoparametric elements near the +boundary, however at the price of considerably higher technicalities. +4.1. A Galerkin formulation. Let s > 1/2 be fixed and assume that ν is any linear form continuous on +Hs(Ω) with norm +(4.1) +Cs := max{ν(v) : ∥v∥Hs(Ω) = 1} +In view of the the definition of the Hs norm, the representer φ ∈ Hs(Ω) of ν for the corresponding inner +product can be defined as +φ = Eψ, +where E is the harmonic extension operator of (4.3) below and where ψ ∈ Hs(Γ) is the solution to the +following variational problem: +(4.2) +⟨ψ, η⟩Hs(Γ) = µ(η) := ν(Eη), +η ∈ Hs(Γ). +Note that this problem admits a unique solution and we have +∥ψ∥Hs(Γ) = ∥φ∥Hs(Ω) = Cs. +Recall that +(4.3) +Eg := argmin{∥∇v∥L2(Ω) : vΓ = g}. +The function Eg is characterized by T (Eg) = g and +� +Ω +∇Eg · ∇v = 0, +v ∈ H1 +0(Ω). +11 + +From the left inequality in (1.3), one has +(4.4) +∥Eg∥H1(Ω) ≤ CE∥g∥H1/2(Γ), +g ∈ H1/2(Γ), +where CE can be taken to be the inverse of the constant c0 in (1.3). +Therefore, one approach to discretizing this problem is the following: consider finite element spaces Vh +associated to a family of meshes {Th}h>0 of Ω, where as usual h denotes the maximum meshsize. We define +Th to be the space obtained by restriction of Vh on the boundary Γ, that is, +Th = T (Vh) +Since we have assumed that Ω is a polygonal or polyhedral domain, the space Th is a standard finite element +space for the boundary mesh. Having also assumed that s < 3/2, when using standard H1 conforming finite +elements such as Pk-Lagrange finite elements, we are ensured that Th ⊂ Hs(Γ). We denote by +Wh := {vh ∈ Vh : T (vh) = 0}, +the finite element space with homogeneous boundary conditions. +We define the discrete harmonic extension operator Eh associated to Vh as follows : for gh ∈ Th, +Ehgh := argmin{∥∇vh∥L2(Ω) : vh ∈ Vh, T (vh) = gh}. +Note that Ehgh is not harmonic. Similar to E, the function Ehgh is characterized by T (Ehgh) = gh and +� +Ω +∇Ehgh · ∇vh = 0, +vh ∈ Wh. +Then, we define the approximation φh ∈ Vh to φ as +φh = Ehψh, +where ψh ∈ Th is the solution to the following variational problem: +(4.5) +⟨ψh, gh⟩Hs(Γ) = µh(gh) := ν(Ehgh), +gh ∈ Th. +Here we are assuming that, in addition to be defined on Hs(Ω), the functional ν is also well defined on the +space Vh. We shall further consider separately two instances where this is the case : (i) ν is a continuous +functional on H1(Ω) and (ii) ν is a point evaluation functional. +Note that (4.5) is not the straightforward Galerkin approximation of (4.2), since µh differs from µ. This +complicates somewhat the further conducted convergence analysis. The numerical method we employ for +computing φh is to numerically solve an equivalent saddle point problem described below. +We apply the strategy (4.5) to ν := λj for each j and thereby obtain the corresponding approximations +ˆφj := φh ∈ Vh. Since Step 2 requires that we guarantee the error ∥φj − ˆφj∥H1 ≤ ε2, our main goal in +this section is to establish a quantitative convergence bound for ∥φ − φh∥H1. We also need to establish a +pointwise convergence bound for |φ(x) − φh(x)| when considering the modified version of Step 2 in the case +that the measurements are point values. +Similar to E, it will be important in our analysis to control the stability of Eh in the sense of a bound +(4.6) +∥Ehgh∥H1(Ω) ≤ DE∥gh∥H1/2(Γ), +gh ∈ Th, +with a constant DE that is independent of h. However, such a uniform bound is not readily inherited from +the stability of E. As observed in [6], its validity is known to depend on the existence of uniformly H1-stable +linear projections onto Vh preserving the homogeneous boundary condition, that is, projectors Ph onto Vh +that satisfy +(4.7) +Ph(H1 +0(Ω)) = Wh +and +∥Phv∥H1(Ω) ≤ B∥v∥H1(Ω), +v ∈ H1(Ω), +for some B independent of h. One straightforward consequence of this is that if v ∈ H1(Ω) with v|Γ ∈ Th +then Ph(v)|Γ = v|Γ. +We next show that the existence of such projectors is sufficient to guarantee the stability of Eh. For this, +suppose (4.7) holds and gh ∈ Th. Then PhEgh ∈ Vh and the trace of PhEgh is equal to gh. It follows that +∥Ehgh − PhEgh∥H1(Ω) +≤ +CP ∥∇Ehgh − ∇PhEgh∥L2(Ω) +≤ +CP ∥∇Ehgh∥L2(Ω) + CP ∥∇PhEgh∥L2(Ω), +≤ +2CP ∥PhEgh∥H1(Ω), +12 + +where CP is the Poincar´e constant for Ω. Here, the last inequality follows from the minimizing property of +Ehgh. Thus, by triangle inequality, one has +∥Ehgh∥H1(Ω) ≤ (1 + 2CP )∥PhEgh∥H1(Ω) ≤ (1 + 2CP )B∥Egh∥H1(Ω) ≤ (1 + 2CP )BCE∥gh∥H1/2(Γ), +which is (4.6) with DE = (1 + 2CP )BCE. +The requirement of uniformly stable projectors Ph with the property (4.7) is satisfied by projectors of +Scott-Zhang type [26] when the family of meshes {Th}h>0 is shape regular, that is, when all elements T +have a uniformly bounded ratio between their diameters h(T ) and the diameter ρ(T ) of their inner circle. +In other words, the shape parameter +(4.8) +σ = σ({Th}h>0) := sup +h>0 +max +T ∈Th +h(T ) +ρ(T ), +is finite. In all that follows in the present paper, we work under such an assumption on the meshes Th. +Therefore, (4.6) holds when Vh is subordinate to such partitions. +4.2. A saddle point formulation. Before attacking the convergence analysis, we need to stress an impor- +tant computational variant of the above described Galerkin method, that leads to the same solution φh. It is +based on imposing harmonicity via a Lagrange multiplier. For this purpose, we introduce the Hilbert space +Xs(Ω) that consists of all v ∈ H1(Ω) such that vΓ ∈ Hs(Γ), and equip it with the norm +∥v∥Xs(Ω) := +� +∥vΓ∥2 +Hs(Γ) + ∥∇v∥2 +L2(Ω) +�1/2 +. +Then, the Riesz representer φ is equivalently determined as the solution of the saddle point problem: find +(φ, π) ∈ Xs(Ω) × H1 +0(Ω) such that +a(φ, v) + b(v, π) += +ν(v), +v ∈ Xs(Ω) +b(φ, z) += +0, +z ∈ H1 +0(Ω), +where the bilinear forms are given by +a(φ, v) := ⟨φΓ, vΓ⟩Hs(Γ) +and +b(v, π) := ⟨∇v, ∇π⟩L2(Ω). +Clearly the second equation in (4.9) means that φ is harmonic and testing the first equation with a v ∈ Hs(Ω) +shows that φ is the Riesz representer of µ. +This saddle point formulation is well-posed: the bilinear forms a and b obviously satisfies the continuity +properties +a(φ, v) ≤ ∥φΓ∥Hs(Γ)∥vΓ∥Hs(Γ) ≤ ∥φ∥Xs(Ω)∥v∥Xs(Ω), +φ, v ∈ Xs(Ω), +and for the standard norm ∥v∥H1 +0 (Ω) = ∥∇v∥L2(Ω), +b(v, π) ≤ ∥∇v∥L2(Ω)∥∇π∥L2(Ω) ≤ ∥v∥Xs(Ω)∥π∥H1 +0(Ω), +v ∈ Xs(Ω), π ∈ H1 +0(Ω). +In addition, for all v ∈ Hs(Ω), one has +∥v∥2 +Xs(Ω) ≤ ∥vΓ∥2 +Hs(Γ) + ∥v∥2 +H1(Ω) ≤ ∥vΓ∥2 +Hs(Γ) + C2 +E∥v∥2 +H1/2(Γ) ≤ (1 + C2 +E)a(v, v), +which shows that a is coercive on the null space of b in Xs(Ω). Finally, the bilinear form b satisfies the +inf-sup condition +inf +π∈H1 +0 (Ω) +sup +v∈Xs(Ω) +b(v, π) +∥v∥Xs(Ω)∥π∥H1 +0(Ω) +≥ +inf +π∈H1 +0 (Ω) +b(π, π) +∥π∥Xs(Ω)∥π∥H1 +0(Ω) += 1. +Therefore the standard LBB theory ensures existence and uniqueness of the solution pair (φ, π). +We now discretize the saddle point problem by searching for (φh, πh) ∈ Vh × Wh such that +a(φh, vh) + b(vh, πh) += ν(vh), +vh ∈ Vh +b(φh, zh) += 0, +zh ∈ Wh. +Remark 4.1. The equivalence with the previous derivation of φh by the Galerkin approach is easily checked: +the second equation tells us that the solution φh is discretely harmonic, and therefore equal to Ehψh for some +ψh ∈ Th. Then taking vh of the form Ehgh for gh ∈ Th gives us exactly the Galerkin formulation (4.5). +13 + +This discrete saddle point problem is uniformly well-posed when we equip the space Wh with the H1 +0 +norm, and the space Vh with the Xs norm. The continuity of a and b, and the inf-sup condition for b follow +by the exact same arguments applied to the finite element spaces, with the same constants. On the other +hand, we need to check the uniform ellipticity of a in the space VH +h ⊂ Vh of discretely harmonic functions, +which can be defined as +VH +h := {vh ∈ Vh : b(vh, zh) = 0, zh ∈ Wh}, +or equivalently as the image of Th by the operator Eh. For all vh ∈ Vh,H and gh = T (uh), we write +∥vh∥2 +Xs(Ω) ≤ ∥gh∥2 +Hs(Γ) + ∥vh∥2 +H1(Ω) ≤ ∥gh∥2 +Hs(Γ) + D2 +E∥gh∥2 +H1/2(Γ) ≤ (1 + D2 +E)a(vh, vh), +where we have used the discrete stability of Eh. +Remark 4.2. In practice, we use this discrete saddle point formulation for the computation of φh rather +than the equivalent Galerkin formulation (4.5) for the following reason. Let Nh := dim Vh, Mh := dim Wh, +and Ph := dim Th = Nh − Mh. +Computing the right hand side load vector in (4.2) requires computing +discretely harmonic extensions of Ph basis functions, which means solving Ph linear systems of dimension +Mh. In addition one has to solve the sparse linear system (4.5) of size Ph followed by another system of size +Mh to compute φh = Ehψh. Using optimal iterative solvers of linear complexity the minimum amount of +work needed to compute one representer scales then like +PhMh ∼ N +1+ d−1 +d +h +. +while solving the saddle point problem requires the order of Nh + Mh ∼ Nh operations. +On the other +hand the characterization of φh through (4.5) appears to be more convenient when deriving error bounds for +∥φ − φh∥H1(Ω). This is the objective of the next sections. +4.3. Preparatory results. In the derivation of error bounds for ∥φ − φh∥H1(Ω), we will need several ingre- +dients. +The first is the following lemma that quantifies the perturbation induced by using Eh in place of E. +Lemma 4.3. For any gh ∈ Th, one has +(4.9) +∥(E − Eh)gh∥H1(Ω) ≤ C2hr−1∥gh∥Hs(Γ). +where C2 depends on r and s, the shape-parameter σ, and on the geometry of Ω. +Proof. From the properties of E and Eh, one has +⟨∇(E − Eh)gh, ∇vh⟩ = 0, +vh ∈ Wh +This orthogonality property shows that +∥∇(Egh − Ehgh)∥L2(Ω) ≤ ∥∇(Egh − Ehgh − vh)∥L2(Ω), +vh ∈ Wh, +and therefore +∥∇(Egh − Ehgh)∥L2(Ω) ≤ +min +vh∈Vh,T (vh)=gh ∥∇(Egh − vh)∥L2(Ω) ≤ ∥∇(Egh − PhEgh)∥L2(Ω), +where Ph is the stable projector that preserves homogeneous boundary condition, see (4.7). It follows that +∥∇(Egh − Ehgh)∥L2(Ω) ≤ (1 + B) min +vh∈Vh ∥Egh − vh∥H1(Ω), +where B is the uniform H1-stability bound on Ph. By standard finite element approximation estimates and +(2.4), we have +min +vh∈Vh ∥Egh − vh∥H1(Ω) ≤ Chr−1∥Egh∥Hr(Ω) ≤ CC1hr−1∥gh∥Hs(Γ), +where the constant C depends on r and on the shape parameter σ. The estimate (4.9) follows by Poincar´e +inequality since Egh − Ehgh ∈ H1 +0(Ω). +The second ingredient concerns the regularity of the solution to the variational problem +(4.10) +⟨κ, v⟩Hs(Γ) = γ(v), +v ∈ Hs(Γ). +14 + +For a general linear functional γ ∈ H−s(Γ), that is, continuous on Hs(Γ), we are only ensured that the +solution κ is bounded in Hs(Γ), with ∥κ∥Hs(Γ) = ∥γ∥H−s(Γ). However, if γ has some extra regularity, this +then translates into additional regularity of κ. +As a simple example, consider the case where γ is in addition +continuous on L2(Γ), that is +(4.11) +γ(v) = ⟨g, v⟩L2(Γ), +for some g ∈ L2(Γ), and assume that we work with s = 1 and a polygonal domain. Then the variational +problem has a solution κ ∈ H1(Γ) and in addition κ ∈ H2(E) for each edge E with weak second derivative +given by +−κ′′ = g − κ ∈ L2(Γ). +In turn, standard finite element approximation estimates yield +min +κh∈Th ∥κ − κh∥H1(Γ) ≤ Ch∥g∥L2(Γ), +with a constant C that depends on the shape parameter σ. +Of course, gain of regularity theorems for elliptic problems are known in various contexts. However, we +have not found a general treatment of gain of regularity that addresses the setting of this paper. In going +forward, we do not wish to systematically explore this gain in regularity and approximability for more general +values of s and smoothness of γ since this would significantly enlarge the scope of this paper. Instead, we +state it as the following general assumption. +Assumption R: for s > +1 +2 and δ > 0, there exists r(s, δ) > 0 such that if γ ∈ H−s+δ(Γ) for some +δ > 0, then the solution κ to (4.10) satisfies +(4.12) +min +κh∈Th ∥κ − κh∥Hs(Γ) ≤ Chr(s,δ)∥γ∥H−s+δ(Γ), +with a constant C that depends on s, δ, and on the shape parameter σ. +The above example shows that r(1, 1) = 1 for a polygonal domain. We expect that this assumption always +holds for the range 1 +2 < s < 3 +2 that is considered here. +4.4. An a priori error estimate for ∥φ − φh∥H1(Ω). In this section, we work under the assumption that +the linear functional ν is continuous on H1(Ω) with norm +Cν := max{ν(v) : ∥v∥H1(Ω) = 1}. +Let us first check that this assumption implies a uniform a priori bound on ∥ψh∥Hs(Γ). Indeed, we may write +∥ψh∥2 +Hs(Γ) = ⟨ψh, ψh⟩Hs(Γ) = ν(Ehψh) ≤ CνDE∥ψh∥H1/2(Γ) ≤ CνDE∥ψh∥Hs(Γ), +where the first inequality used (4.6). Therefore, +(4.13) +∥ψh∥Hs(Γ) ≤ CνDE. +We have seen in §2 that the function φ belongs to the standard Sobolev space Hr(Ω) for r defined in +(2.3). We use this r throughout this section. From (2.4), there exists a constant C1 such that +(4.14) +∥Ew∥Hr(Ω) ≤ C1∥w∥Hs(Γ), +w ∈ Hs(Γ). +As noted in §2, the amount of smoothness r depends both on s and on the geometry of Ω. What is important +for us is that since s > 1/2, we have shown in (2) that r > 1. For example, for smooth domains it is r = s+ 1 +2. +The fact that φ ∈ Hr(Ω) hints that the finite element approximation φh to φ should converge with a certain +rate. +This is indeed the case as given in the following result. +Theorem 4.4. Under Assumption R, we have +(4.15) +∥φ − φh∥H1(Ω) ≤ CCνht, +where t = min{r − 1, r(s, s + 1 +2) + r(s, s − 1 +2)}. The constant C depends on s and on the geometry of Ω, and +on the family of meshes through the shape parameter σ. +15 + +Proof. We use the decomposition +(4.16) +φ − φh = Eψ − Ehψh = E(ψ − ψh) + (E − Eh)ψh, +The second term can be estimated with the help of Lemma 4.3 applied to gh = ψh which gives +∥(E − Eh)ψh∥H1(Ω) ≤ C2hr−1∥ψh∥Hs(Γ) ≤ C2DECνhr−1, +from the a priori estimate (4.13) for ψh. We thus have obtained a bound in O(hr−1) for the H1 norm of the +second term in (4.16). +For the first term, we know that +∥E(ψ − ψh)∥H1(Ω) ≤ CE∥ψ − ψh∥H1/2(Γ), +and so we are led to estimate ψ − ψh in the H1/2(Γ) norm. For this purpose, we introduce the intermediate +solution ψh ∈ Th to the problem +⟨ψh, gh⟩Hs(Γ) = µ(gh) = ν(Egh), +gh ∈ Th, +and we use the decomposition +(4.17) +ψ − ψh = (ψ − ψh) + (ψh − ψh). +We estimate the second term in (4.17) by noting that for any gh ∈ Th, +⟨ψh − ψh, gh⟩Hs(Γ) = ν((E − Eh)gh) ≤ Cν∥(E − Eh)gh∥H1(Ω) ≤ CνC2hr−1∥gh∥Hs(Γ), +where we have again used Lemma 4.3. Taking gh = ψh − ψh we obtain a bound O(hr−1) for its Hs(Γ) norm, +and in turn for its H1/2(Γ) norm. +It remains to estimate ∥ψ − ψh∥H1/2(Γ). Note that ψh is exactly the Galerkin approximation of ψ since +we use the same linear form µ in both problems. In fact, we have +⟨ψ − ψh, gh⟩Hs(Γ) = 0, +gh ∈ Th, +that is ψh is the Hs-orthogonal projection of ψ onto Th and therefore +∥ψ − ψh∥Hs(Γ) = min +κh∈Th ∥ψ − κh∥Hs(Γ). +Since the linear form µ satisfies +|µ(g)| = |ν(Eg)| ≤ Cν∥Eg∥H1(Ω) ≤ CνCE∥g∥H1/2(Γ), +and thus belongs to H−1/2(Γ), we may apply the estimate (4.12) to γ = ν, κ = ψ, δ = s − 1 +2 > 0, to reach +(4.18) +∥ψ − ψh∥H1/2(Γ) ≤ ∥ψ − ψh∥Hs(Γ) ≤ CCνCEhr(s,s− 1 +2 ). +This proves the theorem for the value t = min{r − 1, r(s, s − 1 +2)} > 0. +We finally improve the value of t by using a standard Aubin-Nitsche duality argument as follows. We now +take κ to be the solution of (4.10) with +γ(v) = ⟨ψ − ψh, v⟩H1/2(Γ), +v ∈ H1/2(Γ), +where ⟨., .⟩H1/2(Γ) stands for the H1/2 scalar product associated with the norm ∥.∥H1/2(Γ). We then write +∥ψ − ψh∥2 +H1/2(Γ) = ⟨ψ − ψh, ψ − ψh⟩H1/2(Γ) = ⟨κ, ψ − ψh⟩Hs(Γ) = ⟨κ − κh, ψ − ψh⟩Hs(Γ), +where the last equality comes from Galerkin orthogonality. It follows that +∥ψ − ψh∥2 +H1/2(Γ) ≤ ∥κ − κh∥Hs(Γ)∥ψ − ψh∥Hs(Γ) ≤ Chr(s,s+ 1 +2 )∥ψ − ψh∥H1/2(Γ)∥ψ − ψh∥Hs(Γ), +where we have again used (4.12) now with δ = s+ 1 +2. Using the already established estimate (4.18), it follows +that +∥ψ − ψh∥H1/2(Γ) ≤ CCECνh˜t, +with ˜t := r(s, s + 1 +2) + r(s, s − 1 +2). With all such estimates, the desired convergence bound follows with +t := min{r − 1, ˜t}. +16 + +Remark 4.5. In the case of a polygonal domain and s = 1 which is further considered in our numerical +experiments, we know that r = 3 +2 and r(1, 1) = 1 so that ˜t ≥ r(1, 3 +2) ≥ 1. In turn the convergence bound is +established with t = r − 1 = 1 +2, a rate that we observe in practice, see §5. +4.5. The case of point value evaluations. We discuss now the case where +ν(v) = δz(v) = v(z), +for some z ∈ Ω. In order to guarantee that point evaluation is a continuous functional on Hs, we assume +that +s > d − 1 +2 +, +that is s > 1 +2 for d = 2, and s > 1 for d = 3. We want to find the Riesz representer of such a point evaluation +functional on Hs. Note that our assumption on s ensures the continuous embeddings +Hs(Γ) ⊂ C(Γ), +as well as +Hs(Ω) ⊂ Hr(Ω) ⊂ C(Ω), +since in view of (2.3) +r = min +� +s + 1 +2, r∗� +> d +2. +where in the inequality we recall that r∗ > 3 +2 for polygonal domains. +The point evaluation functional ν is thus continuous on Hs(Ω) with norm Cs bounded independently of +the position of z. Of course, the Galerkin scheme analyzed above for ν ∈ H1(Ω)∗ continues to make sense +since ν is well defined on the space Vh. +As explained in §3.4, the prescriptions in Step 2 of the recovery algorithm need to be strengthened in +the point evaluation setting. Thus, we are interested in bounding the pointwise error |φ(x) − φh(x)| at the +measurement points, in addition to the H1-error ∥φ − φh∥H1(Ω). In what follows, we establish a modified +version of Theorem 4.4 in the point value setting that gives a convergence rate for ∥φ − φh∥H1(Ω), and +in addition for ∥φ − φh∥L∞(Ω) ensuring the pointwise error control. We stress that the numerical method +remains unchanged, that is, φh is defined in the exact same way as previously. The new ingredients that +are needed in our investigation are two classical results on the behavior of the finite element method with +respect to the L∞ norm. +The first one is the so-called weak discrete maximum principle which states that there exists a constant +Cmax such that, for all h > 0, +(4.19) +∥Ehgh∥L∞(Ω) ≤ Cmax∥gh∥L∞(Γ), +gh ∈ Th. +This result was first established in [4] with constant Cmax = 1 for piecewise linear Lagrange finite elements +under acuteness assumptions on the angles of the simplices. The above version with Cmax ≥ 1 is established in +[25] for Lagrange finite elements of any degree on 2d polygonal domains, under the more general assumption +that the meshes {Th}h>0 are quasi-uniform (in addition to shape regularity, all elements of Th have diameters +of order h). A similar result is established in [13] on 3d convex polyhedrons. +The second ingredient we need is a stability property in the L∞ norm of the Galerkin projection Rh : +H1 +0(Ω) → Wh where Rhv, v ∈ H1 +0(Ω), is defined by +� +Ω +∇Rhv · ∇vh = +� +Ω +∇v · ∇vh, +vh ∈ Wh. +Specifically, this result states that there exists a constant Cgal and exponent a ≥ 0 such that, for all h > 0, +(4.20) +∥Rhv∥L∞(Ω) ≤ Cgal(1 + | ln(h)|)a∥v∥L∞(Ω), +v ∈ L∞(Ω) ∩ H1 +0(Ω), +that is, the Ritz projection is stable and quasi-optimal, uniformly in h, up to a logarithmic factor. This +result is established in [25] for Lagrange finite elements on 2d polygonal domains and quasi-uniform partitions, +with a = 1 in the case of piecewise linear elements and a = 0 for higher order elements. A similar result is +established in [13] with a = 0 for convex polygons and polyhedrons. In going further, we assume that the +choice of finite element meshes ensures the validity of (4.19) and (4.20). +17 + +We begin our analysis with the observation that under the additional mesh assumptions, Lemma 4.3 can +be adapted to obtain an estimate on ∥(E − Eh)gh∥L∞(Ω). +Lemma 4.6. For any gh ∈ Th, one has +(4.21) +∥(E − Eh)gh∥L∞(Ω) ≤ C3(1 + | ln(h)|)a)hr− d +2 ∥gh∥Hs(Γ), +where C3 depends on (r, s), the geometry of Ω, and the family of meshes through Cgal. +Proof. For any vh ∈ Vh such that T (vh) = gh, we write +∥(E − Eh)gh∥L∞(Ω) ≤ ∥Egh − vh∥L∞(Ω) + ∥Ehgh − vh∥L∞(Ω). +It is readily seen that Ehgh − vh = Rh(Ehgh − vh) = Rh(Egh − vh). Indeed RhEhgh − RhEgh ∈ Wh and +� +Ω +∇(Rh(Ehgh − Egh)) · ∇vh = +� +Ω +∇(Ehgh − Egh) · ∇vh = 0 for all vh ∈ Wh. Therefore, by (4.20), we obtain +∥(E − Eh)gh∥L∞(Ω) ≤ (1 + Cgal(1 + | ln(h)|)a) +min +vh∈Vh,T (vh)=gh ∥Egh − vh∥L∞(Ω). +On the other hand, we are ensured that Egh belongs to Hr(Ω) where r > +d +2, and therefore has H¨older +smoothness of order r − d +2 > 0 with +∥Egh∥Cr− d +2 (Ω) ≤ Ce∥Egh∥Hr(Ω) ≤ CeC1∥gh∥Hs(Γ), +where Ce is the relevant continuous embedding constant. By standard finite element approximation theory, +min +vh∈Vh,T (vh)=gh ∥Egh − vh∥L∞(Ω) ≤ Chr− d +2 ∥Egh∥Cr− d +2 (Ω), +where C depends on r and the shape-parameter σ and therefore we obtain (4.21). +We are now in position to give an adaptation of Theorem 4.4 to the point value setting. +Theorem 4.7. Under Assumption R, for any t1 < min{r − d +2, r(s, s + 1 +2) + r(s, s − 1 +2)}, one has +(4.22) +∥φ − φh∥H1(Ω) ≤ Cht1, +and for any t2 < min{r − d +2, 2r(s, s − d−1 +2 )}, one has +(4.23) +∥φ − φh∥L∞(Ω) ≤ Cht2. +The constant C depends in both cases on s, t1 and t2, on the geometry of Ω, as well as on the family of +meshes through the constants Cmax and Cgal, and the shape parameter σ. +Proof. We estimate ∥φ − φh∥H1(Ω) by adapting certain steps in the proof of Theorem 4.4. The first change +lies in the a priori estimate of the Hs(Γ) norm of ψh that was previously given by (4.13) which is not valid +anymore since Cν = ∞. Instead, we write +∥ψh∥2 +Hs(Γ) = ⟨ψh, ψh⟩Hs(Γ) = ν(Ehψh) ≤ ∥Ehψh∥L∞(Ω) ≤ Cmax∥ψh∥L∞(Γ) ≤ CmaxBs∥ψh∥Hs(Γ), +where we have used (4.19) and where Bs is the continuous embedding constant between Hs(Γ) and L∞(Γ). +In turn, we find that +(4.24) +∥ψh∥Hs(Γ) ≤ CmaxBs, +which results in the slightly modified estimate +∥(E − Eh)ψh∥H1(Ω) ≤ C2CmaxBshr−1, +for the second term of (4.16). +For the first term E(ψ − ψh), we proceed in a similar manner to the proof of Theorem 4.4. Namely, we +estimate the H1/2(Γ) norms of two summands in (4.17). The estimate of ∥ψh − ψh∥H1/2(Γ) is modified as +follows. We note that for any gh ∈ Th, +⟨ψh − ψh, gh⟩Hs(Γ) = ν((E − Eh)gh) ≤ ∥(E − Eh)gh∥L∞(Ω) ≤ C3(1 + | ln(h)|)a)hr− d +2 ∥gh∥Hs(Γ), +18 + +where we have now used Lemma 4.6. Taking gh = ψh − ψh we obtain a bound of order O(hr− d +2 ) up to +logarithmic factors for its Hs norm, and in turn for its H1/2 norm. The estimate of ∥ψ − ψh∥H1/2(Γ) is +left unchanged and of order O(h˜t). Combining these various estimates, we have established (4.22) for any +t1 < min{r − d +2, ˜t}, with ˜t := r(s, s + 1 +2) + r(s, s − 1 +2). +We next estimate ∥φ − φh∥L∞(Ω) by the following adaptation of the proof of Theorem 4.4. For the first +term (E − Eh)ψh of (4.16) we use Lemma 4.6 combined with the estimate (4.24) of ψh which give us +∥(E − Eh)ψh∥L∞(Ω) ≤ CmaxBsC3(1 + | ln(h)|)a)hr− d +2 . +For the second term E(ψ − ψh), we use the continuous maximum principle to obtain +∥E(ψ − ψh)∥L∞(Ω) ≤ ∥ψ − ψh∥L∞(Γ) ≤ ∥ψh − ψh∥L∞(Γ) + ∥ψ − ψh∥L∞(Γ) +For the first summand, we write +∥ψh − ψh∥L∞(Γ) ≤ Ce∥ψh − ψh∥Hs(Γ), +where Ce is the relevant continuous embedding constant, and we have already observed that ∥ψh − ψh∥Hs(Γ) +satisfies a bound in O(hr− d +2 ) up to logarithmic factors. For the second summand, we may write +∥ψ − ψh∥L∞(Γ) ≤ Ce∥ψ − ψh∥Hs(Γ), +where Ce is the relevant continuous embedding constant. Since ν belongs to H−s+δ(Γ) for all δ < s − d−1 +2 , +we can apply the estimate (4.12) to reach a convergence bound +∥ψ − ψh∥Hs(Γ) ≤ Chr(s,δ), +where C depends on the closeness of δ to s − d−1 +2 , and on the family of meshes through the shape parameter +σ. Combining these estimates then gives (4.23) for any t2 < min{r − d +2, ˜t} where ˜t = r(s, s − d−1 +2 ), since δ +can be picked arbitrarily close to s − d−1 +2 . +We can improve the range of t2 as follows: pick any s such that d−1 +2 +< s < s and write +∥ψ − ψh∥L∞(Γ) ≤ Ce∥ψ − ψh∥Hs(Γ), +where Ce is the relevant continuous embedding constant. We then apply a similar Aubin-Nitsche argument +to derive an estimate +∥ψ − ψh∥Hs(Γ) ≤ Chr(s,δ)+r(s,s−s). +Combining these estimates gives (4.23) for any t2 < min{r − d +2, t}, where t := 2r(s, s − d−1 +2 ) since s can be +picked arbitrarily close to d−1 +2 +and δ arbitrarily close to s − d−1 +2 . +5. Numerical Illustrations +In this section, we implement some examples of our numerical method. For this, we have to specify the +domain Ω, the functionals λj, and a function u ∈ H1(Ω) which gives rise to the data vector w = λ(u). +While our numerical method can be applied to general choices for these quantities, in our illustrations we +make these choices so that the computations are not too involved but yet allow us the flexibility to illustrate +certain features of our algorithm. The specific choices we make for our numerical example are the following. +The domain: In order to simplify the presentation, we restrict ourselves when Ω = (0, 1)2 but point out +again that the algorithm can be extended to more general domains. +The function u: For the function u we choose the harmonic function u = uH where +(5.1) +uH(x, y) = ex cos(y), +(x, y) ∈ Ω := (0, 1)2. +This choice means that u0 = 0 and therefore allows us not to deal with the computation of ˆu0. This choice +corresponds to the right side f = 0. Note that the trace of uH on the boundary Γ is piecewise smooth and +continuous. Therefore, we have T (uH) ∈ H1(Γ). We take s = 1 as our assumption on the value of s. This +means that we shall seek Riesz representor for the functionals given below when viewed as acting on H1(Ω). +19 + +5.1. The case of linear functionals defined on H1(Ω). In this section, we consider numerical experiments +for linear functionals defined on H1(Ω). In our illustrative example, we relabel these functionals by double +indices associated with a regular square grid. More precisely, +(5.2) +λi,j(v) := +1 +√ +2πr2 +� +Ω +v(z)e− 1 +2 +|z−zi,j|2 +r2 +dz, v ∈ H1(Ω), +i, j = 1, ..., √m. +Here, we assume that m is a square integer and r = 0.1 in our simulations. The centers zi,j ∈ Ω are uniformly +distributed +(5.3) +zi,j := +1 +√m + 1(i, j), +i, j = 1, ..., √m. +Recall that our numerical algorithm as described in Section 3.2 is based on finite element methods. +Specifically, we us the finite element spaces +Vh := +� +vh ∈ C0(Ω) : vh|T ∈ Q1, +T ∈ Th +� +, +where Th are subdivisions of Ω made of squares of equal side length h and Q1 denotes the space of polynomials +of degree at most 1 in each direction. In order to study the effect of the mesh-size we specifically consider +h = hn := 2−n, +n = 4, . . . , 9, +that is, bilinear elements on uniformly refined meshes with mesh-size 2−n. +We display in Table 1 the results of our numerical recovery algorithm. The entries in the table are the +recovery errors +e(m, n) := ∥uH − ˆuH∥H1(Ω), +where ˆuH ∈ Vhn is the recovery for the particular values of m and n. +n +m +4 +9 +16 +25 +36 +4 +0.7 +0.28 +0.2 +141.73 +49.43 +5 +0.7 +0.28 +0.18 +16.0 +16.31 +6 +0.7 +0.28 +0.18 +0.2 +1.79 +7 +0.7 +0.28 +0.18 +0.16 +0.11 +8 +0.7 +0.28 +0.18 +0.09 +0.06 +9 +0.7 +0.28 +0.18 +0.09 +0.06 +Table 1. Recovery error e(m, n) for different amounts of Gaussian measurements m and +finite element refinements n. +We have proven in this paper that our numerical recovery algorithm is near optimal with constant C that +can be made arbitrarily close to one by choosing n sufficiently large. This means that the error e(m, n) +satisfies e(m, n) ≤ CR(KH +w )H1(Ω) for n sufficiently large. +Increasing the number m of measurements is +expected to decrease this Chebyshev radius. While one is tempted to think that the entries in each column +of the table provides an upper bound for the Chebyshev radius of KH +w for these measurements, this is not +guaranteed since we are only measuring the error for one function from Kw, namely uH, and not all possible +functions from Kw. However, the entries in any given column provide a lower bound for the Chebyshev +radius of KH +w provided n is sufficiently large. +Increasing the number m of measurements requires a finer resolution, i.e., increasing n, of the finite +element discretization until the perturbation ε in Theorem 3.2 is sufficiently small. This is indeed confirmed +by the results in Table 1 where stagnating error bounds (in each fixed column) indicate the corresponding +tip-over point. We notice in particular that for small values of n, the error becomes very large as m grows. +This is explained by the fact that the Gramian matrix G becomes severely ill-conditioned, and in turn +the prescriptions on ∥G − ˆG∥1 cannot be fulfilled when using finite element approximation of the Riesz +representers on too coarse meshes. An overall convergence of the recovery error to zero can, of course, only +take place when both m and n increase. +20 + +5.2. The case of point value measurements. In this section, we describe our numerical experiments in +the case where the linear functionals λi,j are point evaluations at points from Ω. Recall that while the λi,j are +not defined for general functions in H1(Ω) they are defined for functions in the model class KH := U(Hs(Ω)) +provided s is sufficiently large (s > 1/2 for d = 2 and s > 1 for d = 3). This means that the optimal recovery +problem is well posed in such a case. We have given in §3.4 sufficient conditions on a numerical algorithm to +give near optimal recovery and then we have gone on to show in §4.5 that our proposed numerical algorithm +based on discrete harmonics converges to a near optimal recovery with any constant C > 1 provided that +the finite element spaces are discretized fine enough. +In the numerical experiments of this section, we again take Ω = (0, 1)2, s = 1, and the data to be the +point values of the harmonic function uH defined in (5.1). We choose the evaluation points to be the zi,j of +(5.3). We now provide in Table 2 the recovery error e(m, n). The observed behavior is similar to the case of +Gaussian averages; see Table 1. +n +m +4 +9 +16 +25 +36 +4 +0.70 +0.28 +0.19 +14.43 +15.49 +5 +0.70 +0.28 +0.18 +32.56 +8.02 +6 +0.70 +0.28 +0.18 +1.51 +2.27 +7 +0.70 +0.28 +0.18 +0.53 +0.89 +8 +0.70 +0.28 +0.18 +0.20 +0.14 +9 +0.70 +0.28 +0.18 +0.14 +0.11 +Table 2. Recovery error e(m, n) for different amounts of point evaluation measurements +m and refinements n. +5.3. Additional comments on the approximation of Riesz representers. Finally, we provide a little +more information on the computations that may be of interest to the reader. We work in the same setting +as in the previous sections. Let us begin with the rate of convergence of our numerical approximations to +the Riesz representers. +We first consider the computation of the Riesz representer for the Gaussian measurement functional +centered at z = zi,j := (0.75, 0.5). Let φn ∈ Vhn be the approximation to the Riesz representer φ produced +by the finite element computation. Figure 5.1 shows the error ∥φn − φ9∥H1(Ω), n = 2, . . . , 8. This graph +indicates an error decay Ch1/2 +n +which matches the rate guaranteed by Theorem 4.4, see also Remark 4.5. +Next consider the computation of the Riesz representer for point evaluation at the same z. Figure 5.1 +reports the numerical computations of error in both the H1(Ω) and L∞(Ω) norms. Again, the graph indicates +an error decay Ch1/2 +n +for the H1(Ω) norm which matches the rate guaranteed by Theorem 4.7 and a decay +rate closer to Chn for the L∞(Ω) norm (Theorem 4.7 only guarantees Ch1/2 +n ). +6. Optimal data sites: Gelfand widths and sampling numbers +In this section, we make some comments on the number of measurements m that are needed to guarantee +a prescribed error in the recovery of u. Bounds on m are known to be governed by the Gelfand width for +the case of general linear functionals and by sampling numbers when the functionals are required to be point +evaluations. We explain what is known about these quantities for our specific model classes. As we shall see +these issues are not completely settled for the model classes studied in this paper. The problem of finding +the best choice of functionals, respectively point evaluations, is in need of further research. +We have seen that the accuracy of the optimal recovery of u ∈ Kw is given by the Chebyshev radius +R(Kw) := R(Kw)H1(Ω) or equivalently R(KH +w ) := R(KH +w )H1(Ω) for the harmonic component. The worst +case recovery error R(K) over the class K is defined by +(6.1) +R(K)H1(Ω) := sup +w∈Rm R(Kw)H1(Ω), +Notice that this worst case error fixes the measurement functionals but allows the measurements w to come +from any function in K. Both the individual error R(Kw) and the worst case error R(K) are very dependent +21 + +0.000010 +0.000100 +0.001000 +0.010000 +0.100000 +1.000000 +100 +1000 +dim(Vhn) +Gaussian: H1 error +Point evaluation: L∞ error +Point evaluation: H1 error +order 1 +2 +Figure 5.1. Approximation errors for the Riesz representers of the Gaussian and point +evaluation functionals. +on the choice of the data functionals λj. For example, in the case that these functionals are point evaluations +at points z1, . . . , zm ∈ ¯Ω, then R(Kw) and R(K) will depend very much on the positioning of these points +in ¯Ω. +In the case of general linear functionals, one may fix m and then search for the λ1, . . . , λm that minimize +the worst case recovery error over the class K. This minimal worst case error is called the Gelfand width of +K. If we restrict the linear functionals to be given by point evaluation, we could correspondingly search for +the sampling points x1, . . . , xm minimizing the worst case recovery error. This minimal error is called the +deterministic sampling number of K. +The goal of this section is not to provide new results on Gelfand widths and sampling numbers, since we +regard this as a separate issue in need of a systematic study, but to discuss what is known about them in +our setting and refer the reader to the relevant papers. Let us recall that R(Kw) is equivalent to R(KH +w )H1 +and so we restrict our discussion in what follows to sampling of harmonic functions. +6.1. Optimal choice of functionals. Suppose we fix the number m of observation to be allowed and ask +what is the optimal choice for the λj, j = 1, . . . , m, and what is the optimal error of recovery for this choice. +The answer to the second question is given by the Gelfand width of K. Given a compact set K of a Banach +space X, we define the Gelfand width of K in X by +(6.2) +dm(K)X := +inf +λ1,...,λm R(K)X +where the infimum is taken over the linear functionals defined on X. Let us mention that this definition +differs from that employed in the classical literature [21] where dm(K)X is defined as the infimum over all +spaces F of codimension n of max{∥v∥X : v ∈ K ∩F}. The two definitions are equivalent in the case where +K is a centrally symmetric set such that K + K ⊂ CK for some constant C ≥ 1. +Any set of functionals which attains the infimum in (6.2) would be optimal. The Gelfand width is often +used as a benchmark for performance since it says that no matter how the m functionals λ1, . . . , λm are +chosen, the error of recovery of u ∈ K cannot be better than dm(K)X. +When X is a Hilbert space and K is the ball of a Hilbert space Y with compact embedding in X, it is +known that the Gelfand width coincides with the Kolmogorov width, that is +dm(K)X = dm(K)X := +inf +dim(E)=m dist(K, E)X = +inf +dim(E)=m max{∥v − PEv∥X : v ∈ K}, +where the infimum is taken over all linear spaces E of dimension m. This is precisely our setting as discussed +in §3: taking X = H1 := H1(Ω) and K as in (1.4), we have +(6.3) +dm(K)H1(Ω) = dm(KH)H1(Ω) = dm(KH)H1(Ω) ∼ dm(KB)H1/2(Γ) = dm(KB)H1/2(Γ), +22 + +where the equivalence follows from (1.3). Upper and lower bounds for the Gelfand width of KB in L2(Γ) +are explicitely given in [20]. +We can estimate the rate of decay of the Kolmogorov and Gelfand width of KB in H1/2(Γ) by the following +general argument: as explained in §2.1, for the admissible range of smoothness, the Sobolev spaces Hs(Γ) +have an intrinsic description by locally mapping the boundary onto domains of Rd−1. More precisely, in [17] +and [10], the Hs(Γ) norm of g is defined as +(6.4) +∥g∥Hs(Γ) := +� J +� +j=1 +∥gj∥2 +Hs(Rj) +�1/2 +, +where the Rj are open bounded rectangles of Rd−1 that are mapped by transforms γj into portions Γj that +constitute a covering of Γ, and gj = g ◦ γj are the local pullbacks. +From this it readily follows that the Gelfand and Kolmogorov m-width of the unit ball of Hs(Γ) in the +norm Ht(Γ), with 0 ≤ t < s behaves similar to that of the unit ball of Hs(R) in the norm Ht(R) where R is +a bounded rectangle of Rd−1. The latter is known to behave like m− s−t +d−1 . Therefore, for KH = U(Hs) with +s > 1 +2 in the admissible range allowed by the boundary smoothness, one has +(6.5) +cm− s−1/2 +d−1 ≤ dm(KH)H1(Ω) ≤ Cm− s−1/2 +d−1 , +m ≥ 1, +where c and C are positive constants depending only on Ω and s. +Remark 6.1. We have already observed in §2 that the space Hs(Ω) is continuously embedded in the Sobolev +space Hr(Ω) with r := max{s+ 1 +2, r∗} and in particular r = s+ 1 +2 for smooth domains. However the Gelfand +and Kolmogorov widths of the unit ball of Hr(Ω) in H1(Ω) have the slower decay rate m− r−1 +d += m− s−1/2 +d +compared to (6.5) for Hs(Ω). +This improvement reflects the fact that the functions from Hs(Ω) have d +variables but are in fact determined by functions of d − 1 variables. The reduction in dimension from d to +d − 1 is related to the fact that in our formulation of our problem we have complete knowledge of f. +6.2. Optimal choice of sampling points. We turn to the particular setting where the λj are point +evaluations functionals, +λj(v) = v(xj), +at m points xj ∈ Ω. Similar to the Gelfand width, the deterministic sampling numbers are defined as +(6.6) +ρm(K)X := +inf +x1,...,xm R(K)X, +A variant of this is to measure the worst case expected recovery error when the m points are chosen at +random according to a probabilty distribution and search for the distribution that minimizes this error, +leading to the randomized sampling number of K. Obviously, one has +(6.7) +ρm(K)X ≥ dm(K)X. +In the majority of the literature, deterministic and randomized sampling numbers are studied with error +measured in the L2(Ω) norm. In this setting, concrete strategies for optimal deterministic and randomized +point design have been given when K is the unit ball of a reproducing kernel Hilbert space H defined on Ω. +In particular, the recent results in [16, 12, 18, 5] show that under the assumption +� +m>0 +|dm(K)L2(Ω)|2 < ∞, +then, for all t > 1 +2, +sup +m≥1 +mtdm(K)L2(Ω) < ∞ =⇒ sup +m≥1 +mtρm(K)L2(Ω) < ∞. +In words, under the above assumptions, optimal recovery in L2(Ω) has the same algebraic convergence rate +when using optimally chosen point values compared to an optimal choice of general linear functionals. +While similar general results have not been established for Gelfand width and sampling numbers in the +H1 norm, we argue that they hold in our particular setting where H = Hs(Ω). For simplicity, as in §4, +we consider a domain that is either a polygon when d = 2 or polyhedron when d = 3, and thus consider +the range +d−1 +2 +< s < +3 +2 where the restriction from below ensures that Hs(Ω) ⊂ C(Ω). +Recalling the +23 + +finite element spaces Vh and their traces Th on the boundary, based on quasi-uniform meshes {Th}h>0, we +consider for a given h > 0 the measurement points x1, . . . , xm that are the mesh vertices located on Γ. By +the quasi-uniformity property the number m = m(h) of these points satisfies +ch1−d ≤ m ≤ Ch1−d, +for some c, C > 0 independent of h. If v ∈ Hs(Ω), its trace vΓ belongs to Hs(Γ). Then, denoting by Ih +the piecewise linear interpolant on the boundary, standard finite element approximation theory ensures the +estimate +∥vΓ − IhvΓ∥H1/2(Γ) ≤ Chs− 1 +2 ∥vΓ∥Hs(Γ) = Chs− 1 +2 ∥v∥Hs(Ω), +for some C that only depends on s. Therefore, introducing ˜v := EIhv, one has +∥v − ˜v∥H1(Ω) ≤ CE∥vΓ − IhvΓ∥H1/2(Γ) ≤ CDEm− s−1/2 +d−1 ∥v∥Hs(Ω). +Since ˜v only depends on the value of v at the points x1, . . . , xm, we have thus proved an upper bound of +order m− s−1/2 +d−1 +for ρm(KH)H1(Ω), and in turn for ρm(K)H1(Ω). In view of (6.7) and (6.5), a lower bound of +the same order must hold. In summary, similar to the Gelfand widths, the sampling numbers satisfy +(6.8) +˜cm− s−1/2 +d−1 ≤ ρm(K)H1(Ω) ≤ ˜Cm− s−1/2 +d−1 , +m ≥ 1, +where ˜c and ˜C are positive constants depending only on Ω and s. +References +[1] R. A. Adam and J. F. Fournier, Sobolev spaces, Elsevier, 2003. +[2] G. Auchmuty, Reproducing Kernels for Hilbert Spaces of Real Harmonic Functions, SINUM, 41(5), 2009. +[3] S.L. Brunton, B.R. Noack, P. Koumoutsakos, Machine Learning for Fluid Mechanics, Annual Review of Fluid Mechanics +36(9), 477–508, 2020. +[4] P.G. Ciarlet and P.A. 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Yosida, Functional analysis, Springer Science & Business Media, 2012. +Peter Binev, Department of Mathematics, University of South Carolina, Columbia, SC 29208, binev@math.sc.edu +Andrea Bonito, Department of Mathematics, Texas A&M University, College Station, TX 77843, bonito@math.tamu.edu +Albert +Cohen, +Labortoire +Jacques-Louis +Lions, +Sorbonne +Universi´e, +4, +Place +Jussieu, +75005 +Paris, +France, +albert.cohen@sorbonne-universite.fr +Wolfgang Dahmen, Department of Mathematics, University of South Carolina, Columbia, SC 29208, dahmen@math.sc.edu +Ronald DeVore, Department of Mathematics, Texas A&M University, College Station, TX 77843, rdevore@math.tamu.edu +Guergana +Petrova, +Department +of +Mathematics, +Texas +A&M +University, +College +Station, +TX +77843, +gpetrova@math.tamu.edu +25 + diff --git a/C9E5T4oBgHgl3EQfUA9Q/content/tmp_files/load_file.txt b/C9E5T4oBgHgl3EQfUA9Q/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..08c2c6d6274e476cbef31fddd3eff84f4e1be2c5 --- /dev/null +++ b/C9E5T4oBgHgl3EQfUA9Q/content/tmp_files/load_file.txt @@ -0,0 +1,1157 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf,len=1156 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='05540v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='NA] 13 Jan 2023 SOLVING PDES WITH INCOMPLETE INFORMATION PETER BINEV, ANDREA BONITO, ALBERT COHEN, WOLFGANG DAHMEN RONALD DEVORE, AND GUERGANA PETROVA Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We consider the problem of numerically approximating the solutions to a partial differential equation (PDE) when there is insufficient information to determine a unique solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Our main example is the Poisson boundary value problem, when the boundary data is unknown and instead one observes finitely many linear measurements of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We view this setting as an optimal recovery problem and develop theory and numerical algorithms for its solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The main vehicle employed is the derivation and approximation of the Riesz representers of these functionals with respect to relevant Hilbert spaces of harmonic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Introduction The questions we investigate sit in the broad research area of using measurements to enhance the numer- ical recovery of the solution u to a PDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The particular setting addressed in this paper is to numerically approximate the solution to an elliptic boundary value problem when there is insufficient information on the boundary value to determine a unique solution to the PDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In place of complete boundary information, we have a finite number of data observations of the solution u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This data serves to narrow the set of possible solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We ask what is the optimal accuracy to which we can recover u and what is a near optimal numerical algorithm to approximate u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Problems of this particular type arise in several fields of science and engineering (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' [28, 3, 7] for examples in fluid dynamics), where a lack of full information on boundary conditions arises for various reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For example, the correct physics might not be fully understood [22, 24], or the boundary values are not accessible [11], or they must be appropriately modified in numerical schemes [8, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Other examples of application domains for the results of the present paper can be found in the introduction of [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' A model for PDEs with incomplete data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In this paper, we consider the model elliptic problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1) − ∆u = f in Ω, u = g on Γ := ∂Ω, where Ω ⊂ Rd is a bounded Lipschitz domain with d = 2 or 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The Lax-Milgram theorem [29] implies the existence and uniqueness of a solution u from the Sobolev space H1(Ω) to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1), once f and g are prescribed in H−1(Ω) (the dual of H1 0(Ω)) and in H1/2(Γ) (the image of H1(Ω) by the trace operator), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Recall that the trace operator T is defined on a function w ∈ C(¯Ω) as the restriction of w to Γ and this definition is then generalized to functions in Sobolev spaces by a denseness argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In particular, the trace operator is well defined on H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For any function v in H1(Ω) we denote by vΓ its trace, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2) vΓ := T (v) = v|Γ, v ∈ H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The Lax-Milgram analysis also yields the inequalities (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3) c0∥v∥H1(Ω) ≤ ∥∆v∥H−1(Ω) + ∥vΓ∥H1/2(Γ) ≤ c1∥v∥H1(Ω), v ∈ H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Here the constants c0, c1 depend on Ω and on the particular choice of norms employed on H1(Ω) and H1/2(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Our interest centers on the question of how well we can numerically recover u in the H1 norm when we do not have sufficient knowledge to guarantee a unique solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' There are many possible settings to which our techniques apply, but we shall focus on the following scenario: Date: January 16, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This research was supported by the NSF Grants DMS 2110811 (AB), DMS 2038080 (PB and WD), DMS-2012469 (WD), DMS 21340077 (RD and GP), the MURI ONR Grant N00014-20-1-278 (RD and GP), the ARO Grant W911NF2010318 (PB), and the SFB 1481, funded by the German Research Foundation (WD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 1 (i) We have a complete knowledge of f but we do not know g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' (ii) The function g belongs to a known compact subset KB of H 1 2 (Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Thus, membership in KB describes our knowledge of the boundary data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The function u we wish to recover comes from the set (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='4) K := {u : u solves (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1) for some g ∈ KB}, which is easily seen from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3) to be a compact subset of H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' (iii) We have access to finitely many data observations of the unknown solution u, in terms of a vector (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='5) λ(u) := (λ1(u), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , λm(u)) ∈ Rm, where the λj are fixed and known linear functionals defined on the functions from K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Natural candidates for the compact set KB are balls of Sobolev spaces that are compactly embedded in H 1 2 (Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We thus restrict our attention for the remainder of this paper to the case (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='6) KB := U(Hs(Γ)), for some s > 1 2,where the precise definition of Hs(Γ) and its norm ∥ · ∥Hs(Γ) is described later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Note that U(Y ) denotes the unit ball of a Banach space Y with respect to the norm ∥ · ∥Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The optimal recovery benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Let wj := λj(u), j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , m, and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='7) w := (w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , wm) = λ(u) ∈ Rm, be the vector of data observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Therefore, the totality of information we have about u is that it lies in the compact set (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='8) Kw := {u ∈ K : λ(u) = w}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Our problem is to numerically find a function ˆu ∈ H1(Ω) which approximates simultaneously all the u ∈ Kw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This is a special case of the problem of optimal recovery from data (see [15, 27, 19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The optimal recovery, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' the best choice for ˆu, has the following well known theoretical description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Let B(Kw) be a smallest ball in H1(Ω) which contains Kw and let R(Kw) := R(Kw)H1(Ω) be its radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Then, R(Kw) is the optimal recovery error, that is, the smallest error we can have for recovering u in the norm of H1(Ω), and the center of B(Kw) is an optimal recovery of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We are interested in understanding how small R(Kw) is and what are the numerical algorithms which are near optimal in recovering u from the given data w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We say that an algorithm w �→ ˆu = ˆu(w) delivers near optimal recovery with constant C if (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='9) ∥u − ˆu(w)∥H1(Ω) ≤ CR(Kw), w ∈ Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Of course, we want C to be a reasonable constant independent of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Our results actually deliver a recovery estimate of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='10) ∥u − ˆu(w)∥H1(Ω) ≤ R(Kw) + ε, w ∈ Rm, where ε > 0 can made arbitrarily small at the price of higher computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In this sense, the recovery is near optimal with constant C > 1 in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='9) that can be made arbitrarily close to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' A connection with the recovery of harmonic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' There is a natural restatement of our recovery problem in terms of harmonic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Let f be the right side of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1), where f is a known fixed element of H−1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Let u0 be the function in H1(Ω) which is the solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1) with g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Then, we can write any function u ∈ K as (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='11) u = u0 + uH, where uH is a harmonic function in H1(Ω) which has boundary value g = T (uH) with g ∈ KB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Recall our assumption that KB is the unit ball of Hs(Γ) with s > 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 2 Let Hs(Ω) denote the set of harmonic functions v defined on Ω for which vΓ ∈ Hs(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We refer the reader to [2], where a detailed study of spaces like Hs(Ω) is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We define the norm on Hs(Ω) to be the one induced by the norm on Hs(Γ), namely, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='12) ∥v∥Hs(Ω) := ∥vΓ∥Hs(Γ), v ∈ Hs(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' There exist several equivalent definitions of norms on Hs(Γ), as discussed later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For the moment, observe that from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3) it follows the existence of a constant Cs such that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='13) ∥v∥H1(Ω) ≤ Cs∥v∥Hs(Ω), v ∈ Hs(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Indeed, the space Hs(Ω) is a Hilbert space that is compactly embedded in H1(Ω), as a consequence of the compact embedding of Hs(Γ) in H1/2(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We denote by KH the unit ball of Hs(Ω), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='14) KH := U(Hs(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Since the function u0 in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='11) is fixed, it follows from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='6) that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='15) R(Kw) = R(KH w′)H1(Ω), w′ := λ(uH) = w − λ(u0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' There are two conclusions that can be garnered from this reformulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The first is that the optimal error in recovering u ∈ Kw is the same as that in recovering the harmonic function uH ∈ KH w′ in the H1(Ω) norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The harmonic recovery problem does not involve f except in determining w′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The second point is that one possible numerical algorithm for our original problem is to first construct a sufficiently accurate approximation ˆu0 to u0 and then to numerically implement an optimal recovery of a harmonic function in KH from data observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This numerical approach requires the computation of w′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In theory, u0 is known to us since we have a complete knowledge of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' However, u0 must be computed and any approximation ˆu0 will induce an error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Although this error can be made arbitrarily small, it means that we only know w′ up to a certain numerical accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' One can thus view the harmonic reformulation as an optimal recovery problem with perturbed observations of w′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The numerical algorithm presented here follows this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Its central constituent, namely the recovery of harmonic functions from a finite number of noisy observations, can be readily employed as well in a number of different application scenarios described e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Objectives and outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Our main goal is to create numerical algorithms which are guaranteed to produce a function ˆu which is near optimal and to discuss their practical implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We begin in §2 with some remarks on the definition of the space Hs(Γ) and its norm, which are of importance both in the accuracy analysis and the practical implementation of recovery algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The general approach for optimal recovery that was introduced in [15, 14] is recalled in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We describe a solution algorithm which takes into consideration the effect of numerical perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We first consider the case when the linear functionals λj are defined on all of H1(Ω) and then adapt this algorithm to the case when the linear functionals are point evaluations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='16) λj(u) := u(xj), xj ∈ Ω, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Point evaluations are not defined on all of H1(Ω) when d > 1, however, they are defined on K when the smoothness order s is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The critical ingredient in our proposed algorithm is the numerical computation of the Riesz representers φj of the restrictions of λj to the Hilbert space Hs(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Each of these Riesz representers is characterized as a solution to an elliptic problem and can be computed offline since it does not involve the measurement vector w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Our suggested numerical method for approximating φj is based on finite element discretizations and is discussed in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We establish quantitative error bounds for the numerical approximation in terms of the mesh size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Numerical illustrations of the optimal recovery algorithm are given in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Note that the optimal recovery error over the class K strongly depends on the choice of the linear functionals λj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For example, in the case of point evaluation, this error can be very large if the data sites {xj}m j=1 are poorly positioned, or small if they are optimally positioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This points to the importance of the Gelfand widths and sampling numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' They describe the optimal recovery error over K with optimal choice of functionals in the general case and the point evaluation case, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The numerical behaviour of these quantities in our specific setting is discussed in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The spaces Hs(Γ) and Hs(Ω) In this section, we discuss the definition and basic properties of the spaces Hs(Γ) and Hs(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We refer to [1] for a general treatment of Sobolev spaces on domains D ⊂ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Recall that for fractional orders r > 0, the norm of Hr(D) is defined as ∥v∥2 Hr(D) := ∥v∥2 Hk(D) + � |α|=k � D×D |∂αv(x) − ∂αv(y)|2 |x − y|d+2(r−k) dxdy, where k is the integer such that k < r < k+1, and ∥v∥2 Hk(D) := � |α|≤k ∥∂αv∥2 L2(D) is the standard Hk-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Equivalent definitions of Hs(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Let Ω be any bounded Lipschitz domain in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We recall the trace operator T introduced in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' One first possible definition of the space Hs(Γ), for any s ≥ 1 2, is as the restriction of Hs+ 1 2 (Ω) to Γ, that is, Hs(Γ) = T (Hs+ 1 2 (Ω)), with norm (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1) ∥g∥Hs(Γ) := min � ∥v∥Hs+ 1 2 (Ω) : vΓ = g � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The resulting norm is referred to as the trace norm definition for Hs(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' There is a second, more intrinsic way to define Hs(Γ), by properly adapting the notion of Sobolev smoothness to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This can be done by locally mapping the boundary onto domains of Rd−1 and requiring that the pullback of g by such transformation have Hs smoothness on such domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We refer the reader to [10] and [17] for the complete intrinsic definition, where it is proved to be equivalent to the trace definition for a range of s that depends on the smoothness of the boundary Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For small values of s, Sobolev norms for Hs(Γ) may also be equivalently defined without the help of local parameterizations, as contour integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For example, if 0 < s < 1 and Ω is a Lipschitz domain, we define ∥g∥2 Hs(Γ) := ∥g∥2 L2(Γ) + � Γ×Γ |g(x) − g(y)|2 |x − y|d−1+2s dxdy, and if s = 1 and Ω is a polygonal domain, we define (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2) ∥g∥2 H1(Γ) := ∥g∥2 L2(Γ) + ∥∇Γg∥2 L2(Γ), where ∇Γ is the tangential gradient, and likewise ∥g∥2 Hs(Γ) := ∥g∥2 H1(Γ) + � Γ×Γ |∇Γg(x) − ∇Γg(y)|2 |x − y|d−1+2(s−1) dxdy, for 1 < s < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In the numerical illustration given in §5, we will specifically take the value s = 1 and a square domain, using the definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' When Ω has smooth boundary, it is known that the trace definition and intrinsic definition of the Hs(Γ) norms are equivalent for all s ≥ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' On the other hand, when Ω does not have a smooth boundary, it is easily seen that the two definition are not equivalent unless restrictions are made on s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Consider for example the case of polygonal domains of R2: it is easily seen that the trace vΓ of a smooth function v ∈ C∞(Ω) has a tangential gradient ∇ΓvΓ that generally has jump discontinuities at the corner points and thus does not belong to H1/2(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In turn, the equivalence between the trace and intrinsic norms only holds for s < 3 2 and in such case we limit the value of s to this range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The same restriction s < 3/2 applies to a polyhedral domain in the case d = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The regularity of functions in Hs(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We next give some remarks on the Sobolev smoothness of functions from the space Hs(Ω) when s > 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Clearly such harmonic functions are infinitely smooth inside Ω and also belong to H1(Ω), but one would like to know for which value of r they belong to Hr(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' To answer this question, we consider v ∈ Hs(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' By the definition of Hs(Ω), v is harmonic in Ω and vΓ ∈ Hs(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 4 Having assumed that s in the admissible range where all above definitions of the Hs(Γ) norms are equivalent, and using the first one, we know that there exists a function ˜v ∈ Hs+ 1 2 (Ω) such that ˜vΓ = vΓ ∥˜v∥Hs+ 1 2 (Ω) = ∥vΓ∥Hs(Γ) = ∥v∥Hs(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We define v := v − ˜v so that v = ˜v +v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We are interested in the regularity of v since it will give the regularity of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Notice that vΓ = 0 and −∆v = f := ∆˜v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The function f belongs to the Sobolev space Hs− 3 2 (Ω) and we are left with the classical question of the regularizing effect in Sobolev scales when solving the Laplace equation with Dirichlet boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Obviously, when Ω is smooth, we find that v ∈ Hs+ 1 2 (Ω) and so we have obtained the continuous embedding Hs(Ω) ⊂ Hr(Ω), r = s + 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For less smooth domains, the smoothing effect is limited (in particular by the presence of singularities on the boundary of Ω), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=', v is only guaranteed to be in Hr(Ω) where r may be less than s + 1/2, see [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' More precisely Hs(Ω) ⊂ Hr(Ω), where (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3) r := min � s + 1 2, r∗� , Here, r∗ = r∗(Ω) is the limiting bound for the smoothing effect: (i) For smooth domains r∗ = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' (ii) For convex domains r∗ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' (iii) For non-convex polygonal domains in R2, or a polyhedron in R3, one has 3/2 < r∗ < 2 where the value of r∗ depends on the reentrant angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' (iv) In particular for polygons, we can take r∗ = 1 + π ω − ε, for any ε > 0 where ω is the largest inner angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Note that r∗ could be strictly smaller than s + 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In summary, for an admissible range of r > 1 that depends on s and Ω one has the continuous embedding Hs(Ω) ⊂ Hr(Ω), and so there exists a constant C1 that depends on (r, s) and Ω, such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='4) ∥v∥Hr(Ω) ≤ C1∥v∥Hs(Ω) = C1∥vΓ∥Hs(Γ), v ∈ Hs(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' A near optimal recovery algorithm In this section, we present a numerical algorithm for solving (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1) when the information about the bound- ary value g is incomplete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We first work under the assumption that the λj’s are continuous over H1(Ω), and assumed to be linearly independent (linear independence can be guaranteed by throwing away dependent functionals when necessary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We prove that the proposed numerical recovery algorithm is near optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We then adapt our approach to the case where the λj’s are point evaluations, see (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='16), and therefore not continuous over H1(Ω) when d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Minimum norm data fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' As noted in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3, the problem of recovering u ∈ Kw is directly related to the problem of recovering the harmonic component uH ∈ KH from the given data observations w′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Note that KH is the unit ball of the Hilbert space Hs(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' There is a general approach for optimal recovery from data observations in this Hilbert space setting, as discussed e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We first describe the general principles of this technique and then apply them to our specific setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Let H be any Hilbert space and suppose that λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , λm ∈ H∗ are linearly independent functionals from H∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Let X be a Banach space such that H is continuously embedded in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We are interested in optimal recovery of a function v in the norm ∥ · ∥X, knowing that v ∈ K := U(H), the unit ball of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' If w ∈ Rm is the vector of observations, we define the minimal norm interpolant as v∗(w) = argmin{∥v∥H : v ∈ H and λ(v) = w}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 5 It is easily checked that when Kw is non-empty, the function v∗(w) coincides with the Chebyshev center of Kw in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' To see this, first note that any v ∈ Kw may be written as v = v∗(w) + η where η belongs to the null space N of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Because v∗(w) has minimal norm, v − v∗(w) = η is orthogonal to v∗(w) and hence from the Pythagorean theorem ∥v − v∗∥2 H = ∥v∥2 H − ∥v∗(w)∥2 H ≤ 1 − ∥v∗(w)∥2 H =: r2, because ∥v∥H ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Notice that v∗(w) − η is also in Kw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' It follows that Kw is precisely the ball in the affine space v∗(w) + N centered at v∗(w) and of radius r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In particular, Kw is centrally symmetric around v∗(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Therefore, v∗(w) is the Chebyshev center for Kw for any norm, in particular for the ∥ · ∥X norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Therefore, ∥v − v∗(w)∥X ≤ R(Kw)X, v ∈ Kw, that is, the minimal norm interpolant gives optimal recovery with constant C = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Standard Hilbert space analysis shows that the mapping w �→ v∗(w) is a linear operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' More importantly, it has a natural expression that is useful for numerical computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Namely, from the Riesz representation theorem each λj can be described as λj(v) = ⟨v, φj⟩H, v ∈ H, where φj ∈ H is called the Riesz representer of λj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The minimal norm interpolant has the representation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1) v∗ = m � j=1 a∗ jφj, where a∗ = (a∗ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , a∗ m) solves the system of equations Ga∗ = w, G := (⟨φi, φj⟩H)i,j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=',m, with G being the Gramian matrix associated to φ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , φm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In the case where H is a more general Banach space, we are still ensured that the minimal norm interpolation is a near-optimal recovery with constant C = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' However, its dependence on the data w is no longer linear and the above observation regarding its computation does not apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Let us now apply this general principle to our particular setting in which the Hilbert space H is Hs(Ω) and X = H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Let φj ∈ Hs(Ω) be the Riesz representer of the functional λj when viewed as a functional on Hs(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In other words λj(v) = ⟨v, φj⟩Hs(Ω), v ∈ Hs(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We assume that the λj are linearly independent on Hs(Ω) and thus the Gramian matrix G = � gi,j � i,j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=',m, gi,j := ⟨φi, φj⟩Hs = λj(φi), is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Now, let u = u0 + uH, with uH ∈ KH = U(Hs(Ω)) be the function in K that gave rise to our data observation w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' So, we have w′ = w − λ(u0) = λ(uH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' If a∗ is the vector in Rm which satisfies Ga∗ = w′, then u∗ H := �m j=1 a∗ jφj is the function of minimum Hs(Ω) norm which satisfies the data w′, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=', λ(u∗ H) = w′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We have seen that ∥uH − u∗ H∥H1(Ω) ≤ R(KH w′)H1(Ω), namely, u∗ H is the optimal recovery of the functions in KH w′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Note that the recovery error is measured in H1 not in Hs(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In turn, see (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='15), the function u∗ := u∗ H + u0 is the optimal recovery for functions in Kw: ∥u − u∗∥H1(Ω) ≤ R(Kw)H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The idea behind our proposed numerical method is to numerically construct a function ˆu ∈ H1 that approximates u∗ well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' If, for example, we have for ε > 0 the bound ∥u∗ − ˆu∥H1(Ω) ≤ ε, then for any u ∈ K, we have by the triangle inequality ∥u − ˆu∥H1(Ω) ≤ R(Kw)H1(Ω) + ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 6 Given any C > 1, by taking ε small enough, we have that ˆu is a near best recovery of the functions in Kw with constant C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The numerical recovery algorithm for H1-continuous functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Motivated by the above analysis, we propose the following numerical algorithm for solving our recovery problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The algorithm involves approximations of the function u0 and the Riesz representers φj, typically computed by finite element discretizations, and the application of the linear functionals λj to these approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In order to avoid extra technicalities, we make here the assumption that the applications of the functionals to a known finite element function can be exactly computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We first work under the additional assumption that the linear functionals λj are not only defined on K but that they are continuous over H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We define Λ as the maximum of the norms of the λj on H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In this case (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2) |λj(v)| ≤ Λ∥v∥H1(Ω), v ∈ H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In what follows, throughout this paper, we use the following weighted ℓ2 norm on Rm, ∥z∥ := \uf8eb \uf8ed 1 m m � j=1 |zj|2 \uf8f6 \uf8f8 1/2 = m−1/2∥z∥ℓ2, z = (z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , zm) ∈ Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In particular, we have ∥λ(v)∥ ≤ Λ∥v∥H1(Ω), v ∈ H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Given a user prescribed accuracy ε > 0, our algorithm does the following four steps involving intermediate tolerances (ε1, ε2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Step 1: We numerically find an approximation ˆu0 to u0 which satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3) ∥u0 − ˆu0∥H1(Ω) ≤ ε1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' To find such a ˆu0, we use standard or adaptive FEM methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Given that ˆu0 has been constructed, we define ˆw := w − λ(ˆu0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Then, for w′ := w − λ(u0) we have, see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='4) ∥w′ − ˆw∥ ≤ Λε1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' On the other hand, since |λj(v)| ≤ Λ∥v∥H1(Ω) ≤ Λs∥v∥Hs(Ω) ≤ Λs, where Λs := CsΛ, see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='13) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='14), we derive that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='5) ∥w′∥ ≤ Λs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Thus by triangle inequality, we also find that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='6) ∥ ˆw∥ ≤ Λs + Λε1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Step 2: For each j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , m, we numerically compute an approximation ˆφj ∈ H1(Ω) to φj which satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='7) ∥φj − ˆφj∥H1(Ω) ≤ ε2, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This numerical computation is crucial and is performed during the offline phase of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We detail it in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Note that the ˆφj’s are not assumed to be in Hs(Ω), and in particular not assumed to be harmonic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Step 3: We define and compute the matrix ˆG = (ˆgi,j)i,j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=',m, ˆgi,j := λj(ˆφi), and thus |ˆgi,j − gi,j| ≤ Λε2 for all i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 7 It follows that for the matrix R := G − ˆG we have ∥R∥1 ≤ mΛε2, where we use the shorthand notation ∥ · ∥1 := ∥ · ∥ℓ1→ℓ1 for matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Since G is invertible, we are ensured that ˆG is also invertible for ε2 small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We define M := ∥G−1∥1, ˆ M := ∥ ˆG−1∥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' While these two norms are finite, their size will depend on the nature and the positioning of the linear functionals λj, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , m, as it will be seen in the section on numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' These two numbers are close to one another when ε2 is small since ˆ M converges towards the unknown quantity M as ε2 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In particular, we have |M − ˆ M| = |∥G−1∥1 − ∥ ˆG−1∥1| ≤ ∥G−1 − ˆG−1∥1 = ∥ ˆG−1RG−1∥1 ≤ M ˆ MmΛε2, from which we obtain that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='8) M ≤ ˆ M 1 − m ˆ MΛε2 and ˆ M ≤ M 1 − mMΛε2 , provided that mMΛε2 < 1 and m ˆ MΛε2 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We also have the bound (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='9) ∥ ˆG−1 − G−1∥1 ≤ M 2 1 − mMΛε2 mΛε2 =: δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' It is important to observe that δ can be made arbitrarily small by diminishing ε2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Step 4: We numerically solve the m×m algebraic system ˆGˆa = ˆw, thereby finding a vector ˆa = (ˆa1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , ˆam).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We then define ˆuH := �m j=1 ˆaj ˆφj and our recovery of u is ˆu := ˆu0 + ˆuH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This step can be implemented by standard linear algebra solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' One major advantage of the above algorithm is that Steps 1-2-3 can be performed offline since they do not involve the data w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' That is, we can compute ˆu0, the approximate Riesz representers ˆφj and the approximate Gramian ˆG and its inverse without knowing w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In this way, the computation of ˆu from given data w can be done fast online by Step 4 which only involves solving an m × m linear system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This may be a significant advantage, for example, when having to process a large number of measurements for the same set of sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' A near optimal recovery bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The following theorem shows that a near optimal recovery of u can be reached provided that the tolerances in the above described algorithm are chosen small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For any prescribed ε > 0, if the tolerances (ε1, ε2), are small enough such that mMΛε2 < 1 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='10) ε1 + mMΛsε2 + (C0 + ε2)(mMΛε1 + m(Λs + Λε1)δ) ≤ ε, where C0 := maxj=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=',m ∥φj∥H1(Ω) and δ := M2 1−mMΛε2 mΛε2, then the function ˆu generated by the above algorithm satisfies ∥u − ˆu∥H1(Ω) ≤ R(Kw)H1(Ω) + ε, for every u ∈ Kw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Thus, for any C > 1 it is a near optimal recovery of u with constant C provided ε is taken sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Let u = u0 + v be our target function in Kw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We define w′ = w − λ(u0) and v∗ := v∗(w′) which is the Chebyshev center of KH w′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We recall the algebraic system Ga∗ = w′ associated to the characterization of v∗ (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We write (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='11) ∥u∗ H− ˆuH∥H1(Ω) ≤ ��� m � j=1 a∗ j(φj − ˆφj) ��� H1(Ω) + ��� m � j=1 (a∗ j −ˆaj)ˆφj ��� H1(Ω) ≤ ∥a∗∥ℓ1ε2+∥a∗−ˆa∥ℓ1(C0 +ε2), where we have used (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='7) and the fact that ∥ˆφj∥H1(Ω) ≤ ∥φj∥H1(Ω) + ∥φj − ˆφj∥H1(Ω) ≤ C0 + ε2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 8 Note that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='12) ∥a∗∥ℓ1 = ∥G−1w′∥ℓ1 ≤ M∥w′∥ℓ1 ≤ Mm∥w′∥ ≤ mMΛs, where we have used that ∥w′∥ℓ1 ≤ m∥w′∥ and inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Therefore it follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='11) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='12) that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='13) ∥u∗ H − ˆuH∥H1 ≤ mMΛsε2 + ∥a∗ − ˆa∥ℓ1(C0 + ε2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For the estimation of ∥a∗ − ˆa∥ℓ1, we introduce the intermediate vector ˜a ∈ Rm, which is the solution to the system G˜a = ˆw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Clearly, ∥˜a − a∗∥ℓ1 = ∥G−1( ˆw − w′)∥ℓ1 ≤ M∥ ˆw − w′∥ℓ1 ≤ Mm∥ ˆw − w′∥ ≤ mMΛε1, where we invoked (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' On the other hand, in view of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='9) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='6), we have ∥˜a − ˆa∥ℓ1 = ∥(G−1 − ˆG−1) ˆw∥ℓ1 ≤ δ∥ ˆw∥ℓ1 ≤ mδ∥ ˆw∥ ≤ m(Λs + Λε1)δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Combining these two estimates, we find that ∥a∗ − ˆa∥ℓ1 ≤ mMΛε1 + m(Λs + Λε1)δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We now insert this bound into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='13) to obtain ∥u∗ H − ˆuH∥H1(Ω) ≤ mMΛsε2 + (C0 + ε2)(mMΛε1 + m(Λs + Λε1)δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Thus, for u∗ := u0 + u∗ H and using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3), we have ∥u∗ − ˆu∥H1(Ω) ≤ ∥u0 − ˆu0∥H1(Ω) + ∥u∗ H − ˆuH∥H1(Ω) ≤ ε1 + mMΛsε2 + (C0 + ε2)(mMΛε1 + m(Λs + Λε1)δ) ≤ ε, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='14) Since u = u0 + uH, we have ∥u − u∗∥H1(Ω) = ∥uH − u∗ H∥H1(Ω) ≤ R(KH w′)H1(Ω) = R(Kw)H1(Ω), and the statement of the theorem follows from this inequality and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Note that in numerical computations the quantity ˆ M is available while M is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Thus in practice, in order to achieve the prescribed accuracy ε, we can first impose that ε2 < (2m ˆ MΛ)−1 and derive the inequalities, see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='8), M ≤ ˆ M 1 − m ˆ MΛε2 ≤ 2 ˆ M, ∥G−1 − ˆG−1∥1 ≤ ˆ M 2 1 − m ˆ MΛε2 mΛε2 ≤ 2 ˆ M 2mΛε2 =: ˆδ, where the last inequality is proven in a similar fashion to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' If we then follow the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2, the requirement in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='14) can be substituted by ε1 + 2m ˆ MΛsε2 + (C0 + ε2)(2m ˆ MΛε1 + m(Λs + Λε1)ˆδ) ≤ ε, and thus all participating quantities are computable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The result in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2 can easily be extended to the case of noisy data, that is, to the case when the observations ˜w = w + η, where η is a noise vector of norm ∥η∥ ≤ κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Indeed, the application of the algorithm to this noisy data leads to finding in Step 1 the vector ˆw := w + η − λ(ˆu0) that satisfies ∥w′ − ˆw∥ ≤ Λε1 + κ, and ∥ ˆw∥ ≤ Λs + ε1Λ + κ, where w′ = w − λ(u0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Inspection of the above proof shows that under the same assumption as in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2, one has the recovery bound ∥u − ˆu∥H1(Ω) ≤ R(Kw)H1(Ω) + ε + Cκ, for every u ∈ Kw, where C := (M + δ)m(C0 + ε2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 9 Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For simplicity, we did not introduce in the above analysis the possible errors in the application of the λi to the approximations ˆu0 and ˆφj, and in the numerical solution to the system ˆGˆa = ˆw, which would simply result in similar conditions involving the extra tolerance parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Point evaluation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We now want to extend the numerical algorithm and its analysis to the case when the data functionals λj, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , m, are point evaluations λj(h) := h(xj), xj ∈ Ω, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Of course these functionals are not defined for general functions h from H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' However, we can formulate the recovery problem whenever the functionals λj are well defined on K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We now discuss settings when this is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Recall that any u ∈ K can be written as u = u0 + uH, where u0 is the solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1) with right side f and g = 0 and uH ∈ Hs(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Point evaluation is well defined for the harmonic functions uH ∈ Hs(Ω), provided the points are in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In addition, they are well defined for points on the boundary Γ if the space Hs(Ω) continuously embeds into C(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For d = 2, this is the case when s > 1/2 and when d = 3, this is the case when s > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Concerning u0, we will need some additional assumption to guarantee that point evaluation of u0 makes sense at the data sites xj, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For example, it is enough to assume that u0 is globally continuous or at least in a neighborhood of each of these points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This can be guaranteed by assuming an appropriate regularity of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In this section, we assume that one of these settings holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We then write w′ j := uH(xj) = wj − u0(xj), j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , m, and follow the algorithm of the previous section with the following simple modifications: Modified Step 1: We numerically find an approximation ˆu0 to u0, which in addition to ∥u0 − ˆu0∥H1(Ω) ≤ ε1, satisfies the requirement (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='15) max i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=',m |u0(xi) − ˆu0(xi)| ≤ ε1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' To find such a ˆu0 we use standard or adaptive FEM methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Given that ˆu0 has been constructed, we define ˆwj := wj − ˆu0(xj), j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , m, and thus, using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='15), we have ∥w′ − ˆw∥ ≤ ε1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Modified Step 2: For each j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , m, we numerically compute an approximation ˆφj to φj, which in addition to ∥φj − ˆφj∥H1(Ω) ≤ ε2, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , m, satisfies the condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='16) max i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=',m |φj(xi) − ˆφj(xi)| ≤ ε2, i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='16) ensures that in Step 3 we can choose the entries ˆgi,j of the matrix ˆG as ˆgi,j = ˆφj(xi), i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The Steps 3 and 4 of our algorithm remain the same as in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' With the above modifications, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2 holds with the exact same statement in this point evaluation setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The proof is the same as that of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Finite element approximations of the Riesz representers The computation of an approximation ˆu0 to u0, required in Step 1 of the algorithm, can be carried out by standard finite element Galerkin schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Depending on our knowledge on f one can resort to known a priori estimates for ε1, or may employ standard a posteriori estimates to ensure that the underlying discretization provides a desired target accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Therefore, in the remainder of this section, we focus on a numerical implementation of Step 2 of the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Our proposed numerical algorithm for Step 2 is to use finite element methods to generate the approx- imations ˆφj of the Riesz representers φj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Note that each of the functions φj is harmonic on Ω but we do not require that the sought after numerical approximation ˆφj is itself harmonic but only that it provides an accurate H1(Ω) approximation to φj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This allows us to use finite element approximations which are themselves not harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' However, the ˆφj will necessarily have to be close to being harmonic since they approximate a harmonic function in the H1(Ω) norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Our numerical approach to constructing a ˆφj, discussed in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1, is to use discretely harmonic finite elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Here, ˆφj is a discrete harmonic extension of a finite element approximation to the trace ψj = T (φj) computed by solving a Galerkin problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In order to reduce computational cost (see Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2), we incorporate discrete harmonicity as constraints and introduce in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2 an equivalent saddle point formulation that has the same solution ˆφj, and which is the one that we practically employ in the numerical experiments given in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We give in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='4 an a priori analysis with error bounds for ∥φj − ˆφj∥H1 in terms of the finite element mesh size, in the case where the measurement functionals are continuous on H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' These error bounds can in turn be used to ensure the prescribed accuracy ε2 in Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We finally discuss in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='5 the extensions to the point value case where pointwise error bounds on |ˆφj(xi) − ˆφj(xi)| are also needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In order to simplify notation, we describe these procedures for finding an approximation ˆφ to the Riesz representer φ ∈ Hs = Hs(Ω) of a given linear functional ν on Hs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This numerical procedure is then applied with ν = λj, to find the numerical approximations ˆφj to the Riesz representer φj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For simplicity, throughout this section, we work under the assumption that Ω is a polygonal domain of R2 or polyhedral domain of R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This allows us to define finite element spaces based on triangular or simplicial partitions of Ω that in turn induce similar partitions on the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We assume that 1 2 < s < 3 2, which is the relevant range for such domains, as explained in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Our analysis can be extended to more general domains with smooth or piecewise smooth boundaries, for example by using isoparametric elements near the boundary, however at the price of considerably higher technicalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' A Galerkin formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Let s > 1/2 be fixed and assume that ν is any linear form continuous on Hs(Ω) with norm (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1) Cs := max{ν(v) : ∥v∥Hs(Ω) = 1} In view of the the definition of the Hs norm, the representer φ ∈ Hs(Ω) of ν for the corresponding inner product can be defined as φ = Eψ, where E is the harmonic extension operator of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3) below and where ψ ∈ Hs(Γ) is the solution to the following variational problem: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2) ⟨ψ, η⟩Hs(Γ) = µ(η) := ν(Eη), η ∈ Hs(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Note that this problem admits a unique solution and we have ∥ψ∥Hs(Γ) = ∥φ∥Hs(Ω) = Cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Recall that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3) Eg := argmin{∥∇v∥L2(Ω) : vΓ = g}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The function Eg is characterized by T (Eg) = g and � Ω ∇Eg · ∇v = 0, v ∈ H1 0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 11 From the left inequality in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3), one has (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='4) ∥Eg∥H1(Ω) ≤ CE∥g∥H1/2(Γ), g ∈ H1/2(Γ), where CE can be taken to be the inverse of the constant c0 in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Therefore, one approach to discretizing this problem is the following: consider finite element spaces Vh associated to a family of meshes {Th}h>0 of Ω, where as usual h denotes the maximum meshsize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We define Th to be the space obtained by restriction of Vh on the boundary Γ, that is, Th = T (Vh) Since we have assumed that Ω is a polygonal or polyhedral domain, the space Th is a standard finite element space for the boundary mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Having also assumed that s < 3/2, when using standard H1 conforming finite elements such as Pk-Lagrange finite elements, we are ensured that Th ⊂ Hs(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We denote by Wh := {vh ∈ Vh : T (vh) = 0}, the finite element space with homogeneous boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We define the discrete harmonic extension operator Eh associated to Vh as follows : for gh ∈ Th, Ehgh := argmin{∥∇vh∥L2(Ω) : vh ∈ Vh, T (vh) = gh}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Note that Ehgh is not harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Similar to E, the function Ehgh is characterized by T (Ehgh) = gh and � Ω ∇Ehgh · ∇vh = 0, vh ∈ Wh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Then, we define the approximation φh ∈ Vh to φ as φh = Ehψh, where ψh ∈ Th is the solution to the following variational problem: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='5) ⟨ψh, gh⟩Hs(Γ) = µh(gh) := ν(Ehgh), gh ∈ Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Here we are assuming that, in addition to be defined on Hs(Ω), the functional ν is also well defined on the space Vh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We shall further consider separately two instances where this is the case : (i) ν is a continuous functional on H1(Ω) and (ii) ν is a point evaluation functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Note that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='5) is not the straightforward Galerkin approximation of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2), since µh differs from µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This complicates somewhat the further conducted convergence analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The numerical method we employ for computing φh is to numerically solve an equivalent saddle point problem described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We apply the strategy (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='5) to ν := λj for each j and thereby obtain the corresponding approximations ˆφj := φh ∈ Vh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Since Step 2 requires that we guarantee the error ∥φj − ˆφj∥H1 ≤ ε2, our main goal in this section is to establish a quantitative convergence bound for ∥φ − φh∥H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We also need to establish a pointwise convergence bound for |φ(x) − φh(x)| when considering the modified version of Step 2 in the case that the measurements are point values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Similar to E, it will be important in our analysis to control the stability of Eh in the sense of a bound (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='6) ∥Ehgh∥H1(Ω) ≤ DE∥gh∥H1/2(Γ), gh ∈ Th, with a constant DE that is independent of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' However, such a uniform bound is not readily inherited from the stability of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' As observed in [6], its validity is known to depend on the existence of uniformly H1-stable linear projections onto Vh preserving the homogeneous boundary condition, that is, projectors Ph onto Vh that satisfy (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='7) Ph(H1 0(Ω)) = Wh and ∥Phv∥H1(Ω) ≤ B∥v∥H1(Ω), v ∈ H1(Ω), for some B independent of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' One straightforward consequence of this is that if v ∈ H1(Ω) with v|Γ ∈ Th then Ph(v)|Γ = v|Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We next show that the existence of such projectors is sufficient to guarantee the stability of Eh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For this, suppose (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='7) holds and gh ∈ Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Then PhEgh ∈ Vh and the trace of PhEgh is equal to gh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' It follows that ∥Ehgh − PhEgh∥H1(Ω) ≤ CP ∥∇Ehgh − ∇PhEgh∥L2(Ω) ≤ CP ∥∇Ehgh∥L2(Ω) + CP ∥∇PhEgh∥L2(Ω), ≤ 2CP ∥PhEgh∥H1(Ω), 12 where CP is the Poincar´e constant for Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Here, the last inequality follows from the minimizing property of Ehgh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Thus, by triangle inequality, one has ∥Ehgh∥H1(Ω) ≤ (1 + 2CP )∥PhEgh∥H1(Ω) ≤ (1 + 2CP )B∥Egh∥H1(Ω) ≤ (1 + 2CP )BCE∥gh∥H1/2(Γ), which is (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='6) with DE = (1 + 2CP )BCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The requirement of uniformly stable projectors Ph with the property (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='7) is satisfied by projectors of Scott-Zhang type [26] when the family of meshes {Th}h>0 is shape regular, that is, when all elements T have a uniformly bounded ratio between their diameters h(T ) and the diameter ρ(T ) of their inner circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In other words, the shape parameter (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='8) σ = σ({Th}h>0) := sup h>0 max T ∈Th h(T ) ρ(T ), is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In all that follows in the present paper, we work under such an assumption on the meshes Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Therefore, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='6) holds when Vh is subordinate to such partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' A saddle point formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Before attacking the convergence analysis, we need to stress an impor- tant computational variant of the above described Galerkin method, that leads to the same solution φh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' It is based on imposing harmonicity via a Lagrange multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For this purpose, we introduce the Hilbert space Xs(Ω) that consists of all v ∈ H1(Ω) such that vΓ ∈ Hs(Γ), and equip it with the norm ∥v∥Xs(Ω) := � ∥vΓ∥2 Hs(Γ) + ∥∇v∥2 L2(Ω) �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Then, the Riesz representer φ is equivalently determined as the solution of the saddle point problem: find (φ, π) ∈ Xs(Ω) × H1 0(Ω) such that a(φ, v) + b(v, π) = ν(v), v ∈ Xs(Ω) b(φ, z) = 0, z ∈ H1 0(Ω), where the bilinear forms are given by a(φ, v) := ⟨φΓ, vΓ⟩Hs(Γ) and b(v, π) := ⟨∇v, ∇π⟩L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Clearly the second equation in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='9) means that φ is harmonic and testing the first equation with a v ∈ Hs(Ω) shows that φ is the Riesz representer of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This saddle point formulation is well-posed: the bilinear forms a and b obviously satisfies the continuity properties a(φ, v) ≤ ∥φΓ∥Hs(Γ)∥vΓ∥Hs(Γ) ≤ ∥φ∥Xs(Ω)∥v∥Xs(Ω), φ, v ∈ Xs(Ω), and for the standard norm ∥v∥H1 0 (Ω) = ∥∇v∥L2(Ω), b(v, π) ≤ ∥∇v∥L2(Ω)∥∇π∥L2(Ω) ≤ ∥v∥Xs(Ω)∥π∥H1 0(Ω), v ∈ Xs(Ω), π ∈ H1 0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In addition, for all v ∈ Hs(Ω), one has ∥v∥2 Xs(Ω) ≤ ∥vΓ∥2 Hs(Γ) + ∥v∥2 H1(Ω) ≤ ∥vΓ∥2 Hs(Γ) + C2 E∥v∥2 H1/2(Γ) ≤ (1 + C2 E)a(v, v), which shows that a is coercive on the null space of b in Xs(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Finally, the bilinear form b satisfies the inf-sup condition inf π∈H1 0 (Ω) sup v∈Xs(Ω) b(v, π) ∥v∥Xs(Ω)∥π∥H1 0(Ω) ≥ inf π∈H1 0 (Ω) b(π, π) ∥π∥Xs(Ω)∥π∥H1 0(Ω) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Therefore the standard LBB theory ensures existence and uniqueness of the solution pair (φ, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We now discretize the saddle point problem by searching for (φh, πh) ∈ Vh × Wh such that a(φh, vh) + b(vh, πh) = ν(vh), vh ∈ Vh b(φh, zh) = 0, zh ∈ Wh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The equivalence with the previous derivation of φh by the Galerkin approach is easily checked: the second equation tells us that the solution φh is discretely harmonic, and therefore equal to Ehψh for some ψh ∈ Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Then taking vh of the form Ehgh for gh ∈ Th gives us exactly the Galerkin formulation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 13 This discrete saddle point problem is uniformly well-posed when we equip the space Wh with the H1 0 norm, and the space Vh with the Xs norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The continuity of a and b, and the inf-sup condition for b follow by the exact same arguments applied to the finite element spaces, with the same constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' On the other hand, we need to check the uniform ellipticity of a in the space VH h ⊂ Vh of discretely harmonic functions, which can be defined as VH h := {vh ∈ Vh : b(vh, zh) = 0, zh ∈ Wh}, or equivalently as the image of Th by the operator Eh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For all vh ∈ Vh,H and gh = T (uh), we write ∥vh∥2 Xs(Ω) ≤ ∥gh∥2 Hs(Γ) + ∥vh∥2 H1(Ω) ≤ ∥gh∥2 Hs(Γ) + D2 E∥gh∥2 H1/2(Γ) ≤ (1 + D2 E)a(vh, vh), where we have used the discrete stability of Eh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In practice, we use this discrete saddle point formulation for the computation of φh rather than the equivalent Galerkin formulation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='5) for the following reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Let Nh := dim Vh, Mh := dim Wh, and Ph := dim Th = Nh − Mh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Computing the right hand side load vector in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2) requires computing discretely harmonic extensions of Ph basis functions, which means solving Ph linear systems of dimension Mh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In addition one has to solve the sparse linear system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='5) of size Ph followed by another system of size Mh to compute φh = Ehψh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Using optimal iterative solvers of linear complexity the minimum amount of work needed to compute one representer scales then like PhMh ∼ N 1+ d−1 d h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' while solving the saddle point problem requires the order of Nh + Mh ∼ Nh operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' On the other hand the characterization of φh through (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='5) appears to be more convenient when deriving error bounds for ∥φ − φh∥H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This is the objective of the next sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Preparatory results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In the derivation of error bounds for ∥φ − φh∥H1(Ω), we will need several ingre- dients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The first is the following lemma that quantifies the perturbation induced by using Eh in place of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For any gh ∈ Th, one has (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='9) ∥(E − Eh)gh∥H1(Ω) ≤ C2hr−1∥gh∥Hs(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' where C2 depends on r and s, the shape-parameter σ, and on the geometry of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' From the properties of E and Eh, one has ⟨∇(E − Eh)gh, ∇vh⟩ = 0, vh ∈ Wh This orthogonality property shows that ∥∇(Egh − Ehgh)∥L2(Ω) ≤ ∥∇(Egh − Ehgh − vh)∥L2(Ω), vh ∈ Wh, and therefore ∥∇(Egh − Ehgh)∥L2(Ω) ≤ min vh∈Vh,T (vh)=gh ∥∇(Egh − vh)∥L2(Ω) ≤ ∥∇(Egh − PhEgh)∥L2(Ω), where Ph is the stable projector that preserves homogeneous boundary condition, see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' It follows that ∥∇(Egh − Ehgh)∥L2(Ω) ≤ (1 + B) min vh∈Vh ∥Egh − vh∥H1(Ω), where B is the uniform H1-stability bound on Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' By standard finite element approximation estimates and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='4), we have min vh∈Vh ∥Egh − vh∥H1(Ω) ≤ Chr−1∥Egh∥Hr(Ω) ≤ CC1hr−1∥gh∥Hs(Γ), where the constant C depends on r and on the shape parameter σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='9) follows by Poincar´e inequality since Egh − Ehgh ∈ H1 0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The second ingredient concerns the regularity of the solution to the variational problem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='10) ⟨κ, v⟩Hs(Γ) = γ(v), v ∈ Hs(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 14 For a general linear functional γ ∈ H−s(Γ), that is, continuous on Hs(Γ), we are only ensured that the solution κ is bounded in Hs(Γ), with ∥κ∥Hs(Γ) = ∥γ∥H−s(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' However, if γ has some extra regularity, this then translates into additional regularity of κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' As a simple example, consider the case where γ is in addition continuous on L2(Γ), that is (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='11) γ(v) = ⟨g, v⟩L2(Γ), for some g ∈ L2(Γ), and assume that we work with s = 1 and a polygonal domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Then the variational problem has a solution κ ∈ H1(Γ) and in addition κ ∈ H2(E) for each edge E with weak second derivative given by −κ′′ = g − κ ∈ L2(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In turn, standard finite element approximation estimates yield min κh∈Th ∥κ − κh∥H1(Γ) ≤ Ch∥g∥L2(Γ), with a constant C that depends on the shape parameter σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Of course, gain of regularity theorems for elliptic problems are known in various contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' However, we have not found a general treatment of gain of regularity that addresses the setting of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In going forward, we do not wish to systematically explore this gain in regularity and approximability for more general values of s and smoothness of γ since this would significantly enlarge the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Instead, we state it as the following general assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Assumption R: for s > 1 2 and δ > 0, there exists r(s, δ) > 0 such that if γ ∈ H−s+δ(Γ) for some δ > 0, then the solution κ to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='10) satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='12) min κh∈Th ∥κ − κh∥Hs(Γ) ≤ Chr(s,δ)∥γ∥H−s+δ(Γ), with a constant C that depends on s, δ, and on the shape parameter σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The above example shows that r(1, 1) = 1 for a polygonal domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We expect that this assumption always holds for the range 1 2 < s < 3 2 that is considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' An a priori error estimate for ∥φ − φh∥H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In this section, we work under the assumption that the linear functional ν is continuous on H1(Ω) with norm Cν := max{ν(v) : ∥v∥H1(Ω) = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Let us first check that this assumption implies a uniform a priori bound on ∥ψh∥Hs(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Indeed, we may write ∥ψh∥2 Hs(Γ) = ⟨ψh, ψh⟩Hs(Γ) = ν(Ehψh) ≤ CνDE∥ψh∥H1/2(Γ) ≤ CνDE∥ψh∥Hs(Γ), where the first inequality used (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Therefore, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='13) ∥ψh∥Hs(Γ) ≤ CνDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We have seen in §2 that the function φ belongs to the standard Sobolev space Hr(Ω) for r defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We use this r throughout this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='4), there exists a constant C1 such that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='14) ∥Ew∥Hr(Ω) ≤ C1∥w∥Hs(Γ), w ∈ Hs(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' As noted in §2, the amount of smoothness r depends both on s and on the geometry of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' What is important for us is that since s > 1/2, we have shown in (2) that r > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For example, for smooth domains it is r = s+ 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The fact that φ ∈ Hr(Ω) hints that the finite element approximation φh to φ should converge with a certain rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This is indeed the case as given in the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Under Assumption R, we have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='15) ∥φ − φh∥H1(Ω) ≤ CCνht, where t = min{r − 1, r(s, s + 1 2) + r(s, s − 1 2)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The constant C depends on s and on the geometry of Ω, and on the family of meshes through the shape parameter σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 15 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We use the decomposition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='16) φ − φh = Eψ − Ehψh = E(ψ − ψh) + (E − Eh)ψh, The second term can be estimated with the help of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3 applied to gh = ψh which gives ∥(E − Eh)ψh∥H1(Ω) ≤ C2hr−1∥ψh∥Hs(Γ) ≤ C2DECνhr−1, from the a priori estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='13) for ψh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We thus have obtained a bound in O(hr−1) for the H1 norm of the second term in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For the first term, we know that ∥E(ψ − ψh)∥H1(Ω) ≤ CE∥ψ − ψh∥H1/2(Γ), and so we are led to estimate ψ − ψh in the H1/2(Γ) norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For this purpose, we introduce the intermediate solution ψh ∈ Th to the problem ⟨ψh, gh⟩Hs(Γ) = µ(gh) = ν(Egh), gh ∈ Th, and we use the decomposition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='17) ψ − ψh = (ψ − ψh) + (ψh − ψh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We estimate the second term in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='17) by noting that for any gh ∈ Th, ⟨ψh − ψh, gh⟩Hs(Γ) = ν((E − Eh)gh) ≤ Cν∥(E − Eh)gh∥H1(Ω) ≤ CνC2hr−1∥gh∥Hs(Γ), where we have again used Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Taking gh = ψh − ψh we obtain a bound O(hr−1) for its Hs(Γ) norm, and in turn for its H1/2(Γ) norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' It remains to estimate ∥ψ − ψh∥H1/2(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Note that ψh is exactly the Galerkin approximation of ψ since we use the same linear form µ in both problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In fact, we have ⟨ψ − ψh, gh⟩Hs(Γ) = 0, gh ∈ Th, that is ψh is the Hs-orthogonal projection of ψ onto Th and therefore ∥ψ − ψh∥Hs(Γ) = min κh∈Th ∥ψ − κh∥Hs(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Since the linear form µ satisfies |µ(g)| = |ν(Eg)| ≤ Cν∥Eg∥H1(Ω) ≤ CνCE∥g∥H1/2(Γ), and thus belongs to H−1/2(Γ), we may apply the estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='12) to γ = ν, κ = ψ, δ = s − 1 2 > 0, to reach (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='18) ∥ψ − ψh∥H1/2(Γ) ≤ ∥ψ − ψh∥Hs(Γ) ≤ CCνCEhr(s,s− 1 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This proves the theorem for the value t = min{r − 1, r(s, s − 1 2)} > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We finally improve the value of t by using a standard Aubin-Nitsche duality argument as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We now take κ to be the solution of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='10) with γ(v) = ⟨ψ − ψh, v⟩H1/2(Γ), v ∈ H1/2(Γ), where ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='⟩H1/2(Γ) stands for the H1/2 scalar product associated with the norm ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='∥H1/2(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We then write ∥ψ − ψh∥2 H1/2(Γ) = ⟨ψ − ψh, ψ − ψh⟩H1/2(Γ) = ⟨κ, ψ − ψh⟩Hs(Γ) = ⟨κ − κh, ψ − ψh⟩Hs(Γ), where the last equality comes from Galerkin orthogonality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' It follows that ∥ψ − ψh∥2 H1/2(Γ) ≤ ∥κ − κh∥Hs(Γ)∥ψ − ψh∥Hs(Γ) ≤ Chr(s,s+ 1 2 )∥ψ − ψh∥H1/2(Γ)∥ψ − ψh∥Hs(Γ), where we have again used (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='12) now with δ = s+ 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Using the already established estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='18), it follows that ∥ψ − ψh∥H1/2(Γ) ≤ CCECνh˜t, with ˜t := r(s, s + 1 2) + r(s, s − 1 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' With all such estimates, the desired convergence bound follows with t := min{r − 1, ˜t}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 16 Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In the case of a polygonal domain and s = 1 which is further considered in our numerical experiments, we know that r = 3 2 and r(1, 1) = 1 so that ˜t ≥ r(1, 3 2) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In turn the convergence bound is established with t = r − 1 = 1 2, a rate that we observe in practice, see §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The case of point value evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We discuss now the case where ν(v) = δz(v) = v(z), for some z ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In order to guarantee that point evaluation is a continuous functional on Hs, we assume that s > d − 1 2 , that is s > 1 2 for d = 2, and s > 1 for d = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We want to find the Riesz representer of such a point evaluation functional on Hs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Note that our assumption on s ensures the continuous embeddings Hs(Γ) ⊂ C(Γ), as well as Hs(Ω) ⊂ Hr(Ω) ⊂ C(Ω), since in view of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3) r = min � s + 1 2, r∗� > d 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' where in the inequality we recall that r∗ > 3 2 for polygonal domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The point evaluation functional ν is thus continuous on Hs(Ω) with norm Cs bounded independently of the position of z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Of course, the Galerkin scheme analyzed above for ν ∈ H1(Ω)∗ continues to make sense since ν is well defined on the space Vh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' As explained in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='4, the prescriptions in Step 2 of the recovery algorithm need to be strengthened in the point evaluation setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Thus, we are interested in bounding the pointwise error |φ(x) − φh(x)| at the measurement points, in addition to the H1-error ∥φ − φh∥H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In what follows, we establish a modified version of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='4 in the point value setting that gives a convergence rate for ∥φ − φh∥H1(Ω), and in addition for ∥φ − φh∥L∞(Ω) ensuring the pointwise error control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We stress that the numerical method remains unchanged, that is, φh is defined in the exact same way as previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The new ingredients that are needed in our investigation are two classical results on the behavior of the finite element method with respect to the L∞ norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The first one is the so-called weak discrete maximum principle which states that there exists a constant Cmax such that, for all h > 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='19) ∥Ehgh∥L∞(Ω) ≤ Cmax∥gh∥L∞(Γ), gh ∈ Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This result was first established in [4] with constant Cmax = 1 for piecewise linear Lagrange finite elements under acuteness assumptions on the angles of the simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The above version with Cmax ≥ 1 is established in [25] for Lagrange finite elements of any degree on 2d polygonal domains, under the more general assumption that the meshes {Th}h>0 are quasi-uniform (in addition to shape regularity, all elements of Th have diameters of order h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' A similar result is established in [13] on 3d convex polyhedrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The second ingredient we need is a stability property in the L∞ norm of the Galerkin projection Rh : H1 0(Ω) → Wh where Rhv, v ∈ H1 0(Ω), is defined by � Ω ∇Rhv · ∇vh = � Ω ∇v · ∇vh, vh ∈ Wh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Specifically, this result states that there exists a constant Cgal and exponent a ≥ 0 such that, for all h > 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='20) ∥Rhv∥L∞(Ω) ≤ Cgal(1 + | ln(h)|)a∥v∥L∞(Ω), v ∈ L∞(Ω) ∩ H1 0(Ω), that is, the Ritz projection is stable and quasi-optimal, uniformly in h, up to a logarithmic factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This result is established in [25] for Lagrange finite elements on 2d polygonal domains and quasi-uniform partitions, with a = 1 in the case of piecewise linear elements and a = 0 for higher order elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' A similar result is established in [13] with a = 0 for convex polygons and polyhedrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In going further, we assume that the choice of finite element meshes ensures the validity of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='19) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 17 We begin our analysis with the observation that under the additional mesh assumptions, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3 can be adapted to obtain an estimate on ∥(E − Eh)gh∥L∞(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For any gh ∈ Th, one has (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='21) ∥(E − Eh)gh∥L∞(Ω) ≤ C3(1 + | ln(h)|)a)hr− d 2 ∥gh∥Hs(Γ), where C3 depends on (r, s), the geometry of Ω, and the family of meshes through Cgal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For any vh ∈ Vh such that T (vh) = gh, we write ∥(E − Eh)gh∥L∞(Ω) ≤ ∥Egh − vh∥L∞(Ω) + ∥Ehgh − vh∥L∞(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' It is readily seen that Ehgh − vh = Rh(Ehgh − vh) = Rh(Egh − vh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Indeed RhEhgh − RhEgh ∈ Wh and � Ω ∇(Rh(Ehgh − Egh)) · ∇vh = � Ω ∇(Ehgh − Egh) · ∇vh = 0 for all vh ∈ Wh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Therefore, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='20), we obtain ∥(E − Eh)gh∥L∞(Ω) ≤ (1 + Cgal(1 + | ln(h)|)a) min vh∈Vh,T (vh)=gh ∥Egh − vh∥L∞(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' On the other hand, we are ensured that Egh belongs to Hr(Ω) where r > d 2, and therefore has H¨older smoothness of order r − d 2 > 0 with ∥Egh∥Cr− d 2 (Ω) ≤ Ce∥Egh∥Hr(Ω) ≤ CeC1∥gh∥Hs(Γ), where Ce is the relevant continuous embedding constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' By standard finite element approximation theory, min vh∈Vh,T (vh)=gh ∥Egh − vh∥L∞(Ω) ≤ Chr− d 2 ∥Egh∥Cr− d 2 (Ω), where C depends on r and the shape-parameter σ and therefore we obtain (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We are now in position to give an adaptation of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='4 to the point value setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Under Assumption R, for any t1 < min{r − d 2, r(s, s + 1 2) + r(s, s − 1 2)}, one has (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='22) ∥φ − φh∥H1(Ω) ≤ Cht1, and for any t2 < min{r − d 2, 2r(s, s − d−1 2 )}, one has (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='23) ∥φ − φh∥L∞(Ω) ≤ Cht2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The constant C depends in both cases on s, t1 and t2, on the geometry of Ω, as well as on the family of meshes through the constants Cmax and Cgal, and the shape parameter σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We estimate ∥φ − φh∥H1(Ω) by adapting certain steps in the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The first change lies in the a priori estimate of the Hs(Γ) norm of ψh that was previously given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='13) which is not valid anymore since Cν = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Instead, we write ∥ψh∥2 Hs(Γ) = ⟨ψh, ψh⟩Hs(Γ) = ν(Ehψh) ≤ ∥Ehψh∥L∞(Ω) ≤ Cmax∥ψh∥L∞(Γ) ≤ CmaxBs∥ψh∥Hs(Γ), where we have used (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='19) and where Bs is the continuous embedding constant between Hs(Γ) and L∞(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In turn, we find that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='24) ∥ψh∥Hs(Γ) ≤ CmaxBs, which results in the slightly modified estimate ∥(E − Eh)ψh∥H1(Ω) ≤ C2CmaxBshr−1, for the second term of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For the first term E(ψ − ψh), we proceed in a similar manner to the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Namely, we estimate the H1/2(Γ) norms of two summands in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The estimate of ∥ψh − ψh∥H1/2(Γ) is modified as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We note that for any gh ∈ Th, ⟨ψh − ψh, gh⟩Hs(Γ) = ν((E − Eh)gh) ≤ ∥(E − Eh)gh∥L∞(Ω) ≤ C3(1 + | ln(h)|)a)hr− d 2 ∥gh∥Hs(Γ), 18 where we have now used Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Taking gh = ψh − ψh we obtain a bound of order O(hr− d 2 ) up to logarithmic factors for its Hs norm, and in turn for its H1/2 norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The estimate of ∥ψ − ψh∥H1/2(Γ) is left unchanged and of order O(h˜t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Combining these various estimates, we have established (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='22) for any t1 < min{r − d 2, ˜t}, with ˜t := r(s, s + 1 2) + r(s, s − 1 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We next estimate ∥φ − φh∥L∞(Ω) by the following adaptation of the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For the first term (E − Eh)ψh of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='16) we use Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='6 combined with the estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='24) of ψh which give us ∥(E − Eh)ψh∥L∞(Ω) ≤ CmaxBsC3(1 + | ln(h)|)a)hr− d 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For the second term E(ψ − ψh), we use the continuous maximum principle to obtain ∥E(ψ − ψh)∥L∞(Ω) ≤ ∥ψ − ψh∥L∞(Γ) ≤ ∥ψh − ψh∥L∞(Γ) + ∥ψ − ψh∥L∞(Γ) For the first summand, we write ∥ψh − ψh∥L∞(Γ) ≤ Ce∥ψh − ψh∥Hs(Γ), where Ce is the relevant continuous embedding constant, and we have already observed that ∥ψh − ψh∥Hs(Γ) satisfies a bound in O(hr− d 2 ) up to logarithmic factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For the second summand, we may write ∥ψ − ψh∥L∞(Γ) ≤ Ce∥ψ − ψh∥Hs(Γ), where Ce is the relevant continuous embedding constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Since ν belongs to H−s+δ(Γ) for all δ < s − d−1 2 , we can apply the estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='12) to reach a convergence bound ∥ψ − ψh∥Hs(Γ) ≤ Chr(s,δ), where C depends on the closeness of δ to s − d−1 2 , and on the family of meshes through the shape parameter σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Combining these estimates then gives (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='23) for any t2 < min{r − d 2, ˜t} where ˜t = r(s, s − d−1 2 ), since δ can be picked arbitrarily close to s − d−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We can improve the range of t2 as follows: pick any s such that d−1 2 < s < s and write ∥ψ − ψh∥L∞(Γ) ≤ Ce∥ψ − ψh∥Hs(Γ), where Ce is the relevant continuous embedding constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We then apply a similar Aubin-Nitsche argument to derive an estimate ∥ψ − ψh∥Hs(Γ) ≤ Chr(s,δ)+r(s,s−s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Combining these estimates gives (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='23) for any t2 < min{r − d 2, t}, where t := 2r(s, s − d−1 2 ) since s can be picked arbitrarily close to d−1 2 and δ arbitrarily close to s − d−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Numerical Illustrations In this section, we implement some examples of our numerical method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For this, we have to specify the domain Ω, the functionals λj, and a function u ∈ H1(Ω) which gives rise to the data vector w = λ(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' While our numerical method can be applied to general choices for these quantities, in our illustrations we make these choices so that the computations are not too involved but yet allow us the flexibility to illustrate certain features of our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The specific choices we make for our numerical example are the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The domain: In order to simplify the presentation, we restrict ourselves when Ω = (0, 1)2 but point out again that the algorithm can be extended to more general domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The function u: For the function u we choose the harmonic function u = uH where (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1) uH(x, y) = ex cos(y), (x, y) ∈ Ω := (0, 1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This choice means that u0 = 0 and therefore allows us not to deal with the computation of ˆu0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This choice corresponds to the right side f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Note that the trace of uH on the boundary Γ is piecewise smooth and continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Therefore, we have T (uH) ∈ H1(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We take s = 1 as our assumption on the value of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This means that we shall seek Riesz representor for the functionals given below when viewed as acting on H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 19 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The case of linear functionals defined on H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In this section, we consider numerical experiments for linear functionals defined on H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In our illustrative example, we relabel these functionals by double indices associated with a regular square grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' More precisely, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2) λi,j(v) := 1 √ 2πr2 � Ω v(z)e− 1 2 |z−zi,j|2 r2 dz, v ∈ H1(Ω), i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=', √m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Here, we assume that m is a square integer and r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1 in our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The centers zi,j ∈ Ω are uniformly distributed (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3) zi,j := 1 √m + 1(i, j), i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=', √m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Recall that our numerical algorithm as described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2 is based on finite element methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Specifically, we us the finite element spaces Vh := � vh ∈ C0(Ω) : vh|T ∈ Q1, T ∈ Th � , where Th are subdivisions of Ω made of squares of equal side length h and Q1 denotes the space of polynomials of degree at most 1 in each direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In order to study the effect of the mesh-size we specifically consider h = hn := 2−n, n = 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , 9, that is, bilinear elements on uniformly refined meshes with mesh-size 2−n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We display in Table 1 the results of our numerical recovery algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The entries in the table are the recovery errors e(m, n) := ∥uH − ˆuH∥H1(Ω), where ˆuH ∈ Vhn is the recovery for the particular values of m and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' n m 4 9 16 25 36 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2 141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='73 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='43 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='18 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='0 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='31 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='79 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='11 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='06 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='06 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Recovery error e(m, n) for different amounts of Gaussian measurements m and finite element refinements n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We have proven in this paper that our numerical recovery algorithm is near optimal with constant C that can be made arbitrarily close to one by choosing n sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This means that the error e(m, n) satisfies e(m, n) ≤ CR(KH w )H1(Ω) for n sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Increasing the number m of measurements is expected to decrease this Chebyshev radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' While one is tempted to think that the entries in each column of the table provides an upper bound for the Chebyshev radius of KH w for these measurements, this is not guaranteed since we are only measuring the error for one function from Kw, namely uH, and not all possible functions from Kw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' However, the entries in any given column provide a lower bound for the Chebyshev radius of KH w provided n is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Increasing the number m of measurements requires a finer resolution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=', increasing n, of the finite element discretization until the perturbation ε in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2 is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This is indeed confirmed by the results in Table 1 where stagnating error bounds (in each fixed column) indicate the corresponding tip-over point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We notice in particular that for small values of n, the error becomes very large as m grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This is explained by the fact that the Gramian matrix G becomes severely ill-conditioned, and in turn the prescriptions on ∥G − ˆG∥1 cannot be fulfilled when using finite element approximation of the Riesz representers on too coarse meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' An overall convergence of the recovery error to zero can, of course, only take place when both m and n increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 20 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The case of point value measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In this section, we describe our numerical experiments in the case where the linear functionals λi,j are point evaluations at points from Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Recall that while the λi,j are not defined for general functions in H1(Ω) they are defined for functions in the model class KH := U(Hs(Ω)) provided s is sufficiently large (s > 1/2 for d = 2 and s > 1 for d = 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This means that the optimal recovery problem is well posed in such a case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We have given in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='4 sufficient conditions on a numerical algorithm to give near optimal recovery and then we have gone on to show in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='5 that our proposed numerical algorithm based on discrete harmonics converges to a near optimal recovery with any constant C > 1 provided that the finite element spaces are discretized fine enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In the numerical experiments of this section, we again take Ω = (0, 1)2, s = 1, and the data to be the point values of the harmonic function uH defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We choose the evaluation points to be the zi,j of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We now provide in Table 2 the recovery error e(m, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The observed behavior is similar to the case of Gaussian averages;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' see Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' n m 4 9 16 25 36 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='19 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='43 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='49 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='18 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='56 8.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='14 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='11 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Recovery error e(m, n) for different amounts of point evaluation measurements m and refinements n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Additional comments on the approximation of Riesz representers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Finally, we provide a little more information on the computations that may be of interest to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We work in the same setting as in the previous sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Let us begin with the rate of convergence of our numerical approximations to the Riesz representers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We first consider the computation of the Riesz representer for the Gaussian measurement functional centered at z = zi,j := (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='75, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Let φn ∈ Vhn be the approximation to the Riesz representer φ produced by the finite element computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1 shows the error ∥φn − φ9∥H1(Ω), n = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This graph indicates an error decay Ch1/2 n which matches the rate guaranteed by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='4, see also Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Next consider the computation of the Riesz representer for point evaluation at the same z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1 reports the numerical computations of error in both the H1(Ω) and L∞(Ω) norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Again, the graph indicates an error decay Ch1/2 n for the H1(Ω) norm which matches the rate guaranteed by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='7 and a decay rate closer to Chn for the L∞(Ω) norm (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='7 only guarantees Ch1/2 n ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Optimal data sites: Gelfand widths and sampling numbers In this section, we make some comments on the number of measurements m that are needed to guarantee a prescribed error in the recovery of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Bounds on m are known to be governed by the Gelfand width for the case of general linear functionals and by sampling numbers when the functionals are required to be point evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We explain what is known about these quantities for our specific model classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' As we shall see these issues are not completely settled for the model classes studied in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The problem of finding the best choice of functionals, respectively point evaluations, is in need of further research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We have seen that the accuracy of the optimal recovery of u ∈ Kw is given by the Chebyshev radius R(Kw) := R(Kw)H1(Ω) or equivalently R(KH w ) := R(KH w )H1(Ω) for the harmonic component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The worst case recovery error R(K) over the class K is defined by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1) R(K)H1(Ω) := sup w∈Rm R(Kw)H1(Ω), Notice that this worst case error fixes the measurement functionals but allows the measurements w to come from any function in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Both the individual error R(Kw) and the worst case error R(K) are very dependent 21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='000010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='000100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='001000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='010000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='100000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='000000 100 1000 dim(Vhn) Gaussian: H1 error Point evaluation: L∞ error Point evaluation: H1 error order 1 2 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Approximation errors for the Riesz representers of the Gaussian and point evaluation functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' on the choice of the data functionals λj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For example, in the case that these functionals are point evaluations at points z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , zm ∈ ¯Ω, then R(Kw) and R(K) will depend very much on the positioning of these points in ¯Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In the case of general linear functionals, one may fix m and then search for the λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , λm that minimize the worst case recovery error over the class K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This minimal worst case error is called the Gelfand width of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' If we restrict the linear functionals to be given by point evaluation, we could correspondingly search for the sampling points x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , xm minimizing the worst case recovery error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This minimal error is called the deterministic sampling number of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The goal of this section is not to provide new results on Gelfand widths and sampling numbers, since we regard this as a separate issue in need of a systematic study, but to discuss what is known about them in our setting and refer the reader to the relevant papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Let us recall that R(Kw) is equivalent to R(KH w )H1 and so we restrict our discussion in what follows to sampling of harmonic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Optimal choice of functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Suppose we fix the number m of observation to be allowed and ask what is the optimal choice for the λj, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , m, and what is the optimal error of recovery for this choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The answer to the second question is given by the Gelfand width of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Given a compact set K of a Banach space X, we define the Gelfand width of K in X by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2) dm(K)X := inf λ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=',λm R(K)X where the infimum is taken over the linear functionals defined on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Let us mention that this definition differs from that employed in the classical literature [21] where dm(K)X is defined as the infimum over all spaces F of codimension n of max{∥v∥X : v ∈ K ∩F}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The two definitions are equivalent in the case where K is a centrally symmetric set such that K + K ⊂ CK for some constant C ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Any set of functionals which attains the infimum in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2) would be optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The Gelfand width is often used as a benchmark for performance since it says that no matter how the m functionals λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , λm are chosen, the error of recovery of u ∈ K cannot be better than dm(K)X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' When X is a Hilbert space and K is the ball of a Hilbert space Y with compact embedding in X, it is known that the Gelfand width coincides with the Kolmogorov width, that is dm(K)X = dm(K)X := inf dim(E)=m dist(K, E)X = inf dim(E)=m max{∥v − PEv∥X : v ∈ K}, where the infimum is taken over all linear spaces E of dimension m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This is precisely our setting as discussed in §3: taking X = H1 := H1(Ω) and K as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='4), we have (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3) dm(K)H1(Ω) = dm(KH)H1(Ω) = dm(KH)H1(Ω) ∼ dm(KB)H1/2(Γ) = dm(KB)H1/2(Γ), 22 where the equivalence follows from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Upper and lower bounds for the Gelfand width of KB in L2(Γ) are explicitely given in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We can estimate the rate of decay of the Kolmogorov and Gelfand width of KB in H1/2(Γ) by the following general argument: as explained in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1, for the admissible range of smoothness, the Sobolev spaces Hs(Γ) have an intrinsic description by locally mapping the boundary onto domains of Rd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' More precisely, in [17] and [10], the Hs(Γ) norm of g is defined as (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='4) ∥g∥Hs(Γ) := � J � j=1 ∥gj∥2 Hs(Rj) �1/2 , where the Rj are open bounded rectangles of Rd−1 that are mapped by transforms γj into portions Γj that constitute a covering of Γ, and gj = g ◦ γj are the local pullbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' From this it readily follows that the Gelfand and Kolmogorov m-width of the unit ball of Hs(Γ) in the norm Ht(Γ), with 0 ≤ t < s behaves similar to that of the unit ball of Hs(R) in the norm Ht(R) where R is a bounded rectangle of Rd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The latter is known to behave like m− s−t d−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Therefore, for KH = U(Hs) with s > 1 2 in the admissible range allowed by the boundary smoothness, one has (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='5) cm− s−1/2 d−1 ≤ dm(KH)H1(Ω) ≤ Cm− s−1/2 d−1 , m ≥ 1, where c and C are positive constants depending only on Ω and s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We have already observed in §2 that the space Hs(Ω) is continuously embedded in the Sobolev space Hr(Ω) with r := max{s+ 1 2, r∗} and in particular r = s+ 1 2 for smooth domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' However the Gelfand and Kolmogorov widths of the unit ball of Hr(Ω) in H1(Ω) have the slower decay rate m− r−1 d = m− s−1/2 d compared to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='5) for Hs(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' This improvement reflects the fact that the functions from Hs(Ω) have d variables but are in fact determined by functions of d − 1 variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' The reduction in dimension from d to d − 1 is related to the fact that in our formulation of our problem we have complete knowledge of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Optimal choice of sampling points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' We turn to the particular setting where the λj are point evaluations functionals, λj(v) = v(xj), at m points xj ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Similar to the Gelfand width, the deterministic sampling numbers are defined as (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='6) ρm(K)X := inf x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=',xm R(K)X, A variant of this is to measure the worst case expected recovery error when the m points are chosen at random according to a probabilty distribution and search for the distribution that minimizes this error, leading to the randomized sampling number of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Obviously, one has (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='7) ρm(K)X ≥ dm(K)X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In the majority of the literature, deterministic and randomized sampling numbers are studied with error measured in the L2(Ω) norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In this setting, concrete strategies for optimal deterministic and randomized point design have been given when K is the unit ball of a reproducing kernel Hilbert space H defined on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In particular, the recent results in [16, 12, 18, 5] show that under the assumption � m>0 |dm(K)L2(Ω)|2 < ∞, then, for all t > 1 2, sup m≥1 mtdm(K)L2(Ω) < ∞ =⇒ sup m≥1 mtρm(K)L2(Ω) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In words, under the above assumptions, optimal recovery in L2(Ω) has the same algebraic convergence rate when using optimally chosen point values compared to an optimal choice of general linear functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' While similar general results have not been established for Gelfand width and sampling numbers in the H1 norm, we argue that they hold in our particular setting where H = Hs(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' For simplicity, as in §4, we consider a domain that is either a polygon when d = 2 or polyhedron when d = 3, and thus consider the range d−1 2 < s < 3 2 where the restriction from below ensures that Hs(Ω) ⊂ C(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Recalling the 23 finite element spaces Vh and their traces Th on the boundary, based on quasi-uniform meshes {Th}h>0, we consider for a given h > 0 the measurement points x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , xm that are the mesh vertices located on Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' By the quasi-uniformity property the number m = m(h) of these points satisfies ch1−d ≤ m ≤ Ch1−d, for some c, C > 0 independent of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' If v ∈ Hs(Ω), its trace vΓ belongs to Hs(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Then, denoting by Ih the piecewise linear interpolant on the boundary, standard finite element approximation theory ensures the estimate ∥vΓ − IhvΓ∥H1/2(Γ) ≤ Chs− 1 2 ∥vΓ∥Hs(Γ) = Chs− 1 2 ∥v∥Hs(Ω), for some C that only depends on s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Therefore, introducing ˜v := EIhv, one has ∥v − ˜v∥H1(Ω) ≤ CE∥vΓ − IhvΓ∥H1/2(Γ) ≤ CDEm− s−1/2 d−1 ∥v∥Hs(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' Since ˜v only depends on the value of v at the points x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' , xm, we have thus proved an upper bound of order m− s−1/2 d−1 for ρm(KH)H1(Ω), and in turn for ρm(K)H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In view of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='7) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='5), a lower bound of the same order must hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' In summary, similar to the Gelfand widths, the sampling numbers satisfy (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='8) ˜cm− s−1/2 d−1 ≤ ρm(K)H1(Ω) ≤ ˜Cm− s−1/2 d−1 , m ≥ 1, where ˜c and ˜C are positive constants depending only on Ω and s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content=' References [1] R.' metadata={'source': 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of South Carolina, Columbia, SC 29208, dahmen@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='edu Ronald DeVore, Department of Mathematics, Texas A&M University, College Station, TX 77843, rdevore@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='tamu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='edu Guergana Petrova, Department of Mathematics, Texas A&M University, College Station, TX 77843, gpetrova@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E5T4oBgHgl3EQfUA9Q/content/2301.05540v1.pdf'} +page_content='tamu.' metadata={'source': 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[cond-mat.soft] 12 Jan 2023 +The Dynamics of Fluctuating Thin Sheets Under Random Forcing +Chanania Steinbock and Eytan Katzav +Racah Institute of Physics, The Hebrew University, Jerusalem 9190401, Israel +(Dated: January 13, 2023) +We study the dynamic structure factor of fluctuating elastic thin sheets subject to conservative +(athermal) random forcing. In Steinbock, Katzav & Boudaoud, Phys. Rev. Research 4, 033096 +(2022), the static structure factor of a such a sheet was studied. +In this paper, we recap the +model developed there and investigate its dynamic properties. Using the self-consistent expansion +(SCE), the time dependent two-point function of the height profile is determined and found to decay +exponentially in time. Despite strong nonlinear coupling, the decay rate of the dynamic structure +factor is found to coincide with the effective coupling constant for the static properties which suggests +that the model under investigation exhibits certain quasi-linear behaviour. Confirmation of these +results by numerical simulations is also presented. +I. +INTRODUCTION +Thin sheets and surfaces are ubiquitous in everyday +life yet the theory of their physical properties remains +incomplete. For instance, despite the fact that crumpled +paper balls take but a moment to make, the response of a +thin sheet to random forcing remains poorly understood. +Since randomly driven thin surfaces are relevant to a wide +diversity of fields, ranging from the physics of crumpled +paper to the properties of graphene to the behaviour of +cell membranes, a theory of randomly driven surfaces +derived from first principles would have far reaching con- +sequences. +Loosely speaking, we can distinguish between two +types of random forcing, completely uncorrelated white +noise typical of thermal fluctuations and deliberate cor- +related noise such as the type of forcing applied when +crumpling a sheet of paper. Here, we will focus on the +latter kind of noise. Perhaps the easiest way to probe the +structure of an athermally fluctuating sheet is to study +the properties of the resultant crumpled sheet and in- +deed this is an active field of research in its own right, +both experimentally [1–11] and through mathematical or +numerical modeling [1, 12–14]. The development of the +theory of singular structures supported by thin sheets +such as d-cones and ridges [15–21] has gone some way to +bridging research into crumpled sheets with that of fluc- +tuating sheets however its applicability has been limited +by the impracticality of characterising sheets with more +than a handful of ridges. Additionally, the structure of +crumpled sheets can at most inform us of the static un- +varying properties of fluctuating systems. To obtain in- +sight into the complete dynamic structure of a fluctuating +thin sheet, we must tackle such a system directly. +Previous research into the time-dependent dynamic +structure of fluctuating surfaces is limited but has been +studied in the context of tethered surfaces [22, 23] and +polymerised membranes [24, 25], though this research fo- +cused exclusively on thermally driving white noise. In +the context of tethered surfaces [22, 23], the dynamics +of phantom and self-avoiding flexible sheets was studied +though at the cost of neglecting the elastic properties +of real sheets. In contrast, [24] focuses on the dynamic +character of elastic polymerised membranes coupled to +a random perturbing flowing fluid. Finally, [25] uses a +super-symmetric ε-expansion of a D = 4 − ε dimensional +membrane to obtain the dynamic exponent of an elastic +thermally fluctuating polymerised membrane. +Recently, we showed how the static properties of a fluc- +tuating sheet can be derived directly by applying tech- +niques from out-of-equilibrium statistical mechanics to +the physics of elastic systems [26]. +In particular, we +developed a dynamic variant of the F¨oppl-von K´arm´an +equations which describes the deformations of thin sheets +and used this to obtain the static structure factor of +a fluctuating thin sheet driven by athermal noise. +In +the current paper, we extend this approach to derive +the time-dependent structure factor of the fluctuating +sheet. This dynamic structure should be of direct rel- +evance in understanding many features of the sheet, in- +cluding its acoustic emissions [27–30], optical signature +[31] and dissipative character [32, 33]. Further applica- +tions to diverse fields [34] such as the biophysics of cell +membranes [24, 35], the properties and stability of fluc- +tuating graphene sheets [36–41] and wave turbulence [42] +can also be envisioned. +The paper is organised as follows. In Section II, we +briefly recap the derivation of the overdamped F¨oppl- +von K´arm´an equations developed in [26] and in Section +III, we apply the self-consistent expansion (SCE) to these +equations to determine the dynamic structure factor of +the fluctuating sheet. In Section IV, the accuracy of our +solution is confirmed by comparison with numerical sim- +ulations. +Finally, the implications of these results are +discussed in Section V. +II. +THE OVERDAMPED DYNAMIC +F ¨OPPL-VON K´ARM ´AN EQUATIONS +In [26], we developed a model to describe fluctuating +elastic thin sheets. In this section, we recap the main +ideas and relate them to the dynamic structure factor of +such a system. +The equilibrium out-of-plane displacement ξ (x, y) of +a thin elastic sheet subject to an external pressure Pex + +2 +is given by the well known F¨oppl-von K´arm´an equa- +tions [43] +Pex = +Eh3 +12 (1 − ν2)∇4ξ +− h +� ∂2ξ +∂x2 +∂2χ +∂y2 + ∂2ξ +∂y2 +∂2χ +∂x2 − 2 ∂2ξ +∂x∂y +∂2χ +∂x∂y +� +(1) +0 = ∇4χ + E +� +∂2ξ +∂x2 +∂2ξ +∂y2 − +� ∂2ξ +∂x∂y +�2� +, +(2) +where h, E and ν denote the sheet thickness, Young’s +modulus and Poisson ratio respectively. The scalar field +χ (x, y) denotes the Airy stress potential of the deforma- +tion. By writing the out-of-plane displacement ξ (x, y) in +the Monge parameterisation, ie. as a function of x and +y, we assume that deformations of our sheet are mostly +flat and thus our focus here will be on weak fluctuations. +To explore the dynamics of a driven fluctuating sheet, +we apply Newton’s second law to each element of the +sheet with density ρ +hρ∂2ξ +∂t2 = −Pex + Pdamping + Pdriving . +(3) +where Pdriving and Pdamping describe driving and damp- +ing forces respectively. Though variations of this equa- +tion have been studied in the context of wave turbu- +lence [42, 44–53], here we continue the approach intro- +duced in [26] of a sheet subject to ordinary fluid friction +Pdamping = −α ∂ξ +∂t being driven by conserved Gaussian +noise Pdriving = η (⃗r, t) with noise amplitude D, that is, +⟨η (⃗r, t)⟩ = 0 +(4) +⟨η (⃗r, t) η (⃗r ′, t′)⟩ = −Dδ (t − t′) ∇2δ (⃗r − ⃗r ′) . +(5) +Other driving forces could be considered, but as argued +in [26], there is value in studying the setup where the +sheet’s center of mass does not wander in space and hence +we impose conserved noise on the sheet. More specific +forms of noise which are consistent with the conservation +of center of mass could also be considered but following +the principle of parsimony, we consider only the simplest +possibility here. +Taking the overdamped limit and thus neglecting the +inertia term hρ ∂2ξ +∂t2 , this approach provides a concrete +model for a driven fluctuating elastic sheet, namely, the +overdamped dynamic F¨oppl-von K´arm´an equation +α∂ξ +∂t + +Eh3 +12 (1 − ν2)∇4ξ +− h +� ∂2ξ +∂x2 +∂2χ +∂y2 + ∂2ξ +∂y2 +∂2χ +∂x2 − 2 ∂2ξ +∂x∂y +∂2χ +∂x∂y +� += η (⃗r, t) . +(6) +where the Airy stress potential χ (x, y) is still determined +by equation (2). +The fundamental difference between the problem un- +der study here and the one studied in the wave turbulence +community [42, 44–53] is that they focus on the regime +where inertia is very important, while friction is present +only at the smallest scales. Also, the forcing of the sheet, +which is often modeled as white noise, is applied only at +the largest scales. As a result, the main feature which is +studied is the energy cascade from the large scales (where +the forcing is applied) to the smallest scales (where it +is dissipated). In fact, there exist concrete predictions +regarding this energy cascade depending on the specific +scenario that drives this cascade. In contrast, we focus on +the dynamics of the structure of the sheet under forcing +and friction across all scales. +It is shown in [26] that for a sheet with dimensions +L×L, equations (2) and (6) can be combined into a single +equation for the Fourier components ˜ξ⃗n (t) of ξ (x, y, t) = +� +⃗n ˜ξ⃗n (t) ei 2π +L ⃗n·⃗r where the sum is taken over all lattice +points of Z2. After nondimensionalising, one obtains the +following Langevin equation +∂ ¯ξ⃗n +∂¯t + g |⃗n|4 ¯ξ⃗n ++ 1 +2 +� +⃗ℓ1̸=⃗n +� +⃗ℓ2 +� +⃗ℓ3 +V⃗n,⃗ℓ1,⃗ℓ2,⃗ℓ3 ¯ξ⃗ℓ1 ¯ξ⃗ℓ2 ¯ξ⃗ℓ3 = ¯η⃗n (¯t ) +(7) +containing a single dimensionless parameter +g = +2π +12 (1 − ν2) +� +αh5E +D +. +(8) +The scaled time and Fourier components are given by +¯t = +� +(2π)6 hDE/ +� +α3L8��1/2 +t +(9) +¯ξ⃗n = +� +(2π)2 αhE/D +�1/4 ˜ξ⃗n +(10) +and the dimensionless noise in Fourier space has mean 0 +and variance +⟨¯η⃗n (¯t ) ¯η⃗n′ (¯t ′)⟩ = |⃗n|2 δ⃗n,−⃗n′δ (¯t − ¯t ′) . +(11) +Finally, the kernel V⃗n,⃗ℓ1,⃗ℓ2,⃗ℓ3 is simply the Fourier trans- +form of the transverse projection operator of the sheet +deformation [34, 54] and is given by +V⃗n,⃗ℓ1,⃗ℓ2,⃗ℓ3 = δ⃗n,⃗ℓ1+⃗ℓ2+⃗ℓ3 +���⃗n × ⃗ℓ1 +��� +2 ���⃗ℓ2 × ⃗ℓ3 +��� +2 +���⃗n − ⃗ℓ1 +��� +4 +, +(12) +where we have denoted +���⃗n × ⃗ℓ +��� = nxℓy − nyℓx, thus +equation (7) can be thought of as a type of φ4-field +Langevin equation with a non-trivial spatially varying +kernel V⃗n,⃗ℓ1,⃗ℓ2,⃗ℓ3 [55]. Accordingly, in principle, equation +(7) can be used to find structure factors such as the time- +dependent two-point function +S⃗n (¯t, ¯t ′) = +�¯ξ⃗n (¯t ) ¯ξ−⃗n (¯t ′) +� +(13) + +3 +which in steady-state will only depend on the difference +∆¯t = |¯t − ¯t ′| and thus can be written as a function of a +single argument as +S⃗n (¯t ) = +�¯ξ⃗n (0) ¯ξ−⃗n (¯t ) +� +. +(14) +Unfortunately, the single dimensionless parameter g in +equation (7) is coupled to its linear part and in [26], it +is argued that g is typically small since g ∼ 0.1 for a +typical sheet of aluminum or steel. Indeed, the scaling +g ∼ h5/2 ensures that for any sufficiently thin sheet, g will +be small and thus any expansion around the linear part +of equation (7) which treats the nonlinear part as a mere +correction is guaranteed to fail. Instead, following the +success of [26], we will analyse equation (7) by application +of the self-consistent expansion (SCE). +III. +THE SELF-CONSISTENT EXPANSION +As described in [26], the self-consistent expansion +(SCE) can be thought of as a renormalised perturbation +theory [56] capable of providing series approximations +even in the presence of strong coupling. The method has +found previous application to the KPZ equation and its +variations [57–66], fracture and wetting fronts [67, 68] +and turbulence [69]. More relevant to our system, the +SCE provides an extremely successful solution to the +zero-dimensional φ4-theory giving good results at low +orders and exact convergence at high orders [70, 71]. +Accordingly, the success of the SCE in determining the +static structure factor of a fluctuating sheet in [26] was +not entirely unexpected and since the SCE has a natural +extension to dynamic quantities, we extend the approach +taken in [26] here. +A. +The Fokker-Planck Equation and the SCE +As in [26], we begin by writing the Fokker-Planck equa- +tion corresponding to equation (7) [72] +∂P +∂¯t = 1 +2 +� +⃗n +|⃗n|2 +∂2P +∂ ¯ξ⃗n∂ ¯ξ−⃗n ++ g +� +⃗n +|⃗n|4 +∂ +∂ ¯ξ⃗n +�¯ξ⃗nP +� ++ 1 +2 +� +⃗n +∂ +∂ ¯ξ⃗n + +P +� +⃗ℓ1̸=⃗n +� +⃗ℓ2 +� +⃗ℓ3 +V⃗n,⃗ℓ1,⃗ℓ2,⃗ℓ3 ¯ξ⃗ℓ1 ¯ξ⃗ℓ2 ¯ξ⃗ℓ3 + + , +(15) +where P = P +��¯ξ⃗n (¯t ) +� +, ¯t +� +denotes the probability func- +tional that the system will have a specific configuration, +as prescribed by the Fourier components +�¯ξ⃗n (¯t ) +� +at time +¯t. We can multiply this equation by a function of the +Fourier components F +��¯ξ⃗n (¯t ) +�� +and integrate over all +¯ξ⃗n (¯t ) to obtain the following equation for the expecta- +tions +∂ ⟨F⟩ +∂¯t += 1 +2 +� +⃗n +|⃗n|2 +� +∂2F +∂ ¯ξ⃗n∂ ¯ξ−⃗n +� +− g +� +⃗n +|⃗n|4 +� ∂F +∂ ¯ξ⃗n +¯ξ⃗n +� +− 1 +2 +� +⃗n +� +⃗ℓ1̸=⃗n +� +⃗ℓ2 +� +⃗ℓ3 +V⃗n,⃗ℓ1,⃗ℓ2,⃗ℓ3 +� ∂F +∂ ¯ξ⃗n +¯ξ⃗ℓ1 ¯ξ⃗ℓ2 ¯ξ⃗ℓ3 +� +(16) +where we have defined the expectation values +⟨F⟩ = +� � +⃗n +d¯ξ⃗n F +��¯ξ⃗n +�� +P +��¯ξ⃗n +� +, ¯t +� +. +(17) +Equation (16) can be used to obtain relationships be- +tween various moments. +For instance, in [26], it was +shown that subbing in F = ¯ξ⃗n ¯ξ⃗n′ results in an equation +relating the static two-point function +�¯ξ⃗n ¯ξ⃗n′� +to the static +four-point function +�¯ξ⃗n1 ¯ξ⃗n2 ¯ξ⃗n3 ¯ξ⃗n4 +� +. Similarly, to obtain +relations for dynamic quantities, we can sub in time de- +pendent functions such as F +��¯ξ⃗n (¯t ) +�� += ¯ξ⃗n (0) ¯ξ⃗n′ (¯t ) +which results in the ODE +∂ +�¯ξ⃗n (0) ¯ξ⃗n′ (¯t ) +� +∂¯t += −g |⃗n′|4 �¯ξ⃗n (0) ¯ξ⃗n′ (¯t ) +� +−1 +2 +� +⃗ℓ1̸=⃗n′ +� +⃗ℓ2 +� +⃗ℓ3 +V⃗n′,⃗ℓ1,⃗ℓ2,⃗ℓ3 +� +¯ξ⃗n (0) ¯ξ⃗ℓ1 (¯t ) ¯ξ⃗ℓ2 (¯t ) ¯ξ⃗ℓ3 (¯t ) +� +. +(18) +This first order non-homogeneous ODE provides the +time-dependent +two-point +function +�¯ξ⃗n (0) ¯ξ⃗n′ (¯t ) +� +if +given +the +time-dependent +four-point +function +� +¯ξ⃗n (0) ¯ξ⃗ℓ1 (¯t ) ¯ξ⃗ℓ2 (¯t ) ¯ξ⃗ℓ3 (¯t ) +� +. +An ODE for the four- +point function can of course be obtained by subbing +F +��¯ξ⃗n (¯t ) +�� += ¯ξ⃗n (0) ¯ξ⃗ℓ1 (¯t ) ¯ξ⃗ℓ2 (¯t ) ¯ξ⃗ℓ3 (¯t ) into equation +(16) though the resulting ODE would require knowledge +of the time-dependent six-point function. As observed +in [26], this situation of needing higher moments to +find lower ones is similar to the BBGKY hierarchy +[73–75] and finding closure is in general non-trivial. +The naive approach would be to simply neglect the +non-homogeneous part of equation (18) and then at- +tempt to perturbatively correct for it however since the +small parameter g is coupled to the homogeneous part +of equation (18), the non-homogeneous contribution is +large and non-negligible and thus such an approach is +guaranteed to fail. Since this occurs at every level of the +hierarchy, a more sophisticated approach is required. +Following the approach taken in [26], we apply the SCE +to equation (16) by introducing a free parameter Γ|⃗n| +∂ ⟨F⟩ +∂¯t += 1 +2 +� +⃗n +|⃗n|2 +� +∂2F +∂ ¯ξ⃗n∂ ¯ξ−⃗n +� +− +� +⃗n +Γ|⃗n| +� ∂F +∂ ¯ξ⃗n +¯ξ⃗n +� +− +� +⃗n +� +g |⃗n|4 − Γ|⃗n| +� � ∂F +∂ ¯ξ⃗n +¯ξ⃗n +� +− 1 +2 +� +⃗n +� +⃗ℓ1̸=⃗n +� +⃗ℓ2 +� +⃗ℓ3 +V⃗n,⃗ℓ1,⃗ℓ2,⃗ℓ3 +� ∂F +∂ ¯ξ⃗n +¯ξ⃗ℓ1 ¯ξ⃗ℓ2 ¯ξ⃗ℓ3 +� +. +(19) + +4 +One can think of Γ|⃗n| as an effective coupling constant +such that a perturbative expansion around the linear the- +ory with Γ|⃗n| is valid. The problem of determining its +value will be deferred to later though due to the isotropic +character of our system, we have already assumed that +Γ|⃗n| can only depend on the magnitude of ⃗n and not its +direction. Now if ⟨F⟩(m) denotes an mth order expan- +sion of ⟨F⟩, then by assumption, the latter terms will +contribute at a higher order and thus we can write the +iterative relation +∂ ⟨F⟩(m) +∂¯t += 1 +2 +� +⃗n +|⃗n|2 +� +∂2F +∂ ¯ξ⃗n∂ ¯ξ−⃗n +�(m) +− +� +⃗n +Γ|⃗n| +� ∂F +∂ ¯ξ⃗n +¯ξ⃗n +�(m) +− +� +⃗n +� +g |⃗n|4 − Γ|⃗n| +� � ∂F +∂ ¯ξ⃗n +¯ξ⃗n +�(m−1) +− 1 +2 +� +⃗n +� +⃗ℓ1̸=⃗n +� +⃗ℓ2 +� +⃗ℓ3 +V⃗n,⃗ℓ1,⃗ℓ2,⃗ℓ3 +� ∂F +∂ ¯ξ⃗n +¯ξ⃗ℓ1 ¯ξ⃗ℓ2 ¯ξ⃗ℓ3 +�(m−1) +, +(20) +supplemented with the convention that for m = 0 the +m − 1 terms drop out. +This equation can now be used to obtain any moment +up to any order. For instance, in [26], it was shown that +subbing in F = ¯ξ⃗n ¯ξ⃗n′ or F = ¯ξ⃗n ¯ξ⃗ℓ1 ¯ξ⃗ℓ2 ¯ξ⃗ℓ3 together with +m = 0 directly results in zeroth order expressions for the +static two-point and four-point functions +�¯ξ⃗n ¯ξ⃗n′�(0) = |⃗n|2 +2Γ|⃗n| +δ⃗n,−⃗n′ +(21) +and +� +¯ξ⃗n′ ¯ξ⃗ℓ1 ¯ξ⃗ℓ2 ¯ξ⃗ℓ3 +�(0) += +� +¯ξ⃗n′ ¯ξ⃗ℓ1 +�(0) � +¯ξ⃗ℓ2 ¯ξ⃗ℓ3 +�(0) ++ +� +¯ξ⃗n′ ¯ξ⃗ℓ2 +�(0) � +¯ξ⃗ℓ1 ¯ξ⃗ℓ3 +�(0) ++ +� +¯ξ⃗n′ ¯ξ⃗ℓ3 +�(0) � +¯ξ⃗ℓ1 ¯ξ⃗ℓ2 +�(0) +. +(22) +Upon further substitution of F = ¯ξ⃗n ¯ξ⃗n′ and m = 1, +an expression for the first order static two-point func- +tion +�¯ξ⃗n ¯ξ⃗n′�(1) was obtained in terms of the zeroth order +static two-point and four-point functions. Here, we will +determine what equation (20) has to say about dynamic +time-dependent quantities. +B. +Time-Dependent SCE +Unlike the static quantities described in [26], subbing +time-dependent quantities into equation (20) does not re- +sult in simple expressions for the moments under consid- +eration. Rather, the time derivative on the left-hand side +of equation (20) ensures that time-dependent quantities +are given by ODEs. For instance, subbing F +��¯ξ⃗n (¯t ) +�� += +¯ξ⃗n (0) ¯ξ⃗n′ (¯t ) and m = 0 into equation (20) gives the ho- +mogeneous ODE +∂ +�¯ξ⃗n (0) ¯ξ⃗n′ (¯t ) +�(0) +∂¯t += −Γ|⃗n′| +�¯ξ⃗n (0) ¯ξ⃗n′ (¯t ) +�(0) +(23) +whose solution is simply +�¯ξ⃗n (0) ¯ξ⃗n′ (¯t ) +�(0) = +�¯ξ⃗n ¯ξ⃗n′�(0) e +−Γ|⃗n′|¯t +(24) +where +�¯ξ⃗n ¯ξ⃗n′�(0) is given by equation (21). +Similarly, subbing F = ¯ξ⃗n′ (0) ¯ξ⃗ℓ1 (¯t ) ¯ξ⃗ℓ2 (¯t ) ¯ξ⃗ℓ3 (¯t ) and +m = 0 into equation (20) gives the ODE +∂ +� +¯ξ⃗n′ (0) ¯ξ⃗ℓ1 (¯t ) ¯ξ⃗ℓ2 (¯t ) ¯ξ⃗ℓ3 (¯t ) +�(0) +∂¯t += +− +� +⃗n +Γ|⃗n| +� +¯ξ⃗n′ (0) +∂ +� +¯ξ⃗ℓ1 (¯t ) ¯ξ⃗ℓ2 (¯t ) ¯ξ⃗ℓ3 (¯t ) +� +∂ ¯ξ⃗n (¯t ) +¯ξ⃗n (¯t ) +�(0) ++ 1 +2 +� +⃗n +|⃗n|2 +� +¯ξ⃗n′ (0) +∂2 � +¯ξ⃗ℓ1 (¯t ) ¯ξ⃗ℓ2 (¯t ) ¯ξ⃗ℓ3 (¯t ) +� +∂ ¯ξ⃗n (¯t ) ∂ ¯ξ−⃗n (¯t ) +�(0) +. +(25) +Carrying out the derivatives explicitly and computing the +sums, the first term on the right-hand side is simply pro- +portional to the dynamic four-point function +� +⃗n +Γ|⃗n| +� +¯ξ⃗n (0) +∂ +� +¯ξ⃗ℓ1 (¯t ) ¯ξ⃗ℓ2 (¯t ) ¯ξ⃗ℓ3 (¯t ) +� +∂ ¯ξ⃗n (¯t ) +¯ξ⃗n (¯t ) +�(0) += +� +Γ|⃗ℓ1| + Γ|⃗ℓ2| + Γ|⃗ℓ3| +� � +¯ξ⃗n (0) ¯ξ⃗ℓ1 (¯t ) ¯ξ⃗ℓ2 (¯t ) ¯ξ⃗ℓ3 (¯t ) +�(0) +, +(26) +while the second term is composed of a sum of zeroth +order dynamic two-point functions +� +⃗n +|⃗n|2 +� +¯ξ⃗n′ (0) +∂2 � +¯ξ⃗ℓ1 (¯t ) ¯ξ⃗ℓ2 (¯t ) ¯ξ⃗ℓ3 (¯t ) +� +∂ ¯ξ⃗n (¯t ) ∂ ¯ξ−⃗n (¯t ) +�(0) += +δ⃗ℓ1,−⃗ℓ2 +����⃗ℓ1 +��� +2 ++ +���⃗ℓ2 +��� +2� � +¯ξ⃗n′ (0) ¯ξ⃗ℓ3 (¯t ) +�(0) ++ δ⃗ℓ1,−⃗ℓ3 +����⃗ℓ1 +��� +2 ++ +���⃗ℓ3 +��� +2� � +¯ξ⃗n′ (0) ¯ξ⃗ℓ2 (¯t ) +�(0) ++ δ⃗ℓ2,−⃗ℓ3 +����⃗ℓ2 +��� +2 ++ +���⃗ℓ3 +��� +2� � +¯ξ⃗n′ (0) ¯ξ⃗ℓ1 (¯t ) +�(0) +. +(27) +Since we have already computed the zeroth order +dynamic +two-point +function +in +equation +(24) +and +can rearrange equation (21) to relate |⃗n|2 δ⃗n,−⃗n′ += + +5 +2Γ|⃗n| +�¯ξ⃗n ¯ξ⃗n′�(0), we can observe that +δ⃗ℓ1,−⃗ℓ2 +����⃗ℓ1 +��� +2 ++ +���⃗ℓ2 +��� +2� � +¯ξ⃗n′ (0) ¯ξ⃗ℓ3 (¯t ) +�(0) += +2 +� +Γ|⃗ℓ1| + Γ|⃗ℓ2| + Γ|⃗ℓ3| − Γ|⃗n′| +� +× +× +� +¯ξ⃗ℓ1 ¯ξ⃗ℓ2 +�(0) � +¯ξ⃗ℓ3 ¯ξ⃗n′ +�(0) +e +−Γ|⃗n′|¯t , +(28) +where we have been able to add and subtract Γ|⃗ℓ3| and +Γ|⃗n′| since the factor +� +¯ξ⃗ℓ3 ¯ξ⃗n′ +�(0) +∝ δ⃗ℓ3,−⃗n′ ensures that +Γ|⃗ℓ3| − Γ|⃗n′| = 0. Simplifying each term of equation (27) +in this manner, we obtain +� +⃗n +|⃗n|2 +� +¯ξ⃗n′ (0) +∂2 � +¯ξ⃗ℓ1 (¯t ) ¯ξ⃗ℓ2 (¯t ) ¯ξ⃗ℓ3 (¯t ) +� +∂ ¯ξ⃗n (¯t ) ∂ ¯ξ−⃗n (¯t ) +�(0) += +2 +� +Γ|⃗ℓ1| + Γ|⃗ℓ2| + Γ|⃗ℓ3| − Γ|⃗n′| +� +× +× +� +¯ξ⃗n′ ¯ξ⃗ℓ1 ¯ξ⃗ℓ2 ¯ξ⃗ℓ3 +�(0) +e +−Γ|⃗n′|¯t , +(29) +where +� +¯ξ⃗n′ ¯ξ⃗ℓ1 ¯ξ⃗ℓ2 ¯ξ⃗ℓ3 +�(0) +is the static four-point function +given by equation (22). +Finally, substituting equations (26) and (29) into equa- +tion (25), we obtain the following non-homogeneous ODE +for the dynamic four-point function at zeroth order +∂ +� +¯ξ⃗n′ (0) ¯ξ⃗ℓ1 (¯t ) ¯ξ⃗ℓ2 (¯t ) ¯ξ⃗ℓ3 (¯t ) +�(0) +∂¯t += +− +� +Γ|⃗ℓ1| + Γ|⃗ℓ2| + Γ|⃗ℓ3| +� � +¯ξ⃗n′ (0) ¯ξ⃗ℓ1 (¯t ) ¯ξ⃗ℓ2 (¯t ) ¯ξ⃗ℓ3 (¯t ) +�(0) ++ +� +Γ|⃗ℓ1| + Γ|⃗ℓ2| + Γ|⃗ℓ3| − Γ|⃗n′| +� � +¯ξ⃗n′ ¯ξ⃗ℓ1 ¯ξ⃗ℓ2 ¯ξ⃗ℓ3 +�(0) +e +−Γ|⃗n′|¯t +(30) +and it is straightforward to check, though perhaps un- +surprising, that this is simply solved by +� +¯ξ⃗n′ (0) ¯ξ⃗ℓ1 (¯t ) ¯ξ⃗ℓ2 (¯t ) ¯ξ⃗ℓ3 (¯t ) +�(0) += +� +¯ξ⃗n′ ¯ξ⃗ℓ1 ¯ξ⃗ℓ2 ¯ξ⃗ℓ3 +�(0) +e +−Γ|⃗n′|¯t . +(31) +Now to study the effect of the nonlinearity, we proceed +to higher orders. Subbing F +��¯ξ⃗n (¯t ) +�� += ¯ξ⃗n (0) ¯ξ⃗n′ (¯t ) +and m = 1 into equation (20) gives after some tedious +algebra +∂ +�¯ξ⃗n (0) ¯ξ⃗n′ (¯t ) +�(1) +∂¯t += −Γ|⃗n′| +�¯ξ⃗n (0) ¯ξ⃗n′ (¯t ) +�(1) +− +� +g |⃗n′|4 − Γ|⃗n′| +� �¯ξ⃗n (0) ¯ξ⃗n′ (¯t ) +�(0) +− 1 +2 +� +⃗ℓ1̸=⃗n′ +� +⃗ℓ2 +� +⃗ℓ3 +V⃗n′,⃗ℓ1,⃗ℓ2,⃗ℓ3× +× +� +¯ξ⃗n (0) ¯ξ⃗ℓ1 (¯t ) ¯ξ⃗ℓ2 (¯t ) ¯ξ⃗ℓ3 (¯t ) +�(0) +, +(32) +or after subbing in our expressions for the zeroth order +time-dependent two-point and four-point functions from +Eqs. (24) and (31) respectively +∂ +�¯ξ⃗n (0) ¯ξ⃗n′ (¯t ) +�(1) +∂¯t += −Γ|⃗n′| +�¯ξ⃗n (0) ¯ξ⃗n′ (¯t ) +�(1) +− +�� +g |⃗n′| +4 − Γ|⃗n′| +� �¯ξ⃗n ¯ξ⃗n′�(0) ++ 1 +2 +� +⃗ℓ1̸=⃗n′ +� +⃗ℓ2 +� +⃗ℓ3 +V⃗n′,⃗ℓ1,⃗ℓ2,⃗ℓ3 +� +¯ξ⃗n ¯ξ⃗ℓ1 ¯ξ⃗ℓ2 ¯ξ⃗ℓ3 +�(0) +� +e +−Γ|⃗n′|¯t . +(33) +It is worth noting that being precise, the exponential +decay associated with the four-point function in this ex- +pression should decay with rate Γ|⃗n| instead of Γ|⃗n′|. A +careful analysis of the sum over the kernel V⃗n′,⃗ℓ1,⃗ℓ2,⃗ℓ3 with +the static four-point function reveals however that this +term vanishes unless ⃗n = −⃗n′ and thus no harm is done +by replacing Γ|⃗n| with Γ|⃗n′|. It is now apparent that the +non-homogeneous term in this equation decays at the +natural decay rate Γ|⃗n′| of the ODE and thus its gen- +eral solution will contain a non-physical secular term of +the form ¯te +−Γ|⃗n′|¯t. As in the Poincar´e-Lindstedt method +for perturbatively solving non-linear ODEs [76–78], this +situation can be avoided by setting Γ|⃗n′| such that the +coefficient of the non-homogeneous term vanishes, ie. +0 = +� +g |⃗n′|4 − Γ|⃗n′| +� �¯ξ⃗n ¯ξ⃗n′�(0) ++ 1 +2 +� +⃗ℓ1̸=⃗n′ +� +⃗ℓ2 +� +⃗ℓ3 +V⃗n′,⃗ℓ1,⃗ℓ2,⃗ℓ3 +� +¯ξ⃗n ¯ξ⃗ℓ1 ¯ξ⃗ℓ2 ¯ξ⃗ℓ3 +�(0) +. +(34) +Using this idea to determine the characteristic decay rate +is novel in the context of the SCE method and might be +useful in other problems as well. After subbing in the +static two-point and four-point functions and carrying +out the sums in the last equation, this ultimately simpli- +fies to the following discrete integral equation for Γ|⃗n| +Γ|⃗n| = g |⃗n|4 + 1 +2 +� +⃗ℓ̸=⃗n +���⃗ℓ +��� +2 ���⃗n × ⃗ℓ +��� +4 +Γ|⃗ℓ| +���⃗n − ⃗ℓ +��� +4 . +(35) + +6 +Surprisingly, the above argument for preventing secu- +lar terms occurring in our expression for +�¯ξ⃗n (0) ¯ξ⃗n′ (¯t ) +�(1) +is equivalent to simply imposing that the first order ap- +proximation for the dynamic structure factor equals its +zeroth order approximation, ie. +�¯ξ⃗n (0) ¯ξ⃗n′ (¯t ) +�(1) = +�¯ξ⃗n (0) ¯ξ⃗n′ (¯t ) +�(0) , +(36) +or in other words, we select Γ|⃗n| such that the zeroth +order approximation is exact up to first order. In [26], a +static version of this self-consistent argument was made +by setting the first order approximation for the static +structure factor equal to its zeroth order approximation, +ie. +�¯ξ⃗n ¯ξ⃗n′�(1) = +�¯ξ⃗n ¯ξ⃗n′�(0) , +(37) +and indeed, the same discrete integral equation is ob- +tained from both approaches. It is important to appre- +ciate that these two self-consistent arguments giving the +same discrete integral equation for Γ|⃗n| was by no means +anticipated nor trivial and in fact, it is known that this +does not occur for the ordinary unadorned φ4-model. In +such instances, the fact that the two arguments conflict +suggests that the effective decay rate Γ|⃗n| also needs to +be appropriately expanded in a manner analogous to the +Poincar´e-Lindstedt method [76–78] if we wish our results +to have meaning. Conversely, the situation where the two +arguments result in the same discrete integral equation +implies some degree of quasi-linearity and is indicative +that our approach has self-consistently captured a true +feature of the system. +Equation (35) has been solved in the appendix of [26]. +Here we simply bring the final result +Γ|⃗n| ≃ n4 +� +3π +4 +� +A (g) − ln +� +n +nmax +�� +× +× +� +1 + B (g) +n +nmax +� +, +(38) +where nmax is an upper-frequency cutoff which must be +imposed on the system and A (g) and B (g) are constants +which only depend on g and have the following small g +expansions +A ≃ 0.137 + 0.336g + 0.243g2 + 0.112g3 + O +� +g4� +, +(39) +B ≃ −0.265 + 0.360g − 0.395g2 + 0.311g3 + O +� +g4� +. (40) +It is worth noting that B (g) primarily determines the +behaviour of Γ|⃗n| for large ⃗n, ie. when |⃗n| → nmax, and +can be neglected when |⃗n| is small. Accordingly, for small +⃗n, we have obtained that the decay rate Γ|⃗n| grows like a +logarithmically corrected power law ∼ n4. +To summarise, we have found that at first order, the +dynamic structure factor is given by +�¯ξ⃗n (0) ¯ξ⃗n′ (¯t ) +�(1) = +�¯ξ⃗n ¯ξ⃗n′�(0) e−Γ|⃗n|¯t +(41) += δ⃗n,−⃗n′ +n2 +2Γ|⃗n| e−Γ|⃗n|¯t +(42) +where Γ|⃗n| is given by equation (38) and can be seen to si- +multaneously play the roles of effective coupling constant +and effective decay rate. +IV. +COMPARISON WITH SIMULATIONS +As in [26], our predictions can be validated by nu- +merical integration of equation (7) over a square lattice. +Since we investigate here the dynamic properties of the +simulation rather than its static properties, the simula- +tions must be run for long enough to obtain precise av- +erages of the time-dependent two-point function S⃗n (¯t ) +and thus they must be run for substantially longer than in +[26]. The lattice size imposes a finite maximum frequency +nmax but since our theory necessitates the existence of an +upper-frequency cut-off, this in itself is fine. More press- +ingly, for increasing maximum frequency nmax, the max- +imum size of the time step δ¯t that can be used shrinks if +the simulation is to remain stable. This presents a trade- +off between the maximum resolution in time vs the maxi- +mum resolution in space and since the dynamic structure +factor can only be extracted from long simulation runs, +the size of nmax is sharply constrained by practical con- +siderations. In practice, our simulations were run with a +time-step δ¯t = 10−6 over a 41 × 41 lattice corresponding +to a maximum frequency nmax = 20 +√ +2 ≈ 28.3, though +unlike [26] which only used 106 time steps per simula- +tion, these simulations were run for 107 time steps each. +The consequence of these extended simulations is that +an enormous amount of data is generated though sim- +ulation states which are close to each other in time are +practically indistinguishable except at the very largest +modes (an expectation based on Eq. (38)). Since these +modes equilibrate far more rapidly than the small modes, +they are far less interesting in the current context and +thus in the interest of keeping memory resources man- +ageable, simulation data was only saved every 103 time +steps. Accordingly, the decay rate of modes which decay +faster than ∆¯t = 103 × δ¯t = 10−3 is not presented here. +Finally, it is worth noting that as in [26], efficient calcu- +lation of the quartic interaction term of equation (7) is +non-trivial and as there, was achieved by implementing +the pseudo-spectral method described in [52] in which the +quartic interaction is calculated as the Fourier transform +of its real-space counterpart though this imposes periodic +boundary conditions on the simulation. To achieve pre- +cise results, each simulation was run 10 times for various +values of 0 ≤ g ≤ 1 and the results averaged. +Fig. 1 shows how the dynamic structure factor S⃗n (¯t ) +decays for the first few modes as a function of the time +difference ¯t. +These results are for the simulation per- +formed with g = 0.1 though very similar results are ob- +tained for the other values of g. The solid black lines are +linear fits to the logarithm of the simulation data and +clearly show that the dynamic structure factor S⃗n (¯t ) in- +deed decays exponentially. By performing such fits for +many frequencies, we are able to numerically determine + +7 +0 +0.05 +0.1 +0.15 +0.2 +2-4 +2-3 +2-2 +2-1 +1 +FIG. 1. The dynamic structure factor S⃗n (¯t ) as a function of +the time difference ¯t for g = 0.1. Each data set corresponds to +a different frequency (see legend) and the solid lines are linear +fits to the logarithm of the data. The fact that these fits are +excellent confirms that S⃗n (¯t ) indeed decays exponentially. +how the decay rate, which we will denote Γdyn +|⃗n| , varies as +a function of n. Fig. 2 shows Γdyn +|⃗n| and Γdyn +|⃗n| / |⃗n|4 as a +function of n up to nmax/4 for the various values of g we +used. The solid and dotted lines are fits of the equation +Γdyn +|⃗n| +n4max +≃ C +� +n +nmax +�4 � +A (g) − ln +� +n +nmax +� +(43) +to the numerical results of g = 0.1 (green triangles) and +g = 1 (purple squares) respectively. Here, A (g) was sim- +ply taken from equation (39) while the scaling parameter +C was fitted and, as can be seen, these fits are excel- +lent across the entire range of frequencies. As described +towards the end of section III, the constant B (g) in equa- +tion (38) primarily modifies Γ|⃗n| for large values of n and +thus we have not attempted to determine it from our +data. +Fits for g = 0.01 (red circles) and g = 0 (blue +pluses) can also be carried out but the resulting fits are +so similar to the g = 0.1 case that little insight is gained +by showing them. The dashed line in the top plot of Fig. +2 is a guideline proportional to n4 and together with the +bottom plot of Fig. 2 clearly emphasises that the loga- +rithmic correction is non-negligible. +The fact that these fits of equation (43) so beautifully +capture the behaviour of Γdyn +|⃗n| confirms that our theory +has indeed accurately predicted the functional form of +the dynamic structure S⃗n (¯t ). +It is worth mentioning +that in [26], it was observed that the periodic square +lattice introduces an anisotropy into the simulation and +this required each direction to be treated separately, an +aspect which we have not observed in the temporal de- +cay data. We explain this distinction by observing that +in [26], the anisotropy was primarily accounted for by +scaling the parameter B (g) in equation (38) by some ap- +propriate function f (θ) and thus is only observable for +2-5 +2-4 +2-3 +2-2 +2-20 +2-18 +2-16 +2-14 +2-12 +2-10 + 2-8 + 2-6 +2-5 +2-4 +2-3 +2-2 +0 +1 +2 +3 +FIG. 2. Γdyn +|⃗n| (top) and Γdyn +|⃗n| / |⃗n|4 (bottom) as a function of n +for various values of g. The solid and dotted lines are fits of +equation (43) to the g = 0.1 data (green triangles) and g = 1 +data (purple squares) respectively. +The dashed line in the +top plot is a guideline proportional to n4. The fact that the +fits are excellent over the entire range of frequencies confirms +that our theory correctly predicts the functional form of the +dynamic structure factor S⃗n (¯t ) and the appreciable deviation +between the fits and the guideline shows that the logarithmic +correction of equation (43) is non-negligible. This is further +emphasised in the bottom plot by the fact that Γdyn +|⃗n| / |⃗n|4 is +indeed seen to be non-constant and is well fitted by the theory. +large frequencies. Since we do not present here the large +frequency decay rate, we have been unable to observe this +anisotropy in this data though we presume it too exists. +Finally, it is worth comparing Γdyn +|⃗n| , the decay rate of +the time-dependent structure factor S⃗n (¯t ), with the ef- +fective coupling constant which can be extracted from +the relationship +S⃗n (0) = +�¯ξ⃗n ¯ξ−⃗n +� += +n2 +2Γstat +|⃗n| +. +(44) +Here, we denote the coupling constant by Γstat +|⃗n| +as it is +obtained by only considering static quantities. Accord- + +8 +2-5 +2-4 +2-3 +2-2 +2-20 +2-18 +2-16 +2-14 +2-12 +2-10 + 2-8 + 2-6 +FIG. 3. Comparison between the numerical coupling constant +Γstat +|⃗n| = 1 +2n2/ +�¯ξ⃗n¯ξ−⃗n +� +(blue pluses) and the numerical decay +rate Γdyn +|⃗n| of the time-dependent two-point function S⃗n (¯t ) = +S⃗n (0) e +−Γdyn +|⃗n| ¯t (red circles), for g = 0. These two quantities +are seen to agree over all orders of magnitude. +2-5 +2-4 +2-3 +2-2 +0 +0.5 +1 +1.5 +2 +FIG. 4. Γstat +|⃗n| /Γdyn +|⃗n| as a function of n. For all values of g, this +ratio is seen to be constant and roughly equal to 1 over all +frequency scales. +ing to the theory developed above and in [26], Γstat +|⃗n| and +Γdyn +|⃗n| +should be identical and Fig. 3 and Fig. 4 shows +that these two methods of obtaining Γ|⃗n| are in fact ex- +tremely close. Fig. 3 compares Γstat +|⃗n| and Γdyn +|⃗n| for the case +of g = 0 and shows that even with maximal non-linear +coupling, Γstat +|⃗n| and Γdyn +|⃗n| are practically indistinguishable. +The same result can be obtained for the other values of +g and indeed Fig. 4 shows that the ratio Γstat +|⃗n| /Γdyn +|⃗n| ≈ 1 +for all values of g over all frequency scales. +V. +DISCUSSION +In this work we have extended a previous paper [26] +to study the dynamics of a vibrating thin sheet, gov- +erned by the overdamped F¨oppl-von K´arm´an equations. +Specifically, we applied the self-consistent expansion to +obtain predictions which agree with the results of numer- +ical simulations. Surprisingly, the effective coupling con- +stant Γ|⃗n|, which determines the static structure factor +of the sheet coincides with the decay rate of the dynamic +structure factor. This observation is reminiscent of linear +systems, where these two quantities are indeed the same, +yet it is not obvious that the F¨oppl-von K´arm´an equa- +tions, which are strongly nonlinear, would exhibit such a +phenomena. A way to make sense of this situation is by +using a recent correlation-response inequality in dynam- +ical systems [79, 80]. This inequality was formulated in +the context of Langevin dynamics and shows that when- +ever the equations of motion can be derived from a Hamil- +tonian and hence obey a certain Fluctuation-Dissipation +relation (thus belonging to class I using that classifica- +tion), and are Galilean invariant (that is, the zero-mode +does not affect the dynamics of the higher modes thus +belonging to class II in that classification), the exponent +inequalities become an equality implying quasi-linear dy- +namics. The quasi-linearity refers to the fact that the +scaling of the decay rate of the dynamic correlation func- +tion is related to the static exponents in a simple man- +ner. While the overdamped F¨oppl-von K´arm´an equations +used here are definitely Galilean invariant since the cen- +ter of mass does not drift as a result of the dynamics, the +way they are derived from a Hamiltonian deviates from +the models considered in [79, 80], as they belong to nei- +ther model A nor model B in the classical classification +of Hohenberg and Halperin [81]. Nevertheless, they obey +the quasi-linear property, which may be the result of a +modified Fluctuation-Dissipation relation. +This makes +the overdamped F¨oppl-von K´arm´an equations another +interesting example of a physically motivated model that +obeys quasi-linearity. +From a methodological point of view, imposing the +condition that secular terms should vanish, inspired by +the Poincar´e-Lindstedt method, is a major achievement +in the context of the SCE method. +In the past this +method was only able to yield approximate results for +the dynamical structure factor in certain cases, usually +based on solving another dedicated approximate integral +equation. In this paper, we gained the insight that the +resonant/secular terms that are being generated are actu- +ally non-physical, and just like in the Poincar´e-Linstedt +method, should be equated to zero. This insight might +become useful in other contexts where the SCE method +is applied. +The big surprise here is that, unlike the +Poincar´e-Linstedt method, we do not need to solve a new +equation for the dynamic decay rate, since the new equa- +tion is identical to the integral equation from the static +case. +The results reported here have experimental implica- + +9 +tions. Both the structure of thin sheets, as characterised +by the static structure factor and the characteristic de- +cay rate could be measured directly. There has already +been an attempt to measure the structure of crumpled +sheets using its optical signature [31], and it is reason- +able that with improved imaging techniques leading to +faster analysis of dynamic laser speckle patterns [82], the +full dynamic structure factor could be effectively sam- +pled. The coincidence of the effective coupling constant +and the decay rate due to the quasi-linearity should be +checked experimentally and if confirmed could be used to +enhance the resolution and accuracy of the measurements +by cross validating these two independent sources. +Thin sheets also exhibit a very distinctive acoustic +footprint, often known as crackling noise [27–30]. Clearly, +the dynamic structure factor of the vibrating sheet should +be correlated with its acoustic emissions however the cou- +pling between the two is known to be nontrivial. Accord- +ingly, a verified analytical form of the dynamic structure +factor could provide a solid staring point that may allow +progress in that vein. +Last, since the expression for the dynamic structure +factor depends on the elastic properties of the sheet and +in particular on its elastic moduli, a precise time-resolved +measurement of the dynamics of the sheet could provide +an indirect measurement of the elastic constants of the +material. This might be advantageous when a more di- +rect mechanical test may be destructive or might simply +modify these properties, while vibrating the sheet intro- +duces only a mild perturbation. +An interesting extension of this work would be to con- +sider the regime where inertia is important, which is of +major interest in the context of wave turbulence in thin +sheets [42, 44–53]. In particular, it would be interesting +to see whether the growing knowledge on the energy cas- +cade could be qualitatively and quantitatively connected +to the structure and dynamics of vibrating sheets. Such +a step however would require a major technical advance, +namely a nontrivial extension of the self-consistent ex- +pansion beyond the overdamped regime, or otherwise the +development of an alternative methodology to tackle this +problem. +ACKNOWLEDGMENTS +The authors wish to thank Arezki Boudaoud for use- +ful discussions. This work was supported by the Israel +Science Foundation Grant No. 1682/18. +[1] F. Plourabou´e and S. Roux, Experimental study of the +roughness of crumpled surfaces, Physica A 227, 173 +(1996). +[2] K. +Matan, +R. +B. +Williams, +T. +A. +Witten, +and +S. +R. +Nagel, +Crumpling +a +thin +sheet, +Phys. Rev. Lett. 88, 076101 (2002). +[3] D. L. Blair and A. Kudrolli, Geometry of crumpled paper, +Phys. Rev. Lett. 94, 166107 (2005). +[4] A. S. Balankin, O. S. 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Opt. 60, 172 (2021). + diff --git a/GdE4T4oBgHgl3EQfHgxG/content/tmp_files/load_file.txt b/GdE4T4oBgHgl3EQfHgxG/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..98f77b199aad2a78cdd9b2cd559819780c6c2114 --- /dev/null +++ b/GdE4T4oBgHgl3EQfHgxG/content/tmp_files/load_file.txt @@ -0,0 +1,741 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf,len=740 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content='04903v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content='soft] 12 Jan 2023 The Dynamics of Fluctuating Thin Sheets Under Random Forcing Chanania Steinbock and Eytan Katzav Racah Institute of Physics, The Hebrew University, Jerusalem 9190401, Israel (Dated: January 13, 2023) We study the dynamic structure factor of fluctuating elastic thin sheets subject to conservative (athermal) random forcing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' In Steinbock, Katzav & Boudaoud, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Research 4, 033096 (2022), the static structure factor of a such a sheet was studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' In this paper, we recap the model developed there and investigate its dynamic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Using the self-consistent expansion (SCE), the time dependent two-point function of the height profile is determined and found to decay exponentially in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Despite strong nonlinear coupling, the decay rate of the dynamic structure factor is found to coincide with the effective coupling constant for the static properties which suggests that the model under investigation exhibits certain quasi-linear behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Confirmation of these results by numerical simulations is also presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' INTRODUCTION Thin sheets and surfaces are ubiquitous in everyday life yet the theory of their physical properties remains incomplete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' For instance, despite the fact that crumpled paper balls take but a moment to make, the response of a thin sheet to random forcing remains poorly understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Since randomly driven thin surfaces are relevant to a wide diversity of fields, ranging from the physics of crumpled paper to the properties of graphene to the behaviour of cell membranes, a theory of randomly driven surfaces derived from first principles would have far reaching con- sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Loosely speaking, we can distinguish between two types of random forcing, completely uncorrelated white noise typical of thermal fluctuations and deliberate cor- related noise such as the type of forcing applied when crumpling a sheet of paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Here, we will focus on the latter kind of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Perhaps the easiest way to probe the structure of an athermally fluctuating sheet is to study the properties of the resultant crumpled sheet and in- deed this is an active field of research in its own right, both experimentally [1–11] and through mathematical or numerical modeling [1, 12–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' The development of the theory of singular structures supported by thin sheets such as d-cones and ridges [15–21] has gone some way to bridging research into crumpled sheets with that of fluc- tuating sheets however its applicability has been limited by the impracticality of characterising sheets with more than a handful of ridges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Additionally, the structure of crumpled sheets can at most inform us of the static un- varying properties of fluctuating systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' To obtain in- sight into the complete dynamic structure of a fluctuating thin sheet, we must tackle such a system directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Previous research into the time-dependent dynamic structure of fluctuating surfaces is limited but has been studied in the context of tethered surfaces [22, 23] and polymerised membranes [24, 25], though this research fo- cused exclusively on thermally driving white noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' In the context of tethered surfaces [22, 23], the dynamics of phantom and self-avoiding flexible sheets was studied though at the cost of neglecting the elastic properties of real sheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' In contrast, [24] focuses on the dynamic character of elastic polymerised membranes coupled to a random perturbing flowing fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Finally, [25] uses a super-symmetric ε-expansion of a D = 4 − ε dimensional membrane to obtain the dynamic exponent of an elastic thermally fluctuating polymerised membrane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Recently, we showed how the static properties of a fluc- tuating sheet can be derived directly by applying tech- niques from out-of-equilibrium statistical mechanics to the physics of elastic systems [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' In particular, we developed a dynamic variant of the F¨oppl-von K´arm´an equations which describes the deformations of thin sheets and used this to obtain the static structure factor of a fluctuating thin sheet driven by athermal noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' In the current paper, we extend this approach to derive the time-dependent structure factor of the fluctuating sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' This dynamic structure should be of direct rel- evance in understanding many features of the sheet, in- cluding its acoustic emissions [27–30], optical signature [31] and dissipative character [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Further applica- tions to diverse fields [34] such as the biophysics of cell membranes [24, 35], the properties and stability of fluc- tuating graphene sheets [36–41] and wave turbulence [42] can also be envisioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' The paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' In Section II, we briefly recap the derivation of the overdamped F¨oppl- von K´arm´an equations developed in [26] and in Section III, we apply the self-consistent expansion (SCE) to these equations to determine the dynamic structure factor of the fluctuating sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' In Section IV, the accuracy of our solution is confirmed by comparison with numerical sim- ulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Finally, the implications of these results are discussed in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' THE OVERDAMPED DYNAMIC F ¨OPPL-VON K´ARM ´AN EQUATIONS In [26], we developed a model to describe fluctuating elastic thin sheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' In this section, we recap the main ideas and relate them to the dynamic structure factor of such a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' The equilibrium out-of-plane displacement ξ (x, y) of a thin elastic sheet subject to an external pressure Pex 2 is given by the well known F¨oppl-von K´arm´an equa- tions [43] Pex = Eh3 12 (1 − ν2)∇4ξ − h � ∂2ξ ∂x2 ∂2χ ∂y2 + ∂2ξ ∂y2 ∂2χ ∂x2 − 2 ∂2ξ ∂x∂y ∂2χ ∂x∂y � (1) 0 = ∇4χ + E � ∂2ξ ∂x2 ∂2ξ ∂y2 − � ∂2ξ ∂x∂y �2� , (2) where h, E and ν denote the sheet thickness, Young’s modulus and Poisson ratio respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' The scalar field χ (x, y) denotes the Airy stress potential of the deforma- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' By writing the out-of-plane displacement ξ (x, y) in the Monge parameterisation, ie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' as a function of x and y, we assume that deformations of our sheet are mostly flat and thus our focus here will be on weak fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' To explore the dynamics of a driven fluctuating sheet, we apply Newton’s second law to each element of the sheet with density ρ hρ∂2ξ ∂t2 = −Pex + Pdamping + Pdriving .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' (3) where Pdriving and Pdamping describe driving and damp- ing forces respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Though variations of this equa- tion have been studied in the context of wave turbu- lence [42, 44–53], here we continue the approach intro- duced in [26] of a sheet subject to ordinary fluid friction Pdamping = −α ∂ξ ∂t being driven by conserved Gaussian noise Pdriving = η (⃗r, t) with noise amplitude D, that is, ⟨η (⃗r, t)⟩ = 0 (4) ⟨η (⃗r, t) η (⃗r ′, t′)⟩ = −Dδ (t − t′) ∇2δ (⃗r − ⃗r ′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' (5) Other driving forces could be considered, but as argued in [26], there is value in studying the setup where the sheet’s center of mass does not wander in space and hence we impose conserved noise on the sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' More specific forms of noise which are consistent with the conservation of center of mass could also be considered but following the principle of parsimony, we consider only the simplest possibility here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Taking the overdamped limit and thus neglecting the inertia term hρ ∂2ξ ∂t2 , this approach provides a concrete model for a driven fluctuating elastic sheet, namely, the overdamped dynamic F¨oppl-von K´arm´an equation α∂ξ ∂t + Eh3 12 (1 − ν2)∇4ξ − h � ∂2ξ ∂x2 ∂2χ ∂y2 + ∂2ξ ∂y2 ∂2χ ∂x2 − 2 ∂2ξ ∂x∂y ∂2χ ∂x∂y � = η (⃗r, t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' (6) where the Airy stress potential χ (x, y) is still determined by equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' The fundamental difference between the problem un- der study here and the one studied in the wave turbulence community [42, 44–53] is that they focus on the regime where inertia is very important, while friction is present only at the smallest scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Also, the forcing of the sheet, which is often modeled as white noise, is applied only at the largest scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' As a result, the main feature which is studied is the energy cascade from the large scales (where the forcing is applied) to the smallest scales (where it is dissipated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' In fact, there exist concrete predictions regarding this energy cascade depending on the specific scenario that drives this cascade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' In contrast, we focus on the dynamics of the structure of the sheet under forcing and friction across all scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' It is shown in [26] that for a sheet with dimensions L×L, equations (2) and (6) can be combined into a single equation for the Fourier components ˜ξ⃗n (t) of ξ (x, y, t) = � ⃗n ˜ξ⃗n (t) ei 2π L ⃗n·⃗r where the sum is taken over all lattice points of Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' After nondimensionalising, one obtains the following Langevin equation ∂ ¯ξ⃗n ∂¯t + g |⃗n|4 ¯ξ⃗n + 1 2 � ⃗ℓ1̸=⃗n � ⃗ℓ2 � ⃗ℓ3 V⃗n,⃗ℓ1,⃗ℓ2,⃗ℓ3 ¯ξ⃗ℓ1 ¯ξ⃗ℓ2 ¯ξ⃗ℓ3 = ¯η⃗n (¯t ) (7) containing a single dimensionless parameter g = 2π 12 (1 − ν2) � αh5E D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' (8) The scaled time and Fourier components are given by ¯t = � (2π)6 hDE/ � α3L8��1/2 t (9) ¯ξ⃗n = � (2π)2 αhE/D �1/4 ˜ξ⃗n (10) and the dimensionless noise in Fourier space has mean 0 and variance ⟨¯η⃗n (¯t ) ¯η⃗n′ (¯t ′)⟩ = |⃗n|2 δ⃗n,−⃗n′δ (¯t − ¯t ′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' (11) Finally, the kernel V⃗n,⃗ℓ1,⃗ℓ2,⃗ℓ3 is simply the Fourier trans- form of the transverse projection operator of the sheet deformation [34, 54] and is given by V⃗n,⃗ℓ1,⃗ℓ2,⃗ℓ3 = δ⃗n,⃗ℓ1+⃗ℓ2+⃗ℓ3 ���⃗n × ⃗ℓ1 ��� 2 ���⃗ℓ2 × ⃗ℓ3 ��� 2 ���⃗n − ⃗ℓ1 ��� 4 , (12) where we have denoted ���⃗n × ⃗ℓ ��� = nxℓy − nyℓx, thus equation (7) can be thought of as a type of φ4-field Langevin equation with a non-trivial spatially varying kernel V⃗n,⃗ℓ1,⃗ℓ2,⃗ℓ3 [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Accordingly, in principle, equation (7) can be used to find structure factors such as the time- dependent two-point function S⃗n (¯t, ¯t ′) = �¯ξ⃗n (¯t ) ¯ξ−⃗n (¯t ′) � (13) 3 which in steady-state will only depend on the difference ∆¯t = |¯t − ¯t ′| and thus can be written as a function of a single argument as S⃗n (¯t ) = �¯ξ⃗n (0) ¯ξ−⃗n (¯t ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' (14) Unfortunately, the single dimensionless parameter g in equation (7) is coupled to its linear part and in [26], it is argued that g is typically small since g ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content='1 for a typical sheet of aluminum or steel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Indeed, the scaling g ∼ h5/2 ensures that for any sufficiently thin sheet, g will be small and thus any expansion around the linear part of equation (7) which treats the nonlinear part as a mere correction is guaranteed to fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Instead, following the success of [26], we will analyse equation (7) by application of the self-consistent expansion (SCE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' THE SELF-CONSISTENT EXPANSION As described in [26], the self-consistent expansion (SCE) can be thought of as a renormalised perturbation theory [56] capable of providing series approximations even in the presence of strong coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' The method has found previous application to the KPZ equation and its variations [57–66], fracture and wetting fronts [67, 68] and turbulence [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' More relevant to our system, the SCE provides an extremely successful solution to the zero-dimensional φ4-theory giving good results at low orders and exact convergence at high orders [70, 71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Accordingly, the success of the SCE in determining the static structure factor of a fluctuating sheet in [26] was not entirely unexpected and since the SCE has a natural extension to dynamic quantities, we extend the approach taken in [26] here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' The Fokker-Planck Equation and the SCE As in [26],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' we begin by writing the Fokker-Planck equa- tion corresponding to equation (7) [72] ∂P ∂¯t = 1 2 � ⃗n |⃗n|2 ∂2P ∂ ¯ξ⃗n∂ ¯ξ−⃗n + g � ⃗n |⃗n|4 ∂ ∂ ¯ξ⃗n �¯ξ⃗nP � + 1 2 � ⃗n ∂ ∂ ¯ξ⃗n \uf8ee \uf8f0P � ⃗ℓ1̸=⃗n � ⃗ℓ2 � ⃗ℓ3 V⃗n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content='⃗ℓ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content='⃗ℓ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content='⃗ℓ3 ¯ξ⃗ℓ1 ¯ξ⃗ℓ2 ¯ξ⃗ℓ3 \uf8f9 \uf8fb ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' (15) where P = P ��¯ξ⃗n (¯t ) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' ¯t � denotes the probability func- tional that the system will have a specific configuration,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' as prescribed by the Fourier components �¯ξ⃗n (¯t ) � at time ¯t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' We can multiply this equation by a function of the Fourier components F ��¯ξ⃗n (¯t ) �� and integrate over all ¯ξ⃗n (¯t ) to obtain the following equation for the expecta- tions ∂ ⟨F⟩ ∂¯t = 1 2 � ⃗n |⃗n|2 � ∂2F ∂ ¯ξ⃗n∂ ¯ξ−⃗n � − g � ⃗n |⃗n|4 � ∂F ∂ ¯ξ⃗n ¯ξ⃗n � − 1 2 � ⃗n � ⃗ℓ1̸=⃗n � ⃗ℓ2 � ⃗ℓ3 V⃗n,⃗ℓ1,⃗ℓ2,⃗ℓ3 � ∂F ∂ ¯ξ⃗n ¯ξ⃗ℓ1 ¯ξ⃗ℓ2 ¯ξ⃗ℓ3 � (16) where we have defined the expectation values ⟨F⟩ = � � ⃗n d¯ξ⃗n F ��¯ξ⃗n �� P ��¯ξ⃗n � , ¯t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' (17) Equation (16) can be used to obtain relationships be- tween various moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' For instance, in [26], it was shown that subbing in F = ¯ξ⃗n ¯ξ⃗n′ results in an equation relating the static two-point function �¯ξ⃗n ¯ξ⃗n′� to the static four-point function �¯ξ⃗n1 ¯ξ⃗n2 ¯ξ⃗n3 ¯ξ⃗n4 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Similarly, to obtain relations for dynamic quantities, we can sub in time de- pendent functions such as F ��¯ξ⃗n (¯t ) �� = ¯ξ⃗n (0) ¯ξ⃗n′ (¯t ) which results in the ODE ∂ �¯ξ⃗n (0) ¯ξ⃗n′ (¯t ) � ∂¯t = −g |⃗n′|4 �¯ξ⃗n (0) ¯ξ⃗n′ (¯t ) � −1 2 � ⃗ℓ1̸=⃗n′ � ⃗ℓ2 � ⃗ℓ3 V⃗n′,⃗ℓ1,⃗ℓ2,⃗ℓ3 � ¯ξ⃗n (0) ¯ξ⃗ℓ1 (¯t ) ¯ξ⃗ℓ2 (¯t ) ¯ξ⃗ℓ3 (¯t ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' (18) This first order non-homogeneous ODE provides the time-dependent two-point function �¯ξ⃗n (0) ¯ξ⃗n′ (¯t ) � if given the time-dependent four-point function � ¯ξ⃗n (0) ¯ξ⃗ℓ1 (¯t ) ¯ξ⃗ℓ2 (¯t ) ¯ξ⃗ℓ3 (¯t ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' An ODE for the four- point function can of course be obtained by subbing F ��¯ξ⃗n (¯t ) �� = ¯ξ⃗n (0) ¯ξ⃗ℓ1 (¯t ) ¯ξ⃗ℓ2 (¯t ) ¯ξ⃗ℓ3 (¯t ) into equation (16) though the resulting ODE would require knowledge of the time-dependent six-point function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' As observed in [26], this situation of needing higher moments to find lower ones is similar to the BBGKY hierarchy [73–75] and finding closure is in general non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' The naive approach would be to simply neglect the non-homogeneous part of equation (18) and then at- tempt to perturbatively correct for it however since the small parameter g is coupled to the homogeneous part of equation (18), the non-homogeneous contribution is large and non-negligible and thus such an approach is guaranteed to fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Since this occurs at every level of the hierarchy, a more sophisticated approach is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Following the approach taken in [26], we apply the SCE to equation (16) by introducing a free parameter Γ|⃗n| ∂ ⟨F⟩ ∂¯t = 1 2 � ⃗n |⃗n|2 � ∂2F ∂ ¯ξ⃗n∂ ¯ξ−⃗n � − � ⃗n Γ|⃗n| � ∂F ∂ ¯ξ⃗n ¯ξ⃗n � − � ⃗n � g |⃗n|4 − Γ|⃗n| � � ∂F ∂ ¯ξ⃗n ¯ξ⃗n � − 1 2 � ⃗n � ⃗ℓ1̸=⃗n � ⃗ℓ2 � ⃗ℓ3 V⃗n,⃗ℓ1,⃗ℓ2,⃗ℓ3 � ∂F ∂ ¯ξ⃗n ¯ξ⃗ℓ1 ¯ξ⃗ℓ2 ¯ξ⃗ℓ3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' (19) 4 One can think of Γ|⃗n| as an effective coupling constant such that a perturbative expansion around the linear the- ory with Γ|⃗n| is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' The problem of determining its value will be deferred to later though due to the isotropic character of our system, we have already assumed that Γ|⃗n| can only depend on the magnitude of ⃗n and not its direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Now if ⟨F⟩(m) denotes an mth order expan- sion of ⟨F⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' then by assumption,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' the latter terms will contribute at a higher order and thus we can write the iterative relation ∂ ⟨F⟩(m) ∂¯t = 1 2 � ⃗n |⃗n|2 � ∂2F ∂ ¯ξ⃗n∂ ¯ξ−⃗n �(m) − � ⃗n Γ|⃗n| � ∂F ∂ ¯ξ⃗n ¯ξ⃗n �(m) − � ⃗n � g |⃗n|4 − Γ|⃗n| � � ∂F ∂ ¯ξ⃗n ¯ξ⃗n �(m−1) − 1 2 � ⃗n � ⃗ℓ1̸=⃗n � ⃗ℓ2 � ⃗ℓ3 V⃗n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content='⃗ℓ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content='⃗ℓ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content='⃗ℓ3 � ∂F ∂ ¯ξ⃗n ¯ξ⃗ℓ1 ¯ξ⃗ℓ2 ¯ξ⃗ℓ3 �(m−1) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' (20) supplemented with the convention that for m = 0 the m − 1 terms drop out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' This equation can now be used to obtain any moment up to any order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' For instance, in [26], it was shown that subbing in F = ¯ξ⃗n ¯ξ⃗n′ or F = ¯ξ⃗n ¯ξ⃗ℓ1 ¯ξ⃗ℓ2 ¯ξ⃗ℓ3 together with m = 0 directly results in zeroth order expressions for the static two-point and four-point functions �¯ξ⃗n ¯ξ⃗n′�(0) = |⃗n|2 2Γ|⃗n| δ⃗n,−⃗n′ (21) and � ¯ξ⃗n′ ¯ξ⃗ℓ1 ¯ξ⃗ℓ2 ¯ξ⃗ℓ3 �(0) = � ¯ξ⃗n′ ¯ξ⃗ℓ1 �(0) � ¯ξ⃗ℓ2 ¯ξ⃗ℓ3 �(0) + � ¯ξ⃗n′ ¯ξ⃗ℓ2 �(0) � ¯ξ⃗ℓ1 ¯ξ⃗ℓ3 �(0) + � ¯ξ⃗n′ ¯ξ⃗ℓ3 �(0) � ¯ξ⃗ℓ1 ¯ξ⃗ℓ2 �(0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' (22) Upon further substitution of F = ¯ξ⃗n ¯ξ⃗n′ and m = 1, an expression for the first order static two-point func- tion �¯ξ⃗n ¯ξ⃗n′�(1) was obtained in terms of the zeroth order static two-point and four-point functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Here, we will determine what equation (20) has to say about dynamic time-dependent quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Time-Dependent SCE Unlike the static quantities described in [26], subbing time-dependent quantities into equation (20) does not re- sult in simple expressions for the moments under consid- eration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Rather, the time derivative on the left-hand side of equation (20) ensures that time-dependent quantities are given by ODEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' For instance, subbing F ��¯ξ⃗n (¯t ) �� = ¯ξ⃗n (0) ¯ξ⃗n′ (¯t ) and m = 0 into equation (20) gives the ho- mogeneous ODE ∂ �¯ξ⃗n (0) ¯ξ⃗n′ (¯t ) �(0) ∂¯t = −Γ|⃗n′| �¯ξ⃗n (0) ¯ξ⃗n′ (¯t ) �(0) (23) whose solution is simply �¯ξ⃗n (0) ¯ξ⃗n′ (¯t ) �(0) = �¯ξ⃗n ¯ξ⃗n′�(0) e −Γ|⃗n′|¯t (24) where �¯ξ⃗n ¯ξ⃗n′�(0) is given by equation (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Similarly, subbing F = ¯ξ⃗n′ (0) ¯ξ⃗ℓ1 (¯t ) ¯ξ⃗ℓ2 (¯t ) ¯ξ⃗ℓ3 (¯t ) and m = 0 into equation (20) gives the ODE ∂ � ¯ξ⃗n′ (0) ¯ξ⃗ℓ1 (¯t ) ¯ξ⃗ℓ2 (¯t ) ¯ξ⃗ℓ3 (¯t ) �(0) ∂¯t = − � ⃗n Γ|⃗n| � ¯ξ⃗n′ (0) ∂ � ¯ξ⃗ℓ1 (¯t ) ¯ξ⃗ℓ2 (¯t ) ¯ξ⃗ℓ3 (¯t ) � ∂ ¯ξ⃗n (¯t ) ¯ξ⃗n (¯t ) �(0) + 1 2 � ⃗n |⃗n|2 � ¯ξ⃗n′ (0) ∂2 � ¯ξ⃗ℓ1 (¯t ) ¯ξ⃗ℓ2 (¯t ) ¯ξ⃗ℓ3 (¯t ) � ∂ ¯ξ⃗n (¯t ) ∂ ¯ξ−⃗n (¯t ) �(0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' (25) Carrying out the derivatives explicitly and computing the sums,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' the first term on the right-hand side is simply pro- portional to the dynamic four-point function � ⃗n Γ|⃗n| � ¯ξ⃗n (0) ∂ � ¯ξ⃗ℓ1 (¯t ) ¯ξ⃗ℓ2 (¯t ) ¯ξ⃗ℓ3 (¯t ) � ∂ ¯ξ⃗n (¯t ) ¯ξ⃗n (¯t ) �(0) = � Γ|⃗ℓ1| + Γ|⃗ℓ2| + Γ|⃗ℓ3| � � ¯ξ⃗n (0) ¯ξ⃗ℓ1 (¯t ) ¯ξ⃗ℓ2 (¯t ) ¯ξ⃗ℓ3 (¯t ) �(0) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' (26) while the second term is composed of a sum of zeroth order dynamic two-point functions � ⃗n |⃗n|2 � ¯ξ⃗n′ (0) ∂2 � ¯ξ⃗ℓ1 (¯t ) ¯ξ⃗ℓ2 (¯t ) ¯ξ⃗ℓ3 (¯t ) � ∂ ¯ξ⃗n (¯t ) ∂ ¯ξ−⃗n (¯t ) �(0) = δ⃗ℓ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content='−⃗ℓ2 ����⃗ℓ1 ��� 2 + ���⃗ℓ2 ��� 2� � ¯ξ⃗n′ (0) ¯ξ⃗ℓ3 (¯t ) �(0) + δ⃗ℓ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content='−⃗ℓ3 ����⃗ℓ1 ��� 2 + ���⃗ℓ3 ��� 2� � ¯ξ⃗n′ (0) ¯ξ⃗ℓ2 (¯t ) �(0) + δ⃗ℓ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content='−⃗ℓ3 ����⃗ℓ2 ��� 2 + ���⃗ℓ3 ��� 2� � ¯ξ⃗n′ (0) ¯ξ⃗ℓ1 (¯t ) �(0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' (27) Since we have already computed the zeroth order dynamic two-point function in equation (24) and can rearrange equation (21) to relate |⃗n|2 δ⃗n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content='−⃗n′ = 5 2Γ|⃗n| �¯ξ⃗n ¯ξ⃗n′�(0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' we can observe that δ⃗ℓ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content='−⃗ℓ2 ����⃗ℓ1 ��� 2 + ���⃗ℓ2 ��� 2� � ¯ξ⃗n′ (0) ¯ξ⃗ℓ3 (¯t ) �(0) = 2 � Γ|⃗ℓ1| + Γ|⃗ℓ2| + Γ|⃗ℓ3| − Γ|⃗n′| � × × � ¯ξ⃗ℓ1 ¯ξ⃗ℓ2 �(0) � ¯ξ⃗ℓ3 ¯ξ⃗n′ �(0) e −Γ|⃗n′|¯t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' (28) where we have been able to add and subtract Γ|⃗ℓ3| and Γ|⃗n′| since the factor � ¯ξ⃗ℓ3 ¯ξ⃗n′ �(0) ∝ δ⃗ℓ3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content='−⃗n′ ensures that Γ|⃗ℓ3| − Γ|⃗n′| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Simplifying each term of equation (27) in this manner, we obtain � ⃗n |⃗n|2 � ¯ξ⃗n′ (0) ∂2 � ¯ξ⃗ℓ1 (¯t ) ¯ξ⃗ℓ2 (¯t ) ¯ξ⃗ℓ3 (¯t ) � ∂ ¯ξ⃗n (¯t ) ∂ ¯ξ−⃗n (¯t ) �(0) = 2 � Γ|⃗ℓ1| + Γ|⃗ℓ2| + Γ|⃗ℓ3| − Γ|⃗n′| � × × � ¯ξ⃗n′ ¯ξ⃗ℓ1 ¯ξ⃗ℓ2 ¯ξ⃗ℓ3 �(0) e −Γ|⃗n′|¯t , (29) where � ¯ξ⃗n′ ¯ξ⃗ℓ1 ¯ξ⃗ℓ2 ¯ξ⃗ℓ3 �(0) is the static four-point function given by equation (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' substituting equations (26) and (29) into equa- tion (25),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' we obtain the following non-homogeneous ODE for the dynamic four-point function at zeroth order ∂ � ¯ξ⃗n′ (0) ¯ξ⃗ℓ1 (¯t ) ¯ξ⃗ℓ2 (¯t ) ¯ξ⃗ℓ3 (¯t ) �(0) ∂¯t = − � Γ|⃗ℓ1| + Γ|⃗ℓ2| + Γ|⃗ℓ3| � � ¯ξ⃗n′ (0) ¯ξ⃗ℓ1 (¯t ) ¯ξ⃗ℓ2 (¯t ) ¯ξ⃗ℓ3 (¯t ) �(0) + � Γ|⃗ℓ1| + Γ|⃗ℓ2| + Γ|⃗ℓ3| − Γ|⃗n′| � � ¯ξ⃗n′ ¯ξ⃗ℓ1 ¯ξ⃗ℓ2 ¯ξ⃗ℓ3 �(0) e −Γ|⃗n′|¯t (30) and it is straightforward to check,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' though perhaps un- surprising,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' that this is simply solved by � ¯ξ⃗n′ (0) ¯ξ⃗ℓ1 (¯t ) ¯ξ⃗ℓ2 (¯t ) ¯ξ⃗ℓ3 (¯t ) �(0) = � ¯ξ⃗n′ ¯ξ⃗ℓ1 ¯ξ⃗ℓ2 ¯ξ⃗ℓ3 �(0) e −Γ|⃗n′|¯t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' (31) Now to study the effect of the nonlinearity, we proceed to higher orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Subbing F ��¯ξ⃗n (¯t ) �� = ¯ξ⃗n (0) ¯ξ⃗n′ (¯t ) and m = 1 into equation (20) gives after some tedious algebra ∂ �¯ξ⃗n (0) ¯ξ⃗n′ (¯t ) �(1) ∂¯t = −Γ|⃗n′| �¯ξ⃗n (0) ¯ξ⃗n′ (¯t ) �(1) − � g |⃗n′|4 − Γ|⃗n′| � �¯ξ⃗n (0) ¯ξ⃗n′ (¯t ) �(0) − 1 2 � ⃗ℓ1̸=⃗n′ � ⃗ℓ2 � ⃗ℓ3 V⃗n′,⃗ℓ1,⃗ℓ2,⃗ℓ3× × � ¯ξ⃗n (0) ¯ξ⃗ℓ1 (¯t ) ¯ξ⃗ℓ2 (¯t ) ¯ξ⃗ℓ3 (¯t ) �(0) , (32) or after subbing in our expressions for the zeroth order time-dependent two-point and four-point functions from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' (24) and (31) respectively ∂ �¯ξ⃗n (0) ¯ξ⃗n′ (¯t ) �(1) ∂¯t = −Γ|⃗n′| �¯ξ⃗n (0) ¯ξ⃗n′ (¯t ) �(1) − �� g |⃗n′| 4 − Γ|⃗n′| � �¯ξ⃗n ¯ξ⃗n′�(0) + 1 2 � ⃗ℓ1̸=⃗n′ � ⃗ℓ2 � ⃗ℓ3 V⃗n′,⃗ℓ1,⃗ℓ2,⃗ℓ3 � ¯ξ⃗n ¯ξ⃗ℓ1 ¯ξ⃗ℓ2 ¯ξ⃗ℓ3 �(0) � e −Γ|⃗n′|¯t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' (33) It is worth noting that being precise, the exponential decay associated with the four-point function in this ex- pression should decay with rate Γ|⃗n| instead of Γ|⃗n′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' A careful analysis of the sum over the kernel V⃗n′,⃗ℓ1,⃗ℓ2,⃗ℓ3 with the static four-point function reveals however that this term vanishes unless ⃗n = −⃗n′ and thus no harm is done by replacing Γ|⃗n| with Γ|⃗n′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' It is now apparent that the non-homogeneous term in this equation decays at the natural decay rate Γ|⃗n′| of the ODE and thus its gen- eral solution will contain a non-physical secular term of the form ¯te −Γ|⃗n′|¯t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' As in the Poincar´e-Lindstedt method for perturbatively solving non-linear ODEs [76–78], this situation can be avoided by setting Γ|⃗n′| such that the coefficient of the non-homogeneous term vanishes, ie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' 0 = � g |⃗n′|4 − Γ|⃗n′| � �¯ξ⃗n ¯ξ⃗n′�(0) + 1 2 � ⃗ℓ1̸=⃗n′ � ⃗ℓ2 � ⃗ℓ3 V⃗n′,⃗ℓ1,⃗ℓ2,⃗ℓ3 � ¯ξ⃗n ¯ξ⃗ℓ1 ¯ξ⃗ℓ2 ¯ξ⃗ℓ3 �(0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' (34) Using this idea to determine the characteristic decay rate is novel in the context of the SCE method and might be useful in other problems as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' After subbing in the static two-point and four-point functions and carrying out the sums in the last equation, this ultimately simpli- fies to the following discrete integral equation for Γ|⃗n| Γ|⃗n| = g |⃗n|4 + 1 2 � ⃗ℓ̸=⃗n ���⃗ℓ ��� 2 ���⃗n × ⃗ℓ ��� 4 Γ|⃗ℓ| ���⃗n − ⃗ℓ ��� 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' (35) 6 Surprisingly, the above argument for preventing secu- lar terms occurring in our expression for �¯ξ⃗n (0) ¯ξ⃗n′ (¯t ) �(1) is equivalent to simply imposing that the first order ap- proximation for the dynamic structure factor equals its zeroth order approximation, ie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' �¯ξ⃗n (0) ¯ξ⃗n′ (¯t ) �(1) = �¯ξ⃗n (0) ¯ξ⃗n′ (¯t ) �(0) , (36) or in other words, we select Γ|⃗n| such that the zeroth order approximation is exact up to first order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' In [26], a static version of this self-consistent argument was made by setting the first order approximation for the static structure factor equal to its zeroth order approximation, ie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' �¯ξ⃗n ¯ξ⃗n′�(1) = �¯ξ⃗n ¯ξ⃗n′�(0) , (37) and indeed, the same discrete integral equation is ob- tained from both approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' It is important to appre- ciate that these two self-consistent arguments giving the same discrete integral equation for Γ|⃗n| was by no means anticipated nor trivial and in fact, it is known that this does not occur for the ordinary unadorned φ4-model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' In such instances, the fact that the two arguments conflict suggests that the effective decay rate Γ|⃗n| also needs to be appropriately expanded in a manner analogous to the Poincar´e-Lindstedt method [76–78] if we wish our results to have meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Conversely, the situation where the two arguments result in the same discrete integral equation implies some degree of quasi-linearity and is indicative that our approach has self-consistently captured a true feature of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Equation (35) has been solved in the appendix of [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Here we simply bring the final result Γ|⃗n| ≃ n4 � 3π 4 � A (g) − ln � n nmax �� × × � 1 + B (g) n nmax � , (38) where nmax is an upper-frequency cutoff which must be imposed on the system and A (g) and B (g) are constants which only depend on g and have the following small g expansions A ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content='137 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content='336g + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content='243g2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content='112g3 + O � g4� , (39) B ≃ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content='265 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content='360g − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content='395g2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content='311g3 + O � g4� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' (40) It is worth noting that B (g) primarily determines the behaviour of Γ|⃗n| for large ⃗n, ie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' when |⃗n| → nmax, and can be neglected when |⃗n| is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Accordingly, for small ⃗n, we have obtained that the decay rate Γ|⃗n| grows like a logarithmically corrected power law ∼ n4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' To summarise, we have found that at first order, the dynamic structure factor is given by �¯ξ⃗n (0) ¯ξ⃗n′ (¯t ) �(1) = �¯ξ⃗n ¯ξ⃗n′�(0) e−Γ|⃗n|¯t (41) = δ⃗n,−⃗n′ n2 2Γ|⃗n| e−Γ|⃗n|¯t (42) where Γ|⃗n| is given by equation (38) and can be seen to si- multaneously play the roles of effective coupling constant and effective decay rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' COMPARISON WITH SIMULATIONS As in [26], our predictions can be validated by nu- merical integration of equation (7) over a square lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Since we investigate here the dynamic properties of the simulation rather than its static properties, the simula- tions must be run for long enough to obtain precise av- erages of the time-dependent two-point function S⃗n (¯t ) and thus they must be run for substantially longer than in [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' The lattice size imposes a finite maximum frequency nmax but since our theory necessitates the existence of an upper-frequency cut-off, this in itself is fine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' More press- ingly, for increasing maximum frequency nmax, the max- imum size of the time step δ¯t that can be used shrinks if the simulation is to remain stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' This presents a trade- off between the maximum resolution in time vs the maxi- mum resolution in space and since the dynamic structure factor can only be extracted from long simulation runs, the size of nmax is sharply constrained by practical con- siderations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' In practice, our simulations were run with a time-step δ¯t = 10−6 over a 41 × 41 lattice corresponding to a maximum frequency nmax = 20 √ 2 ≈ 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content='3, though unlike [26] which only used 106 time steps per simula- tion, these simulations were run for 107 time steps each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' The consequence of these extended simulations is that an enormous amount of data is generated though sim- ulation states which are close to each other in time are practically indistinguishable except at the very largest modes (an expectation based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' (38)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Since these modes equilibrate far more rapidly than the small modes, they are far less interesting in the current context and thus in the interest of keeping memory resources man- ageable, simulation data was only saved every 103 time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Accordingly, the decay rate of modes which decay faster than ∆¯t = 103 × δ¯t = 10−3 is not presented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Finally, it is worth noting that as in [26], efficient calcu- lation of the quartic interaction term of equation (7) is non-trivial and as there, was achieved by implementing the pseudo-spectral method described in [52] in which the quartic interaction is calculated as the Fourier transform of its real-space counterpart though this imposes periodic boundary conditions on the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' To achieve pre- cise results, each simulation was run 10 times for various values of 0 ≤ g ≤ 1 and the results averaged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' 1 shows how the dynamic structure factor S⃗n (¯t ) decays for the first few modes as a function of the time difference ¯t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' These results are for the simulation per- formed with g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content='1 though very similar results are ob- tained for the other values of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' The solid black lines are linear fits to the logarithm of the simulation data and clearly show that the dynamic structure factor S⃗n (¯t ) in- deed decays exponentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' By performing such fits for many frequencies, we are able to numerically determine 7 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content='2 2-4 2-3 2-2 2-1 1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' The dynamic structure factor S⃗n (¯t ) as a function of the time difference ¯t for g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Each data set corresponds to a different frequency (see legend) and the solid lines are linear fits to the logarithm of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' The fact that these fits are excellent confirms that S⃗n (¯t ) indeed decays exponentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' how the decay rate, which we will denote Γdyn |⃗n| , varies as a function of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' 2 shows Γdyn |⃗n| and Γdyn |⃗n| / |⃗n|4 as a function of n up to nmax/4 for the various values of g we used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' The solid and dotted lines are fits of the equation Γdyn |⃗n| n4max ≃ C � n nmax �4 � A (g) − ln � n nmax � (43) to the numerical results of g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content='1 (green triangles) and g = 1 (purple squares) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Here, A (g) was sim- ply taken from equation (39) while the scaling parameter C was fitted and, as can be seen, these fits are excel- lent across the entire range of frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' As described towards the end of section III, the constant B (g) in equa- tion (38) primarily modifies Γ|⃗n| for large values of n and thus we have not attempted to determine it from our data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Fits for g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content='01 (red circles) and g = 0 (blue pluses) can also be carried out but the resulting fits are so similar to the g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content='1 case that little insight is gained by showing them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' The dashed line in the top plot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' 2 is a guideline proportional to n4 and together with the bottom plot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' 2 clearly emphasises that the loga- rithmic correction is non-negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' The fact that these fits of equation (43) so beautifully capture the behaviour of Γdyn |⃗n| confirms that our theory has indeed accurately predicted the functional form of the dynamic structure S⃗n (¯t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' It is worth mentioning that in [26], it was observed that the periodic square lattice introduces an anisotropy into the simulation and this required each direction to be treated separately, an aspect which we have not observed in the temporal de- cay data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' We explain this distinction by observing that in [26], the anisotropy was primarily accounted for by scaling the parameter B (g) in equation (38) by some ap- propriate function f (θ) and thus is only observable for 2-5 2-4 2-3 2-2 2-20 2-18 2-16 2-14 2-12 2-10 2-8 2-6 2-5 2-4 2-3 2-2 0 1 2 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Γdyn |⃗n| (top) and Γdyn |⃗n| / |⃗n|4 (bottom) as a function of n for various values of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' The solid and dotted lines are fits of equation (43) to the g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content='1 data (green triangles) and g = 1 data (purple squares) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' The dashed line in the top plot is a guideline proportional to n4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' The fact that the fits are excellent over the entire range of frequencies confirms that our theory correctly predicts the functional form of the dynamic structure factor S⃗n (¯t ) and the appreciable deviation between the fits and the guideline shows that the logarithmic correction of equation (43) is non-negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' This is further emphasised in the bottom plot by the fact that Γdyn |⃗n| / |⃗n|4 is indeed seen to be non-constant and is well fitted by the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' large frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Since we do not present here the large frequency decay rate, we have been unable to observe this anisotropy in this data though we presume it too exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Finally, it is worth comparing Γdyn |⃗n| , the decay rate of the time-dependent structure factor S⃗n (¯t ), with the ef- fective coupling constant which can be extracted from the relationship S⃗n (0) = �¯ξ⃗n ¯ξ−⃗n � = n2 2Γstat |⃗n| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' (44) Here, we denote the coupling constant by Γstat |⃗n| as it is obtained by only considering static quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Accord- 8 2-5 2-4 2-3 2-2 2-20 2-18 2-16 2-14 2-12 2-10 2-8 2-6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Comparison between the numerical coupling constant Γstat |⃗n| = 1 2n2/ �¯ξ⃗n¯ξ−⃗n � (blue pluses) and the numerical decay rate Γdyn |⃗n| of the time-dependent two-point function S⃗n (¯t ) = S⃗n (0) e −Γdyn |⃗n| ¯t (red circles), for g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' These two quantities are seen to agree over all orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' 2-5 2-4 2-3 2-2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content='5 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Γstat |⃗n| /Γdyn |⃗n| as a function of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' For all values of g, this ratio is seen to be constant and roughly equal to 1 over all frequency scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' ing to the theory developed above and in [26], Γstat |⃗n| and Γdyn |⃗n| should be identical and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' 3 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' 4 shows that these two methods of obtaining Γ|⃗n| are in fact ex- tremely close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' 3 compares Γstat |⃗n| and Γdyn |⃗n| for the case of g = 0 and shows that even with maximal non-linear coupling, Γstat |⃗n| and Γdyn |⃗n| are practically indistinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' The same result can be obtained for the other values of g and indeed Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' 4 shows that the ratio Γstat |⃗n| /Γdyn |⃗n| ≈ 1 for all values of g over all frequency scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' DISCUSSION In this work we have extended a previous paper [26] to study the dynamics of a vibrating thin sheet, gov- erned by the overdamped F¨oppl-von K´arm´an equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Specifically, we applied the self-consistent expansion to obtain predictions which agree with the results of numer- ical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Surprisingly, the effective coupling con- stant Γ|⃗n|, which determines the static structure factor of the sheet coincides with the decay rate of the dynamic structure factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' This observation is reminiscent of linear systems, where these two quantities are indeed the same, yet it is not obvious that the F¨oppl-von K´arm´an equa- tions, which are strongly nonlinear, would exhibit such a phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' A way to make sense of this situation is by using a recent correlation-response inequality in dynam- ical systems [79, 80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' This inequality was formulated in the context of Langevin dynamics and shows that when- ever the equations of motion can be derived from a Hamil- tonian and hence obey a certain Fluctuation-Dissipation relation (thus belonging to class I using that classifica- tion),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' and are Galilean invariant (that is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' the zero-mode does not affect the dynamics of the higher modes thus belonging to class II in that classification),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' the exponent inequalities become an equality implying quasi-linear dy- namics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' The quasi-linearity refers to the fact that the scaling of the decay rate of the dynamic correlation func- tion is related to the static exponents in a simple man- ner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' While the overdamped F¨oppl-von K´arm´an equations used here are definitely Galilean invariant since the cen- ter of mass does not drift as a result of the dynamics, the way they are derived from a Hamiltonian deviates from the models considered in [79, 80], as they belong to nei- ther model A nor model B in the classical classification of Hohenberg and Halperin [81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Nevertheless, they obey the quasi-linear property, which may be the result of a modified Fluctuation-Dissipation relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' This makes the overdamped F¨oppl-von K´arm´an equations another interesting example of a physically motivated model that obeys quasi-linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' From a methodological point of view, imposing the condition that secular terms should vanish, inspired by the Poincar´e-Lindstedt method, is a major achievement in the context of the SCE method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' In the past this method was only able to yield approximate results for the dynamical structure factor in certain cases, usually based on solving another dedicated approximate integral equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' In this paper, we gained the insight that the resonant/secular terms that are being generated are actu- ally non-physical, and just like in the Poincar´e-Linstedt method, should be equated to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' This insight might become useful in other contexts where the SCE method is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' The big surprise here is that, unlike the Poincar´e-Linstedt method, we do not need to solve a new equation for the dynamic decay rate, since the new equa- tion is identical to the integral equation from the static case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' The results reported here have experimental implica- 9 tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Both the structure of thin sheets, as characterised by the static structure factor and the characteristic de- cay rate could be measured directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' There has already been an attempt to measure the structure of crumpled sheets using its optical signature [31], and it is reason- able that with improved imaging techniques leading to faster analysis of dynamic laser speckle patterns [82], the full dynamic structure factor could be effectively sam- pled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' The coincidence of the effective coupling constant and the decay rate due to the quasi-linearity should be checked experimentally and if confirmed could be used to enhance the resolution and accuracy of the measurements by cross validating these two independent sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Thin sheets also exhibit a very distinctive acoustic footprint, often known as crackling noise [27–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Clearly, the dynamic structure factor of the vibrating sheet should be correlated with its acoustic emissions however the cou- pling between the two is known to be nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Accord- ingly, a verified analytical form of the dynamic structure factor could provide a solid staring point that may allow progress in that vein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Last, since the expression for the dynamic structure factor depends on the elastic properties of the sheet and in particular on its elastic moduli, a precise time-resolved measurement of the dynamics of the sheet could provide an indirect measurement of the elastic constants of the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' This might be advantageous when a more di- rect mechanical test may be destructive or might simply modify these properties, while vibrating the sheet intro- duces only a mild perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' An interesting extension of this work would be to con- sider the regime where inertia is important, which is of major interest in the context of wave turbulence in thin sheets [42, 44–53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' In particular, it would be interesting to see whether the growing knowledge on the energy cas- cade could be qualitatively and quantitatively connected to the structure and dynamics of vibrating sheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' Such a step however would require a major technical advance, namely a nontrivial extension of the self-consistent ex- pansion beyond the overdamped regime, or otherwise the development of an alternative methodology to tackle this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE4T4oBgHgl3EQfHgxG/content/2301.04903v1.pdf'} +page_content=' ACKNOWLEDGMENTS The authors wish to thank Arezki Boudaoud for use- ful discussions.' metadata={'source': 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/dev/null +++ b/GtE2T4oBgHgl3EQfTgdU/content/tmp_files/2301.03803v1.pdf.txt @@ -0,0 +1,1470 @@ +Enabling Listening Suspension in the Time Slotted +Channel Hopping Protocol +Gianluca Cena, Stefano Scanzio, and Adriano Valenzano +National Research Council of Italy (CNR–IEIIT), Corso Duca degli Abruzzi 24, I-10129 Torino, Italy +Email: {gianluca.cena, stefano.scanzio, adriano.valenzano}@ieiit.cnr.it +Abstract—Time slotted channel hopping provides reliable and +deterministic communication in IEEE 802.15.4 mesh networks. +Although slotted access is able to lower energy consumption +drastically by reducing the duty cycle of the radio module, it +usually leads to significant idle listening experienced by receivers, +which makes it a sub-optimal solution when ultra low-power +wireless is sought for. +In this paper a listening suspension mechanism is described, +which operates at the MAC layer and is part of a more general +approach aimed at cutting down energy consumption by proac- +tively reducing idle listening. Links can be temporarily disabled, +that convey slow-rate data streams whose characteristics, e.g., the +generation period, are either known in advance to some extent +or can be inferred by traffic inspection. +I. INTRODUCTION +Time slotted channel hopping (TSCH) [1], [2] is an en- +hanced access mechanism for IEEE 802.15.4 [3] that, thanks +to scheduled access to the transmission medium, offers high +reliability, determinism, and, due to the reduced duty cycles, +low power consumption [4], [5]. This enables the connection +of battery-powered devices [6] in application scenarios like +process and factory automation, e.g., to retrofit industrial +plants in order to include new features and functions [7]. The +medium access control (MAC) mechanism of TSCH relies on +a periodically recurring slotframe, to which all nodes must be +precisely synchronized. However, an asynchronous transmis- +sion service is offered to the users of the data-link layer, which +is compatible with the Internet of Things (IoT) and Industrial +IoT (IIoT) paradigms, where the Internet Protocol (IP) is +exploited at the network layer. For this reason, some interesting +IoT solutions have appeared recently, such as 6TiSCH [8], [9], +[10], which rely on TSCH for data transmission. +In most wireless sensor networks (WSN) sensing is per- +formed periodically by motes. Packet transmission in TSCH +occurs cyclically too, but the sampling periods selected by ap- +plications are generally uncorrelated to the slotframe duration. +Therefore, network configuration and setup of applications can +be decoupled, and the design and deployment of distributed +functions in heterogeneous systems (e.g., remote diagnostics +and proactive maintenance) made easier. This means that two +distinct kinds of periodicity can be found in TSCH-based +solutions like 6TiSCH. At the MAC level, the duration Tsf +of the slotframe (in terms of number of slots) is selected +network-wide. At the application level, instead, a number of +978-1-6654-2478-3/21/$31.00 ©2021 IEEE +different sampling periods, denoted Tc,i, may be defined, each +one depending on specific physical dynamics. For example, the +temperature of a liquid in a tank may vary more slowly than a +flow through a pipe. Clearly, Tsf must be chosen so that it is +smaller than the shortest period for variables in the network, +Tsf ≤ mini{Tc,i}. +This is not the only constraint Tsf must satisfy. Assuming +that a single transmission opportunity (i.e., cell) is reserved in +every slotframe for each pair of directly communicating nodes, +to prevent network congestion the overall traffic on any link +must never exceed the slotframe repetition rate, not even when +interference and disturbance cause frame retransmissions. Net- +work parameters should be set so as to take into account the +probability that a transmission attempt fails. In practice, a +safety margin has to be provided to avoid unwanted queuing +phenomena in motes. In the current version of OpenWSN for +OpenMote B, the retry limit R is set to 15, the default slot +duration Tslot to 20 ms, and the slotframe size to 101 slots (i.e., +Tsf = 2.02 s). Thus, sampling rates as low as half a minute +can be quietly selected, network- and link-capacity wise. +In many applications, like condition-based monitoring in- +volved in predictive maintenance, sampling periods can be +much longer than the slotframe. If so, a non-negligible amount +of energy is wasted in TSCH, which can shorten the operating +time significantly when motes are powered on batteries. TSCH +does not introduce any waste of energy on the transmitting side +of a link. In fact, if there is no packet queued for transmission +when a slot becomes available, no attempt is performed and +the cell is simply left unused. By contrast, the receiving side +of the link must be enabled in every cell it is associated, since +it must be ready to get each frame potentially transmitted. +As a consequence the receiver mote must be up and listening +for some portion of the slot, in order to establish whether or +not a transmission is being performed by some nearby peer. +Unfortunately, in many real situations listening to the channel +when a cell is unused is comparable to a frame reception from +the power consumption viewpoint. This phenomenon is known +as idle listening, and is the cause of energy wastes. +A simple solution, when all sampling periods are much +larger than the slotframe duration, is the reconfiguration of +the MAC protocol by increasing the number of slots in the +slotframe (Tslot depends on the physical layer and can hardly +be changed). This introduces a drawback affecting cell ⟨0, 0⟩, +that is the cell placed at slot and channel offsets 0 in the TSCH +matrix, which is used by 6TiSCH for neighbor discovery +This is the author’s version of an article that has been published in this journal. +Changes were made to this version by the publisher prior to publication. +The final version of record is available at https://doi.org/10.1109/WFCS46889.2021.9483595 +Copyright (c) 2021 IEEE. Personal use is permitted. +For any other purposes, permission must be obtained from the IEEE by emailing pubs-permissions@ieee.org. +arXiv:2301.03803v1 [cs.NI] 10 Jan 2023 + +and RPL topology construction activities. In fact, network +reconfiguration (e.g., when some motes are moved or switched +on/off, obstacles are placed between them, nearby sources of +interference and noise appear/disappear) becomes slower as +Tsf increases. In many cases this limitation is not severe since, +from a practical viewpoint, having network reconfiguration +times much shorter than the sampling period makes little +sense. However, when sampling periods are very different +(e.g., when they range from few minutes to several hours) +enlarging Tsf may not suffice. This is because Tsf has to be +selected starting with the highest dynamic packet flow, which +makes this approach useful only in part for the slower streams. +Moreover, excessive increases of Tsf reduce the ability of +nodes to keep their time sources synchronized, which is a +basic requisite for correct time slotting operation. +A more effective solution to this problem, which enables +ultra-low power communication, consists of suspending the +receiving mote when no frame is expected to arrive, so that +no energy is wasted because of idle listening. Such a kind +of technique, named Proactive Reduction of Idle Listening +(PRIL), was introduced in [11]. It is worth noting that acting +this way is not trivial, as the involved motes must know in +advance when transmissions occur, and this is in contrast +with the IoT paradigm where packet transmission requests +are driven by applications. A practical method is to explicitly +drive the listening suspension (LS) of the link receiver by the +transmitter, through the inclusion of suitable sleep commands +in the frames being sent. This approach, which requires slight +changes to the TSCH MAC that do not impair backward +compatibility, was envisaged in [11] by exploiting suitable +information elements, embedded in IEEE 802.15.4 frames, to +put the receiver to sleep. In this way the transmitter holds all +the knowledge about data exchanges, while the receiver simply +obeys to sleep commands. Doing so prevents inconsistencies +between link sides and tangibly improves LS reliability. +Mechanisms employed at the data-link layer to disable +listening in selected cells should be clearly separated from +the policies defined at higher levels to reduce the amount of +idle listening. In this paper we focus on the former issue, +by defining commands to be included in the MAC, as well +as a number of relevant strategies. Our goal is to provide +some hints and bounds on the benefits that LS can offer +in realistic scenarios. Characterization of traffic patterns over +links produced by applications and devices, and the analysis +of optimal LS strategies are left for future work. +The paper is structured as follows: Section II introduces +the problem of reducing idle listening, whereas in Section III +sleep commands and some related strategies are described. +Section IV focuses on power saving achievable with differ- +ent strategies, while some numerical results are reported in +Section V. Finally, Section VI concludes the paper. +II. REDUCING IDLE LISTENING +Information about packet transfer timing can be obtained in +two different ways: it is either provided explicitly by the user, +who instructs the data-link layer about its traffic flow, or the +data-link layer tries to implicitly infer the pattern of exchanges +by continuously analyzing the traffic flowing through the +mote. Both approaches are made harder as intermediate nodes +in a mesh WSN forward packets produced by ascendants +and descendants. In general, combining implicit and explicit +data produces best results. Information about traffic can be +obtained from different layers. For example, the network layer +permits to identify streams coming from or directed to specific +endpoints along a route. This is useful to decompose the traffic +flowing through intermediate relays into separate and simpler +contributions. In a similar way, application processes know the +characteristics of the traffic they produce and consume (e.g., +periodicity, minimum inter-arrival times, deadlines). +Effective implementations consist of two separate and co- +operating elements: PRIL and Link Suspension Entity (LSE). +PRIL concerns network management and mechanisms to es- +tablish dynamically at runtime when and how long a link can +be safely suspended, in such a way that application constraints +(e.g., latency, throughput) are not violated for the packet flow. +PRIL can include: +• Link traffic model to describe traffic patterns conveyed +through the link. For example, a single periodic stream +is completely characterized by its period, while a su- +perposition of periodic streams is identified by the set +of related periods and phase offsets. A sporadic stream +can be defined by means of its average and minimum +interarrival times. +• Interfaces to protocol layers. The MAC layer interface +is of utmost importance and enables interactions with +the LSE. Another interface is offered to application pro- +cesses, and is meant to support explicit LS configuration. +For instance, it allows the selection of a specific traffic +model (periodic, sporadic, etc.), the specification of its +parameters (e.g., the sampling period of sensors), and +the definition of additional constraints known by appli- +cations only (e.g., relative deadlines for sporadic mes- +sages). Dissectors are also envisaged for implicit traffic +characterization. They enable analyses of information in +protocol headers, which are encapsulated in frames sent +over the link (e.g., IP source and destination addresses, +UDP source and destination ports). +IEEE 802.15.4 DL layer +(MAC+TSCH) +IEEE 802.15.4 Physical layer +Network layer +Application layer +LSE +PRIL +Application process (TX side) +<0,0> cell for +management +Non-shared TX +cell for link +<0,0> cell for +management +Non-shared RX +cell for link +Application process (RX side) +Repeated slotframe on air (synchronized time grid) +IEEE 802.15.4 DL layer +(MAC+TSCH) +IEEE 802.15.4 Physical layer +Network layer +Application layer +LSE +PRIL +Fig. 1. Block diagram of PRIL techniques (interacting with the LSE). + +• Algorithms (intelligence), possibly based on machine +learning, to (semi-)autonomously detect traffic patterns +from data mentioned above (i.e., finding a traffic model +and its parameter values fitting the link) and optimally +drive the LS mechanism in the MAC. +Instead, LSE is part of the MAC and consists of: +• LS-related commands, to be embedded in exchanged +frames. They include requests issued by the transmitter +to temporarily disable listening on the receiver and other +elements involved in specific PRIL techniques. +• Protocol finite state machines and local state variables +implementing the LS mechanism and supporting a set of +pre-defined LS strategies. +• Data-link layer primitives to enable interactions between +PRIL and LSE. They should be simple, flexible, and +powerful enough to cope with different strategies. +III. LISTENING SUSPENSION MECHANISM +The LS mechanism for TSCH belongs to the MAC layer. +New primitives have to be defined in the MAC sublayer +management entity (MLME) to interact with the LSE. For +example, a generic MLME-LS-SET can be designed to en- +able/disable LS on a given link, to select a specific strategy, +and to specify the strategy operating parameters. A detailed +specification of new primitives cannot abstract from a stable +definition of traffic models and PRIL interfaces, which are not +available yet. In addition, the behavior of data transmission +primitives of the MAC common part sublayer (MCPS), such +as MCPS-DATA, must also be modified. On the one hand, in +fact, procedures involved in listening suspension have to be +executed when needed (e.g., triggered by either transmission +requests or local timers and counters) while, on the other hand, +traffic needs to be analyzed by PRIL starting from transmission +requests (e.g., by setting function hooks to inspect frames on +the link), to infer its characteristics. +In the current proposal sleep commands are defined as a part +of the LSE and sent by the transmitter to the link receiver in +order to force the latter to temporarily disable listening. They +are conveyed in data frames by using information elements +(IE), that are special fields that can be optionally included +in IEEE 802.15.4 frames to carry ancillary information. IEs +extend the basic protocol but preserve backward compatibility +with existing devices and networks. In defining a new IE for +sleep commands, denoted sleep IE, a streamlined encoding +must be envisaged so as not to undermine energy saving these +mechanisms are intended to achieve. In fact, the inclusion of +IEs in frames increases the number of bits sent on air and, +consequently, the power consumption on both the transmitting +and receiving sides. +Every sleep command only concerns the link where the +including frame was received. This is because any intermediate +mote acting as a relay can have multiple links connected to +its children and/or parent neighbors, and suspension must be +managed independently for each one of them. It is worth +noting that the LS mechanism works on both the link sides, +by either putting them on hold or restoring the conventional +TSCH behavior. This prevents the sender from performing +transmission attempts when listening is temporarily disabled +on the receiver. Of course, suitable variables have to be defined +on the transmitting and receiving sides of every link to keep +track of its LS state (they do not refer to the motes as a whole). +In the following subsections some possible options are +briefly described for LS commands and for general-purpose +LS strategies relying on them. +A. Basic Sleep Command +The simplest way to instruct the receiver about the duration +of the listening suspension is by specifying a sleep time in the +sleep IE. This time is encoded as a small integer, denoted Nslp, +representing the number of slotframes where listening has to +be suspended (note that the final decision about the suspension +is left to the receiver, which ensures backward compatibility). +Upon reception, the link is disabled during the following Nslp +slotframes, then it is enabled again, restoring the conventional +TSCH operation. Fig. 2 shows that the time interval before +the next usable cell of the link is (Nslp + 1)Tsf wide. In the +figure, the number of cells that, in every slotframe, precede +and follow the cell allocated to the link is fixed but arbitrary. +The special value Nslp = 0 informs the receiver that it +must be listening in the next slotframe, and it is roughly +equivalent to not including any sleep command. However, +while the absence of sleep commands has clearly no effects, +setting Nslp = 0 can be used to trigger some specific actions. +For example, when multiple cells are reserved for the same +link in the slotframe, this can enforce an intra-slotframe LS +behavior by skipping all the remaining cells in the current +slotframe. This feature, which is useful when overprovisioning +is exploited for high-capacity links [12], is not considered in +the remaining part of this paper. Values Nslp ≥ 1 are instead +meant to enforce inter-slotframe listening suspension. +Another point concerns the maximum extension of the sleep +time interval. In our opinion 6 bits are enough to encode Nslp +for most real situations, leading to link suspension periods up +to about two minutes. Allocating more bits to Nslp does not +bring noticeable advantages from the point of view of power +consumption. In fact, the shared cell ⟨0, 0⟩ remains always ac- +tive in every slotframe, so that longer sleeping periods for other +cells become irrelevant for energy saving (unless the mote is +involved in several links). In addition, a receiving cell disabled +TXreq R +R +R +R +R +D +I +I +I +D +I +I +I +D +I +I +I +D +I +I +I +D +a) TSCH +b) LS mechanism +Cfr +4 3 +2 +1 +0 +2 1 +0 +0 +0 +6 5 +4 +3 +2 +4 3 +2 +1 +0 +4 3 +Ctx +3 +2 +1 +0 +1 +0 +0 +0 +5 +4 +3 +2 +1 +0 +1 +0 +3 +D +3 +X +X +X +D +1 +X +I +I +D +5 +X +X +X +X +X +D +1 +X +D +3 +Crx +3 +2 +1 +0 +1 +0 +0 +0 +5 +4 +3 +2 +1 +0 +1 +0 +3 +D +n +Data frame +(Nslp) +X Disabled +link +I Idle +listening +Legend: +Slotframe +Slot allocated to the link +TX request (TSCH) +TX request (LS) with period +Too short +Too long +Delay +Enabled TX/RX side +Disabled TX/RX side +Fig. 2. Conventional TSCH vs. LS mechanism. + +by a sleep command cannot be (easily) re-enabled before +the re-activation time is reached. Responsiveness to alarms +and operations taking place on demand, such as firmware +upload or configuration over the air, worsens and may become +unacceptable when too long sleep intervals are enforced. +LS works properly when appropriate values are selected +for Nslp at any time. For example, if the link bears a single +periodic packet stream, Nslp should be chosen to match the +packet generation period. As mentioned before, characteriza- +tion of the traffic model and its parameters is up to PRIL, since +the MAC layer has to be kept as simple as possible. Suitable +primitives can drive intelligence in collecting information: for +instance, some parameters (e.g., deadlines) can be provided +by upper layers while others (e.g., periods) can be inferred by +analyzing the traffic over the link. +B. Periodic LS Strategy +Let us first consider a simple LS strategy fitting the needs of +periodic traffic completely defined by its period Tc. Under the +assumption that only one cell is reserved per link, network +stability requires that Tc +> Tsf (strict order applies, as +transmission errors are unavoidable). Whenever a new frame +transmission request is issued for the link (e.g., through +the MCPS-DATA.request primitive), the transmitter starts a +local frame sleep counter Cfr for the frame. The counter +is initialized to Cfr = ⌊τc⌋ ≥ 1, where τc = Tc/Tsf is +the normalized transmission period. It is worth noting that +rounding down allows to re-enable the link and, in particular, +the receiver listening before a new packet becomes available +on the transmitter, thus reducing both the access time and the +average number of queued packets. Of course, if Tc is not +a multiple of Tsf, idle listening may occasionally take place +when the receiver is re-enabled, as depicted in Fig. 3. +When the frame is sent, the value of Nslp in the sleep IE +is set to Cfr, moreover the sleep command is included in the +frame only if Nslp > 0. Reception of a sleep command causes +the receiver to set its own reception sleep counter Crx to Nslp. +Upon ack frame reception, the Cfr value is also loaded into the +transmission sleep counter Ctx. At the very beginning of every +slot allocated to the link Cfr is decreased by one, provided +Cfr +4 3 +2 +1 +0 +4 +3 +2 +1 +0 +4 +3 +2 +1 +0 +4 3 +2 +1 +Ctx +3 +2 +1 +0 +0 +3 +2 +1 +0 +0 +3 +2 +1 +0 +3 +2 +1 +D +3 +X +X +X +I +D +3 +X +X +X +I +D +3 +X +X +X +D +3 +X +X +Crx +3 +2 +1 +0 +0 +3 +2 +1 +0 +0 +3 +2 +1 +0 +3 +2 +1 +Cfr +4 3 +2 +1 +0 +4 3 +2 +1 +0 +4 3 +2 +1 +0 +4 3 +2 +1 +0 +4 3 +Ctx +3 +2 +1 +0 +3 +2 +1 +0 +3 +2 +1 +0 +3 +2 +1 +0 +3 +D +3 +X +X +X +D +3 +X +X +X +D +3 +X +X +X +D +3 +X +X +X +D +3 +Crx +3 +2 +1 +0 +3 +2 +1 +0 +3 +2 +1 +0 +3 +2 +1 +0 +3 +Cfr +4 3 +2 +1 +0 +4 +3 +2 +1 +0 4 +3 +2 +1 +0 +4 3 +2 +1 +0 +Ctx +3 +2 +1 +0 +0 +3 +2 +1 +0 +3 +2 +1 +0 +3 +2 +1 +0 +D +3 +X +X +X +I +D +3 +X +X +X +D +3 +X +X +X +D +3 +X +X +X +Crx +3 +2 +1 +0 +0 +3 +2 +1 +0 +3 +2 +1 +0 +3 +2 +1 +0 +a) TC = 4 Tsf +b) TC = 4.33 Tsf +c) TC = 4.67 Tsf +Fig. 3. Periodic LS strategy (by varying the actual period). +it is greater than 0 (counters contain non-negative values). +The same is true, at the end of the slot, for Ctx and Crx, +assuming that they were not set in the same slot. Each link +side is assumed to be either enabled when the corresponding +counter (Ctx for the transmitter and Crx for the receiver) is +equal to 0, or disabled otherwise. Since slotframes are kept +synchronized all over the network by TSCH, Ctx and Crx are +updated coherently on both sides of the link and their values +are always the same. Should synchronization between motes +be lost, all sleep counters are reset to 0, hence deactivating +any ongoing listening suspension (and restoring conventional +TSCH operation). +Frame delivery for a given transmission request may be +occasionally delayed as shown, for example, to the fourth +request in Fig. 2.b, because of a prior command with an +excessively large sleeping period. This event is dealt with by +the LS mechanism in this way: if, for any reasons, the frame +is not sent at the first opportunity following the transmission +request and it has to wait in the local queue, Cfr is correctly +decreased by the LS algorithm. Consequently, if the sleep +command eventually reaches the receiver, the re-enable instant +is not affected. This also copes with the relevant case of +failed transmission attempts (detected by expiration of the ack +timeout), where Cfr is decreased by one on each retry. In this +way, transmission errors affect the current packet in the same +way as TSCH, whereas the next packet does not suffer from +additional delays with respect to the case when no errors occur. +To help recovering from queuing phenomena, which may +take place for instance when long bursts of errors are experi- +enced, the sleep command is skipped if more than one frame +concerning the same link is pending in the transmission queue, +thus preventing the link capacity from being throttled in these +conditions (which could lead to buffer overruns). +This strategy can also be used for quasi-periodic traffic, +where packet generation takes place cyclically at a nominal +rate but release times are subject to non-negligible jitters. If +the time interval between two consecutive packet transmission +requests exceeds Tc, receiver idle listening occurs. Conversely, +when the interval is shorter than Tc, the early packet experi- +ences some delay since the transmission can start only when +the link is re-enabled (i.e., Ctx = 0 and Crx = 0). +C. Responsiveness +Responsiveness in TSCH is limited by its MAC mechanism, +which is based on a continuously repeating slotframe [13]. +As pointed out in [14], in absence of errors due to frame +corruption on air, one-way transmission latency consists of +two elements, namely access delay and transmission time. +Access delay is the time the transmitter spends waiting for +the slot assigned to the link, and is the largest contribution. +Under the assumption that, at any time, at most one frame is +queued for transmission for the link, it can be modeled as a +random variable uniformly distributed between 0 and Tsf. The +transmission time includes the offset at the beginning of the +slot (macTsTxOffset) as well as the time taken for transmitting +the data frame on air. Since it is always strictly shorter than + +Tslot we will neglect it in the following. In this way the worst- +case transmission latency Twc is approximately the same as +the maximum access delay when no errors occur. +The access delay may grow sensibly when sleep commands +are exploited. With respect to conventional TSCH the period +between usable cells for a link (and access delays) increases +from Tsf to (Nslp + 1)Tsf, while energy spent by the receiver +for listening to the channel decreases by about the same factor. +With strictly periodic patterns of packets and a proper +selection of Nslp (ensuring that the link is re-enabled just be- +fore a new packet becomes available for transmission) delays +are roughly the same as TSCH. However, packets generated +sporadically can experience access delays up to (Nslp + 1)Tsf +when they become ready just after the link suspension. To +improve link responsiveness for sporadic packets, Nslp can +be set based on their relative deadline Td. In particular, to +ensure that Twc ≤ Td, again in the optimistic case of an ideal +channel without errors, Nslp has to be set so that Nslp ≤ τd−1, +where τd = Td/Tsf is the normalized deadline. For example, +with the network parameters we considered, Nslp must not +exceed 13 if latency must be shorter than 30 s. If the period +is sensibly larger than the deadline (Tc ≫ Td), the link is re- +enabled too early most times and the LS mechanism behaves +less effectively. +D. Slow Periodic LS Strategy +In slow periodic streams the period may exceed the maxi- +mum value allowed for the sleep command (64Tsf for Nslp 6- +bit encoding). In such cases the LSE takes care of issuing sleep +commands automatically to repeatedly restart the sleeping +period in the receiver and make the suspension last the desired +amount of time. Nslp is equal to its maximum value (63) in all +the sleep commands in the sequence except for the last one, +which is set to (⌊τc⌋−1) mod 64 and re-enables the link at the +intended time. Overall, the sequence includes ⌈τc/64⌉ frames. +For example, if Tc is equal to 10 minutes (i.e., Nslp = 296), +five frames are sent per packet, spaced by 64Tsf: the first +one is the data frame, which includes the packet and a sleep +command with Nslp = 63, followed by three empty frames +(with no payload) where Nslp = 63, and one final (empty) +frame where Nslp = 39. Should a new transmission request +be issued before the above procedure is completed, the course +of action is interrupted by the transmitter and the new data +frame is sent as soon as the link is re-enabled. +The impact of this approach on power consumption is +relatively low and is roughly equivalent to the energy needed +for sending and receiving one frame every 64 slotframes (about +two minutes). Most of these frames contain no actual data and +only deliver sleep commands to the receiver. To further reduce +power consumption a specific short format can be envisaged +for empty sleep frames that only carry the sleep IE. A new +MAC sleep command frame can be defined to this purpose but +other solutions are possible as well. Note that from the energy +viewpoint this contribution is rather negligible when compared +to cell ⟨0, 0⟩, which is active in every slotframe, nonetheless +it improves network responsiveness by imposing a reasonable +upper bound Twc = 64Tsf on transmission latency. +E. Extended Sleep Command +Energy consumption can be reduced without impairing re- +sponsiveness too much, by means of an extended version of the +sleep command we call xsleep. The relevant IE includes two +parameters: Nslp and Nsnz. Nslp is the number of slotframes +the link must be kept disabled. It specifies the nominal duration +of the sleeping period and is managed the same way as in +basic sleep commands. However, the range of allowed values +for the sleep time can be increased noticeably in this case +without latency is consequently worsened. For instance, Nslp +can be encoded with 12 bits and the sleeping period can last +up to 4096Tsf (more than two hours). Selection of larger field +sizes to enable longer sleeping periods is clearly possible. +The snooze time, denoted Nsnz, specifies how often the link +must be temporarily re-enabled (awakened) during the nominal +sleeping period, in order to flush packets sporadically gener- +ated in the meanwhile and left pending in the transmission +buffer. In this case, on correct reception of the xsleep command +an iterative procedure is started, where the link is disabled for +Nsnz consecutive slotframes before being awakened for one +slotframe. This behavior is automatically repeated by the LSE +until the end of the nominal suspension period (as specified +by Nslp) is reached. Clearly, Nsnz must be strictly lower than +Nslp. Nsnz encoded with 6 bits (e.g., Nsnz ≤ 63) provides +upper bounds for the access time similar to the basic sleep +command. In practice, 3 bytes are enough to encode the xsleep +command into the content of a sleep IE. +Identification of slotframes where the link must be awak- +ened does not consider the instant when the sleep command +reaches the receiver, but proceeds from the end of the sleeping +period moving backwards. In other words, frames can be sent +only when Ctx mod (Nsnz + 1) = 0. In this way delays in +setting Crx caused, for instance, by transmission errors do +not affect the wake-up synchronization on the two link sides. +For example, if Nslp = 58 (Tc ≃ 120 s) and Nsnz = 13 +(Td ≃ 30 s), the receiver temporarily wakes up at the 3-rd, +17-th, 31-st, and 45-th slotframes after the command is issued, +and it is re-enabled at the 59-th slotframe. +S +Cfr +10 9 +8 +7 +6 +5 +4 +3 +2 +1 +0 +10 9 +8 +7 +6 +5 +4 +3 +Ctx +9 +8 +7 +6 +5 +4 +3 +2 +1 +0 +9 +8 +7 +6 +5 +4 +3 +D93 +X +I +X +X +X +D +X +X +X +D93 +X +I +X +X +X +I +Crx +9 +8 +7 +6 +5 +4 +3 +2 +1 +0 +9 +8 +7 +6 +5 +4 +3 +a) Tc = 10 Tsf , Td = 4 Tsf +S +S +S +Cfr +10 9 +8 +7 +6 +5 +4 +3 +2 +1 +0 +10 9 +8 +7 +6 +5 +4 +3 +Ctx +9 +8 +7 +6 +5 +4 +0 +0 +0 +0 +9 +8 +7 +6 +5 +4 +3 +D93 +X +I +X +X +X +D00 +D +D +I +D93 +X +I +X +X +X +I +Crx +9 +8 +7 +6 +5 +4 +0 +0 +0 +0 +9 +8 +7 +6 +5 +4 +3 +b) Tc = 10 Tsf , Td = 4 Tsf (queuing taking place) +Dsz +Data frame [period.] +(Nslp, Nsnz) +X Disabled +link +I Idle +listening +Legend: +Delay +D00 +Data frame [sporad.] +(reset LS) +Sporadic requests +Delay +D Data frame +[sporad.] +LS is reset by transmitter +Link wakes up +Fig. 4. Extended periodic LS strategy (with and without queuing). + +Fig. 4 assumes Nslp = 9 and Nsnz = 3, and shows that +awakening occurs when counters equal 8 and 4. If needed, +the xsleep command can be aborted at those time instants, for +instance when queuing occurs as in Fig. 4b. If, for whatever +reason, a number of sporadic packets become pending in +the transmission buffer (e.g., as a consequence of prolonged +interference on air), the transmitting LSE sends a new sleep +command to reset the behavior of the receiver. A special xsleep +command with Nslp = 0 and Nsnz = 0 can be defined to this +purpose, which reverts the behavior to conventional TSCH. +Then, queued packets can be drained at the maximum link +speed and LS operation resumed when the queue is empty. +LS strategies that rely on xsleep trade energy for responsive- +ness. In fact, the worst-case access delay (and Twc as well) +is equal to the snoozing period (Nsnz + 1)Tsf. Selecting a +low value for Nsnz reduces Twc, but also increases power +consumption on the receiver side consequently. Because of +the large values allowed for Nslp (latency now only depends +on Nsnz) the time between transmissions of xsleep commands +can be as large as about two hours, which makes energy +consumption completely negligible on the transmitter side. +F. Extended Periodic LS Strategy +This strategy is suitable for a traffic pattern consisting of +the superposition of a quasi-periodic stream with (average) +period Tc and sporadic streams with relative deadline Td. For +simplicity we assume that sporadic packets, which should be +served promptly, are issued in consequence of rare events. In +this case, xsleep commands are conveyed inside data frames +carrying periodic packets, and Nsnz and Nslp are selected so +that Nsnz = ⌊τd⌋ − 1 and Nslp = ⌊τc⌋ − 1. +Such a strategy closely resembles periodic LS, but the re- +ceiver wakes up regularly during the sleeping period to ensure +bounded transmission latency to sporadic packets. Since xsleep +enables large sleeping times, high efficiency can be reached. +Moreover, should the inferred period be updated by PRIL, +changes can be enforced by sending a new xsleep command +as soon as the receiver awakens. +IV. PERFORMANCE ANALYSIS +Energy consumed on the transmitting and receiving link +sides can be evaluated for conventional TSCH and LS strate- +gies. In the following we assume that no transmission error +occurs, to simplify the analysis and focus on the intrinsic +energy saving capability of the proposed solutions. +A. Power Consumption Model +Let Etxd and Erxd be the amounts of energy spent for +performing a single data frame transmission and reception at- +tempt, respectively. In the following, we model these quantities +by assuming that they depend on the frame size linearly. In +particular, Etxd = Etx0+etxB·Lphy, where Lphy is the overall +number of bytes transmitted by the physical layer, etxB is +the energy required to send one byte, and Etx0 is a constant +amount of energy additionally spent for every transmission +attempt. For instance, for OpenMote B boards running Open- +WSN and communicating through 6TiSCH, Etx0 = 7 µJ and +etxB = 2 µJ/B (see plots of Fig. 4 in [2]), thus the largest +fraction of power directly depends on the number of bytes +transmitted on air. +A very similar relation holds for reception. From the same +set of experimental data we obtain Erx0 = 65 µJ and erxB = +1.3 µJ/B. Unsurprisingly, the energy per byte is lower than +for transmission. However, a non negligible amount of energy +is spent because listening must be enabled for some time +preceding the instant when transmission is expected to begin. +Erx0 is a mean value, and fluctuations can be experienced +depending on the alignment precision of the transmitter and +receiver time grids at any given time. +Energy needed to send and receive the ack frame (33 B) +is Etxa = 106 µJ and Erxa = 79 µJ, respectively. This also +includes operations to finalize a frame exchange besides the +transmission/reception on air. Finally, the energy spent every +time idle listening occurs for the cell is Elis = 138 µJ. Elis +is about twice Erx0, since the listening interval is centered +around the expected start of frame transmission. +In absence of frame transmission and reception the mote +drains about 31.4 mW (i.e., 628 µJ in every slot). This is due +to the basic activity of the operating system, protocol stack +and application processes, and concerns all board components +besides the microcontroller. It is worth observing that such a +high value is likely due to the fact that both the OpenWSN +software and the OpenMote B hardware have not been opti- +mized yet. Hopefully, this constant consumption will be low- +ered dramatically when industry-grade devices based on this +technology are designed and implemented. For these reasons +this contribution in not considered in the following analysis, +which focuses only on energy spent for communication. +B. Conventional TSCH +Let Λsf = 1/Tsf and Λc = 1/Tc be the slotframe repeti- +tion rate and the packet transmission rate, respectively. The +transmitter power consumption for a packet stream sent over +a conventional TSCH link is +Pt = (Etxd + Erxa)Λc. +(1) +When only reception (and acknowledgment) of frames is +considered, for the receiver we have +Pr0 = (Erxd + Etxa)Λc. +(2) +Pr0 equals the power spent by the receiver when an oracle- +based LS strategy is adopted. Thanks to the oracle, listening is +enabled only when the cell conveys a frame, without the need +to obtain any information from the sender by means of sleep +commands embedded in IEs. As a consequence, Pr0 provides +a lower bound to power consumption for a given stream. The +overall receiver consumption is given by +Pr = Pr0 + Elis[Λsf − Λc], +(3) + +where the rightmost term refers to idle listening, which occurs +when the cell allocated to the link is not used for data (hence, +the difference between Λsf and Λc). +In a TSCH/6TiSCH network operating correctly, additional +frame exchanges are performed automatically in every slot- +frame. This is the case of transmission/reception in the shared +cell ⟨0, 0⟩, which is meant for network management activities. +Despite their contribution to power consumption, they are not +taken into account here because they are not correlated with +data streams produced by applications and affected by the LS +mechanism. Roughly speaking, power consumption for these +frame exchanges can be expressed as P0 = ¯E⟨0,0⟩Λsf, where +¯E⟨0,0⟩ is the average energy spent by the mote to carry out one +(non-confirmed) transmission/reception attempt in cell ⟨0, 0⟩. +In practice, the value of ¯E⟨0,0⟩ does not differ from Etxd, +Erxd, and Elis significantly. +C. Periodic LS Strategy +In periodic LS a basic sleep command is included in every +data frame where Nslp = ⌊τc⌋ − 1. For example, if Tc = +30 s then Nslp = 13. In the following we will assume that +Tc can be inferred precisely by the intelligence of the PRIL +module. Obviously, this represents the best case for the LS +mechanism, and enables the computation of an upper bound +for improvements this approach can offer. +Power consumption on the transmitting side increases +slightly because of the sleep command +P P +t = Pt + LslpetxBΛc, +(4) +where Lslp is the size in bytes of the encapsulating IE (Lslp = +3 B in our case). For typical frame size (90 B) the increase is +negligible (∼ 3%). +Data frames that carry sleep commands suspend listening +for Nslp slotframes when they are received correctly. Thus +every frame prevents idle listening from occurring Nslp = +⌊τc⌋ − 1 times. +Hence +P P +r = Pr0 + LslperxBΛc + Elis [Λsf − ⌊τc⌋ Λc] += Pr + LslperxBΛc − Elis [⌊τc⌋ − 1] Λc. +(5) +Again, the consumption increase due to sleep IEs is typically +negligible. Instead, the contribution due to idle listening (right- +most term in the first line of the equation) is considerably +lower than TSCH, because the multiplicative factor for Elis +is always smaller than Λc, and equal to 0 if Tc is a multiple +of Tsf. In the latter case the difference in terms of power +consumption between the oracle and the periodic LS strategy +is constant and equal, overall, to Lslp(etxB + erxB)Λc, which, +for Lslp = 3 B and Tc = 30.3 s, means 0.33 µW only. +D. Slow Periodic LS Strategy +For slow packet rates (Tc > 64Tsf) empty sleep frames are +automatically sent by the LSE on the link transmitter to enlarge +the sleeping period without worsening responsiveness. Power +consumption on both sides can be derived from the periodic +case by including this additional contribution +P slP +t += P P +t + EtxenempΛc, +(6) +P slP +r += P P +r + ErxenempΛc, +(7) +where nemp = ⌈τc/64⌉ − 1 is the number of empty sleep +frames inserted after every data frame. Because of our as- +sumptions, nempΛc < Λsf/64, which sets an upper bound to +the additional consumption needed to keep LS alive. +E. Extended Periodic LS Strategy +We have shown that when the link conveys both sporadic +streams of packets with relative deadline Td and a periodic +stream with period Tc the best approach is to adopt xsleep +commands where Nslp = ⌊τc⌋ − 1 and Nsnz = ⌊τd⌋ − 1. With +this strategy, the receiver wakes up temporarily nwup times +before listening is re-enabled, and nwup = ⌈(Nslp+1)/(Nsnz+ +1)⌉ − 1 = ⌈⌊τc⌋/⌊τd⌋⌉−1. For example, if Tc = 600 s and +Td = 30 s, then Nslp = 296, Nsnz = 13, and the link is +awakened 21 times during every sleeping period. For the sake +of simplicity, we do not consider cases where Tc > 4096Tsf. +Power consumption on the transmitting side is almost the +same as for the periodic strategy +P xP +t += Pt + LxslpetxBΛc, +(8) +however the size Lxslp of the sleep IE is slightly larger than +the basic sleep command (Lxslp = 5 B). +On the receiving side +P xP +r += Pr0 + LxslperxBΛc + Elis [Λsf − (⌊τc⌋ − nwup) Λc] += P P +r +(Lxslp−Lslp)erxBΛc+ElisnwupΛc. +(9) +As expected, power consumption is higher than for basic +sleep commands with the same value of Nslp, because of +the additional contribution of idle listening introduced by the +awakening mechanism to ensure better responsiveness. +V. PERFORMANCE COMPARISON +To compare strategies we assume that both periodicity and +relative deadlines are known (or correctly estimated by PRIL). +The metric we consider is the power required to deal with +a packet stream that transfers application data on a single +link. Only contributions due to communication are taken into +account, as other kinds of power drain by motes can be reduced +considerably by exploiting careful hardware and software +optimization. The frame size Lphy is selected equal to 90 B, to +resemble ICMP echo requests generated by ping commands +in real networks. Three different packet transmission rates Λc +were considered, corresponding to periods equal to 30 s, 2 +minutes and 10 minutes respectively, while Etxe = 87 µJ and +Erxe = 117 µJ for the slow periodic LS strategy (labeled +“Basic (slow)” in Table I). These values were obtained by +considering empty sleep frames encoded with 40 B. +Results are reported in Table I. TSCH shows noticeably +higher consumption than the oracle on the receiving side. +Tangible saving is possible with the periodic LS strategy + +TABLE I +POWER CONSUMPTION ON BOTH SIDES OF THE LINK AND WORST-CASE +TRANSMISSION LATENCY Twc FOR THE DIFFERENT LS STRATEGIES BY +VARYING PACKET GENERATION PERIOD Tc AND RELATIVE DEADLINE Td. +Tc / Td +Strategy +Nslp +Nsnz +Twc +Pt +Pr +(s) / (s) +(s) +(µW) +(µW) +30 / – +Oracle +– +– +2.02 +8.8667 +9.6000 +30 / – +TSCH +– +– +2.02 +8.8667 +73.3168 +30 / – +Basic +13 +– +28.28 +9.0667 +13.6468 +120 / – +Oracle +– +– +2.02 +2.2167 +2.4000 +120 / – +TSCH +– +– +2.02 +2.2167 +69.5668 +120 / – +Basic +58 +– +119.18 +2.2667 +2.8993 +120 / 10 +eXtended +58 +3 +8.08 +2.3000 +19.0210 +120 / 30 +eXtended +58 +13 +28.28 +2.3000 +7.5210 +600 / – +Oracle +– +– +2.02 +0.4433 +0.4800 +600 / – +TSCH +– +– +2.02 +0.4433 +68.5668 +600 / – +Basic (slow) +296 +– +129.28 +1.0333 +1.2733 +600 / 10 +eXtended +296 +3 +8.08 +0.4600 +17.5177 +600 / 30 +eXtended +296 +13 +28.28 +0.4600 +5.3277 +600 / 120 +eXtended +296 +58 +119.18 +0.4600 +1.6477 +based on sleep commands (“Basic”) and improves with larger +periods. Unfortunately, responsiveness is also impaired, and +this could be a problem if sporadic packets were occasionally +conveyed on the same link. The periodic LS strategy based on +extended sleep commands (“eXtended”) is a reasonable trade- +off and is suited to those cases where the period is much larger +than the relative deadline. By means of two degrees of freedom +(sleep and snooze times), it achieves interesting power saving +yet keeping the MAC complexity at a minimum. In general, it +retains the expected low-power consumption of TSCH on the +transmitting side, whereas responsiveness and energy saving +on the receiver side can be balanced on a per-link basis. +VI. CONCLUSION +By layering time slotting and channel hopping atop IEEE +802.15.4, TSCH offers high reliability, determinism, and low +power consumption, which makes it a good candidate when +mesh WSNs compliant to the IIoT paradigm have to be +deployed in industrial scenarios. Unfortunately, the access +technique of TSCH (but the same also holds for similar +protocols that rely on scheduled access) implies non-negligible +power consumption by receivers due to idle listening, i.e., +when the receiving circuitry is switched on but no one is +sending frames on air. +Details about the PRIL proposal have been presented in +this paper, in order to reduce idle listening phenomena by +exploiting available information about data exchanges. In +particular, we focused on the listening suspension (LS) mech- +anism, to be incorporated in the TSCH MAC, that enables the +receiving side of a link to be put on sleep on demand. Besides +basic strategies, suitable for periodic packet streams, extended +solutions have also been introduced, which offer a reasonable +trade-off between power consumption and responsiveness for +sporadically generated packets thanks to the ability to repeat- +edly snooze the receiver. +Formulas have been derived for power consumption estima- +tion for both sides of a link in 6TiSCH, which were used to +evaluate the behavior of a link in realistic operating conditions, +by leveraging the energy model of a real device (OpenMote B +running OpenWSN). Results confirm that the amount of saved +energy is worth the adoption of LS. Future activities in this +area will focus on the evaluation of LS mechanisms through +simulation of realistic (and more complex) traffic patterns and +the implementation of PRIL algorithms to automatically infer +fitting traffic models. +REFERENCES +[1] D. De Guglielmo, B. Al Nahas, S. Duquennoy, T. Voigt, and G. Anas- +tasi, “Analysis and Experimental Evaluation of IEEE 802.15.4e TSCH +CSMA-CA Algorithm,” IEEE Transactions on Vehicular Technology, +vol. 66, no. 2, pp. 1573–1588, 2017. +[2] S. Scanzio, M. G. Vakili, G. Cena, C. G. Demartini, B. Montrucchio, +A. Valenzano, and C. Zunino, “Wireless Sensor Networks and TSCH: +A Compromise Between Reliability, Power Consumption, and Latency,” +IEEE Access, vol. 8, pp. 167 042–167 058, 2020. +[3] IEEE, “IEEE Standard for Low-Rate Wireless Networks,” IEEE Std +802.15.4-2020 (Revision of IEEE Std 802.15.4-2015), pp. 1–800, 2020. +[4] F. Moreno-Cruz, V. Toral-L´opez, A. Escobar-Molero, V. U. Ru´ız, +A. Rivadeneyra, and D. P. Morales, “treNch: Ultra-Low Power Wireless +Communication Protocol for IoT and Energy Harvesting,” Sensors, +vol. 20, no. 21, 2020. [Online]. Available: https://www.mdpi.com/ +1424-8220/20/21/6156 +[5] V. Venkateswaran and I. O. Kennedy, “How to sleep, control and +transfer data in an energy constrained wireless sensor network,” in +2013 51st Annual Allerton Conference on Communication, Control, and +Computing (Allerton), 2013, pp. 839–846. +[6] H. Yetgin, K. T. K. Cheung, M. El-Hajjar, and L. H. Hanzo, “A Survey +of Network Lifetime Maximization Techniques in Wireless Sensor +Networks,” IEEE Communications Surveys Tutorials, vol. 19, no. 2, pp. +828–854, 2017. +[7] Z. Zhang, A. Mehmood, L. Shu, Z. Huo, Y. Zhang, and M. Mukherjee, +“A Survey on Fault Diagnosis in Wireless Sensor Networks,” IEEE +Access, vol. 6, pp. 11 349–11 364, 2018. +[8] P. Thubert, “An Architecture for IPv6 over the TSCH mode of IEEE +802.15.4,” IETF Std draft-ietf-6tisch-architecture-30, pp. 1–60, Nov +2020. +[9] M. R. Palattella, P. Thubert, X. Vilajosana, T. Watteyne, Q. Wang, and +T. Engel, 6TiSCH Wireless Industrial Networks: Determinism Meets +IPv6. +Cham: Springer International Publishing, 2014, pp. 111–141. +[10] X. Vilajosana, T. Watteyne, T. Chang, M. Vuˇcini´c, S. Duquennoy, and +P. Thubert, “IETF 6TiSCH: A Tutorial,” IEEE Communications Surveys +Tutorials, vol. 22, no. 1, pp. 595–615, 2020. +[11] S. Scanzio, G. Cena, A. Valenzano, and C. Zunino, “Energy Saving in +TSCH Networks by Means of Proactive Reduction of Idle Listening,” +in Ad-Hoc, Mobile, and Wireless Networks, L. A. Grieco, G. Boggia, +G. Piro, Y. Jararweh, and C. Campolo, Eds. +Cham: Springer Interna- +tional Publishing, 2020, pp. 131–144. +[12] G. Cena, S. Scanzio, L. Seno, A. Valenzano, and C. Zunino, “Energy- +Efficient Link Capacity Overprovisioning In Time Slotted Channel +Hopping Networks,” in 16th IEEE International Conference on Factory +Communication Systems (WFCS 2020), 2020, pp. 1–8. +[13] I. Juc, O. Alphand, R. Guizzetti, M. Favre, and A. Duda, “Energy +consumption and performance of IEEE 802.15.4e TSCH and DSME,” +in 2016 IEEE Wireless Communications and Networking Conference, +2016, pp. 1–7. +[14] G. Cena, C. G. Demartini, M. Ghazi Vakili, S. Scanzio, A. Valen- +zano, and C. Zunino, “Evaluating and Modeling IEEE 802.15.4 +TSCH Resilience against Wi-Fi Interference in New-Generation Highly- +Dependable Wireless Sensor Networks,” Ad Hoc Networks, vol. 106, p. +102199, Sep. 2020. + diff --git a/GtE2T4oBgHgl3EQfTgdU/content/tmp_files/load_file.txt b/GtE2T4oBgHgl3EQfTgdU/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..38e7906a80affc8c971a99f99d0960fee1ce10f6 --- /dev/null +++ b/GtE2T4oBgHgl3EQfTgdU/content/tmp_files/load_file.txt @@ -0,0 +1,933 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf,len=932 +page_content='Enabling Listening Suspension in the Time Slotted Channel Hopping Protocol Gianluca Cena, Stefano Scanzio, and Adriano Valenzano National Research Council of Italy (CNR–IEIIT), Corso Duca degli Abruzzi 24, I-10129 Torino, Italy Email: {gianluca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='cena, stefano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='scanzio, adriano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='valenzano}@ieiit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='cnr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='it Abstract—Time slotted channel hopping provides reliable and deterministic communication in IEEE 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='4 mesh networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Although slotted access is able to lower energy consumption drastically by reducing the duty cycle of the radio module, it usually leads to significant idle listening experienced by receivers, which makes it a sub-optimal solution when ultra low-power wireless is sought for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' In this paper a listening suspension mechanism is described, which operates at the MAC layer and is part of a more general approach aimed at cutting down energy consumption by proac- tively reducing idle listening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Links can be temporarily disabled, that convey slow-rate data streams whose characteristics, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=', the generation period, are either known in advance to some extent or can be inferred by traffic inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' INTRODUCTION Time slotted channel hopping (TSCH) [1], [2] is an en- hanced access mechanism for IEEE 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='4 [3] that, thanks to scheduled access to the transmission medium, offers high reliability, determinism, and, due to the reduced duty cycles, low power consumption [4], [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' This enables the connection of battery-powered devices [6] in application scenarios like process and factory automation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=', to retrofit industrial plants in order to include new features and functions [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' The medium access control (MAC) mechanism of TSCH relies on a periodically recurring slotframe, to which all nodes must be precisely synchronized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' However, an asynchronous transmis- sion service is offered to the users of the data-link layer, which is compatible with the Internet of Things (IoT) and Industrial IoT (IIoT) paradigms, where the Internet Protocol (IP) is exploited at the network layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' For this reason, some interesting IoT solutions have appeared recently, such as 6TiSCH [8], [9], [10], which rely on TSCH for data transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' In most wireless sensor networks (WSN) sensing is per- formed periodically by motes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Packet transmission in TSCH occurs cyclically too, but the sampling periods selected by ap- plications are generally uncorrelated to the slotframe duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Therefore, network configuration and setup of applications can be decoupled, and the design and deployment of distributed functions in heterogeneous systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=', remote diagnostics and proactive maintenance) made easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' This means that two distinct kinds of periodicity can be found in TSCH-based solutions like 6TiSCH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' At the MAC level, the duration Tsf of the slotframe (in terms of number of slots) is selected network-wide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' At the application level, instead, a number of 978-1-6654-2478-3/21/$31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='00 ©2021 IEEE different sampling periods, denoted Tc,i, may be defined, each one depending on specific physical dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' For example, the temperature of a liquid in a tank may vary more slowly than a flow through a pipe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Clearly, Tsf must be chosen so that it is smaller than the shortest period for variables in the network, Tsf ≤ mini{Tc,i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' This is not the only constraint Tsf must satisfy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Assuming that a single transmission opportunity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=', cell) is reserved in every slotframe for each pair of directly communicating nodes, to prevent network congestion the overall traffic on any link must never exceed the slotframe repetition rate, not even when interference and disturbance cause frame retransmissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Net- work parameters should be set so as to take into account the probability that a transmission attempt fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' In practice, a safety margin has to be provided to avoid unwanted queuing phenomena in motes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' In the current version of OpenWSN for OpenMote B, the retry limit R is set to 15, the default slot duration Tslot to 20 ms, and the slotframe size to 101 slots (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=', Tsf = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='02 s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Thus, sampling rates as low as half a minute can be quietly selected, network- and link-capacity wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' In many applications, like condition-based monitoring in- volved in predictive maintenance, sampling periods can be much longer than the slotframe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' If so, a non-negligible amount of energy is wasted in TSCH, which can shorten the operating time significantly when motes are powered on batteries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' TSCH does not introduce any waste of energy on the transmitting side of a link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' In fact, if there is no packet queued for transmission when a slot becomes available, no attempt is performed and the cell is simply left unused.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' By contrast, the receiving side of the link must be enabled in every cell it is associated, since it must be ready to get each frame potentially transmitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' As a consequence the receiver mote must be up and listening for some portion of the slot, in order to establish whether or not a transmission is being performed by some nearby peer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Unfortunately, in many real situations listening to the channel when a cell is unused is comparable to a frame reception from the power consumption viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' This phenomenon is known as idle listening, and is the cause of energy wastes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' A simple solution, when all sampling periods are much larger than the slotframe duration, is the reconfiguration of the MAC protocol by increasing the number of slots in the slotframe (Tslot depends on the physical layer and can hardly be changed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' This introduces a drawback affecting cell ⟨0, 0⟩, that is the cell placed at slot and channel offsets 0 in the TSCH matrix, which is used by 6TiSCH for neighbor discovery This is the author’s version of an article that has been published in this journal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Changes were made to this version by the publisher prior to publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' The final version of record is available at https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='1109/WFCS46889.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='9483595 Copyright (c) 2021 IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Personal use is permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' For any other purposes, permission must be obtained from the IEEE by emailing pubs-permissions@ieee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='03803v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='NI] 10 Jan 2023 and RPL topology construction activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' In fact, network reconfiguration (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=', when some motes are moved or switched on/off, obstacles are placed between them, nearby sources of interference and noise appear/disappear) becomes slower as Tsf increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' In many cases this limitation is not severe since, from a practical viewpoint, having network reconfiguration times much shorter than the sampling period makes little sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' However, when sampling periods are very different (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=', when they range from few minutes to several hours) enlarging Tsf may not suffice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' This is because Tsf has to be selected starting with the highest dynamic packet flow, which makes this approach useful only in part for the slower streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Moreover, excessive increases of Tsf reduce the ability of nodes to keep their time sources synchronized, which is a basic requisite for correct time slotting operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' A more effective solution to this problem, which enables ultra-low power communication, consists of suspending the receiving mote when no frame is expected to arrive, so that no energy is wasted because of idle listening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Such a kind of technique, named Proactive Reduction of Idle Listening (PRIL), was introduced in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' It is worth noting that acting this way is not trivial, as the involved motes must know in advance when transmissions occur, and this is in contrast with the IoT paradigm where packet transmission requests are driven by applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' A practical method is to explicitly drive the listening suspension (LS) of the link receiver by the transmitter, through the inclusion of suitable sleep commands in the frames being sent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' This approach, which requires slight changes to the TSCH MAC that do not impair backward compatibility, was envisaged in [11] by exploiting suitable information elements, embedded in IEEE 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='4 frames, to put the receiver to sleep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' In this way the transmitter holds all the knowledge about data exchanges, while the receiver simply obeys to sleep commands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Doing so prevents inconsistencies between link sides and tangibly improves LS reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Mechanisms employed at the data-link layer to disable listening in selected cells should be clearly separated from the policies defined at higher levels to reduce the amount of idle listening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' In this paper we focus on the former issue, by defining commands to be included in the MAC, as well as a number of relevant strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Our goal is to provide some hints and bounds on the benefits that LS can offer in realistic scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Characterization of traffic patterns over links produced by applications and devices, and the analysis of optimal LS strategies are left for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' The paper is structured as follows: Section II introduces the problem of reducing idle listening, whereas in Section III sleep commands and some related strategies are described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Section IV focuses on power saving achievable with differ- ent strategies, while some numerical results are reported in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Finally, Section VI concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' REDUCING IDLE LISTENING Information about packet transfer timing can be obtained in two different ways: it is either provided explicitly by the user, who instructs the data-link layer about its traffic flow, or the data-link layer tries to implicitly infer the pattern of exchanges by continuously analyzing the traffic flowing through the mote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Both approaches are made harder as intermediate nodes in a mesh WSN forward packets produced by ascendants and descendants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' In general, combining implicit and explicit data produces best results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Information about traffic can be obtained from different layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' For example, the network layer permits to identify streams coming from or directed to specific endpoints along a route.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' This is useful to decompose the traffic flowing through intermediate relays into separate and simpler contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' In a similar way, application processes know the characteristics of the traffic they produce and consume (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=', periodicity, minimum inter-arrival times, deadlines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Effective implementations consist of two separate and co- operating elements: PRIL and Link Suspension Entity (LSE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' PRIL concerns network management and mechanisms to es- tablish dynamically at runtime when and how long a link can be safely suspended, in such a way that application constraints (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=', latency, throughput) are not violated for the packet flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' PRIL can include: Link traffic model to describe traffic patterns conveyed through the link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' For example, a single periodic stream is completely characterized by its period, while a su- perposition of periodic streams is identified by the set of related periods and phase offsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' A sporadic stream can be defined by means of its average and minimum interarrival times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Interfaces to protocol layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' The MAC layer interface is of utmost importance and enables interactions with the LSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Another interface is offered to application pro- cesses, and is meant to support explicit LS configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' For instance, it allows the selection of a specific traffic model (periodic, sporadic, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' ), the specification of its parameters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=', the sampling period of sensors), and the definition of additional constraints known by appli- cations only (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=', relative deadlines for sporadic mes- sages).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Dissectors are also envisaged for implicit traffic characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' They enable analyses of information in protocol headers, which are encapsulated in frames sent over the link (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=', IP source and destination addresses, UDP source and destination ports).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' IEEE 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='4 DL layer (MAC+TSCH) IEEE 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='4 Physical layer Network layer Application layer LSE PRIL Application process (TX side) <0,0> cell for management Non-shared TX cell for link <0,0> cell for management Non-shared RX cell for link Application process (RX side) Repeated slotframe on air (synchronized time grid) IEEE 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='4 DL layer (MAC+TSCH) IEEE 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='4 Physical layer Network layer Application layer LSE PRIL Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Block diagram of PRIL techniques (interacting with the LSE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Algorithms (intelligence), possibly based on machine learning, to (semi-)autonomously detect traffic patterns from data mentioned above (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=', finding a traffic model and its parameter values fitting the link) and optimally drive the LS mechanism in the MAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Instead, LSE is part of the MAC and consists of: LS-related commands, to be embedded in exchanged frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' They include requests issued by the transmitter to temporarily disable listening on the receiver and other elements involved in specific PRIL techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Protocol finite state machines and local state variables implementing the LS mechanism and supporting a set of pre-defined LS strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Data-link layer primitives to enable interactions between PRIL and LSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' They should be simple, flexible, and powerful enough to cope with different strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' LISTENING SUSPENSION MECHANISM The LS mechanism for TSCH belongs to the MAC layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' New primitives have to be defined in the MAC sublayer management entity (MLME) to interact with the LSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' For example, a generic MLME-LS-SET can be designed to en- able/disable LS on a given link, to select a specific strategy, and to specify the strategy operating parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' A detailed specification of new primitives cannot abstract from a stable definition of traffic models and PRIL interfaces, which are not available yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' In addition, the behavior of data transmission primitives of the MAC common part sublayer (MCPS), such as MCPS-DATA, must also be modified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' On the one hand, in fact, procedures involved in listening suspension have to be executed when needed (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=', triggered by either transmission requests or local timers and counters) while, on the other hand, traffic needs to be analyzed by PRIL starting from transmission requests (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=', by setting function hooks to inspect frames on the link), to infer its characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' In the current proposal sleep commands are defined as a part of the LSE and sent by the transmitter to the link receiver in order to force the latter to temporarily disable listening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' They are conveyed in data frames by using information elements (IE), that are special fields that can be optionally included in IEEE 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='4 frames to carry ancillary information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' IEs extend the basic protocol but preserve backward compatibility with existing devices and networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' In defining a new IE for sleep commands, denoted sleep IE, a streamlined encoding must be envisaged so as not to undermine energy saving these mechanisms are intended to achieve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' In fact, the inclusion of IEs in frames increases the number of bits sent on air and, consequently, the power consumption on both the transmitting and receiving sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Every sleep command only concerns the link where the including frame was received.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' This is because any intermediate mote acting as a relay can have multiple links connected to its children and/or parent neighbors, and suspension must be managed independently for each one of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' It is worth noting that the LS mechanism works on both the link sides, by either putting them on hold or restoring the conventional TSCH behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' This prevents the sender from performing transmission attempts when listening is temporarily disabled on the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Of course, suitable variables have to be defined on the transmitting and receiving sides of every link to keep track of its LS state (they do not refer to the motes as a whole).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' In the following subsections some possible options are briefly described for LS commands and for general-purpose LS strategies relying on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Basic Sleep Command The simplest way to instruct the receiver about the duration of the listening suspension is by specifying a sleep time in the sleep IE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' This time is encoded as a small integer, denoted Nslp, representing the number of slotframes where listening has to be suspended (note that the final decision about the suspension is left to the receiver, which ensures backward compatibility).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Upon reception, the link is disabled during the following Nslp slotframes, then it is enabled again, restoring the conventional TSCH operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' 2 shows that the time interval before the next usable cell of the link is (Nslp + 1)Tsf wide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' In the figure, the number of cells that, in every slotframe, precede and follow the cell allocated to the link is fixed but arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' The special value Nslp = 0 informs the receiver that it must be listening in the next slotframe, and it is roughly equivalent to not including any sleep command.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' However, while the absence of sleep commands has clearly no effects, setting Nslp = 0 can be used to trigger some specific actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' For example, when multiple cells are reserved for the same link in the slotframe, this can enforce an intra-slotframe LS behavior by skipping all the remaining cells in the current slotframe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' This feature, which is useful when overprovisioning is exploited for high-capacity links [12], is not considered in the remaining part of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Values Nslp ≥ 1 are instead meant to enforce inter-slotframe listening suspension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Another point concerns the maximum extension of the sleep time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' In our opinion 6 bits are enough to encode Nslp for most real situations, leading to link suspension periods up to about two minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Allocating more bits to Nslp does not bring noticeable advantages from the point of view of power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' In fact, the shared cell ⟨0, 0⟩ remains always ac- tive in every slotframe, so that longer sleeping periods for other cells become irrelevant for energy saving (unless the mote is involved in several links).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' In addition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' a receiving cell disabled ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='TXreq R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='a) TSCH ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='Data frame ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='(Nslp) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='X Disabled ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='link ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='I Idle ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='listening ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='Legend: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='Slotframe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='Slot allocated to the link ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='TX request (TSCH) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='TX request (LS) with period ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='Too short ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='Too long ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='Delay ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='Enabled ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='TX/RX side ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='Disabled TX/RX side ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Conventional TSCH vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' LS mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' by a sleep command cannot be (easily) re-enabled before the re-activation time is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Responsiveness to alarms and operations taking place on demand, such as firmware upload or configuration over the air, worsens and may become unacceptable when too long sleep intervals are enforced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' LS works properly when appropriate values are selected for Nslp at any time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' For example, if the link bears a single periodic packet stream, Nslp should be chosen to match the packet generation period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' As mentioned before, characteriza- tion of the traffic model and its parameters is up to PRIL, since the MAC layer has to be kept as simple as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Suitable primitives can drive intelligence in collecting information: for instance, some parameters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=', deadlines) can be provided by upper layers while others (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=', periods) can be inferred by analyzing the traffic over the link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Periodic LS Strategy Let us first consider a simple LS strategy fitting the needs of periodic traffic completely defined by its period Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Under the assumption that only one cell is reserved per link, network stability requires that Tc > Tsf (strict order applies, as transmission errors are unavoidable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Whenever a new frame transmission request is issued for the link (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=', through the MCPS-DATA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='request primitive), the transmitter starts a local frame sleep counter Cfr for the frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' The counter is initialized to Cfr = ⌊τc⌋ ≥ 1, where τc = Tc/Tsf is the normalized transmission period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' It is worth noting that rounding down allows to re-enable the link and, in particular, the receiver listening before a new packet becomes available on the transmitter, thus reducing both the access time and the average number of queued packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Of course, if Tc is not a multiple of Tsf, idle listening may occasionally take place when the receiver is re-enabled, as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' When the frame is sent, the value of Nslp in the sleep IE is set to Cfr, moreover the sleep command is included in the frame only if Nslp > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Reception of a sleep command causes the receiver to set its own reception sleep counter Crx to Nslp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Upon ack frame reception, the Cfr value is also loaded into the transmission sleep counter Ctx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' At the very beginning of every slot allocated to the link Cfr is decreased by one,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' provided ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='Cfr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='4 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='0 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='0 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='4 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='Ctx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='1 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='Crx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='a) TC = 4 Tsf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='b) TC = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='33 Tsf c) TC = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='67 Tsf Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Periodic LS strategy (by varying the actual period).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' it is greater than 0 (counters contain non-negative values).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' The same is true, at the end of the slot, for Ctx and Crx, assuming that they were not set in the same slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Each link side is assumed to be either enabled when the corresponding counter (Ctx for the transmitter and Crx for the receiver) is equal to 0, or disabled otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Since slotframes are kept synchronized all over the network by TSCH, Ctx and Crx are updated coherently on both sides of the link and their values are always the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Should synchronization between motes be lost, all sleep counters are reset to 0, hence deactivating any ongoing listening suspension (and restoring conventional TSCH operation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Frame delivery for a given transmission request may be occasionally delayed as shown, for example, to the fourth request in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='b, because of a prior command with an excessively large sleeping period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' This event is dealt with by the LS mechanism in this way: if, for any reasons, the frame is not sent at the first opportunity following the transmission request and it has to wait in the local queue, Cfr is correctly decreased by the LS algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Consequently, if the sleep command eventually reaches the receiver, the re-enable instant is not affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' This also copes with the relevant case of failed transmission attempts (detected by expiration of the ack timeout), where Cfr is decreased by one on each retry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' In this way, transmission errors affect the current packet in the same way as TSCH, whereas the next packet does not suffer from additional delays with respect to the case when no errors occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' To help recovering from queuing phenomena, which may take place for instance when long bursts of errors are experi- enced, the sleep command is skipped if more than one frame concerning the same link is pending in the transmission queue, thus preventing the link capacity from being throttled in these conditions (which could lead to buffer overruns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' This strategy can also be used for quasi-periodic traffic, where packet generation takes place cyclically at a nominal rate but release times are subject to non-negligible jitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' If the time interval between two consecutive packet transmission requests exceeds Tc, receiver idle listening occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Conversely, when the interval is shorter than Tc, the early packet experi- ences some delay since the transmission can start only when the link is re-enabled (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=', Ctx = 0 and Crx = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Responsiveness Responsiveness in TSCH is limited by its MAC mechanism, which is based on a continuously repeating slotframe [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' As pointed out in [14], in absence of errors due to frame corruption on air, one-way transmission latency consists of two elements, namely access delay and transmission time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Access delay is the time the transmitter spends waiting for the slot assigned to the link, and is the largest contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Under the assumption that, at any time, at most one frame is queued for transmission for the link, it can be modeled as a random variable uniformly distributed between 0 and Tsf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' The transmission time includes the offset at the beginning of the slot (macTsTxOffset) as well as the time taken for transmitting the data frame on air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Since it is always strictly shorter than Tslot we will neglect it in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' In this way the worst- case transmission latency Twc is approximately the same as the maximum access delay when no errors occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' The access delay may grow sensibly when sleep commands are exploited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' With respect to conventional TSCH the period between usable cells for a link (and access delays) increases from Tsf to (Nslp + 1)Tsf, while energy spent by the receiver for listening to the channel decreases by about the same factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' With strictly periodic patterns of packets and a proper selection of Nslp (ensuring that the link is re-enabled just be- fore a new packet becomes available for transmission) delays are roughly the same as TSCH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' However, packets generated sporadically can experience access delays up to (Nslp + 1)Tsf when they become ready just after the link suspension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' To improve link responsiveness for sporadic packets, Nslp can be set based on their relative deadline Td.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' In particular, to ensure that Twc ≤ Td, again in the optimistic case of an ideal channel without errors, Nslp has to be set so that Nslp ≤ τd−1, where τd = Td/Tsf is the normalized deadline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' For example, with the network parameters we considered, Nslp must not exceed 13 if latency must be shorter than 30 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' If the period is sensibly larger than the deadline (Tc ≫ Td), the link is re- enabled too early most times and the LS mechanism behaves less effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Slow Periodic LS Strategy In slow periodic streams the period may exceed the maxi- mum value allowed for the sleep command (64Tsf for Nslp 6- bit encoding).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' In such cases the LSE takes care of issuing sleep commands automatically to repeatedly restart the sleeping period in the receiver and make the suspension last the desired amount of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Nslp is equal to its maximum value (63) in all the sleep commands in the sequence except for the last one, which is set to (⌊τc⌋−1) mod 64 and re-enables the link at the intended time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Overall, the sequence includes ⌈τc/64⌉ frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' For example, if Tc is equal to 10 minutes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=', Nslp = 296), five frames are sent per packet, spaced by 64Tsf: the first one is the data frame, which includes the packet and a sleep command with Nslp = 63, followed by three empty frames (with no payload) where Nslp = 63, and one final (empty) frame where Nslp = 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Should a new transmission request be issued before the above procedure is completed, the course of action is interrupted by the transmitter and the new data frame is sent as soon as the link is re-enabled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' The impact of this approach on power consumption is relatively low and is roughly equivalent to the energy needed for sending and receiving one frame every 64 slotframes (about two minutes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Most of these frames contain no actual data and only deliver sleep commands to the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' To further reduce power consumption a specific short format can be envisaged for empty sleep frames that only carry the sleep IE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' A new MAC sleep command frame can be defined to this purpose but other solutions are possible as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Note that from the energy viewpoint this contribution is rather negligible when compared to cell ⟨0, 0⟩, which is active in every slotframe, nonetheless it improves network responsiveness by imposing a reasonable upper bound Twc = 64Tsf on transmission latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Extended Sleep Command Energy consumption can be reduced without impairing re- sponsiveness too much, by means of an extended version of the sleep command we call xsleep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' The relevant IE includes two parameters: Nslp and Nsnz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Nslp is the number of slotframes the link must be kept disabled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' It specifies the nominal duration of the sleeping period and is managed the same way as in basic sleep commands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' However, the range of allowed values for the sleep time can be increased noticeably in this case without latency is consequently worsened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' For instance, Nslp can be encoded with 12 bits and the sleeping period can last up to 4096Tsf (more than two hours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Selection of larger field sizes to enable longer sleeping periods is clearly possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' The snooze time, denoted Nsnz, specifies how often the link must be temporarily re-enabled (awakened) during the nominal sleeping period, in order to flush packets sporadically gener- ated in the meanwhile and left pending in the transmission buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' In this case, on correct reception of the xsleep command an iterative procedure is started, where the link is disabled for Nsnz consecutive slotframes before being awakened for one slotframe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' This behavior is automatically repeated by the LSE until the end of the nominal suspension period (as specified by Nslp) is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Clearly, Nsnz must be strictly lower than Nslp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Nsnz encoded with 6 bits (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=', Nsnz ≤ 63) provides upper bounds for the access time similar to the basic sleep command.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' In practice, 3 bytes are enough to encode the xsleep command into the content of a sleep IE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Identification of slotframes where the link must be awak- ened does not consider the instant when the sleep command reaches the receiver, but proceeds from the end of the sleeping period moving backwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' In other words, frames can be sent only when Ctx mod (Nsnz + 1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' In this way delays in setting Crx caused, for instance, by transmission errors do not affect the wake-up synchronization on the two link sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' For example, if Nslp = 58 (Tc ≃ 120 s) and Nsnz = 13 (Td ≃ 30 s), the receiver temporarily wakes up at the 3-rd, 17-th, 31-st, and 45-th slotframes after the command is issued, and it is re-enabled at the 59-th slotframe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' S Cfr 10 9 8 7 6 5 4 3 2 1 0 10 9 8 7 6 5 4 3 Ctx 9 8 7 6 5 4 3 2 1 0 9 8 7 6 5 4 3 D93 X I X X X D X X X D93 X I X X X I Crx 9 8 7 6 5 4 3 2 1 0 9 8 7 6 5 4 3 a) Tc = 10 Tsf ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Td = 4 Tsf S S S Cfr 10 9 8 7 6 5 4 3 2 1 0 10 9 8 7 6 5 4 3 Ctx 9 8 7 6 5 4 0 0 0 0 9 8 7 6 5 4 3 D93 X I X X X D00 D D I D93 X I X X X I Crx 9 8 7 6 5 4 0 0 0 0 9 8 7 6 5 4 3 b) Tc = 10 Tsf ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Td = 4 Tsf (queuing taking place) Dsz Data frame [period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='] (Nslp, Nsnz) X Disabled link I Idle listening Legend: Delay D00 Data frame [sporad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='] (reset LS) Sporadic requests Delay D Data frame [sporad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='] LS is reset by transmitter Link wakes up Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Extended periodic LS strategy (with and without queuing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' 4 assumes Nslp = 9 and Nsnz = 3, and shows that awakening occurs when counters equal 8 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' If needed, the xsleep command can be aborted at those time instants, for instance when queuing occurs as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' If, for whatever reason, a number of sporadic packets become pending in the transmission buffer (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=', as a consequence of prolonged interference on air), the transmitting LSE sends a new sleep command to reset the behavior of the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' A special xsleep command with Nslp = 0 and Nsnz = 0 can be defined to this purpose, which reverts the behavior to conventional TSCH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Then, queued packets can be drained at the maximum link speed and LS operation resumed when the queue is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' LS strategies that rely on xsleep trade energy for responsive- ness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' In fact, the worst-case access delay (and Twc as well) is equal to the snoozing period (Nsnz + 1)Tsf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Selecting a low value for Nsnz reduces Twc, but also increases power consumption on the receiver side consequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Because of the large values allowed for Nslp (latency now only depends on Nsnz) the time between transmissions of xsleep commands can be as large as about two hours, which makes energy consumption completely negligible on the transmitter side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Extended Periodic LS Strategy This strategy is suitable for a traffic pattern consisting of the superposition of a quasi-periodic stream with (average) period Tc and sporadic streams with relative deadline Td.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' For simplicity we assume that sporadic packets, which should be served promptly, are issued in consequence of rare events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' In this case, xsleep commands are conveyed inside data frames carrying periodic packets, and Nsnz and Nslp are selected so that Nsnz = ⌊τd⌋ − 1 and Nslp = ⌊τc⌋ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Such a strategy closely resembles periodic LS, but the re- ceiver wakes up regularly during the sleeping period to ensure bounded transmission latency to sporadic packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Since xsleep enables large sleeping times, high efficiency can be reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Moreover, should the inferred period be updated by PRIL, changes can be enforced by sending a new xsleep command as soon as the receiver awakens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' PERFORMANCE ANALYSIS Energy consumed on the transmitting and receiving link sides can be evaluated for conventional TSCH and LS strate- gies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' In the following we assume that no transmission error occurs, to simplify the analysis and focus on the intrinsic energy saving capability of the proposed solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Power Consumption Model Let Etxd and Erxd be the amounts of energy spent for performing a single data frame transmission and reception at- tempt, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' In the following, we model these quantities by assuming that they depend on the frame size linearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' In particular, Etxd = Etx0+etxB·Lphy, where Lphy is the overall number of bytes transmitted by the physical layer, etxB is the energy required to send one byte, and Etx0 is a constant amount of energy additionally spent for every transmission attempt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' For instance, for OpenMote B boards running Open- WSN and communicating through 6TiSCH, Etx0 = 7 µJ and etxB = 2 µJ/B (see plots of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' 4 in [2]), thus the largest fraction of power directly depends on the number of bytes transmitted on air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' A very similar relation holds for reception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' From the same set of experimental data we obtain Erx0 = 65 µJ and erxB = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='3 µJ/B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Unsurprisingly, the energy per byte is lower than for transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' However, a non negligible amount of energy is spent because listening must be enabled for some time preceding the instant when transmission is expected to begin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Erx0 is a mean value, and fluctuations can be experienced depending on the alignment precision of the transmitter and receiver time grids at any given time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Energy needed to send and receive the ack frame (33 B) is Etxa = 106 µJ and Erxa = 79 µJ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' This also includes operations to finalize a frame exchange besides the transmission/reception on air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Finally, the energy spent every time idle listening occurs for the cell is Elis = 138 µJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Elis is about twice Erx0, since the listening interval is centered around the expected start of frame transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' In absence of frame transmission and reception the mote drains about 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='4 mW (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=', 628 µJ in every slot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' This is due to the basic activity of the operating system, protocol stack and application processes, and concerns all board components besides the microcontroller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' It is worth observing that such a high value is likely due to the fact that both the OpenWSN software and the OpenMote B hardware have not been opti- mized yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Hopefully, this constant consumption will be low- ered dramatically when industry-grade devices based on this technology are designed and implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' For these reasons this contribution in not considered in the following analysis, which focuses only on energy spent for communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Conventional TSCH Let Λsf = 1/Tsf and Λc = 1/Tc be the slotframe repeti- tion rate and the packet transmission rate, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' The transmitter power consumption for a packet stream sent over a conventional TSCH link is Pt = (Etxd + Erxa)Λc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' (1) When only reception (and acknowledgment) of frames is considered, for the receiver we have Pr0 = (Erxd + Etxa)Λc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' (2) Pr0 equals the power spent by the receiver when an oracle- based LS strategy is adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Thanks to the oracle, listening is enabled only when the cell conveys a frame, without the need to obtain any information from the sender by means of sleep commands embedded in IEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' As a consequence, Pr0 provides a lower bound to power consumption for a given stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' The overall receiver consumption is given by Pr = Pr0 + Elis[Λsf − Λc], (3) where the rightmost term refers to idle listening, which occurs when the cell allocated to the link is not used for data (hence, the difference between Λsf and Λc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' In a TSCH/6TiSCH network operating correctly, additional frame exchanges are performed automatically in every slot- frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' This is the case of transmission/reception in the shared cell ⟨0, 0⟩, which is meant for network management activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Despite their contribution to power consumption, they are not taken into account here because they are not correlated with data streams produced by applications and affected by the LS mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Roughly speaking, power consumption for these frame exchanges can be expressed as P0 = ¯E⟨0,0⟩Λsf, where ¯E⟨0,0⟩ is the average energy spent by the mote to carry out one (non-confirmed) transmission/reception attempt in cell ⟨0, 0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' In practice, the value of ¯E⟨0,0⟩ does not differ from Etxd, Erxd, and Elis significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Periodic LS Strategy In periodic LS a basic sleep command is included in every data frame where Nslp = ⌊τc⌋ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' For example, if Tc = 30 s then Nslp = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' In the following we will assume that Tc can be inferred precisely by the intelligence of the PRIL module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Obviously, this represents the best case for the LS mechanism, and enables the computation of an upper bound for improvements this approach can offer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Power consumption on the transmitting side increases slightly because of the sleep command P P t = Pt + LslpetxBΛc, (4) where Lslp is the size in bytes of the encapsulating IE (Lslp = 3 B in our case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' For typical frame size (90 B) the increase is negligible (∼ 3%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Data frames that carry sleep commands suspend listening for Nslp slotframes when they are received correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Thus every frame prevents idle listening from occurring Nslp = ⌊τc⌋ − 1 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Hence P P r = Pr0 + LslperxBΛc + Elis [Λsf − ⌊τc⌋ Λc] = Pr + LslperxBΛc − Elis [⌊τc⌋ − 1] Λc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' (5) Again, the consumption increase due to sleep IEs is typically negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Instead, the contribution due to idle listening (right- most term in the first line of the equation) is considerably lower than TSCH, because the multiplicative factor for Elis is always smaller than Λc, and equal to 0 if Tc is a multiple of Tsf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' In the latter case the difference in terms of power consumption between the oracle and the periodic LS strategy is constant and equal, overall, to Lslp(etxB + erxB)Λc, which, for Lslp = 3 B and Tc = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='3 s, means 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='33 µW only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Slow Periodic LS Strategy For slow packet rates (Tc > 64Tsf) empty sleep frames are automatically sent by the LSE on the link transmitter to enlarge the sleeping period without worsening responsiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Power consumption on both sides can be derived from the periodic case by including this additional contribution P slP t = P P t + EtxenempΛc, (6) P slP r = P P r + ErxenempΛc, (7) where nemp = ⌈τc/64⌉ − 1 is the number of empty sleep frames inserted after every data frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Because of our as- sumptions, nempΛc < Λsf/64, which sets an upper bound to the additional consumption needed to keep LS alive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Extended Periodic LS Strategy We have shown that when the link conveys both sporadic streams of packets with relative deadline Td and a periodic stream with period Tc the best approach is to adopt xsleep commands where Nslp = ⌊τc⌋ − 1 and Nsnz = ⌊τd⌋ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' With this strategy, the receiver wakes up temporarily nwup times before listening is re-enabled, and nwup = ⌈(Nslp+1)/(Nsnz+ 1)⌉ − 1 = ⌈⌊τc⌋/⌊τd⌋⌉−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' For example, if Tc = 600 s and Td = 30 s, then Nslp = 296, Nsnz = 13, and the link is awakened 21 times during every sleeping period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' For the sake of simplicity, we do not consider cases where Tc > 4096Tsf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Power consumption on the transmitting side is almost the same as for the periodic strategy P xP t = Pt + LxslpetxBΛc, (8) however the size Lxslp of the sleep IE is slightly larger than the basic sleep command (Lxslp = 5 B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' On the receiving side P xP r = Pr0 + LxslperxBΛc + Elis [Λsf − (⌊τc⌋ − nwup) Λc] = P P r +(Lxslp−Lslp)erxBΛc+ElisnwupΛc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' (9) As expected, power consumption is higher than for basic sleep commands with the same value of Nslp, because of the additional contribution of idle listening introduced by the awakening mechanism to ensure better responsiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' PERFORMANCE COMPARISON To compare strategies we assume that both periodicity and relative deadlines are known (or correctly estimated by PRIL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' The metric we consider is the power required to deal with a packet stream that transfers application data on a single link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Only contributions due to communication are taken into account, as other kinds of power drain by motes can be reduced considerably by exploiting careful hardware and software optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' The frame size Lphy is selected equal to 90 B, to resemble ICMP echo requests generated by ping commands in real networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Three different packet transmission rates Λc were considered, corresponding to periods equal to 30 s, 2 minutes and 10 minutes respectively, while Etxe = 87 µJ and Erxe = 117 µJ for the slow periodic LS strategy (labeled “Basic (slow)” in Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' These values were obtained by considering empty sleep frames encoded with 40 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Results are reported in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' TSCH shows noticeably higher consumption than the oracle on the receiving side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Tangible saving is possible with the periodic LS strategy TABLE I POWER CONSUMPTION ON BOTH SIDES OF THE LINK AND WORST-CASE TRANSMISSION LATENCY Twc FOR THE DIFFERENT LS STRATEGIES BY VARYING PACKET GENERATION PERIOD Tc AND RELATIVE DEADLINE Td.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Tc / Td Strategy Nslp Nsnz Twc Pt Pr (s) / (s) (s) (µW) (µW) 30 / – Oracle – – 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='02 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='8667 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='6000 30 / – TSCH – – 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='02 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='8667 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='3168 30 / – Basic 13 – 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='28 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='0667 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='6468 120 / – Oracle – – 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='02 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='2167 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='4000 120 / – TSCH – – 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='02 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='2167 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='5668 120 / – Basic 58 – 119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='18 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='2667 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='8993 120 / 10 eXtended 58 3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='08 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='3000 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='0210 120 / 30 eXtended 58 13 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='28 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='3000 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='5210 600 / – Oracle – – 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='4433 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='4800 600 / – TSCH – – 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='4433 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='5668 600 / – Basic (slow) 296 – 129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='28 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='0333 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='2733 600 / 10 eXtended 296 3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='4600 17.' metadata={'source': 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+page_content='6477 based on sleep commands (“Basic”) and improves with larger periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Unfortunately, responsiveness is also impaired, and this could be a problem if sporadic packets were occasionally conveyed on the same link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' The periodic LS strategy based on extended sleep commands (“eXtended”) is a reasonable trade- off and is suited to those cases where the period is much larger than the relative deadline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' By means of two degrees of freedom (sleep and snooze times), it achieves interesting power saving yet keeping the MAC complexity at a minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' In general, it retains the expected low-power consumption of TSCH on the transmitting side, whereas responsiveness and energy saving on the receiver side can be balanced on a per-link basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' CONCLUSION By layering time slotting and channel hopping atop IEEE 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='4, TSCH offers high reliability, determinism, and low power consumption, which makes it a good candidate when mesh WSNs compliant to the IIoT paradigm have to be deployed in industrial scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Unfortunately, the access technique of TSCH (but the same also holds for similar protocols that rely on scheduled access) implies non-negligible power consumption by receivers due to idle listening, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=', when the receiving circuitry is switched on but no one is sending frames on air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Details about the PRIL proposal have been presented in this paper, in order to reduce idle listening phenomena by exploiting available information about data exchanges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' In particular, we focused on the listening suspension (LS) mech- anism, to be incorporated in the TSCH MAC, that enables the receiving side of a link to be put on sleep on demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Besides basic strategies, suitable for periodic packet streams, extended solutions have also been introduced, which offer a reasonable trade-off between power consumption and responsiveness for sporadically generated packets thanks to the ability to repeat- edly snooze the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Formulas have been derived for power consumption estima- tion for both sides of a link in 6TiSCH, which were used to evaluate the behavior of a link in realistic operating conditions, by leveraging the energy model of a real device (OpenMote B running OpenWSN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Results confirm that the amount of saved energy is worth the adoption of LS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' Future activities in this area will focus on the evaluation of LS mechanisms through simulation of realistic (and more complex) traffic patterns and the implementation of PRIL algorithms to automatically infer fitting traffic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE2T4oBgHgl3EQfTgdU/content/2301.03803v1.pdf'} +page_content=' REFERENCES [1] D.' metadata={'source': 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Let f = a0 + a1x + · · · + amxm ∈ Z[x] be a primitive polynomial. +Suppose that there exists a positive real number α such that |am|αm > |a0| + +|a1|α + · · · + |am−1|αm−1. We prove that if there exist natural numbers n and +d satisfying n ≥ α + d for which either |f(n)|/d is a prime, or |f(n)|/d is a +prime-power coprime to |f ′(n)|, then f is irreducible in Z[x]. +1. Introduction. +The classical irreducibility criteria due to Sch¨onemann (1846), Eisenstein (1850), +Dumas (1906), and Perron (1907) have become paradigm for testing irreducibility +of polynomials having rational coefficients. The demesne revealing riveting facts +about irreducibility of polynomials over prescribed domains has always been the +cradle of such baroque classical results which for decades have witnessed cogent +extensions and generalizations. Such irreducibility criteria have exhibited a close +affinity to prime numbers and primality as is evident from the illustrious Buni- +akowski’s conjecture of 1854 which asserts that if f is an irreducible polynomial +having integer coefficients such that the elements in the set f(N) have no common +factors other than ±1, then the set f(N) contains infinitely many prime numbers. +The converse of Buniakowski’s conjecture holds affirmatively via primality. +Another classical irreducibility result due to A. Cohn [1, p. 133] states that if a +prime number can be expressed in base 10 as �m +i=0 ai10i for some positive integer +m, then the polynomial �m +i=0 aixi is irreducible in Z[x]. Cohn’s result was then +generalized to arbitrary base by Brillhart et al. [2] and further in Bonciocat et +al. [3]. In [4], Murty provided elementary proof of Cohn’s irreducibility crite- +rion. Interestingly, one of the main results of Murty [4], generalized by Girstmair +[5] apprised of a strong converse of Buniakowski’s conjecture which was further +generalized in [6] and [7] for polynomials having integer coefficients. +Theorem A ([6]). Let f = a0+a1x+· · ·+amxm ∈ Z[x] be a primitive polynomial. +Suppose there exists a positive real number α such that +|am|αm > |a0| + |a1|α + · · · + |am−1|αm−1. +∗Corresponding Author email: sanjeev kumar 19@yahoo.co.in +2010MSC: Primary 12E05; 11C08 +Keywords: Irreducibility of Polynomials; Integer coefficients; Irreducibility Criterion +1 + +2 +JITENDER SINGH1 AND SANJEEV KUMAR2,∗ +If there exist natural numbers n and d satisfying n ≥ α+d for which f(n) = ±pd +for a prime p, then f is irreducible in Z[x]. +Theorem B ([7]). Let f = a0 + a1x + · · · + amxm ∈ Z[x] be primitive, and let +H = +max +0≤i≤m−1{|ai/am|}. +Let f ′(x) denote the formal derivative of f(x) with respect to x. If there exist +natural numbers n, d, k, and a prime p ∤ d such that n ≥ 1+H +d, f(n) = ±pkd, +and for k > 1, also p ∤ f ′(n), then f is irreducible in Z[x]. +In the present note, we generalize Theorem A to the case when |f(n)|/d is a +prime-power with the mild condition of coprimality of |f(n)|/d with |f ′(n)|. More +precisely, we have the following result. +Theorem 1. Let f = a0 + a1x + · · · + amxm ∈ Z[x] be a primitive polynomial. +Suppose that there exists a positive real number α such that +|am|αm > |a0| + |a1|α + · · · + |am−1|αm−1. +If there exist natural numbers n and d satisfying n ≥ α + d for which |f(n)|/d is +prime, or |f(n)|/d is a prime-power coprime to |f ′(n)|, then f is irreducible in +Z[x]. +Example 1. For k ≥ m + 2 ≥ 4 and p ≥ 1 + d, the polynomial +X = −p + x ± (pk−md)xm +satisfies the hypothesis of Theorem 1 with α = 1, a0 = −p, a1 = 1; ai = 0 for +i = 2, 3, . . . , m − 1; am = ±pk−md; and n = p ≥ 1 + d = α + d, since we have +X(p) = ±pkd; X′(p) ≡ 1 mod p so that gcd(|X(p)|/d, |X′(p)|) = 1, and +|am|αm = pk−md ≥ p2 > p + 1 = +m−1 +� +i=0 +|ai|αi. +By Theorem 1, the polynomial X is irreducible in Z[x]. +Example 2. Now consider the polynomial +Y = (x − p) + (x − p)2 + · · · + (x − p)m−1 ± (p2k−1d)xm +for k ≥ m ≥ 2 and p ≥ 1+d. Here, ai = �m−1 +j=i +�j +i +� +(−p)j−i for i = 0, 1, . . . , m−1; +am = ±p2k−1d, α = 1, and n = p ≥ 1 + d. We find that Y (p) = ±p2k+m−1d, +Y ′(p) ≡ 1 mod p. These along with the fact that p2 > 1 + p yield the following: +|am|αm = p2kd +p +≥ (p2)md +p +> (1 + p)m +p +> (1 + p)(1 + p)m−1 − 1 +1 + p − 1 += +m−1 +� +i=0 +|ai|αi. +Since am−1 = 1, it follows that Y is a primitive polynomial. By Theorem 1, the +polynomial Y is irreducible in Z[x]. + +3 +2. Proof of Theorem 1. +Let |f(n)|/d = pk for some prime p and positive integer k. If |x| ≥ α, then in +view of the hypothesis, we have |am|αm > �m−1 +j=0 |aj|αj. Consequently, we have +|f(x)| ≥ |x|m� +|am| − +m−1 +� +i=0 +|ai||x|−(m−i)� +≥ αm� +|am| − +m−1 +� +i=0 +|ai|α−(m−i)� +> 0, +which shows that each zero θ of f satisfies |θ| < α. +Now assume on the contrary that f(x) = f1(x)f2(x) for nonconstant polynomials +f1 and f2 ∈ Z[x]. Since we have +±pkd = f(n) = f1(n)f2(n), +at least one of |f1(n)| and |f2(n)| is divisible by p. Assume that p divides |f2(n)|. +Firstly, let us suppose that p does not divide |f1(n)|. Then pk divides |f2(n)|, and +so, |f1(n)| must divide d so that we have |f1(n)| ≤ d. If β (̸= 0) is the leading +coefficient of f1, then +f1(n) = β +� +θ +(n − θ), +where the product runs over all zeros θ of f1. Observe that each such θ satisfies +|θ| < α. Since +|n − θ| ≥ n − |θ| > n − α ≥ d, +we arrive at the following: +d ≥ |f1(n)| = |β| +� +θ +|n − θ| +> +|β|ddeg f1 ≥ |β|d ≥ d, +leading to a contradiction. +Now assume that p divides |f1(n)|. Since p divides |f2(n)|, we must have k ≥ 2. +Consequently, p divides |f1 +′(n)f2(n) + f1(n)f2 +′(n)|, which in view of the fact that +f1 +′(n)f2(n) + f1(n)f2 +′(n) = f ′(n), +shows that p divides |f ′(n)|. This contradicts the hypothesis. So, f must be +irreducible in Z[x]. +□ +The following remark and examples serve well to make the present idea effica- +ciously comprehensible rendering an advantage over the results already known in +the domain. +Remark. Note that Theorem A is the special case of Theorem 1 with k = 1. The +significance of Theorem 1 lies in the fact that whenever each one of Theorems +A, B, 1 is applicable, Theorems A, B may encounter a tedious factorization of +integers. This is demonstrated in the following explicit examples. + +4 +JITENDER SINGH1 AND SANJEEV KUMAR2,∗ +Example 3. Consider the polynomial +Z = 9 − x + 72x18. +The smallest value of n for which Theorem 1 is applicable for Z is n = 9 with +α = 1, d = 8, and Y (9)/8 = 338, whereas the smallest value of n for which +Theorems A and B are applicable is n = 28 with d = 13 and +Z(28)/13 = 619774506599223645785433953, +which is an 18-digit prime number. +Example 4. Consider the following polynomials Zd as mentioned in [7] +Zd = pk − x ± (pkd)xm, 2 ≤ d ≤ pk − 1, k ≥ 2, +where k, m, d are positive integers and p is a prime number. +Here, a0 = pk, +a1 = −1, ai = 0 for i = 2, . . . , m − 1, and am = ±pkd. Taking α = 1 and n = pk, +we have +|am|αm = pkd > pk + 1 += +m−1 +� +i=0 +|ai|αi; n = pk ≥ 1 + d = α + d, +|Zd(pk)|/d += +pk(1+m); Zd +′(pk) ≡ −1 +mod p, +so that |Zd(pk)|/d is coprime to |Zd +′(pk)|. Thus by Theorem 1, the polynomial +Zd is irreducible in Z[x]. +Here, for the aforementioned value of n and α, Zpk−1 is irreducible by Theorem +1, the irreducibility of which cannot be easily concluded from Theorem A or +Theorem B. +References +1. G. P´olya and G. Szeg¨o, Problems and Theorems in Analysis, vol. 2, Springer-Verlag, New +York, 1976. +2. J. Brillhart, M. Filaseta, and A. Odlyzko, On an irreducibility theorem of A. Cohn. Canadian +J. Math. 33 (1981), 1055–1059. +3. A. I. Bonciocat, N. C. Bonciocat, and A. Zaharescu, On the irreducibility of polynomials that +take a prime power value, Bulletin math´ematique de la Soci´et´e des Sciences Math´ematiques +de Roumanie, Nouvelle S´erie, 54(102), No. 1 (2011), 41–54. +4. M. Ram Murty, Prime numbers and irreducible polynomials. Amer. Math. Monthly. 109:5 +(2002), 452–458. +5. K. Girstmair, On an irreducibility criterion of M. Ram Murty. Amer. Math. Monthly. 112:3 +(2003), 269–270. +6. A. Jakhar, A useful irreducibility result for integer polynomials. Amer. Math. Monthly. +126:10 (2019), 943–944. +7. J. Singh and S. Kumar, A note on Girstmair’s irreducibility criterion, Bull. Aust. Math. Soc. +106:1 (2022), 62–66. + +5 +(1) Department of Mathematics, Guru Nanak Dev University, Amritsar-143005, +India +sonumaths@gmail.com +(2) Department of Mathematics, SGGS College, Sector-26, Chandigarh-160019, +India +sanjeev kumar 19@yahoo.co.in + diff --git a/JtAyT4oBgHgl3EQfTfeL/content/tmp_files/load_file.txt b/JtAyT4oBgHgl3EQfTfeL/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5440d67cf6439f4330694cf8f927d44167eeec26 --- /dev/null +++ b/JtAyT4oBgHgl3EQfTfeL/content/tmp_files/load_file.txt @@ -0,0 +1,154 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf,len=153 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content='00107v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content='NT] 31 Dec 2022 ANOTHER IRREDUCIBILITY CRITERION JITENDER SINGH1 AND SANJEEV KUMAR2,∗ Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' Let f = a0 + a1x + · · · + amxm ∈ Z[x] be a primitive polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' Suppose that there exists a positive real number α such that |am|αm > |a0| + |a1|α + · · · + |am−1|αm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' We prove that if there exist natural numbers n and d satisfying n ≥ α + d for which either |f(n)|/d is a prime, or |f(n)|/d is a prime-power coprime to |f ′(n)|, then f is irreducible in Z[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' The classical irreducibility criteria due to Sch¨onemann (1846), Eisenstein (1850), Dumas (1906), and Perron (1907) have become paradigm for testing irreducibility of polynomials having rational coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' The demesne revealing riveting facts about irreducibility of polynomials over prescribed domains has always been the cradle of such baroque classical results which for decades have witnessed cogent extensions and generalizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' Such irreducibility criteria have exhibited a close affinity to prime numbers and primality as is evident from the illustrious Buni- akowski’s conjecture of 1854 which asserts that if f is an irreducible polynomial having integer coefficients such that the elements in the set f(N) have no common factors other than ±1, then the set f(N) contains infinitely many prime numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' The converse of Buniakowski’s conjecture holds affirmatively via primality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' Another classical irreducibility result due to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' Cohn [1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' 133] states that if a prime number can be expressed in base 10 as �m i=0 ai10i for some positive integer m, then the polynomial �m i=0 aixi is irreducible in Z[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' Cohn’s result was then generalized to arbitrary base by Brillhart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' [2] and further in Bonciocat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' In [4], Murty provided elementary proof of Cohn’s irreducibility crite- rion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' Interestingly, one of the main results of Murty [4], generalized by Girstmair [5] apprised of a strong converse of Buniakowski’s conjecture which was further generalized in [6] and [7] for polynomials having integer coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' Theorem A ([6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' Let f = a0+a1x+· · ·+amxm ∈ Z[x] be a primitive polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' Suppose there exists a positive real number α such that |am|αm > |a0| + |a1|α + · · · + |am−1|αm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' ∗Corresponding Author email: sanjeev kumar 19@yahoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content='co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content='in 2010MSC: Primary 12E05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' 11C08 Keywords: Irreducibility of Polynomials;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' Integer coefficients;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' Irreducibility Criterion 1 2 JITENDER SINGH1 AND SANJEEV KUMAR2,∗ If there exist natural numbers n and d satisfying n ≥ α+d for which f(n) = ±pd for a prime p, then f is irreducible in Z[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' Theorem B ([7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' Let f = a0 + a1x + · · · + amxm ∈ Z[x] be primitive, and let H = max 0≤i≤m−1{|ai/am|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' Let f ′(x) denote the formal derivative of f(x) with respect to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' If there exist natural numbers n, d, k, and a prime p ∤ d such that n ≥ 1+H +d, f(n) = ±pkd, and for k > 1, also p ∤ f ′(n), then f is irreducible in Z[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' In the present note, we generalize Theorem A to the case when |f(n)|/d is a prime-power with the mild condition of coprimality of |f(n)|/d with |f ′(n)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' More precisely, we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' Let f = a0 + a1x + · · · + amxm ∈ Z[x] be a primitive polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' Suppose that there exists a positive real number α such that |am|αm > |a0| + |a1|α + · · · + |am−1|αm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' If there exist natural numbers n and d satisfying n ≥ α + d for which |f(n)|/d is prime, or |f(n)|/d is a prime-power coprime to |f ′(n)|, then f is irreducible in Z[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' For k ≥ m + 2 ≥ 4 and p ≥ 1 + d, the polynomial X = −p + x ± (pk−md)xm satisfies the hypothesis of Theorem 1 with α = 1, a0 = −p, a1 = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' ai = 0 for i = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' , m − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' am = ±pk−md;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' and n = p ≥ 1 + d = α + d, since we have X(p) = ±pkd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' X′(p) ≡ 1 mod p so that gcd(|X(p)|/d, |X′(p)|) = 1, and |am|αm = pk−md ≥ p2 > p + 1 = m−1 � i=0 |ai|αi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' By Theorem 1, the polynomial X is irreducible in Z[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' Now consider the polynomial Y = (x − p) + (x − p)2 + · · · + (x − p)m−1 ± (p2k−1d)xm for k ≥ m ≥ 2 and p ≥ 1+d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' Here, ai = �m−1 j=i �j i � (−p)j−i for i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' , m−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' am = ±p2k−1d, α = 1, and n = p ≥ 1 + d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' We find that Y (p) = ±p2k+m−1d, Y ′(p) ≡ 1 mod p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' These along with the fact that p2 > 1 + p yield the following: |am|αm = p2kd p ≥ (p2)md p > (1 + p)m p > (1 + p)(1 + p)m−1 − 1 1 + p − 1 = m−1 � i=0 |ai|αi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' Since am−1 = 1, it follows that Y is a primitive polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' By Theorem 1, the polynomial Y is irreducible in Z[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' Let |f(n)|/d = pk for some prime p and positive integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' If |x| ≥ α, then in view of the hypothesis, we have |am|αm > �m−1 j=0 |aj|αj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' Consequently, we have |f(x)| ≥ |x|m� |am| − m−1 � i=0 |ai||x|−(m−i)� ≥ αm� |am| − m−1 � i=0 |ai|α−(m−i)� > 0, which shows that each zero θ of f satisfies |θ| < α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' Now assume on the contrary that f(x) = f1(x)f2(x) for nonconstant polynomials f1 and f2 ∈ Z[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' Since we have ±pkd = f(n) = f1(n)f2(n), at least one of |f1(n)| and |f2(n)| is divisible by p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' Assume that p divides |f2(n)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' Firstly, let us suppose that p does not divide |f1(n)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' Then pk divides |f2(n)|, and so, |f1(n)| must divide d so that we have |f1(n)| ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' If β (̸= 0) is the leading coefficient of f1, then f1(n) = β � θ (n − θ), where the product runs over all zeros θ of f1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' Observe that each such θ satisfies |θ| < α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' Since |n − θ| ≥ n − |θ| > n − α ≥ d, we arrive at the following: d ≥ |f1(n)| = |β| � θ |n − θ| > |β|ddeg f1 ≥ |β|d ≥ d, leading to a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' Now assume that p divides |f1(n)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' Since p divides |f2(n)|, we must have k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' Consequently, p divides |f1 ′(n)f2(n) + f1(n)f2 ′(n)|, which in view of the fact that f1 ′(n)f2(n) + f1(n)f2 ′(n) = f ′(n), shows that p divides |f ′(n)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' This contradicts the hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' So, f must be irreducible in Z[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' □ The following remark and examples serve well to make the present idea effica- ciously comprehensible rendering an advantage over the results already known in the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' Note that Theorem A is the special case of Theorem 1 with k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' The significance of Theorem 1 lies in the fact that whenever each one of Theorems A, B, 1 is applicable, Theorems A, B may encounter a tedious factorization of integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' This is demonstrated in the following explicit examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' 4 JITENDER SINGH1 AND SANJEEV KUMAR2,∗ Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' Consider the polynomial Z = 9 − x + 72x18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' The smallest value of n for which Theorem 1 is applicable for Z is n = 9 with α = 1, d = 8, and Y (9)/8 = 338, whereas the smallest value of n for which Theorems A and B are applicable is n = 28 with d = 13 and Z(28)/13 = 619774506599223645785433953, which is an 18-digit prime number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' Consider the following polynomials Zd as mentioned in [7] Zd = pk − x ± (pkd)xm, 2 ≤ d ≤ pk − 1, k ≥ 2, where k, m, d are positive integers and p is a prime number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' Here, a0 = pk, a1 = −1, ai = 0 for i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' , m − 1, and am = ±pkd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' Taking α = 1 and n = pk, we have |am|αm = pkd > pk + 1 = m−1 � i=0 |ai|αi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' n = pk ≥ 1 + d = α + d, |Zd(pk)|/d = pk(1+m);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' Zd ′(pk) ≡ −1 mod p, so that |Zd(pk)|/d is coprime to |Zd ′(pk)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' Thus by Theorem 1, the polynomial Zd is irreducible in Z[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' Here, for the aforementioned value of n and α, Zpk−1 is irreducible by Theorem 1, the irreducibility of which cannot be easily concluded from Theorem A or Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' P´olya and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' Szeg¨o, Problems and Theorems in Analysis, vol.' metadata={'source': 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Girstmair’s irreducibility criterion, Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' Aust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' 106:1 (2022), 62–66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content=' 5 (1) Department of Mathematics, Guru Nanak Dev University, Amritsar-143005, India sonumaths@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content='com (2) Department of Mathematics, SGGS College, Sector-26, Chandigarh-160019, India sanjeev kumar 19@yahoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content='co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} +page_content='in' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfTfeL/content/2301.00107v1.pdf'} diff --git a/KNE3T4oBgHgl3EQfvQuV/content/2301.04692v1.pdf b/KNE3T4oBgHgl3EQfvQuV/content/2301.04692v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..c6e0efe8cc446990586a8763076545eda35a1bae --- /dev/null +++ b/KNE3T4oBgHgl3EQfvQuV/content/2301.04692v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:45d54bedc4dc5c3d81c1835dbcd287d3deb3a4322a7a3424e1c49493f2e170ff +size 14443498 diff --git a/KtE1T4oBgHgl3EQfYwQa/content/tmp_files/2301.03141v1.pdf.txt b/KtE1T4oBgHgl3EQfYwQa/content/tmp_files/2301.03141v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1eb0bd4bd0abadf3fa486cf50070fe40c90007c8 --- /dev/null +++ b/KtE1T4oBgHgl3EQfYwQa/content/tmp_files/2301.03141v1.pdf.txt @@ -0,0 +1,549 @@ +Applying Automated Machine Translation to +Educational Video Courses +Linden Wang | 10 Jan 2023 +Computer Science, University of California, Davis +Abstract +We studied the capability of automated machine translation in the online video education +space by automatically translating Khan Academy videos with state of the art translation +models and applying Text-to-Speech synthesis to build engaging videos in target languages. +We also analyzed and established a reliable translation confidence estimator based on +round-trip translations in order to efficiently manage translation quality and reduce human +translation effort. Finally, we developed a deployable system to deliver translated videos to +end users and collect user corrections for iterative improvement. +I. +Introduction +Online learning has received massive popularity in recent years. The amount of American +students enrolled in online video courses has risen by 30% by 2018, and areas like Asia and the +Middle East are already taking initiatives to improve and broaden educational video content +(Palvia et al., 2018). Since the Coronavirus pandemic, curriculum worldwide has adapted, and +teachers and students are starting to depend more and more on online educational videos +(Nambiar, 2020). +However, language barriers pose a significant challenge to further accessibility for +educational videos (Tahirsylaj et al., 2018; Ruipérez-Valiente et al., 2022; Godwin-Jones, 2014). +English is the most used language in online educational content. But in many non-English +speaking countries, especially underdeveloped ones where free online learning can make the +most impact, young learners cannot fully take advantage of the free educational content. Even in +relatively developed non-English speaking countries, content in local languages can be useful. In +China, the company NetEase translated a subset of Khan Academy videos to Chinese and +estimated more than 200 million page views over a four-year period (Rao et al., 2017). Regional +online video courses, which are frequently being added online, have also been found to attract a +large and diverse population, suggesting the need for educational video translations will only +grow (Ruipérez-Valiente et al., 2022). This highlights the potential of localized online +educational content to alleviate educational resource scarcity in many countries. +Most translation work in this area is done with human translation and human dubbing, +such as the Khan Academy videos translated by NetEase and Khan Academy’s multi-lingual +translation effort (Khan Academy, 2020). Translation to multiple languages at scale can benefit +from automation using machine translation. There have been previous efforts performing +machine translation on online courses such as Khan Academy videos (Bendou, 2021), but the +process is still not quite automatic and scalable. + +There are different approaches in automated translation. Direct speech-to-speech +translation is still relatively new, with challenges such as difficulty in finding appropriate training +datasets (Salesky et al., 2021). Automatic Speech Recognition (ASR) to convert audio to text, +followed by text translation is a more common method of video translation (Alharbi et al., 2021; +Chan & Wang, 2021). In both approaches, background noise can pose a challenge, however most +educational video producers are usually considerate in eliminating background noise to ensure a +pleasant learning experience. +Though reliance on machine translation is increasing, many are still unconvinced on +applying machine translation when it is not perfectly accurate (Way, 2018). TraMOOC, a +systematic way to apply machine translation in education and use multimodal evaluation +techniques to ensure quality, has been proposed but requires a large amount of resources to +control quality (Kordoni et al., 2016; Kordoni et al., 2015). On the other hand, current video +translation options, such as the “translate-caption” feature on websites like YouTube, fail to +provide an engaging learning experience since only translated text instead of speech is made +available to users. +Towards the goal of developing automated translation techniques that can translate +massive online learning content from English to other languages, we report in this paper: 1) a +method to automatically translate Khan Academy videos; 2) a confidence measure to evaluate +translation quality, and flag poorly translated sentences for human correction; 3) a complete +system that delivers videos to end users easily from original Khan Academy video URLs, and +also allows human correction of low-confidence translations via mass collaboration. Transcripts +in Khan Academy videos are easily available, including timestamps for individual sentences. So +we focus on a subsequent workflow that performs text translation at the sentence level, applies +text-to-speech synthesis to generate audio fragments in target languages, and assembles final +audios from fragments to synchronize with original videos. +II. +Course Video Selection +Raw Khan Academy videos were selected and downloaded from YouTube. We chose +videos from two popular subjects (reading and math), and grade levels (primary to high +school/college). Specific categories include second grade reading, fourth grade reading, eighth +grade reading, ninth grade reading, early (kindergarten and first grade) math, second grade math, +fourth grade math, eighth grade math, precalculus, and linear algebra (Table 1). More details in +Table 1 are discussed in the following section. + +Table 1: Selected Video Data +Number of +Videos +Downloaded +Number of +Videos Used +Total Used +Sentences +2nd grade math +50 +42 +1506 +2nd grade reading +20 +10 +588 +4th grade math +171 +58 +2563 +4th grade reading +26 +15 +968 +8th grade math +201 +42 +2542 +8th grade reading +15 +9 +586 +9th grade reading +12 +7 +409 +early math +112 +29 +1041 +linear algebra +131 +20 +2366 +precalculus +268 +29 +1732 +III. +Methods +Collected transcripts for each video consisted of sentences mapped to timestamps of +when the sentences started in the video. Our goal was to translate the sentences while preserving +the mapping of each translated sentence to the timestamp to ensure the translated audio could be +synchronized with the original video, as different languages have different speech tempos. +Individual sentences were translated using two different neural machine translators - Google +Translate and DeepL (DeepL, 2022) - into two languages - Chinese and Spanish, respectively. +Initially, we attempt to translate each sentence, since the original transcript already mapped +sentences to timestamps. However, this could lose the context of neighboring sentences, and +some sentences in the transcript were incomplete, resulting in uninterpretable translations. +Instead we combine the original sentences, translate it as a whole, and split it using a tokenizer. +We then process the tokenized, translated sentences, and match them with their original +timestamps. +However, not every video’s transcript could undergo this matching process due to faulty +transcription, resulting in some loss of data. For example, the transcript may timestamp the +middle of a sentence, splitting it into two parts. This made it impossible to match translated +sentences with timestamps, as we have no way to split a translated sentence to match the split of +the original sentence. This explains the lower numbers of used videos compared to downloaded +videos in Table 1. +Translated sentences were then synthesized to audio fragments, and pauses were inserted +to synchronize speech and video. + +IV. +Translation Confidence Determination +A crucial part of automated machine translation is to automatically determine if +translations are accurate, therefore we established a translation confidence metric for our work. +Many quality estimators have been proposed previously. For example, BLEU (Papineni, 2002) +scores are a popular metric, but require reference text in target languages. Round-trip evaluation +is a process of translating a translated sentence back to the original language and evaluating its +similarity with the original sentence, not requiring reference text. Though initially deemed +inaccurate, it has been shown to work well with neural machine translators like Google +Translate, and round-trip scores are independent of the type of translator (Moon et al., 2020). To +analyze similarity of original sentences and back-translated sentences, we choose quality +estimators sBERT (Reimers & Gurevych, 2019) and BERTScore (Zhang et al., 2019), which +outperform other sentence similarity metrics (Moon et al., 2020). Each BERTScore includes a +triplet of precision, recall, and F1 values and we use its F1 score in our study. +To find the translation confidence threshold, we first take a sample of DeepL-translated, +Chinese videos with 300 random sentences in reading (2nd grade reading, 4th grade reading, 9th +grade reading), and 300 random sentences in math (early math, 4th grade math, linear algebra). +For each sentence, we manually determine if the original and translated sentences were truly +equivalent, and if the original and back-translated sentences were truly equivalent. +We evaluated the amount of sentences where the original versus back-translated sentence +evaluation made a false prediction when taking original versus translated sentence evaluations as +the ground truth (Table 2). Our first observation is that reading videos are less likely to produce +false positive and false negative sentences. This is because math sentences have less context +clues and can sometimes be unpredictable. For example, one original sentence in a math video, +“We'd put the seven in the ones place.”, had a back translation of “We'll put seven people in one +place.”, indicating a clear misunderstanding of the sentence. Second, across both subjects, we +also notice a higher false negative rate which can be partly attributed to homonym confusion. +When a sentence is translated poorly (for example, by using the wrong meaning of a homonym), +the back-translation is more likely to also contain that incorrect definition because +back-translations are independent of forward-translations. It is unlikely for the back-translation +to take a translated sentence with the wrong homonym and produce a back-translated sentence +with the correct homonym (causing a false negative). +Overall, the evaluation results based on back-translated sentences match their +counterparts based on translated sentences in the target language, suggesting confidence +measures using back-translated sentences is feasible. + +Table 2. False Positive and False Negative Percentage Taking Original vs. Translated as Truth +and Original vs. Back-translated as Prediction +Reading +Math +False +Positives +0.0% +2.9% +False +Negatives +2.3% +5.9% +We then try to determine a translation confidence threshold by taking the manually +determined original versus back-translated sentence evaluations as the ground truth, and +sentences which surpass a BERTScore F1 or sBERT threshold as the prediction. For each +threshold, we count the number of false positive and false negative sentences (Table 3). Initially, +we calculated the precision, recall, and F1 values to find the threshold that maximizes F1 score +(which effectively leads to jointly optimized false negative and false positive rates). For example, +DeepL-translated Chinese math videos produced a maximum F1 of 0.938 with 11% of sentences +being false positive, 0.33% of sentences false negative, and BERTScore F1 threshold of 0.907. +However, this is not practical for our application, as threshold scores will be used to flag +potentially-incorrect sentences. Many false positives will result in many incorrect sentences not +being flagged and skipped in the human correction step, while many false negatives mean that +human contributors need to review more translations and mark them as correct in our +contribution system. Therefore, we aim to find a threshold to minimize false positives (as the top +priority) without incurring too many false negatives. +After testing values, we find a false positive percentage of 2-3% is likely optimal. Further +reducing it to below 2% would significantly increase the false negative percentage. As shown in +Table 3, a 0.955 BERTScore F1 or 0.890 sBERT reading threshold and 0.959 BERTScore F1 or +0.931 sBERT math threshold represents the cutoff between accurate and inaccurate sentences +while maintaining a 2% false positive rate. +We notice a stricter BERTScore F1 and sBERT threshold is needed for math sentences, +which is likely due to the unpredictability of math sentences as discussed earlier. Across reading +and math videos, we see the percentages of false negative sentences are comparable between +BERTScore F1 and sBERT, with BERTScore slightly outperforming sBERT, suggesting both to +be effective metrics for determining the translation confidence. We also experimented with a +combination of BERTScore F1 and sBERT thresholds to further reduce false negative rates but +found no improvement. +We use the 2% false positive rate and associated BERTScore F1 threshold for the rest of +the experiment and implementation. + +Table 3. BERTScore F1 and sBERT Thresholds vs. Percentage of False Positive/False Negative +sentences for Reading and Math Videos +BERTScore +F1 +True +Positives +False +Positives +False +Negatives +sBERT +True +Positives +False +Positives +False +Negatives +Reading +0.940 +85.4% +3% +7.94% +0.835 +83.8% +3% +9.60% +0.955 +71.2% +2% +22.2% +0.890 +76.2% +2% +17.2% +0.962 +63.6% +1% +29.8% +0.940 +63.9% +1% +29.5% +Math +0.956 +63.0% +3% +24.6% +0.881 +63.9% +3% +23.6% +0.959 +58.7% +2% +28.9% +0.931 +51.1% +2% +36.4% +0.965 +49.5% +1% +38.0% +0.947 +45.2% +1% +42.3% +V. +Translation Results and Analysis +Understanding translation error sources can provide insight to how machine translation +performs and where it is likely to fail. We first tested the common culprits associated with +unreliable translations: homonyms, interchangeable words, and speaker mistakes. Homonyms +occur frequently in math videos, where, for example, place values can be confused with +quantities. The sentence “We put seven in the ones place” can be correctly interpreted as putting +a seven in the ones digit of a number or incorrectly interpreted as putting a seven in one +(singular) place. We notice the translators handle these situations quite well although there are a +few occurrences where homonyms are mistranslated. Interchangeable words, such as +“magnitude” and “size” and “perpendicular” and “normal” in linear algebra videos, were no +problem for either translator. In the case of speaker mistakes, where the speaker would correct +themselves mid-sentence leaving several extraneous words in the transcript, both translators +correctly ignored the extraneous words, with sBERT scores dropping slightly more. +Translation correctness depends on multiple factors including translation model, target +language, subject, and grade level. To study this, we take each combination of translation model, +target language, and subject (for example, DeepL Spanish Reading) and compute mean +BERTScore F1, median BERTScore F1, standard deviation of BERTScore F1, mean sBERT +score, median sBERT score, standard deviation of sBERT score, and correct translation +percentage based on the BERScore F1 threshold discussed in the previous section. As an +example, Table 5 lists these results for reading videos translated to Spanish for four +representative grade levels. Figure 1 through 4 shows trends of median BERTScore F1 and +Correct Translation Percentage for the four grade levels in both subjects. + +Table 5. DeepL Spanish Reading Video Statistics +Mean +BERTScore +F1 +Median +BERTScore F1 +StdDev +BERTScore +F1 +Mean +sBERT +Median +sBERT +StdDev +sBERT +Correct +Translation +Percentage1 +2nd +grade +reading +0.974 +0.982 +0.021 +0.910 +0.959 +0.132 +0.879 +4th +grade +reading +0.977 +0.981 +0.020 +0.914 +0.956 +0.116 +0.886 +8th +grade +reading +0.978 +0.981 +0.018 +0.920 +0.960 +0.107 +0.884 +9th +grade +reading +0.970 +0.979 +0.017 +0.915 +0.950 +0.107 +0.887 +1 Correct Translation Percentage refers to the percentage of sentences in each category that exceeded the confidence +threshold determined in Section IV (0.955 BERTScore F1 for reading). +Figure 1. Median BERTScore F1 Trends for Reading Videos + +google spanish +google chinese +deeplspanish +deepl chinese +1.000 +0.975 +0.950 +0.925 +0.900 +2nd grade reading +4th grade reading +8th grade reading +9th grade reading +READINGFigure 2. Median BERTScore F1 Trends for Math Videos +Figure 3. Correct Translation Percentage Trends for Reading Videos + +google spanish +google chinese +deepl spanish +deeplchinese +0.990 +0.980 +0.970 +0.960 +0.950 +early math +2nd grade +4th grade +8th grade +precalculus +linear algebra +math +math +mathgoogle spanish +google chinese +deeplspanish +deepl chinese +1.000 +0.920 +0.840 +0.760 +0.680 +0.600 +2nd grade reading +4th grade reading +8th grade reading +9th grade readingFigure 4. Correct Translation Percentage Trends for Math Videos +In general, BERTScore F1 values have a much smaller variation than sBERT scores. +However, we find no obvious differences in variation patterns between BERTScore and sBERT +scores across the selected subjects and grade levels. This is consistent with the observation on the +comparison of the two translation confidence metrics in the previous section. +Looking at Figure 1, 2, 3 and 4, we notice that translation performance differences are +more prominent between target languages than between translation models for the same target +language. For example, Spanish appears to lead to stronger translations than Chinese, regardless +of whether Google Translate or DeepL is used. For Spanish, Google Translator consistently +scores higher than DeepL, whereas for Chinese, DeepL scores slightly higher or equal. +There are relatively small variations across grade levels for each subject. Looking at +trends across grade levels, reading videos were expected to slightly lose accuracy as the material +became more difficult. However, the opposite was true; lower level reading videos often +contained more colloquial terms and deviating content in attempts to engage a younger audience. +A particular lower-scoring couple of videos teaching second graders word roots featured an +exercise where the instructor was creating imaginary words. +Overall, we are able to automatically discriminate between correct (TRUE) and poor +(FALSE) translations at an estimated 2% false positive rate, and a manageable false negative +rate. In the case of Spanish translation, we can achieve a correct translation percentage of about +86%–95%. For Chinese translation, the correct translation percentage is lower (62%–77%). +Among all sentences marked as incorrect (FALSE) that contributors are likely to review, false +negatives are estimated to be ~70% for math and ~80% for reading, as inferred from Table 3. +During the review process, these correct false negative sentences can be quickly confirmed as +correct by contributors with low effort. + +google spanish +google chinese +deeplspanish +deeplchinese +1.000 +0.900 +0.800 +0.700 +0.600 +early math +2nd grade math 4th grade math 8th grade math +precalculus +linear algebraVI. +Implementation +We also developed a system to deliver the translated videos to end users, and also receive +user contributions. Users first find a Khan Academy video on the Khan Academy website or +YouTube. Using the chrome extension (Wang, 2022a), they have the option to view the translated +video which is hosted on YouTube (Wang, 2022b). The user can also correct a translation error +on a contribution webpage. On the contribution page, they will first authenticate themselves +before gaining access to a sentence list. Sentences which are likely to be incorrect (flagged by +our translation confidence score ), will be highlighted, and sentence corrections are submitted to +our database. The interaction between chrome extension, backend system, and video processing +is shown in Figure 5. With each iteration, videos will be re-assembled with the more accurate +sentences to replace older versions. This contribution system allows us to catch occasional +mistranslated sentences while reducing human labor. +Figure 5. Video Translation Application System Design +VII. +Discussion +Our results suggest modern automatic machine translation is an interesting approach to +localize education content at scale. Our translation confidence measure does not require +reference text, allowing efficient detection of poor translations to save human labor. The +proposed translation contribution system allows human contributors to correct the occasional +machine translation inaccuracies iteratively. Future work includes further increasing the + +Chrome Extension +Viewtranslation +Contribute +Contribution +YouTube +BackendSystem +Page +Video Processing +Database +Khan +Academy +Videoseffectiveness of the confidence measures, increasing the contributing user base to enhance +quality of translated videos, comparing confidence scores before and after contribution iterations, +and extending the technique to other subjects, grade levels and target languages. + +REFERENCES +Alharbi, S., Alrazgan, M., Alrashed, A., Alnomasi, T., Almojel, R., Alharbi, R., ... & Almojil, M. +(2021). Automatic speech recognition: Systematic literature review. IEEE Access, 9, +131858-131876. +Bendou, I. (2021). Automatic Arabic Translation of English Educational Content Online using +Neural Machine Translation: the Case of Khan Academy (Doctoral dissertation, Carnegie +Mellon University). +Chan, J. Y. & Wang, H. H. (2021). Speech Recorder and Translator using Google Cloud +Speech-to-Text and Translation. Journal of IT in Asia, 9(1), 11-28. +DeepL. (2022). DeepL Translator [Software]. Retrieved from https://www.deepl.com/ +Godwin-Jones, R. (2014). Global reach and local practice: The promise of MOOCS. Language +Learning & Technology, 18(3), 5-15. +Khan Academy. (2020). Contribute [Web page]. Retrieved from https://www.khanacademy +.org/contribute +Kordoni, V., Cholakov, K., Egg, M., Way, A., Birch, L., Kermanidis, K. L., ... & Orlic, D. (2015). +TraMOOC: Translation for Massive Open Online Courses. In Proceedings of the 18th +Annual Conference of the European Association for Machine Translation. +Kordoni, V., Van den Bosch, A., Kermanidis, K. L., Sosoni, V., Cholakov, K., Hendrickx, I., ... & +Way, A. (2016, May). Enhancing access to online education: Quality machine translation of +MOOC content. In Proceedings of the Tenth International Conference on Language +Resources and Evaluation (LREC'16) (pp. 16-22). +Linden Wang. (2022a). Khan Academy Video Translator [Software]. https://chrome.google.com/ +webstore/detail/khan-academy-video-transl/gbpgbjnhccemhkjedfadjbekpmaoembh +Linden Wang. (2022b). Khan Academy Videos Translated [YouTube Channel]. https://www. +youtube.com/@khanacademyvideostranslate4164 +Moon, J., Cho, H., & Park, E. L. (2020). Revisiting round-trip translation for quality estimation. +European Association for Machine Translation. +Nambiar, D. (2020). The impact of online learning during COVID-19: students’ and teachers’ +perspective. The International Journal of Indian Psychology, 8(2), 783-793. +Palvia, S., Aeron, P., Gupta, P., Mahapatra, D., Parida, R., Rosner, R., & Sindhi, S. (2018). +Online education: Worldwide status, challenges, trends, and implications. Journal of Global +Information Technology Management, 21(4), 233-241. +Papineni, K., Roukos, S., Ward, T., & Zhu, W. J. (2002, July). Bleu: a method for automatic +evaluation of machine translation. In Proceedings of the 40th annual meeting of the +Association for Computational Linguistics (pp. 311-318). +Rao, A., Hilton III, J., & Harper, S. (2017). Khan Academy videos in Chinese: A case study in +OER revision. The International Review of Research in Open and Distributed Learning, +18(5). +Reimers, N., & Gurevych, I. (2019). Sentence-bert: Sentence embeddings using siamese +bert-networks. The 2019 Conference on Empirical Methods in Natural Language Processing. + +Ruipérez-Valiente, J. A., Staubitz, T., Jenner, M., Halawa, S., Zhang, J., Despujol, I., ... & Reich, +J. (2022). Large scale analytics of global and regional MOOC providers: Differences in +learners’ demographics, preferences, and perceptions. Computers & Education, 180, 104426. +Salesky, E., Mäder, J., & Klinger, S. (2021, December). Assessing Evaluation Metrics for +Speech-to-Speech Translation. In 2021 IEEE Automatic Speech Recognition and +Understanding Workshop (ASRU) (pp. 733-740). IEEE. +Tahirsylaj, A., Mann, B., & Matson, J. (2018). Teaching creativity at scale: Overcoming +language barriers in a MOOC. International Journal of Innovation, Creativity and Change, +4(2), 1-19. +Way, A. (2018). Quality expectations of machine translation. In Translation quality assessment +(pp. 159-178). Springer, Cham. +Zhang, T., Kishore, V., Wu, F., Weinberger, K. Q., & Artzi, Y. (2019). Bertscore: Evaluating text +generation with bert. International Conference on Learning Representations + diff --git a/KtE1T4oBgHgl3EQfYwQa/content/tmp_files/load_file.txt b/KtE1T4oBgHgl3EQfYwQa/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..75b364ee41a5ebfde240f045b4711795f6f39e2d --- /dev/null +++ b/KtE1T4oBgHgl3EQfYwQa/content/tmp_files/load_file.txt @@ -0,0 +1,404 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf,len=403 +page_content='Applying Automated Machine Translation to Educational Video Courses Linden Wang | 10 Jan 2023 Computer Science, University of California, Davis Abstract We studied the capability of automated machine translation in the online video education space by automatically translating Khan Academy videos with state of the art translation models and applying Text-to-Speech synthesis to build engaging videos in target languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' We also analyzed and established a reliable translation confidence estimator based on round-trip translations in order to efficiently manage translation quality and reduce human translation effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Finally, we developed a deployable system to deliver translated videos to end users and collect user corrections for iterative improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Introduction Online learning has received massive popularity in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' The amount of American students enrolled in online video courses has risen by 30% by 2018, and areas like Asia and the Middle East are already taking initiatives to improve and broaden educational video content (Palvia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Since the Coronavirus pandemic, curriculum worldwide has adapted, and teachers and students are starting to depend more and more on online educational videos (Nambiar, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' However, language barriers pose a significant challenge to further accessibility for educational videos (Tahirsylaj et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Ruipérez-Valiente et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Godwin-Jones, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' English is the most used language in online educational content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' But in many non-English speaking countries, especially underdeveloped ones where free online learning can make the most impact, young learners cannot fully take advantage of the free educational content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Even in relatively developed non-English speaking countries, content in local languages can be useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' In China, the company NetEase translated a subset of Khan Academy videos to Chinese and estimated more than 200 million page views over a four-year period (Rao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Regional online video courses, which are frequently being added online, have also been found to attract a large and diverse population, suggesting the need for educational video translations will only grow (Ruipérez-Valiente et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' This highlights the potential of localized online educational content to alleviate educational resource scarcity in many countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Most translation work in this area is done with human translation and human dubbing, such as the Khan Academy videos translated by NetEase and Khan Academy’s multi-lingual translation effort (Khan Academy, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Translation to multiple languages at scale can benefit from automation using machine translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' There have been previous efforts performing machine translation on online courses such as Khan Academy videos (Bendou, 2021), but the process is still not quite automatic and scalable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' There are different approaches in automated translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Direct speech-to-speech translation is still relatively new, with challenges such as difficulty in finding appropriate training datasets (Salesky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Automatic Speech Recognition (ASR) to convert audio to text, followed by text translation is a more common method of video translation (Alharbi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Chan & Wang, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' In both approaches, background noise can pose a challenge, however most educational video producers are usually considerate in eliminating background noise to ensure a pleasant learning experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Though reliance on machine translation is increasing, many are still unconvinced on applying machine translation when it is not perfectly accurate (Way, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' TraMOOC, a systematic way to apply machine translation in education and use multimodal evaluation techniques to ensure quality, has been proposed but requires a large amount of resources to control quality (Kordoni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Kordoni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' On the other hand, current video translation options, such as the “translate-caption” feature on websites like YouTube, fail to provide an engaging learning experience since only translated text instead of speech is made available to users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Towards the goal of developing automated translation techniques that can translate massive online learning content from English to other languages, we report in this paper: 1) a method to automatically translate Khan Academy videos;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' 2) a confidence measure to evaluate translation quality, and flag poorly translated sentences for human correction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' 3) a complete system that delivers videos to end users easily from original Khan Academy video URLs, and also allows human correction of low-confidence translations via mass collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Transcripts in Khan Academy videos are easily available, including timestamps for individual sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' So we focus on a subsequent workflow that performs text translation at the sentence level, applies text-to-speech synthesis to generate audio fragments in target languages, and assembles final audios from fragments to synchronize with original videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Course Video Selection Raw Khan Academy videos were selected and downloaded from YouTube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' We chose videos from two popular subjects (reading and math), and grade levels (primary to high school/college).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Specific categories include second grade reading, fourth grade reading, eighth grade reading, ninth grade reading, early (kindergarten and first grade) math, second grade math, fourth grade math, eighth grade math, precalculus, and linear algebra (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' More details in Table 1 are discussed in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Table 1: Selected Video Data Number of Videos Downloaded Number of Videos Used Total Used Sentences 2nd grade math 50 42 1506 2nd grade reading 20 10 588 4th grade math 171 58 2563 4th grade reading 26 15 968 8th grade math 201 42 2542 8th grade reading 15 9 586 9th grade reading 12 7 409 early math 112 29 1041 linear algebra 131 20 2366 precalculus 268 29 1732 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Methods Collected transcripts for each video consisted of sentences mapped to timestamps of when the sentences started in the video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Our goal was to translate the sentences while preserving the mapping of each translated sentence to the timestamp to ensure the translated audio could be synchronized with the original video, as different languages have different speech tempos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Individual sentences were translated using two different neural machine translators - Google Translate and DeepL (DeepL, 2022) - into two languages - Chinese and Spanish, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Initially, we attempt to translate each sentence, since the original transcript already mapped sentences to timestamps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' However, this could lose the context of neighboring sentences, and some sentences in the transcript were incomplete, resulting in uninterpretable translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Instead we combine the original sentences, translate it as a whole, and split it using a tokenizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' We then process the tokenized, translated sentences, and match them with their original timestamps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' However, not every video’s transcript could undergo this matching process due to faulty transcription, resulting in some loss of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' For example, the transcript may timestamp the middle of a sentence, splitting it into two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' This made it impossible to match translated sentences with timestamps, as we have no way to split a translated sentence to match the split of the original sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' This explains the lower numbers of used videos compared to downloaded videos in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Translated sentences were then synthesized to audio fragments, and pauses were inserted to synchronize speech and video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Translation Confidence Determination A crucial part of automated machine translation is to automatically determine if translations are accurate, therefore we established a translation confidence metric for our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Many quality estimators have been proposed previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' For example, BLEU (Papineni, 2002) scores are a popular metric, but require reference text in target languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Round-trip evaluation is a process of translating a translated sentence back to the original language and evaluating its similarity with the original sentence, not requiring reference text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Though initially deemed inaccurate, it has been shown to work well with neural machine translators like Google Translate, and round-trip scores are independent of the type of translator (Moon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' To analyze similarity of original sentences and back-translated sentences, we choose quality estimators sBERT (Reimers & Gurevych, 2019) and BERTScore (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=', 2019), which outperform other sentence similarity metrics (Moon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Each BERTScore includes a triplet of precision, recall, and F1 values and we use its F1 score in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' To find the translation confidence threshold, we first take a sample of DeepL-translated, Chinese videos with 300 random sentences in reading (2nd grade reading, 4th grade reading, 9th grade reading), and 300 random sentences in math (early math, 4th grade math, linear algebra).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' For each sentence, we manually determine if the original and translated sentences were truly equivalent, and if the original and back-translated sentences were truly equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' We evaluated the amount of sentences where the original versus back-translated sentence evaluation made a false prediction when taking original versus translated sentence evaluations as the ground truth (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Our first observation is that reading videos are less likely to produce false positive and false negative sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' This is because math sentences have less context clues and can sometimes be unpredictable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=" For example, one original sentence in a math video, “We'd put the seven in the ones place.”, had a back translation of “We'll put seven people in one place.”, indicating a clear misunderstanding of the sentence." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Second, across both subjects, we also notice a higher false negative rate which can be partly attributed to homonym confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' When a sentence is translated poorly (for example, by using the wrong meaning of a homonym), the back-translation is more likely to also contain that incorrect definition because back-translations are independent of forward-translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' It is unlikely for the back-translation to take a translated sentence with the wrong homonym and produce a back-translated sentence with the correct homonym (causing a false negative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Overall, the evaluation results based on back-translated sentences match their counterparts based on translated sentences in the target language, suggesting confidence measures using back-translated sentences is feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' False Positive and False Negative Percentage Taking Original vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Translated as Truth and Original vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Back-translated as Prediction Reading Math False Positives 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='0% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='9% False Negatives 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='3% 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='9% We then try to determine a translation confidence threshold by taking the manually determined original versus back-translated sentence evaluations as the ground truth, and sentences which surpass a BERTScore F1 or sBERT threshold as the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' For each threshold, we count the number of false positive and false negative sentences (Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Initially, we calculated the precision, recall, and F1 values to find the threshold that maximizes F1 score (which effectively leads to jointly optimized false negative and false positive rates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' For example, DeepL-translated Chinese math videos produced a maximum F1 of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='938 with 11% of sentences being false positive, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='33% of sentences false negative, and BERTScore F1 threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' However, this is not practical for our application, as threshold scores will be used to flag potentially-incorrect sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Many false positives will result in many incorrect sentences not being flagged and skipped in the human correction step, while many false negatives mean that human contributors need to review more translations and mark them as correct in our contribution system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Therefore, we aim to find a threshold to minimize false positives (as the top priority) without incurring too many false negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' After testing values, we find a false positive percentage of 2-3% is likely optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Further reducing it to below 2% would significantly increase the false negative percentage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' As shown in Table 3, a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='955 BERTScore F1 or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='890 sBERT reading threshold and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='959 BERTScore F1 or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='931 sBERT math threshold represents the cutoff between accurate and inaccurate sentences while maintaining a 2% false positive rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' We notice a stricter BERTScore F1 and sBERT threshold is needed for math sentences, which is likely due to the unpredictability of math sentences as discussed earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Across reading and math videos, we see the percentages of false negative sentences are comparable between BERTScore F1 and sBERT, with BERTScore slightly outperforming sBERT, suggesting both to be effective metrics for determining the translation confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' We also experimented with a combination of BERTScore F1 and sBERT thresholds to further reduce false negative rates but found no improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' We use the 2% false positive rate and associated BERTScore F1 threshold for the rest of the experiment and implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' BERTScore F1 and sBERT Thresholds vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Percentage of False Positive/False Negative sentences for Reading and Math Videos BERTScore F1 True Positives False Positives False Negatives sBERT True Positives False Positives False Negatives Reading 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='940 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='4% 3% 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='94% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='835 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='8% 3% 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='60% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='955 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='2% 2% 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='2% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='890 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='2% 2% 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='2% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='962 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='6% 1% 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='8% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='940 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='9% 1% 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='5% Math 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='956 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='0% 3% 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='6% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='881 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='9% 3% 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='6% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='959 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='7% 2% 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='9% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='931 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='1% 2% 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='4% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='965 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='5% 1% 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='947 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='2% 1% 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='3% V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Translation Results and Analysis Understanding translation error sources can provide insight to how machine translation performs and where it is likely to fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' We first tested the common culprits associated with unreliable translations: homonyms, interchangeable words, and speaker mistakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Homonyms occur frequently in math videos, where, for example, place values can be confused with quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' The sentence “We put seven in the ones place” can be correctly interpreted as putting a seven in the ones digit of a number or incorrectly interpreted as putting a seven in one (singular) place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' We notice the translators handle these situations quite well although there are a few occurrences where homonyms are mistranslated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Interchangeable words, such as “magnitude” and “size” and “perpendicular” and “normal” in linear algebra videos, were no problem for either translator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' In the case of speaker mistakes, where the speaker would correct themselves mid-sentence leaving several extraneous words in the transcript, both translators correctly ignored the extraneous words, with sBERT scores dropping slightly more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Translation correctness depends on multiple factors including translation model, target language, subject, and grade level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' To study this, we take each combination of translation model, target language, and subject (for example, DeepL Spanish Reading) and compute mean BERTScore F1, median BERTScore F1, standard deviation of BERTScore F1, mean sBERT score, median sBERT score, standard deviation of sBERT score, and correct translation percentage based on the BERScore F1 threshold discussed in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' As an example, Table 5 lists these results for reading videos translated to Spanish for four representative grade levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Figure 1 through 4 shows trends of median BERTScore F1 and Correct Translation Percentage for the four grade levels in both subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' DeepL Spanish Reading Video Statistics Mean BERTScore F1 Median BERTScore F1 StdDev BERTScore F1 Mean sBERT Median sBERT StdDev sBERT Correct Translation Percentage1 2nd grade reading 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='974 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='982 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='021 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='910 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='959 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='132 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='879 4th grade reading 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='977 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='981 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='914 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='956 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='116 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='886 8th grade reading 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='978 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='981 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='920 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='960 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='107 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='884 9th grade reading 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='970 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='979 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='915 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='950 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='107 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='887 1 Correct Translation Percentage refers to the percentage of sentences in each category that exceeded the confidence threshold determined in Section IV (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='955 BERTScore F1 for reading).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Median BERTScore F1 Trends for Reading Videos google spanish google chinese deeplspanish deepl chinese 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='975 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='950 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='925 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='900 2nd grade reading 4th grade reading 8th grade reading 9th grade reading READINGFigure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Median BERTScore F1 Trends for Math Videos Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Correct Translation Percentage Trends for Reading Videos google spanish google chinese deepl spanish deeplchinese 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='990 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='980 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='970 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='960 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='950 early math 2nd grade 4th grade 8th grade precalculus linear algebra math math mathgoogle spanish google chinese deeplspanish deepl chinese 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='920 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='840 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='760 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='680 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='600 2nd grade reading 4th grade reading 8th grade reading 9th grade readingFigure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Correct Translation Percentage Trends for Math Videos In general, BERTScore F1 values have a much smaller variation than sBERT scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' However, we find no obvious differences in variation patterns between BERTScore and sBERT scores across the selected subjects and grade levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' This is consistent with the observation on the comparison of the two translation confidence metrics in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Looking at Figure 1, 2, 3 and 4, we notice that translation performance differences are more prominent between target languages than between translation models for the same target language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' For example, Spanish appears to lead to stronger translations than Chinese, regardless of whether Google Translate or DeepL is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' For Spanish, Google Translator consistently scores higher than DeepL, whereas for Chinese, DeepL scores slightly higher or equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' There are relatively small variations across grade levels for each subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Looking at trends across grade levels, reading videos were expected to slightly lose accuracy as the material became more difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' However, the opposite was true;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' lower level reading videos often contained more colloquial terms and deviating content in attempts to engage a younger audience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' A particular lower-scoring couple of videos teaching second graders word roots featured an exercise where the instructor was creating imaginary words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Overall, we are able to automatically discriminate between correct (TRUE) and poor (FALSE) translations at an estimated 2% false positive rate, and a manageable false negative rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' In the case of Spanish translation, we can achieve a correct translation percentage of about 86%–95%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' For Chinese translation, the correct translation percentage is lower (62%–77%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Among all sentences marked as incorrect (FALSE) that contributors are likely to review, false negatives are estimated to be ~70% for math and ~80% for reading, as inferred from Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' During the review process, these correct false negative sentences can be quickly confirmed as correct by contributors with low effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' google spanish google chinese deeplspanish deeplchinese 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='900 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='700 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content='600 early math 2nd grade math 4th grade math 8th grade math precalculus linear algebraVI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Implementation We also developed a system to deliver the translated videos to end users, and also receive user contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Users first find a Khan Academy video on the Khan Academy website or YouTube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Using the chrome extension (Wang, 2022a), they have the option to view the translated video which is hosted on YouTube (Wang, 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' The user can also correct a translation error on a contribution webpage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' On the contribution page, they will first authenticate themselves before gaining access to a sentence list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Sentences which are likely to be incorrect (flagged by our translation confidence score ), will be highlighted, and sentence corrections are submitted to our database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' The interaction between chrome extension, backend system, and video processing is shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' With each iteration, videos will be re-assembled with the more accurate sentences to replace older versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' This contribution system allows us to catch occasional mistranslated sentences while reducing human labor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Video Translation Application System Design VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Discussion Our results suggest modern automatic machine translation is an interesting approach to localize education content at scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Our translation confidence measure does not require reference text, allowing efficient detection of poor translations to save human labor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' The proposed translation contribution system allows human contributors to correct the occasional machine translation inaccuracies iteratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' Future work includes further increasing the Chrome Extension Viewtranslation Contribute Contribution YouTube BackendSystem Page Video Processing Database Khan Academy Videoseffectiveness of the confidence measures, increasing the contributing user base to enhance quality of translated videos, comparing confidence scores before and after contribution iterations, and extending the technique to other subjects, grade levels and target languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=' REFERENCES Alharbi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=', Alrazgan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=', Alrashed, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=', Alnomasi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfYwQa/content/2301.03141v1.pdf'} +page_content=', Almojel, R.' metadata={'source': 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Given a pair of content +and style images, a stylized one is hoped that retains the +content from the former while catching style patterns from +the latter. However, it is difficult to simultaneously keep +well the trade-off between the content details and the style +features. To stylize the image with sufficient style patterns, +the content details may be damaged and sometimes the ob- +jects of images can not be distinguished clearly. For this +reason, we present a new transformer-based method named +STT for image style transfer and an edge loss which can +enhance the content details apparently to avoid generat- +ing blurred results for excessive rendering on style features. +Qualitative and quantitative experiments demonstrate that +STT achieves comparable performance to state-of-the-art +image style transfer methods while alleviating the content +leak problem. +1. Introduction +Rendering a content image into the artistic style of a +referenced image is the main purpose of the image style +transfer. Image style transfer is an interesting topic of com- +puter vision with a long history. About 2 decades ago, re- +searchers [1,2] utilize techniques, such as texture synthesis +and style transfer functions, to achieve the stylizing pro- +cess. But they only focus on low-level features. Thereafter +Gatys et al. [3, 4] prove that the correlation between fea- +tures extracted from a pre-trained VGG can be treated as +the representation of style patterns, which opens the gate of +neural style transfer (NST). Iterative methods [4–10] ren- +der the content images gradually by applying gradients on +the input images or the noise images while the feed-forward +networks [11–19] can complete the stylizing process in one +feed-forward manner after training. Vivid results though the +iterative and feed-forward methods may produce, are still +limited to a certain number of styles or achieve inadequate +style quality. Thanks to the encoder-transfer-decoder archi- +Content +Style +w/o Edge Loss +w/ Edge Loss (Ours) +Figure 1. Visual effects of edge loss. Compared to the results +from the model without edge loss (column 3), the objects of styl- +ized images with edge loss, especially the small ones like letters or +windows, are apparently more clear and distinguishable (column +4). For the convenience of comparison, the model used above is +STT. The results from other methods are also blurred in this case. +tecture, arbitrary style transfer methods [20–47] are capable +of rendering images into any styles. However, these models +may not work well in some cases due to the limited ability +to merge the content and style features. To cope with this +problem, the attention mechanism [48,49] is introduced by +a few methods [37–42] to enhance the fusion effects. +Recently, An et al. [43] discover the content leak prob- +lem that the structure of results from the CNN-based meth- +ods will be dramatically changed after a few rounds of +repetitive stylization process. Deng et al. [45] and Zhang +et al. [47] then prove that the transformer-based methods +are capable of alleviating the problem. Different from the +previous methods, IEST [42] and CAST [46] leverage the +contrastive learning strategy to enhance the visual quality. +However, in some cases, the structure of results from previ- +ous methods still could be blurred and the objects in images +are difficult to be distinguished (see Fig. 1). +Thanks to the flexibility, scalability, and ability to cap- +ture long-range dependencies, Transformers [49–51] have +been widely used in all kinds of vision tasks. Owing to +the self-attention mechanism, Transformer can efficiently +gather global information which is important for preserv- +ing the structure of input images. Compared to the archi- +tecture of typical CNN-based methods, Transformer evades +the multi-time downsample operations which may lead to +1 +arXiv:2301.00592v1 [cs.CV] 2 Jan 2023 + +EXPRESSEXPRESS +NUSBRILLIA +E +MONEthe content leak problem [43]. Therefore, the Transformer +structure has a good effect on the image style transfer task. +In this work, we propose a new Transformer-based im- +age style transfer algorithm that is capable of producing +stylized results with high visual quality while preserving +fine content details. +We call it STT (Style Transfer on +Transformers). +Different from StyTr2 [45] and S2WAT +[47], neither does STT choose to encode content and style +features in different encoders as StyTr2 did, nor does STT +adopts the hierarchal structure as S2WAT did. A tiny CNN- +based module is equipped as a positional encoding layer +(Conv PE) which extracts the positional encoding (PE) ac- +cording to the semantic information. Furthermore, to en- +hance the content details, a novel edge loss is applied as an +extra restriction when stylized images are blurred due to the +overdose of style features imposed on the inputs. +The main contributions of our work are as follows: +• A new image style transfer network name STT which +can stylize images with high quality while preserving +fine content details. +• A novel edge loss to enhance the content details, which +improves the picture clarity of the stylized images ob- +viously. +• Extensive experiments demonstrate that STT achieves +outstanding effects and is capable of generating favor- +able results while preserving fine content details. +2. Related Work +Image Style Transfer. Starting from Gatys et al. [3, 4], +the number of methods in NST is increasingly growing +with time forward and the stylizing effects have been more +and more colorful. Here we make a rough classification +of these models with respect to their generalization abili- +ties, the backbone architecture, and training strategies. In +generalization abilities, the categories can be divided into +single style transfer [4–15], multiple style transfer [16–18], +and arbitrary style transfer [20–47]. The single style trans- +fer encodes the fixed style features into models while the +multiple style transfer utilizes certain tricks, such as con- +ditional instance normalization [16] and StyleBank [17], to +support a number of styles. With a certain module to merge +the features of content and style, the arbitrary style trans- +fer is capable of handling any style transfer. As the tech- +niques of upstream tasks like image classification and im- +age generation have been rapidly developing these years, an +increasing number of sorceries have been introduced into +image style transfer. +Except for the typical CNN-based +methods, the Flow-based [43] and the Transformer-based +[45, 47] methods also appear in recent years. The Flow- +based ArtFlow [43] is proposed to solve the problem of +the content leak and the Transformer-based StyTr2 [45] and +S2WAT [47] are able to alleviate the problem. The encoders +of StyTr2 have the traditional Transformer structure where +the shape of representations will not be changed in pro- +cessing while S2WAT adopts hierarchal architecture which +means the features will be downscaled gradually. Recently, +the contrastive learning strategy is introduced by IEST [42] +and CAST [46]. Different from other methods trained with +perceptual losses or identity losses, IEST and CAST treat +the contrastive loss and the adversarial loss as the opti- +mization targets to achieve satisfying effects. However, in +some cases, the results from the existing image style trans- +fer methods may still be blurred due to no restriction on the +edge of the main objects in inputs. +Vison Transformer. Inherited the ability to capture the +long-range dependencies from Transformers in natural lan- +guage processing (NLP), vision Transformers have been de- +veloped in a wide variety of vision tasks, including image +classification [52–64], object detection [65–69], semantic +segmentation [70, 71], and image generation [72, 73]. In +image style transfer, StyTr2 and S2WAT have demonstrated +that both the traditional structure and the hierarchical ar- +chitecture have a favorable effect on style transfer. In this +paper, we leverage several convolutional operations to ful- +fill the positional encoding instead of parametric positional +encoding [52] or the one with pooling operations [45]. +Utilization of Edge Maps in Style Transfer. The oper- +ators like Laplacian, Canny, and Sobel are widely used in +edge and contour detection. In image style transfer, Li et +al. discover the correspondences between Laplacian devia- +tions and image distortions and then propose Lapstyle [8], +an iterative image style transfer method based on a Lapla- +cian loss. Subsequently, Li et al. apply the Laplacian fil- +ter on the drafting and revision network and then present +LapStyle [36], a feed-forward image style transfer method +based on the Laplacian filter. However, the above meth- +ods are all applied to the CNN-based models. In this work, +we leverage the edge maps to enhance the results from the +Transformer-based STT which is capable of producing styl- +ized images with fine content details and colorful artistic +features. +3. Method +As shown in Fig. 2, the proposed STT has the architec- +ture of encoder-transfer-decoder. The positional encoding +(PE) is first extracted from both the content images Ic and +the style image Is by a module name Conv PE. Then after +splitting the content images Ic and style image Is into non- +overlapping patches, a linear projection is equipped to trans- +form the patches into sequences. The sum of the sequences +and the PE will be treated as the inputs of the Transformer +encoder. Generated from the encoder, the content features +fc and style features fs then will be merged in the transfer +module which is based on a Transformer decoder. Finally, +2 + +Patch Partition +Linear Projection +Conv PE +Conv PE +Layer 1 +Layer 2 +Layer 3 +Layer 1 +Layer 2 +Layer 3 +Transformer Encoder +Transformer Decoder +Decoder +Edge Extractor +Edge Extractor +Edge Loss +LN +MSA +MHA +LN +MLP +LN +fc +fs +Q +K +V +(a) Net Architecture +(b) Transformer Decoder Layer +Figure 2. The net architecture of the proposed STT. +the outputs can be obtained by decoding the fused features +fcs in a CNN-based decoder. In addition, to calculate the +edge loss during the training step, the edge maps of the +content images and the stylized images need to be extracted +by a fine-designed edge extractor which will be introduced +later. +In this part, we plan to present the overall architecture +of the proposed STT first in Section 3.1 and in Section 3.2 +introduce the edge extractor which is used to calculate the +edge loss. Finally, the optimization strategy will be dis- +cussed in Section 3.3. +3.1. Overall Architecture +Encoder. Before the process of the encoder, a tiny CNN- +based module named Conv PE is applied to the content +images and style images to extract the content-aware po- +sitional encoding. +As depicted in Fig. 3, Conv PE is +composed by three convolutional layers, two reflections +(padding) layers, and one ReLU activation layer. The main +role of the reflection layers is to ensure that the size of re- +sults is consistent before and after processing while the con- +volutional layers are used to extract the content-aware posi- +tional encoding. As shown in Fig. 6, we find that the results +from the model with Conv PE are obviously better than that +of the one without. +Different from the design of StyTr2 [45] which has two +independent domain-specific encoders for the content im- +ages and style images, STT treats them as normal pic- +tures with content & style features and encodes them in +one Transformer-based encoder. Given the input image in +the shape of H × W × 3, the input will first be split into +patches by the patch partition layer and then embedded into +sequences linearly with the shape of HW +8×8 × C (768 is the +default value of C). Adding the positional encoding from +Conv PE, the resulting sequences will be fed to a three- +layer Transformer encoder. +The computation process of +each layer can be defined as: +ˆcl = MSA(LN(cl−1)) + cl−1 +(1) +cl = MLP(LN(ˆcl)) + ˆcl +(2) +where ˆcl and cl denote the outputs of MSA and MLP for +layer l respectively; MSA represents the module of multi- +head self-attention while MLP denotes the module of multi- +layer perceptron; and LN means LayerNorm. After the +three layers of the encoder, we obtain the content features +fc and style features fs with the shape consistent before and +after processing. +Input +Conv1x1 +Reflect +Conv3x3 +Relu +Reflect +Conv3x3 +Output +Figure 3. The illustration of Conv PE. +Decoder. Instead of upsampling the stylized features fcs +to the original size in a single projection as [74] once did, +STT follows [21, 37, 41] to utilize a mirrored VGG to de- +code the stylized features gradually. Before the step of de- +coding, the stylized features need to be reshaped first for +the sequence-like shape HW +8×8 × C. After the three stages of +upsampling and refining, we obtain the stylized images Ics +with the shape of H × W × 3. +3 + +Transfer Module. The transfer module is used to merge +the content features fc and style features fs. We introduce +the transfer module of S2WAT [47] as the means to fuse +the features. As depicted in Fig. 2 (b), the transfer module +consists of three layers of the Transformer decoder layers, +and each layer is mainly composed by an MSA module, an +MHA module, and an MLP module. The computational +process can be defined as: +ˆxl = MSA(LN(xl−1)) + xl−1 +Q = LN(ˆxl) · WQ +K, V = y · WK, y · WV +˜xl = MHA(Q, K, V ) + ˆxl +xl = MLP(LN(˜xl)) + ˜xl +(3) +where ˆxl, ˜xl, and xl represent the results of MSA, MHA, +and MLP for layer l, respectively; y denotes the style fea- +tures; WQ, WK, and WV are the projection matrices for Q, +K, and V ; Q, K, and V denote the query, key, and value +vectors. Leveraging the fusion effects of the cross attention, +the stylized features fcs can be received. +3.2. Edge Extractor +To make the content structure of the stylized images +clear, we design a novel edge loss to enhance the edge of the +objects in output images. Before calculating the edge loss, +the edge maps suitable for style transfer need to be captured +by a fine-designed edge extractor. Different from the tasks +like edge detection or contour extraction, the content de- +tails of the outputs in image style transfer are probably not +the same as that of the content images, especially the back- +ground which may has the artistic patterns from the style +images. The results will be blurred if we take the similar- +ity between the edge maps from the content images and the +stylized images as the optimization target directly. There- +fore one of the problems that need to be solved is to filter +out the place where the main structure of the content im- +ages does not exist. A mask operation is introduced to cope +with this problem. As shown in (6), all of the edges in the +edge maps of the stylized images (edg′-Ics) that are not ex- +ist in the corresponding place of the edge maps of content +images (edg-Ic) will be masked out. Furthermore, we also +set a threshold to exclude the weak responses of edge maps +which may play a role as noise. The overall computational +process can be defined as: +edg-Ic = threshold(lap(Ic), τ) +(4) +edg′-Ics = threshold(lap(Ics), τ) +(5) +edg-Ics = mask(edg′-Ics, edg-Ic) +(6) +where edg-Ic and edg-Ics are the edge maps of the content +and stylized images respectively; lap denotes the Laplacian +operator and threshold represents the function that sets 0 +to the responses where the value is smaller than the thresh- +old parameter τ (0.2 is set as the default value). After the +above steps, we obtain the refined edge maps to be used in +calculating the edge loss. +3.3. Network Optimization +The main purpose of image style transfer is to maintain +the structure of the content images while transferring the +artistic patterns to the stylized results from the style images. +To achieve this target, we follow [21] to construct two per- +ceptual losses to measure the content differences between +the stylized images and the content images as well as the +style differences between the stylized images and the style +images. Furthermore, we also adopt the identity losses [37] +to enrich the content details and style patterns of the styl- +ized images. Finally, the proposed edge loss is equipped to +enhance the content structure further. As shown in (7), the +whole loss function can be defined as: +Ltotal =λcLcontent + λsLstyle+ +λid1Lid1 + λid2Lid2 + λedgLedg +(7) +where the λc, λs, λid1, λid2, and λedg are the weights of +losses; Lcontent and Lstyle denote the perceptual losses; +Lid1 and Lid2 are the identity losses; Ledg represents the +edge loss and we only apply the edge loss in the situation +when the results are apparently blurred. We set λc, λs, λid1, +λid2, and λedg to 1, 3, 50, 1, and 5000 to alleviate the impact +of magnitude differences. +Perceptual Loss. Similar to [21], we leverage a pretrained +VGG19 to extract the feature maps of the content and style +images which are used to calculate the perceptual losses. In +our model, the layer Relu 4 1 and Relu 5 1 are used to cal- +culate the content perceptual loss while the layer Relu 1 1, +Relu 2 1, Relu 3 1, Relu 4 1, and Relu 5 1 are used to +calculate the style perceptual loss. One thing that needs to +be attended to is that the mean-variance channel-wise nor- +malization is applied on the feature maps before the calcu- +lation of the content perceptual loss. The perceptual losses +can be defined as: +Lcontent = +� +l∈C +∥φl(Ics) − φl(Ic)∥2 +(8) +Lstyle = +� +l∈L +∥µ(φl(Ics)) − µ(φl(Is))∥2+ +∥σ(φl(Ics)) − σ(φl(Is))∥2 +(9) +where the C and L are the layers of the pretrained VGG +which are concerned to calculate the content and style per- +ceptual losses respectively; φl denotes the feature maps of +the l-th layer in the pretrained VGG; µ and σ are the mean +and variance of the features; and the overline represents the +mean-variance channel-wise normalization. +4 + +Identity Loss. Following the work of [37], a pair of identity +losses are constructed to learn the relationship between the +content and style representations. The identity losses are +defined as : +Lid1 = ∥Icc − Ic∥2 + ∥Iss − Is∥2 +(10) +Lid2 = +� +l∈L +∥φl(Icc) − φl(Ic)∥2 + ∥φl(Iss) − φl(Is)∥2 +(11) +where Icc (Iss) denotes the stylized images from a com- +mon pair of content (style) images. Specifically, the origi- +nal content (style) image is expected when we feed two of +the same content (style) images to the model. As shown in +(11), this operation is also applied on feature maps from +the pretrained VGG. And the layer Relu 1 1, Relu 2 1, +Relu 3 1, Relu 4 1, and Relu 5 1 are used to calculate +the second identity loss. +Edge Loss. To enhance the edge of the objects when the +original results from STT are obviously blurred, we design +an edge loss to cope with this problem. As depicted in Sec- +tion 3.2, the edge maps are computed by the Laplacian op- +erator first and then refined by a threshold function and a +mask operation successively. After we obtain the refined +edge maps, the edge loss can be computed in the following +process: +Ledg = ∥edg-Ic − edg-Ics∥2 +(12) +where edg-Ic and edg-Ics are the refined edge maps of the +content and stylized images respectively. As shown in Fig. 1 +and Fig. 7 (columns 3 and 6), applying the edge loss on STT +can obviously improve the edges of blurred results. +4. Experiments +4.1. Implementation Details +Datasets. MS-COCO [75] is used as the content dataset +while WikiArt [76] is used as the style dataset. We ran- +domly select 80000 images of each dataset to build the train- +ing datasets. During the process of training, the input image +will be resized to 512 on the shorter side first and then ran- +domly cropped into 224 × 224. While in the process of +testing, inputs of any size are accepted. +Training Information. Pytorch framework is used to im- +plement STT and 40000 iterations are taken to complete the +training. With a batch size of 4 and an initial learning rate of +1e-4, we use an Adam optimizer [77] to train the network +and the warmup strategy [78] to adjust the learning rate. +The training step is taken about 10 hours on a single Tesla +V100 GPU. We also calculate the reference time (see the +last row of Table 1) of different image style transfer models +with one Tesla P100 GPU. +4.2. Style Transfer Results +In order to demonstrate the style transfer effect of the +proposed STT, we make a comparison between the results +from the proposed STT and the state-of-the-art arbitrary +style transfer methods, including AdaIn [21], WCT [22], +SANet [37], MCC [41], ArtFlow [43], IEST [42], CAST +[46], StyTr2 [45], and S2WAT [47] . +Qualitative Comparison. +The results of the qualitative +comparison are presented in Fig. 4. Although the different +methods fulfill the image style transfer in different ways, +they all achieve colorful results. Due to the over-simplified +alignment of the second-order statistics, AdaIN can not +draw sufficient style patterns on the content images. By +applying the alignment process on the style feature space +with whitening and coloring operations, WCT attracts more +artistic characteristics but damages the content details. In- +spired by the attention mechanism, SANet transfers ade- +quate style features to the content images but the structure is +not ideal sometimes. MCC suffers from an overflow issue +for the lack of linear operations. In conjunction with the +projection flow network, ArtFlow is capable of producing +content-unbiased results but sometimes may generate unde- +sired patterns on the borders. Different from other methods +which train the models with perceptual losses or identity +losses, IEST and CAST adopt the contrastive learning strat- +egy and make favorable effects sometimes. But in some +cases, the results fail to obtain plentiful style representa- +tions. Transformer-based methods find a better balance be- +tween content and style. With the Transformer-based en- +coder and transfer module, StyTr2 and S2WAT both achieve +satisfying effects while S2WAT may lose some style pat- +terns and StyTr2 drops content details in some places. As +shown in the last column of Fig. 4, STT preserves the fine +content details while sufficient artistic characteristics are +transferred. +Quantitative Comparison. In this part, the content differ- +ences between the stylized images and the content images +are computed as an indirect metric to measure the content +quality while the style differences between the results and +the style images are calculated as an implicit metric to eval- +uate the style quality. The identity losses are also taken into +consideration playing a role as the auxiliary metrics to judge +the ability to preserve content/style features. As shown in +Table 1, S2WAT achieves the lowest content loss while STT +and SANet outperform the other methods on style quality. +Compared with the CNN-based models, the Transformer- +based methods have obvious advantages in identity losses. +Due to the ability of completely reversible transformation, +ArtFlow does not use identity losses. Although ArtFlow +can produce content-unbiased results, STT outperforms it +on style quality. In summary, STT can preserve both the +5 + +(a) Content +(b) Style +(c) AdaIN +(d) WCT +(f) SANet +(g) MCC +(h) ArtFlow +(i) IEST +(j) CAST +(k) StyTr2 +(l) S2WAT +(m) Ours +Figure 4. The visual comparison of the state-of-the-art arbitrary style transfer algorithms. +content details from the content image as well as the style +patterns from the style images. +4.3. Content Leak +After repeated stylization with the same pair of content +and style images, CNN-based methods will suffer the prob- +lem of content leak that the content structure will drop grad- +ually as the number of experimental rounds grows. An et +al. [43] utilize the projection flow network, a kind of net- +work which is able to achieve completely reversible trans- +formation, to settle the content leak problem. +However, +strict reversibility may have an undesired impact on the styl- +izing process. With the ability to capture long-range depen- +dencies, StyTr2 [45] and S2WAT [47] are demonstrated to +be capable of alleviating the content leak problem. +To examine the stylizing effects on the content leak is- +sue, we make a comparison with the CNN-based method +[21,22,37,41,42,46], the Flow-based method [43], and the +Transformer-based methods [45,47]. As depicted in Fig. 5, +the results from the 1st and the 20th rounds of repeated styl- +ization have been presented. All the methods can keep the +content details well after the 1st stylizing process except +that the results from AdaIN and ArtFlow are to some degree +lack of style features. However, after the 20th round of the +stylizing process, the CNN-based methods fail to preserve +the content structure and the results are apparently blurred. +Compared to the completely content-unbiased ArtFlow, the +Transformer-based StyTr2, S2WAT, and the proposed STT +still drop the content details slightly but the results are ob- +viously superior to that of the CNN-based methods. There- +fore, the proposed STT can preserve both the content struc- +ture and the style features while capable of alleviating the +content leak problem. +4.4. Ablation Study +Conv PE. Positional encodings (PE) are important for +Transformer-based models, which provide information on +locations. There are two types of absolute positional encod- +ing (APE) that are widely used: functional [49] and para- +metric [51] positional encoding. As Deng et al. [45] have +discussed in StyTr2, the functional APE, such as the sinu- +soidal APE, will result in vertical track artifacts due to the +large positional deviation. And we examine the parametric +APE whose results are shown in Fig. 6 (column 4). Some +undesired patterns that do not vary substantially with the +inputs appear on the outputs. Due to the unsatisfactory per- +formance of the functional APE and parametric APE, we +propose a positional encoding based on convolutional oper- +ations (Conv PE), and the results are presented in Fig. 6 +(column 5). Because the CAPE needs to work with the +transfer module while the transfer module of STT does not +have the interface of PE, we do not conduct the experiments +on CAPE. +As depicted in Fig. 6, the strokes of the results from the +model without PE are obviously thicker than that from the +model with Conv PE. Furthermore, there are a few verti- +cal track artifacts on the edge of objects in images (row 1 +column 3). For the results from the model with parametric +APE, the background is blurry and a sort of undesired pat- +tern makes the pictures unsightly. By contrast, the results +from STT fix these problems and preserve both the content +details and style features. +Edge Loss. When the results of image style transfer are +blurred, applying the edge loss on STT can improve picture +clarity obviously. As depicted in Fig. 1, the model without +the edge loss erases the majority of content details in the +6 + +4749310047493L474934749347493Method +Ours +S2WAT +StyTr2 +CAST +IEST +ArtFlow +MCC +SANet +WCT +AdaIN +Content Loss ↓ +2.18 +1.66 +1.66 +1.66 +1.83 +2.07 +1.81 +1.93 +1.92 +2.16 +2.56 +1.71 +Style Loss ↓ +1.35 +1.74 +1.52 +4.33 +2.72 +1.90 +1.70 +1.11 +1.11 +1.11 +2.23 +3.50 +Identity Loss 1 ↓ +0.16 +0.16 +0.16 +0.16 +0.16 +0.16 +0.26 +1.94 +0.91 +0.00 +1.07 +0.81 +3.01 +2.54 +Identity Loss 2 ↓ +1.55 +1.38 +1.38 +1.38 +3.10 +18.72 +7.16 +0.00 +7.72 +6.03 +21.88 +17.97 +Time(seconds) ↓ +0.270 +0.558 +0.237 +0.042 +0.042 +0.042 +0.061 +0.325 +0.078 +0.061 +0.590 +0.042 +0.042 +0.042 +Table 1. Quantitative comparison between the results from different image style transfer methods. The loss values above are all computed +on 400 random samples average and the reference time is calculated on a hundred random samples in a resolution of 512 × 512. The bold +font marks the best values while the underline shows the second-best values. +Content +Content +Style +Style +Ours +S2WAT +StyTr2 +ArtFlow +CAST +IEST +MCC +SANet +WCT +AdaIN +Round 1 +Round 20 +Round 1 +Round 20 +Figure 5. Visualization of the content leak problem. +Content +Style +No PE +APE +Conv PE (Ours) +Figure 6. Comparison between the results from different types of +PE. +content images, such as the windows on buildings (row 1 +column 3) and the letters on the billboards (row 2 column +3). In contrast, these details are well preserved when the +edge loss is equipped (column 4). +Besides the comparison between the models with and +without the edge loss, we also compare the operators to ex- +tract the edge maps which are the important step to form the +edge loss. As depicted in Fig. 7 (a), the operator of Canny, +Sobel, and Laplacian are taken into consideration. A kind +of hollow stroke appears on the results based on the Canny +operator (see column 4) while the results on the Laplacian +operator can produce natural and fine strokes. The clear- +est result though the model based on the Sobel operator can +generate, unpleasant patterns, such as the vertical/horizontal +tracks and the blurred strokes, appears in the stylized im- +ages (see column 5). For the outputs based on the Laplacian +operator which is applied to the edge loss, the strokes are +natural and the structure of objects is clear which demon- +strate the performance of the edge loss. +In addition, we also provide the edge maps calculated by +the edge extractor where a phenomenon can be easily found +that the edges of the results from the model applying the +edge loss will be much richer than that of the results from +models without the edge loss. +5. Conclusion +In this work, we proposed a Transformer-based method +named STT for arbitrary image style transfer. The proposed +STT has a Transformer-based encoder that can encode both +the content and style images capturing the long-range in- +formation between them. A content-aware positional en- +coding scheme (Conv PE) based on the convolutional op- +erations is applied to the encoder to provide the positional +information. To overcome the problem that the results of +image style transfer are blurred in some cases, a novel edge +loss is presented to improve the clarity of the stylized im- +ages. As another new method based on Transformer, STT +is capable of producing vivid stylized images with fine con- +tent details and sufficient style features while alleviating the +content leak problem. +7 + +漕江分MAContent +Style +w/o Edge Loss +Canny +Sobel +Laplacian (Ours) +(a) Stylized Images +Content +w/o Edge Loss +Canny +Sobel +Laplacian (Ours) +(b) Edge Images +Figure 7. Comparison between the results using different edge detection operators. +References +[1] Alexei A Efros and William T Freeman. 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PMLR, 2020. 5 +11 + diff --git a/ONAyT4oBgHgl3EQftPmP/content/tmp_files/load_file.txt b/ONAyT4oBgHgl3EQftPmP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4c9a45dc564e40712f08a5b89f26c92f58b91572 --- /dev/null +++ b/ONAyT4oBgHgl3EQftPmP/content/tmp_files/load_file.txt @@ -0,0 +1,606 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf,len=605 +page_content='Edge Enhanced Image Style Transfer via Transformers Chiyu Zhang1, Jun Yang2, Zaiyan Dai3, Peng Cao4 Sichuan Normal University 1alienzhang19961005@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='com 2jkxy yjun@sicnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='cn {3daizaiyan, 4pc}@stu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='sicnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='cn Abstract In recent years, arbitrary image style transfer has at- tracted more and more attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Given a pair of content and style images, a stylized one is hoped that retains the content from the former while catching style patterns from the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' However, it is difficult to simultaneously keep well the trade-off between the content details and the style features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' To stylize the image with sufficient style patterns, the content details may be damaged and sometimes the ob- jects of images can not be distinguished clearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' For this reason, we present a new transformer-based method named STT for image style transfer and an edge loss which can enhance the content details apparently to avoid generat- ing blurred results for excessive rendering on style features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Qualitative and quantitative experiments demonstrate that STT achieves comparable performance to state-of-the-art image style transfer methods while alleviating the content leak problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Introduction Rendering a content image into the artistic style of a referenced image is the main purpose of the image style transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Image style transfer is an interesting topic of com- puter vision with a long history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' About 2 decades ago, re- searchers [1,2] utilize techniques, such as texture synthesis and style transfer functions, to achieve the stylizing pro- cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' But they only focus on low-level features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Thereafter Gatys et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' [3, 4] prove that the correlation between fea- tures extracted from a pre-trained VGG can be treated as the representation of style patterns, which opens the gate of neural style transfer (NST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Iterative methods [4–10] ren- der the content images gradually by applying gradients on the input images or the noise images while the feed-forward networks [11–19] can complete the stylizing process in one feed-forward manner after training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Vivid results though the iterative and feed-forward methods may produce, are still limited to a certain number of styles or achieve inadequate style quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Thanks to the encoder-transfer-decoder archi- Content Style w/o Edge Loss w/ Edge Loss (Ours) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Visual effects of edge loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Compared to the results from the model without edge loss (column 3), the objects of styl- ized images with edge loss, especially the small ones like letters or windows, are apparently more clear and distinguishable (column 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' For the convenience of comparison, the model used above is STT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' The results from other methods are also blurred in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' tecture, arbitrary style transfer methods [20–47] are capable of rendering images into any styles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' However, these models may not work well in some cases due to the limited ability to merge the content and style features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' To cope with this problem, the attention mechanism [48,49] is introduced by a few methods [37–42] to enhance the fusion effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Recently, An et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' [43] discover the content leak prob- lem that the structure of results from the CNN-based meth- ods will be dramatically changed after a few rounds of repetitive stylization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' [45] and Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' [47] then prove that the transformer-based methods are capable of alleviating the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Different from the previous methods, IEST [42] and CAST [46] leverage the contrastive learning strategy to enhance the visual quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' However, in some cases, the structure of results from previ- ous methods still could be blurred and the objects in images are difficult to be distinguished (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Thanks to the flexibility, scalability, and ability to cap- ture long-range dependencies, Transformers [49–51] have been widely used in all kinds of vision tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Owing to the self-attention mechanism, Transformer can efficiently gather global information which is important for preserv- ing the structure of input images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Compared to the archi- tecture of typical CNN-based methods, Transformer evades the multi-time downsample operations which may lead to 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='00592v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='CV] 2 Jan 2023 EXPRESSEXPRESS NUSBRILLIA E MONEthe content leak problem [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Therefore, the Transformer structure has a good effect on the image style transfer task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' In this work, we propose a new Transformer-based im- age style transfer algorithm that is capable of producing stylized results with high visual quality while preserving fine content details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' We call it STT (Style Transfer on Transformers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Different from StyTr2 [45] and S2WAT [47], neither does STT choose to encode content and style features in different encoders as StyTr2 did, nor does STT adopts the hierarchal structure as S2WAT did.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' A tiny CNN- based module is equipped as a positional encoding layer (Conv PE) which extracts the positional encoding (PE) ac- cording to the semantic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Furthermore, to en- hance the content details, a novel edge loss is applied as an extra restriction when stylized images are blurred due to the overdose of style features imposed on the inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' The main contributions of our work are as follows: A new image style transfer network name STT which can stylize images with high quality while preserving fine content details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' A novel edge loss to enhance the content details, which improves the picture clarity of the stylized images ob- viously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Extensive experiments demonstrate that STT achieves outstanding effects and is capable of generating favor- able results while preserving fine content details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Related Work Image Style Transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Starting from Gatys et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' [3, 4], the number of methods in NST is increasingly growing with time forward and the stylizing effects have been more and more colorful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Here we make a rough classification of these models with respect to their generalization abili- ties, the backbone architecture, and training strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' In generalization abilities, the categories can be divided into single style transfer [4–15], multiple style transfer [16–18], and arbitrary style transfer [20–47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' The single style trans- fer encodes the fixed style features into models while the multiple style transfer utilizes certain tricks, such as con- ditional instance normalization [16] and StyleBank [17], to support a number of styles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' With a certain module to merge the features of content and style, the arbitrary style trans- fer is capable of handling any style transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' As the tech- niques of upstream tasks like image classification and im- age generation have been rapidly developing these years, an increasing number of sorceries have been introduced into image style transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Except for the typical CNN-based methods, the Flow-based [43] and the Transformer-based [45, 47] methods also appear in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' The Flow- based ArtFlow [43] is proposed to solve the problem of the content leak and the Transformer-based StyTr2 [45] and S2WAT [47] are able to alleviate the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' The encoders of StyTr2 have the traditional Transformer structure where the shape of representations will not be changed in pro- cessing while S2WAT adopts hierarchal architecture which means the features will be downscaled gradually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Recently, the contrastive learning strategy is introduced by IEST [42] and CAST [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Different from other methods trained with perceptual losses or identity losses, IEST and CAST treat the contrastive loss and the adversarial loss as the opti- mization targets to achieve satisfying effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' However, in some cases, the results from the existing image style trans- fer methods may still be blurred due to no restriction on the edge of the main objects in inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Vison Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Inherited the ability to capture the long-range dependencies from Transformers in natural lan- guage processing (NLP), vision Transformers have been de- veloped in a wide variety of vision tasks, including image classification [52–64], object detection [65–69], semantic segmentation [70, 71], and image generation [72, 73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' In image style transfer, StyTr2 and S2WAT have demonstrated that both the traditional structure and the hierarchical ar- chitecture have a favorable effect on style transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' In this paper, we leverage several convolutional operations to ful- fill the positional encoding instead of parametric positional encoding [52] or the one with pooling operations [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Utilization of Edge Maps in Style Transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' The oper- ators like Laplacian, Canny, and Sobel are widely used in edge and contour detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' In image style transfer, Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' discover the correspondences between Laplacian devia- tions and image distortions and then propose Lapstyle [8], an iterative image style transfer method based on a Lapla- cian loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Subsequently, Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' apply the Laplacian fil- ter on the drafting and revision network and then present LapStyle [36], a feed-forward image style transfer method based on the Laplacian filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' However, the above meth- ods are all applied to the CNN-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' In this work, we leverage the edge maps to enhance the results from the Transformer-based STT which is capable of producing styl- ized images with fine content details and colorful artistic features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Method As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' 2, the proposed STT has the architec- ture of encoder-transfer-decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' The positional encoding (PE) is first extracted from both the content images Ic and the style image Is by a module name Conv PE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Then after splitting the content images Ic and style image Is into non- overlapping patches, a linear projection is equipped to trans- form the patches into sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' The sum of the sequences and the PE will be treated as the inputs of the Transformer encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Generated from the encoder, the content features fc and style features fs then will be merged in the transfer module which is based on a Transformer decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Finally, 2 Patch Partition Linear Projection Conv PE Conv PE Layer 1 Layer 2 Layer 3 Layer 1 Layer 2 Layer 3 Transformer Encoder Transformer Decoder Decoder Edge Extractor Edge Extractor Edge Loss LN MSA MHA LN MLP LN fc fs Q K V (a) Net Architecture (b) Transformer Decoder Layer Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' The net architecture of the proposed STT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' the outputs can be obtained by decoding the fused features fcs in a CNN-based decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' In addition, to calculate the edge loss during the training step, the edge maps of the content images and the stylized images need to be extracted by a fine-designed edge extractor which will be introduced later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' In this part, we plan to present the overall architecture of the proposed STT first in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='1 and in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='2 introduce the edge extractor which is used to calculate the edge loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Finally, the optimization strategy will be dis- cussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Overall Architecture Encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Before the process of the encoder, a tiny CNN- based module named Conv PE is applied to the content images and style images to extract the content-aware po- sitional encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' As depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' 3, Conv PE is composed by three convolutional layers, two reflections (padding) layers, and one ReLU activation layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' The main role of the reflection layers is to ensure that the size of re- sults is consistent before and after processing while the con- volutional layers are used to extract the content-aware posi- tional encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' 6, we find that the results from the model with Conv PE are obviously better than that of the one without.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Different from the design of StyTr2 [45] which has two independent domain-specific encoders for the content im- ages and style images, STT treats them as normal pic- tures with content & style features and encodes them in one Transformer-based encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Given the input image in the shape of H × W × 3, the input will first be split into patches by the patch partition layer and then embedded into sequences linearly with the shape of HW 8×8 × C (768 is the default value of C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Adding the positional encoding from Conv PE, the resulting sequences will be fed to a three- layer Transformer encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' The computation process of each layer can be defined as: ˆcl = MSA(LN(cl−1)) + cl−1 (1) cl = MLP(LN(ˆcl)) + ˆcl (2) where ˆcl and cl denote the outputs of MSA and MLP for layer l respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' MSA represents the module of multi- head self-attention while MLP denotes the module of multi- layer perceptron;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' and LN means LayerNorm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' After the three layers of the encoder, we obtain the content features fc and style features fs with the shape consistent before and after processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Input Conv1x1 Reflect Conv3x3 Relu Reflect Conv3x3 Output Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' The illustration of Conv PE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Instead of upsampling the stylized features fcs to the original size in a single projection as [74] once did, STT follows [21, 37, 41] to utilize a mirrored VGG to de- code the stylized features gradually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Before the step of de- coding, the stylized features need to be reshaped first for the sequence-like shape HW 8×8 × C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' After the three stages of upsampling and refining, we obtain the stylized images Ics with the shape of H × W × 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' 3 Transfer Module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' The transfer module is used to merge the content features fc and style features fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' We introduce the transfer module of S2WAT [47] as the means to fuse the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' As depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' 2 (b), the transfer module consists of three layers of the Transformer decoder layers, and each layer is mainly composed by an MSA module, an MHA module, and an MLP module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' The computational process can be defined as: ˆxl = MSA(LN(xl−1)) + xl−1 Q = LN(ˆxl) · WQ K, V = y · WK, y · WV ˜xl = MHA(Q, K, V ) + ˆxl xl = MLP(LN(˜xl)) + ˜xl (3) where ˆxl, ˜xl, and xl represent the results of MSA, MHA, and MLP for layer l, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' y denotes the style fea- tures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' WQ, WK, and WV are the projection matrices for Q, K, and V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Q, K, and V denote the query, key, and value vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Leveraging the fusion effects of the cross attention, the stylized features fcs can be received.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Edge Extractor To make the content structure of the stylized images clear, we design a novel edge loss to enhance the edge of the objects in output images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Before calculating the edge loss, the edge maps suitable for style transfer need to be captured by a fine-designed edge extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Different from the tasks like edge detection or contour extraction, the content de- tails of the outputs in image style transfer are probably not the same as that of the content images, especially the back- ground which may has the artistic patterns from the style images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' The results will be blurred if we take the similar- ity between the edge maps from the content images and the stylized images as the optimization target directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' There- fore one of the problems that need to be solved is to filter out the place where the main structure of the content im- ages does not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' A mask operation is introduced to cope with this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' As shown in (6), all of the edges in the edge maps of the stylized images (edg′-Ics) that are not ex- ist in the corresponding place of the edge maps of content images (edg-Ic) will be masked out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Furthermore, we also set a threshold to exclude the weak responses of edge maps which may play a role as noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' The overall computational process can be defined as: edg-Ic = threshold(lap(Ic), τ) (4) edg′-Ics = threshold(lap(Ics), τ) (5) edg-Ics = mask(edg′-Ics, edg-Ic) (6) where edg-Ic and edg-Ics are the edge maps of the content and stylized images respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' lap denotes the Laplacian operator and threshold represents the function that sets 0 to the responses where the value is smaller than the thresh- old parameter τ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='2 is set as the default value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' After the above steps, we obtain the refined edge maps to be used in calculating the edge loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Network Optimization The main purpose of image style transfer is to maintain the structure of the content images while transferring the artistic patterns to the stylized results from the style images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' To achieve this target, we follow [21] to construct two per- ceptual losses to measure the content differences between the stylized images and the content images as well as the style differences between the stylized images and the style images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Furthermore, we also adopt the identity losses [37] to enrich the content details and style patterns of the styl- ized images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Finally, the proposed edge loss is equipped to enhance the content structure further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' As shown in (7), the whole loss function can be defined as: Ltotal =λcLcontent + λsLstyle+ λid1Lid1 + λid2Lid2 + λedgLedg (7) where the λc, λs, λid1, λid2, and λedg are the weights of losses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Lcontent and Lstyle denote the perceptual losses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Lid1 and Lid2 are the identity losses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Ledg represents the edge loss and we only apply the edge loss in the situation when the results are apparently blurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' We set λc, λs, λid1, λid2, and λedg to 1, 3, 50, 1, and 5000 to alleviate the impact of magnitude differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Perceptual Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Similar to [21], we leverage a pretrained VGG19 to extract the feature maps of the content and style images which are used to calculate the perceptual losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' In our model, the layer Relu 4 1 and Relu 5 1 are used to cal- culate the content perceptual loss while the layer Relu 1 1, Relu 2 1, Relu 3 1, Relu 4 1, and Relu 5 1 are used to calculate the style perceptual loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' One thing that needs to be attended to is that the mean-variance channel-wise nor- malization is applied on the feature maps before the calcu- lation of the content perceptual loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' The perceptual losses can be defined as: Lcontent = � l∈C ∥φl(Ics) − φl(Ic)∥2 (8) Lstyle = � l∈L ∥µ(φl(Ics)) − µ(φl(Is))∥2+ ∥σ(φl(Ics)) − σ(φl(Is))∥2 (9) where the C and L are the layers of the pretrained VGG which are concerned to calculate the content and style per- ceptual losses respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' φl denotes the feature maps of the l-th layer in the pretrained VGG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' µ and σ are the mean and variance of the features;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' and the overline represents the mean-variance channel-wise normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' 4 Identity Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Following the work of [37], a pair of identity losses are constructed to learn the relationship between the content and style representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' The identity losses are defined as : Lid1 = ∥Icc − Ic∥2 + ∥Iss − Is∥2 (10) Lid2 = � l∈L ∥φl(Icc) − φl(Ic)∥2 + ∥φl(Iss) − φl(Is)∥2 (11) where Icc (Iss) denotes the stylized images from a com- mon pair of content (style) images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Specifically, the origi- nal content (style) image is expected when we feed two of the same content (style) images to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' As shown in (11), this operation is also applied on feature maps from the pretrained VGG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' And the layer Relu 1 1, Relu 2 1, Relu 3 1, Relu 4 1, and Relu 5 1 are used to calculate the second identity loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Edge Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' To enhance the edge of the objects when the original results from STT are obviously blurred, we design an edge loss to cope with this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' As depicted in Sec- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='2, the edge maps are computed by the Laplacian op- erator first and then refined by a threshold function and a mask operation successively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' After we obtain the refined edge maps, the edge loss can be computed in the following process: Ledg = ∥edg-Ic − edg-Ics∥2 (12) where edg-Ic and edg-Ics are the refined edge maps of the content and stylized images respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' 1 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' 7 (columns 3 and 6), applying the edge loss on STT can obviously improve the edges of blurred results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Experiments 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Implementation Details Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' MS-COCO [75] is used as the content dataset while WikiArt [76] is used as the style dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' We ran- domly select 80000 images of each dataset to build the train- ing datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' During the process of training, the input image will be resized to 512 on the shorter side first and then ran- domly cropped into 224 × 224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' While in the process of testing, inputs of any size are accepted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Training Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Pytorch framework is used to im- plement STT and 40000 iterations are taken to complete the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' With a batch size of 4 and an initial learning rate of 1e-4, we use an Adam optimizer [77] to train the network and the warmup strategy [78] to adjust the learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' The training step is taken about 10 hours on a single Tesla V100 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' We also calculate the reference time (see the last row of Table 1) of different image style transfer models with one Tesla P100 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Style Transfer Results In order to demonstrate the style transfer effect of the proposed STT, we make a comparison between the results from the proposed STT and the state-of-the-art arbitrary style transfer methods, including AdaIn [21], WCT [22], SANet [37], MCC [41], ArtFlow [43], IEST [42], CAST [46], StyTr2 [45], and S2WAT [47] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Qualitative Comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' The results of the qualitative comparison are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Although the different methods fulfill the image style transfer in different ways, they all achieve colorful results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Due to the over-simplified alignment of the second-order statistics, AdaIN can not draw sufficient style patterns on the content images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' By applying the alignment process on the style feature space with whitening and coloring operations, WCT attracts more artistic characteristics but damages the content details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' In- spired by the attention mechanism, SANet transfers ade- quate style features to the content images but the structure is not ideal sometimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' MCC suffers from an overflow issue for the lack of linear operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' In conjunction with the projection flow network, ArtFlow is capable of producing content-unbiased results but sometimes may generate unde- sired patterns on the borders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Different from other methods which train the models with perceptual losses or identity losses, IEST and CAST adopt the contrastive learning strat- egy and make favorable effects sometimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' But in some cases, the results fail to obtain plentiful style representa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Transformer-based methods find a better balance be- tween content and style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' With the Transformer-based en- coder and transfer module, StyTr2 and S2WAT both achieve satisfying effects while S2WAT may lose some style pat- terns and StyTr2 drops content details in some places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' As shown in the last column of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' 4, STT preserves the fine content details while sufficient artistic characteristics are transferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Quantitative Comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' In this part, the content differ- ences between the stylized images and the content images are computed as an indirect metric to measure the content quality while the style differences between the results and the style images are calculated as an implicit metric to eval- uate the style quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' The identity losses are also taken into consideration playing a role as the auxiliary metrics to judge the ability to preserve content/style features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' As shown in Table 1, S2WAT achieves the lowest content loss while STT and SANet outperform the other methods on style quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Compared with the CNN-based models, the Transformer- based methods have obvious advantages in identity losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Due to the ability of completely reversible transformation, ArtFlow does not use identity losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Although ArtFlow can produce content-unbiased results, STT outperforms it on style quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' In summary, STT can preserve both the 5 (a) Content (b) Style (c) AdaIN (d) WCT (f) SANet (g) MCC (h) ArtFlow (i) IEST (j) CAST (k) StyTr2 (l) S2WAT (m) Ours Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' The visual comparison of the state-of-the-art arbitrary style transfer algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' content details from the content image as well as the style patterns from the style images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Content Leak After repeated stylization with the same pair of content and style images, CNN-based methods will suffer the prob- lem of content leak that the content structure will drop grad- ually as the number of experimental rounds grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' An et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' [43] utilize the projection flow network, a kind of net- work which is able to achieve completely reversible trans- formation, to settle the content leak problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' However, strict reversibility may have an undesired impact on the styl- izing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' With the ability to capture long-range depen- dencies, StyTr2 [45] and S2WAT [47] are demonstrated to be capable of alleviating the content leak problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' To examine the stylizing effects on the content leak is- sue, we make a comparison with the CNN-based method [21,22,37,41,42,46], the Flow-based method [43], and the Transformer-based methods [45,47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' As depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' 5, the results from the 1st and the 20th rounds of repeated styl- ization have been presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' All the methods can keep the content details well after the 1st stylizing process except that the results from AdaIN and ArtFlow are to some degree lack of style features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' However, after the 20th round of the stylizing process, the CNN-based methods fail to preserve the content structure and the results are apparently blurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Compared to the completely content-unbiased ArtFlow, the Transformer-based StyTr2, S2WAT, and the proposed STT still drop the content details slightly but the results are ob- viously superior to that of the CNN-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' There- fore, the proposed STT can preserve both the content struc- ture and the style features while capable of alleviating the content leak problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Ablation Study Conv PE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Positional encodings (PE) are important for Transformer-based models, which provide information on locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' There are two types of absolute positional encod- ing (APE) that are widely used: functional [49] and para- metric [51] positional encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' As Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' [45] have discussed in StyTr2, the functional APE, such as the sinu- soidal APE, will result in vertical track artifacts due to the large positional deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' And we examine the parametric APE whose results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' 6 (column 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Some undesired patterns that do not vary substantially with the inputs appear on the outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Due to the unsatisfactory per- formance of the functional APE and parametric APE, we propose a positional encoding based on convolutional oper- ations (Conv PE), and the results are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' 6 (column 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Because the CAPE needs to work with the transfer module while the transfer module of STT does not have the interface of PE, we do not conduct the experiments on CAPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' As depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' 6, the strokes of the results from the model without PE are obviously thicker than that from the model with Conv PE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Furthermore, there are a few verti- cal track artifacts on the edge of objects in images (row 1 column 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' For the results from the model with parametric APE, the background is blurry and a sort of undesired pat- tern makes the pictures unsightly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' By contrast, the results from STT fix these problems and preserve both the content details and style features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Edge Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' When the results of image style transfer are blurred, applying the edge loss on STT can improve picture clarity obviously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' As depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' 1, the model without the edge loss erases the majority of content details in the 6 4749310047493L474934749347493Method Ours S2WAT StyTr2 CAST IEST ArtFlow MCC SANet WCT AdaIN Content Loss ↓ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='66 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='66 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='66 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='83 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='81 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='93 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='92 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='56 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='71 Style Loss ↓ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='74 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='52 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='33 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='72 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='90 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='70 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='23 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='50 Identity Loss 1 ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='26 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='81 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='54 Identity Loss 2 ↓ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='55 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='38 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='38 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='38 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='10 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='72 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='00 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='72 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='03 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='88 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='97 Time(seconds) ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='270 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='558 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='061 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='590 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='042 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='042 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content='042 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Quantitative comparison between the results from different image style transfer methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' The loss values above are all computed on 400 random samples average and the reference time is calculated on a hundred random samples in a resolution of 512 × 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' The bold font marks the best values while the underline shows the second-best values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Content Content Style Style Ours S2WAT StyTr2 ArtFlow CAST IEST MCC SANet WCT AdaIN Round 1 Round 20 Round 1 Round 20 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Visualization of the content leak problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Content Style No PE APE Conv PE (Ours) Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Comparison between the results from different types of PE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' content images, such as the windows on buildings (row 1 column 3) and the letters on the billboards (row 2 column 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' In contrast, these details are well preserved when the edge loss is equipped (column 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Besides the comparison between the models with and without the edge loss, we also compare the operators to ex- tract the edge maps which are the important step to form the edge loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' As depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' 7 (a), the operator of Canny, Sobel, and Laplacian are taken into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' A kind of hollow stroke appears on the results based on the Canny operator (see column 4) while the results on the Laplacian operator can produce natural and fine strokes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' The clear- est result though the model based on the Sobel operator can generate, unpleasant patterns, such as the vertical/horizontal tracks and the blurred strokes, appears in the stylized im- ages (see column 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' For the outputs based on the Laplacian operator which is applied to the edge loss, the strokes are natural and the structure of objects is clear which demon- strate the performance of the edge loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' In addition, we also provide the edge maps calculated by the edge extractor where a phenomenon can be easily found that the edges of the results from the model applying the edge loss will be much richer than that of the results from models without the edge loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' Conclusion In this work, we proposed a Transformer-based method named STT for arbitrary image style transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' The proposed STT has a Transformer-based encoder that can encode both the content and style images capturing the long-range in- formation between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' A content-aware positional en- coding scheme (Conv PE) based on the convolutional op- erations is applied to the encoder to provide the positional information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' To overcome the problem that the results of image style transfer are blurred in some cases, a novel edge loss is presented to improve the clarity of the stylized im- ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' As another new method based on Transformer, STT is capable of producing vivid stylized images with fine con- tent details and sufficient style features while alleviating the content leak problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQftPmP/content/2301.00592v1.pdf'} +page_content=' 7 漕江分MAContent Style w/o Edge Loss Canny Sobel Laplacian (Ours) (a) Stylized Images Content w/o Edge Loss Canny Sobel Laplacian (Ours) (b) Edge Images Figure 7.' metadata={'source': 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e-learning platforms +Maksimjeet Chowdhary∗ +maksimjeet20566@iiitd.ac.in +IIIT Delhi +Delhi, India +Sanyam Goyal∗ +sanyam20116@iiitd.ac.in +IIIT Delhi +Delhi, India +Venktesh V +venkteshv@iiitd.ac.in +IIIT Delhi +Delhi, India +Mukesh Mohania +mukesh@iiitd.ac.in +IIIT Delhi +Delhi, India +Vikram Goyal +vikram@iiitd.ac.in +IIIT Delhi +Delhi, India +ABSTRACT +Online learning platforms provide diverse questions to gauge the +learners’ understanding of different concepts. The repository of +questions has to be constantly updated to ensure a diverse pool +of questions to conduct assessments for learners. However, it is +impossible for the academician to manually skim through the large +repository of questions to check for duplicates when onboarding +new questions from external sources. Hence, we propose a tool +QDup in this paper that can surface near-duplicate and semanti- +cally related questions without any supervised data. The proposed +tool follows an unsupervised hybrid pipeline of statistical and neu- +ral approaches for incorporating different nuances in similarity +for the task of question duplicate detection. We demonstrate that +QDup can detect near-duplicate questions and also suggest related +questions for practice with remarkable accuracy and speed from +a large repository of questions. The demo video of the tool can be +found at https://www.youtube.com/watch?v=loh0_-7XLW4. +CCS CONCEPTS +• Information systems → Information retrieval; • Applied com- +puting → Document searching. +KEYWORDS +semantic similarity, duplicate detection +ACM Reference Format: +Maksimjeet Chowdhary, Sanyam Goyal, Venktesh V, Mukesh Mohania, +and Vikram Goyal. 2023. Unsupervised Question Duplicate and Related +Questions Detection in e-learning platforms. In Proceedings of the Sixteenth +ACM International Conference on Web Search and Data Mining (WSDM ’23), +February 27-March 3, 2023, Singapore, Singapore. ACM, New York, NY, USA, +4 pages. https://doi.org/10.1145/3539597.3573035 +∗Both authors contributed equally to this research. +Permission to make digital or hard copies of all or part of this work for personal or +classroom use is granted without fee provided that copies are not made or distributed +for profit or commercial advantage and that copies bear this notice and the full citation +on the first page. Copyrights for components of this work owned by others than ACM +must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, +to post on servers or to redistribute to lists, requires prior specific permission and/or a +fee. Request permissions from permissions@acm.org. +WSDM ’23, February 27-March 3, 2023, Singapore, Singapore +© 2023 Association for Computing Machinery. +ACM ISBN 978-1-4503-9407-9/23/02...$15.00 +https://doi.org/10.1145/3539597.3573035 +1 +INTRODUCTION +The e-learning platforms usually curate a large repository of ques- +tions across subjects, chapters, and topics for conducting assess- +ments to test the understanding of the learner. These reposito- +ries are constantly augmented with new questions. The new ques- +tions could be collected in batches from other platforms or external +sources. They could also be added manually by the academicians. +When new questions are added, there are cases of them being near- +duplicates or related to existing questions in the data repository. +It is impossible for the academicians to manually skim through +the entire repository to check for duplicates. Hence, in this work, +we propose a tool with support for bulk on-boarding of questions +while surfacing duplicate questions already present in the database. +The duplicate question detection task, particularly in the context +of e-learning platforms, is a significant challenge due to the nature +of the questions. Two questions can differ in entities or techni- +cal concepts though their verbiage and the rest of the semantics +could be similar. In certain cases, though the questions are centered +around the same entity and have mostly similar verbiage, the an- +swers could be different. For example, the questions What is GDP? +and What is the significance of GDP? might have high Jaccard or co- +sine similarity but are not duplicate questions. Hence, to encompass +the mentioned scenarios, we define two questions to be duplicates +of each other if they satisfy all of the following conditions: +• The questions are lexically similar and have synonymous +keyphrases or entities. +• The questions are semantically related. +• The correct answers to both the questions are equivalent +We also recommend related questions to aid the academicians +in generating diverse questions for assessments. For instance, the +questions What is the strongest bone in the body? and What is the +weakest bone in the body? are related questions. +The duplicate text detection [2–4, 7, 10] is a well explored prob- +lem. These approaches range from comparing topics obtained through +topic modelling [10], comparing syntactic structure [4] to neural +IR based methods [3]. However, these approaches consider only +uni-dimensional aspects of similarity, as mentioned earlier, and +fail to identify duplicates in other scenarios where the questions +only differ in entities. They also require significant amounts of +the labeled dataset where questions are labeled as duplicates like +arXiv:2301.05150v1 [cs.CL] 20 Dec 2022 + +WSDM ’23, February 27-March 3, 2023, Singapore, Singapore +Maksimjeet and Sanyam, et al. +Figure 1: Duplicate Question Detection Pipleine +CQADupStack [6, 8], which is not available in the problem setting +explained in this paper. +The pipeline proposed in this paper is unsupervised and efficient +in that it does not require any training. Since our approach is hybrid +and uses a combination of classical and neural IR approaches, it +is also efficient at inference time. The overview of the proposed +pipeline can be seen in Figure 1. In summary, our core contributions +are: +• We propose an unsupervised approach for near-duplicate +detection in online learning platforms to enable smooth on- +boarding of new questions. We also recommend related ques- +tions for serving diverse questions. +• We develop and release an easy-to-use tool that can support +both individual and bulk on-boarding of questions at +https://github.com/ADS-AI/QDup. +2 +SYSTEM DESIGN +In this section, we describe the methodology used for searching for +duplicate questions with respect to a large existing question repos- +itory. Given an input question 𝑞𝑛𝑒𝑤 = {𝑥1,𝑥2...𝑥𝑛} of sequence +length 𝑛 our goal is to surface exact duplicate questions 𝑞𝑑𝑢𝑝𝑒𝑥𝑎𝑐𝑡, +near-duplicates 𝑞𝑑𝑢𝑝 and related questions 𝑞𝑟𝑒𝑙. We present an +unsupervised pipeline that uses an iterative elimination approach, +removing questions that are certainly non-duplicates and retaining +exact or near-duplicate questions. The proposed approach is differ- +ent from existing paraphrase identification or duplicate detection +approaches as it covers different aspects of similarity in a single +pipeline with no supervised data. The pipeline proposed is shown +in Figure 1. The pipeline consists of the following stages : +(1) Preprocessing and hierarchical learning taxonomy tagging +(2) Jaccard similarity between questions tagged with similar +learning taxonomy. +(3) Named Entity Recognition for computing entity differences. +(4) Overlap of key concepts obtained through concept extraction +algorithm and negation detection. +2.1 +Preprocessing and Indexing by +Hierarchical Learning Taxonomy +Given a question 𝑞𝑛𝑒𝑤 as input, we preprocess the question, such +as sentence level tokenization, removing HTML tags, and non- +alphanumeric characters, and removing punctuation marks. The +database includes questions asked in high school and belongs to +various subjects, including chemistry, physics etc. +Therefore we normalize chemical element abbreviations and +symbols to their complete form (Cl → chlorine, pi → 𝜋 etc.) using +a dictionary 𝑑𝑖𝑐𝑡𝑠𝑦𝑚 to ensure consistency resulting in 𝑞𝑛𝑜𝑟𝑚. +𝑞𝑛𝑜𝑟𝑚 = 𝑓𝑛𝑜𝑟𝑚(𝑞𝑛𝑒𝑤) +𝑆 ← 𝑡𝑜𝑘𝑒𝑛𝑖𝑧𝑒(𝑞𝑛𝑒𝑤) +𝑓𝑛𝑜𝑟𝑚 = 𝑑𝑖𝑐𝑡𝑠𝑦𝑚[𝑠𝑖] 𝑓 𝑜𝑟 𝑠𝑖 𝑖𝑛 𝑆 +After preprocessing the input question, we tag the input question +to its standardized hierarchical learning taxonomy of form subject +- chapter - topic using the TagRec [9] model. The TagRec approach +follows a two-tower transformers-based architecture that aligns +the vector subspaces of the input question and the hierarchical +learning taxonomy using a contrastive learning approach. We use +this trained model to tag our database of questions and 𝑞𝑛𝑒𝑤 in a +zero-shot setting and index the questions according to the tags. +We extract the subject portion of the taxonomy to which the +question belongs and query the complete database to return the +candidate set 𝑆𝑐𝑎𝑛𝑑 = { 𝑞1 , 𝑞2 , . . . . 𝑞𝑛 } of all the questions in the +database that belong to the same subject. Our dataset primarily +consists of questions from the subjects: Physics, Chemistry, Social +Science, etc. For example, the question How many 𝜋 bonds are +present in ferrocene? belongs to the subject Chemistry. +2.2 +Token level comparison +After getting the set 𝑆𝑐𝑎𝑛𝑑 of the questions belonging to the same +subject as from the same hierarchy as the input question 𝑞𝑛𝑒𝑤, the +model iterates over 𝑆𝑐𝑎𝑛𝑑 and checks for the 𝐽𝑎𝑐𝑐𝑎𝑟𝑑𝑆𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦 +measure. + +Search space pruning +Preprocessing +Subject prediction using +NEs and Keywords +Removing +TagRec(e.g.chemistry) +Comparison +punctuations +Input +Preprocessed +Chemicalsymbols +Question x +Input +Input +question +Input ++name (H +belonging to +question and +question +question +same subject +set of +Input +Potential +hydrogen) +potential +Special +question +candidates +candidates +characters +words (TT-pi) +Candidates with high +word share +Give one example of a reaction where order and molecularity are equal +Candidates witha high +Relatedguestions +keyword share & same NEs +Duplicate questions +Duplicate +Represent +the question +questions +Give four differences between Order and Molecularity of a reaction +embedding +Only leave questions with +asan +same negation +Closest questions +For which type of reactions order and molecularity have the same valu +from DB in +embedding space +An example of questions with the closest embeddings i.e, related questionsUnsupervised Question Duplicate and Related Questions Detection in e-learning platforms +WSDM ’23, February 27-March 3, 2023, Singapore, Singapore +More formally, Let 𝑞1 , 𝑞2 be two lists of tokens for the input +questions, then Jaccard similarity between these two questions can +be calculated as +𝐽 (𝑞1,𝑞2) = #(𝑞1 ∩ 𝑞2) +#(𝑞1 ∪ 𝑞2) +(1) +If the Jaccard similarity (𝐽 (𝑞𝑛𝑒𝑤,𝑞𝑖)) between the input ques- +tion and a question from 𝑆𝑐𝑎𝑛𝑑 is less than a certain threshold +(𝐽 (𝑞𝑛𝑒𝑤,𝑞𝑖) < 0.4) we remove that question from our search space +𝑆𝑐𝑎𝑛𝑑. The threshold value of 0.4 was chosen after multiple itera- +tions and validation of the results for the dataset that we worked +on. +𝑆𝑐𝑎𝑛𝑑 ← 𝑆𝑐𝑎𝑛𝑑 − 𝑞𝑖 (𝑖𝑓 𝐽 (𝑞𝑛𝑒𝑤,𝑞𝑖) < 0.4) +If the Jaccard similarity is 1 we directly add that question to our +exact duplicate question set (𝑞𝑑𝑢𝑝𝑒𝑥𝑎𝑐𝑡). +2.3 +NER and comparison +To further partition 𝑆𝑐𝑎𝑛𝑑, we remove questions with different +named entities than those in 𝑞𝑛𝑒𝑤. For extracting the set of named +entities (𝑁𝐸𝑞) of 𝑞𝑛𝑒𝑤 we use spaCy, an implementation in Python. +𝑁𝐸𝑞 = 𝑁𝐸𝑅(𝑞𝑛𝑒𝑤) +Examples : Who is the CEO of Google ? → ’Google’: ORG , Who is +the CEO of Apple ? → ‘Apple’ : ORG +The Named Entity Recognition step performs a sequence labeling +task where the noun phrases are tagged with ’PERSON’, ’ORG’, +’LOC’, etc as applicable. Once extracted, the set of entities for 𝑞𝑛𝑒𝑤 +is compared to the set of entities 𝑁𝐸𝑖 for question 𝑞𝑖, where i = 1... +|𝑆𝑐𝑎𝑛𝑑|. All those questions which have a non-empty difference set +between 𝑁𝐸𝑞 and 𝑁𝐸𝑖 are removed from the search space (𝑆𝑐𝑎𝑛𝑑). +𝑆𝑐𝑎𝑛𝑑 ← 𝑆𝑐𝑎𝑛𝑑 − 𝑞𝑖 𝑖𝑓 𝑁𝐸𝑞 ∩ 𝑁𝐸𝑖 ≠ ∅ +2.4 +Keyphrase extraction and calculating the +overlap +Following the previous stages of the pipeline, the set 𝑆𝑐𝑎𝑛𝑑 = { 𝑞1 +, 𝑞2 , . . . . 𝑞𝑛 } is left of potential candidates for a duplicate ques- +tion. The next stage of the pipeline (as shown in Figure 1) is to run +an unsupervised method to automatically extract concept terms +(keyphrases) from the input question 𝑞𝑛𝑒𝑤 into a set 𝐾𝑊𝑖. We lever- +age the EmbedRank algorithm [1] for extracting keyphrases. The +proposed approach first extracts candidate phrases using POS tags +and projects them and the original question 𝑞𝑛𝑒𝑤 to a continuous +vector space. It then computes the semantic relatedness between +the question and the phrase representations and retrieves the top 𝑘 +keyphrases. For all the questions, we pre-compute the keyphrases +and index them. We run a comparison to determine the percentage +of keyphrases shared between 𝐾𝑊𝑖, and the set of keyphrases ex- +tracted for each of the questions in set 𝑆𝑐𝑎𝑛𝑑. Questions that have +keyphrases sharing score of less than 0.7 (chosen after multiple +validations) are eliminated from 𝑆𝑐𝑎𝑛𝑑. +𝐾𝑊𝑖 ← 𝐸𝑚𝑏𝑒𝑑𝑅𝑎𝑛𝑘(𝑞𝑛𝑒𝑤) +𝑆𝑐𝑎𝑛𝑑 ← 𝑆𝑐𝑎𝑛𝑑 − 𝑞𝑖 𝑖𝑓 𝐾𝑊𝑠ℎ𝑎𝑟𝑒 < 0.7 +2.5 +Negation detection +As a result of the previous steps, the set 𝑆𝑐𝑎𝑛𝑑 is much smaller in +size and has questions very similar to 𝑞𝑛𝑒𝑤. However, we observed +that multiple questions with similar verbiage exist though they +differ by a negation resulting them having different answers. For +example: What is an example of a metal ? and What is not an example +of a metal ? +Similar cases might still be left in 𝑆𝑐𝑎𝑛𝑑, and hence we check for +the difference in negation. We compare 𝑞𝑛𝑒𝑤 against each question +in 𝑆𝑐𝑎𝑛𝑑 and eliminate any questions that may be the negation +of 𝑞𝑛𝑒𝑤 by ensuring that standard negation constructions, if any, +are present in both samples being compared. This ensures that +questions with high levels of Jaccard similarity and overlapping +keyphrases shares but differing by a single negation are not identi- +fied as duplicates (false positives). After this stage, we assign the +remaining questions to be duplicates. +2.6 +Related Questions +The above-mentioned pipeline focuses on a higher recall by sacri- +ficing precision since the problem statement focuses on e-learning +platforms being able to rid their database of duplicates. An addi- +tional property of our tool is that these platforms can test students +on their knowledge of the topic by retrieving related questions +which center around the same or similar topics. +Such questions are referred to as related questions in this paper +and are computed by utilizing the architecture of𝑎𝑙𝑙−𝑚𝑝𝑛𝑒𝑡−𝑏𝑎𝑠𝑒− +𝑣2 sentence transformers model1. In the approach, for a duplicate +question 𝑞𝑑𝑢𝑝 of an input question 𝑞𝑛𝑒𝑤, we find the questions that +have embeddings closest to 𝑞𝑑𝑢𝑝𝑒𝑥𝑎𝑐𝑡 or 𝑞𝑑𝑢𝑝 (pre-computed in +the database), measured by the cosine similarity between the two +embedding vectors and return the 3 closest neighbors. +The results demonstrated that nearest neighbors search over a +large database gave slow performance, which led us to leverage +𝑆𝑐𝑎𝑁𝑁 (an efficient searching technique developed by [5]. The set +of embeddings for all the questions in the database is precalculated +(using the same 𝑎𝑙𝑙 − 𝑚𝑝𝑛𝑒𝑡 − 𝑏𝑎𝑠𝑒 − 𝑣2 model) and stored locally +for higher efficiency during running. +For every input 𝑞𝑛𝑒𝑤, we have 𝑞𝑑𝑢𝑝𝑒𝑥𝑎𝑐𝑡, 𝑞𝑑𝑢𝑝 and related ques- +tions 𝑞𝑑𝑢𝑝𝑟𝑒𝑙 (𝑞𝑟𝑒𝑙 is non-empty only if 𝑞𝑑𝑢𝑝 is non-empty). +3 +DEMONSTRATION +We demo our tool from the perspective of it’s ability to perform +near-duplicate detection, analysis to gauge usability of the tool. +3.1 +Dataset +The dataset we used consists of 114804 secondary high school +questions from the CBSE (India) curated with the help of a leading +e-learning platform. The dataset statistics are shown in Figure 3. +3.2 +Evaluation +The tool was evaluated on a set of 100 input questions by two in- +dependent researchers. The tool was provided with 100 random +1 + +WSDM ’23, February 27-March 3, 2023, Singapore, Singapore +Maksimjeet and Sanyam, et al. +Figure 2: Screenshot of the tool QDup +Figure 3: Dataset Statistics +Method +Accuracy (%) +QDup +81.5 +keyphrases based +76.5 +Closest neighbours +51.5 +Table 1: Performance Evaluation for Duplicate Detection +questions across domains and the researchers were requested to la- +bel the correct duplicates as 1 or 0 in all other conditions. Similarly, +the outputs from other approaches were also provided to the re- +searchers for labeling. These approaches included nearest neighbor +search for embeddings extracted with all-mpnet-base-v2 sentence +embeddings model and comparison of keyphrases extracted using +EmbedRank. We observed a Cohen’s kappa of 0.60, 0.72 and 0.65 +in the three scenarios, respectively indicating substantial agree- +ment between annotators. We report the accuracy in Table 1. We +observe that the proposed approach QDup outperforms classical +keyphrases only or vector based nearest neighbor search methods. +3.3 +Tool Ease of Use +We also conducted a user study with 14 well trained academicians. +A screenshot of the tool is shown in Figure 2. We asked the users +to rate the tool on a scale of 1-3 (lowest to highest) from aspects of +intuitiveness, responsiveness and relevance of output. The intuitive- +ness metric indicates how intuitive and easy to use the interface is +without external help. The responsiveness measures the response +time and relevance measures how much the users think the output +for the given questions are accurate duplicates. We observed that +the average intuitiveness score is 2.46 and average responsiveness +score is 2.78. The average relevance score is 2.68. We observe that +the majority of the users find the tool easy to use. +4 +CONCLUSION AND FUTURE WORK +In this paper, we propose a tool to find duplicates and related ques- +tions in a large repository. The proposed approach is resource and +time efficient, and the interface is easy to use. In the future, we plan +to use the data collected from this tool as weakly supervised data to +train a bi-encoder transformer-based model in a contrastive setting +to identify duplicate and related questions in one stage. We also +plan to explore knowledge distillation and quantization approaches +for efficient deployment of the model. +5 +ACKNOWLEDGMENTS +We thank Extramarks and SERB-FICCI for the support. +REFERENCES +[1] Kamil Bennani-Smires, Claudiu Musat, Andreea Hossmann, Michael Baeriswyl, +and Martin Jaggi. 2018. 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CQADupStack: +A Benchmark Data Set for Community Question-Answering Research (ADCS +’15). +[7] Di Liang, Fubao Zhang, Weidong Zhang, Qi Zhang, Jinlan Fu, Minlong Peng, Tao +Gui, and Xuanjing Huang. 2019. Adaptive Multi-Attention Network Incorporating +Answer Information for Duplicate Question Detection (SIGIR’19). 95–104. +[8] Preslav Nakov, Lluís Màrquez, Alessandro Moschitti, Walid Magdy, Hamdy +Mubarak, Abed Alhakim Freihat, Jim Glass, and Bilal Randeree. 2016. SemEval- +2016 Task 3: Community Question Answering. ACL, San Diego, California. +[9] Venktesh V, Mukesh Mohania, and Vikram Goyal. 2021. TagRec: Automated +Tagging of Questions with Hierarchical Learning Taxonomy. +[10] Kai Zhang, Wei Wu, Haocheng Wu, Zhoujun Li, and Ming Zhou. 2014. Question +Retrieval with High Quality Answers in Community Question Answering (CIKM +’14). Association for Computing Machinery, New York, NY, USA, 371–380. + +Online Question Duplicity Finder +An application to prevent academicians from creating duplicate questions by suggesting near duplicates of a question present in the database +Definition of Duplicacy : +Add Question +Add CSV File +Question +Question Duplicates +ferric chloride stops bleeding because +1. Ferric chloride is applied to stop bleeding because +Answer +Related questions +Enter Answer or Leave it empty. +1. The aluminium salt commonly used to stop bleeding is +2. A blood red colour is obtained when ferric chloride solution reacts +with +3. The salt of aluminium commonly used to stop bleeding is +SubmitSubject Tag +Science +Social Science +Computer Science +Chemistry +Physics +Political Science +0 +10,000 +20,000 +30,000 +40,000 +Count \ No newline at end of file diff --git a/PdE4T4oBgHgl3EQfkQ3K/content/tmp_files/load_file.txt b/PdE4T4oBgHgl3EQfkQ3K/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4e29990a71a98f644ff4949ddd5b152bf4ded33c --- /dev/null +++ b/PdE4T4oBgHgl3EQfkQ3K/content/tmp_files/load_file.txt @@ -0,0 +1,295 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf,len=294 +page_content='Unsupervised Question Duplicate and Related Questions Detection in e-learning platforms Maksimjeet Chowdhary∗ maksimjeet20566@iiitd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='in IIIT Delhi Delhi, India Sanyam Goyal∗ sanyam20116@iiitd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='in IIIT Delhi Delhi, India Venktesh V venkteshv@iiitd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='in IIIT Delhi Delhi, India Mukesh Mohania mukesh@iiitd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='in IIIT Delhi Delhi, India Vikram Goyal vikram@iiitd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='in IIIT Delhi Delhi, India ABSTRACT Online learning platforms provide diverse questions to gauge the learners’ understanding of different concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' The repository of questions has to be constantly updated to ensure a diverse pool of questions to conduct assessments for learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' However, it is impossible for the academician to manually skim through the large repository of questions to check for duplicates when onboarding new questions from external sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' Hence, we propose a tool QDup in this paper that can surface near-duplicate and semanti- cally related questions without any supervised data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' The proposed tool follows an unsupervised hybrid pipeline of statistical and neu- ral approaches for incorporating different nuances in similarity for the task of question duplicate detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' We demonstrate that QDup can detect near-duplicate questions and also suggest related questions for practice with remarkable accuracy and speed from a large repository of questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' The demo video of the tool can be found at https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='youtube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='com/watch?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='v=loh0_-7XLW4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' CCS CONCEPTS Information systems → Information retrieval;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' • Applied com- puting → Document searching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' KEYWORDS semantic similarity, duplicate detection ACM Reference Format: Maksimjeet Chowdhary, Sanyam Goyal, Venktesh V, Mukesh Mohania, and Vikram Goyal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' Unsupervised Question Duplicate and Related Questions Detection in e-learning platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' In Proceedings of the Sixteenth ACM International Conference on Web Search and Data Mining (WSDM ’23), February 27-March 3, 2023, Singapore, Singapore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' ACM, New York, NY, USA, 4 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='1145/3539597.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='3573035 ∗Both authors contributed equally to this research.' metadata={'source': 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requires prior specific permission and/or a fee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' Request permissions from permissions@acm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' WSDM ’23, February 27-March 3, 2023, Singapore, Singapore © 2023 Association for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' ACM ISBN 978-1-4503-9407-9/23/02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='$15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='00 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='1145/3539597.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='3573035 1 INTRODUCTION The e-learning platforms usually curate a large repository of ques- tions across subjects, chapters, and topics for conducting assess- ments to test the understanding of the learner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' These reposito- ries are constantly augmented with new questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' The new ques- tions could be collected in batches from other platforms or external sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' They could also be added manually by the academicians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' When new questions are added, there are cases of them being near- duplicates or related to existing questions in the data repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' It is impossible for the academicians to manually skim through the entire repository to check for duplicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' Hence, in this work, we propose a tool with support for bulk on-boarding of questions while surfacing duplicate questions already present in the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' The duplicate question detection task, particularly in the context of e-learning platforms, is a significant challenge due to the nature of the questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' Two questions can differ in entities or techni- cal concepts though their verbiage and the rest of the semantics could be similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' In certain cases, though the questions are centered around the same entity and have mostly similar verbiage, the an- swers could be different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' For example, the questions What is GDP?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' and What is the significance of GDP?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' might have high Jaccard or co- sine similarity but are not duplicate questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' Hence, to encompass the mentioned scenarios, we define two questions to be duplicates of each other if they satisfy all of the following conditions: The questions are lexically similar and have synonymous keyphrases or entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' The questions are semantically related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' The correct answers to both the questions are equivalent We also recommend related questions to aid the academicians in generating diverse questions for assessments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' For instance, the questions What is the strongest bone in the body?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' and What is the weakest bone in the body?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' are related questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' The duplicate text detection [2–4, 7, 10] is a well explored prob- lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' These approaches range from comparing topics obtained through topic modelling [10], comparing syntactic structure [4] to neural IR based methods [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' However, these approaches consider only uni-dimensional aspects of similarity, as mentioned earlier, and fail to identify duplicates in other scenarios where the questions only differ in entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' They also require significant amounts of the labeled dataset where questions are labeled as duplicates like arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='05150v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='CL] 20 Dec 2022 WSDM ’23, February 27-March 3, 2023, Singapore, Singapore Maksimjeet and Sanyam, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' Figure 1: Duplicate Question Detection Pipleine CQADupStack [6, 8], which is not available in the problem setting explained in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' The pipeline proposed in this paper is unsupervised and efficient in that it does not require any training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' Since our approach is hybrid and uses a combination of classical and neural IR approaches, it is also efficient at inference time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' The overview of the proposed pipeline can be seen in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' In summary, our core contributions are: We propose an unsupervised approach for near-duplicate detection in online learning platforms to enable smooth on- boarding of new questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' We also recommend related ques- tions for serving diverse questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' We develop and release an easy-to-use tool that can support both individual and bulk on-boarding of questions at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='com/ADS-AI/QDup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' 2 SYSTEM DESIGN In this section, we describe the methodology used for searching for duplicate questions with respect to a large existing question repos- itory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' Given an input question 𝑞𝑛𝑒𝑤 = {𝑥1,𝑥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='𝑥𝑛} of sequence length 𝑛 our goal is to surface exact duplicate questions 𝑞𝑑𝑢𝑝𝑒𝑥𝑎𝑐𝑡, near-duplicates 𝑞𝑑𝑢𝑝 and related questions 𝑞𝑟𝑒𝑙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' We present an unsupervised pipeline that uses an iterative elimination approach, removing questions that are certainly non-duplicates and retaining exact or near-duplicate questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' The proposed approach is differ- ent from existing paraphrase identification or duplicate detection approaches as it covers different aspects of similarity in a single pipeline with no supervised data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' The pipeline proposed is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' The pipeline consists of the following stages : (1) Preprocessing and hierarchical learning taxonomy tagging (2) Jaccard similarity between questions tagged with similar learning taxonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' (3) Named Entity Recognition for computing entity differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' (4) Overlap of key concepts obtained through concept extraction algorithm and negation detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='1 Preprocessing and Indexing by Hierarchical Learning Taxonomy Given a question 𝑞𝑛𝑒𝑤 as input, we preprocess the question, such as sentence level tokenization, removing HTML tags, and non- alphanumeric characters, and removing punctuation marks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' The database includes questions asked in high school and belongs to various subjects, including chemistry, physics etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' Therefore we normalize chemical element abbreviations and symbols to their complete form (Cl → chlorine, pi → 𝜋 etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=') using a dictionary 𝑑𝑖𝑐𝑡𝑠𝑦𝑚 to ensure consistency resulting in 𝑞𝑛𝑜𝑟𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' 𝑞𝑛𝑜𝑟𝑚 = 𝑓𝑛𝑜𝑟𝑚(𝑞𝑛𝑒𝑤) 𝑆 ← 𝑡𝑜𝑘𝑒𝑛𝑖𝑧𝑒(𝑞𝑛𝑒𝑤) 𝑓𝑛𝑜𝑟𝑚 = 𝑑𝑖𝑐𝑡𝑠𝑦𝑚[𝑠𝑖] 𝑓 𝑜𝑟 𝑠𝑖 𝑖𝑛 𝑆 After preprocessing the input question, we tag the input question to its standardized hierarchical learning taxonomy of form subject chapter - topic using the TagRec [9] model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' The TagRec approach follows a two-tower transformers-based architecture that aligns the vector subspaces of the input question and the hierarchical learning taxonomy using a contrastive learning approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' We use this trained model to tag our database of questions and 𝑞𝑛𝑒𝑤 in a zero-shot setting and index the questions according to the tags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' We extract the subject portion of the taxonomy to which the question belongs and query the complete database to return the candidate set 𝑆𝑐𝑎𝑛𝑑 = { 𝑞1 , 𝑞2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' 𝑞𝑛 } of all the questions in the database that belong to the same subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' Our dataset primarily consists of questions from the subjects: Physics, Chemistry, Social Science, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' For example, the question How many 𝜋 bonds are present in ferrocene?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' belongs to the subject Chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='2 Token level comparison After getting the set 𝑆𝑐𝑎𝑛𝑑 of the questions belonging to the same subject as from the same hierarchy as the input question 𝑞𝑛𝑒𝑤, the model iterates over 𝑆𝑐𝑎𝑛𝑑 and checks for the 𝐽𝑎𝑐𝑐𝑎𝑟𝑑𝑆𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦 measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' Search space pruning Preprocessing Subject prediction using NEs and Keywords Removing TagRec(e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='chemistry) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='Comparison ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='punctuations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='Input ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='Preprocessed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='Chemicalsymbols ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='Question x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='Input ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='Input ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='question ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='Input ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='+name (H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='belonging to ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='question and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='question ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='question ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='same subject ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='set of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='Input ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='Potential ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='hydrogen) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='potential ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='Special ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='question ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='candidates ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='candidates ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='characters ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='words (TT-pi) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='Candidates with high ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='word share ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='Give one example of a reaction where order and molecularity are equal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='Candidates witha high ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='Relatedguestions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='keyword share & same NEs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='Duplicate questions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='Duplicate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='Represent ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='the question ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='questions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='Give four differences between Order and Molecularity of a reaction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='embedding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='Only leave questions with ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='asan ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='same negation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='Closest questions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='For which type of reactions order and molecularity have the same valu ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='from DB in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='embedding space ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='An example of questions with the closest embeddings i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' related questionsUnsupervised Question Duplicate and Related Questions Detection in e-learning platforms WSDM ’23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' February 27-March 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' 2023,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' Singapore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' Singapore More formally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' Let 𝑞1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' 𝑞2 be two lists of tokens for the input questions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' then Jaccard similarity between these two questions can be calculated as 𝐽 (𝑞1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='𝑞2) = #(𝑞1 ∩ 𝑞2) #(𝑞1 ∪ 𝑞2) (1) If the Jaccard similarity (𝐽 (𝑞𝑛𝑒𝑤,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='𝑞𝑖)) between the input ques- tion and a question from 𝑆𝑐𝑎𝑛𝑑 is less than a certain threshold (𝐽 (𝑞𝑛𝑒𝑤,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='𝑞𝑖) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='4) we remove that question from our search space 𝑆𝑐𝑎𝑛𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' The threshold value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='4 was chosen after multiple itera- tions and validation of the results for the dataset that we worked on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' 𝑆𝑐𝑎𝑛𝑑 ← 𝑆𝑐𝑎𝑛𝑑 − 𝑞𝑖 (𝑖𝑓 𝐽 (𝑞𝑛𝑒𝑤,𝑞𝑖) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='4) If the Jaccard similarity is 1 we directly add that question to our exact duplicate question set (𝑞𝑑𝑢𝑝𝑒𝑥𝑎𝑐𝑡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='3 NER and comparison To further partition 𝑆𝑐𝑎𝑛𝑑, we remove questions with different named entities than those in 𝑞𝑛𝑒𝑤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' For extracting the set of named entities (𝑁𝐸𝑞) of 𝑞𝑛𝑒𝑤 we use spaCy, an implementation in Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' 𝑁𝐸𝑞 = 𝑁𝐸𝑅(𝑞𝑛𝑒𝑤) Examples : Who is the CEO of Google ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' → ’Google’: ORG , Who is the CEO of Apple ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' → ‘Apple’ : ORG The Named Entity Recognition step performs a sequence labeling task where the noun phrases are tagged with ’PERSON’, ’ORG’, ’LOC’, etc as applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' Once extracted, the set of entities for 𝑞𝑛𝑒𝑤 is compared to the set of entities 𝑁𝐸𝑖 for question 𝑞𝑖, where i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' |𝑆𝑐𝑎𝑛𝑑|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' All those questions which have a non-empty difference set between 𝑁𝐸𝑞 and 𝑁𝐸𝑖 are removed from the search space (𝑆𝑐𝑎𝑛𝑑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' 𝑆𝑐𝑎𝑛𝑑 ← 𝑆𝑐𝑎𝑛𝑑 − 𝑞𝑖 𝑖𝑓 𝑁𝐸𝑞 ∩ 𝑁𝐸𝑖 ≠ ∅ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='4 Keyphrase extraction and calculating the overlap Following the previous stages of the pipeline, the set 𝑆𝑐𝑎𝑛𝑑 = { 𝑞1 , 𝑞2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' 𝑞𝑛 } is left of potential candidates for a duplicate ques- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' The next stage of the pipeline (as shown in Figure 1) is to run an unsupervised method to automatically extract concept terms (keyphrases) from the input question 𝑞𝑛𝑒𝑤 into a set 𝐾𝑊𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' We lever- age the EmbedRank algorithm [1] for extracting keyphrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' The proposed approach first extracts candidate phrases using POS tags and projects them and the original question 𝑞𝑛𝑒𝑤 to a continuous vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' It then computes the semantic relatedness between the question and the phrase representations and retrieves the top 𝑘 keyphrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' For all the questions, we pre-compute the keyphrases and index them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' We run a comparison to determine the percentage of keyphrases shared between 𝐾𝑊𝑖, and the set of keyphrases ex- tracted for each of the questions in set 𝑆𝑐𝑎𝑛𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' Questions that have keyphrases sharing score of less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='7 (chosen after multiple validations) are eliminated from 𝑆𝑐𝑎𝑛𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' 𝐾𝑊𝑖 ← 𝐸𝑚𝑏𝑒𝑑𝑅𝑎𝑛𝑘(𝑞𝑛𝑒𝑤) 𝑆𝑐𝑎𝑛𝑑 ← 𝑆𝑐𝑎𝑛𝑑 − 𝑞𝑖 𝑖𝑓 𝐾𝑊𝑠ℎ𝑎𝑟𝑒 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='5 Negation detection As a result of the previous steps, the set 𝑆𝑐𝑎𝑛𝑑 is much smaller in size and has questions very similar to 𝑞𝑛𝑒𝑤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' However, we observed that multiple questions with similar verbiage exist though they differ by a negation resulting them having different answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' For example: What is an example of a metal ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' and What is not an example of a metal ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' Similar cases might still be left in 𝑆𝑐𝑎𝑛𝑑, and hence we check for the difference in negation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' We compare 𝑞𝑛𝑒𝑤 against each question in 𝑆𝑐𝑎𝑛𝑑 and eliminate any questions that may be the negation of 𝑞𝑛𝑒𝑤 by ensuring that standard negation constructions, if any, are present in both samples being compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' This ensures that questions with high levels of Jaccard similarity and overlapping keyphrases shares but differing by a single negation are not identi- fied as duplicates (false positives).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' After this stage, we assign the remaining questions to be duplicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='6 Related Questions The above-mentioned pipeline focuses on a higher recall by sacri- ficing precision since the problem statement focuses on e-learning platforms being able to rid their database of duplicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' An addi- tional property of our tool is that these platforms can test students on their knowledge of the topic by retrieving related questions which center around the same or similar topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' Such questions are referred to as related questions in this paper and are computed by utilizing the architecture of𝑎𝑙𝑙−𝑚𝑝𝑛𝑒𝑡−𝑏𝑎𝑠𝑒− 𝑣2 sentence transformers model1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' In the approach, for a duplicate question 𝑞𝑑𝑢𝑝 of an input question 𝑞𝑛𝑒𝑤, we find the questions that have embeddings closest to 𝑞𝑑𝑢𝑝𝑒𝑥𝑎𝑐𝑡 or 𝑞𝑑𝑢𝑝 (pre-computed in the database), measured by the cosine similarity between the two embedding vectors and return the 3 closest neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' The results demonstrated that nearest neighbors search over a large database gave slow performance, which led us to leverage 𝑆𝑐𝑎𝑁𝑁 (an efficient searching technique developed by [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' The set of embeddings for all the questions in the database is precalculated (using the same 𝑎𝑙𝑙 − 𝑚𝑝𝑛𝑒𝑡 − 𝑏𝑎𝑠𝑒 − 𝑣2 model) and stored locally for higher efficiency during running.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' For every input 𝑞𝑛𝑒𝑤, we have 𝑞𝑑𝑢𝑝𝑒𝑥𝑎𝑐𝑡, 𝑞𝑑𝑢𝑝 and related ques- tions 𝑞𝑑𝑢𝑝𝑟𝑒𝑙 (𝑞𝑟𝑒𝑙 is non-empty only if 𝑞𝑑𝑢𝑝 is non-empty).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' 3 DEMONSTRATION We demo our tool from the perspective of it’s ability to perform near-duplicate detection, analysis to gauge usability of the tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='1 Dataset The dataset we used consists of 114804 secondary high school questions from the CBSE (India) curated with the help of a leading e-learning platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' The dataset statistics are shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='2 Evaluation The tool was evaluated on a set of 100 input questions by two in- dependent researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' The tool was provided with 100 random 1 WSDM ’23, February 27-March 3, 2023, Singapore, Singapore Maksimjeet and Sanyam, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' Figure 2: Screenshot of the tool QDup Figure 3: Dataset Statistics Method Accuracy (%) QDup 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='5 keyphrases based 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='5 Closest neighbours 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='5 Table 1: Performance Evaluation for Duplicate Detection questions across domains and the researchers were requested to la- bel the correct duplicates as 1 or 0 in all other conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' Similarly, the outputs from other approaches were also provided to the re- searchers for labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' These approaches included nearest neighbor search for embeddings extracted with all-mpnet-base-v2 sentence embeddings model and comparison of keyphrases extracted using EmbedRank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' We observed a Cohen’s kappa of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='60, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='72 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='65 in the three scenarios, respectively indicating substantial agree- ment between annotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' We report the accuracy in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' We observe that the proposed approach QDup outperforms classical keyphrases only or vector based nearest neighbor search methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='3 Tool Ease of Use We also conducted a user study with 14 well trained academicians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' A screenshot of the tool is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' We asked the users to rate the tool on a scale of 1-3 (lowest to highest) from aspects of intuitiveness, responsiveness and relevance of output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' The intuitive- ness metric indicates how intuitive and easy to use the interface is without external help.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' The responsiveness measures the response time and relevance measures how much the users think the output for the given questions are accurate duplicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' We observed that the average intuitiveness score is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='46 and average responsiveness score is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' The average relevance score is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content='68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' We observe that the majority of the users find the tool easy to use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' 4 CONCLUSION AND FUTURE WORK In this paper, we propose a tool to find duplicates and related ques- tions in a large repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' The proposed approach is resource and time efficient, and the interface is easy to use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' In the future, we plan to use the data collected from this tool as weakly supervised data to train a bi-encoder transformer-based model in a contrastive setting to identify duplicate and related questions in one stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' We also plan to explore knowledge distillation and quantization approaches for efficient deployment of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' 5 ACKNOWLEDGMENTS We thank Extramarks and SERB-FICCI for the support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' REFERENCES [1] Kamil Bennani-Smires, Claudiu Musat, Andreea Hossmann, Michael Baeriswyl, and Martin Jaggi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' Simple Unsupervised Keyphrase Extraction using Sen- tence Embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' [2] Giovanni Da San Martino, Salvatore Romeo, Alberto Barroón-Cedeño, Shafiq Joty, Lluís Maàrquez, Alessandro Moschitti, and Preslav Nakov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' Cross-Language Question Re-Ranking (SIGIR ’17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' [3] Arpita Das, Harish Yenala, Manoj Chinnakotla, and Manish Shrivastava.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' Together we stand: Siamese Networks for Similar Question Retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' In ACL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' Berlin, Germany, 378–387.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' [4] Simone Filice and Alessandro Moschitti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' Learning pairwise patterns in Community Question Answering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' Intelligenza Artificiale 12 (2018), 49–65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' [5] Ruiqi Guo, Philip Sun, Erik Lindgren, Quan Geng, David Simcha, Felix Chern, and Sanjiv Kumar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' Accelerating Large-Scale Inference with Anisotropic Vector Quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' [6] Doris Hoogeveen, Karin M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' Verspoor, and Timothy Baldwin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' CQADupStack: A Benchmark Data Set for Community Question-Answering Research (ADCS ’15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' [7] Di Liang, Fubao Zhang, Weidong Zhang, Qi Zhang, Jinlan Fu, Minlong Peng, Tao Gui, and Xuanjing Huang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' Adaptive Multi-Attention Network Incorporating Answer Information for Duplicate Question Detection (SIGIR’19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' 95–104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' [8] Preslav Nakov, Lluís Màrquez, Alessandro Moschitti, Walid Magdy, Hamdy Mubarak, Abed Alhakim Freihat, Jim Glass, and Bilal Randeree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' SemEval- 2016 Task 3: Community Question Answering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' ACL, San Diego, California.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' [9] Venktesh V, Mukesh Mohania, and Vikram Goyal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' TagRec: Automated Tagging of Questions with Hierarchical Learning Taxonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' [10] Kai Zhang, Wei Wu, Haocheng Wu, Zhoujun Li, and Ming Zhou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' Question Retrieval with High Quality Answers in Community Question Answering (CIKM ’14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' Association for Computing Machinery, New York, NY, USA, 371–380.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' Online Question Duplicity Finder An application to prevent academicians from creating duplicate questions by suggesting near duplicates of a question present in the database Definition of Duplicacy : Add Question Add CSV File Question Question Duplicates ferric chloride stops bleeding because 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' Ferric chloride is applied to stop bleeding because Answer Related questions Enter Answer or Leave it empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' The aluminium salt commonly used to stop bleeding is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' A blood red colour is obtained when ferric chloride solution reacts with 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE4T4oBgHgl3EQfkQ3K/content/2301.05150v1.pdf'} +page_content=' The salt of aluminium commonly 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University, UK +• Emanuele Catalano Univ. Grenoble Alpes, 3SR lab. +• Robert Caulk Univ. Grenoble Alpes, 3SR lab. +• Bruno Chareyre Univ. Grenoble Alpes, 3SR lab. +• William Chèvremont Univ. Grenoble Alpes, LRP +• Sergei Dorofeenko IPCP RAS, Chernogolovka +• Jérôme Duriez INRAE, Aix Marseille Univ, RECOVER, Aix-en-Provence, France +• Nolan Dyck Univ. of Western Ontario +• Jan Eliáš Brno University of Technology +• Burak Er Bursa Technical University +• Alexander Eulitz TU Berlin / Institute for Machine Tools and Factory Management +• Anton Gladky TU Bergakademie Freiberg +• Ning Guo Hong Kong Univ. of Science and Tech. +• Christian Jakob TU Bergakademie Freiberg +• François Kneib Univ. Grenoble Alpes, 3SR lab. / Irstea Grenoble +• Janek Kozicki Gdansk University of Technology +• Donia Marzougui Univ. Grenoble Alpes, 3SR lab. +• Raphaël Maurin Irstea Grenoble +• Chiara Modenese University of Oxford +• Gerald Pekmezi University of Alabama at Birmingham +• Luc Scholtès Univ. Grenoble Alpes, 3SR lab. +• Luc Sibille University of Nantes, lab. GeM +• Jan Stránský CVUT Prague +• Thomas Sweijen Utrecht University +• Klaus Thoeni The University of Newcastle (Australia) +• Chao Yuan Univ. Grenoble Alpes, 3SR lab. +Citing this document +When referring to Yade-DEM software in scientific publication please cite it ”by DOI” as follows: +Šmilauer V. et al. (2021) Yade Documentation 3rd ed. The Yade Project. DOI:10.5281/zenodo.5705394. +http://yade-dem.org +See also http://yade-dem.org/doc/citing.html. +i + +ii + +Contents +1 +Guided tour +1 +1.1 +Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +1 +1.1.1 +Getting started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +1 +1.1.2 +Architecture overview +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +6 +1.2 +Tutorial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +15 +1.2.1 +Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +15 +1.2.2 +Hands-on +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +15 +1.2.3 +Data mining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +25 +1.2.4 +Setting up a simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +30 +1.2.5 +Advanced & more . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +34 +1.2.6 +Examples with tutorial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +35 +1.2.7 +More examples +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +45 +2 +Yade for users +51 +2.1 +DEM formulation +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +51 +2.1.1 +Collision detection +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +51 +2.1.2 +Creating interaction between particles +. . . . . . . . . . . . . . . . . . . . . . . . +55 +2.1.3 +Kinematic variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +57 +2.1.4 +Contact model (example) +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +60 +2.1.5 +Motion integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +60 +2.1.6 +Periodic boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +68 +2.1.7 +Computational aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +72 +2.2 +User’s manual +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +73 +2.2.1 +Scene construction +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +73 +2.2.2 +Controlling simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +93 +2.2.3 +Postprocessing +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 +2.2.4 +Python specialties and tricks +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 +2.2.5 +Extending Yade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 +2.2.6 +Troubleshooting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 +2.3 +Yade wrapper class reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 +2.3.1 +Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 +2.3.2 +Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 +2.3.3 +Global engines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 +2.3.4 +Partial engines +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 +2.3.5 +Dispatchers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 +2.3.6 +Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 +2.3.7 +Bounding volume creation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 +2.3.8 +Interaction Geometry creation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 +2.3.9 +Interaction Physics creation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 +2.3.10 +Constitutive laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 +2.3.11 +Internal forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 +2.3.12 +Callbacks +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 +2.3.13 +Preprocessors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 +2.3.14 +Rendering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 +2.3.15 +Simulation data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 +2.3.16 +Other classes +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 +iii + +2.4 +Yade modules reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 +2.4.1 +yade.bodiesHandling module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 +2.4.2 +yade.export module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 +2.4.3 +yade.geom module +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 +2.4.4 +yade.gridpfacet module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 +2.4.5 +yade.libVersions module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 +2.4.6 +yade.linterpolation module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 +2.4.7 +yade.log module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 +2.4.8 +yade.math module +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404 +2.4.9 +yade.minieigenHP module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 +2.4.10 +yade.mpy module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474 +2.4.11 +yade.pack module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479 +2.4.12 +yade.plot module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490 +2.4.13 +yade.polyhedra_utils module +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494 +2.4.14 +yade.post2d module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494 +2.4.15 +yade.qt module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 +2.4.16 +yade.timing module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499 +2.4.17 +yade.utils module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500 +2.4.18 +yade.ymport module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516 +2.5 +Installation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520 +2.5.1 +Packages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521 +2.5.2 +Docker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522 +2.5.3 +Source code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523 +2.5.4 +Speed-up compilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529 +2.5.5 +Cloud Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530 +2.5.6 +GPU Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530 +2.5.7 +Special builds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530 +2.5.8 +Yubuntu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531 +2.6 +Acknowledging Yade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531 +3 +Yade for programmers +533 +3.1 +Programmer’s manual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 +3.1.1 +Build system +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 +3.1.2 +Development tools +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534 +3.1.3 +Debugging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535 +3.1.4 +Regression tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541 +3.1.5 +Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543 +3.1.6 +Support framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548 +3.1.7 +Simulation framework +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572 +3.1.8 +Runtime structure +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578 +3.1.9 +Python framework +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579 +3.1.10 +Adding a new python/C++ module +. . . . . . . . . . . . . . . . . . . . . . . . . 581 +3.1.11 +Maintaining compatibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583 +3.2 +Yade on GitLab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584 +3.2.1 +Fast checkout (read-only) +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584 +3.2.2 +Branches on GitLab +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584 +3.2.3 +Merge requests +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587 +3.2.4 +Guidelines for pushing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588 +4 +Theoretical background and extensions +589 +4.1 +DEM formulation +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589 +4.2 +CFD-DEM coupled simulations with Yade and OpenFOAM . . . . . . . . . . . . . . . . . 589 +4.2.1 +Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589 +4.2.2 +Setting up a case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592 +4.2.3 +Post-Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594 +4.3 +FEM-DEM hierarchical multiscale modeling with Yade and Escript +. . . . . . . . . . . . 594 +4.3.1 +Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594 +4.3.2 +Finite element formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595 +iv + +4.3.3 +Multiscale solution procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595 +4.3.4 +Work on the YADE side . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596 +4.3.5 +Work on the Escript side . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596 +4.3.6 +Example tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597 +4.3.7 +Disclaim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597 +4.4 +Simulating Acoustic Emissions in Yade +. . . . . . . . . . . . . . . . . . . . . . . . . . . . 598 +4.4.1 +Summary +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598 +4.4.2 +Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598 +4.4.3 +Activating the algorithm within Yade . . . . . . . . . . . . . . . . . . . . . . . . . 599 +4.4.4 +Visualizing and post processing acoustic emissions +. . . . . . . . . . . . . . . . . 600 +4.4.5 +Consideration of rock heterogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . 601 +4.5 +Using YADE 1D vertical VANS fluid resolution . . . . . . . . . . . . . . . . . . . . . . . . 601 +4.5.1 +DEM-fluid coupling and fluid resolution in YADE . . . . . . . . . . . . . . . . . . 603 +4.5.2 +Application of drag and buoyancy forces (HydroForceEngine::action) . . . . . . . 603 +4.5.3 +Solid phase averaging (HydroForceEngine::averageProfile) +. . . . . . . . . . . . . 604 +4.5.4 +Fluid resolution\HydroForceEngine::fluidResolution . . . . . . . . . . . . . . . . . 605 +4.6 +Potential Particles and Potential Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606 +4.6.1 +Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606 +4.6.2 +Potential Particles code (PP) +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606 +4.6.3 +Potential Blocks code (PB) +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607 +4.6.4 +Engines +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609 +4.6.5 +Contact Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610 +4.6.6 +Shape definition of a PP and a PB . . . . . . . . . . . . . . . . . . . . . . . . . . 611 +4.6.7 +Body definition of a PP and a PB . . . . . . . . . . . . . . . . . . . . . . . . . . . 612 +4.6.8 +Boundary Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613 +4.6.9 +Visualization +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614 +4.6.10 +Axis-Aligned Bounding Box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615 +4.6.11 +Block Generation algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616 +4.6.12 +Examples +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616 +4.6.13 +Disclaimer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616 +4.6.14 +References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616 +5 +Performance enhancements +617 +5.1 +Accelerating Yade’s FlowEngine with GPU . . . . . . . . . . . . . . . . . . . . . . . . . . 617 +5.1.1 +Summary +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617 +5.1.2 +Hardware, Software, and Model Requirements . . . . . . . . . . . . . . . . . . . . 617 +5.1.3 +Install CUDA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618 +5.1.4 +Install OpenBlas, and Lapack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618 +5.1.5 +Install SuiteSparse +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618 +5.1.6 +Compile Yade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618 +5.1.7 +Controlling the GPU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619 +5.1.8 +Performance increase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619 +5.2 +MPI parallelization +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 620 +5.2.1 +Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 620 +5.2.2 +Walkthrough +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621 +5.2.3 +MPI initialization and communications . . . . . . . . . . . . . . . . . . . . . . . . 623 +5.2.4 +Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629 +5.2.5 +Merging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632 +5.2.6 +Hints and problems to expect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632 +5.2.7 +Control variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633 +5.2.8 +Benchmark +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633 +5.3 +Using YADE with cloud computing on Amazon EC2 . . . . . . . . . . . . . . . . . . . . . 634 +5.3.1 +Summary +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634 +5.3.2 +Launching an EC2 instance +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634 +5.3.3 +Installing YADE and managing files +. . . . . . . . . . . . . . . . . . . . . . . . . 636 +5.3.4 +Plotting output in the terminal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637 +5.3.5 +Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638 +5.4 +High precision calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638 +v + +5.4.1 +Installation +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639 +5.4.2 +Supported modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639 +5.4.3 +Double, quadruple and higher precisions . . . . . . . . . . . . . . . . . . . . . . . 640 +5.4.4 +Compatibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641 +5.4.5 +Debugging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645 +6 +Literature +647 +6.1 +Yade Technical Archive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647 +6.1.1 +About . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647 +6.1.2 +Contribute . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647 +6.1.3 +Contact +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647 +6.1.4 +Archive +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647 +6.2 +Publications on Yade +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648 +6.2.1 +Citing Yade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648 +6.2.2 +Journal articles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648 +6.2.3 +Conference materials and book chapters . . . . . . . . . . . . . . . . . . . . . . . 648 +6.2.4 +Master and PhD theses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648 +6.2.5 +Yade Technical Archive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648 +6.3 +References +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648 +7 +Indices and tables +649 +Bibliography +651 +Python Module Index +679 +vi + +Chapter 1 +Guided tour +1.1 Introduction +1.1.1 Getting started +Before you start moving around in Yade, you should have some prior knowledge. +• Basics of command line in your Linux system are necessary for running yade. Look on the web for +tutorials. +• Python language; we recommend the official Python tutorial. Reading further documents on the +topic, such as Dive into Python will certainly not hurt either. +You are advised to try all commands described yourself. Don’t be afraid to experiment. +Hint: +Sometimes reading this documentation in a .pdf format can be more comfortable. For example +in okular pdf viewer clicking links is faster than a page refresh in the web browser and to go back press +the shortcut Alt Shift ←. To try it have a look at the inheritance graph of PartialEngine then go back. +Starting yade +Yade is being run primarily from terminal; the name of command is yade.1 (In case you did not install +from package, you might need to give specific path to the command2): +$ yade +Welcome to Yade +TCP python prompt on localhost:9001, auth cookie `sdksuy' +TCP info provider on localhost:21000 +(continues on next page) +1 The executable name can carry a suffix, such as version number (yade-0.20), depending on compilation options. +Packaged versions on Debian systems always provide the plain yade alias, by default pointing to latest stable version (or +latest snapshot, if no stable version is installed). You can use update-alternatives to change this. +2 In general, Unix shell (command line) has environment variable PATH defined, which determines directories searched +for executable files if you give name of the file without path. Typically, $PATH contains /usr/bin/, /usr/local/bin, /bin +and others; you can inspect your PATH by typing echo $PATH in the shell (directories are separated by :). +If Yade executable is not in directory contained in PATH, you have to specify it by hand, i.e. by typing the path in front +of the filename, such as in /home/user/bin/yade and similar. You can also navigate to the directory itself (cd ~/bin/yade, +where ~ is replaced by your home directory automatically) and type ./yade then (the . is the current directory, so ./ +specifies that the file is to be found in the current directory). +To save typing, you can add the directory where Yade is installed to your PATH, typically by editing ~/.profile (in +normal cases automatically executed when shell starts up) file adding line like export PATH=/home/user/bin:$PATH. You +can also define an alias by saying alias yade="/home/users/bin/yade" in that file. +Details depend on what shell you use (bash, zsh, tcsh, …) and you will find more information in introductory material +on Linux/Unix. +1 + +Yade Documentation, Release 3rd ed. +(continued from previous page) +[[ ^L clears screen, ^U kills line. F12 controller, F11 3d view, F10 both, F9 generator, F8␣ +�→plot. ]] +Yade [1]: +These initial lines give you some information about +• some information for Remote control, which you are unlikely to need now; +• basic help for the command-line that just appeared (Yade [1]:). +Type quit(), exit() or simply press ^D (^ is a commonly used written shortcut for pressing the Ctrl +key, so here ^D means Ctrl D) to quit Yade. +The command-line is ipython, python shell with enhanced interactive capabilities; it features persistent +history (remembers commands from your last sessions), searching and so on. See ipython’s documentation +for more details. +Typically, you will not type Yade commands by hand, but use scripts, python programs describing and +running your simulations. Let us take the most simple script that will just print “Hello world!”: +print("Hello world!") +Saving such script as hello.py, it can be given as argument to Yade: +$ yade hello.py +Welcome to Yade +TCP python prompt on localhost:9001, auth cookie `askcsu' +TCP info provider on localhost:21000 +Running script hello.py +## the script is being run +Hello world! +## output from the script +[[ ^L clears screen, ^U kills line. F12 controller, F11 3d view, F10 both, F9 generator, F8␣ +�→plot. ]] +Yade [1]: +Yade will run the script and then drop to the command-line again.3 If you want Yade to quit immediately +after running the script, use the -x switch: +$ yade -x script.py +There is more command-line options than just -x, run yade -h to see all of them. +Options: +-v, --version +show program’s version number and exit +-h, --help +show this help message and exit +-j THREADS, --threads=THREADS +Number of OpenMP threads +to run; defaults to 1. Equivalent to setting OMP_- +NUM_THREADS environment variable. +--cores=CORES +Set number of OpenMP threads (as –threads) and in +addition set affinity of threads to the cores given. +--update +Update deprecated class names in given script(s) using +text search & replace. Changed files will be backed up +with ~ suffix. Exit when done without running any +simulation. +--nice=NICE +Increase nice level (i.e. +decrease priority) by given +number. +3 Plain Python interpreter exits once it finishes running the script. The reason why Yade does the contrary is that most +of the time script only sets up simulation and lets it run; since computation typically runs in background thread, the script +is technically finished, but the computation is running. +2 +Chapter 1. +Guided tour + +Yade Documentation, Release 3rd ed. +-x +Exit when the script finishes +-f +Set logging verbosity, default is -f3 (yade.log.WARN) +for all classes +-n +Run without graphical interface (equivalent to unset- +ting the DISPLAY environment variable) +--test +Run regression test suite and exit; the exists status is 0 +if all tests pass, 1 if a test fails and 2 for an unspecified +exception. +--check +Run a series of user-defined check tests as described +in scripts/checks-and-tests/checks/README and Re- +gression tests +--performance +Starts a test to measure the productivity. +--stdperformance +Starts a standardized test to measure the productiv- +ity, which will keep retrying to run the benchmark +until standard deviation of the performance is below +1%. A common type of simulation is done: the spheres +fall down in a box and are given enough time to settle +in there. Note: better to use this with argument -j +THREADS (explained above). +--quickperformance +Starts a quick test to measure the productivity. +Same as above, but only two short runs are performed, +without the attempts to find the computer perfor- +mance with small error. +--no-gdb +Do not show backtrace when yade crashes (only effec- +tive with –debug)4. +Quick inline help +All of functions callable from ipython shell have a quickly accessible help by appending ? to the function +name, or calling help(…) command on them: +Yade [1]: O.run? +Docstring: +run( (Omega)arg1 [, (int)nSteps=-1 [, (bool)wait=False]]) -> None : +Run the simulation. *nSteps* how many steps to run, then stop (if positive); *wait* will␣ +�→cause not returning to python until simulation will have stopped. +Type: +method +Yade [2]: help(O.pause) +Help on method pause: +pause(...) method of yade.wrapper.Omega instance +pause( (Omega)arg1) -> None : +Stop simulation execution. (May be called from within the loop, and it will stop after␣ +�→the current step). +A quick way to discover available functions is by using the tab-completion mechanism, e.g. type O. then +press tab. +Creating simulation +To create simulation, one can either use a specialized class of type FileGenerator to create full scene, +possibly receiving some parameters. Generators are written in C++ and their role is limited to well- +4 On some linux systems stack trace will produce Operation not permitted error. See debugging section for solution. +1.1. +Introduction +3 + +Yade Documentation, Release 3rd ed. +defined scenarios. For instance, to create triaxial test scene: +Yade [3]: TriaxialTest(numberOfGrains=200).load() +Yade [4]: len(O.bodies) +Out[4]: 206 +Generators are regular yade objects that support attribute access. +It is also possible to construct the scene by a python script; this gives much more flexibility and speed of +development and is the recommended way to create simulation. Yade provides modules for streamlined +body construction, import of geometries from files and reuse of common code. Since this topic is more +involved, it is explained in the User’s manual. +Running simulation +As explained below, the loop consists in running defined sequence of engines. Step number can be queried +by O.iter and advancing by one step is done by O.step(). Every step advances virtual time by current +timestep, O.dt that can be directly assigned or, which is usually better, automatically determined by a +GlobalStiffnessTimeStepper, if present: +Yade [5]: O.iter +Out[5]: 0 +Yade [6]: O.time +Out[6]: 0.0 +Yade [7]: O.dt=1e-4 +Yade [8]: O.dynDt=False #else it would be adjusted automaticaly during first iteration +Yade [9]: O.step() +Yade [10]: O.iter +Out[10]: 1 +Yade [11]: O.time +Out[11]: 0.0001 +Normal simulations, however, are run continuously. Starting/stopping the loop is done by O.run() and +O.pause(); note that O.run() returns control to Python and the simulation runs in background; if +you want to wait for it to finish, use O.wait(). Fixed number of steps can be run with O.run(1000), +O.run(1000,True) will run and wait. To stop at absolute step number, O.stopAtIter can be set and +O.run() called normally. +Yade [12]: O.run() +Yade [13]: O.pause() +Yade [14]: O.iter +Out[14]: 1715 +Yade [15]: O.run(100000,True) +Yade [16]: O.iter +Out[16]: 101715 +Yade [17]: O.stopAtIter=500000 +Yade [18]: O.run() +(continues on next page) +4 +Chapter 1. +Guided tour + +Yade Documentation, Release 3rd ed. +(continued from previous page) +Yade [19]: O.wait() +Yade [20]: O.iter +Out[20]: 500000 +Saving and loading +Simulation can be saved at any point to a binary file (optionaly compressed if the filename has extensions +such as “.gz” or “.bz2”). Saving to a XML file is also possible though resulting in larger files and slower +save/load, it is used when the filename contains “xml”. With some limitations, it is generally possible to +load the scene later and resume the simulation as if it were not interrupted. Note that since the saved +scene is a dump of Yade’s internal objects, it might not (probably will not) open with different Yade +version. This problem can be sometimes solved by migrating the saved file using “.xml” format. +Yade [21]: O.save('/tmp/a.yade.bz2') +Yade [22]: O.reload() +Yade [23]: O.load('/tmp/another.yade.bz2') +The principal use of saving the simulation to XML is to use it as temporary in-memory storage for +checkpoints in simulation, e.g. for reloading the initial state and running again with different parameters +(think tension/compression test, where each begins from the same virgin state). +The functions O. +saveTmp() and O.loadTmp() can be optionally given a slot name, under which they will be found in +memory: +Yade [24]: O.saveTmp() +Yade [25]: O.loadTmp() +Yade [26]: O.saveTmp('init') ## named memory slot +Yade [27]: O.loadTmp('init') +Simulation can be reset to empty state by O.reset(). +It can be sometimes useful to run different simulation, while the original one is temporarily suspended, +e.g. when dynamically creating packing. O.switchWorld() toggles between the primary and secondary +simulation. +Graphical interface +Yade can be optionally compiled with QT based graphical interface (qt4 and qt5 are supported). It can +be started by pressing F12 in the command-line, and also is started automatically when running a script. +1.1. +Introduction +5 + +Yade Documentation, Release 3rd ed. +The control window on the left (fig. imgQtGui) is called Controller (can be invoked by yade.qt. +Controller() from python or by pressing F12 key in terminal): +1. The Simulation tab is mostly self-explanatory, and permits basic simulation control. +2. The Display tab has various rendering-related options, which apply to all opened views (they can +be zero or more, new one is opened by the New 3D button). +3. The Python tab has only a simple text entry area; it can be useful to enter python commands while +the command-line is blocked by running script, for instance. +Inside the Inspect window (on the right in fig. imgQtGui) all simulation data can be examined and +modified in realtime. +1. Clicking left mouse button on any of the blue hyperlinks will open documentation. +2. Clicking middle mouse button will copy the fully qualified python name into clipboard, which can +be pasted into terminal by clicking middle mouse button in the terminal (or pressing Ctrl-V). +3d views can be controlled using mouse and keyboard shortcuts; help is displayed if you press the h key +while in the 3d view. Note that having the 3d view open can slow down running simulation significantly, +it is meant only for quickly checking whether the simulation runs smoothly. Advanced post-processing +is described in dedicated section Data mining. +1.1.2 Architecture overview +In the following, a high-level overview of Yade architecture will be given. As many of the features are +directly represented in simulation scripts, which are written in Python, being familiar with this language +will help you follow the examples. For the rest, this knowledge is not strictly necessary and you can +ignore code examples. +6 +Chapter 1. +Guided tour + +Yade +-oX +Simulation +Display +Generate +Python +Load +Save +Inspect +Primary view +real 00:02:20 +virt 000s671m552μ639n +iter : +#3243, 23.0/s +At +O fixed +O timestepper +Simulation Inspection +0.000207077594613 +Engines +Bodies +Interactions +Cell +:memory: +56 +V +0 +56+0 +Body0x466fd80 +bound +Aabb 0x4684670 +color +1.0 +1.0 +1.0 +clumpld +-1 +flags +1 +groupMask +1 +id +56 +material +FrictMat "defaultMat" +density +1000.0 +New 3D +Reference +Center +Ni +X +frictionAngle +0.5 +id +0 +label +defaultMat +#3243 +poisson +0.3 +cl0ck 02:20 +671m552u639n +young +10000000.0 +shape +Sphere 0x46845e0 +Left mouse button +open documentation +color +656201236 +882652506303750713 +Middle mouse +buttor +copy +to clipboard full +highlight +python name +radius +0.0316463982726Yade Documentation, Release 3rd ed. +Data and functions +To assure flexibility of software design, yade makes clear distinction of 2 families of classes: data com- +ponents and functional components. The former only store data without providing functionality, while +the latter define functions operating on the data. In programming, this is known as visitor pattern (as +functional components “visit” the data, without being bound to them explicitly). +Entire simulation, i.e. both data and functions, are stored in a single Scene object. It is accessible +through the Omega class in python (a singleton), which is by default stored in the O global variable: +Yade [28]: O.bodies +# some data components +Out[28]: +Yade [29]: len(O.bodies) +# there are no bodies as of yet +Out[29]: 0 +Yade [30]: O.engines +# functional components, empty at the moment +Out[30]: [] +Data components +Bodies +Yade simulation (class Scene, but hidden inside Omega in Python) is represented by Bodies, their Inter- +actions and resultant generalized forces (all stored internally in special containers). +Each Body comprises the following: +Shape represents particle’s geometry (neutral with regards to its spatial orientation), such as Sphere, +Facet or inifinite Wall; it usually does not change during simulation. +Material stores characteristics pertaining to mechanical behavior, such as Young’s modulus or density, +which are independent on particle’s shape and dimensions; usually constant, might be shared +amongst multiple bodies. +State contains state variables, in particular spatial position and orientation, linear and angular velocity; +it is updated by the integrator at every step. The derived classes would contain other information +related to current state of this body, e.g. its temperature, averaged damage or broken links between +components. +Bound is used for approximate (“pass 1”) contact detection; updated as necessary following body’s +motion. Currently, Aabb is used most often as Bound. Some bodies may have no Bound, in which +case they are exempt from contact detection. +(In addition to these 4 components, bodies have several more minor data associated, such as Body::id or +Body::mask.) +All these four properties can be of different types, derived from their respective base types. +Yade +frequently makes decisions about computation based on those types: Sphere + Sphere collision has to be +treated differently than Facet + Sphere collision. Objects making those decisions are called Dispatchers +and are essential to understand Yade’s functioning; they are discussed below. +Explicitly assigning all 4 properties to each particle by hand would be not practical; there are utility +functions defined to create them with all necessary ingredients. +For example, we can create sphere +particle using utils.sphere: +Yade [31]: s=utils.sphere(center=[0,0,0],radius=1) +Yade [32]: s.shape, s.state, s.mat, s.bound +Out[32]: +(, +(continues on next page) +1.1. +Introduction +7 + +Yade Documentation, Release 3rd ed. +Fig. 1: Examples of concrete classes that might be used to describe a Body: State, CpmState, Chained- +State, Material, ElastMat, FrictMat, FrictViscoMat, Shape, Polyhedra, PFacet, GridConnection, Bound, +Aabb. +8 +Chapter 1. +Guided tour + +Yade Documentation, Release 3rd ed. +(continued from previous page) +, +, +None) +Yade [33]: s.state.pos +Out[33]: Vector3(0,0,0) +Yade [34]: s.shape.radius +Out[34]: 1.0 +We see that a sphere with material of type FrictMat (default, unless you provide another Material) and +bounding volume of type Aabb (axis-aligned bounding box) was created. Its position is at the origin and +its radius is 1.0. Finally, this object can be inserted into the simulation; and we can insert yet one sphere +as well. +Yade [35]: O.bodies.append(s) +Out[35]: 0 +Yade [36]: O.bodies.append(utils.sphere([0,0,2],.5)) +Out[36]: 1 +In each case, return value is Body.id of the body inserted. +Since till now the simulation was empty, its id is 0 for the first sphere and 1 for the second one. Saving +the id value is not necessary, unless you want to access this particular body later; it is remembered +internally in Body itself. You can address bodies by their id: +Yade [37]: O.bodies[1].state.pos +Out[37]: Vector3(0,0,2) +Yade [38]: O.bodies[100] +# error because there are only two bodies +--------------------------------------------------------------------------- +IndexError +Traceback (most recent call last) +~/yade/lib/x86_64-linux-gnu/yadeflip/py/yade/__init__.py in +----> 1 O.bodies[100] +# error because there are only two bodies +IndexError: Body id out of range. +Adding the same body twice is, for reasons of the id uniqueness, not allowed: +Yade [39]: O.bodies.append(s) +# error because this sphere was already added +--------------------------------------------------------------------------- +IndexError +Traceback (most recent call last) +~/yade/lib/x86_64-linux-gnu/yadeflip/py/yade/__init__.py in +----> 1 O.bodies.append(s) +# error because this sphere was already added +IndexError: Body already has id 0 set; appending such body (for the second time) is not␣ +�→allowed. +Bodies can be iterated over using standard python iteration syntax: +Yade [40]: for b in O.bodies: +....: +print(b.id,b.shape.radius) +....: +0 1.0 +1 0.5 +Interactions +Interactions are always between pair of bodies; usually, they are created by the collider based on spatial +1.1. +Introduction +9 + +Yade Documentation, Release 3rd ed. +proximity; they can, however, be created explicitly and exist independently of distance. Each interaction +has 2 components: +IGeom holding geometrical configuration of the two particles in collision; it is updated automatically +as the particles in question move and can be queried for various geometrical characteristics, such +as penetration distance or shear strain. +Based on combination of types of Shapes of the particles, there might be different storage require- +ments; for that reason, a number of derived classes exists, e.g. for representing geometry of contact +between Sphere+Sphere, Cylinder+Sphere etc. Note, however, that it is possible to represent many +type of contacts with the basic sphere-sphere geometry (for instance in Ig2_Wall_Sphere_Sc- +Geom). +IPhys representing non-geometrical features of the interaction; some are computed from Materials of +the particles in contact using some averaging algorithm (such as contact stiffness from Young’s +moduli of particles), others might be internal variables like damage. +Fig. 2: Examples of concrete classes that might be used to describe an Interaction: IGeom, Generic- +SpheresContact, PolyhedraGeom, CylScGeom, IPhys, NormPhys, NormShearPhys, FrictPhys. +Suppose now interactions have been already created. We can access them by the id pair: +Yade [41]: O.interactions[0,1] +Out[41]: +Yade [42]: O.interactions[1,0] +# order of ids is not important +Out[42]: +Yade [43]: i=O.interactions[0,1] +Yade [44]: i.id1,i.id2 +Out[44]: (0, 1) +Yade [45]: i.geom +Out[45]: +Yade [46]: i.phys +Out[46]: +Yade [47]: O.interactions[100,10111] +# asking for non existing interaction throws exception +--------------------------------------------------------------------------- +IndexError +Traceback (most recent call last) +~/yade/lib/x86_64-linux-gnu/yadeflip/py/yade/__init__.py in +(continues on next page) +10 +Chapter 1. +Guided tour + +Yade Documentation, Release 3rd ed. +(continued from previous page) +----> 1 O.interactions[100,10111] +# asking for non existing interaction throws exception +IndexError: No such interaction +Generalized forces +Generalized forces include force, torque and forced displacement and rotation; they are stored only tem- +porariliy, during one computation step, and reset to zero afterwards. For reasons of parallel computation, +they work as accumulators, i.e. only can be added to, read and reset. +Yade [48]: O.forces.f(0) +Out[48]: Vector3(0,0,0) +Yade [49]: O.forces.addF(0,Vector3(1,2,3)) +Yade [50]: O.forces.f(0) +Out[50]: Vector3(1,2,3) +You will only rarely modify forces from Python; it is usually done in c++ code and relevant documen- +tation can be found in the Programmer’s manual. +Function components +In a typical DEM simulation, the following sequence is run repeatedly: +• reset forces on bodies from previous step +• approximate collision detection (pass 1) +• detect exact collisions of bodies, update interactions as necessary +• solve interactions, applying forces on bodies +• apply other external conditions (gravity, for instance). +• change position of bodies based on forces, by integrating motion equations. +Each of these actions is represented by an Engine, functional element of simulation. The sequence of +engines is called simulation loop. +Engines +Simulation loop, shown at fig. img-yade-iter-loop, can be described as follows in Python (details will be +explained later); each of the O.engines items is instance of a type deriving from Engine: +O.engines=[ +# reset forces +ForceResetter(), +# approximate collision detection, create interactions +InsertionSortCollider([Bo1_Sphere_Aabb(),Bo1_Facet_Aabb()]), +# handle interactions +InteractionLoop( +[Ig2_Sphere_Sphere_ScGeom(),Ig2_Facet_Sphere_ScGeom()], +[Ip2_FrictMat_FrictMat_FrictPhys()], +[Law2_ScGeom_FrictPhys_CundallStrack()], +), +# apply other conditions +GravityEngine(gravity=(0,0,-9.81)), +(continues on next page) +1.1. +Introduction +11 + +Yade Documentation, Release 3rd ed. +bodies +Shape +Material +State +Bound + +interactions +geometry + collision detection pass 2 + strain evaluation + +physics + properties of new interactions + +constitutive law + compute forces from strains +forces +(generalized) +update +bounds + +collision +detection +pass 1 + +other forces +(gravity, BC, ...) +miscellaneous engines +(recorders, ...) +reset forces +forces → acceleration +velocity update +position update +simulation +loop + +increment +time by Δt +Fig. 3: Typical simulation loop; each step begins at body-centered bit at 11 o’clock, continues with +interaction bit, force application bit, miscellanea and ends with time update. +(continued from previous page) +# update positions using Newton's equations +NewtonIntegrator() +] +There are 3 fundamental types of Engines: +GlobalEngines operating on the whole simulation (e.g. ForceResetter which zeroes forces acting on +bodies or GravityEngine looping over all bodies and applying force based on their mass) +PartialEngine operating only on some pre-selected bodies (e.g. ForceEngine applying constant force +to some selected bodies) +Dispatchers do not perform any computation themselves; they merely call other functions, represented +by function objects, Functors. Each functor is specialized, able to handle certain object types, and +will be dispatched if such obejct is treated by the dispatcher. +Dispatchers and functors +For approximate collision detection (pass 1), we want to compute bounds for all bodies in the simulation; +suppose we want bound of type axis-aligned bounding box. Since the exact algorithm is different depend- +ing on particular shape, we need to provide functors for handling all specific cases. In the O.engines=[…] +declared above, the line: +InsertionSortCollider([Bo1_Sphere_Aabb(),Bo1_Facet_Aabb()]) +creates InsertionSortCollider (it internally uses BoundDispatcher, but that is a detail). It traverses all +bodies and will, based on shape type of each body, dispatch one of the functors to create/update bound +for that particular body. In the case shown, it has 2 functors, one handling spheres, another facets. +The name is composed from several parts: Bo (functor creating Bound), which accepts 1 type Sphere +and creates an Aabb (axis-aligned bounding box; it is derived from Bound). The Aabb objects are used +by InsertionSortCollider itself. All Bo1 functors derive from BoundFunctor. +The next part, reading +12 +Chapter 1. +Guided tour + +Yade Documentation, Release 3rd ed. +Fig. 4: Example bound functors producing Aabb accepting various different types, such as Sphere, Facet +or Cylinder. In the case shown, the Bo1 functors produce Aabb instances from single specific Shape, +hence the number 1 in the functor name. Each of those functors uses specific geometry of the Shape i.e. +position of nodes in Facet or radius of sphere to calculate the Aabb. +InteractionLoop( +[Ig2_Sphere_Sphere_ScGeom(),Ig2_Facet_Sphere_ScGeom()], +[Ip2_FrictMat_FrictMat_FrictPhys()], +[Law2_ScGeom_FrictPhys_CundallStrack()], +), +hides 3 internal dispatchers within the InteractionLoop engine; they all operate on interactions and are, +for performance reasons, put together: +IGeomDispatcher which uses IGeomFunctor uses the first set of functors (Ig2), which are dis- +patched based on combination of 2 Shapes objects. Dispatched functor resolves exact collision +configuration and creates an Interaction Geometry IGeom (whence Ig in the name) associated +with the interaction, if there is collision. The functor might as well determine that there is no real +collision even if they did overlap in the approximate collision detection (e.g. the Aabb did overlap, +but the shapes did not). In that case the attribute is set to false and interaction is scheduled for +removal. +1. The first functor, Ig2_Sphere_Sphere_ScGeom, is called on interaction of 2 Spheres and +creates ScGeom instance, if appropriate. +2. The second functor, Ig2_Facet_Sphere_ScGeom, is called for interaction of Facet with Sphere +and might create (again) a ScGeom instance. +All Ig2 functors derive from IGeomFunctor (they are documented at the same place). +Fig. 5: Example interaction geometry functors producing ScGeom or ScGridCoGeom accepting two +various different types (hence 2 in their name Ig2), such as Sphere, Wall or PFacet. Each of those +functors uses specific geometry of the Shape i.e. +position of nodes in PFacet or radius of sphere to +calculate the interaction geometry. +1.1. +Introduction +13 + +Yade Documentation, Release 3rd ed. +IPhysDispatcher which uses IPhysFunctor dispatches to the second set of functors based on com- +bination of 2 Materials; these functors return return IPhys instance (the Ip prefix). In our case, +there is only 1 functor used, Ip2_FrictMat_FrictMat_FrictPhys, which create FrictPhys from 2 +FrictMat’s. +Ip2 functors are derived from IPhysFunctor. +Fig. 6: Example interaction physics functors (Ip2_FrictMat_CpmMat_FrictPhys, Ip2_FrictMat_Frict- +Mat_FrictPhys and Ip2_FrictMat_FrictViscoMat_FrictViscoPhys) producing FrictPhys or FrictVisco- +Phys accepting two various different types of Material (hence Ip2), such as CpmMat, FrictMat or +FrictViscoMat. +LawDispatcher which uses LawFunctor dispatches to the third set of functors, based on combina- +tions of IGeom and IPhys (wherefore 2 in their name again) of each particular interaction, created +by preceding functors. The Law2 functors represent constitutive law; they resolve the interaction +by computing forces on the interacting bodies (repulsion, attraction, shear forces, …) or otherwise +update interaction state variables. +Law2 functors all inherit from LawFunctor. +Fig. 7: +Example LawFunctors (Law2_CylScGeom_FrictPhys_CundallStrack, Law2_ScGeom_Frict- +Phys_CundallStrack and Law2_ScGridCoGeom_FrictPhys_CundallStrack) each of them performing +calcuation of forces according to selected constitutive law. +There is chain of types produced by earlier functors and accepted by later ones; the user is responsible +to satisfy type requirement (see img. img-dispatch-loop). An exception (with explanation) is raised in +the contrary case. +Note: +When Yade starts, O.engines is filled with a reasonable default list, so that it is not strictly +14 +Chapter 1. +Guided tour + +Yade Documentation, Release 3rd ed. +Fig. 8: Chain of functors producing and accepting certain types. In the case shown, the Ig2 functors +produce ScGeom instances from all handled Shapes combinations; the Ig2 functor produces FrictMat. +The constitutive law functor Law2 accepts the combination of types produced. Note that the types are +stated in the functor’s class names. +necessary to redefine it when trying simple things. The default scene will handle spheres, boxes, and +facets with frictional properties correctly, and adjusts the timestep dynamically. You can find an example +in examples/simple-scene/simple-scene-default-engines.py. +1.2 Tutorial +This tutorial originated as handout for a course held at Technische Universität Dresden / Fakultät +Bauingenieurwesen / Institut für Geotechnik in Jaunary 2011. The focus was to give quick and rather +practical introduction to people without prior modeling experience, but with knowledge of mechanics. +Some computer literacy was assumed, though basics are reviewed in the Hands-on section. +The course did not in reality follow this document, but was based on interactive writing and commenting +simple Examples, which were mostly suggested by participants; many thanks to them for their ideas and +suggestions. +1.2.1 Introduction +The chapter Introduction is summarized in following presentation Yade: past, present and future with +some additional different examples. This presentation is from year 2011 and does not include latest +additions. As of year 2019 it is factually correct. +1.2.2 Hands-on +Shell basics +Directory tree +Directory tree is hierarchical way to organize files in operating systems. A typical (reduced) tree in linux +looks like this: +1.2. +Tutorial +15 + +Yade Documentation, Release 3rd ed. +/ +Root +￿￿￿boot +System startup +￿￿￿bin +Low-level programs +￿￿￿lib +Low-level libraries +￿￿￿dev +Hardware access +￿￿￿sbin +Administration programs +￿￿￿proc +System information +￿￿￿var +Files modified by system services +￿￿￿root +Root (administrator) home directory +￿￿￿etc +Configuration files +￿￿￿media +External drives +￿￿￿tmp +Temporary files +￿￿￿usr +Everything for normal operation (usr = UNIX system resources) +￿ +￿￿￿bin +User programs +￿ +￿￿￿sbin +Administration programs +￿ +￿￿￿include +Header files for c/c++ +￿ +￿￿￿lib +Libraries +￿ +￿￿￿local +Locally installed software +￿ +￿￿￿doc +Documentation +￿￿￿home +Contains the user's home directories +￿￿￿user +Home directory for user +￿￿￿user1 +Home directory for user1 +Note that there is a single root /; all other disks (such as USB sticks) attach to some point in the tree +(e.g. in /media). +Shell navigation +Shell is the UNIX command-line, interface for conversation with the machine. Don’t be afraid. +Moving around +The shell is always operated by some user, at some concrete machine; these two are constant. We can +move in the directory structure, and the current place where we are is current directory. By default, it +is the home directory which contains all files belonging to the respective user: +user@machine:~$ +# user operating at machine, in the directory ~ (= user +�→'s home directory) +user@machine:~$ ls . +# list contents of the current directory +user@machine:~$ ls foo +# list contents of directory foo, relative to the␣ +�→dcurrent directory ~ (= ls ~/foo = ls /home/user/foo) +user@machine:~$ ls /tmp +# list contents of /tmp +user@machine:~$ cd foo +# change directory to foo +user@machine:~/foo$ ls ~ +# list home directory (= ls /home/user) +user@machine:~/foo$ cd bar +# change to bar (= cd ~/foo/bar) +user@machine:~/foo/bar$ cd ../../foo2 +# go to the parent directory twice, then to foo2 (cd ~/ +�→foo/bar/../../foo2 = cd ~/foo2 = cd /home/user/foo2) +user@machine:~/foo2$ cd +# go to the home directory (= ls ~ = ls /home/user) +user@machine:~$ +Users typically have only permissions to write (i.e. modify files) only in their home directory (abbreviated +~, usually is /home/user) and /tmp, and permissions to read files in most other parts of the system: +user@machine:~$ ls /root +# see what files the administrator has +ls: cannot open directory /root: Permission denied +16 +Chapter 1. +Guided tour + +Yade Documentation, Release 3rd ed. +Keys +Useful keys on the command-line are: + +show possible completions of what is being typed (use abundantly!) +^C (=Ctrl+C) +delete current line +^D +exit the shell +↑↓ +move up and down in the command history +^C +interrupt currently running program +^\ +kill currently running program +Shift-PgUp +scroll the screen up (show past output) +Shift-PgDown +scroll the screen down (show future output; works only on quantum computers) +Running programs +When a program is being run (without giving its full path), several directories are searched for program +of that name; those directories are given by $PATH: +user@machine:~$ echo $PATH +# show the value of $PATH +/usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/sbin:/bin:/usr/games +user@machine:~$ which ls +# say what is the real path of ls +The first part of the command-line is the program to be run (which), the remaining parts are arguments +(ls in this case). It is up to the program which arguments it understands. Many programs can take +special arguments called options starting with - (followed by a single letter) or -- (followed by words); +one of the common options is -h or --help, which displays how to use the program (try ls --help). +Full documentation for each program usually exists as manual page (or man page), which can be shown +using e.g. man ls (q to exit) +Starting yade +If yade is installed on the machine, it can be (roughly speaking) run as any other program; without any +arguments, it runs in the “dialog mode”, where a command-line is presented: +user@machine:~$ yade +Welcome to Yade 2019.01a +TCP python prompt on localhost:9002, auth cookie `adcusk' +XMLRPC info provider on http://localhost:21002 +[[ ^L clears screen, ^U kills line. F12 controller, F11 3d view, F10 both, F9 generator, F8␣ +�→plot. ]] +Yade [1]: +#### hit ^D to exit +Do you really want to exit ([y]/n)? +Yade: normal exit. +The command-line is in fact python, enriched with some yade-specific features. (Pure python interpreter +can be run with python or ipython commands). +Instead of typing commands on-by-one on the command line, they can be be written in a file (with the +.py extension) and given as argument to Yade: +user@machine:~$ yade simulation.py +For a complete help, see man yade +1.2. +Tutorial +17 + +Yade Documentation, Release 3rd ed. +Exercises +1. Open the terminal, navigate to your home directory +2. Create a new empty file and save it in ~/first.py +3. Change directory to /tmp; delete the file ~/first.py +4. Run program xeyes +5. Look at the help of Yade. +6. Look at the manual page of Yade +7. Run Yade, exit and run it again. +Python basics +We assume the reader is familar with Python tutorial and only briefly review some of the basic capabili- +ties. The following will run in pure-python interpreter (python or ipython), but also inside Yade, which +is a super-set of Python. +Numerical operations and modules: +Yade [1]: (1+3*4)**2 +# usual rules for operator precedence, ** is exponentiation +Out[1]: 169 +Yade [2]: import math +# gain access to "module" of functions +Yade [3]: math.sqrt(2) +# use a function from that module +Out[3]: 1.4142135623730951 +Yade [4]: import math as m +# use the module under a different name +Yade [5]: m.cos(m.pi) +Out[5]: -1.0 +Yade [6]: from math import * +# import everything so that it can be used without module name +Yade [7]: cos(pi) +Out[7]: -1.0 +Variables: +Yade [8]: a=1; b,c=2,3 +# multiple commands separated with ;, multiple assignment +Yade [9]: a+b+c +Out[9]: 6 +Sequences +Lists +Lists are variable-length sequences, which can be modified; they are written with braces [...], and their +elements are accessed with numerical indices: +Yade [10]: a=[1,2,3] +# list of numbers +Yade [11]: a[0] +# first element has index 0 +Out[11]: 1 +(continues on next page) +18 +Chapter 1. +Guided tour + +Yade Documentation, Release 3rd ed. +(continued from previous page) +Yade [12]: a[-1] +# negative counts from the end +Out[12]: 3 +Yade [13]: a[3] +# error +--------------------------------------------------------------------------- +IndexError +Traceback (most recent call last) +~/yade/lib/x86_64-linux-gnu/yadeflip/py/yade/__init__.py in +----> 1 a[3] +# error +IndexError: list index out of range +Yade [14]: len(a) +# number of elements +Out[14]: 3 +Yade [15]: a[1:] +# from second element to the end +Out[15]: [2, 3] +Yade [16]: a+=[4,5] +# extend the list +Yade [17]: a+=[6]; a.append(7) # extend with single value, both have the same effect +Yade [18]: 9 in a +# test presence of an element +Out[18]: False +Lists can be created in various ways: +Yade [19]: range(10) +Out[19]: range(0, 10) +Yade [20]: range(10)[-1] +Out[20]: 9 +List of squares of even number smaller than 20, i.e. +� +a2 ∀a ∈ {0, · · · , 19} +�� 2∥a +� +(note the similarity): +Yade [21]: [a**2 for a in range(20) if a%2==0] +Out[21]: [0, 4, 16, 36, 64, 100, 144, 196, 256, 324] +Tuples +Tuples are constant sequences: +Yade [22]: b=(1,2,3) +Yade [23]: b[0] +Out[23]: 1 +Yade [24]: b[0]=4 +# error +--------------------------------------------------------------------------- +TypeError +Traceback (most recent call last) +~/yade/lib/x86_64-linux-gnu/yadeflip/py/yade/__init__.py in +----> 1 b[0]=4 +# error +TypeError: 'tuple' object does not support item assignment +Dictionaries +Mapping from keys to values: +1.2. +Tutorial +19 + +Yade Documentation, Release 3rd ed. +Yade [25]: ende={'one':'ein' , 'two':'zwei' , 'three':'drei'} +Yade [26]: de={1:'ein' , 2:'zwei' , 3:'drei'}; en={1:'one' , 2:'two' , 3:'three'} +Yade [27]: ende['one'] +## access values +Out[27]: 'ein' +Yade [28]: de[1], en[2] +Out[28]: ('ein', 'two') +Functions, conditionals +Yade [29]: 4==5 +Out[29]: False +Yade [30]: a=3.1 +Yade [31]: if a<10: +....: +b=-2 +# conditional statement +....: else: +....: +b=3 +....: +Yade [32]: c=0 if a<1 else 1 +# trenary conditional expression +Yade [33]: b,c +Out[33]: (-2, 1) +Yade [34]: def square(x): return x**2 +# define a new function +....: +Yade [35]: square(2) +# and call that function +Out[35]: 4 +Exercises +1. Read the following code and say what wil be the values of a and b: +a=range(5) +b=[(aa**2 if aa%2==0 else -aa**2) for aa in a] +Yade basics +Yade objects are constructed in the following manner (this process is also called “instantiation”, since we +create concrete instances of abstract classes: one individual sphere is an instance of the abstract Sphere, +like Socrates is an instance of “man”): +Yade [36]: Sphere +# try also Sphere? +Out[36]: yade.wrapper.Sphere +Yade [37]: s=Sphere() +# create a Sphere, without specifying any attributes +Yade [38]: s.radius +# 'nan' is a special value meaning "not a number" (i.e. not␣ +�→defined) +Out[38]: nan +(continues on next page) +20 +Chapter 1. +Guided tour + +Yade Documentation, Release 3rd ed. +(continued from previous page) +Yade [39]: s.radius=2 +# set radius of an existing object +Yade [40]: s.radius +Out[40]: 2.0 +Yade [41]: ss=Sphere(radius=3) +# create Sphere, giving radius directly +Yade [42]: s.radius, ss.radius +# also try typing s. to see defined attributes +Out[42]: (2.0, 3.0) +Particles +Particles are the “data” component of simulation; they are the objects that will undergo some processes, +though do not define those processes yet. +Singles +There is a number of pre-defined functions to create particles of certain type; in order to create a sphere, +one has to (see the source of utils.sphere for instance): +1. Create Body +2. Set Body.shape to be an instance of Sphere with some given radius +3. Set Body.material (last-defined material is used, otherwise a default material is created) +4. Set position and orientation in Body.state, compute mass and moment of inertia based on Material +and Shape +In order to avoid such tasks, shorthand functions are defined in the utils module; to mention a few of +them, they are utils.sphere, utils.facet, utils.wall. +Yade [43]: s=utils.sphere((0,0,0),radius=1) +# create sphere particle centered at (0,0,0)␣ +�→with radius=1 +Yade [44]: s.shape +# s.shape describes the geometry of the particle +Out[44]: +Yade [45]: s.shape.radius +# we already know the Sphere class +Out[45]: 1.0 +Yade [46]: s.state.mass, s.state.inertia # inertia is computed from density and geometry +Out[46]: +(4188.790204786391, +Vector3(1675.516081914556253,1675.516081914556253,1675.516081914556253)) +Yade [47]: s.state.pos +# position is the one we prescribed +Out[47]: Vector3(0,0,0) +Yade [48]: s2=utils.sphere((-2,0,0),radius=1,fixed=True) +# explanation below +In the last example, the particle was fixed in space by the fixed=True parameter to utils.sphere; such a +particle will not move, creating a primitive boundary condition. +A particle object is not yet part of the simulation; in order to do so, a special function O.bodies.append +(also see Omega::bodies and Scene) is called: +Yade [49]: O.bodies.append(s) +# adds particle s to the simulation; returns id of␣ +�→the particle(s) added +Out[49]: 24 +1.2. +Tutorial +21 + +Yade Documentation, Release 3rd ed. +Packs +There are functions to generate a specific arrangement of particles in the pack module; for instance, cloud +(random loose packing) of spheres can be generated with the pack.SpherePack class: +Yade [50]: from yade import pack +Yade [51]: sp=pack.SpherePack() +# create an empty cloud; SpherePack contains␣ +�→only geometrical information +Yade [52]: sp.makeCloud((1,1,1),(2,2,2),rMean=.2) # put spheres with defined radius inside box␣ +�→given by corners (1,1,1) and (2,2,2) +Out[52]: 6 +Yade [53]: for c,r in sp: print(c,r) +# print center and radius of all particles␣ +�→(SpherePack is a sequence which can be iterated over) +....: +Vector3(1.274443445943540087,1.367880531586413095,1.275812677797829142) 0.2 +Vector3(1.395144492365041344,1.682654071894964964,1.504498259316015663) 0.2 +Vector3(1.691901039056097789,1.646021131505697621,1.775464815231047933) 0.2 +Vector3(1.698276069049370784,1.239026483132536161,1.307143271925819583) 0.2 +Vector3(1.546530002061732745,1.248592120112199888,1.780182798004842359) 0.2 +Vector3(1.77882382544176032,1.662755981140189299,1.256899030310328236) 0.2 +Yade [54]: sp.toSimulation() +# create particles and add them to the␣ +�→simulation +Out[54]: [25, 26, 27, 28, 29, 30] +Boundaries +utils.facet (triangle Facet) and utils.wall (infinite axes-aligned plane Wall) geometries are typically used +to define boundaries. For instance, a “floor” for the simulation can be created like this: +Yade [55]: O.bodies.append(utils.wall(-1,axis=2)) +Out[55]: 31 +There are other conveinence functions (like utils.facetBox for creating closed or open rectangular box, or +family of ymport functions) +Look inside +The simulation can be inspected in several ways. All data can be accessed from python directly: +Yade [56]: len(O.bodies) +Out[56]: 32 +Yade [57]: O.bodies[10].shape.radius +# radius of body #10 (will give error if not sphere,␣ +�→since only spheres have radius defined) +Out[57]: 0.16 +Yade [58]: O.bodies[12].state.pos +# position of body #12 +Out[58]: Vector3(1.224408970802669305,1.614102010700998235,1.392374433964128411) +Besides that, Yade says this at startup (the line preceding the command-line): +[[ ^L clears screen, ^U kills line. F12 controller, F11 3d view, F10 both, F9 generator, F8␣ +�→plot. ]] +22 +Chapter 1. +Guided tour + +Yade Documentation, Release 3rd ed. +Controller Pressing F12 brings up a window for controlling the simulation. +Although typically no +human intervention is done in large simulations (which run “headless”, without any graphical +interaction), it can be handy in small examples. There are basic information on the simulation +(will be used later). +3d view The 3d view can be opened with F11 (or by clicking on button in the Controller – see below). +There is a number of keyboard shortcuts to manipulate it (press h to get basic help), and it can +be moved, rotated and zoomed using mouse. Display-related settings can be set in the “Display” +tab of the controller (such as whether particles are drawn). +Inspector Inspector is opened by clicking on the appropriate button in the Controller. It shows (and +updates) internal data of the current simulation. In particular, one can have a look at engines, +particles (Bodies) and interactions (Interactions). Clicking at each of the attribute names links to +the appropriate section in the documentation. +Exercises +1. What is this code going to do? +Yade [59]: O.bodies.append([utils.sphere((2*i,0,0),1) for i in range(1,20)]) +Out[59]: [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50] +2. Create a simple simulation with cloud of spheres enclosed in the box (0,0,0) and (1,1,1) with +mean radius .1. (hint: pack.SpherePack.makeCloud) +3. Enclose the cloud created above in box with corners (0,0,0) and (1,1,1); keep the top of the +box open. (hint: utils.facetBox; type utils.facetBox? or utils.facetBox?? to get help on the +command line) +4. Open the 3D view, try zooming in/out; position axes so that z is upwards, y goes to the right and +x towards you. +Engines +Engines define processes undertaken by particles. As we know from the theoretical introduction, the +sequence of engines is called simulation loop. Let us define a simple interaction loop: +Yade [60]: O.engines=[ +# newlines and indentations are not important until␣ +�→the brace is closed +....: +ForceResetter(), +....: +InsertionSortCollider([Bo1_Sphere_Aabb(),Bo1_Wall_Aabb()]), +....: +InteractionLoop( +# dtto for the parenthesis here +....: +[Ig2_Sphere_Sphere_ScGeom(),Ig2_Wall_Sphere_ScGeom()], +....: +[Ip2_FrictMat_FrictMat_FrictPhys()], +....: +[Law2_ScGeom_FrictPhys_CundallStrack()] +....: +), +....: +NewtonIntegrator(damping=.2,label='newtonCustomLabel') +# define a label␣ +�→newtonCustomLabel under which we can access this engine easily +....: ] +....: +Yade [61]: O.engines +Out[61]: +[, +, +, +] +Yade [62]: O.engines[-1]==newtonCustomLabel +# is it the same object? +Out[62]: True +(continues on next page) +1.2. +Tutorial +23 + +Yade Documentation, Release 3rd ed. +(continued from previous page) +Yade [63]: newtonCustomLabel.damping +Out[63]: 0.2 +Instead of typing everything into the command-line, one can describe simulation in a file (script) and +then run yade with that file as an argument. We will therefore no longer show the command-line unless +necessary; instead, only the script part will be shown. Like this: +O.engines=[ +# newlines and indentations are not important until the brace is␣ +�→closed +ForceResetter(), +InsertionSortCollider([Bo1_Sphere_Aabb(),Bo1_Wall_Aabb()]), +InteractionLoop( +# dtto for the parenthesis here +[Ig2_Sphere_Sphere_ScGeom(),Ig2_Wall_Sphere_ScGeom()], +[Ip2_FrictMat_FrictMat_FrictPhys()], +[Law2_ScGeom_FrictPhys_CundallStrack()] +), +GravityEngine(gravity=(0,0,-9.81)), +# 9.81 is the gravity␣ +�→acceleration, and we say that +NewtonIntegrator(damping=.2,label='newtonCustomLabel') # define a label under which we␣ +�→can access this engine easily +] +Besides engines being run, it is likewise important to define how often they will run. Some engines can +run only sometimes (we will see this later), while most of them will run always; the time between two +successive runs of engines is timestep (∆t). There is a mathematical limit on the timestep value, called +critical timestep, which is computed from properties of particles. Since there is a function for that, we +can just set timestep using utils.PWaveTimeStep: +O.dt=utils.PWaveTimeStep() +Each time when the simulation loop finishes, time O.time is advanced by the timestep O.dt: +Yade [64]: O.dt=0.01 +Yade [65]: O.time +Out[65]: 0.0 +Yade [66]: O.step() +Yade [67]: O.time +Out[67]: 0.01 +For experimenting with a single simulations, it is handy to save it to memory; this can be achieved, once +everything is defined, with: +O.saveTmp() +Exercises +1. Define engines as in the above example, run the Inspector and click through the engines to see +their sequence. +2. Write a simple script which will +1. define particles as in the previous exercise (cloud of spheres inside a box open from the top) +2. define a simple simulation loop, as the one given above +3. set ∆t equal to the critical P-Wave ∆t +24 +Chapter 1. +Guided tour + +Yade Documentation, Release 3rd ed. +4. save the initial simulation state to memory +3. Run the previously-defined simulation multiple times, while changing the value of timestep (use +the ￿ button to reload the initial configuration). +1. See what happens as you increase ∆t above the P-Wave value. +2. Try changing the gravity parameter, before running the simulation. +3. Try changing damping +4. Reload the simulation, open the 3d view, open the Inspector, select a particle in the 3d view (shift- +click). Then run the simulation and watch how forces on that particle change; pause the simulation +somewhere in the middle, look at interactions of this particle. +5. At which point can we say that the deposition is done, so that the simulation can be stopped? +See also: +The Bouncing sphere example shows a basic simulation. +1.2.3 Data mining +Read +Local data +All data of the simulation are accessible from python; when you open the Inspector, blue labels of various +data can be clicked – left button for getting to the documentation, middle click to copy the name of the +object (use Ctrl-V or middle-click to paste elsewhere). The interesting objects are among others (see +Omega for a full list): +1. O.engines +Engines are accessed by their index (position) in the simulation loop: +O.engines[0] +# first engine +O.engines[-1] +# last engine +Note: +The index can change if O.engines is modified. Labeling introduced in the section below +is a better solution for reliable access to a particular engine. +2. O.bodies +Bodies are identified by their id, which is guaranteed to not change during the whole simulation: +O.bodies[0] +# first body +[b.shape.radius for b in O.bodies if isinstance(b.shape,Sphere)] +# list of radii of␣ +�→all spherical bodies +sum([b.state.mass for b in O.bodies]) +# sum of masses of␣ +�→all bodies +numpy.average([b.state.vel[0] for b in O.bodies]) +# average velocity in␣ +�→x direction +Note: +Uniqueness of Body.id is not guaranteed, since newly created bodies might recycle ids of +deleted ones. +3. O.forces +1.2. +Tutorial +25 + +Yade Documentation, Release 3rd ed. +Generalized forces (forces, torques) acting on each particle. They are (usually) reset at the begin- +ning of each step with ForceResetter, subsequently forces from individual interactions are accumu- +lated in InteractionLoop. To access the data, use: +O.forces.f(0) +# force on #0 +O.forces.t(1) +# torque on #1 +4. O.interactions +Interactions are identified by ids of the respective interacting particles (they are created and deleted +automatically during the simulation): +O.interactions[0,1] +# interactions of #0 with #1 +O.interactions[1,0] +# the same object +O.bodies[0].intrs() +# all interactions of body #0 +for i in O.bodies[12].intrs(): print (i.isReal,i.id1,i.id2) +# get some info about␣ +�→interactions of body #12 +[(i.isReal,i.id1,i.id2) for i in O.bodies[12].intrs()] +# same thing, but make a␣ +�→list +Labels +Engines and functors can be labeled, which means that python variable of that name is automatically +created. +Yade [1]: O.engines=[ +...: +NewtonIntegrator(damping=.2,label='newtonCustomLabel') +...: ] +...: +Yade [2]: newtonCustomLabel.damping=.4 +Yade [3]: O.engines[0].damping +# O.engines[0] and newtonCustomLabel are the same␣ +�→objects +Out[3]: 0.4 +Yade [4]: newtonCustomLabel==O.engines[0] +# O.engines[0] and newtonCustomLabel are the same␣ +�→objects +Out[4]: True +Exercises +1. Find meaning of this expression: +max([b.state.vel.norm() for b in O.bodies]) +2. Run the Gravity deposition script, pause after a few seconds of simulation. Write expressions that +compute +1. kinetic energy � 1 +2mi|vi|2 +2. average mass (hint: use numpy.average) +3. maximum z-coordinate of all particles +4. number of interactions of body #1 +Global data +Useful measures of what happens in the simulation globally: +26 +Chapter 1. +Guided tour + +Yade Documentation, Release 3rd ed. +unbalanced force ratio of maximum contact force and maximum per-body force; measure of staticity, +computed with unbalancedForce. +porosity ratio of void volume and total volume; computed with porosity. +coordination number average number of interactions per particle, avgNumInteractions +stress tensor (periodic boundary conditions) averaged force in interactions, computed with nor- +malShearStressTensors +fabric tensor distribution of contacts in space (not yet implemented); can be visualized with plotDi- +rections +Energies +Evaluating energy data for all components in the simulation (such as gravity work, kinetic energy, plastic +dissipation, damping dissipation) can be enabled with +O.trackEnergy=True +Subsequently, energy values are accessible in the O.energy; it is a dictionary where its entries can be +retrived with keys() and their values with O.energy[key]. +Save +PyRunner +To save data that we just learned to access, we need to call Python from within the simulation loop. +PyRunner is created just for that; it inherits periodicy control from PeriodicEngine and takes the code +to run as text (must be quoted, i.e. inside '...') attribute called command. For instance, adding this +to O.engines will print the current step number every one second wall clock time: +O.engines=O.engines+[ PyRunner(command='print(O.iter)',realPeriod=1) ] +Writing complicated code inside command is awkward; in such case, we define a function that will be +called: +def myFunction(): +'''Print step number, and pause the simulation is unbalanced force is smaller than 0. +�→05.''' +print(O.iter) +if utils.unbalancedForce()<0.05: +print('Unbalanced force is smaller than 0.05, pausing.') +O.pause() +Now this function can be added to O.engines: +O.engines+=[PyRunner(command='myFunction()',iterPeriod=100)] +or, in general, like that: +O.engines=[ +# ... +PyRunner(command='myFunction()',iterPeriod=100) # call myFunction every 100 steps +] +Warning: +If a function was declared inside a live yade session (ipython) and PyRunner attribute +updateGlobals is set to False then an error NameError: name 'myFunction' is not defined will +occur unless python globals() are updated with command +1.2. +Tutorial +27 + +Yade Documentation, Release 3rd ed. +globals().update(locals()) +Exercises +1. Run the Gravity deposition simulation, but change it such that: +1. utils.unbalancedForce is printed every 2 seconds. +2. check every 1000 steps the value of unbalanced force +• if smaller than 0.2, set damping to 0.8 (hint: use labels) +• if smaller than 0.1, pause the simulation +Keeping history +Yade provides the plot module used for storing and plotting variables (plotting itself will be discussed +later). Let us start by importing this module and declare variable names that will be plotted: +from yade import plot +plot.plots={'t':('coordNum','unForce',None,'Ek')} +# kinetic energy will have␣ +�→legend on the right as indicated by None separator. +Periodic storing of data is done with PyRunner and the plot.addData function. Also let’s enable energy +tracking: +O.trackEnergy=True +def addPlotData(): +# this function adds current values to the history of data, under the names specified +plot.addData(t=O.time,Ek=utils.kineticEnergy(),coordNum=utils.avgNumInteractions(), +�→unForce=utils.unbalancedForce()) +Now this function can be added to O.engines: +O.engines+=[PyRunner(command='addPlotData()',iterPeriod=20)] +or, in general, like that: +O.engines=[ +# ..., +PyRunner(command='addPlotData()',iterPeriod=20) +# call the addPlotData␣ +�→function every 20 iterations +] +History is stored in plot.data, and can be accessed using the variable name, e.g. plot.data['Ek'], and +saved to text file (for post-processing outside yade) with plot.saveDataTxt. +Plot +plot provides facilities for plotting history saved with plot.addData as 2d plots. Data to be plotted are +specified using dictionary plot.plots +plot.plots={'t':('coordNum','unForce',None,'Ek')} +History of all values is given as the name used for plot.addData; keys of the dictionary are x-axis values, +and values are sequence of data on the y axis; the None separates data on the left and right axes (they +are scaled independently). The plot itself is created with +28 +Chapter 1. +Guided tour + +Yade Documentation, Release 3rd ed. +plot.plot() +# on the command line, F8 can be used as shorthand +While the plot is open, it will be updated periodically, so that simulation evolution can be seen in +real-time. +Energy plots +Plotting all energy contributions would be difficult, since names of all energies might not be known in +advance. Fortunately, there is a way to handle that in Yade. It consists in two parts: +1. plot.addData is given all the energies that are currently defined: +plot.addData(i=O.iter,total=O.energy.total(),**O.energy) +The O.energy.total functions, which sums all energies together. The **O.energy is special python +syntax for converting dictionary (remember that O.energy is a dictionary) to named functions +arguments, so that the following two commands are identical: +function(a=3,b=34) +# give arguments as arguments +function(**{'a':3,'b':34}) +# create arguments from dictionary +2. Data to plot are specified using a function that gives names of data to plot, rather than providing +the data names directly: +plot.plots={'i':['total']+O.energy.keys()} +where total is the name we gave to O.energy.total() above, while O.energy.keys() will always +return list of currently defined energies. +Energy plot example +Plotting energies inside a live yade session, for example by launching examples/test/triax-basic-without- +plots.py would look following: +from yade import plot +O.trackEnergy=True +O.step() +# performing a single simulation step is necessary to␣ +�→populate O.energy.keys() +plot.plots={'t':O.energy.keys()+['total']} +def addPlotData(): +# this function adds current values to the history of data, under the names specified +plot.addData( t=O.time , total=O.energy.total() , **O.energy ) +O.engines+=[PyRunner(command='addPlotData()',iterPeriod=20)] +globals().update(locals()) +# do this only because this is an example of a live yade␣ +�→session +Press F8 to show plot window and F11 to show 3D view, then press ￿ to start simulation. +Using multiple plots +It is also possible to make several separate plots, for example like this: +plot.plots={ 't':('total','kinetic') , 't ':['elastPotential','gravWork'] , 't +':('nonviscDamp +�→') } +1.2. +Tutorial +29 + +Yade Documentation, Release 3rd ed. +Warning: +There cannot be duplicate names declared in separate plots. This is why spaces were +used above to indicate the same variable t. +With the caveat above, a following example inside a live yade session launched on examples/test/triax- +basic-without-plots.py would look following: +from yade import plot +O.trackEnergy=True +plot.plots={ 't':('total','kinetic') , 't ':['elastPotential','gravWork'] , 't +':('nonviscDamp +�→') } +def addPlotData(): +# assign value to all three: 't', 't ' and 't +' with single t=... assignment +plot.addData( t=O.time , total=O.energy.total() , **O.energy ) +O.engines+=[PyRunner(command='addPlotData()',iterPeriod=20)] +globals().update(locals()) +# do this only because this is an example of a live yade␣ +�→session +plot.plot(subPlots=False) +# show plots in separate windows +plot.plot(subPlots=True) +# same as pressing F8: close current plot windows and reopen␣ +�→a single new one +Press F8 to show plot window and F11 to show 3D view, then press ￿ to start simulation, see video below: +Exercises +1. Calculate average momentum in y direction. +2. Run the Gravity deposition script, plotting unbalanced force and kinetic energy. +3. While the script is running, try changing the NewtonIntegrator.damping parameter (do it from both +Inspector and from the command-line). What influence does it have on the evolution of unbalanced +force and kinetic energy? +4. Think about and write down all energy sources (input); write down also all energy sinks (dissipa- +tion). +5. Simulate Gravity deposition and plot all energies as they evolve during the simulation. +See also: +Most Examples with tutorial use plotting facilities of Yade, some of them also track energy of the +simulation. +1.2.4 Setting up a simulation +See also: +Examples Gravity deposition, Oedometric test, Periodic simple shear, Periodic triaxial test deal with +topics discussed here. +Parametric studies +Input parameters of the simulation (such as size distribution, damping, various contact parameters, …) +influence the results, but frequently an analytical relationship is not known. To study such influence, +similar simulations differing only in a few parameters can be run and results compared. Yade can be run +in batch mode, where one simulation script is used in conjunction with parameter table, which specifies +30 +Chapter 1. +Guided tour + +Yade Documentation, Release 3rd ed. +parameter values for each run of the script. Batch simulation are run non-interactively, i.e. without user +intervention; the user must therefore start and stop the simulation explicitly. +Suppose we want to study the influence of damping on the evolution of kinetic energy. The script has to +be adapted at several places: +1. We have to make sure the script reads relevant parameters from the parameter table. This is done +using utils.readParamsFromTable; the parameters which are read are created as variables in the +yade.params.table module: +readParamsFromTable(damping=.2) +# yade.params.table.damping variable will be created +from yade.params import table +# typing table.damping is easier than yade. +�→params.table.damping +Note that utils.readParamsFromTable takes default values of its parameters, which are used if the +script is not run in non-batch mode. +2. Parameters from the table are used at appropriate places: +NewtonIntegrator(damping=table.damping), +3. The simulation is run non-interactively; we must therefore specify at which point it should stop: +O.engines+=[PyRunner(iterPeriod=1000,command='checkUnbalancedForce()')] +# call our␣ +�→function defined below periodically +def checkUnbalancedForce(): +if unbalancedForce<0.05: +# exit Yade if unbalanced force␣ +�→drops below 0.05 +plot.saveDataTxt(O.tags['d.id']+'.data.bz2') +# save all data into a unique file␣ +�→before exiting +import sys +sys.exit(0) +# exit the program +4. Finally, we must start the simulation at the very end of the script: +O.run() +# run forever, until stopped by checkUnbalancedForce() +waitIfBatch() +# do not finish the script until the simulation ends; does nothing␣ +�→in non-batch mode +The parameter table is a simple text-file (e.g. params.txt ), where each line specifies a simulation to +run: +# comments start with # as in python +damping +# first non-comment line is variable name +.2 +.4 +.6 +Finally, the simulation is run using the special batch command: +user@machine:~$ yade-batch params.txt simulation.py +Exercises +1. Run the Gravity deposition script in batch mode, varying damping to take values of .2, .4, .6. +2. See the http://localhost:9080 overview page while the batch is running (fig. imgBatchExample). +1.2. +Tutorial +31 + +Yade Documentation, Release 3rd ed. +Boundary +Particles moving in infinite space usually need some constraints to make the simulation meaningful. +Supports +So far, supports (unmovable particles) were providing necessary boundary: in the Gravity deposition +script the geom.facetBox is internally composed of facets (triangulation elements), which are fixed in +space; facets are also used for arbitrary triangulated surfaces (see relevant sections of the User’s manual). +Another frequently used boundary is utils.wall (infinite axis-aligned plane). +Periodic +Periodic boundary is a “boundary” created by using periodic (rather than infinite) space. Such boundary +is activated by O.periodic=True , and the space configuration is decribed by O.cell . It is well suited for +studying bulk material behavior, as boundary effects are avoided, leading to smaller number of particles. +On the other hand, it might not be suitable for studying localization, as any cell-level effects (such as +shear bands) have to satisfy periodicity as well. +32 +Chapter 1. +Guided tour + +r! +Yade-batch overview +口x +File Edit +View +History +BookmarksToolsHelp +http:/localhost:9080 +32coresavailable,3used+29free +Jobs +3total,3 +running +odone +id +status +info +cores +plots +le1 +1e2 +1.4 +unbalanced +elastPotential +gravWork. +kinetic +1.2 +nonviscDamp +2 +nonviscDamp, plastDissip +plastDissip +1 +1.0 +Q0:00:04 +step12929 +0.8 +speed 3405.9/sec +YADE_BATCH= +damping=.2 +352bodies +examples/bin/ya +Stop +1459intrs +0.4 +0.2 +0.0 + 0.0 +s'0 +1.0 +2.0 +2.5 +1e3 +1e2 +1.5 +unbalanced +elastPotential +gravork +kinetic +nonviscDamp +1.4 +plastDissip +0.5 +localhost (::1)Yade Documentation, Release 3rd ed. +The periodic cell is described by its reference size of box aligned with global axes, and current transfor- +mation, which can capture stretch, shear and rotation. Deformation is prescribed via velocity gradient, +which updates the transformation before the next step. +Homothetic deformation can smear velocity +gradient accross the cell, making the boundary dissolve in the whole cell. +Stress and strains can be controlled with PeriTriaxController; it is possible to prescribe mixed +strain/stress goal state using PeriTriaxController.stressMask. +The following creates periodic cloud of spheres and compresses to achieve σx=-10 kPa, σy=-10kPa and +εz=-0.1. Since stress is specified for y and z, stressMask is binary 0b011 (x→1, y→2, z→4, in decimal +1+2=3). +Yade [1]: sp=pack.SpherePack() +Yade [2]: sp.makeCloud((1,1,1),(2,2,2),rMean=.16,periodic=True) +Out[2]: 20 +Yade [3]: sp.toSimulation() +# implicitly sets O.periodic=True, and O.cell.refSize␣ +�→to the packing period size +Out[3]: [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23] +Yade [4]: O.engines+=[PeriTriaxController(goal=(-1e4,-1e4,-.1),stressMask=0b011,maxUnbalanced=. +�→2,doneHook='functionToRunWhenFinished()')] +When the simulation runs, PeriTriaxController takes over the control and calls doneHook when goal is +reached. A full simulation with PeriTriaxController might look like the following: +from __future__ import print_function +from yade import pack, plot +sp = pack.SpherePack() +rMean = .05 +sp.makeCloud((0, 0, 0), (1, 1, 1), rMean=rMean, periodic=True) +sp.toSimulation() +O.engines = [ +ForceResetter(), +InsertionSortCollider([Bo1_Sphere_Aabb()], verletDist=.05 * rMean), +InteractionLoop([Ig2_Sphere_Sphere_ScGeom()], [Ip2_FrictMat_FrictMat_FrictPhys()],␣ +�→[Law2_ScGeom_FrictPhys_CundallStrack()]), +NewtonIntegrator(damping=.6), +PeriTriaxController( +goal=(-1e6, -1e6, -.1), stressMask=0b011, maxUnbalanced=.2, doneHook= +�→'goalReached()', label='triax', maxStrainRate=(.1, .1, .1), dynCell=True +), +PyRunner(iterPeriod=100, command='addPlotData()') +] +O.dt = .5 * utils.PWaveTimeStep() +O.trackEnergy = True +def goalReached(): +print('Goal reached, strain', triax.strain, ' stress', triax.stress) +O.pause() +def addPlotData(): +plot.addData( +sx=triax.stress[0], +sy=triax.stress[1], +sz=triax.stress[2], +ex=triax.strain[0], +ey=triax.strain[1], +ez=triax.strain[2], +i=O.iter, +(continues on next page) +1.2. +Tutorial +33 + +Yade Documentation, Release 3rd ed. +(continued from previous page) +unbalanced=utils.unbalancedForce(), +totalEnergy=O.energy.total(), +**O.energy +# plot all energies +) +plot.plots = { +'i': (('unbalanced', 'go'), None, 'kinetic'), +' i': ('ex', 'ey', 'ez', None, 'sx', 'sy', 'sz'), +'i ': (O.energy.keys, None, ('totalEnergy', 'bo')) +} +plot.plot() +O.saveTmp() +O.run() +1.2.5 Advanced & more +Particle size distribution +See Periodic triaxial test and examples/test/psd.py +Clumps +Clump; see Periodic triaxial test +Testing laws +LawTester, scripts/checks-and-tests/law-test.py +New law +Visualization +See the example 3d-postprocessing and video recording +• VTKRecorder & Paraview +• makeVideo +• SnapshotEngine +• doc/sphinx/tutorial/05-3d-postprocessing.py +• examples/test/force-network-video.py +• doc/sphinx/tutorial/make-simulation-video.py +Convert python 2 scripts to python 3 +Below is a non-exhaustive list of common things to do to convert your scripts to python 3. +34 +Chapter 1. +Guided tour + +Yade Documentation, Release 3rd ed. +Mandatory: +• print ... becomes print(...), +• myDict.iterkeys(), +myDict.itervalues(), +myDict.iteritems() becomes myDict.keys(), +myDict.values(), myDict.items(), +• import cPickle becomes import pickle, +• ‘‘ and <> operators are no longer recognized, +• inconsistent use of tabs and spaces in indentation is prohibited, for this reason all scripts in yade +use tabs for indentation. +Should be checked, but not always mandatory: +• (euclidian) division of two integers: i1/i2 becomes i1//i2, +• myDict.keys(), myDict.values(), myDict.items() becomes sometimes list(myDict.keys()), +list(myDict.values()), list(myDict.items()) (depending on your usage), +• map(), filter(), zip() becomes sometimes list(map()), list(filter()), list(zip()) (de- +pending on your usage), +• string encoding is now UTF8 everywhere, it may cause problems on user inputs/outputs (keyboard, +file…) with special chars. +Optional: +• # encoding: utf-8 no longer needed +1.2.6 Examples with tutorial +The online version of this tutorial contains embedded videos. +Bouncing sphere +Following example is in file doc/sphinx/tutorial/01-bouncing-sphere.py. +# basic simulation showing sphere falling ball gravity, +# bouncing against another sphere representing the support +# DATA COMPONENTS +# add 2 particles to the simulation +# they the default material (utils.defaultMat) +O.bodies.append( +[ +# fixed: particle's position in space will not change (support) +sphere(center=(0, 0, 0), radius=.5, fixed=True), +# this particles is free, subject to dynamics +sphere((0, 0, 2), .5) +] +) +# FUNCTIONAL COMPONENTS +# simulation loop -- see presentation for the explanation +O.engines = [ +(continues on next page) +1.2. +Tutorial +35 + +Yade Documentation, Release 3rd ed. +(continued from previous page) +ForceResetter(), +InsertionSortCollider([Bo1_Sphere_Aabb()]), +InteractionLoop( +[Ig2_Sphere_Sphere_ScGeom()], +# collision geometry +[Ip2_FrictMat_FrictMat_FrictPhys()], +# collision "physics" +[Law2_ScGeom_FrictPhys_CundallStrack()] +# contact law -- apply forces +), +# Apply gravity force to particles. damping: numerical dissipation of energy. +NewtonIntegrator(gravity=(0, 0, -9.81), damping=0.1) +] +# set timestep to a fraction of the critical timestep +# the fraction is very small, so that the simulation is not too fast +# and the motion can be observed +O.dt = .5e-4 * PWaveTimeStep() +# save the simulation, so that it can be reloaded later, for experimentation +O.saveTmp() +Gravity deposition +Following example is in file doc/sphinx/tutorial/02-gravity-deposition.py. +# gravity deposition in box, showing how to plot and save history of data, +# and how to control the simulation while it is running by calling +# python functions from within the simulation loop +# import yade modules that we will use below +from yade import pack, plot +# create rectangular box from facets +O.bodies.append(geom.facetBox((.5, .5, .5), (.5, .5, .5), wallMask=31)) +# create empty sphere packing +# sphere packing is not equivalent to particles in simulation, it contains only the pure␣ +�→geometry +sp = pack.SpherePack() +# generate randomly spheres with uniform radius distribution +sp.makeCloud((0, 0, 0), (1, 1, 1), rMean=.05, rRelFuzz=.5) +# add the sphere pack to the simulation +sp.toSimulation() +O.engines = [ +ForceResetter(), +InsertionSortCollider([Bo1_Sphere_Aabb(), Bo1_Facet_Aabb()]), +InteractionLoop( +# handle sphere+sphere and facet+sphere collisions +[Ig2_Sphere_Sphere_ScGeom(), Ig2_Facet_Sphere_ScGeom()], +[Ip2_FrictMat_FrictMat_FrictPhys()], +[Law2_ScGeom_FrictPhys_CundallStrack()] +), +NewtonIntegrator(gravity=(0, 0, -9.81), damping=0.4), +# call the checkUnbalanced function (defined below) every 2 seconds +PyRunner(command='checkUnbalanced()', realPeriod=2), +# call the addPlotData function every 200 steps +PyRunner(command='addPlotData()', iterPeriod=100) +] +O.dt = .5 * PWaveTimeStep() +# enable energy tracking; any simulation parts supporting it +(continues on next page) +36 +Chapter 1. +Guided tour + +Yade Documentation, Release 3rd ed. +(continued from previous page) +# can create and update arbitrary energy types, which can be +# accessed as O.energy['energyName'] subsequently +O.trackEnergy = True +# if the unbalanced forces goes below .05, the packing +# is considered stabilized, therefore we stop collected +# data history and stop +def checkUnbalanced(): +if unbalancedForce() < .05: +O.pause() +plot.saveDataTxt('bbb.txt.bz2') +# plot.saveGnuplot('bbb') is also possible +# collect history of data which will be plotted +def addPlotData(): +# each item is given a names, by which it can be the unsed in plot.plots +# the **O.energy converts dictionary-like O.energy to plot.addData arguments +plot.addData(i=O.iter, unbalanced=unbalancedForce(), **O.energy) +# define how to plot data: 'i' (step number) on the x-axis, unbalanced force +# on the left y-axis, all energies on the right y-axis +# (O.energy.keys is function which will be called to get all defined energies) +# None separates left and right y-axis +plot.plots = {'i': ('unbalanced', None, O.energy.keys)} +# show the plot on the screen, and update while the simulation runs +plot.plot() +O.saveTmp() +Oedometric test +Following example is in file doc/sphinx/tutorial/03-oedometric-test.py. +# gravity deposition, continuing with oedometric test after stabilization +# shows also how to run parametric studies with yade-batch +# The components of the batch are: +# 1. table with parameters, one set of parameters per line (ccc.table) +# 2. readParamsFromTable which reads respective line from the parameter file +# 3. the simulation muse be run using yade-batch, not yade +# +# $ yade-batch --job-threads=1 03-oedometric-test.table 03-oedometric-test.py +# +# load parameters from file if run in batch +# default values are used if not run from batch +readParamsFromTable(rMean=.05, rRelFuzz=.3, maxLoad=1e6, minLoad=1e4) +# make rMean, rRelFuzz, maxLoad accessible directly as variables later +from yade.params.table import * +# create box with free top, and ceate loose packing inside the box +from yade import pack, plot +O.bodies.append(geom.facetBox((.5, .5, .5), (.5, .5, .5), wallMask=31)) +sp = pack.SpherePack() +sp.makeCloud((0, 0, 0), (1, 1, 1), rMean=rMean, rRelFuzz=rRelFuzz) +sp.toSimulation() +(continues on next page) +1.2. +Tutorial +37 + +Yade Documentation, Release 3rd ed. +(continued from previous page) +O.engines = [ +ForceResetter(), +# sphere, facet, wall +InsertionSortCollider([Bo1_Sphere_Aabb(), Bo1_Facet_Aabb(), Bo1_Wall_Aabb()]), +InteractionLoop( +# the loading plate is a wall, we need to handle sphere+sphere, sphere+facet,␣ +�→sphere+wall +[Ig2_Sphere_Sphere_ScGeom(), Ig2_Facet_Sphere_ScGeom(), Ig2_Wall_Sphere_ +�→ScGeom()], +[Ip2_FrictMat_FrictMat_FrictPhys()], +[Law2_ScGeom_FrictPhys_CundallStrack()] +), +NewtonIntegrator(gravity=(0, 0, -9.81), damping=0.5), +# the label creates an automatic variable referring to this engine +# we use it below to change its attributes from the functions called +PyRunner(command='checkUnbalanced()', realPeriod=2, label='checker'), +] +O.dt = .5 * PWaveTimeStep() +# the following checkUnbalanced, unloadPlate and stopUnloading functions are all called by the +�→'checker' +# (the last engine) one after another; this sequence defines progression of different stages␣ +�→of the +# simulation, as each of the functions, when the condition is satisfied, updates 'checker' to␣ +�→call +# the next function when it is run from within the simulation next time +# check whether the gravity deposition has already finished +# if so, add wall on the top of the packing and start the oedometric test +def checkUnbalanced(): +# at the very start, unbalanced force can be low as there is only few contacts, but it␣ +�→does not mean the packing is stable +if O.iter < 5000: +return +# the rest will be run only if unbalanced is < .1 (stabilized packing) +if unbalancedForce() > .1: +return +# add plate at the position on the top of the packing +# the maximum finds the z-coordinate of the top of the topmost particle +O.bodies.append(wall(max([b.state.pos[2] + b.shape.radius for b in O.bodies if␣ +�→isinstance(b.shape, Sphere)]), axis=2, sense=-1)) +global plate +# without this line, the plate variable would only exist inside this␣ +�→function +plate = O.bodies[-1] +# the last particles is the plate +# Wall objects are "fixed" by default, i.e. not subject to forces +# prescribing a velocity will therefore make it move at constant velocity (downwards) +plate.state.vel = (0, 0, -.1) +# start plotting the data now, it was not interesting before +O.engines = O.engines + [PyRunner(command='addPlotData()', iterPeriod=200)] +# next time, do not call this function anymore, but the next one (unloadPlate) instead +checker.command = 'unloadPlate()' +def unloadPlate(): +# if the force on plate exceeds maximum load, start unloading +if abs(O.forces.f(plate.id)[2]) > maxLoad: +plate.state.vel *= -1 +# next time, do not call this function anymore, but the next one␣ +�→(stopUnloading) instead +(continues on next page) +38 +Chapter 1. +Guided tour + +Yade Documentation, Release 3rd ed. +(continued from previous page) +checker.command = 'stopUnloading()' +def stopUnloading(): +if abs(O.forces.f(plate.id)[2]) < minLoad: +# O.tags can be used to retrieve unique identifiers of the simulation +# if running in batch, subsequent simulation would overwrite each other's␣ +�→output files otherwise +# d (or description) is simulation description (composed of parameter values) +# while the id is composed of time and process number +plot.saveDataTxt(O.tags['d.id'] + '.txt') +O.pause() +def addPlotData(): +if not isinstance(O.bodies[-1].shape, Wall): +plot.addData() +return +Fz = O.forces.f(plate.id)[2] +plot.addData(Fz=Fz, w=plate.state.pos[2] - plate.state.refPos[2],␣ +�→unbalanced=unbalancedForce(), i=O.iter) +# besides unbalanced force evolution, also plot the displacement-force diagram +plot.plots = {'i': ('unbalanced',), 'w': ('Fz',)} +plot.plot() +O.run() +# when running with yade-batch, the script must not finish until the simulation is done fully +# this command will wait for that (has no influence in the non-batch mode) +waitIfBatch() +Batch table +To run the same script doc/sphinx/tutorial/03-oedometric-test.py in batch mode to test different param- +eters, execute command yade-batch 03-oedometric-test.table 03-oedometric-test.py, also visit +page http://localhost:9080 to see the batch simulation progress. +rMean rRelFuzz maxLoad +.05 .1 1e6 +.05 .2 1e6 +.05 .3 1e6 +Periodic simple shear +Following example is in file doc/sphinx/tutorial/04-periodic-simple-shear.py. +# encoding: utf-8 +# script for periodic simple shear test, with periodic boundary +# first compresses to attain some isotropic stress (checkStress), +# then loads in shear (checkDistorsion) +# +# the initial packing is either regular (hexagonal), with empty bands along the boundary, +# or periodic random cloud of spheres +# +# material friction angle is initially set to zero, so that the resulting packing is dense +(continues on next page) +1.2. +Tutorial +39 + +Yade Documentation, Release 3rd ed. +(continued from previous page) +# (sphere rearrangement is easier if there is no friction) +# +# setup the periodic boundary +from __future__ import print_function +O.periodic = True +O.cell.hSize = Matrix3(2, 0, 0, 0, 2, 0, 0, 0, 2) +from yade import pack, plot +# the "if 0:" block will be never executed, therefore the "else:" block will be +# to use cloud instead of regular packing, change to "if 1:" or something similar +if 0: +# create cloud of spheres and insert them into the simulation +# we give corners, mean radius, radius variation +sp = pack.SpherePack() +sp.makeCloud((0, 0, 0), (2, 2, 2), rMean=.1, rRelFuzz=.6, periodic=True) +# insert the packing into the simulation +sp.toSimulation(color=(0, 0, 1)) +# pure blue +else: +# in this case, add dense packing +O.bodies.append(pack.regularHexa(pack.inAlignedBox((0, 0, 0), (2, 2, 2)), radius=.1,␣ +�→gap=0, color=(0, 0, 1))) +# create "dense" packing by setting friction to zero initially +O.materials[0].frictionAngle = 0 +# simulation loop (will be run at every step) +O.engines = [ +ForceResetter(), +InsertionSortCollider([Bo1_Sphere_Aabb()]), +InteractionLoop( +# interaction loop +[Ig2_Sphere_Sphere_ScGeom()], +[Ip2_FrictMat_FrictMat_FrictPhys()], +[Law2_ScGeom_FrictPhys_CundallStrack()] +), +NewtonIntegrator(damping=.4), +# run checkStress function (defined below) every second +# the label is arbitrary, and is used later to refer to this engine +PyRunner(command='checkStress()', realPeriod=1, label='checker'), +# record data for plotting every 100 steps; addData function is defined below +PyRunner(command='addData()', iterPeriod=100) +] +# set the integration timestep to be 1/2 of the "critical" timestep +O.dt = .5 * PWaveTimeStep() +# prescribe isotropic normal deformation (constant strain rate) +# of the periodic cell +O.cell.velGrad = Matrix3(-.1, 0, 0, 0, -.1, 0, 0, 0, -.1) +# when to stop the isotropic compression (used inside checkStress) +limitMeanStress = -5e5 +# called every second by the PyRunner engine +def checkStress(): +# stress tensor as the sum of normal and shear contributions +# Matrix3.Zero is the intial value for sum(...) +stress = getStress().trace() / 3. +(continues on next page) +40 +Chapter 1. +Guided tour + +Yade Documentation, Release 3rd ed. +(continued from previous page) +print('mean stress', stress) +# if mean stress is below (bigger in absolute value) limitMeanStress, start shearing +if stress < limitMeanStress: +# apply constant-rate distorsion on the periodic cell +O.cell.velGrad = Matrix3(0, 0, .1, 0, 0, 0, 0, 0, 0) +# change the function called by the checker engine +# (checkStress will not be called anymore) +checker.command = 'checkDistorsion()' +# block rotations of particles to increase tanPhi, if desired +# disabled by default +if 0: +for b in O.bodies: +# block X,Y,Z rotations, translations are free +b.state.blockedDOFs = 'XYZ' +# stop rotations if any, as blockedDOFs block accelerations␣ +�→really +b.state.angVel = (0, 0, 0) +# set friction angle back to non-zero value +# tangensOfFrictionAngle is computed by the Ip2_* functor from material +# for future contacts change material (there is only one material for all␣ +�→particles) +O.materials[0].frictionAngle = .5 +# radians +# for existing contacts, set contact friction directly +for i in O.interactions: +i.phys.tangensOfFrictionAngle = tan(.5) +# called from the 'checker' engine periodically, during the shear phase +def checkDistorsion(): +# if the distorsion value is >.3, exit; otherwise do nothing +if abs(O.cell.trsf[0, 2]) > .5: +# save data from addData(...) before exiting into file +# use O.tags['id'] to distinguish individual runs of the same simulation +plot.saveDataTxt(O.tags['id'] + '.txt') +# exit the program +#import sys +#sys.exit(0) # no error (0) +O.pause() +# called periodically to store data history +def addData(): +# get the stress tensor (as 3x3 matrix) +stress = sum(normalShearStressTensors(), Matrix3.Zero) +# give names to values we are interested in and save them +plot.addData(exz=O.cell.trsf[0, 2], szz=stress[2, 2], sxz=stress[0, 2],␣ +�→tanPhi=(stress[0, 2] / stress[2, 2]) if stress[2, 2] != 0 else 0, i=O.iter) +# color particles based on rotation amount +for b in O.bodies: +# rot() gives rotation vector between reference and current position +b.shape.color = scalarOnColorScale(b.state.rot().norm(), 0, pi / 2.) +# define what to plot (3 plots in total) +## exz(i), [left y axis, separate by None:] szz(i), sxz(i) +## szz(exz), sxz(exz) +## tanPhi(i) +# note the space in 'i ' so that it does not overwrite the 'i' entry +plot.plots = {'i': ('exz', None, 'szz', 'sxz'), 'exz': ('szz', 'sxz'), 'i ': ('tanPhi',)} +# better show rotation of particles +(continues on next page) +1.2. +Tutorial +41 + +Yade Documentation, Release 3rd ed. +(continued from previous page) +Gl1_Sphere.stripes = True +# open the plot on the screen +plot.plot() +O.saveTmp() +3d postprocessing +Following example is in file doc/sphinx/tutorial/05-3d-postprocessing.py. +This example will run for +20000 iterations, saving *.png snapshots, then it will make a video 3d.mpeg out of those snapshots. +# demonstrate 3d postprocessing with yade +# +# 1. qt.SnapshotEngine saves images of the 3d view as it appears on the screen periodically +# +makeVideo is then used to make real movie from those images +# 2. VTKRecorder saves data in files which can be opened with Paraview +# +see the User's manual for an intro to Paraview +# generate loose packing +from yade import pack, qt +sp = pack.SpherePack() +sp.makeCloud((0, 0, 0), (2, 2, 2), rMean=.1, rRelFuzz=.6, periodic=True) +# add to scene, make it periodic +sp.toSimulation() +O.engines = [ +ForceResetter(), +InsertionSortCollider([Bo1_Sphere_Aabb()]), +InteractionLoop( +# interaction loop +[Ig2_Sphere_Sphere_ScGeom()], +[Ip2_FrictMat_FrictMat_FrictPhys()], +[Law2_ScGeom_FrictPhys_CundallStrack()] +), +NewtonIntegrator(damping=.4), +# save data for Paraview +VTKRecorder(fileName='3d-vtk-', recorders=['all'], iterPeriod=1000), +# save data from Yade's own 3d view +qt.SnapshotEngine(fileBase='3d-', iterPeriod=200, label='snapshot'), +# this engine will be called after 20000 steps, only once +PyRunner(command='finish()', iterPeriod=20000) +] +O.dt = .5 * PWaveTimeStep() +# prescribe constant-strain deformation of the cell +O.cell.velGrad = Matrix3(-.1, 0, 0, 0, -.1, 0, 0, 0, -.1) +# we must open the view explicitly (limitation of the qt.SnapshotEngine) +qt.View() +# this function is called when the simulation is finished +def finish(): +# snapshot is label of qt.SnapshotEngine +# the 'snapshots' attribute contains list of all saved files +makeVideo(snapshot.snapshots, '3d.mpeg', fps=10, bps=10000) +O.pause() +(continues on next page) +42 +Chapter 1. +Guided tour + +Yade Documentation, Release 3rd ed. +(continued from previous page) +# set parameters of the renderer, to show network chains rather than particles +# these settings are accessible from the Controller window, on the second tab ("Display") as␣ +�→well +rr = yade.qt.Renderer() +rr.shape = False +rr.intrPhys = True +Periodic triaxial test +Following example is in file doc/sphinx/tutorial/06-periodic-triaxial-test.py. +# encoding: utf-8 +# periodic triaxial test simulation +# +# The initial packing is either +# +# 1. random cloud with uniform distribution, or +# 2. cloud with specified granulometry (radii and percentages), or +# 3. cloud of clumps, i.e. rigid aggregates of several particles +# +# The triaxial consists of 2 stages: +# +# 1. isotropic compaction, until sigmaIso is reached in all directions; +# +this stage is ended by calling compactionFinished() +# 2. constant-strain deformation along the z-axis, while maintaining +# +constant stress (sigmaIso) laterally; this stage is ended by calling +# +triaxFinished() +# +# Controlling of strain and stresses is performed via PeriTriaxController, +# of which parameters determine type of control and also stability +# condition (maxUnbalanced) so that the packing is considered stabilized +# and the stage is done. +# +from __future__ import print_function +sigmaIso = -1e5 +#import matplotlib +#matplotlib.use('Agg') +# generate loose packing +from yade import pack, qt, plot +O.periodic = True +sp = pack.SpherePack() +if 0: +## uniform distribution +sp.makeCloud((0, 0, 0), (2, 2, 2), rMean=.1, rRelFuzz=.3, periodic=True) +else: +## create packing from clumps +# configuration of one clump +c1 = pack.SpherePack([((0, 0, 0), .03333), ((.03, 0, 0), .017), ((0, .03, 0), .017)]) +# make cloud using the configuration c1 (there could c2, c3, ...; selection between␣ +�→them would be random) +sp.makeClumpCloud((0, 0, 0), (2, 2, 2), [c1], periodic=True, num=500) +# setup periodic boundary, insert the packing +sp.toSimulation() +(continues on next page) +1.2. +Tutorial +43 + +Yade Documentation, Release 3rd ed. +(continued from previous page) +O.engines = [ +ForceResetter(), +InsertionSortCollider([Bo1_Sphere_Aabb()]), +InteractionLoop([Ig2_Sphere_Sphere_ScGeom()], [Ip2_FrictMat_FrictMat_FrictPhys()],␣ +�→[Law2_ScGeom_FrictPhys_CundallStrack()]), +PeriTriaxController( +label='triax', +# specify target values and whether they are strains or stresses +goal=(sigmaIso, sigmaIso, sigmaIso), +stressMask=7, +# type of servo-control +dynCell=True, +maxStrainRate=(10, 10, 10), +# wait until the unbalanced force goes below this value +maxUnbalanced=.1, +relStressTol=1e-3, +# call this function when goal is reached and the packing is stable +doneHook='compactionFinished()' +), +NewtonIntegrator(damping=.2), +PyRunner(command='addPlotData()', iterPeriod=100), +] +O.dt = .5 * PWaveTimeStep() +def addPlotData(): +plot.addData( +unbalanced=unbalancedForce(), +i=O.iter, +sxx=triax.stress[0], +syy=triax.stress[1], +szz=triax.stress[2], +exx=triax.strain[0], +eyy=triax.strain[1], +ezz=triax.strain[2], +# save all available energy data +Etot=O.energy.total(), +**O.energy +) +# enable energy tracking in the code +O.trackEnergy = True +# define what to plot +plot.plots = { +'i': ('unbalanced',), +'i ': ('sxx', 'syy', 'szz'), +' i': ('exx', 'eyy', 'ezz'), +# energy plot +' i ': (O.energy.keys, None, 'Etot'), +} +# show the plot +plot.plot() +def compactionFinished(): +# set the current cell configuration to be the reference one +O.cell.trsf = Matrix3.Identity +# change control type: keep constant confinement in x,y, 20% compression in z +(continues on next page) +44 +Chapter 1. +Guided tour + +Yade Documentation, Release 3rd ed. +(continued from previous page) +triax.goal = (sigmaIso, sigmaIso, -.2) +triax.stressMask = 3 +# allow faster deformation along x,y to better maintain stresses +triax.maxStrainRate = (1., 1., .1) +# next time, call triaxFinished instead of compactionFinished +triax.doneHook = 'triaxFinished()' +# do not wait for stabilization before calling triaxFinished +triax.maxUnbalanced = 10 +def triaxFinished(): +print('Finished') +O.pause() +1.2.7 More examples +The same list with embedded videos is available online, but not recommended for viewing on slow internet +connection. +A full list of examples is in file examples/list_of_examples.txt. Videos of some of those examples are +listed below. +FluidCouplingLBM +• refFastBuoyancy, source file, video. +FluidCouplingPFV +• refFastOedometer, source file, video. +HydroForceEngine +• refFastBuoyantParticles, source file, video. +• refFastFluidizedBed, source file, video. +• refFastSedimentTransportExample, source file, video. +• refFastLaminarShearFlow, source file, video. +• refFastPostProcessValidMaurin2015, source file, video. +• refFastValidMaurin2015, source file, video. +PeriodicBoundaries +• refFastCellFlipping, source file, video. +• refFastPeri3dController-example1, source file, video. +• refFastPeri3dController-shear, source file, video. +• refFastPeri3dController-triaxialCompression, source file, video. +• refFastPeriodic-compress, source file, video. +• refFastPeriodic-shear, source file, video. +• refFastPeriodic-simple-shear, source file, video. +1.2. +Tutorial +45 + +Yade Documentation, Release 3rd ed. +• refFastPeriodic-simple, source file, video. +• refFastPeriodic-triax-settingHsize, source file, video. +• refFastPeriodic-triax, source file, video. +• refFastPeriodicSandPile, source file, video. +PotentialBlocks +• refFastWedgeYADE, source file, video. +• refFastCubePBscaled, source file, video. +PotentialParticles +• refFastCubePPscaled, source file, video. +WireMatPM +• refFastWirecontacttest, source file, video. +• refFastWirepackings, source file, video. +• refFastWiretensiltest, source file, video. +Adaptiveintegrator +• refFastSimple-scene-plot-NewtonIntegrator, source file, video. +• refFastSimple-scene-plot-RungeKuttaCashKarp54, source file, video. +Agglomerate +• refFastCompress, source file, video. +• refFastSimulation, source file, video. +Baraban +• refFastBicyclePedalEngine, source file, video. +• refFastBaraban, source file, video. +• refFastRotating-cylinder, source file, video. +Bulldozer +• refFastBulldozer, source file, video. +Capillary +• refFastCapillar, source file, video. +46 +Chapter 1. +Guided tour + +Yade Documentation, Release 3rd ed. +CapillaryLaplaceYoung +• refFastCapillaryPhys-example, source file, video. +• refFastCapillaryBridge, source file, video. +Chained-cylinders +• refFastCohesiveCylinderSphere, source file, video. +• refFastChained-cylinder-roots, source file, video. +• refFastChained-cylinder-spring, source file, video. +Clumps +• refFastAddToClump-example, source file, video. +• refFastApply-buoyancy-clumps, source file, video. +• refFastClump-hopper-test, source file, video. +• refFastClump-hopper-viscoelastic, source file, video. +• refFastClump-inbox-viscoelastic, source file, video. +• refFastClump-viscoelastic, source file, video. +• refFastReleaseFromClump-example, source file, video. +• refFastReplaceByClumps-example, source file, video. +• refFastTriax-basic-with-clumps, source file, video. +Concrete +• refFastBrazilian, source file, video. +• refFastInteraction-histogram, source file, video. +• refFastPeriodic, source file, video. +• refFastTriax, source file, video. +• refFastUniax-post, source file, video. +• refFastUniax, source file, video. +Conveyor +• refFastConveyor, source file, video. +Cylinders +• refFastBendingbeams, source file, video. +• refFastCylinder-cylinder, source file, video. +• refFastCylinderconnection-roots, source file, video. +• refFastMikado, source file, video. +1.2. +Tutorial +47 + +Yade Documentation, Release 3rd ed. +Deformableelem +• refFastMinimalTensileTest, source file, video. +• refFastTestDeformableBodies, source file, video. +• refFastTestDeformableBodies-pressure, source file, video. +Grids +• refFastCohesiveGridConnectionSphere, source file, video. +• refFastGridConnection-Spring, source file, video. +• refFastSimple-GridConnection-Falling, source file, video. +• refFastSimple-Grid-Falling, source file, video. +Gts-horse +• refFastGts-horse, source file, video. +• refFastGts-operators, source file, video. +• refFastGts-random-pack-obb, source file, video. +• refFastGts-random-pack, source file, video. +Hourglass +• refFastHourglass, source file, video. +Packs +• refFastPacks, source file, video. +Pfacet +• refFastGts-pfacet, source file, video. +• refFastMesh-pfacet, source file, video. +• refFastPFacets-grids-spheres-interacting, source file, video. +• refFastPfacetcreators, source file, video. +Polyhedra +• refFastBall, source file, video. +• refFastHorse, source file, video. +• refFastIrregular, source file, video. +• refFastSphere-interaction, source file, video. +• refFastSplitter, source file, video. +• refFastInteractinDetectionFactor, source file, video. +• refFastScGeom, source file, video. +• refFastTextExport, source file, video. +48 +Chapter 1. +Guided tour + +Yade Documentation, Release 3rd ed. +PolyhedraBreak +• refFastUniaxial-compression, source file, video. +Ring2d +• refFastRingCundallDamping, source file, video. +• refFastRingSimpleViscoelastic, source file, video. +Rod-penetration +• refFastModel, source file, video. +Simple-scene +• refFast2SpheresNormVisc, source file, video. +• refFastSave-then-reload, source file, video. +• refFastSimple-scene-default-engines, source file, video. +• refFastSimple-scene-energy-tracking, source file, video. +• refFastSimple-scene-plot, source file, video. +• refFastSimple-scene, source file, video. +Stl-gts +• refFastGts-stl, source file, video. +Tesselationwrapper +• refFastTesselationWrapper, source file, video. +Test +• refFastNet-2part-displ-unloading, source file, video. +• refFastNet-2part-displ, source file, video. +• refFastBeam-l6geom, source file, video. +• refFastClump-facet, source file, video. +• refFastClumpPack, source file, video. +• refFastCollider-stride-triax, source file, video. +• refFastCollider-stride, source file, video. +• refFastCombined-kinematic-engine, source file, video. +• refFastEnergy, source file, video. +• refFastFacet-box, source file, video. +• refFastFacet-sphere-ViscElBasic-peri, source file, video. +• refFastFacet-sphere-ViscElBasic, source file, video. +• refFastFacet-sphere, source file, video. +1.2. +Tutorial +49 + +Yade Documentation, Release 3rd ed. +• refFastHelix, source file, video. +• refFastInterpolating-force, source file, video. +• refFastKinematic, source file, video. +• refFastMindlin, source file, video. +• refFastMulti, source file, video. +• refFastPack-cloud, source file, video. +• refFastPack-inConvexPolyhedron, source file, video. +• refFastPv-section, source file, video. +• refFastPeriodic-geom-compare, source file, video. +• refFastPsd, source file, video. +• refFastSphere-sphere-ViscElBasic-peri, source file, video. +• refFastSubdomain-balancer, source file, video. +• refFastTest-sphere-facet-corner, source file, video. +• refFastTest-sphere-facet, source file, video. +• refFastTriax-basic, source file, video. +• refFastTriax-basic-without-plots, source file, video. +• refFastUnvRead, source file, video. +Tetra +• refFastOneTetra, source file, video. +• refFastOneTetraPoly, source file, video. +• refFastTwoTetras, source file, video. +• refFastTwoTetrasPoly, source file, video. +50 +Chapter 1. +Guided tour + +Chapter 2 +Yade for users +2.1 DEM formulation +In this chapter, we mathematically describe general features of explicit DEM simulations, with some +reference to Yade implementation of these algorithms. +They are given roughly in the order as they +appear in simulation; first, two particles might establish a new interaction, which consists in +1. detecting collision between particles; +2. creating new interaction and determining its properties (such as stiffness); they are either precom- +puted or derived from properties of both particles; +Then, for already existing interactions, the following is performed: +1. strain evaluation; +2. stress computation based on strains; +3. force application to particles in interaction. +This simplified description serves only to give meaning to the ordering of sections within this chapter. +A more detailed description of this simulation loop is given later. +In this chapter we refer to kinematic variables of the contacts as ‘‘strains‘‘, although at this scale it +is also common to speak of ‘‘displacements‘‘. +Which semantic is more appropriate depends on the +conceptual model one is starting from, and therefore it cannot be decided independently of specific +problems. The reader familiar with displacements can mentaly replace normal strain and shear strain by +normal displacement and shear displacement, respectively, without altering the meaning of what follows. +2.1.1 Collision detection +Generalities +Exact computation of collision configuration between two particles can be relatively expensive (for in- +stance between Sphere and Facet). Taking a general pair of bodies i and j and their ‘‘exact‘‘ (In the +sense of precision admissible by numerical implementation.) spatial predicates (called Shape in Yade) +represented by point sets Pi, Pj the detection generally proceeds in 2 passes: +1. fast collision detection using approximate predicate ˜Pi and ˜Pj; they are pre-constructed in such a +way as to abstract away individual features of Pi and Pj and satisfy the condition +∀x ∈ R3 : x ∈ Pi ⇒ x ∈ ˜Pi +(2.1) +(likewise for Pj). The approximate predicate is called ‘‘bounding volume’’ (Bound in Yade) since it +bounds any particle’s volume from outside (by virtue of the implication). It follows that (Pi ∩Pj) ̸= +51 + +Yade Documentation, Release 3rd ed. +∅ ⇒ (˜Pi ∩ ˜Pj) ̸= ∅ and, by applying modus tollens, +�˜Pi ∩ ˜Pj +� += ∅ ⇒ +� +Pi ∩ Pj +� += ∅ +(2.2) +which is a candidate exclusion rule in the proper sense. +2. By filtering away impossible collisions in (2.2), a more expensive, exact collision detection algo- +rithms can be run on possible interactions, filtering out remaining spurious couples (˜Pi ∩ ˜Pj) ̸= +∅∧ +� +Pi ∩Pj +� += ∅. These algorithms operate on Pi and Pj and have to be able to handle all possible +combinations of shape types. +It is only the first step we are concerned with here. +Algorithms +Collision evaluation algorithms have been the subject of extensive research in fields such as robotics, +computer graphics and simulations. They can be roughly divided in two groups: +Hierarchical algorithms which recursively subdivide space and restrict the number of approximate +checks in the first pass, knowing that lower-level bounding volumes can intersect only if they +are part of the same higher-level bounding volume. Hierarchy elements are bounding volumes of +different kinds: octrees [Jung1997], bounding spheres [Hubbard1996], k-DOP’s [Klosowski1998]. +Flat algorithms work directly with bounding volumes without grouping them in hierarchies first; let +us only mention two kinds commonly used in particle simulations: +Sweep and prune algorithm operates on axis-aligned bounding boxes, which overlap +if and only if they overlap along all axes. These algorithms have roughly O(n log n) +complexity, where n is number of particles as long as they exploit temporal coherence +of the simulation. +Grid algorithms represent continuous R3 space by a finite set of regularly spaced +points, leading to very fast neighbor search; they can reach the O(n) complexity +[Munjiza1998] and recent research suggests ways to overcome one of the major draw- +backs of this method, which is the necessity to adjust grid cell size to the largest +particle in the simulation ([Munjiza2006], the ‘‘multistep’’ extension). +Temporal coherence expresses the fact that motion of particles in simulation is not arbitrary but +governed by physical laws. This knowledge can be exploited to optimize performance. +Numerical stability of integrating motion equations dictates an upper limit on ∆t (sect. Stability consid- +erations) and, by consequence, on displacement of particles during one step. This consideration is taken +into account in [Munjiza2006], implying that any particle may not move further than to a neighboring +grid cell during one step allowing the O(n) complexity; it is also explored in the periodic variant of the +sweep and prune algorithm described below. +On a finer level, it is common to enlarge ˜Pi predicates in such a way that they satisfy the (2.1) condition +during several timesteps; the first collision detection pass might then be run with stride, speeding up +the simulation considerably. The original publication of this optimization by Verlet [Verlet1967] used +enlarged list of neighbors, giving this technique the name Verlet list. In general cases, however, where +neighbor lists are not necessarily used, the term Verlet distance is employed. +Sweep and prune +Let us describe in detail the sweep and prune algorithm used for collision detection in Yade (class +InsertionSortCollider). Axis-aligned bounding boxes (Aabb) are used as ˜Pi; each Aabb is given by lower +and upper corner ∈ R3 (in the following, ˜Px0 +i , ˜Px1 +i +are minimum/maximum coordinates of ˜Pi along the +x-axis and so on). Construction of Aabb from various particle Shape’s (such as Sphere, Facet, Wall) is +straightforward, handled by appropriate classes deriving form BoundFunctor (Bo1_Sphere_Aabb, Bo1_- +Facet_Aabb, …). +52 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +Presence of overlap of two Aabb’s can be determined from conjunction of separate overlaps of intervals +along each axis (fig-sweep-and-prune): +� +˜Pi ∩ ˜Pj +� +̸= ∅ ⇔ +� +w∈{x,y,z} +��� +˜Pw0 +i +, ˜Pw1 +i +� +∩ +� +˜Pw0 +j +, ˜Pw1 +j +�� +̸= ∅ +� +where (a, b) denotes interval in R. +P1 +P2 +P3 +˜Px0 +1 ++x ++y +˜Px1 +1 +˜Px0 +2 +˜Px0 +3 +˜Px1 +2 +˜Px1 +3 +˜Py1 +3 +˜Py0 +3 +˜Py1 +2 +˜Py0 +2 +˜Py1 +1 +˜Py0 +1 +˜P3 +˜P2 +˜P1 +Fig. 1: Sweep and prune algorithm (shown in 2D), where Aabb of each sphere is represented by minimum +and maximum value along each axis. Spatial overlap of Aabb’s is present if they overlap along all axes. +In this case, ˜P1 ∩ ˜P2 ̸= ∅ (but note that P1 ∩ P2 = ∅) and ˜P2 ∩ ˜P3 ̸= ∅.} +The collider keeps 3 separate lists (arrays) Lw for each axis w ∈ {x, y, z} +Lw = +� +i +� +˜Pw0 +i +, ˜Pw1 +i +� +where i traverses all particles. Lw arrays (sorted sets) contain respective coordinates of minimum and +maximum corners for each Aabb (we call these coordinates bound in the following); besides bound, each +of list elements further carries id referring to particle it belongs to, and a flag whether it is lower or +upper bound. +In the initial step, all lists are sorted (using quicksort, average O(n log n)) and one axis is used to create +initial interactions: the range between lower and upper bound for each body is traversed, while bounds +in-between indicate potential Aabb overlaps which must be checked on the remaining axes as well. +At each successive step, lists are already pre-sorted. Inversions occur where a particle’s coordinate has +just crossed another particle’s coordinate; this number is limited by numerical stability of simulation and +its physical meaning (giving spatio-temporal coherence to the algorithm). The insertion sort algorithm +swaps neighboring elements if they are inverted, and has complexity between O(n) and O(n2), for pre- +sorted and unsorted lists respectively. For our purposes, we need only to handle inversions, which by +nature of the sort algorithm are detected inside the sort loop. An inversion might signify: +• overlap along the current axis, if an upper bound inverts (swaps) with a lower bound (i.e. that the +upper bound with a higher coordinate was out of order in coming before the lower bound with a +lower coordinate). Overlap along the other 2 axes is checked and if there is overlap along all axes, +a new potential interaction is created. +• End of overlap along the current axis, if lower bound inverts (swaps) with an upper bound. If there +is only potential interaction between the two particles in question, it is deleted. +• Nothing if both bounds are upper or both lower. +2.1. +DEM formulation +53 + +Yade Documentation, Release 3rd ed. +Aperiodic insertion sort +Let us show the sort algorithm on a sample sequence of numbers: +Elements are traversed from left to right; each of them keeps inverting (swapping) with neighbors to the +left, moving left itself, until any of the following conditions is satisfied: +(≤) +the sorting order with the left neighbor is correct, or +(||) +the element is at the beginning of the sequence. +We start at the leftmost element (the current element is marked i ) +It obviously immediately satisfies (||), and we move to the next element: +Condition (≤) holds, therefore we move to the right. The 2 is not in order (violating (≤)) and two +inversions take place; after that, (||) holds: +The last element 4 first violates (≤), but satisfies it after one inversion +All elements having been traversed, the sequence is now sorted. +It is obvious that if the initial sequence were sorted, elements only would have to be traversed without +any inversion to handle (that happens in O(n) time). +54 +Chapter 2. +Yade for users + +1 3 +7 +2 +4=3 +7 +411. +2= +3 +7 +2 +411.7 +2 +4 l, +3 +/ +二 +3 +2 +7 +4 l, +2 +3 +7 +4 2 +3 +7 +4 +2 +3 +4 +7 +1I.Yade Documentation, Release 3rd ed. +For each inversion during the sort in simulation, the function that investigates change in Aabb overlap is +invoked, creating or deleting interactions. +The periodic variant of the sort algorithm is described in Periodic insertion sort algorithm, along with +other periodic-boundary related topics. +Optimization with Verlet distances +As noted above, [Verlet1967] explored the possibility of running the collision detection only sparsely by +enlarging predicates ˜Pi. +In Yade, this is achieved by enlarging Aabb of particles by fixed relative length (or Verlet’s distance) in all +dimensions ∆L (InsertionSortCollider.sweepLength). Suppose the collider run last time at step m and the +current step is n. NewtonIntegrator tracks the cummulated distance traversed by each particle between +m and n by comparing the current position with the reference position from time n (Bound::refPos), +Lmn = |Xn − Xm| +(2.3) +triggering the collider re-run as soon as one particle gives: +Lmn > ∆L. +(2.4) +∆L is defined primarily by the parameter InsertionSortCollider.verletDist. +It can be set directly by +assigning a positive value, or indirectly by assigning negative value (which defines ∆L in proportion of +the smallest particle radius). In addition, InsertionSortCollider.targetInterv can be used to adjust ∆L +independently for each particle. Larger ∆L will be assigned to the fastest ones, so that all particles would +ideally reach the edge of their bounds after this “target” number of iterations. Results of using Verlet +distance depend highly on the nature of simulation and choice of InsertionSortCollider.targetInterv. +Adjusting the sizes independently for each particle is especially efficient if some parts of a problem have +high-speed particles will others are not moving. If it is not the case, no significant gain should be expected +as compared to targetInterv=0 (assigning the same ∆L to all particles). +The number of particles and the number of available threads is also to be considered for choosing an +appropriate Verlet’s distance. A larger distance will result in less time spent in the collider (which runs +single-threaded) and more time in computing interactions (multi-threaded). Typically, large ∆L will be +used for large simulations with more than 105 particles on multi-core computers. On the other hand +simulations with less than 104 particles on single processor will probably benefit from smaller ∆L. Users +benchmarks may be found on Yade’s wiki (see e.g. https://yade-dem.org/wiki/Colliders_performace). +2.1.2 Creating interaction between particles +Collision detection described above is only approximate. Exact collision detection depends on the ge- +ometry of individual particles and is handled separately. In Yade terminology, the Collider creates only +potential interactions; potential interactions are evaluated exactly using specialized algorithms for colli- +sion of two spheres or other combinations. Exact collision detection must be run at every timestep since +it is at every step that particles can change their mutual position (the collider is only run sometimes +if the Verlet distance optimization is in use). Some exact collision detection algorithms are described +in Kinematic variables; in Yade, they are implemented in classes deriving from IGeomFunctor (prefixed +with Ig2). +Besides detection of geometrical overlap (which corresponds to IGeom in Yade), there are also non- +geometrical properties of the interaction to be determined (IPhys). In Yade, they are computed for +every new interaction by calling a functor deriving from IPhysFunctor (prefixed with Ip2) which accepts +the given combination of Material types of both particles. +Stiffnesses +Basic DEM interaction defines two stiffnesses: normal stiffness KN and shear (tangent) stiffness KT. +It is desirable that KN be related to fictitious Young’s modulus of the particles’ material, while KT is +2.1. +DEM formulation +55 + +Yade Documentation, Release 3rd ed. +typically determined as a given fraction of computed KN. The KT/KN ratio determines macroscopic +Poisson’s ratio of the arrangement, which can be shown by dimensional analysis: elastic continuum has +two parameters (E and ν) and basic DEM model also has 2 parameters with the same dimensions KN and +KT/KN; macroscopic Poisson’s ratio is therefore determined solely by KT/KN and macroscopic Young’s +modulus is then proportional to KN and affected by KT/KN. +Naturally, such analysis is highly simplifying and does not account for particle radius distribution, packing +configuration and other possible parameters such as the interaction radius introduced later. +Normal stiffness +The algorithm commonly used in Yade computes normal interaction stiffness as stiffness of two springs +in serial configuration with lengths equal to the sphere radii (fig-spheres-contact-stiffness). +E1 +E2 +l1 = r1 +l2 = r2 +l = l1 + l2 +Fig. 2: Series of 2 springs representing normal stiffness of contact between 2 spheres. +Let us define distance l = l1 +l2, where li are distances between contact point and sphere centers, which +are initially (roughly speaking) equal to sphere radii. Change of distance between the sphere centers ∆l +is distributed onto deformations of both spheres ∆l = ∆l1 + ∆l2 proportionally to their compliances. +Displacement change ∆li generates force Fi = Ki∆li, where Ki assures proportionality and has physical +meaning and dimension of stiffness; Ki is related to the sphere material modulus Ei and some length ˜li +proportional to ri. +∆l = ∆l1 + ∆l2 +Ki = Ei˜li +KN∆l = F = F1 = F2 +KN (∆l1 + ∆l2) = F +KN +� F +K1 ++ F +K2 +� += F +K−1 +1 ++ K−1 +2 += K−1 +N +KN = +K1K2 +K1 + K2 +KN = +E1˜l1E2˜l2 +E1˜l1 + E2˜l2 +The most used class computing interaction properties Ip2_FrictMat_FrictMat_FrictPhys uses ˜li = 2ri. +Some formulations define an equivalent cross-section Aeq, which in that case appears in the ˜li term as +Ki = Ei˜li = Ei +Aeq +li . Such is the case for the concrete model (Ip2_CpmMat_CpmMat_CpmPhys), where +Aeq = min(r1, r2). +For reasons given above, no pretense about equality of particle-level Ei and macroscopic modulus E should +be made. Some formulations, such as [Hentz2003], introduce parameters to match them numerically. +This is not appropriate, in our opinion, since it binds those values to particular features of the sphere +arrangement that was used for calibration. +56 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +Other parameters +Non-elastic parameters differ for various material models. +Usually, though, they are averaged from +the particles’ material properties, if it makes sense. For instance, Ip2_CpmMat_CpmMat_CpmPhys +averages most quantities, while Ip2_FrictMat_FrictMat_FrictPhys computes internal friction angle as +φ = min(φ1, φ2) to avoid friction with bodies that are frictionless. +2.1.3 Kinematic variables +In the general case, mutual configuration of two particles has 6 degrees of freedom (DoFs) just like a +beam in 3D space: both particles have 6 DoFs each, but the interaction itself is free to move and rotate +in space (with both spheres) having 6 DoFs itself; then 12 − 6 = 6. They are shown at fig-spheres-dofs. +initial configuration +twisting (1DoF) +normal straining (1DoF) +shearing (2 DoFs) +bending (2 DoFs) +Fig. 3: Degrees of freedom of configuration of two spheres. Normal motion appears if there is a difference +of linear velocity along the interaction axis (n); shearing originates from the difference of linear velocities +perpendicular to n and from the part of ω1 + ω2 perpendicular to n; twisting is caused by the part of +ω1 − ω2 parallel with n; bending comes from the part of ω1 − ω2 perpendicular to n. +We will only describe normal and shear components of the relative movement in the following, leaving +torsion and bending aside. The reason is that most constitutive laws for contacts do not use the latter +two. +Normal deformation +Constants +Let us consider two spheres with initial centers ¯ +C1, ¯C2 and radii r1, r2 that enter into contact. The +order of spheres within the contact is arbitrary and has no influence on the behavior. Then we define +lengths +d0 = |¯C2 − ¯C1| +d1 = r1 + d0 − r1 − r2 +2 +, +d2 = d0 − d1. +These quantities are constant throughout the life of the interaction and are computed only once when +the interaction is established. The distance d0 is the reference distance and is used for the conversion +of absolute displacements to dimensionless strain, for instance. It is also the distance where (for usual +contact laws) there is neither repulsive nor attractive force between the spheres, whence the name +equilibrium distance. +Distances d1 and d2 define reduced (or expanded) radii of spheres; geometrical radii r1 and r2 are used +only for collision detection and may not be the same as d1 and d2, as shown in fig. fig-sphere-sphere. +2.1. +DEM formulation +57 + +Yade Documentation, Release 3rd ed. +d0 = d1 + d2 +¯C2 +¯C1 +d1 +d2 +r1 +r2 +¯C +Fig. 4: Geometry of the initial contact of 2 spheres; this case pictures spheres which already overlap +when the contact is created (which can be the case at the beginning of a simulation) for the sake of +generality. The initial contact point ¯C is in the middle of the overlap zone. +This difference is exploited in cases where the average number of contacts between spheres should be +increased, e.g. +to influence the response in compression or to stabilize the packing. +In such case, +interactions will be created also for spheres that do not geometrically overlap based on the interaction +radius RI, a dimensionless parameter determining „non-locality“ of contact detection. For RI = 1, only +spheres that touch are considered in contact; the general condition reads +d0 ≤ RI(r1 + r2). +(2.5) +The value of RI directly influences the average number of interactions per sphere (percolation), which +for some models is necessary in order to achieve realistic results. In such cases, Aabb (or ˜Pi predicates +in general) must be enlarged accordingly (Bo1_Sphere_Aabb.aabbEnlargeFactor). +Contact cross-section +Some constitutive laws are formulated with strains and stresses (Law2_ScGeom_CpmPhys_Cpm, the +concrete model described later, for instance); in that case, equivalent cross-section of the contact must +be introduced for the sake of dimensionality. The exact definition is rather arbitrary; the CPM model +(Ip2_CpmMat_CpmMat_CpmPhys) uses the relation +Aeq = π min(r1, r2)2 +(2.6) +which will be used to convert stresses to forces, if the constitutive law used is formulated in terms of +stresses and strains. Note that other values than π can be used; it will merely scale macroscopic packing +stiffness; it is only for the intuitive notion of a truss-like element between the particle centers that we +choose Aeq representing the circle area. Besides that, another function than min(r1, r2) can be used, +although the result should depend linearly on r1 and r2 so that the equation gives consistent results if +the particle dimensions are scaled. +Variables +The following state variables are updated as spheres undergo motion during the simulation (as C◦ +1 and +C◦ +2 change): +n◦ = C◦ +2 − C◦ +1 +|C◦ +2 − C◦ +1| ≡ +� +C◦ +2 − C◦ +1 +(2.7) +and +C◦ = C◦ +1 + +� +d1 − d0 − |C◦ +2 − C◦ +1| +2 +� +n. +(2.8) +58 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +The contact point C◦ is always in the middle of the spheres’ overlap zone (even if the overlap is neg- +ative, when it is in the middle of the empty space between the spheres). The contact plane is always +perpendicular to the contact plane normal n◦ and passes through C◦. +Normal displacement and strain can be defined as +uN = |C◦ +2 − C◦ +1| − d0, +εN = uN +d0 += |C◦ +2 − C◦ +1| +d0 +− 1. +Since uN is always aligned with n, it can be stored as a scalar value multiplied by n if necessary. +For massively compressive simulations, it might be beneficial to use the logarithmic strain, such that the +strain tends to −∞ (rather than −1) as centers of both spheres approach. Otherwise, repulsive force +would remain finite and the spheres could penetrate through each other. Therefore, we can adjust the +definition of normal strain as follows: +εN = +� +log +� +|C◦ +2−C◦ +1| +d0 +� +if |C◦ +2 − C◦ +1| < d0 +|C◦ +2−C◦ +1| +d0 +− 1 +otherwise. +Such definition, however, has the disadvantage of effectively increasing rigidity (up to infinity) of contacts, +requiring ∆t to be adjusted, lest the simulation becomes unstable. Such dynamic adjustment is possible +using a stiffness-based time-stepper (GlobalStiffnessTimeStepper in Yade). +Shear deformation +In order to keep uT consistent (e.g. that uT must be constant if two spheres retain mutually constant +configuration but move arbitrarily in space), then either uT must track spheres’ spatial motion or must +(somehow) rely on sphere-local data exclusively. +Geometrical meaning of shear strain is shown in fig-shear-2d. +uT +C +n +Fig. 5: Evolution of shear displacement uT due to mutual motion of spheres, both linear and rotational. +Left configuration is the initial contact, right configuration is after displacement and rotation of one +particle. +The classical incremental algorithm is widely used in DEM codes and is described frequently +([Luding2008], [Alonso2004]). +Yade implements this algorithm in the ScGeom class. +At each step, +shear displacement uT is updated; the update increment can be decomposed in 2 parts: motion of the +interaction (i.e. C and n) in global space and mutual motion of spheres. +1. Contact moves dues to changes of the spheres’ positions C1 and C2, which updates current C◦ +and n◦ as per (2.8) and (2.7). u− +T is perpendicular to the contact plane at the previous step n− +and must be updated so that u− +T + (∆uT) = u◦ +T ⊥ n◦; this is done by perpendicular projection to +the plane first (which might decrease |uT|) and adding what corresponds to spatial rotation of the +2.1. +DEM formulation +59 + +Yade Documentation, Release 3rd ed. +interaction instead: +(∆uT)1 = −u− +T × (n− × n◦) +(∆uT)2 = −u− +T × +�∆t +2 n◦ · (ω⊖ +1 + ω⊖ +2 ) +� +n◦ +2. Mutual movement of spheres, using only its part perpendicular to n◦; v12 denotes mutual velocity +of spheres at the contact point: +v12 = +� +v⊖ +2 + ω⊖ +2 × (−d2n◦) +� +− +� +v⊖ +1 + ω⊖ +1 × (d1n◦) +� +v⊥ +12 = v12 − (n◦ · v12)n◦ +(∆uT)3 = −∆tv⊥ +12 +Finally, we compute +u◦ +T = u− +T + (∆uT)1 + (∆uT)2 + (∆uT)3. +2.1.4 Contact model (example) +The kinematic variables of an interaction are used to determine the forces acting on both spheres via +a constitutive law. In DEM generally, some constitutive laws are expressed using strains and stresses +while others prefer displacement/force formulation. The law described here falls in the latter category. +The constitutive law presented here is the most common in DEM, originally proposed by Cundall. While +the kinematic variables are described in the previous section regardless of the contact model, the force +evaluation depends on the nature of the material being modeled. The constitutive law presented here is +the simplest non-cohesive elastic-frictional contact model, which Yade implements in Law2_ScGeom_- +FrictPhys_CundallStrack (all constitutive laws derive from base class LawFunctor). +When new contact is established (discussed in Engines) it has its properties (IPhys) computed from +Materials associated with both particles. +In the simple case of frictional material FrictMat, Ip2_- +FrictMat_FrictMat_FrictPhys creates a new FrictPhys instance, which defines normal stiffness KN, +shear stiffness KT and friction angle φ. +At each step, given normal and shear displacements uN, uT, normal and shear forces are computed (if +uN > 0, the contact is deleted without generating any forces): +FN = KNuNn, +Ft +T = KTuT +where FN is normal force and Ft +T is trial shear force. A simple non-associated stress return algorithm is +applied to compute final shear force +FT = +� +Ft +T +|FN| tan φ +|Ft +T | +if |Ft +T| > |FN| tan φ, +Ft +T +otherwise. +Summary force F = FN + FT is then applied to both particles – each particle accumulates forces and +torques acting on it in the course of each step. Because the force computed acts at contact point C, +which is difference from spheres’ centers, torque generated by F must also be considered. +F1+ = F +F2+ = −F +T 1+ = d1(−n) × F +T 2+ = d2n × F. +2.1.5 Motion integration +Each particle accumulates generalized forces (forces and torques) from the contacts in which it partici- +pates. These generalized forces are then used to integrate motion equations for each particle separately; +therefore, we omit i indices denoting the i-th particle in this section. +60 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +The customary leapfrog scheme (also known as the Verlet scheme) is used, with some adjustments for +rotation of non-spherical particles, as explained below. The “leapfrog” name comes from the fact that +even derivatives of position/orientation are known at on-step points, whereas odd derivatives are known +at mid-step points. Let us recall that we use a−, a◦, a+ for on-step values of a at t − ∆t, t and t + ∆t +respectively; and a⊖, a⊕ for mid-step values of a at t − ∆t/2, t + ∆t/2. +Described integration algorithms are implemented in the NewtonIntegrator class in Yade. +Position +Integrating motion consists in using current acceleration ¨u◦ on a particle to update its position from the +current value u◦ to its value at the next timestep u+. Computation of acceleration, knowing current +forces F acting on the particle in question and its mass m, is simply +¨u◦ = F/m. +Using the 2nd order finite difference with step ∆t, we obtain +¨u◦ ∼= u− − 2u◦ + u+ +∆t2 +from which we express +u+ = 2u◦ − u− + ¨u◦∆t2 = += u◦ + ∆t +�u◦ − u− +∆t ++ ¨u◦∆t +� +� +�� +� +(†) +. +Typically, u− is already not known (only u◦ is); we notice, however, that +˙u⊖ ≃ u◦ − u− +∆t +, +i.e. the mean velocity during the previous step, which is known. Plugging this approximate into the (†) +term, we also notice that mean velocity during the current step can be approximated as +˙u⊕ ≃ ˙u⊖ + ¨u◦∆t, +which is (†); we arrive finally at +u+ = u◦ + ∆t +� ˙u⊖ + ¨u◦∆t +� +. +The algorithm can then be written down by first computing current mean velocity ˙u⊕ which we need to +store for the next step (just as we use its old value ˙u⊖ now), then computing the position for the next +time step u+: +˙u⊕ = ˙u⊖ + ¨u◦∆t +u+ = u◦ + ˙u⊕∆t. +Positions are known at times i∆t (if ∆t is constant) while velocities are known at i∆t+ ∆t +2 . The fact that +they interleave (jump over each other) in such way gave rise to the colloquial name “leapfrog” scheme. +Orientation (spherical) +Updating particle orientation q◦ proceeds in an analogous way to position update. First, we compute +current angular acceleration ˙ω◦ from known current torque T. For spherical particles where the inertia +tensor is diagonal in any orientation (therefore also in current global orientation), satisfying I11 = I22 = +I33, we can write +˙ω◦ +i = T i/I11, +2.1. +DEM formulation +61 + +Yade Documentation, Release 3rd ed. +We use the same approximation scheme, obtaining an equation analogous to (2.1.5) +ω⊕ = ω⊖ + ∆t ˙ω◦. +The quaternion ∆q representing rotation vector ω⊕∆t is constructed, i.e. such that +(∆q)ϑ = |ω⊕|, +(∆q)u = � +ω⊕ +Finally, we compute the next orientation q+ by rotation composition +q+ = ∆qq◦. +Orientation (aspherical) +Integrating rotation of aspherical particles is considerably more complicated than their position, as their +local reference frame is not inertial. Rotation of rigid body in the local frame, where inertia matrix I is +diagonal, is described in the continuous form by Euler’s equations (i ∈ {1, 2, 3} and i, j, k are subsequent +indices): +T i = Iii ˙ωi + (Ikk − Ijj)ωjωk. +Due to the presence of the current values of both ω and ˙ω, they cannot be solved using the standard +leapfrog algorithm (that was the case for translational motion and also for the spherical bodies’ rotation +where this equation reduced to T = I ˙ω). +The algorithm presented here is described by [Allen1989] (pg. 84–89) and was designed by Fincham +for molecular dynamics problems; it is based on extending the leapfrog algorithm by mid-step/on-step +estimators of quantities known at on-step/mid-step points in the basic formulation. Although it has +received criticism and more precise algorithms are known ([Omelyan1999], [Neto2006], [Johnson2008]), +this one is currently implemented in Yade for its relative simplicity. +Each body has its local coordinate system based on the principal axes of inertia for that body. We use �• to +denote vectors in local coordinates. The orientation of the local system is given by the current particle’s +orientation q◦ as a quaternion; this quaternion can be expressed as the (current) rotation matrix A. +Therefore, every vector a is transformed as �a = qaq∗ = Aa. Since A is a rotation (orthogonal) matrix, +the inverse rotation A−1 = AT. +For given particle in question, we know +• �I +◦ (constant) inertia matrix; diagonal, since in local, principal coordinates, +• T ◦ external torque, +• q◦ current orientation (and its equivalent rotation matrix A), +• ω⊖ mid-step angular velocity, +• L⊖ mid-step angular momentum; this is an auxiliary variable that must be tracked in addition for +use in this algorithm. It will be zero in the initial step. +Our goal is to compute new values of the latter three, that is L⊕, q+, ω⊕. We first estimate current +angular momentum and compute current local angular velocity: +L◦ = L⊖ + T ◦ ∆t +2 , +�L +◦ = AL◦, +L⊕ = L⊖ + T ◦∆t, +�L +⊕ = AL⊕, +�ω◦ = �I +◦−1�L +◦, +�ω⊕ = �I +◦−1�L +⊕. +62 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +Then we compute ˙q◦, using q◦ and �ω◦: +� +� +� +� +˙q◦ +w +˙q◦ +x +˙q◦ +y +˙q◦ +z +� +� +� +� = 1 +2 +� +� +� +� +q◦ +w +−q◦ +x +−q◦ +y +−q◦ +z +q◦ +x +q◦ +w +−q◦ +z +q◦ +y +q◦ +y +q◦ +z +q◦ +w +−q◦ +x +q◦ +z +−q◦ +y +q◦ +x +q◦ +w +� +� +� +� +� +� +� +� +0 +�ω◦ +x +�ω◦ +y +�ω◦ +z +� +� +� +� , +q⊕ = q◦ + ˙q◦ ∆t +2 . +We evaluate ˙q⊕ from q⊕ and �ω⊕ in the same way as in (2.1.5) but shifted by ∆t/2 ahead. Then we can +finally compute the desired values +q+ = q◦ + ˙q⊕∆t, +ω⊕ = A−1 �ω⊕ +Clumps (rigid aggregates) +DEM simulations frequently make use of rigid aggregates of particles to model complex shapes [Price2007] +called clumps, typically composed of many spheres. Dynamic properties of clumps are computed from +the properties of its members: +• For non-overlapping clump members the clump’s mass mc is summed over members, the inertia +tensor Ic is computed using the parallel axes theorem: Ic = � +i(mi ∗d2 +i +Ii), where mi is the mass +of clump member i, di is the distance from center of clump member i to clump’s centroid and Ii +is the inertia tensor of the clump member i. +• For overlapping clump members the clump’s mass mc is summed over cells using a regular grid +spacing inside axis-aligned bounding box (Aabb) of the clump, the inertia tensor is computed using +the parallel axes theorem: Ic = � +j(mj ∗ d2 +j + Ij), where mj is the mass of cell j, dj is the distance +from cell center to clump’s centroid and Ij is the inertia tensor of the cell j. +Local axes are oriented such that they are principal and inertia tensor is diagonal and clump’s orientation +is changed to compensate rotation of the local system, as to not change the clump members’ positions +in global space. Initial positions and orientations of all clump members in local coordinate system are +stored. +In Yade (class Clump), clump members behave as stand-alone particles during simulation for purposes of +collision detection and contact resolution, except that they have no contacts created among themselves +within one clump. It is at the stage of motion integration that they are treated specially. Instead of inte- +grating each of them separately, forces/torques on those particles Fi, T i are converted to forces/torques +on the clump itself. Let us denote ri relative position of each particle with regards to clump’s centroid, +in global orientation. Then summary force and torque on the clump are +Fc = +� +Fi, +T c = +� +ri × Fi + Ti. +Motion of the clump is then integrated, using aspherical rotation integration. Afterwards, clump members +are displaced in global space, to keep their initial positions and orientations in the clump’s local coordinate +system. In such a way, relative positions of clump members are always the same, resulting in the behavior +of a rigid aggregate. +Numerical damping +In simulations of quasi-static phenomena, it it desirable to dissipate kinetic energy of particles. Since most +constitutive laws (including Law_ScGeom_FrictPhys_Basic shown above, Contact model (example)) do +not include velocity-based damping (such as one in [Addetta2001]), it is possible to use artificial numerical +damping. The formulation is described in [Pfc3dManual30], although our version is slightly adapted. The +2.1. +DEM formulation +63 + +Yade Documentation, Release 3rd ed. +basic idea is to decrease forces which increase the particle velocities and vice versa by (∆F)d, comparing +the current acceleration sense and particle velocity sense. This is done by component, which makes the +damping scheme clearly non-physical, as it is not invariant with respect to coordinate system rotation; +on the other hand, it is very easy to compute. Cundall proposed the form (we omit particle indices i +since it applies to all of them separately): +(∆F)dw +Fw += −λd sgn(Fw ˙u⊖ +w), +w ∈ {x, y, z} +where λd is the damping coefficient. This formulation has several advantages [Hentz2003]: +• it acts on forces (accelerations), not constraining uniform motion; +• it is independent of eigenfrequencies of particles, they will be all damped equally; +• it needs only the dimensionless parameter λd which does not have to be scaled. +In Yade, we use the adapted form +(∆F)dw +Fw += −λd sgn Fw +� +˙u⊖ +w + ¨u◦ +w∆t +2 +� +� +�� +� +≃ ˙u◦w +, +(2.9) +where we replaced the previous mid-step velocity ˙u⊖ by its on-step estimate in parentheses. This is to +avoid locked-in forces that appear if the velocity changes its sign due to force application at each step, +i.e. when the particle in question oscillates around the position of equilibrium with 2∆t period. +In Yade, damping (2.9) is implemented in the NewtonIntegrator engine; the damping coefficient λd is +NewtonIntegrator.damping. +Stability considerations +Critical timestep +In order to ensure stability for the explicit integration sceheme, an upper limit is imposed on ∆t: +∆tcr = +2 +ωmax +(2.10) +where ωmax is the highest eigenfrequency within the system. +Single mass-spring system +Single 1D mass-spring system with mass m and stiffness K is governed by the equation +m¨x = −Kx +where x is displacement from the mean (equilibrium) position. The solution of harmonic oscillation is +x(t) = A cos(ωt + φ) where phase φ and amplitude A are determined by initial conditions. The angular +frequency +ω(1) = +� +K +m +(2.11) +does not depend on initial conditions. Since there is one single mass, ω(1) +max = ω(1). Plugging (2.11) into +(2.10), we obtain +∆t(1) +cr = 2/ω(1) +max = 2 +� +m/K +for a single oscillator. +64 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +General mass-spring system +In a general mass-spring system, the highest frequency occurs if two connected masses mi, mj are in +opposite motion; let us suppose they have equal velocities (which is conservative) and they are connected +by a spring with stiffness Ki: displacement ∆xi of mi will be accompained by ∆xj = −∆xi of mj, giving +∆Fi = −Ki(∆xi − (−∆xi)) = −2Ki∆xi. That results in apparent stiffness K(2) +i += 2Ki, giving maximum +eigenfrequency of the whole system +ωmax = max +i +� +K(2) +i +/mi. +The overall critical timestep is then +∆tcr = +2 +ωmax += min +i +2 +� +mi +K(2) +i += min +i +2 +� mi +2Ki += min +i +√ +2 +�mi +Ki +. +(2.12) +This equation can be used for all 6 degrees of freedom (DOF) in translation and rotation, by considering +generalized mass and stiffness matrices M and K, and replacing fractions mi +Ki by eigen values of M.K−1. +The critical timestep is then associated to the eigen mode with highest frequency : +∆tcr = min ∆tcrk, +k ∈ {1, ..., 6}. +(2.13) +DEM simulations +In DEM simulations, per-particle stiffness Kij is determined from the stiffnesses of contacts in which it +participates. Suppose each contact has normal stiffness KNk, shear stiffness KTk = ξKNk and is oriented +by normal nk. A translational stiffness matrix Kij can be defined as the sum of contributions of all +contacts in which it participates (indices k), as [Chareyre2005]. +Kij = +� +k +(KNk − KTk)ninj + KTk = +� +j +KNk ((1 − ξ)ninj + ξ) +(2.14) +with i and j ∈ {x, y, z}. +Equations (2.13) and (2.14) determine ∆tcr in a simulation. +A similar ap- +proach generalized to all 6 DOFs is implemented by the GlobalStiffnessTimeStepper engine in Yade. +The derivation of generalized stiffness including rotational terms is very similar and can be found in +[AboulHosn2017]. +Note that for computation efficiency reasons, eigenvalues of the stiffness matrices are not computed. They +are only approximated assuming than DOF’s are uncoupled, and using the diagonal terms of K.M−1. +They give good approximates in typical mechanical systems. +There is one important condition that ωmax > 0: if there are no contacts between particles and ωmax = 0, +we would obtain value ∆tcr = ∞. While formally correct, this value is numerically erroneous: we were +silently supposing that stiffness remains constant during each timestep, which is not true if contacts are +created as particles collide. In case of no contact, therefore, stiffness must be pre-estimated based on +future interactions, as shown in the next section. +Estimation of ∆tcr by wave propagation speed +Estimating timestep in absence of interactions is based on the connection between interaction stiffnesses +and the particle’s properties. Note that in this section, symbols E and ρ refer exceptionally to Young’s +modulus and density of particles, not of macroscopic arrangement. +In Yade, particles have associated Material which defines density ρ (Material.density), and also may +define (in ElastMat and derived classes) particle’s “Young’s modulus” E (ElastMat.young). ρ is used +when particle’s mass m is initially computed from its ρ, while E is taken in account when creating new +interaction between particles, affecting stiffness KN. Knowing m and KN, we can estimate (2.14) for +each particle; we obviously neglect +2.1. +DEM formulation +65 + +Yade Documentation, Release 3rd ed. +• number of interactions per particle Ni; for a “reasonable” radius distribution, however, there is a +geometrically imposed upper limit (12 for a packing of spheres with equal radii, for instance); +• the exact relationship the between particles’ rigidities Ei, Ej, supposing only that KN is somehow +proportional to them. +By defining E and ρ, particles have continuum-like quantities. Explicit integration schemes for continuum +equations impose a critical timestep based on sonic speed +� +E/ρ; the elastic wave must not propagate +farther than the minimum distance of integration points lmin during one step. Since E, ρ are parameters +of the elastic continuum and lmin is fixed beforehand, we obtain +∆t(c) +cr = lmin +� +ρ +E. +For our purposes, we define E and ρ for each particle separately; lmin can be replaced by the sphere’s +radius Ri; technically, lmin = 2Ri could be used, but because of possible interactions of spheres and facets +(which have zero thickness), we consider lmin = Ri instead. Then +∆t(p) +cr += min +i +Ri +�ρi +Ei +. +This algorithm is implemented in the utils.PWaveTimeStep function. +Let us compare this result to (2.12); this necessitates making several simplifying hypotheses: +• all particles are spherical and have the same radius R; +• the sphere’s material has the same E and ρ; +• the average number of contacts per sphere is N; +• the contacts have sufficiently uniform spatial distribution around each particle; +• the ξ = KN/KT ratio is constant for all interactions; +• contact stiffness KN is computed from E using a formula of the form +KN = Eπ′R′, +(2.15) +where π′ is some constant depending on the algorithm in usefootnote{For example, π′ = π/2 in the +concrete particle model (Ip2_CpmMat_CpmMat_CpmPhys), while π′ = 2 in the classical DEM +model (Ip2_FrictMat_FrictMat_FrictPhys) as implemented in Yade.} +and R′ is half-distance +between spheres in contact, equal to R for the case of interaction radius RI = 1. If RI = 1 (and +R′ ≡ R by consequence), all interactions will have the same stiffness KN. In other cases, we will +consider KN as the average stiffness computed from average R′ (see below). +As all particles have the same parameters, we drop the i index in the following formulas. +We try to express the average per-particle stiffness from (2.14). It is a sum over all interactions where KN +and ξ are scalars that will not rotate with interaction, while nw is w-th component of unit interaction +normal n. Since we supposed uniform spatial distribution, we can replace n2 +w by its average value n2 +w. +Recognizing components of n as direction cosines, the average values of n2 +w is 1/3. We find the average +value by integrating over all possible orientations, which are uniformly distributed in space: +Moreover, since all directions are equal, we can write the per-body stiffness as K = Kw for all w ∈ {x, y, z}. +We obtain +K = +� +KN +� +(1 − ξ)1 +3 + ξ +� += +� +KN +1 + 2ξ +3 +and can put constant terms (everything) in front of the summation. � 1 equals the number of contacts +per sphere, i.e. N. Arriving at +K = NKN +1 − 2ξ +3 +, +66 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +we substitute K into (2.12) using (2.15): +∆tcr = +√ +2 +� +m +K = +√ +2 +� +4 +3πR3ρ +NEπ′R 1−2ξ +3 += R +� +ρ +E +� �� � +∆t(p) +cr +2 +� +π/π′ +N(1 − 2ξ). +The ratio of timestep ∆t(p) +cr +predicted by the p-wave velocity and numerically stable timestep ∆tcr is the +inverse value of the last (dimensionless) term: +∆t(p) +cr +∆tcr += 2 +� +N(1 + ξ) +π/π′ +. +Actual values of this ratio depend on characteristics of packing N, KN/KT = ξ ratio and the way of +computing contact stiffness from particle rigidity. Let us show it for two models in Yade: +Concrete particle model computes contact stiffness from the equivalent area Aeq first (2.6), +Aeq = πR2KN += AeqE +d0 +. +d0 is the initial contact length, which will be, for interaction radius (2.5) RI > 1, in average larger +than 2R. For RI = 1.5 ,we can roughly estimate d0 = 1.25 · 2R = 5 +2R, getting +KN = E +�2 +5π +� +R +where 2 +5π = π′ by comparison with (2.15). +Interaction radius RI = 1.5 leads to average N ≈ 12 interactions per sphere for dense packing of +spheres with the same radius R. ξ = 0.2 is calibrated to match the desired macroscopic Poisson’s +ratio ν = 0.2. +Finally, we obtain the ratio +∆t(p) +cr +∆tcr += 2 +� +12(1 − 2 · 0.2) +π +(2/5)π += 3.39, +showing significant overestimation by the p-wave algorithm. +Non-cohesive dry friction model is the basic model proposed by Cundall explained in Contact model +(example). Supposing almost-constant sphere radius R and rather dense packing, each sphere will +have N = 6 interactions on average (that corresponds to maximally dense packing of spheres with +a constant radius). If we use the Ip2_FrictMat_FrictMat_FrictPhys class, we have π′ = 2, as +KN = E2R; we again use ξ = 0.2 (for lack of a more significant value). In this case, we obtain the +result +∆t(p) +cr +∆tcr += 2 +� +6(1 − 2 · 0.2) +π/2 += 3.02 +which again overestimates the numerical critical timestep. +To conclude, p-wave timestep gives estimate proportional to the real ∆tcr, but in the cases shown, the +value of about ∆t = 0.3∆t(p) +cr +should be used to guarantee stable simulation. +Non-elastic ∆t constraints +Let us note at this place that not only ∆tcr assuring numerical stability of motion integration is a +constraint. In systems where particles move at relatively high velocities, position change during one +timestep can lead to non-elastic irreversible effects such as damage. The ∆t needed for reasonable result +can be lower ∆tcr. We have no rigorously derived rules for such cases. +2.1. +DEM formulation +67 + +Yade Documentation, Release 3rd ed. +2.1.6 Periodic boundary conditions +While most DEM simulations happen in R3 space, it is frequently useful to avoid boundary effects by +using periodic space instead. +In order to satisfy periodicity conditions, periodic space is created by +repetition of parallelepiped-shaped cell. In Yade, periodic space is implemented in the Cell class. The +geometry of the cell in the reference coordinates system is defined by three edges of the parallepiped. +The corresponding base vectors are stored in the columns of matrix H (Cell.hSize). +The initial H can be explicitly defined as a 3x3 matrix at the beginning of the simulation. There are no +restricitions on the possible shapes: any parallelepiped is accepted as the initial cell. If the base vectors +are axis-aligned, defining only their sizes can be more convenient than defining the full H matrix; in that +case it is enough to define the norms of columns in H (see Cell.size). +After the definition of the initial cell’s geometry, H should generally not be modified by direct assignment. +Instead, its deformation rate will be defined via the velocity gradient Cell.velGrad described below. It +is the only variable that let the period deformation be correctly accounted for in constitutive laws and +Newton integrator (NewtonIntegrator). +Deformations handling +The deformation of the cell over time is defined via a tensor representing the gradient of an homoge- +neous velocity field ∇v (Cell.velGrad). This gradient represents arbitrary combinations of rotations and +stretches. It can be imposed externaly or updated by boundary controllers (see PeriTriaxController or +Peri3dController) in order to reach target strain values or to maintain some prescribed stress. +The velocity gradient is integrated automatically over time, and the cumulated transformation is re- +flected in the transformation matrix F (Cell.trsf) and the current shape of the cell H. The per-step +transformation update reads (it is similar for H), with I the identity matrix: +F+ = (I + ∇v∆t)F◦. +F can be set back to identity at any point in simulations, in order to define the current state as reference +for strains definition in boundary controllers. It will have no effect on H. +Along with the automatic integration of cell transformation, there is an option to homothetically displace +all particles so that ∇v is applied over the whole simulation (enabled via Cell.homoDeform). This avoids +all boundary effects coming from change of the velocity gradient. +Collision detection in periodic cell +In usual implementations, particle positions are forced to be inside the cell by wrapping their positions +if they get over the boundary (so that they appear on the other side). As we wanted to avoid abrupt +changes of position (it would make particle’s velocity inconsistent with step displacement change), a +different method was chosen. +Approximate collision detection +Pass 1 collision detection (based on sweep and prune algorithm, sect. Sweep and prune) operates on +axis-aligned bounding boxes (Aabb) of particles. +During the collision detection phase, bounds of all +Aabb’s are wrapped inside the cell in the first step. At subsequent runs, every bound remembers by how +many cells it was initially shifted from coordinate given by the Aabb and uses this offset repeatedly as +it is being updated from Aabb during particle’s motion. Bounds are sorted using the periodic insertion +sort algorithm (sect. Periodic insertion sort algorithm), which tracks periodic cell boundary ||. +Upon inversion of two Aabb’s, their collision along all three axes is checked, wrapping real coordinates +inside the cell for that purpose. +68 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +This algorithm detects collisions as if all particles were inside the cell but without the need of constructing +“ghost particles” (to represent periodic image of a particle which enters the cell from the other side) or +changing the particle’s positions. +It is required by the implementation (and partly by the algorithm itself) that particles do not span more +than half of the current cell size along any axis; the reason is that otherwise two (or more) contacts +between both particles could appear, on each side. Since Yade identifies contacts by Body.id of both +bodies, they would not be distinguishable. +In presence of shear, the sweep-and-prune collider could not sort bounds independently along three axes: +collision along x axis depends on the mutual position of particles on the y axis. Therefore, bounding +boxes are expressed in transformed coordinates which are perpendicular in the sense of collision detection. +This requires some extra computation: Aabb of sphere in transformed coordinates will no longer be cube, +but cuboid, as the sphere itself will appear as ellipsoid after transformation. Inversely, the sphere in +simulation space will have a parallelepiped bounding “box”, which is cuboid around the ellipsoid in +transformed axes (the Aabb has axes aligned with transformed cell basis). This is shown in fig. fig-cell- +shear-aabb. +The restriction of a single particle not spanning more than half of the transformed axis becomes stringent +as Aabb is enlarged due to shear. Considering Aabb of a sphere with radius r in the cell where x′ ≡ x, +z′ ≡ z, but ∠(y, y′) = φ, the x-span of the Aabb will be multiplied by 1/ cos φ. For the infinite shear +φ → π/2, which can be desirable to simulate, we have 1/ cos φ → ∞. Fortunately, this limitation can be +easily circumvented by realizing the quasi-identity of all periodic cells which, if repeated in space, create +the same grid with their corners: the periodic cell can be flipped, keeping all particle interactions intact, +as shown in fig. fig-cell-flip. It only necessitates adjusting the Interaction.cellDist of interactions and +re-initialization of the collider (Collider::invalidatePersistentData). Cell flipping is implemented +in the utils.flipCell function. +y′ +1 +y′ +2 +x′ +1 +x′ +2 ≡ x′ +1 +ϕ1 +y +y +ϕ2 +Fig. 6: Flipping cell (utils.flipCell) to avoid infinite stretch of the bounding boxes’ spans with growing φ. +Cell flip does not affect interactions from the point of view of the simulation. The periodic arrangement +on the left is the same as the one on the right, only the cell is situated differently between identical grid +points of repetition; at the same time |φ2| < |φ1| and sphere bounding box’s x-span stretched by 1/ cos φ +becomes smaller. Flipping can be repeated, making effective infinite shear possible. +This algorithm is implemented in InsertionSortCollider and is used whenever simulation is periodic +(Omega.isPeriodic); individual BoundFunctor’s are responsible for computing sheared Aabb’s; currently +it is implemented for spheres and facets (in Bo1_Sphere_Aabb and Bo1_Facet_Aabb respectively). +Exact collision detection +When the collider detects approximate contact (on the Aabb level) and the contact does not yet exist, +it creates potential contact, which is subsequently checked by exact collision algorithms (depending on +the combination of Shapes). Since particles can interact over many periodic cells (recall we never change +2.1. +DEM formulation +69 + +Yade Documentation, Release 3rd ed. +y ≡ y′ +y +y′ +x ≡ x′ +x +x′ +Fig. 7: Constructing axis-aligned bounding box (Aabb) of a sphere in simulation space coordinates +(without periodic cell – left) and transformed cell coordinates (right), where collision detection axes x′, +y′ are not identical with simulation space axes x, y. Bounds’ projection to axes is shown by orange lines. +their positions in simulation space), the collider embeds the relative cell coordinate of particles in the +interaction itself (Interaction.cellDist) as an integer vector c. +Multiplying current cell size Ts by c +component-wise, we obtain particle offset ∆x in aperiodic R3; this value is passed (from InteractionLoop) +to the functor computing exact collision (IGeomFunctor), which adds it to the position of the particle +Interaction.id2. +By storing the integral offset c, ∆x automatically updates as cell parameters change. +Periodic insertion sort algorithm +The extension of sweep and prune algorithm (described in Sweep and prune) to periodic boundary +conditions is non-trivial. +Its cornerstone is a periodic variant of the insertion sort algorithm, which +involves keeping track of the “period” of each boundary; e.g. taking period ⟨0, 10), then 81 ≡ −22 < 22 +(subscript indicating period). Doing so efficiently (without shuffling data in memory around as bound +wraps from one period to another) requires moving period boundary rather than bounds themselves and +making the comparison work transparently at the edge of the container. +This algorithm was also extended to handle non-orthogonal periodic Cell boundaries by working in trans- +formed rather than Cartesian coordinates; this modifies computation of Aabb from Cartesian coordinates +in which bodies are positioned (treated in detail in Approximate collision detection). +The sort algorithm is tracking Aabb extrema along all axes. At the collider’s initialization, each value is +assigned an integral period, i.e. its distance from the cell’s interior expressed in the cell’s dimension along +its respective axis, and is wrapped to a value inside the cell. We put the period number in subscript. +Let us give an example of coordinate sequence along x axis (in a real case, the number of elements would +be even, as there is maximum and minimum value couple for each particle; this demonstration only +shows the sorting algorithm, however.) +with cell x-size sx = 10. +The 41 value then means that the real coordinate xi of this extremum is +xi + 1 · 10 = 4, i.e. xi = −4. The || symbol denotes the periodic cell boundary. +Sorting starts from the first element in the cell, i.e. right of ||, and inverts elements as in the aperiodic +variant. The rules are, however, more complicated due to the presence of the boundary ||: +70 +Chapter 2. +Yade for users + +41 +122 +2 =-12 +-24 +50Yade Documentation, Release 3rd ed. +(≤) +stop inverting if neighbors are ordered; +(||•) +current element left of || is below 0 (lower period boundary); in this case, decrement element’s +period, decrease its coordinate by sx and move || right; +(•||) +current element right of || is above sx (upper period boundary); increment element’s period, +increase its coordinate by sx and move || left; +(||<) +inversion across || must subtract sx from the left coordinate during comparison. If the elements +are not in order, they are swapped, but they must have their periods changed as they traverse +||. Apply (||◦) if necessary; +(||◦) +if after (||<) the element that is now right of || has xi < sx, decrease its coordinate by sx and +decrement its period. Do not move ||. +In the first step, (||•) is applied, and inversion with 122 happens; then we stop because of (≤): +We move to next element −24 ; first, we apply (||•), then invert until (≤): +The next element is 50 ; we satisfy (||<), therefore instead of comparing 122 > 50, we must do (122−sx) = +23 ≤ 5; we adjust periods when swapping over || and apply (||◦), turning 122 into 23; then we keep +inverting, until (≤): +2.1. +DEM formulation +71 + +122 +-12 +-24 +41 +11 +50, +41 +122 +91 +-24 +50, +41 +91 +122 +-24 +50:41 +91 +122 +-24 +50, +41 +91 +122 +83 +50, +41 +91 +83 +122 +1 +50, +41 +83 +91 +122 +50.Yade Documentation, Release 3rd ed. +We move (wrapping around) to 41 , which is ordered: +and so is the last element +2.1.7 Computational aspects +Cost +The DEM computation using an explicit integration scheme demands a relatively high number of steps +during simulation, compared to implicit scehemes. The total computation time Z of simulation spanning +T seconds (of simulated time), containing N particles in volume V depends on: +• linearly, the number of steps i = T/(st∆tcr), where st is timestep safety factor; ∆tcr can be +estimated by p-wave velocity using E and ρ (sect. Estimation of \Dtcr by wave propagation speed) +as ∆t(p) +cr += r +� ρ +E. Therefore +i = T +str +� +E +ρ. +• the number of particles N; for fixed value of simulated domain volume V and particle radius r +N = p +V +4 +3πr3 , +where p is packing porosity, roughly 1 +2 for dense irregular packings of spheres of similar radius. +72 +Chapter 2. +Yade for users + +41 +83 +91 +122 +50 +41 +83 +91 +5-1 +23, +41 +83 +5-1 +91 +23 +41 +5-1 +83 +91 +11 +23.41 +5-1 +83 +91I 2341 +5-1 +83 +91l23.Yade Documentation, Release 3rd ed. +The dependency is not strictly linear (which would be the best case), as some algorithms do not +scale linearly; a case in point is the sweep and prune collision detection algorithm introduced in +sect. Sweep and prune, with scaling roughly O(N log N). +The number of interactions scales with N, as long as packing characteristics are the same. +• the number of computational cores ncpu; in the ideal case, the dependency would be inverse-linear +were all algorithms parallelized (in Yade, collision detection is not). +Let us suppose linear scaling. Additionally, let us suppose that the material to be simulated (E, ρ) and +the simulation setup (V, T) are given in advance. Finally, dimensionless constants st, p and ncpu will +have a fixed value. This leaves us with one last degree of freedom, r. We may write +Z ∝ iN +1 +ncpu += T +str +� +E +ρp +V +4 +3πr3 +1 +ncpu +∝ 1 +r +1 +r3 = 1 +r4 . +This (rather trivial) result is essential to realize DEM scaling; if we want to have finer results, refining +the “mesh” by halving r, the computation time will grow 24 = 16 times. +For very crude estimates, one can use a known simulation to obtain a machine “constant” +µ = Z +Ni +with the meaning of time per particle and per timestep (in the order of 10−6 s for current machines). +µ will be only useful if simulation characteristics are similar and non-linearities in scaling do not have +major influence, i.e. N should be in the same order of magnitude as in the reference case. +Result indeterminism +It is naturally expected that running the same simulation several times will give exactly the same results: +although the computation is done with finite precision, round-off errors would be deterministically the +same at every run. While this is true for single-threaded computation where exact order of all operations +is given by the simulation itself, it is not true anymore in multi-threaded computation which is described +in detail in later sections. +The straight-forward manner of parallel processing in explicit DEM is given by the possibility of treating +interactions in arbitrary order. Strain and stress is evaluated for each interaction independently, but +forces from interactions have to be summed up. If summation order is also arbitrary (in Yade, forces are +accumulated for each thread in the order interactions are processed, then summed together), then the +results can be slightly different. For instance +(1/10.)+(1/13.)+(1/17.)=0.23574660633484162 +(1/17.)+(1/13.)+(1/10.)=0.23574660633484165 +As forces generated by interactions are assigned to bodies in quasi-random order, summary force Fi on +the body can be different between single-threaded and multi-threaded computations, but also between +different runs of multi-threaded computation with exactly the same parameters. Exact thread scheduling +by the kernel is not predictable since it depends on asynchronous events (hardware interrupts) and other +unrelated tasks running on the system; and it is thread scheduling that ultimately determines summation +order of force contributions from interactions. +2.2 User’s manual +2.2.1 Scene construction +Adding particles +The BodyContainer holds Body objects in the simulation; it is accessible as O.bodies. +2.2. +User’s manual +73 + +Yade Documentation, Release 3rd ed. +Creating Body objects +Body objects are only rarely constructed by hand by their components (Shape, Bound, State, Material); +instead, convenience functions sphere, facet and wall are used to create them. Using these functions also +ensures better future compatibility, if internals of Body change in some way. These functions receive +geometry of the particle and several other characteristics. See their documentation for details. If the +same Material is used for several (or many) bodies, it can be shared by adding it in O.materials, as +explained below. +Defining materials +The O.materials object (instance of Omega.materials) holds defined shared materials for bodies. It +only supports addition, and will typically hold only a few instances (though there is no limit). +label given to each material is optional, but can be passed to sphere and other functions for constructing +body. The value returned by O.materials.append is an id of the material, which can be also passed to +sphere – it is a little bit faster than using label, though not noticeable for small number of particles and +perhaps less convenient. +If no Material is specified when calling sphere, the last defined material is used; that is a convenient +default. If no material is defined yet (hence there is no last material), a default material will be created: +FrictMat(density=1e3,young=1e7,poisson=.3,frictionAngle=.5). This should not happen for serious sim- +ulations, but is handy in simple scripts, where exact material properties are more or less irrelevant. +Yade [1]: len(O.materials) +Out[1]: 0 +Yade [2]: idConcrete=O.materials.append(FrictMat(young=30e9,poisson=.2,frictionAngle=.6,label= +�→"concrete")) +Yade [3]: O.materials[idConcrete] +Out[3]: +# uses the last defined material +Yade [4]: O.bodies.append(sphere(center=(0,0,0),radius=1)) +Out[4]: 0 +# material given by id +Yade [5]: O.bodies.append(sphere((0,0,2),1,material=idConcrete)) +Out[5]: 1 +# material given by label +Yade [6]: O.bodies.append(sphere((0,2,0),1,material="concrete")) +Out[6]: 2 +Yade [7]: idSteel=O.materials.append(FrictMat(young=210e9,poisson=.25,frictionAngle=.8,label= +�→"steel")) +Yade [8]: len(O.materials) +Out[8]: 2 +# implicitly uses "steel" material, as it is the last one now +Yade [9]: O.bodies.append(facet([(1,0,0),(0,1,0),(-1,-1,0)])) +Out[9]: 3 +Adding multiple particles +As shown above, bodies are added one by one or several at the same time using the append method: +74 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +Yade [10]: O.bodies.append(sphere((0,10,0),1)) +Out[10]: 0 +Yade [11]: O.bodies.append(sphere((0,0,2),1)) +Out[11]: 1 +# this is the same, but in one function call +Yade [12]: O.bodies.append([ +....: +sphere((0,0,0),1), +....: +sphere((1,1,3),1) +....: ]) +....: +Out[12]: [2, 3] +Many functions introduced in next sections return list of bodies which can be readily added to the +simulation, including +• packing generators, such as pack.randomDensePack, pack.regularHexa +• surface function pack.gtsSurface2Facets +• import functions ymport.gmsh, ymport.stl, … +As those functions use sphere and facet internally, they accept additional arguments passed to those +functions. In particular, material for each body is selected following the rules above (last one if not +specified, by label, by index, etc.). +Clumping particles together +In some cases, you might want to create rigid aggregate of individual particles (i.e. particles will retain +their mutual position during simulation). This we call a clump. A clump is internally represented by a +special body, referenced by clumpId of its members (see also isClump, isClumpMember and isStandalone). +Like every body a clump has a position, which is the (mass) balance point between all members. A +clump body itself has no interactions with other bodies. Interactions between clumps is represented by +interactions between clump members. There are no interactions between clump members of the same +clump. +YADE supports different ways of creating clumps: +• Create clumps and spheres (clump members) directly with one command: +The function appendClumped() is designed for this task. +For instance, we might add 2 spheres tied +together: +Yade [13]: O.bodies.appendClumped([ +....: +sphere([0,0,0],1), +....: +sphere([0,0,2],1) +....: ]) +....: +Out[13]: (2, [0, 1]) +Yade [14]: len(O.bodies) +Out[14]: 3 +Yade [15]: O.bodies[1].isClumpMember, O.bodies[2].clumpId +Out[15]: (True, 2) +Yade [16]: O.bodies[2].isClump, O.bodies[2].clumpId +Out[16]: (True, 2) +-> appendClumped() returns a tuple of ids (clumpId,[memberId1,memberId2,...]) +• Use existing spheres and clump them together: +2.2. +User’s manual +75 + +Yade Documentation, Release 3rd ed. +For this case the function clump() can be applied on a list of existing bodies: +Yade [17]: bodyList = [] +Yade [18]: for ii in range(0,5): +....: +bodyList.append(O.bodies.append(sphere([ii,0,1],.5)))#create a "chain" of 5 spheres +....: +Yade [19]: print(bodyList) +[0, 1, 2, 3, 4] +Yade [20]: idClump=O.bodies.clump(bodyList) +-> clump() returns clumpId +• Another option is to replace standalone spheres from a given packing (see SpherePack and make- +Cloud) by clumps using clump templates. +This is done by a function called replaceByClumps(). This function takes a list of clumpTemplates() and +a list of amounts and replaces spheres by clumps. The volume of a new clump will be the same as the +volume of the sphere, that was replaced (clump volume/mass/inertia is accounting for overlaps assuming +that there are only pair overlaps). +-> replaceByClumps() returns a list of tuples: [(clumpId1,[memberId1,memberId2,...]),(clumpId2, +[memberId1,memberId2,...]),...] +It is also possible to add bodies to a clump and release bodies from a clump. Also you can erase the +clump (clump members will become standalone). +Additionally YADE allows to achieve the roundness of a clump or roundness coefficient of a packing. +Parts of the packing can be excluded from roundness measurement via exclude list. +Yade [21]: bodyList = [] +Yade [22]: for ii in range(1,5): +....: +bodyList.append(O.bodies.append(sphere([ii,ii,ii],.5))) +....: +Yade [23]: O.bodies.clump(bodyList) +Out[23]: 4 +Yade [24]: RC=O.bodies.getRoundness() +Yade [25]: print(RC) +0.25619141423166986 +-> getRoundness() returns roundness coefficient RC of a packing or a part of the packing +Note: +Have a look at examples/clumps/ folder. There you will find some examples, that show usage +of different functions for clumps. +Sphere packings +Representing a solid of an arbitrary shape by arrangement of spheres presents the problem of sphere +packing, i.e. spatial arrangement of spheres such that a given solid is approximately filled with them. +For the purposes of DEM simulation, there can be several requirements. +1. Distribution of spheres’ radii. Arbitrary volume can be filled completely with spheres provided +there are no restrictions on their radius; in such case, number of spheres can be infinite and their +radii approach zero. +Since both number of particles and minimum sphere radius (via critical +timestep) determine computation cost, radius distribution has to be given mandatorily. The most +76 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +typical distribution is uniform: mean±dispersion; if dispersion is zero, all spheres will have the +same radius. +2. Smooth boundary. Some algorithms treat boundaries in such way that spheres are aligned on them, +making them smoother as surface. +3. Packing density, or the ratio of spheres volume and solid size. +It is closely related to radius +distribution. +4. Coordination number, (average) number of contacts per sphere. +5. Isotropy (related to regularity/irregularity); packings with preferred directions are usually not +desirable, unless the modeled solid also has such preference. +6. Permissible Spheres’ overlap; some algorithms might create packing where spheres slightly overlap; +since overlap usually causes forces in DEM, overlap-free packings are sometimes called “stress-free￿. +Volume representation +There are 2 methods for representing exact volume of the solid in question in Yade: boundary repre- +sentation and constructive solid geometry. Despite their fundamental differences, they are abstracted in +Yade in the Predicate class. Predicate provides the following functionality: +1. defines axis-aligned bounding box for the associated solid (optionally defines oriented bounding +box); +2. can decide whether given point is inside or outside the solid; most predicates can also (exactly or +approximately) tell whether the point is inside and satisfies some given padding distance from the +represented solid boundary (so that sphere of that volume doesn’t stick out of the solid). +Constructive Solid Geometry (CSG) +CSG approach describes volume by geometric primitives or primitive solids (sphere, cylinder, box, cone, +…) and boolean operations on them. Primitives defined in Yade include inCylinder, inSphere, inEllipsoid, +inHyperboloid, notInNotch. +For instance, hyperboloid (dogbone) specimen for tension-compression test can be constructed in this +way (shown at img. img-hyperboloid): +from yade import pack +## construct the predicate first +pred=pack.inHyperboloid(centerBottom=(0,0,-.1),centerTop=(0,0,.1),radius=.05,skirt=.03) +## alternatively: pack.inHyperboloid((0,0,-.1),(0,0,.1),.05,.03) +## pack the predicate with spheres (will be explained later) +spheres=pack.randomDensePack(pred,spheresInCell=2000,radius=3.5e-3) +## add spheres to simulation +O.bodies.append(spheres) +Boundary representation (BREP) +Representing a solid by its boundary is much more flexible than CSG volumes, but is mostly only ap- +proximate. Yade interfaces to GNU Triangulated Surface Library (GTS) to import surfaces readable by +GTS, but also to construct them explicitly from within simulation scripts. This makes possible para- +metric construction of rather complicated shapes; there are functions to create set of 3d polylines from +2d polyline (pack.revolutionSurfaceMeridians), to triangulate surface between such set of 3d polylines +(pack.sweptPolylines2gtsSurface). +2.2. +User’s manual +77 + +Yade Documentation, Release 3rd ed. +Fig. +8: +Specimen +constructed +with +the +pack.inHyperboloid +predicate, +packed +with +pack.randomDensePack. +For example, we can construct a simple funnel (examples/funnel.py, shown at img-funnel): +from numpy import linspace +from yade import pack +# angles for points on circles +thetas=linspace(0,2*pi,num=16,endpoint=True) +# creates list of polylines in 3d from list of 2d projections +# turned from 0 to π +meridians=pack.revolutionSurfaceMeridians( +[[(3+rad*sin(th),10*rad+rad*cos(th)) for th in thetas] for rad in linspace(1,2, +�→num=10)], +linspace(0,pi,num=10) +) +# create surface +surf=pack.sweptPolylines2gtsSurface( +meridians ++[[Vector3(5*sin(-th),-10+5*cos(-th),30) for th in thetas]] +# add funnel top +) +# add to simulation +O.bodies.append(pack.gtsSurface2Facets(surf)) +GTS surface objects can be used for 2 things: +1. pack.gtsSurface2Facets function can create the triangulated surface (from Facet particles) in the +simulation itself, as shown in the funnel example. +(Triangulated surface can also be imported +directly from a STL file using ymport.stl.) +2. pack.inGtsSurface predicate can be created, using the surface as boundary representation of the +enclosed volume. +The examples/gts-horse/gts-horse.py (img. img-horse) shows both possibilities; first, a GTS surface is +imported: +import gts +surf=gts.read(open('horse.coarse.gts')) +That surface object is used as predicate for packing: +78 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +Fig. 9: Triangulated funnel, constructed with the examples/funnel.py script. +pred=pack.inGtsSurface(surf) +aabb=pred.aabb() +radius=(aabb[1][0]-aabb[0][0])/40 +O.bodies.append(pack.regularHexa(pred,radius=radius,gap=radius/4.)) +and then, after being translated, as base for triangulated surface in the simulation itself: +surf.translate(0,0,-(aabb[1][2]-aabb[0][2])) +O.bodies.append(pack.gtsSurface2Facets(surf,wire=True)) +Fig. 10: Imported GTS surface (horse) used as packing predicate (top) and surface constructed from +facets (bottom). See http://www.youtube.com/watch?v=PZVruIlUX1A for movie of this simulation. +2.2. +User’s manual +79 + +1:05Yade Documentation, Release 3rd ed. +Boolean operations on predicates +Boolean operations on pair of predicates (noted A and B) are defined: +• intersection A & B (conjunction): point must be in both predicates involved. +• union A | B (disjunction): point must be in the first or in the second predicate. +• difference A - B (conjunction with second predicate negated): the point must be in the first pred- +icate and not in the second one. +• symmetric difference A ^ B (exclusive disjunction): point must be in exactly one of the two pred- +icates. +Composed predicates also properly define their bounding box. For example, we can take box and remove +cylinder from inside, using the A - B operation (img. img-predicate-difference): +pred=pack.inAlignedBox((-2,-2,-2),(2,2,2))-pack.inCylinder((0,-2,0),(0,2,0),1) +spheres=pack.randomDensePack(pred,spheresInCell=2000,radius=.1,rRelFuzz=.4, +�→returnSpherePack=True) +spheres.toSimulation() +Fig. 11: Box with cylinder removed from inside, using difference of these two predicates. +Packing algorithms +Algorithms presented below operate on geometric spheres, defined by their center and radius. With a +few exception documented below, the procedure is as follows: +1. Sphere positions and radii are computed (some functions use volume predicate for this, some do +not) +2. sphere is called for each position and radius computed; it receives extra keyword arguments of the +packing function (i.e. arguments that the packing function doesn’t specify in its definition; they +are noted **kw). Each sphere call creates actual Body objects with Sphere shape. List of Body +objects is returned. +3. List returned from the packing function can be added to simulation using toSimulation(). Legacy +code used a call to O.bodies.append. +Taking the example of pierced box: +80 +Chapter 2. +Yade for users + +0.01 +#口 +clock 02:50Yade Documentation, Release 3rd ed. +pred=pack.inAlignedBox((-2,-2,-2),(2,2,2))-pack.inCylinder((0,-2,0),(0,2,0),1) +spheres=pack.randomDensePack(pred,spheresInCell=2000,radius=.1,rRelFuzz=.4,wire=True,color=(0, +�→0,1),material=1,returnSpherePack=True) +Keyword arguments wire, color and material are not declared in pack.randomDensePack, therefore +will be passed to sphere, where they are also documented. spheres is now a SpherePack object.: +spheres.toSimulation() +Packing algorithms described below produce dense packings. If one needs loose packing, SpherePack +class provides functions for generating loose packing, via its makeCloud() method. It is used internally +for generating initial configuration in dynamic algorithms. For instance: +from yade import pack +sp=pack.SpherePack() +sp.makeCloud(minCorner=(0,0,0),maxCorner=(3,3,3),rMean=.2,rRelFuzz=.5) +will fill given box with spheres, until no more spheres can be placed. The object can be used to add +spheres to simulation: +sp.toSimulation() +Geometric +Geometric algorithms compute packing without performing dynamic simulation; among their advantages +are +• speed; +• spheres touch exactly, there are no overlaps (what some people call “stress-free” packing); +their chief disadvantage is that radius distribution cannot be prescribed exactly, save in specific cases +(regular packings); sphere radii are given by the algorithm, which already makes the system determined. +If exact radius distribution is important for your problem, consider dynamic algorithms instead. +Regular +Yade defines packing generators for spheres with constant radii, which can be used with volume predicates +as described above. They are dense orthogonal packing (pack.regularOrtho) and dense hexagonal packing +(pack.regularHexa). The latter creates so-called “hexagonal close packing”, which achieves maximum +density (http://en.wikipedia.org/wiki/Close-packing_of_spheres). +Clear disadvantage of regular packings is that they have very strong directional preferences, which might +not be an issue in some cases. +Irregular +Random geometric algorithms do not integrate at all with volume predicates described above; rather, +they take their own boundary/volume definition, which is used during sphere positioning. On the other +hand, this makes it possible for them to respect boundary in the sense of making spheres touch it at +appropriate places, rather than leaving empty space in-between. +GenGeo is library (python module) for packing generation developed with ESyS-Particle. It creates +packing by random insertion of spheres with given radius range. Inserted spheres touch each other +exactly and, more importantly, they also touch the boundary, if in its neighbourhood. Boundary +is represented as special object of the GenGeo library (Sphere, cylinder, box, convex polyhedron, +…). Therefore, GenGeo cannot be used with volume represented by yade predicates as explained +above. +2.2. +User’s manual +81 + +Yade Documentation, Release 3rd ed. +Packings generated by this module can be imported directly via ymport.gengeo, or from saved file via +ymport.gengeoFile. There is an example script examples/test/genCylLSM.py. Full documentation +for GenGeo can be found at ESyS documentation website. +There are debian packages esys-particle and python-demgengeo. +Dynamic +The most versatile algorithm for random dense packing is provided by pack.randomDensePack. Initial +loose packing of non-overlapping spheres is generated by randomly placing them in cuboid volume, +with radii given by requested (currently only uniform) radius distribution. When no more spheres can +be inserted, the packing is compressed and then uncompressed (see py/pack/pack.py for exact values +of these “stresses”) by running a DEM simulation; Omega.switchScene is used to not affect existing +simulation). Finally, resulting packing is clipped using provided predicate, as explained above. +By its nature, this method might take relatively long; and there are 2 provisions to make the computation +time shorter: +• If number of spheres using the spheresInCell parameter is specified, only smaller specimen with +periodic boundary is created and then repeated as to fill the predicate. This can provide high- +quality packing with low regularity, depending on the spheresInCell parameter (value of several +thousands is recommended). +• Providing memoizeDb parameter will make pack.randomDensePack first look into provided file +(SQLite database) for packings with similar parameters. On success, the packing is simply read +from database and returned. If there is no similar pre-existent packing, normal procedure is run, +and the result is saved in the database before being returned, so that subsequent calls with same +parameters will return quickly. +If you need to obtain full periodic packing (rather than packing clipped by predicate), you can use +pack.randomPeriPack. +In case of specific needs, you can create packing yourself, “by hand”. For instance, packing boundary +can be constructed from facets, letting randomly positioned spheres in space fall down under gravity. +Triangulated surfaces +Yade integrates with the the GNU Triangulated Surface library, exposed in python via GTS module. GTS +provides variety of functions for surface manipulation (coarsening, tesselation, simplification, import), +to be found in its documentation. +GTS surfaces are geometrical objects, which can be inserted into simulation as set of particles whose +Body.shape is of type Facet – single triangulation elements. pack.gtsSurface2Facets can be used to convert +GTS surface triangulation into list of bodies ready to be inserted into simulation via O.bodies.append. +Facet particles are created by default as non-Body.dynamic (they have zero inertial mass). That means +that they are fixed in space and will not move if subject to forces. You can however +• prescribe arbitrary movement to facets using a PartialEngine (such as TranslationEngine or Rota- +tionEngine); +• assign explicitly mass and inertia to that particle; +• make that particle part of a clump and assign mass and inertia of the clump itself (described +below). +Note: +Facets can only (currently) interact with spheres, not with other facets, even if they are dynamic. +Collision of 2 facets will not create interaction, therefore no forces on facets. +82 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +Import +Yade currently offers 3 formats for importing triangulated surfaces from external files, in the ymport +module: +ymport.gts text file in native GTS format. +ymport.stl STereoLitography format, in either text or binary form; exported from Blender, but from +many CAD systems as well. +ymport.gmsh. text file in native format for GMSH, popular open-source meshing program. +If you need to manipulate surfaces before creating list of facets, you can study the py/ymport.py file +where the import functions are defined. They are rather simple in most cases. +Parametric construction +The GTS module provides convenient way of creating surface by vertices, edges and triangles. +Frequently, though, the surface can be conveniently described as surface between polylines in space. For +instance, cylinder is surface between two polygons (closed polylines). The pack.sweptPolylines2gtsSurface +offers the functionality of connecting several polylines with triangulation. +Note: +The implementation of pack.sweptPolylines2gtsSurface is rather simplistic: all polylines must be +of the same length, and they are connected with triangles between points following their indices within +each polyline (not by distance). On the other hand, points can be co-incident, if the threshold parameter +is positive: degenerate triangles with vertices closer that threshold are automatically eliminated. +Manipulating lists efficiently (in terms of code length) requires being familiar with list comprehensions +in python. +Another examples can be found in examples/mill.py (fully parametrized) or examples/funnel.py (with +hardcoded numbers). +Creating interactions +In typical cases, interactions are created during simulations as particles collide. This is done by a Collider +detecting approximate contact between particles and then an IGeomFunctor detecting exact collision. +Some material models (such as the concrete model) rely on initial interaction network which is denser +than geometrical contact of spheres: sphere’s radii as “enlarged” by a dimensionless factor called inter- +action radius (or interaction ratio) to create this initial network. This is done typically in this way (see +examples/concrete/uniax.py for an example): +1. Approximate collision detection is adjusted so that approximate contacts are detected also be- +tween particles within the interaction radius. +This consists in setting value of Bo1_Sphere_- +Aabb.aabbEnlargeFactor to the interaction radius value. +2. The geometry functor (Ig2) would normally say that “there is no contact” if given 2 spheres that +are not in contact. Therefore, the same value as for Bo1_Sphere_Aabb.aabbEnlargeFactor must +be given to it (Ig2_Sphere_Sphere_ScGeom.interactionDetectionFactor ). +Note that only Sphere + Sphere interactions are supported; there is no parameter analogous to +distFactor in Ig2_Facet_Sphere_ScGeom. This is on purpose, since the interaction radius is mean- +ingful in bulk material represented by sphere packing, whereas facets usually represent boundary +conditions which should be exempt from this dense interaction network. +3. Run one single step of the simulation so that the initial network is created. +4. Reset interaction radius in both Bo1 and Ig2 functors to their default value again. +2.2. +User’s manual +83 + +Yade Documentation, Release 3rd ed. +5. Continue the simulation; interactions that are already established will not be deleted (the Law2 +functor in use permitting). +In code, such scenario might look similar to this one (labeling is explained in Labeling things): +intRadius=1.5 +damping=0.05 +O.engines=[ +ForceResetter(), +InsertionSortCollider([ +# enlarge here +Bo1_Sphere_Aabb(aabbEnlargeFactor=intRadius,label='bo1s'), +Bo1_Facet_Aabb(), +]), +InteractionLoop( +[ +# enlarge here +Ig2_Sphere_Sphere_ScGeom(interactionDetectionFactor=intRadius,label='ig2ss'), +Ig2_Facet_Sphere_ScGeom(), +], +[Ip2_CpmMat_CpmMat_CpmPhys()], +[Law2_ScGeom_CpmPhys_Cpm(epsSoft=0)], # deactivated +), +NewtonIntegrator(damping=damping,label='damper'), +] +# run one single step +O.step() +# reset interaction radius to the default value +bo1s.aabbEnlargeFactor=1.0 +ig2ss.interactionDetectionFactor=1.0 +# now continue simulation +O.run() +Individual interactions on demand +It is possible to create an interaction between a pair of particles independently of collision detection using +createInteraction. This function looks for and uses matching Ig2 and Ip2 functors. Interaction will be +created regardless of distance between given particles (by passing a special parameter to the Ig2 functor +to force creation of the interaction even without any geometrical contact). Appropriate constitutive law +should be used to avoid deletion of the interaction at the next simulation step. +Yade [26]: O.materials.append(FrictMat(young=3e10,poisson=.2,density=1000)) +Out[26]: 0 +Yade [27]: O.bodies.append([ +....: +sphere([0,0,0],1), +....: +sphere([0,0,1000],1) +....: ]) +....: +Out[27]: [0, 1] +# only add InteractionLoop, no other engines are needed now +Yade [28]: O.engines=[ +....: +InteractionLoop( +....: +[Ig2_Sphere_Sphere_ScGeom(),], +....: +[Ip2_FrictMat_FrictMat_FrictPhys()], +....: +[] # not needed now +(continues on next page) +84 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +(continued from previous page) +....: +) +....: ] +....: +Yade [29]: i=createInteraction(0,1) +# created by functors in InteractionLoop +Yade [30]: i.geom, i.phys +Out[30]: (, ) +This method will be rather slow if many interactions are to be created (the functor lookup will be repeated +for each of them). +In such case, ask on yade-dev@lists.launchpad.net to have the createInteraction +function accept list of pairs id’s as well. +Base engines +A typical DEM simulation in Yade does at least the following at each step (see Function components for +details): +1. Reset forces from previous step +2. Detect new collisions +3. Handle interactions +4. Apply forces and update positions of particles +Each of these points corresponds to one or several engines: +O.engines=[ +ForceResetter(), +# reset forces +InsertionSortCollider([...]), +# approximate collision detection +InteractionLoop([...],[...],[...]) # handle interactions +NewtonIntegrator() +# apply forces and update positions +] +The order of engines is important. In majority of cases, you will put any additional engine after Inter- +actionLoop: +• if it applies force, it should come before NewtonIntegrator, otherwise the force will never be effective. +• if it makes use of bodies’ positions, it should also come before NewtonIntegrator, otherwise, posi- +tions at the next step will be used (this might not be critical in many cases, such as output for +visualization with VTKRecorder). +The O.engines sequence must be always assigned at once (the reason is in the fact that although engines +themselves are passed by reference, the sequence is copied from c++ to Python or from Python to c++). +This includes modifying an existing O.engines; therefore +O.engines.append(SomeEngine()) # wrong +will not work; +O.engines=O.engines+[SomeEngine()] # ok +must be used instead. For inserting an engine after position #2 (for example), use python slice notation: +O.engines=O.engines[:2]+[SomeEngine()]+O.engines[2:] +Note: +When Yade starts, O.engines is filled with a reasonable default list, so that it is not strictly +necessary to redefine it when trying simple things. The default scene will handle spheres, boxes, and +2.2. +User’s manual +85 + +Yade Documentation, Release 3rd ed. +facets with frictional properties correctly, and adjusts the timestep dynamically. You can find an example +in examples/simple-scene/simple-scene-default-engines.py. +Functors choice +In the above example, we omited functors, only writing ellipses ... instead. As explained in Dispatchers +and functors, there are 4 kinds of functors and associated dispatchers. User can choose which ones to +use, though the choice must be consistent. +Bo1 functors +Bo1 functors must be chosen depending on the collider in use; they are given directly to the collider +(which internally uses BoundDispatcher). +At this moment (January 2019), the most common choice is InsertionSortCollider, which uses Aabb; +functors creating Aabb must be used in that case. Depending on particle shapes in your simulation, +choose appropriate functors: +O.engines=[..., +InsertionSortCollider([Bo1_Sphere_Aabb(),Bo1_Facet_Aabb()]), +... +] +Using more functors than necessary (such as Bo1_Facet_Aabb if there are no facets in the simulation) +has no performance penalty. On the other hand, missing functors for existing shapes will cause those +bodies to not collide with other bodies (they will freely interpenetrate). +There are other colliders as well, though their usage is only experimental: +• SpatialQuickSortCollider is correctness-reference collider operating on Aabb; it is significantly +slower than InsertionSortCollider. +• PersistentTriangulationCollider only works on spheres; it does not use a BoundDispatcher, as it +operates on spheres directly. +• FlatGridCollider is proof-of-concept grid-based collider, which computes grid positions internally +(no BoundDispatcher either) +Ig2 functors +Ig2 functor choice (all of them derive from IGeomFunctor) depends on +1. shape combinations that should collide; for instance: +InteractionLoop([Ig2_Sphere_Sphere_ScGeom()],[],[]) +will handle collisions for Sphere + Sphere, but not for Facet + Sphere – if that is desired, an +additional functor must be used: +InteractionLoop([ +Ig2_Sphere_Sphere_ScGeom(), +Ig2_Facet_Sphere_ScGeom() +],[],[]) +Again, missing combination will cause given shape combinations to freely interpenetrate one an- +other. There are several possible choices of a functor for each pair, hence they cannot be put into +InsertionSortCollider by default. A common mistake for bodies going through each other is that +the necessary functor was not added. +86 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +2. IGeom type accepted by the Law2 functor (below); it is the first part of functor’s name after Law2 +(for instance, Law2_ScGeom_CpmPhys_Cpm accepts ScGeom). +Ip2 functors +Ip2 functors (deriving from IPhysFunctor) must be chosen depending on +1. Material combinations within the simulation. In most cases, Ip2 functors handle 2 instances of the +same Material class (such as Ip2_FrictMat_FrictMat_FrictPhys for 2 bodies with FrictMat) +2. IPhys accepted by the constitutive law (Law2 functor), which is the second part of the Law2 functor’s +name (e.g. Law2_ScGeom_FrictPhys_CundallStrack accepts FrictPhys) +Note: +Unlike with Bo1 and Ig2 functors, unhandled combination of Materials is an error condition +signaled by an exception. +Law2 functor(s) +Law2 functor was the ultimate criterion for the choice of Ig2 and Ip2 functors; there are no restrictions +on its choice in itself, as it only applies forces without creating new objects. +In most simulations, only one Law2 functor will be in use; it is possible, though, to have several of them, +dispatched based on combination of IGeom and IPhys produced previously by Ig2 and Ip2 functors +respectively (in turn based on combination of Shapes and Materials). +Note: +As in the case of Ip2 functors, receiving a combination of IGeom and IPhys which is not handled +by any Law2 functor is an error. +Warning: +Many Law2 exist in Yade, and new ones can appear at any time. In some cases different +functors are only different implementations of the same contact law (e.g. Law2_ScGeom_FrictPhys_- +CundallStrack and Law2_L3Geom_FrictPhys_ElPerfPl). Also, sometimes, the peculiarity of one +functor may be reproduced as a special case of a more general one. Therefore, for a given constitutive +behavior, the user may have the choice between different functors. It is strongly recommended to +favor the most used and most validated implementation when facing such choice. A list of available +functors classified from mature to unmaintained is updated here to guide this choice. +Examples +Let us give several examples of the chain of created and accepted types. +Basic DEM model +Suppose we want to use the Law2_ScGeom_FrictPhys_CundallStrack constitutive law. We see that +1. the Ig2 functors must create ScGeom. If we have for instance spheres and boxes in the simulation, +we will need functors accepting Sphere + Sphere and Box + Sphere combinations. We don’t want +interactions between boxes themselves (as a matter of fact, there is no such functor anyway). That +gives us Ig2_Sphere_Sphere_ScGeom and Ig2_Box_Sphere_ScGeom. +2. the Ip2 functors should create FrictPhys. Looking at InteractionPhysicsFunctors, there is only +Ip2_FrictMat_FrictMat_FrictPhys. That obliges us to use FrictMat for particles. +The result will be therefore: +2.2. +User’s manual +87 + +Yade Documentation, Release 3rd ed. +InteractionLoop( +[Ig2_Sphere_Sphere_ScGeom(),Ig2_Box_Sphere_ScGeom()], +[Ip2_FrictMat_FrictMat_FrictPhys()], +[Law2_ScGeom_FrictPhys_CundallStrack()] +) +Concrete model +In this case, our goal is to use the Law2_ScGeom_CpmPhys_Cpm constitutive law. +• We use spheres and facets in the simulation, which selects Ig2 functors accepting those types and +producing ScGeom: Ig2_Sphere_Sphere_ScGeom and Ig2_Facet_Sphere_ScGeom. +• We have to use Material which can be used for creating CpmPhys. +We find that CpmPhys is +only created by Ip2_CpmMat_CpmMat_CpmPhys, which determines the choice of CpmMat for +all particles. +Therefore, we will use: +InteractionLoop( +[Ig2_Sphere_Sphere_ScGeom(),Ig2_Facet_Sphere_ScGeom()], +[Ip2_CpmMat_CpmMat_CpmPhys()], +[Law2_ScGeom_CpmPhys_Cpm()] +) +Imposing conditions +In most simulations, it is not desired that all particles float freely in space. There are several ways of +imposing boundary conditions that block movement of all or some particles with regard to global space. +Motion constraints +• Body.dynamic determines whether a body will be accelerated by NewtonIntegrator; it is mandatory +to make it false for bodies with zero mass, where applying non-zero force would result in infinite +displacement. +Facets are case in the point: facet makes them non-dynamic by default, as they have zero volume +and zero mass (this can be changed, by passing dynamic=True to facet or setting Body.dynamic; +setting State.mass to a non-zero value must be done as well). The same is true for wall. +Making sphere non-dynamic is achieved simply by: +b = sphere([x,y,z],radius,dynamic=False) +b.dynamic=True #revert the previous +• State.blockedDOFs permits selective blocking of any of 6 degrees of freedom in global space. For +instance, a sphere can be made to move only in the xy plane by saying: +Yade [31]: O.bodies.append(sphere((0,0,0),1)) +Out[31]: 0 +Yade [32]: O.bodies[0].state.blockedDOFs='zXY' +In contrast to Body.dynamic, blockedDOFs will only block forces (and acceleration) in se- +lected directions. +Actually, +b.dynamic=False is nearly only a shorthand for b.state. +blockedDOFs=='xyzXYZ' . A subtle difference is that the former does reset the velocity components +automaticaly, while the latest does not. If you prescribed linear or angular velocity, they will be +applied regardless of blockedDOFs. It also implies that if the velocity is not zero when degrees of +88 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +freedom are blocked via blockedDOFs assignements, the body will keep moving at the velocity it +has at the time of blocking. The differences are shown below: +Yade [33]: b1 = sphere([0,0,0],1,dynamic=True) +Yade [34]: b1.state.blockedDOFs +Out[34]: '' +Yade [35]: b1.state.vel = Vector3(1,0,0) #we want it to move... +Yade [36]: b1.dynamic = False #... at a constant velocity +Yade [37]: print(b1.state.blockedDOFs, b1.state.vel) +xyzXYZ Vector3(0,0,0) +Yade [38]: # oops, velocity has been reset when setting dynamic=False +Yade [39]: b1.state.vel = (1,0,0) # we can still assign it now +Yade [40]: print(b1.state.blockedDOFs, b1.state.vel) +xyzXYZ Vector3(1,0,0) +Yade [41]: b2 = sphere([0,0,0],1,dynamic=True) #another try +Yade [42]: b2.state.vel = (1,0,0) +Yade [43]: b2.state.blockedDOFs = "xyzXYZ" #this time we assign blockedDOFs directly,␣ +�→velocity is unchanged +Yade [44]: print(b2.state.blockedDOFs, b2.state.vel) +xyzXYZ Vector3(1,0,0) +It might be desirable to constrain motion of some particles constructed from a generated sphere packing, +following some condition, such as being at the bottom of a specimen; this can be done by looping over +all bodies with a conditional: +for b in O.bodies: +# block all particles with z coord below .5: +if b.state.pos[2]<.5: b.dynamic=False +Arbitrary spatial predicates introduced above can be expoited here as well: +from yade import pack +pred=pack.inAlignedBox(lowerCorner,upperCorner) +for b in O.bodies: +if not isinstance(b.shape,Sphere): continue # skip non-spheres +# ask the predicate if we are inside +if pred(b.state.pos,b.shape.radius): b.dynamic=False +Imposing motion and forces +Imposed velocity +If a degree of freedom is blocked and a velocity is assigned along that direction (translational or rotational +velocity), then the body will move at constant velocity. This is the simpler and recommended method +to impose the motion of a body. This, for instance, will result in a constant velocity along x (it can still +be freely accelerated along y and z): +2.2. +User’s manual +89 + +Yade Documentation, Release 3rd ed. +O.bodies.append(sphere((0,0,0),1)) +O.bodies[0].state.blockedDOFs='x' +O.bodies[0].state.vel=(10,0,0) +Conversely, modifying the position directly is likely to break Yade’s algorithms, especially those related +to collision detection and contact laws, as they are based on bodies velocities. Therefore, unless you +really know what you are doing, don’t do that for imposing a motion: +O.bodies.append(sphere((0,0,0),1)) +O.bodies[0].state.blockedDOFs='x' +O.bodies[0].state.pos=10*O.dt #REALLY BAD! Don't assign position +Imposed force +Applying a force or a torque on a body is done via functions of the ForceContainer. It is as simple as +this: +O.forces.addF(0,(1,0,0)) #applies for one step +This way, the force applies for one time step only, and is resetted at the beginning of each step. For this +reason, imposing a force at the begining of one step will have no effect at all, since it will be immediatly +resetted. The only way is to place a PyRunner inside the simulation loop. +Applying the force permanently is possible with another function (in this case it does not matter if the +command comes at the begining of the time step): +O.forces.setPermF(0,(1,0,0)) #applies permanently +The force will persist across iterations, until it is overwritten by another call to O.forces.setPermF(id, +f) or erased by O.forces.reset(resetAll=True). The permanent force on a body can be checked with +O.forces.permF(id). +Boundary controllers +Engines deriving from BoundaryController impose boundary conditions during simulation, either di- +rectly, or by influencing several bodies. You are referred to their individual documentation for details, +though you might find interesting in particular +• UniaxialStrainer for applying strain along one axis at constant rate; useful for plotting strain-stress +diagrams for uniaxial loading case. See examples/concrete/uniax.py for an example. +• TriaxialStressController which applies prescribed stress/strain along 3 perpendicular axes on +cuboid-shaped packing using 6 walls (Box objects) +• PeriTriaxController for applying stress/strain along 3 axes independently, for simulations using +periodic boundary conditions (Cell) +Field appliers +Engines deriving from FieldApplier are acting on all particles. The one most used is GravityEngine +applying uniform acceleration field (GravityEngine is deprecated, use NewtonIntegrator.gravity instead). +Partial engines +Engines deriving from PartialEngine define the ids attribute determining bodies which will be affected. +Several of them warrant explicit mention here: +90 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +• TranslationEngine and RotationEngine for applying constant speed linear and rotational motion +on subscribers. +• ForceEngine and TorqueEngine applying given values of force/torque on subscribed bodies at every +step. +• StepDisplacer for applying generalized displacement delta at every timestep; designed for precise +control of motion when testing constitutive laws on 2 particles. +The real value of partial engines is when you need to prescribe a complex type of force or displacement +field. For moving a body at constant velocity or for imposing a single force, the methods explained in +Imposing motion and forces are much simpler. There are several interpolating engines (InterpolatingDi- +rectedForceEngine for applying force with varying magnitude, InterpolatingHelixEngine for applying spi- +ral displacement with varying angular velocity; see examples/test/helix.py and possibly others); writing +a new interpolating engine is rather simple using examples of those that already exist. +Convenience features +Labeling things +Engines and functors can define a label attribute. Whenever the O.engines sequence is modified, python +variables of those names are created/updated; since it happens in the __builtins__ namespaces, these +names are immediately accessible from anywhere. +This was used in Creating interactions to change +interaction radius in multiple functors at once. +Warning: +Make sure you do not use label that will overwrite (or shadow) an object that you already +use under that variable name. Take care not to use syntactically wrong names, such as “er*452” or +“my engine”; only variable names permissible in Python can be used. +Simulation tags +Omega.tags is a dictionary (it behaves like a dictionary, although the implementation in C++ is different) +mapping keys to labels. Contrary to regular python dictionaries that you could create, +• O.tags is saved and loaded with simulation; +• O.tags has some values pre-initialized. +After Yade startup, O.tags contains the following: +Yade [45]: dict(O.tags) # convert to real dictionary +Out[45]: +{'author': 'bchareyre~(bchareyre@HP-ZBook-15-G3)', +'isoTime': '20220726T141512', +'id': '20220726T141512p61547', +'d.id': '20220726T141512p61547', +'id.d': '20220726T141512p61547'} +author Real name, username and machine as obtained from your system at simulation creation +id Unique identifier of this Yade instance (or of the instance which created a loaded simulation). It is +composed of date, time and process number. Useful if you run simulations in parallel and want +to avoid overwriting each other’s outputs; embed O.tags['id'] in output filenames (either as +directory name, or as part of the file’s name itself) to avoid it. This is explained in Separating +output files from jobs in detail. +isoTime Time when simulation was created (with second resolution). +d.id, id.d Simulation description and id joined by period (and vice-versa). Description is used in batch +jobs; in non-batch jobs, these tags are identical to id. +2.2. +User’s manual +91 + +Yade Documentation, Release 3rd ed. +You can add your own tags by simply assigning value, with the restriction that the left-hand side object +must be a string and must not contain =. +Yade [46]: O.tags['anythingThat I lik3']='whatever' +Yade [47]: O.tags['anythingThat I lik3'] +Out[47]: 'whatever' +Saving python variables +Python variable lifetime is limited; in particular, if you save simulation, variables will be lost after +reloading. Yade provides limited support for data persistence for this reason (internally, it uses special +values of O.tags). The functions in question are saveVars and loadVars. +saveVars takes dictionary (variable names and their values) and a mark (identification string for the +variable set); it saves the dictionary inside the simulation. These variables can be re-created (after the +simulation was loaded from a XML file, for instance) in the yade.params.mark namespace by calling +loadVars with the same identification mark: +Yade [48]: a=45; b=pi/3 +Yade [49]: saveVars('ab',a=a,b=b) +# save simulation (we could save to disk just as well) +Yade [49]: O.saveTmp() +Yade [51]: O.loadTmp() +Yade [52]: loadVars('ab') +Yade [53]: yade.params.ab.a +Out[53]: 45 +# import like this +Yade [54]: from yade.params import ab +Yade [55]: ab.a, ab.b +Out[55]: (45, 1.0471975511965976) +# also possible +Yade [56]: from yade.params import * +Yade [57]: ab.a, ab.b +Out[57]: (45, 1.0471975511965976) +Enumeration of variables can be tedious if they are many; creating local scope (which is a function +definition in Python, for instance) can help: +def setGeomVars(): +radius=4 +thickness=22 +p_t=4/3*pi +dim=Vector3(1.23,2.2,3) +# +# define as much as you want here +# it all appears in locals() (and nothing else does) +# +saveVars('geom',loadNow=True,**locals()) +setGeomVars() +(continues on next page) +92 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +(continued from previous page) +from yade.params.geom import * +# use the variables now +Note: +Only types that can be pickled can be passed to saveVars. +2.2.2 Controlling simulation +Tracking variables +Running python code +A special engine PyRunner can be used to periodically call python code, specified via the command +parameter. Periodicity can be controlled by specifying computation time (realPeriod), virtual time +(virtPeriod) or iteration number (iterPeriod). +For instance, to print kinetic energy (using kineticEnergy) every 5 seconds, the following engine will be +put to O.engines: +PyRunner(command="print('kinetic energy',kineticEnergy())",realPeriod=5) +For running more complex commands, it is convenient to define an external function and only call it +from within the engine. Since the command is run in the script’s namespace, functions defined within +scripts can be called. Let us print information on interaction between bodies 0 and 1 periodically: +def intrInfo(id1,id2): +try: +i=O.interactions[id1,id2] +# assuming it is a CpmPhys instance +print (d1,id2,i.phys.sigmaN) +except: +# in case the interaction doesn't exist (yet?) +print("No interaction between",id1,id2) +O.engines=[..., +PyRunner(command="intrInfo(0,1)",realPeriod=5) +] +Warning: +If a function was declared inside a live yade session (ipython) then an error NameError: +name 'intrInfo' is not defined will occur unless python globals() are updated with command +globals().update(locals()) +More useful examples will be given below. +The plot module provides simple interface and storage for tracking various data. Although originally +conceived for plotting only, it is widely used for tracking variables in general. +The data are in plot.data dictionary, which maps variable names to list of their values; the plot.addData +function is used to add them. +Yade [58]: from yade import plot +Yade [59]: plot.data +Out[59]: {} +Yade [60]: plot.addData(sigma=12,eps=1e-4) +(continues on next page) +2.2. +User’s manual +93 + +Yade Documentation, Release 3rd ed. +(continued from previous page) +# not adding sigma will add a NaN automatically +# this assures all variables have the same number of records +Yade [61]: plot.addData(eps=1e-3) +# adds NaNs to already existing sigma and eps columns +Yade [62]: plot.addData(force=1e3) +Yade [63]: plot.data +Out[63]: +{'sigma': [12, nan, nan], +'eps': [0.0001, 0.001, nan], +'force': [nan, nan, 1000.0]} +# retrieve only one column +Yade [64]: plot.data['eps'] +Out[64]: [0.0001, 0.001, nan] +# get maximum eps +Yade [65]: max(plot.data['eps']) +Out[65]: 0.001 +New record is added to all columns at every time plot.addData is called; this assures that lines in different +columns always match. The special value nan or NaN (Not a Number) is inserted to mark the record +invalid. +Note: +It is not possible to have two columns with the same name, since data are stored as a dictionary. +To record data periodically, use PyRunner. This will record the z coordinate and velocity of body #1, +iteration number and simulation time (every 20 iterations): +O.engines=O.engines+[PyRunner(command='myAddData()', iterPeriod=20)] +from yade import plot +def myAddData(): +b=O.bodies[1] +plot.addData(z1=b.state.pos[2], v1=b.state.vel.norm(), i=O.iter, t=O.time) +Note: +Arbitrary string can be used as a column label for plot.data. However if the name has spaces +inside (e.g. my funny column) or is a reserved python keyword (e.g. for) the only way to pass it to +plot.addData is to use a dictionary: +plot.addData(**{'my funny column':1e3, 'for':0.3}) +An exception are columns having leading of trailing whitespaces. They are handled specially in plot.plots +and should not be used (see below). +Labels can be conveniently used to access engines in the myAddData function: +O.engines=[..., +UniaxialStrainer(...,label='strainer') +] +def myAddData(): +plot.addData(sigma=strainer.avgStress,eps=strainer.strain) +In that case, naturally, the labeled object must define attributes which are used (UniaxialStrainer.strain +and UniaxialStrainer.avgStress in this case). +94 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +Plotting variables +Above, we explained how to track variables by storing them using plot.addData. These data can be +readily used for plotting. Yade provides a simple, quick to use, plotting in the plot module. Naturally, +since direct access to underlying data is possible via plot.data, these data can be processed in any other +way. +The plot.plots dictionary is a simple specification of plots. +Keys are x-axis variable, and values are +tuple of y-axis variables, given as strings that were used for plot.addData; each entry in the dictionary +represents a separate figure: +plot.plots={ +'i':('t',), +# plot t(i) +'t':('z1','v1') # z1(t) and v1(t) +} +Actual plot using data in plot.data and plot specification of plot.plots can be triggered by invoking the +plot.plot function. +Live updates of plots +Yade features live-updates of figures during calculations. It is controlled by following settings: +• plot.live - By setting yade.plot.live=True you can watch the plot being updated while the cal- +culations run. Set to False otherwise. +• plot.liveInterval - This is the interval in seconds between the plot updates. +• plot.autozoom - When set to True the plot will be automatically rezoomed. +Controlling line properties +In this subsection let us use a basic complete script like examples/simple-scene/simple-scene-plot.py, +which we will later modify to make the plots prettier. Line of interest from that file is, and generates a +picture presented below: +plot.plots={'i':('t'),'t':('z_sph',None,('v_sph','go-'),'z_sph_half')} +The line plots take an optional second string argument composed of a line color (eg. +'r', 'g' or +'b'), a line style (eg. +'-', '–-' or ':') and a line marker ('o', 's' or 'd'). +A red dotted line +with circle markers is created with ‘ro:’ argument. For a listing of all options please have a look at +http://matplotlib.sourceforge.net/api/pyplot_api.html#matplotlib.pyplot.plot +For example using following plot.plots() command, will produce a following graph: +plot.plots={'i':(('t','xr:'),),'t':(('z_sph','r:'),None,('v_sph','g--'),('z_sph_half','b-.'))} +And this one will produce a following graph: +plot.plots={'i':(('t','xr:'),),'t':(('z_sph','Hr:'),None,('v_sph','+g--'),('z_sph_half','*b-. +�→'))} +Note: +You +can +learn +more +in +matplotlib +tutorial +http://matplotlib.sourceforge.net/users/ +pyplot_tutorial.html +and +documentation +http://matplotlib.sourceforge.net/users/pyplot_tutorial. +html#controlling-line-properties +2.2. +User’s manual +95 + +Yade Documentation, Release 3rd ed. +0.0 +0.5 +1.0 +1.5 +2.0 +1.4 +1.5 +1.6 +1.7 +1.8 +1.9 +2.0 +z_sph +z_sph +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +v_sph,z_sph_half +v_sph +z_sph_half +Fig. 12: Figure generated by examples/simple-scene/simple-scene-plot.py. +96 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +0.0 +0.5 +1.0 +1.5 +2.0 +1.4 +1.5 +1.6 +1.7 +1.8 +1.9 +2.0 +z_sph +z_sph +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +v_sph,z_sph_half +v_sph +z_sph_half +Fig. 13: Figure generated by changing parameters to plot.plots as above. +2.2. +User’s manual +97 + +Yade Documentation, Release 3rd ed. +0.0 +0.5 +1.0 +1.5 +2.0 +1.4 +1.5 +1.6 +1.7 +1.8 +1.9 +2.0 +z_sph +z_sph +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +v_sph,z_sph_half +v_sph +z_sph_half +Fig. 14: Figure generated by changing parameters to plot.plots as above. +98 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +Note: +Please note that there is an extra , in 'i':(('t','xr:'),), otherwise the 'xr:' wouldn’t be +recognized as a line style parameter, but would be treated as an extra data to plot. +Controlling text labels +It is possible to use TeX syntax in plot labels. For example using following two lines in examples/simple- +scene/simple-scene-plot.py, will produce a following picture: +plot.plots={'i':(('t','xr:'),),'t':(('z_sph','r:'),None,('v_sph','g--'),('z_sph_half','b-.'))} +plot.labels={'z_sph':'$z_{sph}$' , 'v_sph':'$v_{sph}$' , 'z_sph_half':'$z_{sph}/2$'} +0.0 +0.5 +1.0 +1.5 +2.0 +1.4 +1.5 +1.6 +1.7 +1.8 +1.9 +2.0 +zsph +zsph +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +vsph,zsph/2 +vsph +zsph/2 +Fig. 15: Figure generated by examples/simple-scene/simple-scene-plot.py, with TeX labels. +Greek letters are simply a '$\alpha$', '$\beta$' etc. in those labels. To change the font style a +following command could be used: +yade.plot.matplotlib.rc('mathtext', fontset='stixsans') +But this is not part of yade, but a part of matplotlib, and if you want something more complex you really +should have a look at matplotlib users manual http://matplotlib.sourceforge.net/users/index.html +Multiple figures +Since plot.plots is a dictionary, multiple entries with the same key (x-axis variable) would not be possible, +since they overwrite each other: +2.2. +User’s manual +99 + +Yade Documentation, Release 3rd ed. +Yade [66]: plot.plots={ +....: +'i':('t',), +....: +'i':('z1','v1') +....: } +....: +Yade [67]: plot.plots +Out[67]: {'i': ('z1', 'v1')} +You can, however, distinguish them by prepending/appending space to the x-axis variable, which will be +removed automatically when looking for the variable in plot.data – both x-axes will use the i column: +Yade [68]: plot.plots={ +....: +'i':('t',), +....: +'i ':('z1','v1') # note the space in 'i ' +....: } +....: +Yade [69]: plot.plots +Out[69]: {'i': ('t',), 'i ': ('z1', 'v1')} +Split y1 y2 axes +To avoid big range differences on the y axis, it is possible to have left and right y axes separate (like +axes x1y2 in gnuplot). This is achieved by inserting None to the plot specifier; variables coming before +will be plot normally (on the left y-axis), while those after will appear on the right: +plot.plots={'i':('z1',None,'v1')} +Exporting +Plots +and +data +can +be +exported +to +external +files +for +later +post-processing +in +Gnuplot + via that plot.saveGnuplot function. +Note that all data you added via +plot.addData is saved - even data that you don’t plot live during simulation. By editing the gener- +ated .gnuplot file you can plot any of the added Data afterwards. +• Data file is saved (compressed using bzip2) separately from the gnuplot file, so any other programs +can be used to process them. In particular, the numpy.genfromtxt (documented here) can be +useful to import those data back to python; the decompression happens automatically. +• The gnuplot file can be run through gnuplot to produce the figure; see plot.saveGnuplot documen- +tation for details. +For post-processing with other tools than gnuplot, saved data can also be exported in another kind of +text file with plot.saveDataTxt. +Stop conditions +For simulations with a pre-determined number of steps, it can be prescribed: +# absolute iteration number +O.stopAtIter=35466 +O.run() +O.wait() +or +100 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +# number of iterations to run from now +O.run(35466,True) # wait=True +causes the simulation to run 35466 iterations, then stopping. +Frequently, decisions have to be made based on evolution of the simulation itself, which is not yet known. +In such case, a function checking some specific condition is called periodically; if the condition is satisfied, +O.pause or other functions can be called to stop the stimulation. See documentation for Omega.run, +Omega.pause, Omega.step, Omega.stopAtIter for details. +For simulations that seek static equilibrium, the unbalancedForce can provide a useful metrics (see its +documentation for details); for a desired value of 1e-2 or less, for instance, we can use: +def checkUnbalanced(): +if unbalancedForce<1e-2: O.pause() +O.engines=O.engines+[PyRunner(command="checkUnbalanced()",iterPeriod=100)] +# this would work as well, without the function defined apart: +# +PyRunner(command="if unablancedForce<1e-2: O.pause()",iterPeriod=100) +O.run(); O.wait() +# will continue after O.pause() will have been called +Arbitrary functions can be periodically checked, and they can also use history of variables tracked via +plot.addData. For example, this is a simplified version of damage control in examples/concrete/uniax.py; +it stops when current stress is lower than half of the peak stress: +O.engines=[..., +UniaxialStrainer=(...,label='strainer'), +PyRunner(command='myAddData()',iterPeriod=100), +PyRunner(command='stopIfDamaged()',iterPeriod=100) +] +def myAddData(): +plot.addData(t=O.time,eps=strainer.strain,sigma=strainer.stress) +def stopIfDamaged(): +currSig=plot.data['sigma'][-1] # last sigma value +maxSig=max(plot.data['sigma']) # maximum sigma value +# print something in any case, so that we know what is happening +print(plot.data['eps'][-1],currSig) +if currSig<.5*maxSig: +print("Damaged, stopping") +print('gnuplot',plot.saveGnuplot(O.tags['id'])) +import sys +sys.exit(0) +O.run(); O.wait() +# this place is never reached, since we call sys.exit(0) directly +Checkpoints +Occasionally, it is useful to revert to simulation at some past point and continue from it with different +parameters. For instance, tension/compression test will use the same initial state but load it in 2 different +directions. Two functions, Omega.saveTmp and Omega.loadTmp are provided for this purpose; memory +is used as storage medium, which means that saving is faster, and also that the simulation will disappear +when Yade finishes. +2.2. +User’s manual +101 + +Yade Documentation, Release 3rd ed. +O.saveTmp() +# do something +O.saveTmp('foo') +O.loadTmp() +# loads the first state +O.loadTmp('foo') # loads the second state +Warning: +O.loadTmp cannot be called from inside an engine, since before loading a simulation, the +old one must finish the current iteration; it would lead to deadlock, since O.loadTmp would wait for +the current iteration to finish, while the current iteration would be blocked on O.loadTmp. +A special trick must be used: a separate function to be run after the current iteration is defined and +is invoked from an independent thread launched only for that purpose: +O.engines=[...,PyRunner('myFunc()',iterPeriod=345)] +def myFunc(): +if someCondition: +import thread +# the () are arguments passed to the function +thread.start_new_thread(afterIterFunc,()) +def afterIterFunc(): +O.pause(); O.wait() # wait till the iteration really finishes +O.loadTmp() +O.saveTmp() +O.run() +Remote control +Yade can be controlled remotely over network. At yade startup, the following lines appear, among other +messages: +TCP python prompt on localhost:9000, auth cookie `dcekyu' +TCP info provider on localhost:21000 +They inform about 2 ports on which connection of 2 different kind is accepted. +Python prompt +TCP python prompt is telnet server with authenticated connection, providing full python command-line. +It listens on port 9000, or higher if already occupied (by another yade instance, for example). +Using the authentication cookie, connection can be made using telnet: +$ telnet localhost 9000 +Trying 127.0.0.1... +Connected to localhost. +Escape character is '^]'. +Enter auth cookie: dcekyu +__ +__ +____ +__ +_____ ____ ____ +\ \ / /_ _| +_ \ +___ +___ +/ / |_ +_/ ___| +_ \ +\ V / _` | | | |/ _ \ +/ _ \ / / +| || | +| |_) | +| | (_| | |_| | +__/ | (_) / / +| || |___| +__/ +|_|\__,_|____/ \___| +\___/_/ +|_| \____|_| +(connected from 127.0.0.1:40372) +>>> +102 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +The python pseudo-prompt >>> lets you write commands to manipulate simulation in variety of ways as +usual. Two things to notice: +1. The new python interpreter (>>>) lives in a namespace separate from Yade [1]: command-line. +For your convenience, from yade import * is run in the new python instance first, but local and +global variables are not accessible (only builtins are). +2. The (fake) >>> interpreter does not have rich interactive feature of IPython, which handles the +usual command-line Yade [1]:; therefore, you will have no command history, ? help and so on. +Note: +By giving access to python interpreter, full control of the system (including reading user’s files) +is possible. For this reason, connection is only allowed from localhost, not over network remotely. +Of course you can log into the system via SSH over network to get remote access. +Warning: +Authentication cookie is trivial to crack via bruteforce attack. Although the listener +stalls for 5 seconds after every failed login attempt (and disconnects), the cookie could be guessed by +trial-and-error during very long simulations on a shared computer. +Info provider +TCP Info provider listens at port 21000 (or higher) and returns some basic information about current +simulation upon connection; the connection terminates immediately afterwards. +The information is +python dictionary represented as string (serialized) using standard pickle module. +This functionality is used by the batch system (described below) to be informed about individual sim- +ulation progress and estimated times. If you want to access this information yourself, you can study +core/main/yade-batch.in for details. +Batch queuing and execution (yade-batch) +Yade features light-weight system for running one simulation with different parameters; it handles as- +signment of parameter values to python variables in simulation script, scheduling jobs based on number +of available and required cores and more. The whole batch consists of 2 files: +simulation script regular Yade script, which calls readParamsFromTable to obtain parameters from +parameter table. In order to make the script runnable outside the batch, readParamsFromTable +takes default values of parameters, which might be overridden from the parameter table. +readParamsFromTable knows which parameter file and which line to read by inspecting the PARAM_- +TABLE environment variable, set by the batch system. +parameter table simple text file, each line representing one parameter set. This file is read by read- +ParamsFromTable (using TableParamReader class), called from simulation script, as explained +above. For better reading of the text file you can make use of tabulators, these will be ignored +by readParamsFromTable. Parameters are not restricted to numerical values. You can also make +use of strings by "quoting" them (' ' may also be used instead of " "). This can be useful for +nominal parameters. +The batch can be run as +yade-batch parameters.table simulation.py +and it will intelligently run one simulation for each parameter table line. A minimal example is found in +examples/test/batch/params.table and examples/test/batch/sim.py, another example follows. +2.2. +User’s manual +103 + +Yade Documentation, Release 3rd ed. +Example +Suppose we want to study influence of parameters density and initialVelocity on position of a sphere +falling on fixed box. We create parameter table like this: +description density initialVelocity # first non-empty line are column headings +reference +2400 +10 +hi_v += +20 +# = to use value from previous line +lo_v += +5 +# comments are allowed +hi_rho +5000 +10 +# blank lines as well: +hi_rho_v += +20 +hi_rh0_lo_v += +5 +Each line give one combination of these 2 parameters and assigns (which is optional) a description of +this simulation. +In the simulation file, we read parameters from table, at the beginning of the script; each parameter has +default value, which is used if not specified in the parameters file: +readParamsFromTable( +gravity=-9.81, +density=2400, +initialVelocity=20, +noTableOk=True +# use default values if not run in batch +) +from yade.params.table import * +print(gravity, density, initialVelocity) +after the call to readParamsFromTable, corresponding python variables are created in the yade.params. +table module and can be readily used in the script, e.g. +GravityEngine(gravity=(0,0,gravity)) +Let us see what happens when running the batch: +$ yade-batch batch.table batch.py +Will run `/usr/local/bin/yade-trunk' on `batch.py' with nice value 10, output redirected to␣ +�→`batch.@.log', 4 jobs at a time. +Will use table `batch.table', with available lines 2, 3, 4, 5, 6, 7. +Will use lines +2 (reference), 3 (hi_v), 4 (lo_v), 5 (hi_rho), 6 (hi_rho_v), 7 (hi_rh0_lo_v). +Master process pid 7030 +These lines inform us about general batch information: nice level, log file names, how many cores will be +used (4); table name, and line numbers that contain parameters; finally, which lines will be used; master +PID is useful for killing (stopping) the whole batch with the kill command. +Job summary: +#0 (reference/4): PARAM_TABLE=batch.table:2 DISPLAY= +/usr/local/bin/yade-trunk --threads=4␣ +�→--nice=10 -x batch.py > batch.reference.log 2>&1 +#1 (hi_v/4): PARAM_TABLE=batch.table:3 DISPLAY= +/usr/local/bin/yade-trunk --threads=4 -- +�→nice=10 -x batch.py > batch.hi_v.log 2>&1 +#2 (lo_v/4): PARAM_TABLE=batch.table:4 DISPLAY= +/usr/local/bin/yade-trunk --threads=4 -- +�→nice=10 -x batch.py > batch.lo_v.log 2>&1 +#3 (hi_rho/4): PARAM_TABLE=batch.table:5 DISPLAY= +/usr/local/bin/yade-trunk --threads=4 -- +�→nice=10 -x batch.py > batch.hi_rho.log 2>&1 +#4 (hi_rho_v/4): PARAM_TABLE=batch.table:6 DISPLAY= +/usr/local/bin/yade-trunk --threads=4 - +�→-nice=10 -x batch.py > batch.hi_rho_v.log 2>&1 +#5 (hi_rh0_lo_v/4): PARAM_TABLE=batch.table:7 DISPLAY= +/usr/local/bin/yade-trunk -- +�→threads=4 --nice=10 -x batch.py > batch.hi_rh0_lo_v.log 2>&1 +104 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +displays all jobs with command-lines that will be run for each of them. At this moment, the batch starts +to be run. +#0 (reference/4) started on Tue Apr 13 13:59:32 2010 +#0 (reference/4) done +(exit status 0), duration 00:00:01, log batch.reference.log +#1 (hi_v/4) started on Tue Apr 13 13:59:34 2010 +#1 (hi_v/4) done +(exit status 0), duration 00:00:01, log batch.hi_v.log +#2 (lo_v/4) started on Tue Apr 13 13:59:35 2010 +#2 (lo_v/4) done +(exit status 0), duration 00:00:01, log batch.lo_v.log +#3 (hi_rho/4) started on Tue Apr 13 13:59:37 2010 +#3 (hi_rho/4) done +(exit status 0), duration 00:00:01, log batch.hi_rho.log +#4 (hi_rho_v/4) started on Tue Apr 13 13:59:38 2010 +#4 (hi_rho_v/4) done +(exit status 0), duration 00:00:01, log batch.hi_rho_v.log +#5 (hi_rh0_lo_v/4) started on Tue Apr 13 13:59:40 2010 +#5 (hi_rh0_lo_v/4) done +(exit status 0), duration 00:00:01, log batch.hi_rh0_lo_v.log +information about job status changes is being printed, until: +All jobs finished, total time +00:00:08 +Log files: +batch.reference.log batch.hi_v.log batch.lo_v.log batch.hi_rho.log batch.hi_rho_v.log batch.hi_ +�→rh0_lo_v.log +Bye. +Separating output files from jobs +As one might output data to external files during simulation (using classes such as VTKRecorder), it is +important to name files in such way that they are not overwritten by next (or concurrent) job in the same +batch. A special tag O.tags['id'] is provided for such purposes: it is comprised of date, time and PID, +which makes it always unique (e.g. 20100413T144723p7625); additional advantage is that alphabetical +order of the id tag is also chronological. To add the used parameter set or the description of the job, if +set, you could add O.tags[‘params’] to the filename. +For smaller simulations, prepending all output file names with O.tags['id'] can be sufficient: +saveGnuplot(O.tags['id']) +For larger simulations, it is advisable to create separate directory of that name first, putting all files +inside afterwards: +os.mkdir(O.tags['id']) +O.engines=[ +# … +VTKRecorder(fileName=O.tags['id']+'/'+'vtk'), +# … +] +# … +O.saveGnuplot(O.tags['id']+'/'+'graph1') +Controlling parallel computation +Default total number of available cores is determined from /proc/cpuinfo (provided by Linux kernel); +in addition, if OMP_NUM_THREADS environment variable is set, minimum of these two is taken. +The +-j/--jobs option can be used to override this number. +By default, each job uses all available cores for itself, which causes jobs to be effectively run in parallel. +Number of cores per job can be globally changed via the --job-threads option. +Table column named !OMP_NUM_THREADS (! prepended to column generally means to assign environment +variable, rather than python variable) controls number of threads for each job separately, if it exists. +2.2. +User’s manual +105 + +Yade Documentation, Release 3rd ed. +If number of cores for a job exceeds total number of cores, warning is issued and only the total number +of cores is used instead. +Merging gnuplot from individual jobs +Frequently, it is desirable to obtain single figure for all jobs in the batch, for comparison purposes. +Somewhat heuristic way for this functionality is provided by the batch system. yade-batch must be run +with the --gnuplot option, specifying some file name that will be used for the merged figure: +yade-trunk --gnuplot merged.gnuplot batch.table batch.py +Data are collected in usual way during the simulation (using plot.addData) and saved to gnuplot file via +plot.saveGnuplot (it creates 2 files: gnuplot command file and compressed data file). The batch system +scans, once the job is finished, log file for line of the form gnuplot [something]. Therefore, in order to +print this magic line we put: +print('gnuplot',plot.saveGnuplot(O.tags['id'])) +and the end of the script (even after waitIfBatch()) , which prints: +gnuplot 20100413T144723p7625.gnuplot +to the output (redirected to log file). +This file itself contains single graph: +Fig. 16: Figure from single job in the batch. +At the end, the batch system knows about all gnuplot files and tries to merge them together, by assembling +the merged.gnuplot file. +106 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +Fig. 17: Merged figure from all jobs in the batch. Note that labels are prepended by job description to +make lines distinguishable. +2.2. +User’s manual +107 + +Yade Documentation, Release 3rd ed. +HTTP overview +While job is running, the batch system presents progress via simple HTTP server running at port 9080, +which can be acessed from a regular web browser (or e.g. lynx for a terminal usage) by requesting the +http://localhost:9080 URL. This page can be accessed remotely over network as well. +Fig. 18: Summary page available at port 9080 as batch is processed (updates every 5 seconds automati- +cally). Possible job statuses are pending, running, done, failed. +Batch execution on Job-based clusters (OAR) +On High Performance Computation clusters with a scheduling system, the following script might be use- +ful. Exactly like yade-batch, it handles assignemnt of parameters value to python variables in simulation +script from a parameter table, and job submission. This script is written for oar-based system , and may +be extended to others ones. On those system, usually, a job can’t run forever and has a specific duration +allocation. The whole job submission consists of 3 files: +Simulation script: Regular Yade script, which calls readParamsFromTable to obtain parameters from +parameter table. In order to make the script runnable outside the batch, readParamsFromTable +takes default values of parameters, which might be overridden from the parameter table. +readParamsFromTable knows which parameter file and which line to read by inspecting the PARAM_- +TABLE environment variable, set by the batch system. +Parameter table: Simple text file, each line representing one parameter set. +This file is read by +readParamsFromTable (using TableParamReader class), called from simulation script, as explained +above. For better reading of the text file you can make use of tabulators, these will be ignored +by readParamsFromTable. Parameters are not restricted to numerical values. You can also make +use of strings by "quoting" them (' ' may also be used instead of " "). This can be useful for +nominal parameters. +108 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +Job script: Bash script, which calls yade on computing nodes. This script eventually creates temp +folders, save data to storage server etc. The script must be formatted as a template where some +tags will be replaced by specific values at the execution time: +• __YADE_COMMAND__ will be replaced by the actual yade run command +• __YADE_LOGFILE__ will be replaced by the log file path (output to stdout) +• __YADE_ERRFILE__ will be replaced by the error file path (output to stderr) +• __YADE_JOBNO__ will be replaced by an identifier composed as (launch script pid)-(job order) +• __YADE_JOBID__ will be replaced by an identifier composed of all parameters values +The batch can be run as +yade-oar --oar-project= --oar-script=job.sh --oar-walltime=hh:mm:ss␣ +�→parameters.table simulation.py +and it will generate one launch script and submit one job for each parameter table line. A minimal +example is found in examples/oar/params.table examples/oar/job.sh and examples/oar/sim.py. +Note: You have to specify either –oar-walltime or a !WALLTIME column in params.table. !WALLTIME +will override –oar-walltime +Warning: +yade-oar is not compiled by default. +Use -DENABLE_OAR=1 option to cmake to +enable it. +2.2.3 Postprocessing +3d rendering & videos +There are multiple ways to produce a video of simulation: +1. Capture screen output (the 3d rendering window) during the simulation − there are tools available +for that (such as Istanbul or RecordMyDesktop, which are also packaged for most Linux distribu- +tions). The output is “what you see is what you get”, with all the advantages and disadvantages. +2. Periodic frame snapshot using SnapshotEngine (see examples/test/force-network-video.py, exam- +ples/bulldozer/bulldozer.py or examples/test/beam-l6geom.py for a complete example): +O.engines=[ +#... +SnapshotEngine(iterPeriod=100,fileBase='/tmp/bulldozer-',viewNo=0,label='snapshooter') +] +which will save numbered files like /tmp/bulldozer-0000.png. These files can be processed ex- +ternally (with mencoder and similar tools) or directly with the makeVideo: +makeVideo(frameSpec,out,renameNotOverwrite=True,fps=24,kbps=6000,bps=None) +The video is encoded using the default mencoder codec (mpeg4). +3. Specialized post-processing tools, notably Paraview. This is described in more detail in the follow- +ing section. +2.2. +User’s manual +109 + +Yade Documentation, Release 3rd ed. +Paraview +Saving data during the simulation +Paraview is based on the Visualization Toolkit, which defines formats for saving various types of data. +One of them (with the .vtu extension) can be written by a special engine VTKRecorder. It is added to +the simulation loop: +O.engines=[ +# ... +VTKRecorder(iterPeriod=100,recorders=['spheres','facets','colors'],fileName='/tmp/p1-') +] +• iterPeriod determines how often to save simulation data (besides iterPeriod, you can also use +virtPeriod or realPeriod). If the period is too high (and data are saved only few times), the video +will have few frames. +• fileName is the prefix for files being saved. +In this case, output files will be named /tmp/ +p1-spheres.0.vtu and /tmp/p1-facets.0.vtu, where the number is the number of iteration; +many files are created, putting them in a separate directory is advisable. +• recorders determines what data to save +export.VTKExporter plays a similar role, with the difference that it is more flexible. It will save any user +defined variable associated to the bodies. +Loading data into Paraview +All sets of files (spheres, facets, …) must be opened one-by-one in Paraview. +The open dialogue +automatically collapses numbered files in one, making it easy to select all of them: +Click on the “Apply” button in the “Object inspector” sub-window to make loaded objects visible. You +can see tree of displayed objects in the “Pipeline browser”: +Rendering spherical particles. Glyphs +Spheres will only appear as points. To make them look as spheres, you have to add “glyph” to the +p1-spheres.* item in the pipeline using the +icon. Then set (in the Object inspector) +110 +Chapter 2. +Yade for users + +Look in: +/tmp/ +Home +Filename +orbit-vaclav +pulse-cCpZoyohzlBC +ssh-ngGvMp1467 +virtual-vaclav.ezXPXx +after-O.periodic=False.png +田 +pl-facets...vtu +甲 +pl-spheres...vtu +periodic-interactions.png +File name: +p1-facet...vtu +OK +Files of type: +ParaView Files (*.d3plot *.k *.Isdyna *.pvd *.vtp *.vtu +CancelYade Documentation, Release 3rd ed. +2.2. +User’s manual +111 + +File +View +Sources +Filters +Animatic +产 +? +珍 + Solid Color +中 +Pipeline Browser +区 + builtin: +p1-facets.* +pl-spheres.* +object Inspector +回区 +Properties +Display +Information +APPK +Reset +× Delete +× Cell/Point Array Status +x oradii +x o colorYade Documentation, Release 3rd ed. +• “Glyph type” to Sphere +• “Radius” to 1 +• “Scale mode” to Scalar (Scalar is set above to be the radii value saved in the file, therefore spheres +with radius 1 will be scaled by their true radius) +• “Set scale factor” to 1 +• optionally uncheck “Mask points” and “Random mode” (they make some particles not to be ren- +dered for performance reasons, controlled by the “Maximum Number of Points”) +After clicking “Apply”, spheres will appear. They will be rendered over the original white points, which +you can disable by clicking on the eye icon next to p1-spheres.* in the Pipeline browser. +Rendering spherical particles. PointSprite +Another opportunity to display spheres is by using PointSprite plugin. This technique requires much +less RAM in comparison to Glyphs. +• “Tools -> Manage Plugins” +• “PointSprite_Plugin -> Load selected -> Close” +• Load VTU-files +• “Representation -> Point Sprite” +• “Point Sprite -> Scale By -> radii” +• “Edit Radius Transfer Function -> Proportional -> Multiplier = 1.0 -> Close” +Rendering interactions as force chain +Data saved by VTKRecorder (the steps below generates cones rather than tubes) or export. +VTKExporter(...).exportInteractions(what=dict(forceN='i.phys.normalForce.norm()')) (the +steps below generates per interaction tubes with constant radius): +• Load interactions VTP or VTK files +• Filters -> Cell Data To Point Data +• Filters -> Tube +• Set color by “forceN” +• Set “Vary Radius” to “By Scalar” +• Set “Radius” and “Radius Factor” such that the result looks OK (in 3D postprocessing tutorial +script, Radius=0.0005 and Radius Factor=100 looks reasonably) +Facet transparency +If you want to make facet objects transparent, select p1-facets.* in the Pipeline browser, then go to +the Object inspector on the Display tab. Under “Style”, you can set the “Opacity” value to something +smaller than 1. +Animation +You can move between frames (snapshots that were saved) via the “Animation” menu. After setting the +view angle, zoom etc to your satisfaction, the animation can be saved with File/Save animation. +112 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +Micro-stress and micro-strain +It is sometimes useful to visualize a DEM simulation through equivalent strain fields or stress fields. This +is possible with TesselationWrapper. This class handles the triangulation of spheres in a scene, build +tesselation on request, and give access to computed quantities: volume, porosity and local deformation for +each sphere. The definition of microstrain and microstress is at the scale of particle-centered subdomains +shown below, as explained in [Catalano2014a] . +Micro-strain +Below is an output of the defToVtk function visualized with paraview (in this case Yade’s Tessela- +tionWrapper was used to process experimental data obtained on sand by Edward Ando at Grenoble +University, 3SR lab.). The output is visualized with paraview, as explained in the previous section. +Similar results can be generated from simulations: +tt=TriaxialTest() +tt.generate("test.yade") +O.load("test.yade") +O.run(100,True) +TW=TesselationWrapper() +TW.triangulate() +#compute regular Delaunay triangulation, don’t construct tesselation +TW.computeVolumes() +#will silently tesselate the packing, then compute volume of each␣ +�→Voronoi cell +TW.volume(10) +#get volume associated to sphere of id 10 +TW.setState(0) +#store current positions internaly for later use as the "0" state +O.run(100,True) +#make particles move a little (let's hope they will!) +TW.setState(1) +#store current positions internaly in the "1" (deformed) state +#Now we can define strain by comparing states 0 and 1, and average them at the particles scale +TW.defToVtk("strain.vtk") +Micro-stress +Stress fields can be generated by combining the volume returned by TesselationWrapper to per-particle +stress given by bodyStressTensors. Since the stress σ from bodyStressTensor implies a division by the +2.2. +User’s manual +113 + +(a +(b)Yade Documentation, Release 3rd ed. +volume Vb of the solid particle, one has to re-normalize it in order to obtain the micro-stress as defined +in [Catalano2014a] (equation 39 therein), i.e. σk = σk × Vk +b/Vk +σ where Vk +σ is the volume assigned to +particle k in the tesselation. For instance: +#"b" being a body +TW=TesselationWrapper() +TW.setState() +TW.computeVolumes() +s=bodyStressTensors() +stress = s[b.id]*4.*pi/3.*b.shape.radius**3/TW.volume(b.id) +As any other value, the stress can be exported to a vtk file for display in Paraview using ex- +port.VTKExporter. +2.2.4 Python specialties and tricks +Importing Yade in other Python applications +Yade can be imported in other Python applications. To do so, you need somehow to make yade executable +.py extended. The easiest way is to create a symbolic link, i.e. (suppose your Yade executable file is +called “yade-trunk” and you want make it “yadeimport.py”): +$ cd /path/where/you/want/yadeimport +$ ln -s /path/to/yade/executable/yade-trunk yadeimport.py +Then you need to make your yadeimport.py findable by Python. +You can export PYTHONPATH +environment variable, or simply use sys.path directly in Python script: +import sys +sys.path.append('/path/where/you/want/yadeimport') +from yadeimport import * +print(Matrix3(1,2,3, 4,5,6, 7,8,9)) +print(O.bodies) +# any other Yade code +2.2.5 Extending Yade +• new particle shape +114 +Chapter 2. +Yade for users + +Strain_devia +0.478597 +0.4 +0.3 +0.2 +0.000247Yade Documentation, Release 3rd ed. +• new constitutive law +2.2.6 Troubleshooting +Crashes +It is possible that you encounter crash of Yade, i.e. Yade terminates with error message such as +Segmentation fault (core dumped) +without further explanation. Frequent causes of such conditions are +• program error in Yade itself; +• fatal condition in your particular simulation (such as impossible dispatch); +• problem with graphics card driver. +Try to reproduce the error (run the same script) with debug-enabled version of Yade. Debugger will +be automatically launched at crash, showing backtrace of the code (in this case, we triggered crash by +hand): +Yade [1]: import os,signal +Yade [2]: os.kill(os.getpid(),signal.SIGSEGV) +SIGSEGV/SIGABRT handler called; gdb batch file is `/tmp/yade-YwtfRY/tmp-0' +GNU gdb (GDB) 7.1-ubuntu +Copyright (C) 2010 Free Software Foundation, Inc. +License GPLv3+: GNU GPL version 3 or later +This is free software: you are free to change and redistribute it. +There is NO WARRANTY, to the extent permitted by law. +Type "show copying" +and "show warranty" for details. +This GDB was configured as "x86_64-linux-gnu". +For bug reporting instructions, please see: +. +[Thread debugging using libthread_db enabled] +[New Thread 0x7f0fb1268710 (LWP 16471)] +[New Thread 0x7f0fb29f2710 (LWP 16470)] +[New Thread 0x7f0fb31f3710 (LWP 16469)] +… +What looks as cryptic message is valuable information for developers to locate source of the bug. In +particular, there is (usually) line ; lines below it are source of the bug (at +least very likely so): +Thread 1 (Thread 0x7f0fcee53700 (LWP 16465)): +#0 +0x00007f0fcd8f4f7d in __libc_waitpid (pid=16497, stat_loc=,␣ +�→options=0) at ../sysdeps/unix/sysv/linux/waitpid.c:41 +#1 +0x00007f0fcd88c7e9 in do_system (line=) at ../sysdeps/posix/system. +�→c:149 +#2 +0x00007f0fcd88cb20 in __libc_system (line=) at ../sysdeps/posix/ +�→system.c:190 +#3 +0x00007f0fcd0b4b23 in crashHandler (sig=11) at core/main/pyboot.cpp:45 +#4 + +#5 +0x00007f0fcd87ed57 in kill () at ../sysdeps/unix/syscall-template.S:82 +#6 +0x000000000051336d in posix_kill (self=, args=)␣ +�→at ../Modules/posixmodule.c:4046 +#7 +0x00000000004a7c5e in call_function (f=Frame 0x1c54620, for file , line 1, +�→ in (), throwflag=) at ../Python/ceval.c:3750 +#8 +PyEval_EvalFrameEx (f=Frame 0x1c54620, for file , line 1, in (),␣ +�→throwflag=) at ../Python/ceval.c:2412 +2.2. +User’s manual +115 + +Yade Documentation, Release 3rd ed. +If you think this might be error in Yade, file a bug report as explained below. Do not forget to attach full +yade output from terminal, including startup messages and debugger output – select with right mouse +button, with middle button paste the bugreport to a file and attach it. Attach your simulation script as +well. +Reporting bugs +Bugs are general name for defects (functionality shortcomings, misdocumentation, crashes) or feature +requests. They are tracked at https://gitlab.com/yade-dev/trunk/issues. +When reporting a new bug, be as specific as possible; state version of yade you use, system version and +the output of printAllVersions(), as explained in the above section on crashes. +Getting help +Questions and answers +Hint: +Please use Launchpad interface at https://answers.launchpad.net/yade/ for asking questions +about Yade. +In case you’re not familiar with computer oriented discussion lists, please read this wiki page (a Yade- +oriented and shortened version of How To Ask Questions The Smart Way) before posting, in order to +increase your chances getting help. Do not forget to state what version of Yade you use (shown when you +start Yade, or even better as printed by function libVersions.printAllVersions), whether you installed it +from source code or a package, what operating system (such as Ubuntu 18.04), and if you have done any +local modifications to source code in case of compiled version. +Mailing lists +In addition to the Q&A Launchpad interface, Yade has two mailing-lists. +Both are hosted at http: +//www.launchpad.net and before posting, you must register to Launchpad and subscribe to the list by +adding yourself to “team” of the same name running the list. +yade-users@lists.launchpad.net is a general discussion list for all Yade users. Add yourself to yade- +users team so that you can post messages. List archives: +• https://lists.launchpad.net/yade-users/ +• http://www.mail-archive.com/yade-users@lists.launchpad.net/ +yade-dev@lists.launchpad.net is for discussions about Yade development; you must be member of +yade-dev team to post. This list is archived in two places: +• https://lists.launchpad.net/yade-dev/ +• http://www.mail-archive.com/yade-dev@lists.launchpad.net/ +Wiki +http://www.yade-dem.org/wiki/ +Private and/or paid support +You might contact developers by their private mail (rather than by mailing list) if you do not want to +disclose details on the mailing list. This is also a suitable method for proposing financial reward for +116 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +implementation of a substantial feature that is not yet in Yade – typically, though, we will request this +feature to be part of the public codebase once completed, so that the rest of the community can benefit +from it as well. +2.3 Yade wrapper class reference +2.3.1 Bodies +Body +class yade.wrapper.Body(inherits Serializable) +A particle, basic element of simulation; interacts with other bodies. +aspherical(=false) +Whether this body has different inertia along principal axes; NewtonIntegrator makes use of +this flag to call rotation integration routine for aspherical bodies, which is more expensive. +bound(=uninitalized) +Bound, approximating volume for the purposes of collision detection. +bounded(=true) +Whether this body should have Body.bound created. Note that bodies without a bound +do +not participate in collision detection. (In c++, use Body::isBounded/Body::setBounded) +chain +Returns Id of chain to which the body belongs. +clumpId +Id of clump this body makes part of; +invalid number if not part of clump; +see +Body::isStandalone, Body::isClump, Body::isClumpMember properties. +Not meant to be modified directly from Python, use O.bodies.appendClumped instead. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dynamic(=true) +Whether +this +body +will +be +moved +by +forces. +(In +c++, +use +Body::isDynamic/Body::setDynamic) +flags(=FLAG_BOUNDED) +Bits of various body-related flags. +Do +not +access +directly. +In c++, +use isDy- +namic/setDynamic, isBounded/setBounded, isAspherical/setAspherical. +In python, use +Body.dynamic, Body.bounded, Body.aspherical. +groupMask(=1) +Bitmask for interaction detection purposes: it is required that two bodies have at least one +bit in common in their groupMask for their interaction to be possible from the Collider point +of view. +id(=Body::ID_NONE) +Unique id of this body. +intrs((Body)arg1) → list : +Return list of all real interactions in which this body participates. +isClump +True if this body is clump itself, false otherwise. +isClumpMember +True if this body is clump member, false otherwise. +2.3. +Yade wrapper class reference +117 + +Yade Documentation, Release 3rd ed. +isStandalone +True if this body is neither clump, nor clump member; false otherwise. +iterBorn(=-1) +Step number at which the body was added to simulation. +mask +Shorthand for Body::groupMask +mat +Shorthand for Body::material +material(=uninitalized) +Material instance associated with this body. +shape(=uninitalized) +Geometrical Shape. +state(=new State) +Physical state. +timeBorn(=-1) +Time at which the body was added to simulation. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +Shape +Shape +PFacet +Tetra +Clump +Sphere +GridConnection +Cylinder +GridNode +Box +Wall +Facet +ChainedCylinder +Fig. 19: Inheritance graph of Shape. See also: Box, ChainedCylinder, Clump, Cylinder, Facet, GridCon- +nection, GridNode, PFacet, Sphere, Tetra, Wall. +class yade.wrapper.Shape(inherits Serializable) +Geometry of a body +color(=Vector3r(1, 1, 1)) +Color for rendering (normalized RGB). +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((Shape)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +118 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +dispIndex +Return class index of this instance. +highlight(=false) +Whether this Shape will be highlighted when rendered. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +wire(=false) +Whether this Shape is rendered using color surfaces, or only wireframe (can still be overridden +by global config of the renderer). +class yade.wrapper.Box(inherits Shape → Serializable) +color(=Vector3r(1, 1, 1)) +Color for rendering (normalized RGB). +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((Shape)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +extents(=uninitalized) +Half-size of the cuboid +highlight(=false) +Whether this Shape will be highlighted when rendered. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +wire(=false) +Whether this Shape is rendered using color surfaces, or only wireframe (can still be overridden +by global config of the renderer). +class yade.wrapper.ChainedCylinder(inherits Cylinder → Sphere → Shape → Serializable) +Geometry of a deformable chained cylinder, using geometry Cylinder. +chainedOrientation(=Quaternionr::Identity()) +Deviation of node1 orientation from node-to-node vector +color(=Vector3r(1, 1, 1)) +Color for rendering (normalized RGB). +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((Shape)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +highlight(=false) +Whether this Shape will be highlighted when rendered. +initLength(=0) +tensile-free length, used as reference for tensile strain +2.3. +Yade wrapper class reference +119 + +Yade Documentation, Release 3rd ed. +length(=NaN) +Length [m] +radius(=NaN) +Radius [m] +segment(=Vector3r::Zero()) +Length vector +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +wire(=false) +Whether this Shape is rendered using color surfaces, or only wireframe (can still be overridden +by global config of the renderer). +class yade.wrapper.Clump(inherits Shape → Serializable) +Rigid aggregate of bodies +color(=Vector3r(1, 1, 1)) +Color for rendering (normalized RGB). +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((Shape)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +highlight(=false) +Whether this Shape will be highlighted when rendered. +ids(=uninitalized) +Ids of constituent particles (only informative; direct modifications will have no effect). +members +Return clump members as {‘id1’:(relPos,relOri),…} +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +wire(=false) +Whether this Shape is rendered using color surfaces, or only wireframe (can still be overridden +by global config of the renderer). +class yade.wrapper.Cylinder(inherits Sphere → Shape → Serializable) +Geometry of a cylinder, as Minkowski sum of line and sphere. +color(=Vector3r(1, 1, 1)) +Color for rendering (normalized RGB). +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((Shape)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +highlight(=false) +Whether this Shape will be highlighted when rendered. +120 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +length(=NaN) +Length [m] +radius(=NaN) +Radius [m] +segment(=Vector3r::Zero()) +Length vector +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +wire(=false) +Whether this Shape is rendered using color surfaces, or only wireframe (can still be overridden +by global config of the renderer). +class yade.wrapper.Facet(inherits Shape → Serializable) +Facet (triangular particle) geometry. +area(=NaN) +Facet’s area +color(=Vector3r(1, 1, 1)) +Color for rendering (normalized RGB). +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((Shape)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +highlight(=false) +Whether this Shape will be highlighted when rendered. +normal(=Vector3r(NaN, NaN, NaN)) +Facet’s normal (in local coordinate system) +setVertices((Facet)arg1, (Vector3)arg2, (Vector3)arg3, (Vector3)arg4) → None : +TODO +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +vertices(=vector(3, Vector3r(NaN, NaN, NaN))) +Vertex positions in local coordinates. +wire(=false) +Whether this Shape is rendered using color surfaces, or only wireframe (can still be overridden +by global config of the renderer). +class yade.wrapper.GridConnection(inherits Sphere → Shape → Serializable) +GridConnection shape (see [Effeindzourou2016], [Bourrier2013]). Component of a grid designed to +link two GridNodes. It is highly recommended to use gridpfacet.gridConnection to generate correct +GridConnections. +addPFacet((GridConnection)arg1, (Body)Body) → None : +Add a PFacet to the GridConnection. +cellDist(=Vector3i(0, 0, 0)) +Distance of bodies in cell size units, if using periodic boundary conditions. Note that periodic +boundary conditions for GridConnections have not yet been fully implemented. +2.3. +Yade wrapper class reference +121 + +Yade Documentation, Release 3rd ed. +color(=Vector3r(1, 1, 1)) +Color for rendering (normalized RGB). +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((Shape)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +getPFacets((GridConnection)arg1) → object : +get list of linked PFacets. +highlight(=false) +Whether this Shape will be highlighted when rendered. +node1(=uninitalized) +First Body the GridConnection is connected to. +node2(=uninitalized) +Second Body the GridConnection is connected to. +periodic(=false) +true if two nodes from different periods are connected. +radius(=NaN) +Radius [m] +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +wire(=false) +Whether this Shape is rendered using color surfaces, or only wireframe (can still be overridden +by global config of the renderer). +class yade.wrapper.GridNode(inherits Sphere → Shape → Serializable) +GridNode shape, component of a grid. To create a Grid, place the nodes first, they will define the +spacial discretisation of it. It is highly recommended to use gridpfacet.gridNode to generate correct +GridNodes. Note that the GridNodes should only be in an Interaction with other GridNodes. The +Sphere-Grid contact is only handled by the GridConnections. +addConnection((GridNode)arg1, (Body)Body) → None : +Add a GridConnection to the GridNode. +addPFacet((GridNode)arg1, (Body)Body) → None : +Add a PFacet to the GridNode. +color(=Vector3r(1, 1, 1)) +Color for rendering (normalized RGB). +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((Shape)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +getConnections((GridNode)arg1) → object : +get list of linked GridConnection’s. +122 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +getPFacets((GridNode)arg1) → object : +get list of linked PFacet’s. +highlight(=false) +Whether this Shape will be highlighted when rendered. +radius(=NaN) +Radius [m] +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +wire(=false) +Whether this Shape is rendered using color surfaces, or only wireframe (can still be overridden +by global config of the renderer). +class yade.wrapper.PFacet(inherits Shape → Serializable) +PFacet (particle facet) geometry (see [Effeindzourou2016], [Effeindzourou2015a]). It is highly rec- +ommended to use the helper functions in gridpfacet (e.g., gridpfacet.pfacetCreator1-4) to generate +correct PFacet elements. +area(=NaN) +PFacet’s area +cellDist(=Vector3i(0, 0, 0)) +Distance of bodies in cell size units, if using periodic boundary conditions. Note that periodic +boundary conditions for PFacets have not yet been fully implemented. +color(=Vector3r(1, 1, 1)) +Color for rendering (normalized RGB). +conn1(=uninitalized) +First Body the Pfacet is connected to. +conn2(=uninitalized) +Second Body the Pfacet is connected to. +conn3(=uninitalized) +third Body the Pfacet is connected to. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((Shape)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +highlight(=false) +Whether this Shape will be highlighted when rendered. +node1(=uninitalized) +First Body the Pfacet is connected to. +node2(=uninitalized) +Second Body the Pfacet is connected to. +node3(=uninitalized) +third Body the Pfacet is connected to. +normal(=Vector3r(NaN, NaN, NaN)) +PFacet’s normal (in local coordinate system) +radius(=-1) +PFacet’s radius +2.3. +Yade wrapper class reference +123 + +Yade Documentation, Release 3rd ed. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +wire(=false) +Whether this Shape is rendered using color surfaces, or only wireframe (can still be overridden +by global config of the renderer). +class yade.wrapper.Sphere(inherits Shape → Serializable) +Geometry of spherical particle. +color(=Vector3r(1, 1, 1)) +Color for rendering (normalized RGB). +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((Shape)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +highlight(=false) +Whether this Shape will be highlighted when rendered. +radius(=NaN) +Radius [m] +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +wire(=false) +Whether this Shape is rendered using color surfaces, or only wireframe (can still be overridden +by global config of the renderer). +class yade.wrapper.Tetra(inherits Shape → Serializable) +Tetrahedron geometry. +color(=Vector3r(1, 1, 1)) +Color for rendering (normalized RGB). +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((Shape)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +highlight(=false) +Whether this Shape will be highlighted when rendered. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +v(=std::vector(4)) +Tetrahedron vertices (in local coordinate system). +wire(=false) +Whether this Shape is rendered using color surfaces, or only wireframe (can still be overridden +by global config of the renderer). +124 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +class yade.wrapper.Wall(inherits Shape → Serializable) +Object representing infinite plane aligned with the coordinate system (axis-aligned wall). +axis(=0) +Axis of the normal; can be 0,1,2 for +x, +y, +z respectively (Body’s orientation is disregarded +for walls) +color(=Vector3r(1, 1, 1)) +Color for rendering (normalized RGB). +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((Shape)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +highlight(=false) +Whether this Shape will be highlighted when rendered. +sense(=0) +Which side of the wall interacts: -1 for negative only, 0 for both, +1 for positive only +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +wire(=false) +Whether this Shape is rendered using color surfaces, or only wireframe (can still be overridden +by global config of the renderer). +State +State +WireState +JCFpmState +ChainedState +CpmState +Fig. 20: Inheritance graph of State. See also: ChainedState, CpmState, JCFpmState, WireState. +class yade.wrapper.State(inherits Serializable) +State of a body (spatial configuration, internal variables). +angMom(=Vector3r::Zero()) +Current angular momentum +angVel(=Vector3r::Zero()) +Current angular velocity +blockedDOFs +Degress of freedom where linear/angular velocity will be always constant (equal to zero, or to +an user-defined value), regardless of applied force/torque. String that may contain ‘xyzXYZ’ +(translations and rotations). +2.3. +Yade wrapper class reference +125 + +Yade Documentation, Release 3rd ed. +densityScaling(=-1) +(auto-updated) see GlobalStiffnessTimeStepper::targetDt. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((State)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +displ((State)arg1) → Vector3 : +Displacement from reference position (pos - refPos) +inertia(=Vector3r::Zero()) +Inertia of associated body, in local coordinate system. +isDamped(=true) +Damping in NewtonIntegrator can be deactivated for individual particles by setting this vari- +able to FALSE. E.g. damping is inappropriate for particles in free flight under gravity but it +might still be applicable to other particles in the same simulation. +mass(=0) +Mass of this body +ori +Current orientation. +pos +Current position. +refOri(=Quaternionr::Identity()) +Reference orientation +refPos(=Vector3r::Zero()) +Reference position +rot((State)arg1) → Vector3 : +Rotation from reference orientation (as rotation vector) +se3(=Se3r(Vector3r::Zero(), Quaternionr::Identity())) +Position and orientation as one object. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +vel(=Vector3r::Zero()) +Current linear velocity. +class yade.wrapper.ChainedState(inherits State → Serializable) +State of a chained bodies, containing information on connectivity in order to track contacts jumping +over contiguous elements. Chains are 1D lists from which id of chained bodies are retrieved via +rank and chainNumber. +addToChain((ChainedState)arg1, (int)bodyId) → None : +Add body to current active chain +angMom(=Vector3r::Zero()) +Current angular momentum +angVel(=Vector3r::Zero()) +Current angular velocity +bId(=-1) +id of the body containing - for postLoad operations only. +126 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +blockedDOFs +Degress of freedom where linear/angular velocity will be always constant (equal to zero, or to +an user-defined value), regardless of applied force/torque. String that may contain ‘xyzXYZ’ +(translations and rotations). +chainNumber(=0) +chain id. +currentChain = 0 +densityScaling(=-1) +(auto-updated) see GlobalStiffnessTimeStepper::targetDt. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((State)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +displ((State)arg1) → Vector3 : +Displacement from reference position (pos - refPos) +inertia(=Vector3r::Zero()) +Inertia of associated body, in local coordinate system. +isDamped(=true) +Damping in NewtonIntegrator can be deactivated for individual particles by setting this vari- +able to FALSE. E.g. damping is inappropriate for particles in free flight under gravity but it +might still be applicable to other particles in the same simulation. +mass(=0) +Mass of this body +ori +Current orientation. +pos +Current position. +rank(=0) +rank in the chain. +refOri(=Quaternionr::Identity()) +Reference orientation +refPos(=Vector3r::Zero()) +Reference position +rot((State)arg1) → Vector3 : +Rotation from reference orientation (as rotation vector) +se3(=Se3r(Vector3r::Zero(), Quaternionr::Identity())) +Position and orientation as one object. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +vel(=Vector3r::Zero()) +Current linear velocity. +class yade.wrapper.CpmState(inherits State → Serializable) +State information about body use by cpm-model. +2.3. +Yade wrapper class reference +127 + +Yade Documentation, Release 3rd ed. +None of that is used for computation (at least not now), only for post-processing. +angMom(=Vector3r::Zero()) +Current angular momentum +angVel(=Vector3r::Zero()) +Current angular velocity +blockedDOFs +Degress of freedom where linear/angular velocity will be always constant (equal to zero, or to +an user-defined value), regardless of applied force/torque. String that may contain ‘xyzXYZ’ +(translations and rotations). +damageTensor(=Matrix3r::Zero()) +Damage tensor computed with microplane theory averaging. state.damageTensor.trace() = +state.normDmg +densityScaling(=-1) +(auto-updated) see GlobalStiffnessTimeStepper::targetDt. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((State)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +displ((State)arg1) → Vector3 : +Displacement from reference position (pos - refPos) +epsVolumetric(=0) +Volumetric strain around this body (unused for now) +inertia(=Vector3r::Zero()) +Inertia of associated body, in local coordinate system. +isDamped(=true) +Damping in NewtonIntegrator can be deactivated for individual particles by setting this vari- +able to FALSE. E.g. damping is inappropriate for particles in free flight under gravity but it +might still be applicable to other particles in the same simulation. +mass(=0) +Mass of this body +normDmg(=0) +Average damage including already deleted contacts (it is really not damage, but 1- +relResidualStrength now) +numBrokenCohesive(=0) +Number of (cohesive) contacts that damaged completely +numContacts(=0) +Number of contacts with this body +ori +Current orientation. +pos +Current position. +refOri(=Quaternionr::Identity()) +Reference orientation +128 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +refPos(=Vector3r::Zero()) +Reference position +rot((State)arg1) → Vector3 : +Rotation from reference orientation (as rotation vector) +se3(=Se3r(Vector3r::Zero(), Quaternionr::Identity())) +Position and orientation as one object. +stress(=Matrix3r::Zero()) +Stress tensor of the spherical particle (under assumption that particle volume = pi*r*r*r*4/3.) +for packing fraction 0.62 +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +vel(=Vector3r::Zero()) +Current linear velocity. +class yade.wrapper.JCFpmState(inherits State → Serializable) +JCFpm state information about each body. +angMom(=Vector3r::Zero()) +Current angular momentum +angVel(=Vector3r::Zero()) +Current angular velocity +blockedDOFs +Degress of freedom where linear/angular velocity will be always constant (equal to zero, or to +an user-defined value), regardless of applied force/torque. String that may contain ‘xyzXYZ’ +(translations and rotations). +damageIndex(=0) +Ratio of broken bonds over initial bonds. [-] +densityScaling(=-1) +(auto-updated) see GlobalStiffnessTimeStepper::targetDt. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((State)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +displ((State)arg1) → Vector3 : +Displacement from reference position (pos - refPos) +inertia(=Vector3r::Zero()) +Inertia of associated body, in local coordinate system. +isDamped(=true) +Damping in NewtonIntegrator can be deactivated for individual particles by setting this vari- +able to FALSE. E.g. damping is inappropriate for particles in free flight under gravity but it +might still be applicable to other particles in the same simulation. +joint(=0) +Indicates the number of joint surfaces to which the particle belongs (0-> no joint, 1->1 joint, +etc..). [-] +jointNormal1(=Vector3r::Zero()) +Specifies the normal direction to the joint plane 1. Rk: the ideal here would be to create a +2.3. +Yade wrapper class reference +129 + +Yade Documentation, Release 3rd ed. +vector of vector wich size is defined by the joint integer (as much joint normals as joints). +However, it needs to make the pushback function works with python since joint detection is +done through a python script. lines 272 to 312 of cpp file should therefore be adapted. [-] +jointNormal2(=Vector3r::Zero()) +Specifies the normal direction to the joint plane 2. [-] +jointNormal3(=Vector3r::Zero()) +Specifies the normal direction to the joint plane 3. [-] +mass(=0) +Mass of this body +nbBrokenBonds(=0) +Number of broken bonds. [-] +nbInitBonds(=0) +Number of initial bonds. [-] +onJoint(=false) +Identifies if the particle is on a joint surface. +ori +Current orientation. +pos +Current position. +refOri(=Quaternionr::Identity()) +Reference orientation +refPos(=Vector3r::Zero()) +Reference position +rot((State)arg1) → Vector3 : +Rotation from reference orientation (as rotation vector) +se3(=Se3r(Vector3r::Zero(), Quaternionr::Identity())) +Position and orientation as one object. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +vel(=Vector3r::Zero()) +Current linear velocity. +class yade.wrapper.WireState(inherits State → Serializable) +Wire state information of each body. +None of that is used for computation (at least not now), only for post-processing. +angMom(=Vector3r::Zero()) +Current angular momentum +angVel(=Vector3r::Zero()) +Current angular velocity +blockedDOFs +Degress of freedom where linear/angular velocity will be always constant (equal to zero, or to +an user-defined value), regardless of applied force/torque. String that may contain ‘xyzXYZ’ +(translations and rotations). +densityScaling(=-1) +(auto-updated) see GlobalStiffnessTimeStepper::targetDt. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +130 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +dispHierarchy((State)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +displ((State)arg1) → Vector3 : +Displacement from reference position (pos - refPos) +inertia(=Vector3r::Zero()) +Inertia of associated body, in local coordinate system. +isDamped(=true) +Damping in NewtonIntegrator can be deactivated for individual particles by setting this vari- +able to FALSE. E.g. damping is inappropriate for particles in free flight under gravity but it +might still be applicable to other particles in the same simulation. +mass(=0) +Mass of this body +numBrokenLinks(=0) +Number of broken links (e.g. number of wires connected to the body which are broken). [-] +ori +Current orientation. +pos +Current position. +refOri(=Quaternionr::Identity()) +Reference orientation +refPos(=Vector3r::Zero()) +Reference position +rot((State)arg1) → Vector3 : +Rotation from reference orientation (as rotation vector) +se3(=Se3r(Vector3r::Zero(), Quaternionr::Identity())) +Position and orientation as one object. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +vel(=Vector3r::Zero()) +Current linear velocity. +Material +class yade.wrapper.Material(inherits Serializable) +Material properties of a body. +density(=1000) +Density of the material [kg/m3] +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((Material)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +2.3. +Yade wrapper class reference +131 + +Yade Documentation, Release 3rd ed. +Material +ViscElMat +FrictMat +ElastMat +BubbleMat +LudingMat +CohFrictMat +WireMat +FrictViscoMat +MortarMat +InelastCohFrictMat +FrictMatCDM +JCFpmMat +ViscElCapMat +CpmMat +Fig. 21: Inheritance graph of Material. See also: BubbleMat, CohFrictMat, CpmMat, ElastMat, Frict- +Mat, FrictMatCDM, FrictViscoMat, InelastCohFrictMat, JCFpmMat, LudingMat, MortarMat, ViscEl- +CapMat, ViscElMat, WireMat. +dispIndex +Return class index of this instance. +id(=-1, not shared) +Numeric id of this material; is non-negative only if this Material is shared (i.e. in O.materials), +-1 otherwise. This value is set automatically when the material is inserted to the simulation +via O.materials.append. +(This id was necessary since before boost::serialization was used, +shared pointers were not tracked properly; it might disappear in the future) +label(=uninitalized) +Textual identifier for this material; can be used for shared materials lookup in MaterialCon- +tainer. +newAssocState((Material)arg1) → State : +Return new State instance, which is associated with this Material. +Some materials have +special requirement on Body::state type and calling this function when the body is created +will ensure that they match. (This is done automatically if you use utils.sphere, … functions +from python). +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.BubbleMat(inherits Material → Serializable) +material for bubble interactions, for use with other Bubble classes +density(=1000) +Density of the material [kg/m3] +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((Material)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +132 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +id(=-1, not shared) +Numeric id of this material; is non-negative only if this Material is shared (i.e. in O.materials), +-1 otherwise. This value is set automatically when the material is inserted to the simulation +via O.materials.append. +(This id was necessary since before boost::serialization was used, +shared pointers were not tracked properly; it might disappear in the future) +label(=uninitalized) +Textual identifier for this material; can be used for shared materials lookup in MaterialCon- +tainer. +newAssocState((Material)arg1) → State : +Return new State instance, which is associated with this Material. +Some materials have +special requirement on Body::state type and calling this function when the body is created +will ensure that they match. (This is done automatically if you use utils.sphere, … functions +from python). +surfaceTension(=0.07197) +The surface tension in the fluid surrounding the bubbles. The default value is that of water +at 25 degrees Celcius. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.CohFrictMat(inherits FrictMat → ElastMat → Material → Serializable) +Material description extending FrictMat with cohesive properties and rotational stiffnesses. For +use e.g. with Law2_ScGeom6D_CohFrictPhys_CohesionMoment. +alphaKr(=2.0) +Dimensionless rolling stiffness. +alphaKtw(=2.0) +Dimensionless twist stiffness. +density(=1000) +Density of the material [kg/m3] +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((Material)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +etaRoll(=-1.) +Dimensionless rolling (aka ‘bending’) strength. If negative, rolling moment will be elastic. +etaTwist(=-1.) +Dimensionless twisting strength. If negative, twist moment will be elastic. +fragile(=true) +do cohesion disappear when contact strength is exceeded +frictionAngle(=.5) +Contact friction angle (in radians). Hint : use ‘radians(degreesValue)’ in python scripts. +id(=-1, not shared) +Numeric id of this material; is non-negative only if this Material is shared (i.e. in O.materials), +-1 otherwise. This value is set automatically when the material is inserted to the simulation +via O.materials.append. +(This id was necessary since before boost::serialization was used, +shared pointers were not tracked properly; it might disappear in the future) +2.3. +Yade wrapper class reference +133 + +Yade Documentation, Release 3rd ed. +isCohesive(=true) +Whether this body can form possibly cohesive interactions (if true and depending on other +parameters such as Ip2_CohFrictMat_CohFrictMat_CohFrictPhys.setCohesionNow). +label(=uninitalized) +Textual identifier for this material; can be used for shared materials lookup in MaterialCon- +tainer. +momentRotationLaw(=false) +Use bending/twisting moment at contact. The contact may have moments only if both bodies +have this flag true. +See Law2_ScGeom6D_CohFrictPhys_CohesionMoment.always_use_- +moment_law for details. +newAssocState((Material)arg1) → State : +Return new State instance, which is associated with this Material. +Some materials have +special requirement on Body::state type and calling this function when the body is created +will ensure that they match. (This is done automatically if you use utils.sphere, … functions +from python). +normalCohesion(=-1) +Tensile strength, homogeneous to a pressure. If negative the normal force is purely elastic. +poisson(=.25) +Poisson’s ratio or the ratio between shear and normal stiffness [-]. It has different meanings +depending on the Ip functor. +shearCohesion(=-1) +Shear strength, homogeneous to a pressure. If negative the shear force is purely elastic. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +young(=1e9) +elastic modulus [Pa]. It has different meanings depending on the Ip functor. +class yade.wrapper.CpmMat(inherits FrictMat → ElastMat → Material → Serializable) +Concrete material, for use with other Cpm classes. +Note: +Density is initialized to 4800 kgm￿3automatically, which gives approximate 2800 kgm￿3 on +0.5 density packing. +Concrete Particle Model (CPM) +CpmMat is particle material, Ip2_CpmMat_CpmMat_CpmPhys averages two particles’ materials, +creating CpmPhys, which is then used in interaction resultion by Law2_ScGeom_CpmPhys_Cpm. +CpmState is associated to CpmMat and keeps state defined on particles rather than interactions +(such as number of completely damaged interactions). +The model is contained in externally defined macro CPM_MATERIAL_MODEL, which features +damage in tension, plasticity in shear and compression and rate-dependence. For commercial rea- +sons, rate-dependence and compression-plasticity is not present in reduced version of the model, +used when CPM_MATERIAL_MODEL is not defined. The full model will be described in de- +tail in my (Václav Šmilauer) thesis along with calibration procedures (rigidity, poisson’s ratio, +compressive/tensile strength ratio, fracture energy, behavior under confinement, rate-dependent +behavior). +Even the public model is useful enough to run simulation on concrete samples, such as uniaxial +tension-compression test. +damLaw(=1) +Law for damage evolution in uniaxial tension. 0 for linear stress-strain softening branch, 1 +(default) for exponential damage evolution law +134 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +density(=1000) +Density of the material [kg/m3] +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((Material)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +dmgRateExp(=0) +Exponent for normal viscosity function. [-] +dmgTau(=-1, deactivated if negative) +Characteristic time for normal viscosity. [s] +epsCrackOnset(=NaN) +Limit elastic strain [-] +equivStrainShearContrib(=0) +Coefficient of shear contribution to equivalent strain +frictionAngle(=.5) +Contact friction angle (in radians). Hint : use ‘radians(degreesValue)’ in python scripts. +id(=-1, not shared) +Numeric id of this material; is non-negative only if this Material is shared (i.e. in O.materials), +-1 otherwise. This value is set automatically when the material is inserted to the simulation +via O.materials.append. +(This id was necessary since before boost::serialization was used, +shared pointers were not tracked properly; it might disappear in the future) +isoPrestress(=0) +Isotropic prestress of the whole specimen. [Pa] +label(=uninitalized) +Textual identifier for this material; can be used for shared materials lookup in MaterialCon- +tainer. +neverDamage(=false) +If true, no damage will occur (for testing only). +newAssocState((Material)arg1) → State : +Return new State instance, which is associated with this Material. +Some materials have +special requirement on Body::state type and calling this function when the body is created +will ensure that they match. (This is done automatically if you use utils.sphere, … functions +from python). +plRateExp(=0) +Exponent for visco-plasticity function. [-] +plTau(=-1, deactivated if negative) +Characteristic time for visco-plasticity. [s] +poisson(=.25) +Poisson’s ratio or the ratio between shear and normal stiffness [-]. It has different meanings +depending on the Ip functor. +relDuctility(=NaN) +relative ductility of bonds in normal direction +sigmaT(=NaN) +Initial cohesion [Pa] +2.3. +Yade wrapper class reference +135 + +Yade Documentation, Release 3rd ed. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +young(=1e9) +elastic modulus [Pa]. It has different meanings depending on the Ip functor. +class yade.wrapper.ElastMat(inherits Material → Serializable) +Purely elastic material. The material parameters may have different meanings depending on the +IPhysFunctor used : true Young and Poisson in Ip2_FrictMat_FrictMat_MindlinPhys, or contact +stiffnesses in Ip2_FrictMat_FrictMat_FrictPhys. +density(=1000) +Density of the material [kg/m3] +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((Material)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +id(=-1, not shared) +Numeric id of this material; is non-negative only if this Material is shared (i.e. in O.materials), +-1 otherwise. This value is set automatically when the material is inserted to the simulation +via O.materials.append. +(This id was necessary since before boost::serialization was used, +shared pointers were not tracked properly; it might disappear in the future) +label(=uninitalized) +Textual identifier for this material; can be used for shared materials lookup in MaterialCon- +tainer. +newAssocState((Material)arg1) → State : +Return new State instance, which is associated with this Material. +Some materials have +special requirement on Body::state type and calling this function when the body is created +will ensure that they match. (This is done automatically if you use utils.sphere, … functions +from python). +poisson(=.25) +Poisson’s ratio or the ratio between shear and normal stiffness [-]. It has different meanings +depending on the Ip functor. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +young(=1e9) +elastic modulus [Pa]. It has different meanings depending on the Ip functor. +class yade.wrapper.FrictMat(inherits ElastMat → Material → Serializable) +Elastic material with contact friction. See also ElastMat. +density(=1000) +Density of the material [kg/m3] +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((Material)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +136 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +dispIndex +Return class index of this instance. +frictionAngle(=.5) +Contact friction angle (in radians). Hint : use ‘radians(degreesValue)’ in python scripts. +id(=-1, not shared) +Numeric id of this material; is non-negative only if this Material is shared (i.e. in O.materials), +-1 otherwise. This value is set automatically when the material is inserted to the simulation +via O.materials.append. +(This id was necessary since before boost::serialization was used, +shared pointers were not tracked properly; it might disappear in the future) +label(=uninitalized) +Textual identifier for this material; can be used for shared materials lookup in MaterialCon- +tainer. +newAssocState((Material)arg1) → State : +Return new State instance, which is associated with this Material. +Some materials have +special requirement on Body::state type and calling this function when the body is created +will ensure that they match. (This is done automatically if you use utils.sphere, … functions +from python). +poisson(=.25) +Poisson’s ratio or the ratio between shear and normal stiffness [-]. It has different meanings +depending on the Ip functor. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +young(=1e9) +elastic modulus [Pa]. It has different meanings depending on the Ip functor. +class yade.wrapper.FrictMatCDM(inherits FrictMat → ElastMat → Material → Serializable) +Material to be used for extended Hertz-Mindlin contact law. Normal direction: parameters for +Conical Damage Model (Harkness et al. 2016, Suhr & Six 2017). Tangential direction: parameters +for stress dependent interparticle friction coefficient (Suhr & Six 2016). +Both models can be +switched on/off separately. +alpha(=1e-6) +[rad] angle of conical asperities, alpha in (0, pi/2) +c1(=0.0) +[-] parameter of pressure dependent friction model c1, choose 0 for constant interparticle +friction coefficient +c2(=0.0) +[-] parameter of pressure dependent friction model c2, choose 0 for constant interparticle +friction coefficient +density(=1000) +Density of the material [kg/m3] +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((Material)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +frictionAngle(=.5) +Contact friction angle (in radians). Hint : use ‘radians(degreesValue)’ in python scripts. +2.3. +Yade wrapper class reference +137 + +Yade Documentation, Release 3rd ed. +id(=-1, not shared) +Numeric id of this material; is non-negative only if this Material is shared (i.e. in O.materials), +-1 otherwise. This value is set automatically when the material is inserted to the simulation +via O.materials.append. +(This id was necessary since before boost::serialization was used, +shared pointers were not tracked properly; it might disappear in the future) +label(=uninitalized) +Textual identifier for this material; can be used for shared materials lookup in MaterialCon- +tainer. +newAssocState((Material)arg1) → State : +Return new State instance, which is associated with this Material. +Some materials have +special requirement on Body::state type and calling this function when the body is created +will ensure that they match. (This is done automatically if you use utils.sphere, … functions +from python). +poisson(=.25) +Poisson’s ratio or the ratio between shear and normal stiffness [-]. It has different meanings +depending on the Ip functor. +sigmaMax(=1e99) +>0 [Pa] max compressive strength of material, choose 1e99 to switch off conical damage model +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +young(=1e9) +elastic modulus [Pa]. It has different meanings depending on the Ip functor. +class yade.wrapper.FrictViscoMat(inherits FrictMat → ElastMat → Material → Serializable) +Material for use with the FrictViscoPM classes +betan(=0.) +Fraction of the viscous damping coefficient in normal direction equal to +cn +Cn,crit . +density(=1000) +Density of the material [kg/m3] +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((Material)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +frictionAngle(=.5) +Contact friction angle (in radians). Hint : use ‘radians(degreesValue)’ in python scripts. +id(=-1, not shared) +Numeric id of this material; is non-negative only if this Material is shared (i.e. in O.materials), +-1 otherwise. This value is set automatically when the material is inserted to the simulation +via O.materials.append. +(This id was necessary since before boost::serialization was used, +shared pointers were not tracked properly; it might disappear in the future) +label(=uninitalized) +Textual identifier for this material; can be used for shared materials lookup in MaterialCon- +tainer. +newAssocState((Material)arg1) → State : +Return new State instance, which is associated with this Material. +Some materials have +special requirement on Body::state type and calling this function when the body is created +138 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +will ensure that they match. (This is done automatically if you use utils.sphere, … functions +from python). +poisson(=.25) +Poisson’s ratio or the ratio between shear and normal stiffness [-]. It has different meanings +depending on the Ip functor. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +young(=1e9) +elastic modulus [Pa]. It has different meanings depending on the Ip functor. +class yade.wrapper.InelastCohFrictMat(inherits FrictMat → ElastMat → Material → Serial- +izable) +alphaKr(=2.0) +Dimensionless coefficient used for the rolling stiffness. +alphaKtw(=2.0) +Dimensionless coefficient used for the twist stiffness. +compressionModulus(=0.0) +Compresion elasticity modulus +creepBending(=0.0) +Bending creeping coefficient. Usual values between 0 and 1. +creepTension(=0.0) +Tension/compression creeping coefficient. Usual values between 0 and 1. +creepTwist(=0.0) +Twist creeping coefficient. Usual values between 0 and 1. +density(=1000) +Density of the material [kg/m3] +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((Material)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +epsilonMaxCompression(=0.0) +Maximal plastic strain compression +epsilonMaxTension(=0.0) +Maximal plastic strain tension +etaMaxBending(=0.0) +Maximal plastic bending strain +etaMaxTwist(=0.0) +Maximal plastic twist strain +frictionAngle(=.5) +Contact friction angle (in radians). Hint : use ‘radians(degreesValue)’ in python scripts. +id(=-1, not shared) +Numeric id of this material; is non-negative only if this Material is shared (i.e. in O.materials), +-1 otherwise. This value is set automatically when the material is inserted to the simulation +via O.materials.append. +(This id was necessary since before boost::serialization was used, +shared pointers were not tracked properly; it might disappear in the future) +2.3. +Yade wrapper class reference +139 + +Yade Documentation, Release 3rd ed. +label(=uninitalized) +Textual identifier for this material; can be used for shared materials lookup in MaterialCon- +tainer. +newAssocState((Material)arg1) → State : +Return new State instance, which is associated with this Material. +Some materials have +special requirement on Body::state type and calling this function when the body is created +will ensure that they match. (This is done automatically if you use utils.sphere, … functions +from python). +nuBending(=0.0) +Bending elastic stress limit +nuTwist(=0.0) +Twist elastic stress limit +poisson(=.25) +Poisson’s ratio or the ratio between shear and normal stiffness [-]. It has different meanings +depending on the Ip functor. +shearCohesion(=0.0) +Shear elastic stress limit +shearModulus(=0.0) +shear elasticity modulus +sigmaCompression(=0.0) +Compression elastic stress limit +sigmaTension(=0.0) +Tension elastic stress limit +tensionModulus(=0.0) +Tension elasticity modulus +unloadBending(=0.0) +Bending plastic unload coefficient. Usual values between 0 and +infinity. +unloadTension(=0.0) +Tension/compression plastic unload coefficient. Usual values between 0 and +infinity. +unloadTwist(=0.0) +Twist plastic unload coefficient. Usual values between 0 and +infinity. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +young(=1e9) +elastic modulus [Pa]. It has different meanings depending on the Ip functor. +class yade.wrapper.JCFpmMat(inherits FrictMat → ElastMat → Material → Serializable) +Possibly jointed, cohesive frictional material, for use with other JCFpm classes +cohesion(=0.) +Defines the maximum admissible tangential force in shear, for Fn=0, in the matrix (FsMax += cohesion * crossSection). [Pa] +density(=1000) +Density of the material [kg/m3] +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((Material)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +140 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +dispIndex +Return class index of this instance. +frictionAngle(=.5) +Contact friction angle (in radians). Hint : use ‘radians(degreesValue)’ in python scripts. +id(=-1, not shared) +Numeric id of this material; is non-negative only if this Material is shared (i.e. in O.materials), +-1 otherwise. This value is set automatically when the material is inserted to the simulation +via O.materials.append. +(This id was necessary since before boost::serialization was used, +shared pointers were not tracked properly; it might disappear in the future) +jointCohesion(=0.) +Defines the maximum admissible tangential force in shear, for Fn=0, on the joint surface. [Pa] +jointDilationAngle(=0) +Defines the dilatancy of the joint surface (only valid for smooth contact logic). [rad] +jointFrictionAngle(=-1) +Defines Coulomb friction on the joint surface. [rad] +jointNormalStiffness(=0.) +Defines the normal stiffness on the joint surface. [Pa/m] +jointShearStiffness(=0.) +Defines the shear stiffness on the joint surface. [Pa/m] +jointTensileStrength(=0.) +Defines the maximum admissible normal force in traction on the joint surface. [Pa] +label(=uninitalized) +Textual identifier for this material; can be used for shared materials lookup in MaterialCon- +tainer. +newAssocState((Material)arg1) → State : +Return new State instance, which is associated with this Material. +Some materials have +special requirement on Body::state type and calling this function when the body is created +will ensure that they match. (This is done automatically if you use utils.sphere, … functions +from python). +poisson(=.25) +Poisson’s ratio or the ratio between shear and normal stiffness [-]. It has different meanings +depending on the Ip functor. +residualFrictionAngle(=-1.) +Defines the residual friction angle (when contacts are not cohesive). +residualFrictionAn- +gle=frictionAngle if not specified. [rad] +tensileStrength(=0.) +Defines the maximum admissible normal force in traction in the matrix (FnMax = ten- +sileStrength * crossSection). [Pa] +type(=0) +If particles of two different types interact, it will be with friction only (no cohesion).[-] +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +young(=1e9) +elastic modulus [Pa]. It has different meanings depending on the Ip functor. +class yade.wrapper.LudingMat(inherits Material → Serializable) +Material for simple Luding‘s model of contact [Luding2008] ,[Singh2013]_ . +G0(=NaN) +Viscous damping +2.3. +Yade wrapper class reference +141 + +Yade Documentation, Release 3rd ed. +PhiF(=NaN) +Dimensionless plasticity depth +density(=1000) +Density of the material [kg/m3] +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((Material)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +frictionAngle(=NaN) +Friction angle [rad] +id(=-1, not shared) +Numeric id of this material; is non-negative only if this Material is shared (i.e. in O.materials), +-1 otherwise. This value is set automatically when the material is inserted to the simulation +via O.materials.append. +(This id was necessary since before boost::serialization was used, +shared pointers were not tracked properly; it might disappear in the future) +k1(=NaN) +Slope of loading plastic branch +kc(=NaN) +Slope of irreversible, tensile adhesive branch +kp(=NaN) +Slope of unloading and reloading limit elastic branch +ks(=NaN) +Shear stiffness +label(=uninitalized) +Textual identifier for this material; can be used for shared materials lookup in MaterialCon- +tainer. +newAssocState((Material)arg1) → State : +Return new State instance, which is associated with this Material. +Some materials have +special requirement on Body::state type and calling this function when the body is created +will ensure that they match. (This is done automatically if you use utils.sphere, … functions +from python). +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.MortarMat(inherits FrictMat → ElastMat → Material → Serializable) +Material for mortar interface, used in Ip2_MortarMat_MortarMat_MortarPhys and Law2_Sc- +Geom_MortarPhys_Lourenco. Default values according to +cohesion(=1e6) +cohesion [Pa] +compressiveStrength(=10e6) +compressiveStrength [Pa] +density(=1000) +Density of the material [kg/m3] +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +142 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +dispHierarchy((Material)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +ellAspect(=3) +aspect ratio of elliptical ‘cap’. Value >1 means the ellipse is longer along normal stress axis. +frictionAngle(=.25) +Friction angle +id(=-1, not shared) +Numeric id of this material; is non-negative only if this Material is shared (i.e. in O.materials), +-1 otherwise. This value is set automatically when the material is inserted to the simulation +via O.materials.append. +(This id was necessary since before boost::serialization was used, +shared pointers were not tracked properly; it might disappear in the future) +label(=uninitalized) +Textual identifier for this material; can be used for shared materials lookup in MaterialCon- +tainer. +neverDamage(=false) +If true, interactions remain elastic regardless stresses +newAssocState((Material)arg1) → State : +Return new State instance, which is associated with this Material. +Some materials have +special requirement on Body::state type and calling this function when the body is created +will ensure that they match. (This is done automatically if you use utils.sphere, … functions +from python). +poisson(=1) +Shear to normal modulus ratio +tensileStrength(=1e6) +tensileStrength [Pa] +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +young(=1e9) +Normal elastic modulus [Pa] +class yade.wrapper.ViscElCapMat(inherits ViscElMat → FrictMat → ElastMat → Material → +Serializable) +Material for extended viscoelastic model of contact with capillary parameters. +Capillar(=false) +True, if capillar forces need to be added. +CapillarType(=””) +Different types of capillar interaction: +Willett_numeric, Willett_analytic [Willett2000] , +Weigert [Weigert1999] , Rabinovich [Rabinov2005] , Lambert (simplified, corrected Rabinovich +model) [Lambert2008] +Vb(=0.0) +Liquid bridge volume [m^3] +cn(=NaN) +Normal viscous constant. Attention, this parameter cannot be set if tc, en or es is defined! +cs(=NaN) +Shear viscous constant. Attention, this parameter cannot be set if tc, en or es is defined! +2.3. +Yade wrapper class reference +143 + +Yade Documentation, Release 3rd ed. +dcap(=0.0) +Damping coefficient for the capillary phase [-] +density(=1000) +Density of the material [kg/m3] +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((Material)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +en(=NaN) +Restitution coefficient in normal direction +et(=NaN) +Restitution coefficient in tangential direction +frictionAngle(=.5) +Contact friction angle (in radians). Hint : use ‘radians(degreesValue)’ in python scripts. +gamma(=0.0) +Surface tension [N/m] +id(=-1, not shared) +Numeric id of this material; is non-negative only if this Material is shared (i.e. in O.materials), +-1 otherwise. This value is set automatically when the material is inserted to the simulation +via O.materials.append. +(This id was necessary since before boost::serialization was used, +shared pointers were not tracked properly; it might disappear in the future) +kn(=NaN) +Normal elastic stiffness. Attention, this parameter cannot be set if tc, en or es is defined! +ks(=NaN) +Shear elastic stiffness. Attention, this parameter cannot be set if tc, en or es is defined! +label(=uninitalized) +Textual identifier for this material; can be used for shared materials lookup in MaterialCon- +tainer. +lubrication(=false) +option to apply lubrication forces when material is defined from young, poisson and en (resti- +tution coefficient). +mR(=0.0) +Rolling resistance, see [Zhou1999536]. +mRtype(=1) +Rolling resistance type, see [Zhou1999536]. mRtype=1 - equation (3) in [Zhou1999536]; mR- +type=2 - equation (4) in [Zhou1999536]. +newAssocState((Material)arg1) → State : +Return new State instance, which is associated with this Material. +Some materials have +special requirement on Body::state type and calling this function when the body is created +will ensure that they match. (This is done automatically if you use utils.sphere, … functions +from python). +poisson(=.25) +Poisson’s ratio or the ratio between shear and normal stiffness [-]. It has different meanings +depending on the Ip functor. +144 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +roughnessScale(=1e-3) +if lubrication is activated, roughness scale considered for the particles to evaluate the effective +restitution coefficient. +tc(=NaN) +Contact time +theta(=0.0) +Contact angle [°] +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +viscoDyn(=1e-3) +if lubrication is activated, surrounding fluid dynamic viscosity considered to evaluate the +effective restitution coefficient as a function of the local Stokes number of the collision. +young(=1e9) +elastic modulus [Pa]. It has different meanings depending on the Ip functor. +class yade.wrapper.ViscElMat(inherits FrictMat → ElastMat → Material → Serializable) +Material for simple viscoelastic model of contact from analytical solution of a pair spheres inter- +action problem [Pournin2001] . +cn(=NaN) +Normal viscous constant. Attention, this parameter cannot be set if tc, en or es is defined! +cs(=NaN) +Shear viscous constant. Attention, this parameter cannot be set if tc, en or es is defined! +density(=1000) +Density of the material [kg/m3] +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((Material)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +en(=NaN) +Restitution coefficient in normal direction +et(=NaN) +Restitution coefficient in tangential direction +frictionAngle(=.5) +Contact friction angle (in radians). Hint : use ‘radians(degreesValue)’ in python scripts. +id(=-1, not shared) +Numeric id of this material; is non-negative only if this Material is shared (i.e. in O.materials), +-1 otherwise. This value is set automatically when the material is inserted to the simulation +via O.materials.append. +(This id was necessary since before boost::serialization was used, +shared pointers were not tracked properly; it might disappear in the future) +kn(=NaN) +Normal elastic stiffness. Attention, this parameter cannot be set if tc, en or es is defined! +ks(=NaN) +Shear elastic stiffness. Attention, this parameter cannot be set if tc, en or es is defined! +2.3. +Yade wrapper class reference +145 + +Yade Documentation, Release 3rd ed. +label(=uninitalized) +Textual identifier for this material; can be used for shared materials lookup in MaterialCon- +tainer. +lubrication(=false) +option to apply lubrication forces when material is defined from young, poisson and en (resti- +tution coefficient). +mR(=0.0) +Rolling resistance, see [Zhou1999536]. +mRtype(=1) +Rolling resistance type, see [Zhou1999536]. mRtype=1 - equation (3) in [Zhou1999536]; mR- +type=2 - equation (4) in [Zhou1999536]. +newAssocState((Material)arg1) → State : +Return new State instance, which is associated with this Material. +Some materials have +special requirement on Body::state type and calling this function when the body is created +will ensure that they match. (This is done automatically if you use utils.sphere, … functions +from python). +poisson(=.25) +Poisson’s ratio or the ratio between shear and normal stiffness [-]. It has different meanings +depending on the Ip functor. +roughnessScale(=1e-3) +if lubrication is activated, roughness scale considered for the particles to evaluate the effective +restitution coefficient. +tc(=NaN) +Contact time +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +viscoDyn(=1e-3) +if lubrication is activated, surrounding fluid dynamic viscosity considered to evaluate the +effective restitution coefficient as a function of the local Stokes number of the collision. +young(=1e9) +elastic modulus [Pa]. It has different meanings depending on the Ip functor. +class yade.wrapper.WireMat(inherits FrictMat → ElastMat → Material → Serializable) +Material for use with the Wire classes. In conjunction with the corresponding functors it can be +used to model steel wire meshes [Thoeni2014], geotextiles [Cheng2016] and more. +as(=0.) +Cross-section area of a single wire used to transform stress into force. [m2] +density(=1000) +Density of the material [kg/m3] +diameter(=0.0027) +Diameter of the single wire in [m] (the diameter is used to compute the cross-section area of +the wire). +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((Material)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +146 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +frictionAngle(=.5) +Contact friction angle (in radians). Hint : use ‘radians(degreesValue)’ in python scripts. +id(=-1, not shared) +Numeric id of this material; is non-negative only if this Material is shared (i.e. in O.materials), +-1 otherwise. This value is set automatically when the material is inserted to the simulation +via O.materials.append. +(This id was necessary since before boost::serialization was used, +shared pointers were not tracked properly; it might disappear in the future) +isDoubleTwist(=false) +Type of the mesh. If true two particles of the same material which body ids differ by one will +be considered as double-twisted interaction. +label(=uninitalized) +Textual identifier for this material; can be used for shared materials lookup in MaterialCon- +tainer. +lambdaEps(=0.47) +Parameter between 0 and 1 to reduce strain at failure of a double-twisted wire (as used by +[Bertrand2008]). [-] +lambdaF(=1.0) +Parameter between 0 and 1 introduced by [Thoeni2013] which defines where the shifted force- +displacement curve intersects with the new initial stiffness: F∗ = λFFelastic. [-] +lambdak(=0.73) +Parameter between 0 and 1 to compute the elastic stiffness of a double-twisted wire (as used +by [Bertrand2008]): kD = 2(λkkh + (1 − λk)kS). [-] +lambdau(=0.2) +Parameter between 0 and 1 introduced by [Thoeni2013] which defines the maximum shift +of the force-displacement curve in order to take an additional initial elongation (e.g. wire +distortion/imperfections, slipping, system flexibility) into account: ∆l∗ = λul0rnd(seed). [-] +newAssocState((Material)arg1) → State : +Return new State instance, which is associated with this Material. +Some materials have +special requirement on Body::state type and calling this function when the body is created +will ensure that they match. (This is done automatically if you use utils.sphere, … functions +from python). +poisson(=.25) +Poisson’s ratio or the ratio between shear and normal stiffness [-]. It has different meanings +depending on the Ip functor. +seed(=12345) +Integer used to initialize the random number generator for the calculation of the distortion. +If the integer is equal to 0 a internal seed number based on the time is computed. [-] +strainStressValues(=uninitalized) +Piecewise linear definition of the stress-strain curve by set of points (strain[-]>0,stress[Pa]>0) +for one single wire. Tension only is considered and the point (0,0) is not needed! NOTE: +Vector needs to be initialized! +strainStressValuesDT(=uninitalized) +Piecewise linear definition of the stress-strain curve by set of points (strain[-]>0,stress[Pa]>0) +for the double twist. Tension only is considered and the point (0,0) is not needed! If this value +is given the calculation will be based on two different stress-strain curves without considering +the parameter introduced by [Bertrand2008] (see [Thoeni2013]). +type +Three different types are considered: +2.3. +Yade wrapper class reference +147 + +Yade Documentation, Release 3rd ed. +0 +Corresponds to Bertrand’s approach (see [Bertrand2008]): only one stress-strain curve +is used +1 +New approach: two separate stress-strain curves can be used (see [Thoeni2013]) +2 +New approach with stochastically distorted contact model: two separate stress-strain +curves with changed initial stiffness and horizontal shift (shift is random if seed ≥ 0, +for more details see [Thoeni2013]) +By default the type is 0. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +young(=1e9) +elastic modulus [Pa]. It has different meanings depending on the Ip functor. +Bound +Bound +Aabb +Fig. 22: Inheritance graph of Bound. See also: Aabb. +class yade.wrapper.Bound(inherits Serializable) +Object bounding part of space taken by associated body; might be larger, used to optimalize +collision detection +color(=Vector3r(1, 1, 1)) +Color for rendering this object +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((Bound)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +lastUpdateIter(=0) +record iteration of last reference position update (auto-updated) +max(=Vector3r(NaN, NaN, NaN)) +Upper corner of box containing this bound (and the Body as well) +min(=Vector3r(NaN, NaN, NaN)) +Lower corner of box containing this bound (and the Body as well) +refPos(=Vector3r(NaN, NaN, NaN)) +Reference position, updated at current body position each time the bound dispatcher update +bounds (auto-updated) +sweepLength(=0) +The length used to increase the bounding boxe size, can be adjusted on the basis of previous +displacement if BoundDispatcher::targetInterv>0. (auto-updated) +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +148 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +class yade.wrapper.Aabb(inherits Bound → Serializable) +Axis-aligned bounding box, for use with InsertionSortCollider. (This class is quasi-redundant since +min,max are already contained in Bound itself. That might change at some point, though.) +color(=Vector3r(1, 1, 1)) +Color for rendering this object +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((Bound)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +lastUpdateIter(=0) +record iteration of last reference position update (auto-updated) +max(=Vector3r(NaN, NaN, NaN)) +Upper corner of box containing this bound (and the Body as well) +min(=Vector3r(NaN, NaN, NaN)) +Lower corner of box containing this bound (and the Body as well) +refPos(=Vector3r(NaN, NaN, NaN)) +Reference position, updated at current body position each time the bound dispatcher update +bounds (auto-updated) +sweepLength(=0) +The length used to increase the bounding boxe size, can be adjusted on the basis of previous +displacement if BoundDispatcher::targetInterv>0. (auto-updated) +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +2.3.2 Interactions +Interaction +class yade.wrapper.Interaction(inherits Serializable) +Interaction between pair of bodies. +cellDist +Distance of bodies in cell size units, if using periodic boundary conditions; id2 is shifted by +this number of cells from its State::pos coordinates for this interaction to exist. Assigned by +the collider. +Warning: +(internal) cellDist must survive Interaction::reset(), it is only initialized in +ctor. Interaction that was cancelled by the constitutive law, was reset() and became only +potential must have the period information if the geometric functor again makes it real. +Good to know after few days of debugging that :-) +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +geom(=uninitalized) +Geometry part of the interaction. +2.3. +Yade wrapper class reference +149 + +Yade Documentation, Release 3rd ed. +id1(=0) +Id of the first body in this interaction. +id2(=0) +Id of the second body in this interaction. +isActive +True if this interaction is active. Otherwise the forces from this interaction will not be taken +into account. True by default. +isReal +True if this interaction has both geom and phys; False otherwise. +iterBorn(=-1) +Step number at which the interaction was added to simulation. +iterMadeReal(=-1) +Step number at which the interaction was fully (in the sense of geom and phys) created. +(Should be touched only by IPhysDispatcher and InteractionLoop, therefore they are made +friends of Interaction +phys(=uninitalized) +Physical (material) part of the interaction. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +IGeom +IGeom +GenericSpheresContact +GridNodeGeom6D +ScGeom6D +ScGeom +ChCylGeom6D +TTetraSimpleGeom +TTetraGeom +ScGridCoGeom +CylScGeom +GridCoGridCoGeom +L3Geom +L6Geom +CylScGeom6D +Fig. 23: Inheritance graph of IGeom. +See also: ChCylGeom6D, CylScGeom, CylScGeom6D, Gener- +icSpheresContact, GridCoGridCoGeom, GridNodeGeom6D, L3Geom, L6Geom, ScGeom, ScGeom6D, +ScGridCoGeom, TTetraGeom, TTetraSimpleGeom. +class yade.wrapper.IGeom(inherits Serializable) +Geometrical configuration of interaction +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((IGeom)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +150 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +class yade.wrapper.ChCylGeom6D(inherits ScGeom6D → ScGeom → GenericSpheresContact → +IGeom → Serializable) +Test +bending(=Vector3r::Zero()) +Bending at contact as a vector defining axis of rotation and angle (angle=norm). +contactPoint(=uninitalized) +some reference point for the interaction (usually in the middle). (auto-computed) +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((IGeom)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +incidentVel((ScGeom)arg1, (Interaction)i[, (bool)avoidGranularRatcheting=True]) → Vec- +tor3 : +Return +incident +velocity +of +the +interaction +(see +also +Ig2_Sphere_Sphere_Sc- +Geom.avoidGranularRatcheting for explanation of the ratcheting argument). +initialOrientation1(=Quaternionr(1.0, 0.0, 0.0, 0.0)) +Orientation of body 1 one at initialisation time (auto-updated) +initialOrientation2(=Quaternionr(1.0, 0.0, 0.0, 0.0)) +Orientation of body 2 one at initialisation time (auto-updated) +normal(=uninitalized) +Unit vector oriented along the interaction, from particle #1, towards particle #2. +(auto- +updated) +penetrationDepth(=NaN) +Penetration distance of spheres (positive if overlapping) +refR1(=uninitalized) +Reference radius of particle #1. (auto-computed) +refR2(=uninitalized) +Reference radius of particle #2. (auto-computed) +relAngVel((ScGeom)arg1, (Interaction)i) → Vector3 : +Return relative angular velocity of the interaction. +shearInc(=Vector3r::Zero()) +Shear displacement increment in the last step +twist(=0) +Elastic twist angle (around normal axis) of the contact. +twistCreep(=Quaternionr(1.0, 0.0, 0.0, 0.0)) +Stored creep, substracted from total relative rotation for computation of elastic moment (auto- +updated) +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.CylScGeom(inherits ScGeom → GenericSpheresContact → IGeom → Seri- +alizable) +Geometry of a cylinder-sphere contact. +contactPoint(=uninitalized) +some reference point for the interaction (usually in the middle). (auto-computed) +2.3. +Yade wrapper class reference +151 + +Yade Documentation, Release 3rd ed. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((IGeom)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +end(=Vector3r::Zero()) +position of 2nd node (auto-updated) +id3(=0) +id of next chained cylinder (auto-updated) +incidentVel((ScGeom)arg1, (Interaction)i[, (bool)avoidGranularRatcheting=True]) → Vec- +tor3 : +Return +incident +velocity +of +the +interaction +(see +also +Ig2_Sphere_Sphere_Sc- +Geom.avoidGranularRatcheting for explanation of the ratcheting argument). +isDuplicate(=0) +this flag is turned true (1) automatically if the contact is shared between two chained cylinders. +A duplicated interaction will be skipped once by the constitutive law, so that only one contact +at a time is effective. If isDuplicate=2, it means one of the two duplicates has no longer +geometric interaction, and should be erased by the constitutive laws. +normal(=uninitalized) +Unit vector oriented along the interaction, from particle #1, towards particle #2. +(auto- +updated) +onNode(=false) +contact on node? +penetrationDepth(=NaN) +Penetration distance of spheres (positive if overlapping) +refR1(=uninitalized) +Reference radius of particle #1. (auto-computed) +refR2(=uninitalized) +Reference radius of particle #2. (auto-computed) +relAngVel((ScGeom)arg1, (Interaction)i) → Vector3 : +Return relative angular velocity of the interaction. +relPos(=0) +position of the contact on the cylinder (0: node-, 1:node+) (auto-updated) +shearInc(=Vector3r::Zero()) +Shear displacement increment in the last step +start(=Vector3r::Zero()) +position of 1st node (auto-updated) +trueInt(=-1) +Defines the body id of the cylinder where the contact is real, when CylScGeom::isDuplicate>0. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.CylScGeom6D(inherits ScGeom6D → ScGeom → GenericSpheresContact → +IGeom → Serializable) +Class representing geometry of two bodies in contact. The contact has 6 DOFs (normal, 2×shear, +twist, 2xbending) and uses ScGeom incremental algorithm for updating shear. +152 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +bending(=Vector3r::Zero()) +Bending at contact as a vector defining axis of rotation and angle (angle=norm). +contactPoint(=uninitalized) +some reference point for the interaction (usually in the middle). (auto-computed) +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((IGeom)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +end(=Vector3r::Zero()) +position of 2nd node (auto-updated) +id3(=0) +id of next chained cylinder (auto-updated) +incidentVel((ScGeom)arg1, (Interaction)i[, (bool)avoidGranularRatcheting=True]) → Vec- +tor3 : +Return +incident +velocity +of +the +interaction +(see +also +Ig2_Sphere_Sphere_Sc- +Geom.avoidGranularRatcheting for explanation of the ratcheting argument). +initialOrientation1(=Quaternionr(1.0, 0.0, 0.0, 0.0)) +Orientation of body 1 one at initialisation time (auto-updated) +initialOrientation2(=Quaternionr(1.0, 0.0, 0.0, 0.0)) +Orientation of body 2 one at initialisation time (auto-updated) +isDuplicate(=0) +this flag is turned true (1) automatically if the contact is shared between two chained cylinders. +A duplicated interaction will be skipped once by the constitutive law, so that only one contact +at a time is effective. If isDuplicate=2, it means one of the two duplicates has no longer +geometric interaction, and should be erased by the constitutive laws. +normal(=uninitalized) +Unit vector oriented along the interaction, from particle #1, towards particle #2. +(auto- +updated) +onNode(=false) +contact on node? +penetrationDepth(=NaN) +Penetration distance of spheres (positive if overlapping) +refR1(=uninitalized) +Reference radius of particle #1. (auto-computed) +refR2(=uninitalized) +Reference radius of particle #2. (auto-computed) +relAngVel((ScGeom)arg1, (Interaction)i) → Vector3 : +Return relative angular velocity of the interaction. +relPos(=0) +position of the contact on the cylinder (0: node-, 1:node+) (auto-updated) +shearInc(=Vector3r::Zero()) +Shear displacement increment in the last step +start(=Vector3r::Zero()) +position of 1st node (auto-updated) +2.3. +Yade wrapper class reference +153 + +Yade Documentation, Release 3rd ed. +trueInt(=-1) +Defines the body id of the cylinder where the contact is real, when CylScGeom::isDuplicate>0. +twist(=0) +Elastic twist angle (around normal axis) of the contact. +twistCreep(=Quaternionr(1.0, 0.0, 0.0, 0.0)) +Stored creep, substracted from total relative rotation for computation of elastic moment (auto- +updated) +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.GenericSpheresContact(inherits IGeom → Serializable) +Class uniting ScGeom and L3Geom, for the purposes of GlobalStiffnessTimeStepper. (It might be +removed in the future). Do not use this class directly. +contactPoint(=uninitalized) +some reference point for the interaction (usually in the middle). (auto-computed) +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((IGeom)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +normal(=uninitalized) +Unit vector oriented along the interaction, from particle #1, towards particle #2. +(auto- +updated) +refR1(=uninitalized) +Reference radius of particle #1. (auto-computed) +refR2(=uninitalized) +Reference radius of particle #2. (auto-computed) +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.GridCoGridCoGeom(inherits ScGeom → GenericSpheresContact → IGeom +→ Serializable) +Geometry of a GridConnection-GridConnection contact. +contactPoint(=uninitalized) +some reference point for the interaction (usually in the middle). (auto-computed) +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((IGeom)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +incidentVel((ScGeom)arg1, (Interaction)i[, (bool)avoidGranularRatcheting=True]) → Vec- +tor3 : +Return +incident +velocity +of +the +interaction +(see +also +Ig2_Sphere_Sphere_Sc- +Geom.avoidGranularRatcheting for explanation of the ratcheting argument). +154 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +normal(=uninitalized) +Unit vector oriented along the interaction, from particle #1, towards particle #2. +(auto- +updated) +penetrationDepth(=NaN) +Penetration distance of spheres (positive if overlapping) +refR1(=uninitalized) +Reference radius of particle #1. (auto-computed) +refR2(=uninitalized) +Reference radius of particle #2. (auto-computed) +relAngVel((ScGeom)arg1, (Interaction)i) → Vector3 : +Return relative angular velocity of the interaction. +relPos1(=0) +position of the contact on the first connection (0: node-, 1:node+) (auto-updated) +relPos2(=0) +position of the contact on the first connection (0: node-, 1:node+) (auto-updated) +shearInc(=Vector3r::Zero()) +Shear displacement increment in the last step +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.GridNodeGeom6D(inherits ScGeom6D → ScGeom → GenericSpheresCon- +tact → IGeom → Serializable) +Geometry of a GridNode-GridNode contact. Inherits almost everything from ScGeom6D. +bending(=Vector3r::Zero()) +Bending at contact as a vector defining axis of rotation and angle (angle=norm). +connectionBody(=uninitalized) +Reference to the GridNode Body who is linking the two GridNodes. +contactPoint(=uninitalized) +some reference point for the interaction (usually in the middle). (auto-computed) +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((IGeom)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +incidentVel((ScGeom)arg1, (Interaction)i[, (bool)avoidGranularRatcheting=True]) → Vec- +tor3 : +Return +incident +velocity +of +the +interaction +(see +also +Ig2_Sphere_Sphere_Sc- +Geom.avoidGranularRatcheting for explanation of the ratcheting argument). +initialOrientation1(=Quaternionr(1.0, 0.0, 0.0, 0.0)) +Orientation of body 1 one at initialisation time (auto-updated) +initialOrientation2(=Quaternionr(1.0, 0.0, 0.0, 0.0)) +Orientation of body 2 one at initialisation time (auto-updated) +normal(=uninitalized) +Unit vector oriented along the interaction, from particle #1, towards particle #2. +(auto- +updated) +2.3. +Yade wrapper class reference +155 + +Yade Documentation, Release 3rd ed. +penetrationDepth(=NaN) +Penetration distance of spheres (positive if overlapping) +refR1(=uninitalized) +Reference radius of particle #1. (auto-computed) +refR2(=uninitalized) +Reference radius of particle #2. (auto-computed) +relAngVel((ScGeom)arg1, (Interaction)i) → Vector3 : +Return relative angular velocity of the interaction. +shearInc(=Vector3r::Zero()) +Shear displacement increment in the last step +twist(=0) +Elastic twist angle (around normal axis) of the contact. +twistCreep(=Quaternionr(1.0, 0.0, 0.0, 0.0)) +Stored creep, substracted from total relative rotation for computation of elastic moment (auto- +updated) +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.L3Geom(inherits GenericSpheresContact → IGeom → Serializable) +Geometry of contact given in local coordinates with 3 degress of freedom: normal and two in shear +plane. [experimental] +F(=Vector3r::Zero()) +Applied force in local coordinates [debugging only, will be removed] +contactPoint(=uninitalized) +some reference point for the interaction (usually in the middle). (auto-computed) +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((IGeom)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +normal(=uninitalized) +Unit vector oriented along the interaction, from particle #1, towards particle #2. +(auto- +updated) +refR1(=uninitalized) +Reference radius of particle #1. (auto-computed) +refR2(=uninitalized) +Reference radius of particle #2. (auto-computed) +trsf(=Matrix3r::Identity()) +Transformation (rotation) from global to local coordinates. (the translation part is in Gener- +icSpheresContact.contactPoint) +u(=Vector3r::Zero()) +Displacement components, in local coordinates. (auto-updated) +u0 +Zero displacement value; u0 should be always subtracted from the geometrical displacement +u computed by appropriate IGeomFunctor, resulting in u. This value can be changed for +instance +156 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +1. by IGeomFunctor, e.g. to take in account large shear displacement value unrepresentable +by underlying geomeric algorithm based on quaternions) +2. by LawFunctor, to account for normal equilibrium position different from zero geometric +overlap (set once, just after the interaction is created) +3. by LawFunctor to account for plastic slip. +Note: +Never set an absolute value of u0, only increment, since both IGeomFunctor and +LawFunctor use it. If you need to keep track of plastic deformation, store it in IPhys isntead +(this might be changed: have u0 for LawFunctor exclusively, and a separate value stored +(when that is needed) inside classes deriving from L3Geom. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.L6Geom(inherits L3Geom → GenericSpheresContact → IGeom → Serializ- +able) +Geometric of contact in local coordinates with 6 degrees of freedom. [experimental] +F(=Vector3r::Zero()) +Applied force in local coordinates [debugging only, will be removed] +contactPoint(=uninitalized) +some reference point for the interaction (usually in the middle). (auto-computed) +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((IGeom)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +normal(=uninitalized) +Unit vector oriented along the interaction, from particle #1, towards particle #2. +(auto- +updated) +phi(=Vector3r::Zero()) +Rotation components, in local coordinates. (auto-updated) +phi0(=Vector3r::Zero()) +Zero rotation, should be always subtracted from phi to get the value. See L3Geom.u0. +refR1(=uninitalized) +Reference radius of particle #1. (auto-computed) +refR2(=uninitalized) +Reference radius of particle #2. (auto-computed) +trsf(=Matrix3r::Identity()) +Transformation (rotation) from global to local coordinates. (the translation part is in Gener- +icSpheresContact.contactPoint) +u(=Vector3r::Zero()) +Displacement components, in local coordinates. (auto-updated) +u0 +Zero displacement value; u0 should be always subtracted from the geometrical displacement +u computed by appropriate IGeomFunctor, resulting in u. This value can be changed for +instance +2.3. +Yade wrapper class reference +157 + +Yade Documentation, Release 3rd ed. +1. by IGeomFunctor, e.g. to take in account large shear displacement value unrepresentable +by underlying geomeric algorithm based on quaternions) +2. by LawFunctor, to account for normal equilibrium position different from zero geometric +overlap (set once, just after the interaction is created) +3. by LawFunctor to account for plastic slip. +Note: +Never set an absolute value of u0, only increment, since both IGeomFunctor and +LawFunctor use it. If you need to keep track of plastic deformation, store it in IPhys isntead +(this might be changed: have u0 for LawFunctor exclusively, and a separate value stored +(when that is needed) inside classes deriving from L3Geom. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.ScGeom(inherits GenericSpheresContact → IGeom → Serializable) +Class representing geometry of a contact point between two bodies. It is more general than sphere- +sphere contact even though it is primarily focused on spheres contact interactions (reason for +the ‘Sc’ naming); it is also used for representing contacts of a Sphere with non-spherical bodies +(Facet, Plane, Box, ChainedCylinder), or between two non-spherical bodies (ChainedCylinder). +The contact has 3 DOFs (normal and 2×shear) and uses incremental algorithm for updating shear. +We use symbols x, v, ω respectively for position, linear and angular velocities (all in global +coordinates) and r for particles radii; subscripted with 1 or 2 to distinguish 2 spheres in contact. +Then we define branch length and unit contact normal +l = ||x2 − x1||, n = +x2 − x1 +||x2 − x1|| +The relative velocity of the spheres is then +v12 = r1 + r2 +l +(v2 − v1) − (r2ω2 + r1ω1) × n +where the fraction multiplying translational velocities is to make the definition objective and avoid +ratcheting effects (see Ig2_Sphere_Sphere_ScGeom.avoidGranularRatcheting). The shear compo- +nent is +vs +12 = v12 − (n · v12)n. +Tangential displacement increment over last step then reads +∆xs +12 = ∆tvs +12. +contactPoint(=uninitalized) +some reference point for the interaction (usually in the middle). (auto-computed) +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((IGeom)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +incidentVel((ScGeom)arg1, (Interaction)i[, (bool)avoidGranularRatcheting=True]) → Vec- +tor3 : +Return +incident +velocity +of +the +interaction +(see +also +Ig2_Sphere_Sphere_Sc- +Geom.avoidGranularRatcheting for explanation of the ratcheting argument). +158 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +normal(=uninitalized) +Unit vector oriented along the interaction, from particle #1, towards particle #2. +(auto- +updated) +penetrationDepth(=NaN) +Penetration distance of spheres (positive if overlapping) +refR1(=uninitalized) +Reference radius of particle #1. (auto-computed) +refR2(=uninitalized) +Reference radius of particle #2. (auto-computed) +relAngVel((ScGeom)arg1, (Interaction)i) → Vector3 : +Return relative angular velocity of the interaction. +shearInc(=Vector3r::Zero()) +Shear displacement increment in the last step +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.ScGeom6D(inherits ScGeom → GenericSpheresContact → IGeom → Serial- +izable) +Class representing geometry of two bodies in contact. The contact has 6 DOFs (normal, 2×shear, +twist, 2xbending) and uses ScGeom incremental algorithm for updating shear. +bending(=Vector3r::Zero()) +Bending at contact as a vector defining axis of rotation and angle (angle=norm). +contactPoint(=uninitalized) +some reference point for the interaction (usually in the middle). (auto-computed) +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((IGeom)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +incidentVel((ScGeom)arg1, (Interaction)i[, (bool)avoidGranularRatcheting=True]) → Vec- +tor3 : +Return +incident +velocity +of +the +interaction +(see +also +Ig2_Sphere_Sphere_Sc- +Geom.avoidGranularRatcheting for explanation of the ratcheting argument). +initialOrientation1(=Quaternionr(1.0, 0.0, 0.0, 0.0)) +Orientation of body 1 one at initialisation time (auto-updated) +initialOrientation2(=Quaternionr(1.0, 0.0, 0.0, 0.0)) +Orientation of body 2 one at initialisation time (auto-updated) +normal(=uninitalized) +Unit vector oriented along the interaction, from particle #1, towards particle #2. +(auto- +updated) +penetrationDepth(=NaN) +Penetration distance of spheres (positive if overlapping) +refR1(=uninitalized) +Reference radius of particle #1. (auto-computed) +refR2(=uninitalized) +Reference radius of particle #2. (auto-computed) +2.3. +Yade wrapper class reference +159 + +Yade Documentation, Release 3rd ed. +relAngVel((ScGeom)arg1, (Interaction)i) → Vector3 : +Return relative angular velocity of the interaction. +shearInc(=Vector3r::Zero()) +Shear displacement increment in the last step +twist(=0) +Elastic twist angle (around normal axis) of the contact. +twistCreep(=Quaternionr(1.0, 0.0, 0.0, 0.0)) +Stored creep, substracted from total relative rotation for computation of elastic moment (auto- +updated) +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.ScGridCoGeom(inherits ScGeom6D → ScGeom → GenericSpheresContact +→ IGeom → Serializable) +Geometry of a GridConnection-Sphere contact. +bending(=Vector3r::Zero()) +Bending at contact as a vector defining axis of rotation and angle (angle=norm). +contactPoint(=uninitalized) +some reference point for the interaction (usually in the middle). (auto-computed) +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((IGeom)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +id3(=0) +id of the first GridNode. (auto-updated) +id4(=0) +id of the second GridNode. (auto-updated) +id5(=-1) +id of the third GridNode. (auto-updated) +incidentVel((ScGeom)arg1, (Interaction)i[, (bool)avoidGranularRatcheting=True]) → Vec- +tor3 : +Return +incident +velocity +of +the +interaction +(see +also +Ig2_Sphere_Sphere_Sc- +Geom.avoidGranularRatcheting for explanation of the ratcheting argument). +initialOrientation1(=Quaternionr(1.0, 0.0, 0.0, 0.0)) +Orientation of body 1 one at initialisation time (auto-updated) +initialOrientation2(=Quaternionr(1.0, 0.0, 0.0, 0.0)) +Orientation of body 2 one at initialisation time (auto-updated) +isDuplicate(=0) +this flag is turned true (1) automatically if the contact is shared between two Connections. A +duplicated interaction will be skipped once by the constitutive law, so that only one contact +at a time is effective. If isDuplicate=2, it means one of the two duplicates has no longer +geometric interaction, and should be erased by the constitutive laws. +normal(=uninitalized) +Unit vector oriented along the interaction, from particle #1, towards particle #2. +(auto- +updated) +160 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +penetrationDepth(=NaN) +Penetration distance of spheres (positive if overlapping) +refR1(=uninitalized) +Reference radius of particle #1. (auto-computed) +refR2(=uninitalized) +Reference radius of particle #2. (auto-computed) +relAngVel((ScGeom)arg1, (Interaction)i) → Vector3 : +Return relative angular velocity of the interaction. +relPos(=0) +position of the contact on the connection (0: node-, 1:node+) (auto-updated) +shearInc(=Vector3r::Zero()) +Shear displacement increment in the last step +trueInt(=-1) +Defines the body id of the GridConnection where the contact is real, when ScGridCo- +Geom::isDuplicate>0. +twist(=0) +Elastic twist angle (around normal axis) of the contact. +twistCreep(=Quaternionr(1.0, 0.0, 0.0, 0.0)) +Stored creep, substracted from total relative rotation for computation of elastic moment (auto- +updated) +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +weight(=Vector3r(0, 0, 0)) +barycentric coordinates of the projection point (auto-updated) +class yade.wrapper.TTetraGeom(inherits IGeom → Serializable) +Geometry of interaction between 2 tetrahedra, including volumetric characteristics +contactPoint(=uninitalized) +Contact point (global coords) +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((IGeom)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +equivalentCrossSection(=NaN) +Cross-section of the overlap (perpendicular to the axis of least inertia +equivalentPenetrationDepth(=NaN) +?? +maxPenetrationDepthA(=NaN) +?? +maxPenetrationDepthB(=NaN) +?? +normal(=uninitalized) +Normal of the interaction, directed in the sense of least inertia of the overlap volume +2.3. +Yade wrapper class reference +161 + +Yade Documentation, Release 3rd ed. +penetrationVolume(=NaN) +Volume of overlap [m3] +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.TTetraSimpleGeom(inherits IGeom → Serializable) +EXPERIMENTAL. Geometry of interaction between 2 tetrahedra +contactPoint(=uninitalized) +Contact point (global coords) +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((IGeom)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +flag(=0) +TODO +normal(=uninitalized) +Normal of the interaction TODO +penetrationVolume(=NaN) +Volume of overlap [m3] +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +IPhys +IPhys +MindlinCapillaryPhys +MindlinPhys +RotStiffFrictPhys +BubblePhys +InelastCohFrictPhys +FrictPhys +JCFpmPhys +NormShearPhys +NormPhys +FrictViscoPhys +MortarPhys +CohFrictPhys +LudingPhys +ViscElPhys +MindlinPhysCDM +CapillaryPhys +CpmPhys +ViscoFrictPhys +ViscElCapPhys +LubricationPhys +WirePhys +Fig. 24: Inheritance graph of IPhys. See also: BubblePhys, CapillaryPhys, CohFrictPhys, CpmPhys, +FrictPhys, FrictViscoPhys, InelastCohFrictPhys, JCFpmPhys, LubricationPhys, LudingPhys, Mindlin- +CapillaryPhys, MindlinPhys, MindlinPhysCDM, MortarPhys, NormPhys, NormShearPhys, RotStiff- +FrictPhys, ViscElCapPhys, ViscElPhys, ViscoFrictPhys, WirePhys. +class yade.wrapper.IPhys(inherits Serializable) +Physical (material) properties of interaction. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +162 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +dispHierarchy((IPhys)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.BubblePhys(inherits IPhys → Serializable) +Physics of bubble-bubble interactions, for use with BubbleMat +Dmax(=NaN) +Maximum penetrationDepth of the bubbles before the force displacement curve changes to +an artificial exponential curve. Setting this value will have no effect. See Law2_ScGeom_- +BubblePhys_Bubble::pctMaxForce for more information +static computeForce((float)arg1, (float)arg2, (float)arg3, (int)arg4, (float)arg5, (float)arg6, +(float)arg7, (BubblePhys)arg8) → float : +Computes the normal force acting between the two interacting bubbles using the Newton- +Rhapson method +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((IPhys)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +fN(=NaN) +Contact normal force +newtonIter(=50) +Maximum number of force iterations allowed +newtonTol(=1e-6) +Convergence criteria for force iterations +normalForce(=Vector3r::Zero()) +Normal force +rAvg(=NaN) +Average radius of the two interacting bubbles +surfaceTension(=NaN) +Surface tension of the surrounding liquid +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.CapillaryPhys(inherits FrictPhys → NormShearPhys → NormPhys → +IPhys → Serializable) +Physics (of interaction) for Law2_ScGeom_CapillaryPhys_Capillarity. +Delta1(=0.) +Defines the surface area wetted by the meniscus on the smallest grains of radius R1 (R1= -FnMax. [N] +FsMax(=0.) +computed from cohesion (or jointCohesion) to define the maximum admissible tangential force +in shear, for Fn=0. [N] +checkedForCluster(=false) +Have we checked if this int belongs in cluster? +clusterInts(=uninitalized) +vector of pointers to the broken interactions nearby constituting a cluster +2.3. +Yade wrapper class reference +171 + +Yade Documentation, Release 3rd ed. +clusteredEvent(=false) +is this interaction part of a cluster? +computedCentroid(=false) +Flag for moment calculation +crackJointAperture(=0.) +Relative displacement between 2 spheres (in case of a crack it is equivalent of the crack +aperture) +crossSection(=0.) +crossSection=pi*Rmin^2. [m2] +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dilation(=0.) +defines the normal displacement in the joint after sliding treshold. [m] +dispHierarchy((IPhys)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +elapsedIter(=0) +number of elapsed iterations for moment calculation +eventBeginTime(=0) +The time at which event initiated +eventNumber(=0) +cluster event number +firstMomentCalc(=true) +Flag for moment calculation (auto-updated) +initD(=0.) +equilibrium distance for interacting particles. Computed as the interparticular distance at +first contact detection. +interactionsAdded(=false) +have we added the ints associated with this event? +isBroken(=false) +flag for broken interactions +isCohesive(=false) +If false, particles interact in a frictional way. If true, particles are bonded regarding the given +cohesion and tensile strength (or their jointed variants). +isOnJoint(=false) +defined as true when both interacting particles are on joint and are in opposite sides of the +joint surface. In this case, mechanical parameters of the interaction are derived from the +‘’joint…’’ material properties of the particles. Furthermore, the normal of the interaction may +be re-oriented (see Law2_ScGeom_JCFpmPhys_JointedCohesiveFrictionalPM.smoothJoint). +isOnSlot(=false) +defined as true when interaction is located in the perforation slot (surface). +jointCumulativeSliding(=0.) +sliding distance for particles interacting on a joint. Used, when is true, to take into account +dilatancy due to shearing. [-] +172 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +jointNormal(=Vector3r::Zero()) +normal direction to the joint, deduced from e.g. . +kineticEnergy(=0) +kinetic energy of the two spheres participating in the interaction (easiest to store this value +with interaction instead of spheres since we are using this information for moment magnitude +estimations and associated interaction searches) +kn(=0) +Normal stiffness +ks(=0) +Shear stiffness +momentBroken(=false) +Flag for moment calculation +momentCalculated(=false) +Flag for moment calculation to avoid repeating twice the operations (auto-updated) +momentCentroid(=Vector3r::Zero()) +centroid of the AE event (avg location of clustered breaks) +momentEnergy(=0) +reference strain (or kinetic) energy of surrounding interactions (particles) +momentEnergyChange(=0) +storage of the maximum strain (or kinetic) energy change for surrounding interactions (par- +ticles) +momentMagnitude(=0) +Moment magnitude of a failed interaction +more(=false) +specifies if the interaction is crossed by more than 3 joints. If true, interaction is deleted +(temporary solution). +nearbyFound(=0) +Count used to debug moment calc +nearbyInts(=uninitalized) +vector of pointers to the nearby ints used for moment calc +normalForce(=Vector3r::Zero()) +Normal force after previous step (in global coordinates), as sustained by particle #2 (from +particle #1). +originalClusterEvent(=false) +the original AE event for a cluster +originalEvent(=uninitalized) +pointer to the original interaction of a cluster +shearForce(=Vector3r::Zero()) +Shear force after previous step (in global coordinates), as sustained by particle #2 (from +particle #1). +strainEnergy(=0) +strain energy of interaction +tanDilationAngle(=0.) +tangent of the angle defining the dilatancy of the joint surface (auto. computed from JCFp- +mMat.jointDilationAngle). [-] +tanFrictionAngle(=0.) +tangent of Coulomb friction angle for this interaction (auto. computed). [-] +2.3. +Yade wrapper class reference +173 + +Yade Documentation, Release 3rd ed. +temporalWindow(=0) +temporal window for the clustering algorithm +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.LubricationPhys(inherits ViscElPhys → FrictPhys → NormShearPhys → +NormPhys → IPhys → Serializable) +IPhys class for Lubrication w/o FlowEngine. Used by Law2_ScGeom_ImplicitLubricationPhys. +Fn(=0.0) +Normal force of the contact +Fv(=0.0) +Viscous force of the contact +a(=0.) +Mean radius [m] +cn(=NaN) +Normal viscous constant +contact(=false) +The spheres are in contact +cs(=NaN) +Shear viscous constant +delta(=0) +log(u) - used for scheme with δ = log(u) variable change +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((IPhys)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +eps(=0.001) +Roughness: fraction of radius used as roughness [-] +eta(=1) +Fluid viscosity [Pa.s] +keps(=1) +stiffness coefficient of the asperities [N/m]. Only used with resolution method=0, with reso- +lution>0 it is always equal to kn. +kn(=0) +Normal stiffness +kno(=0.0) +Coefficient for normal stiffness (Hertzian-like contact) [N/m^(3/2)] +ks(=0) +Shear stiffness +mR(=0.0) +Rolling resistance, see [Zhou1999536]. +mRtype(=1) +Rolling resistance type, see [Zhou1999536]. mRtype=1 - equation (3) in [Zhou1999536]; mR- +type=2 - equation (4) in [Zhou1999536] +174 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +mum(=0.3) +Friction coefficient [-] +normalContactForce(=Vector3r::Zero()) +Normal contact force [N] +normalForce(=Vector3r::Zero()) +Normal force after previous step (in global coordinates), as sustained by particle #2 (from +particle #1). +normalLubricationForce(=Vector3r::Zero()) +Normal lubrication force [N] +normalPotentialForce(=Vector3r::Zero()) +Normal force from potential other than contact [N] +nun(=0.0) +Coefficient for normal lubrication [N.s] +prevDotU(=0) +du/dt from previous integration - used for trapezoidal scheme (see Law2_ScGeom_Implic- +itLubricationPhys::resolution for choosing resolution scheme) +prev_un(=0) +Nondeformed distance (un) at t-dt [m] +shearContactForce(=Vector3r::Zero()) +Frictional contact force [N] +shearForce(=Vector3r::Zero()) +Shear force after previous step (in global coordinates), as sustained by particle #2 (from +particle #1). +shearLubricationForce(=Vector3r::Zero()) +Shear lubrication force [N] +slip(=false) +The contact is slipping +tangensOfFrictionAngle(=NaN) +tan of angle of friction +u(=-1) +Interfacial distance (u) at t-dt [m] +ue(=0.) +Surface deflection (ue) at t-dt [m] +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.LudingPhys(inherits FrictPhys → NormShearPhys → NormPhys → IPhys +→ Serializable) +IPhys created from LudingMat, for use with Law2_ScGeom_LudingPhys_Basic. +DeltMax(=NaN) +Maximum overlap between particles for a collision +DeltMin(=NaN) +MinimalDelta value of delta +DeltNull(=NaN) +Force free overlap, plastic contact deformation +DeltPMax(=NaN) +Maximum overlap between particles for the limit case +DeltPNull(=NaN) +Max force free overlap, plastic contact deformation +2.3. +Yade wrapper class reference +175 + +Yade Documentation, Release 3rd ed. +DeltPrev(=NaN) +Previous value of delta +G0(=NaN) +Viscous damping +PhiF(=NaN) +Dimensionless plasticity depth +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((IPhys)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +k1(=NaN) +Slope of loading plastic branch +k2(=NaN) +Slope of unloading and reloading elastic branch +kc(=NaN) +Slope of irreversible, tensile adhesive branch +kn(=0) +Normal stiffness +kp(=NaN) +Slope of unloading and reloading limit elastic branch +ks(=0) +Shear stiffness +normalForce(=Vector3r::Zero()) +Normal force after previous step (in global coordinates), as sustained by particle #2 (from +particle #1). +shearForce(=Vector3r::Zero()) +Shear force after previous step (in global coordinates), as sustained by particle #2 (from +particle #1). +tangensOfFrictionAngle(=NaN) +tan of angle of friction +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.MindlinCapillaryPhys(inherits +MindlinPhys +→ +RotStiffFrictPhys +→ +FrictPhys → NormShearPhys → NormPhys → +IPhys → Serializable) +Adds capillary physics to Mindlin’s interaction physics. +Delta1(=0.) +Defines the surface area wetted by the meniscus on the smallest grains of radius R1 (R11 means the ellipse is longer along normal stress axis. +failureCondition((MortarPhys)arg1, (float)arg2, (float)arg3) → bool : +Failure condition from normal stress and norm of shear stress (false=elastic, true=damaged) +kn(=0) +Normal stiffness +ks(=0) +Shear stiffness +neverDamage(=false) +If true, interactions remain elastic regardless stresses +normalForce(=Vector3r::Zero()) +Normal force after previous step (in global coordinates), as sustained by particle #2 (from +particle #1). +shearForce(=Vector3r::Zero()) +Shear force after previous step (in global coordinates), as sustained by particle #2 (from +particle #1). +sigmaN +Current normal stress (auto-updated) +sigmaT +Current shear stress (auto-updated) +tangensOfFrictionAngle(=NaN) +tan of angle of friction +tensileStrength(=NaN) +tensileStrength [Pa] +182 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.NormPhys(inherits IPhys → Serializable) +Abstract class for interactions that have normal stiffness. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((IPhys)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +kn(=0) +Normal stiffness +normalForce(=Vector3r::Zero()) +Normal force after previous step (in global coordinates), as sustained by particle #2 (from +particle #1). +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.NormShearPhys(inherits NormPhys → IPhys → Serializable) +Abstract class for interactions that have shear stiffnesses, in addition to normal stiffness. This class +is used in the PFC3d-style stiffness timestepper. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((IPhys)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +kn(=0) +Normal stiffness +ks(=0) +Shear stiffness +normalForce(=Vector3r::Zero()) +Normal force after previous step (in global coordinates), as sustained by particle #2 (from +particle #1). +shearForce(=Vector3r::Zero()) +Shear force after previous step (in global coordinates), as sustained by particle #2 (from +particle #1). +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.RotStiffFrictPhys(inherits FrictPhys → NormShearPhys → NormPhys +→ IPhys → Serializable) +Version of FrictPhys with a rotational stiffness +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +2.3. +Yade wrapper class reference +183 + +Yade Documentation, Release 3rd ed. +dispHierarchy((IPhys)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +kn(=0) +Normal stiffness +kr(=0) +rotational stiffness [N.m/rad] +ks(=0) +Shear stiffness +ktw(=0) +twist stiffness [N.m/rad] +normalForce(=Vector3r::Zero()) +Normal force after previous step (in global coordinates), as sustained by particle #2 (from +particle #1). +shearForce(=Vector3r::Zero()) +Shear force after previous step (in global coordinates), as sustained by particle #2 (from +particle #1). +tangensOfFrictionAngle(=NaN) +tan of angle of friction +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.ViscElCapPhys(inherits ViscElPhys → FrictPhys → NormShearPhys → +NormPhys → IPhys → Serializable) +IPhys created from ViscElCapMat, for use with Law2_ScGeom_ViscElCapPhys_Basic. +Capillar(=false) +True, if capillar forces need to be added. +CapillarType(=None_Capillar) +Different types of capillar interaction: Willett_numeric, Willett_analytic, Weigert, Rabi- +novich, Lambert, Soulie +Fn(=0.0) +Normal force of the contact +Fv(=0.0) +Viscous force of the contact +Vb(=0.0) +Liquid bridge volume [m^3] +cn(=NaN) +Normal viscous constant +cs(=NaN) +Shear viscous constant +dcap(=0.0) +Damping coefficient for the capillary phase [-] +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((IPhys)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +184 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +gamma(=0.0) +Surface tension [N/m] +kn(=0) +Normal stiffness +ks(=0) +Shear stiffness +liqBridgeActive(=false) +Whether liquid bridge is active at the moment +liqBridgeCreated(=false) +Whether liquid bridge was created, only after a normal contact of spheres +mR(=0.0) +Rolling resistance, see [Zhou1999536]. +mRtype(=1) +Rolling resistance type, see [Zhou1999536]. mRtype=1 - equation (3) in [Zhou1999536]; mR- +type=2 - equation (4) in [Zhou1999536] +normalForce(=Vector3r::Zero()) +Normal force after previous step (in global coordinates), as sustained by particle #2 (from +particle #1). +sCrit(=false) +Critical bridge length [m] +shearForce(=Vector3r::Zero()) +Shear force after previous step (in global coordinates), as sustained by particle #2 (from +particle #1). +tangensOfFrictionAngle(=NaN) +tan of angle of friction +theta(=0.0) +Contact angle [rad] +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.ViscElPhys(inherits FrictPhys → NormShearPhys → NormPhys → IPhys +→ Serializable) +IPhys created from ViscElMat, for use with Law2_ScGeom_ViscElPhys_Basic. +Fn(=0.0) +Normal force of the contact +Fv(=0.0) +Viscous force of the contact +cn(=NaN) +Normal viscous constant +cs(=NaN) +Shear viscous constant +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +2.3. +Yade wrapper class reference +185 + +Yade Documentation, Release 3rd ed. +dispHierarchy((IPhys)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +kn(=0) +Normal stiffness +ks(=0) +Shear stiffness +mR(=0.0) +Rolling resistance, see [Zhou1999536]. +mRtype(=1) +Rolling resistance type, see [Zhou1999536]. mRtype=1 - equation (3) in [Zhou1999536]; mR- +type=2 - equation (4) in [Zhou1999536] +normalForce(=Vector3r::Zero()) +Normal force after previous step (in global coordinates), as sustained by particle #2 (from +particle #1). +shearForce(=Vector3r::Zero()) +Shear force after previous step (in global coordinates), as sustained by particle #2 (from +particle #1). +tangensOfFrictionAngle(=NaN) +tan of angle of friction +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.ViscoFrictPhys(inherits FrictPhys → NormShearPhys → NormPhys → +IPhys → Serializable) +Temporary version of FrictPhys for compatibility reasons +creepedShear(=Vector3r(0, 0, 0)) +Creeped force (parallel) +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((IPhys)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +kn(=0) +Normal stiffness +ks(=0) +Shear stiffness +normalForce(=Vector3r::Zero()) +Normal force after previous step (in global coordinates), as sustained by particle #2 (from +particle #1). +shearForce(=Vector3r::Zero()) +Shear force after previous step (in global coordinates), as sustained by particle #2 (from +particle #1). +186 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +tangensOfFrictionAngle(=NaN) +tan of angle of friction +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.WirePhys(inherits FrictPhys → NormShearPhys → NormPhys → IPhys → +Serializable) +Representation of a single interaction of the WirePM type, storage for relevant parameters +dL(=0.) +Additional wire length for considering the distortion for WireMat type=2 (see [Thoeni2013]). +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispHierarchy((IPhys)arg1[, (bool)names=True]) → list : +Return list of dispatch classes (from down upwards), starting with the class instance itself, +top-level indexable at last. If names is true (default), return class names rather than numerical +indices. +dispIndex +Return class index of this instance. +displForceValues(=uninitalized) +Defines the values for force-displacement curve. +initD(=0.) +Equilibrium distance for particles. Computed as the initial inter-particular distance when +particle are linked. +isDoubleTwist(=false) +If true the properties of the interaction will be defined as a double-twisted wire. +isLinked(=false) +If true particles are linked and will interact. Interactions are linked automatically by the +definition of the corresponding interaction radius. The value is false if the wire breaks (no +more interaction). +isShifted(=false) +If true WireMat type=2 and the force-displacement curve will be shifted. +kn(=0) +Normal stiffness +ks(=0) +Shear stiffness +limitFactor(=0.) +This value indicates on how far from failing the wire is, e.g. +actual normal displacement +divided by admissible normal displacement. +normalForce(=Vector3r::Zero()) +Normal force after previous step (in global coordinates), as sustained by particle #2 (from +particle #1). +plastD +Plastic part of the inter-particular distance of the previous step. +Note: +Only elastic displacements are reversible (the elastic stiffness is used for unloading) +and compressive forces are inadmissible. The compressive stiffness is assumed to be equal to +zero. +2.3. +Yade wrapper class reference +187 + +Yade Documentation, Release 3rd ed. +shearForce(=Vector3r::Zero()) +Shear force after previous step (in global coordinates), as sustained by particle #2 (from +particle #1). +stiffnessValues(=uninitalized) +Defines the values for the various stiffnesses (the elastic stiffness is stored as kn). +tangensOfFrictionAngle(=NaN) +tan of angle of friction +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +2.3.3 Global engines +GlobalEngine +GlobalEngine +ElasticContactLaw +Collider +BoundaryController +TimeStepper +FieldApplier +GlobalStiffnessTimeStepper +CircularFactory +SpheresFactory +InteractionLoop +NewtonIntegrator +PeriodicEngine +BoxFactory +RungeKuttaCashKarp54Integrator +Integrator +ForceResetter +TetraVolumetricLaw +Law2_ScGeom_CapillaryPhys_Capillarity +CohesiveFrictionalContactLaw +FacetTopologyAnalyzer +Fig. 25: Inheritance graph of GlobalEngine, gray dashed classes are discussed in their own sections: +Collider, BoundaryController, FieldApplier, PeriodicEngine. +See also: BoxFactory, CircularFactory, +CohesiveFrictionalContactLaw, ElasticContactLaw, FacetTopologyAnalyzer, ForceResetter, GlobalStiff- +nessTimeStepper, Integrator, InteractionLoop, Law2_ScGeom_CapillaryPhys_Capillarity, NewtonInte- +grator, RungeKuttaCashKarp54Integrator, SpheresFactory, TetraVolumetricLaw, TimeStepper. +class yade.wrapper.GlobalEngine(inherits Engine → Serializable) +Engine that will generally affect the whole simulation (contrary to PartialEngine). +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +188 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.BoxFactory(inherits SpheresFactory → GlobalEngine → Engine → Serial- +izable) +Box geometry of the SpheresFactory region, given by extents and center +PSDcalculateMass(=true) +PSD-Input is in mass (true), otherwise the number of particles will be considered. +PSDcum(=uninitalized) +PSD-dispersion, cumulative procent meanings [-] +PSDsizes(=uninitalized) +PSD-dispersion, sizes of cells, Diameter [m] +blockedDOFs(=””) +Blocked degress of freedom +center(=Vector3r(NaN, NaN, NaN)) +Center of the region +color(=Vector3r(-1, -1, -1)) +Use the color for newly created particles, if specified +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +exactDiam(=true) +If true, the particles only with the defined in PSDsizes diameters will be created. Otherwise +the diameter will be randomly chosen in the range [PSDsizes[i-1]:PSDsizes[i]], in this case the +length of PSDsizes should be more on 1, than the length of PSDcum. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +extents(=Vector3r(NaN, NaN, NaN)) +Extents of the region +goalMass(=0) +Total mass that should be attained at the end of the current step. (auto-updated) +ids(=uninitalized) +ids of created bodies +2.3. +Yade wrapper class reference +189 + +Yade Documentation, Release 3rd ed. +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +mask(=-1) +groupMask to apply for newly created spheres +massFlowRate(=NaN) +Mass flow rate [kg/s] +materialId(=-1) +Shared material id to use for newly created spheres (can be negative to count from the end) +maxAttempt(=5000) +Maximum number of attempts to position a new sphere randomly. +maxMass(=-1) +Maximal mass at which to stop generating new particles regardless of massFlowRate. +if +maxMass=-1 - this parameter is ignored. +maxParticles(=100) +The number of particles at which to stop generating new ones regardless of massFlowRate. if +maxParticles=-1 - this parameter is ignored . +normal(=Vector3r(NaN, NaN, NaN)) +Orientation of the region’s geometry, direction of particle’s velocites if normalVel is not set. +normalVel(=Vector3r(NaN, NaN, NaN)) +Direction of particle’s velocites. +numParticles(=0) +Cummulative number of particles produces so far (auto-updated) +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +rMax(=NaN) +Maximum radius of generated spheres (uniform distribution) +rMin(=NaN) +Minimum radius of generated spheres (uniform distribution) +silent(=false) +If true no complain about excessing maxAttempt but disable the factory (by set mass- +FlowRate=0). +stopIfFailed(=true) +If true, the SpheresFactory stops (sets massFlowRate=0), when maximal number of attempts +to insert particle exceed. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +totalMass(=0) +Mass of spheres that was produced so far. (auto-updated) +totalVolume(=0) +Volume of spheres that was produced so far. (auto-updated) +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +190 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +vAngle(=NaN) +Maximum angle by which the initial sphere velocity deviates from the normal. +vMax(=NaN) +Maximum velocity norm of generated spheres (uniform distribution) +vMin(=NaN) +Minimum velocity norm of generated spheres (uniform distribution) +class yade.wrapper.CircularFactory(inherits SpheresFactory → GlobalEngine → Engine → +Serializable) +Circular geometry of the SpheresFactory region. It can be disk (given by radius and center), or +cylinder (given by radius, length and center). +PSDcalculateMass(=true) +PSD-Input is in mass (true), otherwise the number of particles will be considered. +PSDcum(=uninitalized) +PSD-dispersion, cumulative procent meanings [-] +PSDsizes(=uninitalized) +PSD-dispersion, sizes of cells, Diameter [m] +blockedDOFs(=””) +Blocked degress of freedom +center(=Vector3r(NaN, NaN, NaN)) +Center of the region +color(=Vector3r(-1, -1, -1)) +Use the color for newly created particles, if specified +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +exactDiam(=true) +If true, the particles only with the defined in PSDsizes diameters will be created. Otherwise +the diameter will be randomly chosen in the range [PSDsizes[i-1]:PSDsizes[i]], in this case the +length of PSDsizes should be more on 1, than the length of PSDcum. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +goalMass(=0) +Total mass that should be attained at the end of the current step. (auto-updated) +ids(=uninitalized) +ids of created bodies +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +length(=0) +Length of the cylindrical region (0 by default) +mask(=-1) +groupMask to apply for newly created spheres +2.3. +Yade wrapper class reference +191 + +Yade Documentation, Release 3rd ed. +massFlowRate(=NaN) +Mass flow rate [kg/s] +materialId(=-1) +Shared material id to use for newly created spheres (can be negative to count from the end) +maxAttempt(=5000) +Maximum number of attempts to position a new sphere randomly. +maxMass(=-1) +Maximal mass at which to stop generating new particles regardless of massFlowRate. +if +maxMass=-1 - this parameter is ignored. +maxParticles(=100) +The number of particles at which to stop generating new ones regardless of massFlowRate. if +maxParticles=-1 - this parameter is ignored . +normal(=Vector3r(NaN, NaN, NaN)) +Orientation of the region’s geometry, direction of particle’s velocites if normalVel is not set. +normalVel(=Vector3r(NaN, NaN, NaN)) +Direction of particle’s velocites. +numParticles(=0) +Cummulative number of particles produces so far (auto-updated) +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +rMax(=NaN) +Maximum radius of generated spheres (uniform distribution) +rMin(=NaN) +Minimum radius of generated spheres (uniform distribution) +radius(=NaN) +Radius of the region +silent(=false) +If true no complain about excessing maxAttempt but disable the factory (by set mass- +FlowRate=0). +stopIfFailed(=true) +If true, the SpheresFactory stops (sets massFlowRate=0), when maximal number of attempts +to insert particle exceed. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +totalMass(=0) +Mass of spheres that was produced so far. (auto-updated) +totalVolume(=0) +Volume of spheres that was produced so far. (auto-updated) +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +vAngle(=NaN) +Maximum angle by which the initial sphere velocity deviates from the normal. +192 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +vMax(=NaN) +Maximum velocity norm of generated spheres (uniform distribution) +vMin(=NaN) +Minimum velocity norm of generated spheres (uniform distribution) +class yade.wrapper.CohesiveFrictionalContactLaw(inherits GlobalEngine → Engine → Se- +rializable) +[DEPRECATED] Loop over interactions applying Law2_ScGeom6D_CohFrictPhys_CohesionMo- +ment on all interactions. +Note: +Use InteractionLoop and Law2_ScGeom6D_CohFrictPhys_CohesionMoment instead of +this class for performance reasons. +always_use_moment_law(=false) +If true, use bending/twisting moments at all contacts. If false, compute moments only for +cohesive contacts. +creep_viscosity(=false) +creep viscosity [Pa.s/m]. probably should be moved to Ip2_CohFrictMat_CohFrictMat_- +CohFrictPhys… +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +neverErase(=false) +Keep interactions even if particles go away from each other (only in case another constitutive +law is in the scene, e.g. Law2_ScGeom_CapillaryPhys_Capillarity) +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +shear_creep(=false) +activate creep on the shear force, using CohesiveFrictionalContactLaw::creep_viscosity. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +twist_creep(=false) +activate creep on the twisting moment, using CohesiveFrictionalContactLaw::creep_viscosity. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +2.3. +Yade wrapper class reference +193 + +Yade Documentation, Release 3rd ed. +class yade.wrapper.ElasticContactLaw(inherits GlobalEngine → Engine → Serializable) +[DEPRECATED] Loop over interactions applying Law2_ScGeom_FrictPhys_CundallStrack on all +interactions. +Note: +Use InteractionLoop and Law2_ScGeom_FrictPhys_CundallStrack instead of this class +for performance reasons. +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +neverErase(=false) +Keep interactions even if particles go away from each other (only in case another constitutive +law is in the scene, e.g. Law2_ScGeom_CapillaryPhys_Capillarity) +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.FacetTopologyAnalyzer(inherits GlobalEngine → Engine → Serializable) +Initializer for filling adjacency geometry data for facets. +Common vertices and common edges are identified and mutual angle between facet faces is written +to Facet instances. If facets don’t move with respect to each other, this must be done only at the +beginng. +commonEdgesFound(=0) +how many common edges were identified during last run. (auto-updated) +commonVerticesFound(=0) +how many common vertices were identified during last run. (auto-updated) +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +194 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +projectionAxis(=Vector3r::UnitX()) +Axis along which to do the initial vertex sort +relTolerance(=1e-4) +maximum distance of ‘identical’ vertices, relative to minimum facet size +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.ForceResetter(inherits GlobalEngine → Engine → Serializable) +Reset all forces stored in Scene::forces (O.forces in python). Typically, this is the first engine to +be run at every step. In addition, reset those energies that should be reset, if energy tracing is +enabled. +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +2.3. +Yade wrapper class reference +195 + +Yade Documentation, Release 3rd ed. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.GlobalStiffnessTimeStepper(inherits TimeStepper → GlobalEngine → +Engine → Serializable) +An engine assigning the time-step as a fraction of the minimum eigen-period in the problem. The +derivation is detailed in the chapter on DEM formulation. The viscEl option enables to evaluate +the timestep in a similar way for the visco-elastic contact law Law2_ScGeom_ViscElPhys_Basic, +more detail in GlobalStiffnessTimestepper::viscEl. +active(=true) +is the engine active? +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +defaultDt(=-1) +used as the initial value of the timestep (especially useful in the first steps when no contact +exist). +If negative, it will be defined by utils.PWaveTimeStep * GlobalStiffnessTimeStep- +per::timestepSafetyCoefficient +densityScaling(=false) +(auto-updated) don’t modify this value if you don’t plan to modify the scaling factor manually +for some bodies. In most cases, it is enough to set NewtonIntegrator::densityScaling and let +this one be adjusted automatically. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +maxDt(=Mathr::MAX_REAL) +if positive, used as max value of the timestep whatever the computed value +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +previousDt(=Mathr::MAX_REAL) +last computed dt (auto-updated) +targetDt(=1) +if NewtonIntegrator::densityScaling is active, this value will be used as the simulation timestep +and the scaling will use this value of dt as the target value. The value of targetDt is arbitrary +and should have no effect in the result in general. However if some bodies have imposed +velocities, for instance, they will move more or less per each step depending on this value. +timeStepUpdateInterval(=1) +dt update interval +timestepSafetyCoefficient(=0.8) +safety factor between the minimum eigen-period and the final assigned dt (less than 1) +196 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +viscEl(=false) +To use with ViscElPhys. if True, evaluate separetly the minimum eigen-period in the problem +considering only the elastic contribution on one hand (spring only), and only the viscous +contribution on the other hand (dashpot only). +Take then the minimum of the two and +use the safety coefficient GlobalStiffnessTimestepper::timestepSafetyCoefficient to take into +account the possible coupling between the two contribution. +class yade.wrapper.Integrator(inherits TimeStepper → GlobalEngine → Engine → Serializ- +able) +Integration Engine Interface. +active(=true) +is the engine active? +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +integrationsteps(=uninitalized) +all integrationsteps count as all succesfull substeps +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +maxVelocitySq(=NaN) +store square of max. velocity, for informative purposes; computed again at every step. (auto- +updated) +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +slaves +List of lists of Engines to calculate the force acting on the particles; to obtain the derivatives +of the states, engines inside will be run sequentially. +timeStepUpdateInterval(=1) +dt update interval +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +2.3. +Yade wrapper class reference +197 + +Yade Documentation, Release 3rd ed. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.InteractionLoop(inherits GlobalEngine → Engine → Serializable) +Unified dispatcher for handling interaction loop at every step, for parallel performance reasons. +Special constructor +Constructs from 3 lists of Ig2, Ip2, Law2 functors respectively; they will be passed to internal dis- +patchers, which you might retrieve as geomDispatcher, physDispatcher, lawDispatcher respectively. +callbacks(=uninitalized) +Callbacks which will be called for every Interaction, if activated. +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +geomDispatcher(=new IGeomDispatcher) +IGeomDispatcher object that is used for dispatch. +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +lawDispatcher(=new LawDispatcher) +LawDispatcher object used for dispatch. +loopOnSortedInteractions(=false) +If true, the main interaction loop will occur on a sorted list of interactions. This is SLOW +but useful to workaround floating point force addition non reproducibility when debugging +parallel implementations of yade. +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +physDispatcher(=new IPhysDispatcher) +IPhysDispatcher object used for dispatch. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Law2_ScGeom_CapillaryPhys_Capillarity(inherits GlobalEngine → En- +gine → Serializable) +This law allows one to take into account capillary forces/effects between spheres coming from the +presence of interparticular liquid bridges (menisci). +198 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +The control parameter is the capillary pressure (or suction) Uc = Ugas - Uliquid. Liquid bridges +properties (volume V, extent over interacting grains delta1 and delta2) are computed as a result +of the defined capillary pressure and of the interacting geometry (spheres radii and interparticular +distance). +References: in english [Scholtes2009b]; more detailed, but in french [Scholtes2009d]. +The law needs ascii files M(r=i) with i=R1/R2 to work (see https://yade-dem.org/wiki/ +CapillaryTriaxialTest). These ASCII files contain a set of results from the resolution of the Laplace- +Young equation for different configurations of the interacting geometry, assuming a null wetting +angle. +In order to allow capillary forces between distant spheres, it is necessary to enlarge the bounding +boxes using Bo1_Sphere_Aabb::aabbEnlargeFactor and make the Ig2 define define distant inter- +actions via interactionDetectionFactor. It is also necessary to disable interactions removal by the +constitutive law (Law2). +The only combinations of laws supported are currently capillary law ++ Law2_ScGeom_FrictPhys_CundallStrack and capillary law + Law2_ScGeom_MindlinPhys_- +Mindlin (and the other variants of Hertz-Mindlin). +See CapillaryPhys-example.py for an example script. +binaryFusion(=true) +If true, capillary forces are set to zero as soon as, at least, 1 overlap (menisci fusion) is detected. +Otherwise fCap = fCap / (fusionNumber + 1 ) +capillaryPressure(=0.) +Value of the capillary pressure Uc defined as Uc=Ugas-Uliquid +createDistantMeniscii(=false) +Generate meniscii between distant spheres? Else only maintain the existing ones. For modeling +a wetting path this flag should always be false. For a drying path it should be true for one +step (initialization) then false, as in the logic of [Scholtes2009c] +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +fusionDetection(=false) +If true potential menisci overlaps are checked, computing fusionNumber for each capillary +interaction, and reducing fCap according to binaryFusion +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +suffCapFiles(=””) +Capillary files suffix: M(r=X)suffCapFiles +2.3. +Yade wrapper class reference +199 + +Yade Documentation, Release 3rd ed. +surfaceTension(=0.073) +Value of considered surface tension +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.NewtonIntegrator(inherits GlobalEngine → Engine → Serializable) +Engine integrating newtonian motion equations. +dampGravity(=true) +By default, numerical damping applies to ALL forces, even gravity. If this option is set to +false, then the gravity forces calculated based on NewtonIntegrator.gravity are excluded from +the damping calculation. This option has no effect on gravity forces added by GravityEngine. +damping(=0.2) +damping coefficient for Cundall’s non viscous damping (see Numerical damping and +[Chareyre2005]) +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +densityScaling +if True, then density scaling [Pfc3dManual30] will be applied in order to have a critical timestep +equal to GlobalStiffnessTimeStepper::targetDt for all bodies. This option makes the simulation +unrealistic from a dynamic point of view, but may speedup quasistatic simulations. In rare +situations, it could be useful to not set the scalling factor automatically for each body (which +the time-stepper does). In such case revert GlobalStiffnessTimeStepper.densityScaling to False. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +exactAsphericalRot(=true) +Enable more exact body rotation integrator for aspherical bodies only, using formulation from +[Allen1989], pg. 89. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +gravity(=Vector3r::Zero()) +Gravitational acceleration (effectively replaces GravityEngine). +kinSplit(=false) +Whether to separately track translational and rotational kinetic energy. +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +mask(=-1) +If mask defined and the bitwise AND between mask and body‘s groupMask gives 0, the body +will not move/rotate. Velocities and accelerations will be calculated not paying attention to +this parameter. +maxVelocitySq(=0) +stores max. displacement, based on which we trigger collision detection. (auto-updated) +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +200 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +prevVelGrad(=Matrix3r::Zero()) +Store previous velocity gradient (Cell::velGrad) to track acceleration. (auto-updated) +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +warnNoForceReset(=true) +Warn when forces were not resetted in this step by ForceResetter; this mostly points to +ForceResetter being forgotten incidentally and should be disabled only with a good reason. +class yade.wrapper.RungeKuttaCashKarp54Integrator(inherits Integrator → TimeStepper → +GlobalEngine → Engine → Serializ- +able) +RungeKuttaCashKarp54Integrator engine. +__init__((object)arg1) → None +object __init__(tuple args, dict kwds) +__init__( (object)arg1, (list)arg2) -> object : Construct from (possibly nested) list +of slaves. +a_dxdt(=1.0) +a_x(=1.0) +abs_err(=1e-6) +Relative integration tolerance +active(=true) +is the engine active? +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +integrationsteps(=uninitalized) +all integrationsteps count as all succesfull substeps +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +maxVelocitySq(=NaN) +store square of max. velocity, for informative purposes; computed again at every step. (auto- +updated) +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +2.3. +Yade wrapper class reference +201 + +Yade Documentation, Release 3rd ed. +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +rel_err(=1e-6) +Absolute integration tolerance +slaves +List of lists of Engines to calculate the force acting on the particles; to obtain the derivatives +of the states, engines inside will be run sequentially. +stepsize(=1e-6) +It is not important for an adaptive integration but important for the observer for setting the +found states after integration +timeStepUpdateInterval(=1) +dt update interval +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.SpheresFactory(inherits GlobalEngine → Engine → Serializable) +Engine for spitting spheres based on mass flow rate, particle size distribution etc. Initial velocity +of particles is given by vMin, vMax, the massFlowRate determines how many particles to generate +at each step. +When goalMass is attained or positive maxParticles is reached, the engine does +not produce particles anymore. Geometry of the region should be defined in a derived engine by +overridden SpheresFactory::pickRandomPosition(). +A sample script for this engine is in scripts/spheresFactory.py. +PSDcalculateMass(=true) +PSD-Input is in mass (true), otherwise the number of particles will be considered. +PSDcum(=uninitalized) +PSD-dispersion, cumulative procent meanings [-] +PSDsizes(=uninitalized) +PSD-dispersion, sizes of cells, Diameter [m] +blockedDOFs(=””) +Blocked degress of freedom +color(=Vector3r(-1, -1, -1)) +Use the color for newly created particles, if specified +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +exactDiam(=true) +If true, the particles only with the defined in PSDsizes diameters will be created. Otherwise +the diameter will be randomly chosen in the range [PSDsizes[i-1]:PSDsizes[i]], in this case the +length of PSDsizes should be more on 1, than the length of PSDcum. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +202 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +goalMass(=0) +Total mass that should be attained at the end of the current step. (auto-updated) +ids(=uninitalized) +ids of created bodies +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +mask(=-1) +groupMask to apply for newly created spheres +massFlowRate(=NaN) +Mass flow rate [kg/s] +materialId(=-1) +Shared material id to use for newly created spheres (can be negative to count from the end) +maxAttempt(=5000) +Maximum number of attempts to position a new sphere randomly. +maxMass(=-1) +Maximal mass at which to stop generating new particles regardless of massFlowRate. +if +maxMass=-1 - this parameter is ignored. +maxParticles(=100) +The number of particles at which to stop generating new ones regardless of massFlowRate. if +maxParticles=-1 - this parameter is ignored . +normal(=Vector3r(NaN, NaN, NaN)) +Orientation of the region’s geometry, direction of particle’s velocites if normalVel is not set. +normalVel(=Vector3r(NaN, NaN, NaN)) +Direction of particle’s velocites. +numParticles(=0) +Cummulative number of particles produces so far (auto-updated) +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +rMax(=NaN) +Maximum radius of generated spheres (uniform distribution) +rMin(=NaN) +Minimum radius of generated spheres (uniform distribution) +silent(=false) +If true no complain about excessing maxAttempt but disable the factory (by set mass- +FlowRate=0). +stopIfFailed(=true) +If true, the SpheresFactory stops (sets massFlowRate=0), when maximal number of attempts +to insert particle exceed. +2.3. +Yade wrapper class reference +203 + +Yade Documentation, Release 3rd ed. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +totalMass(=0) +Mass of spheres that was produced so far. (auto-updated) +totalVolume(=0) +Volume of spheres that was produced so far. (auto-updated) +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +vAngle(=NaN) +Maximum angle by which the initial sphere velocity deviates from the normal. +vMax(=NaN) +Maximum velocity norm of generated spheres (uniform distribution) +vMin(=NaN) +Minimum velocity norm of generated spheres (uniform distribution) +class yade.wrapper.TetraVolumetricLaw(inherits GlobalEngine → Engine → Serializable) +Calculate physical response of 2 tetrahedra in interaction, based on penetration configuration given +by TTetraGeom. +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.TimeStepper(inherits GlobalEngine → Engine → Serializable) +Engine defining time-step (fundamental class) +active(=true) +is the engine active? +204 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +timeStepUpdateInterval(=1) +dt update interval +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +PeriodicEngine +PeriodicEngine +Recorder +MeasureCapStress +CapillaryStressRecorder +DomainLimiter +ForceRecorder +ResetRandomPosition +TriaxialStateRecorder +SnapshotEngine +LubricationPDFEngine +PDFEngine +CpmStateUpdater +TorqueRecorder +PyRunner +Fig. 26: Inheritance graph of PeriodicEngine. +See also: CapillaryStressRecorder, CpmStateUpdater, +DomainLimiter, ForceRecorder, LubricationPDFEngine, MeasureCapStress, PDFEngine, PyRunner, +Recorder, ResetRandomPosition, SnapshotEngine, TorqueRecorder, TriaxialStateRecorder. +class yade.wrapper.PeriodicEngine(inherits GlobalEngine → Engine → Serializable) +Run Engine::action with given fixed periodicity real time (=wall clock time, computation time), +virtual time (simulation time), iteration number), by setting any of those criteria (virtPeriod, +realPeriod, iterPeriod) to a positive value. They are all negative (inactive) by default. +2.3. +Yade wrapper class reference +205 + +Yade Documentation, Release 3rd ed. +The number of times this engine is activated can be limited by setting nDo>0. If the number of +activations will have been already reached, no action will be called even if an active period has +elapsed. +If initRun is set (false by default), the engine will run when called for the first time; otherwise +it will only start counting period (realLast, etc, interval variables) from that point, but without +actually running, and will run only once a period has elapsed since the initial run. +This class should not be used directly; rather, derive your own engine which you want to be run +periodically. +Derived engines should override Engine::action(), which will be called periodically. If the derived +Engine overrides also Engine::isActivated, it should also take in account return value from Periodi- +cEngine::isActivated, since otherwise the periodicity will not be functional. +Example with PyRunner, which derives from PeriodicEngine; likely to be encountered in python +scripts: +PyRunner(realPeriod=5,iterPeriod=10000,command='print O.iter') +will print iteration number every 10000 iterations or every 5 seconds of wall clock time, whichever +comes first since it was last run. +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +firstIterRun(=0) +Sets the step number, at each an engine should be executed for the first time (disabled by +default). +initRun(=false) +Run the first time we are called as well. +iterLast(=0) +Tracks step number of last run (auto-updated). +iterPeriod(=0, deactivated) +Periodicity criterion using step number (deactivated if <= 0) +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +nDo(=-1, deactivated) +Limit number of executions by this number (deactivated if negative) +nDone(=0) +Track number of executions (cummulative) (auto-updated). +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +206 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +realLast(=0) +Tracks real time of last run (auto-updated). +realPeriod(=0, deactivated) +Periodicity criterion using real (wall clock, computation, human) time in seconds (deactivated +if <=0) +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +virtLast(=0) +Tracks virtual time of last run (auto-updated). +virtPeriod(=0, deactivated) +Periodicity criterion using virtual (simulation) time (deactivated if <= 0) +class yade.wrapper.CapillaryStressRecorder(inherits Recorder → PeriodicEngine → Glob- +alEngine → Engine → Serializable) +Records information from capillary meniscii on samples submitted to triaxial compressions. Clas- +sical sign convention (tension positiv) is used for capillary stresses. -> New formalism needs to be +tested!!! +addIterNum(=false) +Adds an iteration number to the file name, when the file was created. Useful for creating new +files at each call (false by default) +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +file(=uninitalized) +Name of file to save to; must not be empty. +firstIterRun(=0) +Sets the step number, at each an engine should be executed for the first time (disabled by +default). +initRun(=false) +Run the first time we are called as well. +iterLast(=0) +Tracks step number of last run (auto-updated). +iterPeriod(=0, deactivated) +Periodicity criterion using step number (deactivated if <= 0) +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +2.3. +Yade wrapper class reference +207 + +Yade Documentation, Release 3rd ed. +nDo(=-1, deactivated) +Limit number of executions by this number (deactivated if negative) +nDone(=0) +Track number of executions (cummulative) (auto-updated). +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +realLast(=0) +Tracks real time of last run (auto-updated). +realPeriod(=0, deactivated) +Periodicity criterion using real (wall clock, computation, human) time in seconds (deactivated +if <=0) +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +truncate(=false) +Whether to delete current file contents, if any, when opening (false by default) +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +virtLast(=0) +Tracks virtual time of last run (auto-updated). +virtPeriod(=0, deactivated) +Periodicity criterion using virtual (simulation) time (deactivated if <= 0) +class yade.wrapper.CpmStateUpdater(inherits PeriodicEngine → GlobalEngine → Engine → +Serializable) +Update CpmState of bodies based on state variables in CpmPhys of interactions with this bod. In +particular, bodies’ colors and CpmState::normDmg depending on average damage of their interac- +tions and number of interactions that were already fully broken and have disappeared is updated. +This engine contains its own loop (2 loops, more precisely) over all bodies and should be run +periodically to update colors during the simulation, if desired. +avgRelResidual(=NaN) +Average residual strength at last run. +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +firstIterRun(=0) +Sets the step number, at each an engine should be executed for the first time (disabled by +default). +208 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +initRun(=false) +Run the first time we are called as well. +iterLast(=0) +Tracks step number of last run (auto-updated). +iterPeriod(=0, deactivated) +Periodicity criterion using step number (deactivated if <= 0) +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +maxOmega(=NaN) +Globally maximum damage parameter at last run. +nDo(=-1, deactivated) +Limit number of executions by this number (deactivated if negative) +nDone(=0) +Track number of executions (cummulative) (auto-updated). +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +realLast(=0) +Tracks real time of last run (auto-updated). +realPeriod(=0, deactivated) +Periodicity criterion using real (wall clock, computation, human) time in seconds (deactivated +if <=0) +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +virtLast(=0) +Tracks virtual time of last run (auto-updated). +virtPeriod(=0, deactivated) +Periodicity criterion using virtual (simulation) time (deactivated if <= 0) +class yade.wrapper.DomainLimiter(inherits PeriodicEngine → GlobalEngine → Engine → Se- +rializable) +Delete particles that are out of axis-aligned box given by lo and hi. +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +2.3. +Yade wrapper class reference +209 + +Yade Documentation, Release 3rd ed. +firstIterRun(=0) +Sets the step number, at each an engine should be executed for the first time (disabled by +default). +hi(=Vector3r(0, 0, 0)) +Upper corner of the domain. +initRun(=false) +Run the first time we are called as well. +iterLast(=0) +Tracks step number of last run (auto-updated). +iterPeriod(=0, deactivated) +Periodicity criterion using step number (deactivated if <= 0) +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +lo(=Vector3r(0, 0, 0)) +Lower corner of the domain. +mDeleted(=0) +Mass of deleted particles. +mask(=-1) +If mask is defined, only particles with corresponding groupMask will be deleted. +nDeleted(=0) +Cummulative number of particles deleted. +nDo(=-1, deactivated) +Limit number of executions by this number (deactivated if negative) +nDone(=0) +Track number of executions (cummulative) (auto-updated). +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +realLast(=0) +Tracks real time of last run (auto-updated). +realPeriod(=0, deactivated) +Periodicity criterion using real (wall clock, computation, human) time in seconds (deactivated +if <=0) +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +vDeleted(=0) +Volume of deleted spheres (clumps not counted, in that case check mDeleted) +virtLast(=0) +Tracks virtual time of last run (auto-updated). +virtPeriod(=0, deactivated) +Periodicity criterion using virtual (simulation) time (deactivated if <= 0) +210 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +class yade.wrapper.ForceRecorder(inherits Recorder → PeriodicEngine → GlobalEngine → +Engine → Serializable) +Engine saves the resultant force affecting to bodies, listed in ids. For instance, can be useful for +defining the forces, which affects to _buldozer_ during its work. +addIterNum(=false) +Adds an iteration number to the file name, when the file was created. Useful for creating new +files at each call (false by default) +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +file(=uninitalized) +Name of file to save to; must not be empty. +firstIterRun(=0) +Sets the step number, at each an engine should be executed for the first time (disabled by +default). +ids(=uninitalized) +List of bodies whose state will be measured +initRun(=false) +Run the first time we are called as well. +iterLast(=0) +Tracks step number of last run (auto-updated). +iterPeriod(=0, deactivated) +Periodicity criterion using step number (deactivated if <= 0) +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +nDo(=-1, deactivated) +Limit number of executions by this number (deactivated if negative) +nDone(=0) +Track number of executions (cummulative) (auto-updated). +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +realLast(=0) +Tracks real time of last run (auto-updated). +realPeriod(=0, deactivated) +Periodicity criterion using real (wall clock, computation, human) time in seconds (deactivated +if <=0) +2.3. +Yade wrapper class reference +211 + +Yade Documentation, Release 3rd ed. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +totalForce(=Vector3r::Zero()) +Resultant force, returning by the function. +truncate(=false) +Whether to delete current file contents, if any, when opening (false by default) +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +virtLast(=0) +Tracks virtual time of last run (auto-updated). +virtPeriod(=0, deactivated) +Periodicity criterion using virtual (simulation) time (deactivated if <= 0) +class yade.wrapper.LubricationPDFEngine(inherits PDFEngine → PeriodicEngine → Glob- +alEngine → Engine → Serializable) +Implementation of PDFEngine for Lubrication law +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +filename(=”PDF.txt”) +Filename +firstIterRun(=0) +Sets the step number, at each an engine should be executed for the first time (disabled by +default). +initRun(=false) +Run the first time we are called as well. +iterLast(=0) +Tracks step number of last run (auto-updated). +iterPeriod(=0, deactivated) +Periodicity criterion using step number (deactivated if <= 0) +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +nDo(=-1, deactivated) +Limit number of executions by this number (deactivated if negative) +nDone(=0) +Track number of executions (cummulative) (auto-updated). +numDiscretizeAnglePhi(=20) +Number of sector for phi-angle +numDiscretizeAngleTheta(=20) +Number of sector for theta-angle +212 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +realLast(=0) +Tracks real time of last run (auto-updated). +realPeriod(=0, deactivated) +Periodicity criterion using real (wall clock, computation, human) time in seconds (deactivated +if <=0) +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +virtLast(=0) +Tracks virtual time of last run (auto-updated). +virtPeriod(=0, deactivated) +Periodicity criterion using virtual (simulation) time (deactivated if <= 0) +warnedOnce(=false) +For one-time warning. May trigger usefull warnings +class yade.wrapper.MeasureCapStress(inherits PeriodicEngine → GlobalEngine → Engine → +Serializable) +Post-processing engine giving the capillary stress tensor (the fluids mixture contribution to the to- +tal stress in unsaturated, i.e. triphasic, conditions) according to the µUNSAT expression detailled +in [Duriez2017c]. Although this expression differs in nature from the one of utils.getCapillaryStress +(consideration of distributed integrals herein, vs resultant capillary force therein), both are equiv- +alent [Duriez2016b], [Duriez2017], [Duriez2017c]. The REV volume V entering the expression is +automatically measured, from the Cell for periodic conditions, or from utils.aabbExtrema function +otherwise. +capillaryPressure(=0) +Capillary pressure uc, to be defined equal to Law2_ScGeom_CapillaryPhys_Capillar- +ity.capillaryPressure. +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +debug(=0) +To output some debugging messages. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +firstIterRun(=0) +Sets the step number, at each an engine should be executed for the first time (disabled by +default). +2.3. +Yade wrapper class reference +213 + +Yade Documentation, Release 3rd ed. +initRun(=false) +Run the first time we are called as well. +iterLast(=0) +Tracks step number of last run (auto-updated). +iterPeriod(=0, deactivated) +Periodicity criterion using step number (deactivated if <= 0) +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +muGamma(=Matrix3r::Zero()) +Tensorial contribution to sigmaCap from the contact lines Γ: µΓ = +� +Γ νnw ⊗ x dl with νnw +the fluid-fluid interface conormal [Duriez2017c], and x the position. (auto-updated) +muSnw(=Matrix3r::Zero()) +Tensorial contribution to sigmaCap from the wetting/non-wetting (e.g. liquid/gas) interface +Snw: µSnw = +� +Snw(δ − n ⊗ n)dS with n the outward normal and δ the identity tensor. +(auto-updated) +muSsw(=Matrix3r::Zero()) +Tensorial contribution to sigmaCap from the wetted solid surfaces Ssw: µSsw = +� +Ssw n⊗xdS +with n the outward normal and x the position. (auto-updated) +muVw(=Matrix3r::Zero()) +Tensorial contribution (spherical i.e. isotropic) to sigmaCap from the wetting fluid volume: +µVw = Vw δ with Vw = vW and δ the identity tensor. (auto-updated) +nDo(=-1, deactivated) +Limit number of executions by this number (deactivated if negative) +nDone(=0) +Track number of executions (cummulative) (auto-updated). +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +realLast(=0) +Tracks real time of last run (auto-updated). +realPeriod(=0, deactivated) +Periodicity criterion using real (wall clock, computation, human) time in seconds (deactivated +if <=0) +sigmaCap(=Matrix3r::Zero()) +The capillary stress tensor σcap itself, expressed as σcap = 1/V [uc(µVw + µSsw) + +γnw(µSnw + µΓ)] where the four microstructure tensors µVw, µSsw, µSnw, µΓ correspond +to muVw, muSsw, muSnw and muGamma attributes. (auto-updated) +surfaceTension(=0.073) +Fluid-fluid surface tension γnw, to be defined equal to Law2_ScGeom_CapillaryPhys_Cap- +illarity.surfaceTension. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +214 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +vW(=0) +Wetting fluid volume, summing menisci volumes (faster here than through python loops). +(auto-updated) +virtLast(=0) +Tracks virtual time of last run (auto-updated). +virtPeriod(=0, deactivated) +Periodicity criterion using virtual (simulation) time (deactivated if <= 0) +wettAngle(=0) +Wetting, i.e. contact, angle value (radians). To be defined consistently with the value upon +which the capillary files (used by Law2_ScGeom_CapillaryPhys_Capillarity) rely. +class yade.wrapper.PDFEngine(inherits PeriodicEngine → GlobalEngine → Engine → Serializ- +able) +Base class for spectrums calculations. Compute Probability Density Functions of normalStress, +shearStress, distance, velocity and interactions in spherical coordinates and write result to a file. +Column name format is: Data(theta, phi). Convention used: x: phi = 0, y: theta = 0, z: phi = +pi/2 +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +filename(=”PDF.txt”) +Filename +firstIterRun(=0) +Sets the step number, at each an engine should be executed for the first time (disabled by +default). +initRun(=false) +Run the first time we are called as well. +iterLast(=0) +Tracks step number of last run (auto-updated). +iterPeriod(=0, deactivated) +Periodicity criterion using step number (deactivated if <= 0) +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +nDo(=-1, deactivated) +Limit number of executions by this number (deactivated if negative) +nDone(=0) +Track number of executions (cummulative) (auto-updated). +numDiscretizeAnglePhi(=20) +Number of sector for phi-angle +numDiscretizeAngleTheta(=20) +Number of sector for theta-angle +2.3. +Yade wrapper class reference +215 + +Yade Documentation, Release 3rd ed. +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +realLast(=0) +Tracks real time of last run (auto-updated). +realPeriod(=0, deactivated) +Periodicity criterion using real (wall clock, computation, human) time in seconds (deactivated +if <=0) +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +virtLast(=0) +Tracks virtual time of last run (auto-updated). +virtPeriod(=0, deactivated) +Periodicity criterion using virtual (simulation) time (deactivated if <= 0) +warnedOnce(=false) +For one-time warning. May trigger usefull warnings +class yade.wrapper.PyRunner(inherits PeriodicEngine → GlobalEngine → Engine → Serializ- +able) +Execute a python command periodically, with defined (and adjustable) periodicity. See Periodi- +cEngine documentation for details. +command(=””) +Command to be run by python interpreter. Not run if empty. +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +firstIterRun(=0) +Sets the step number, at each an engine should be executed for the first time (disabled by +default). +ignoreErrors(=false) +Debug only: set this value to true to tell PyRunner to ignore any errors encountered during +command execution. +initRun(=false) +Run the first time we are called as well. +iterLast(=0) +Tracks step number of last run (auto-updated). +216 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +iterPeriod(=0, deactivated) +Periodicity criterion using step number (deactivated if <= 0) +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +nDo(=-1, deactivated) +Limit number of executions by this number (deactivated if negative) +nDone(=0) +Track number of executions (cummulative) (auto-updated). +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +realLast(=0) +Tracks real time of last run (auto-updated). +realPeriod(=0, deactivated) +Periodicity criterion using real (wall clock, computation, human) time in seconds (deactivated +if <=0) +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +updateGlobals +Whether to workaround ipython not recognizing local variables by calling globals(). +update(locals()). If true then PyRunner is able to call functions declared later locally in +a running live yade session. The PyRunner call is a bit slower because it updates globals() +with recently declared python functions. +Warning: +When updateGlobals==False and a function was declared inside a live +yade session (ipython) then an error NameError: name 'command' is not +defined will occur unless python globals() are updated with command +globals().update(locals()) +virtLast(=0) +Tracks virtual time of last run (auto-updated). +virtPeriod(=0, deactivated) +Periodicity criterion using virtual (simulation) time (deactivated if <= 0) +class yade.wrapper.Recorder(inherits PeriodicEngine → GlobalEngine → Engine → Serializ- +able) +Engine periodically storing some data to (one) external file. In addition PeriodicEngine, it handles +opening the file as needed. See PeriodicEngine for controlling periodicity. +addIterNum(=false) +Adds an iteration number to the file name, when the file was created. Useful for creating new +files at each call (false by default) +2.3. +Yade wrapper class reference +217 + +Yade Documentation, Release 3rd ed. +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +file(=uninitalized) +Name of file to save to; must not be empty. +firstIterRun(=0) +Sets the step number, at each an engine should be executed for the first time (disabled by +default). +initRun(=false) +Run the first time we are called as well. +iterLast(=0) +Tracks step number of last run (auto-updated). +iterPeriod(=0, deactivated) +Periodicity criterion using step number (deactivated if <= 0) +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +nDo(=-1, deactivated) +Limit number of executions by this number (deactivated if negative) +nDone(=0) +Track number of executions (cummulative) (auto-updated). +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +realLast(=0) +Tracks real time of last run (auto-updated). +realPeriod(=0, deactivated) +Periodicity criterion using real (wall clock, computation, human) time in seconds (deactivated +if <=0) +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +truncate(=false) +Whether to delete current file contents, if any, when opening (false by default) +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +virtLast(=0) +Tracks virtual time of last run (auto-updated). +218 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +virtPeriod(=0, deactivated) +Periodicity criterion using virtual (simulation) time (deactivated if <= 0) +class yade.wrapper.ResetRandomPosition(inherits PeriodicEngine → GlobalEngine → Engine +→ Serializable) +Creates spheres during simulation, placing them at random positions. Every time called, one new +sphere will be created and inserted in the simulation. +angularVelocity(=Vector3r::Zero()) +Mean angularVelocity of spheres. +angularVelocityRange(=Vector3r::Zero()) +Half size of a angularVelocity distribution interval. New sphere will have random angularVe- +locity within the range angularVelocity±angularVelocityRange. +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +factoryFacets(=uninitalized) +The geometry of the section where spheres will be placed; they will be placed on facets or in +volume between them depending on volumeSection flag. +firstIterRun(=0) +Sets the step number, at each an engine should be executed for the first time (disabled by +default). +initRun(=false) +Run the first time we are called as well. +iterLast(=0) +Tracks step number of last run (auto-updated). +iterPeriod(=0, deactivated) +Periodicity criterion using step number (deactivated if <= 0) +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +maxAttempts(=20) +Max attempts to place sphere. If placing the sphere in certain random position would cause +an overlap with any other physical body in the model, SpheresFactory will try to find another +position. +nDo(=-1, deactivated) +Limit number of executions by this number (deactivated if negative) +nDone(=0) +Track number of executions (cummulative) (auto-updated). +normal(=Vector3r(0, 1, 0)) +?? +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +2.3. +Yade wrapper class reference +219 + +Yade Documentation, Release 3rd ed. +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +point(=Vector3r::Zero()) +?? +realLast(=0) +Tracks real time of last run (auto-updated). +realPeriod(=0, deactivated) +Periodicity criterion using real (wall clock, computation, human) time in seconds (deactivated +if <=0) +subscribedBodies(=uninitalized) +Affected bodies. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +velocity(=Vector3r::Zero()) +Mean velocity of spheres. +velocityRange(=Vector3r::Zero()) +Half size of a velocities distribution interval. New sphere will have random velocity within the +range velocity±velocityRange. +virtLast(=0) +Tracks virtual time of last run (auto-updated). +virtPeriod(=0, deactivated) +Periodicity criterion using virtual (simulation) time (deactivated if <= 0) +volumeSection(=false, define factory by facets.) +Create new spheres inside factory volume rather than on its surface. +class yade.wrapper.SnapshotEngine(inherits PeriodicEngine → GlobalEngine → Engine → +Serializable) +Periodically save snapshots of GLView(s) as .png files. Files are named fileBase + counter + ‘.png’ +(counter is left-padded by 0s, i.e. snap00004.png). +counter(=0) +Number that will be appended to fileBase when the next snapshot is saved (incremented at +every save). (auto-updated) +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +deadTimeout(=3) +Timeout for 3d operations (opening new view, saving snapshot); after timing out, throw +exception (or only report error if ignoreErrors) and make myself dead. [s] +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +220 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +fileBase(=””) +Basename for snapshots +firstIterRun(=0) +Sets the step number, at each an engine should be executed for the first time (disabled by +default). +format(=”PNG”) +Format of snapshots (one of JPEG, PNG, EPS, PS, PPM, BMP) QGLViewer documentation. +File extension will be lowercased format. Validity of format is not checked. +ignoreErrors(=true) +Only report errors instead of throwing exceptions, in case of timeouts. +initRun(=false) +Run the first time we are called as well. +iterLast(=0) +Tracks step number of last run (auto-updated). +iterPeriod(=0, deactivated) +Periodicity criterion using step number (deactivated if <= 0) +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +msecSleep(=0) +number of msec to sleep after snapshot (to prevent 3d hw problems) [ms] +nDo(=-1, deactivated) +Limit number of executions by this number (deactivated if negative) +nDone(=0) +Track number of executions (cummulative) (auto-updated). +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +plot(=uninitalized) +Name of field in plot.imgData to which taken snapshots will be appended automatically. +realLast(=0) +Tracks real time of last run (auto-updated). +realPeriod(=0, deactivated) +Periodicity criterion using real (wall clock, computation, human) time in seconds (deactivated +if <=0) +snapshots(=uninitalized) +Files that have been created so far +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +virtLast(=0) +Tracks virtual time of last run (auto-updated). +2.3. +Yade wrapper class reference +221 + +Yade Documentation, Release 3rd ed. +virtPeriod(=0, deactivated) +Periodicity criterion using virtual (simulation) time (deactivated if <= 0) +class yade.wrapper.TorqueRecorder(inherits Recorder → PeriodicEngine → GlobalEngine → +Engine → Serializable) +Engine saves the total torque according to the given axis and ZeroPoint, the force is taken from +bodies, listed in ids For instance, can be useful for defining the torque, which affects on ball mill +during its work. +addIterNum(=false) +Adds an iteration number to the file name, when the file was created. Useful for creating new +files at each call (false by default) +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +file(=uninitalized) +Name of file to save to; must not be empty. +firstIterRun(=0) +Sets the step number, at each an engine should be executed for the first time (disabled by +default). +ids(=uninitalized) +List of bodies whose state will be measured +initRun(=false) +Run the first time we are called as well. +iterLast(=0) +Tracks step number of last run (auto-updated). +iterPeriod(=0, deactivated) +Periodicity criterion using step number (deactivated if <= 0) +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +nDo(=-1, deactivated) +Limit number of executions by this number (deactivated if negative) +nDone(=0) +Track number of executions (cummulative) (auto-updated). +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +realLast(=0) +Tracks real time of last run (auto-updated). +222 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +realPeriod(=0, deactivated) +Periodicity criterion using real (wall clock, computation, human) time in seconds (deactivated +if <=0) +rotationAxis(=Vector3r::UnitX()) +Rotation axis +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +totalTorque(=0) +Resultant torque, returning by the function. +truncate(=false) +Whether to delete current file contents, if any, when opening (false by default) +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +virtLast(=0) +Tracks virtual time of last run (auto-updated). +virtPeriod(=0, deactivated) +Periodicity criterion using virtual (simulation) time (deactivated if <= 0) +zeroPoint(=Vector3r::Zero()) +Point of rotation center +class yade.wrapper.TriaxialStateRecorder(inherits Recorder → PeriodicEngine → Glob- +alEngine → Engine → Serializable) +Engine recording triaxial variables (see the variables list in the first line of the output file). This +recorder needs TriaxialCompressionEngine or ThreeDTriaxialEngine present in the simulation). +addIterNum(=false) +Adds an iteration number to the file name, when the file was created. Useful for creating new +files at each call (false by default) +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +file(=uninitalized) +Name of file to save to; must not be empty. +firstIterRun(=0) +Sets the step number, at each an engine should be executed for the first time (disabled by +default). +initRun(=false) +Run the first time we are called as well. +iterLast(=0) +Tracks step number of last run (auto-updated). +iterPeriod(=0, deactivated) +Periodicity criterion using step number (deactivated if <= 0) +2.3. +Yade wrapper class reference +223 + +Yade Documentation, Release 3rd ed. +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +nDo(=-1, deactivated) +Limit number of executions by this number (deactivated if negative) +nDone(=0) +Track number of executions (cummulative) (auto-updated). +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +porosity(=1) +porosity of the packing [-] +realLast(=0) +Tracks real time of last run (auto-updated). +realPeriod(=0, deactivated) +Periodicity criterion using real (wall clock, computation, human) time in seconds (deactivated +if <=0) +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +truncate(=false) +Whether to delete current file contents, if any, when opening (false by default) +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +virtLast(=0) +Tracks virtual time of last run (auto-updated). +virtPeriod(=0, deactivated) +Periodicity criterion using virtual (simulation) time (deactivated if <= 0) +BoundaryController +BoundaryController +KinemCNSEngine +KinemSimpleShearBox +PeriTriaxController +KinemCTDEngine +TriaxialStressController +TriaxialCompressionEngine +UniaxialStrainer +Peri3dController +KinemCNDEngine +KinemCNLEngine +ThreeDTriaxialEngine +Disp2DPropLoadEngine +PeriIsoCompressor +Fig. 27: Inheritance graph of BoundaryController. See also: Disp2DPropLoadEngine, KinemCNDEngine, +KinemCNLEngine, KinemCNSEngine, KinemCTDEngine, KinemSimpleShearBox, Peri3dController, +PeriIsoCompressor, PeriTriaxController, ThreeDTriaxialEngine, TriaxialCompressionEngine, Triaxial- +StressController, UniaxialStrainer. +224 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +class yade.wrapper.BoundaryController(inherits GlobalEngine → Engine → Serializable) +Base for engines controlling boundary conditions of simulations. Not to be used directly. +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Disp2DPropLoadEngine(inherits BoundaryController → GlobalEngine → +Engine → Serializable) +Disturbs a simple shear sample in a given displacement direction +This engine allows one to apply, on a simple shear sample, a loading controlled by du/dgamma = +cste, which is equivalent to du + cste’ * dgamma = 0 (proportionnal path loadings). To do so, +the upper plate of the simple shear box is moved in a given direction (corresponding to a given +du/dgamma), whereas lateral plates are moved so that the box remains closed. This engine can +easily be used to perform directionnal probes, with a python script launching successivly the same +.xml which contains this engine, after having modified the direction of loading (see theta attribute). +That’s why this Engine contains a saveData procedure which can save data on the state of the +sample at the end of the loading (in case of successive loadings - for successive directions - through +a python script, each line would correspond to one direction of loading). +Key(=””) +string to add at the names of the saved files, and of the output file filled by saveData +LOG(=false) +boolean controling the output of messages on the screen +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +2.3. +Yade wrapper class reference +225 + +Yade Documentation, Release 3rd ed. +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +id_boxback(=4) +the id of the wall at the back of the sample +id_boxbas(=1) +the id of the lower wall +id_boxfront(=5) +the id of the wall in front of the sample +id_boxleft(=0) +the id of the left wall +id_boxright(=2) +the id of the right wall +id_topbox(=3) +the id of the upper wall +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +nbre_iter(=0) +the number of iterations of loading to perform +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +theta(=0.0) +the angle, in a (gamma,h=-u) plane from the gamma - axis to the perturbation vector (trigo +wise) [degrees] +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +v(=0.0) +the speed at which the perturbation is imposed. In case of samples which are more sensitive +to normal loadings than tangential ones, one possibility is to take v = V_shear - | (V_shear- +V_comp)*sin(theta) | => v=V_shear in shear; V_comp in compression [m/s] +class yade.wrapper.KinemCNDEngine(inherits KinemSimpleShearBox → BoundaryController → +GlobalEngine → Engine → Serializable) +To apply a Constant Normal Displacement (CND) shear for a parallelogram box +This engine, designed for simulations implying a simple shear box (SimpleShear Preprocessor or +scripts/simpleShear.py), allows one to perform a constant normal displacement shear, by translat- +ing horizontally the upper plate, while the lateral ones rotate so that they always keep contact +with the lower and upper walls. +Key(=””) +string to add at the names of the saved files +LOG(=false) +boolean controling the output of messages on the screen +226 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +alpha(=Mathr::PI/2.0) +the angle from the lower box to the left box (trigo wise). Measured by this Engine. Has to +be saved, but not to be changed by the user. +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +f0(=0.0) +the (vertical) force acting on the upper plate on the very first time step (determined by the +Engine). Controls of the loadings in case of KinemCNSEngine or KinemCNLEngine will be +done according to this initial value [N]. Has to be saved, but not to be changed by the user. +firstRun(=true) +boolean set to false as soon as the engine has done its job one time : useful to know if initial +height of, and normal force sustained by, the upper box are known or not (and thus if they +have to be initialized). Has to be saved, but not to be changed by the user. +gamma(=0.0) +the current value of the tangential displacement +gamma_save(=uninitalized) +vector with the values of gamma at which a save of the simulation is performed [m] +gammalim(=0.0) +the value of the tangential displacement at wich the displacement is stopped [m] +id_boxback(=4) +the id of the wall at the back of the sample +id_boxbas(=1) +the id of the lower wall +id_boxfront(=5) +the id of the wall in front of the sample +id_boxleft(=0) +the id of the left wall +id_boxright(=2) +the id of the right wall +id_topbox(=3) +the id of the upper wall +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +max_vel(=1.0) +to limit the speed of the vertical displacements done to control σ (CNL or CNS cases) [m/s] +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +2.3. +Yade wrapper class reference +227 + +Yade Documentation, Release 3rd ed. +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +shearSpeed(=0.0) +the speed at which the shear is performed : speed of the upper plate [m/s] +temoin_save(=uninitalized) +vector (same length as ‘gamma_save’ for ex), with 0 or 1 depending whether the save for the +corresponding value of gamma has been done (1) or not (0). Has to be saved, but not to be +changed by the user. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +wallDamping(=0.2) +the vertical displacements done to to control σ (CNL or CNS cases) are in fact damped, +through this wallDamping +y0(=0.0) +the height of the upper plate at the very first time step : the engine finds its value [m]. Has +to be saved, but not to be changed by the user. +class yade.wrapper.KinemCNLEngine(inherits KinemSimpleShearBox → BoundaryController → +GlobalEngine → Engine → Serializable) +To apply a constant normal stress shear (i.e. Constant Normal Load : CNL) for a parallelogram +box (simple shear box : SimpleShear Preprocessor or scripts/simpleShear.py) +This engine allows one to translate horizontally the upper plate while the lateral ones rotate so +that they always keep contact with the lower and upper walls. +In fact the upper plate can move not only horizontally but also vertically, so that the normal stress +acting on it remains constant (this constant value is not chosen by the user but is the one that +exists at the beginning of the simulation) +The right vertical displacements which will be allowed are computed from the rigidity Kn of the +sample over the wall (so to cancel a deltaSigma, a normal dplt deltaSigma*S/(Kn) is set) +The movement is moreover controlled by the user via a shearSpeed which will be the speed of the +upper wall, and by a maximum value of horizontal displacement gammalim, after which the shear +stops. +Note: +Not only the positions of walls are updated but also their speeds, which is all but useless +considering the fact that in the contact laws these velocities of bodies are used to compute values +of tangential relative displacements. +Warning: +Because of this last point, if you want to use later saves of simulations executed +with this Engine, but without that stopMovement was executed, your boxes will keep their +speeds => you will have to cancel them ‘by hand’ in the .xml. +Key(=””) +string to add at the names of the saved files +LOG(=false) +boolean controling the output of messages on the screen +228 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +alpha(=Mathr::PI/2.0) +the angle from the lower box to the left box (trigo wise). Measured by this Engine. Has to +be saved, but not to be changed by the user. +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +f0(=0.0) +the (vertical) force acting on the upper plate on the very first time step (determined by the +Engine). Controls of the loadings in case of KinemCNSEngine or KinemCNLEngine will be +done according to this initial value [N]. Has to be saved, but not to be changed by the user. +firstRun(=true) +boolean set to false as soon as the engine has done its job one time : useful to know if initial +height of, and normal force sustained by, the upper box are known or not (and thus if they +have to be initialized). Has to be saved, but not to be changed by the user. +gamma(=0.0) +current value of tangential displacement [m] +gamma_save(=uninitalized) +vector with the values of gamma at which a save of the simulation is performed [m] +gammalim(=0.0) +the value of tangential displacement (of upper plate) at wich the shearing is stopped [m] +id_boxback(=4) +the id of the wall at the back of the sample +id_boxbas(=1) +the id of the lower wall +id_boxfront(=5) +the id of the wall in front of the sample +id_boxleft(=0) +the id of the left wall +id_boxright(=2) +the id of the right wall +id_topbox(=3) +the id of the upper wall +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +max_vel(=1.0) +to limit the speed of the vertical displacements done to control σ (CNL or CNS cases) [m/s] +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +2.3. +Yade wrapper class reference +229 + +Yade Documentation, Release 3rd ed. +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +shearSpeed(=0.0) +the speed at wich the shearing is performed : speed of the upper plate [m/s] +temoin_save(=uninitalized) +vector (same length as ‘gamma_save’ for ex), with 0 or 1 depending whether the save for the +corresponding value of gamma has been done (1) or not (0). Has to be saved, but not to be +changed by the user. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +wallDamping(=0.2) +the vertical displacements done to to control σ (CNL or CNS cases) are in fact damped, +through this wallDamping +y0(=0.0) +the height of the upper plate at the very first time step : the engine finds its value [m]. Has +to be saved, but not to be changed by the user. +class yade.wrapper.KinemCNSEngine(inherits KinemSimpleShearBox → BoundaryController → +GlobalEngine → Engine → Serializable) +To apply a Constant Normal Stifness (CNS) shear for a parallelogram box (simple shear) +This engine, useable in simulations implying one deformable parallelepipedic box, allows one to +translate horizontally the upper plate while the lateral ones rotate so that they always keep contact +with the lower and upper walls. The upper plate can move not only horizontally but also vertically, +so that the normal rigidity defined by DeltaF(upper plate)/DeltaU(upper plate) = constant (= KnC +defined by the user). +The movement is moreover controlled by the user via a shearSpeed which is the horizontal speed +of the upper wall, and by a maximum value of horizontal displacement gammalim (of the upper +plate), after which the shear stops. +Note: +not only the positions of walls are updated but also their speeds, which is all but useless +considering the fact that in the contact laws these velocities of bodies are used to compute values +of tangential relative displacements. +Warning: +But, because of this last point, if you want to use later saves of simulations +executed with this Engine, but without that stopMovement was executed, your boxes will keep +their speeds => you will have to cancel them by hand in the .xml +Key(=””) +string to add at the names of the saved files +KnC(=10.0e6) +the normal rigidity chosen by the user [MPa/mm] - the conversion in Pa/m will be made +LOG(=false) +boolean controling the output of messages on the screen +alpha(=Mathr::PI/2.0) +the angle from the lower box to the left box (trigo wise). Measured by this Engine. Has to +be saved, but not to be changed by the user. +230 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +f0(=0.0) +the (vertical) force acting on the upper plate on the very first time step (determined by the +Engine). Controls of the loadings in case of KinemCNSEngine or KinemCNLEngine will be +done according to this initial value [N]. Has to be saved, but not to be changed by the user. +firstRun(=true) +boolean set to false as soon as the engine has done its job one time : useful to know if initial +height of, and normal force sustained by, the upper box are known or not (and thus if they +have to be initialized). Has to be saved, but not to be changed by the user. +gamma(=0.0) +current value of tangential displacement [m] +gammalim(=0.0) +the value of tangential displacement (of upper plate) at wich the shearing is stopped [m] +id_boxback(=4) +the id of the wall at the back of the sample +id_boxbas(=1) +the id of the lower wall +id_boxfront(=5) +the id of the wall in front of the sample +id_boxleft(=0) +the id of the left wall +id_boxright(=2) +the id of the right wall +id_topbox(=3) +the id of the upper wall +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +max_vel(=1.0) +to limit the speed of the vertical displacements done to control σ (CNL or CNS cases) [m/s] +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +shearSpeed(=0.0) +the speed at wich the shearing is performed : speed of the upper plate [m/s] +2.3. +Yade wrapper class reference +231 + +Yade Documentation, Release 3rd ed. +temoin_save(=uninitalized) +vector (same length as ‘gamma_save’ for ex), with 0 or 1 depending whether the save for the +corresponding value of gamma has been done (1) or not (0). Has to be saved, but not to be +changed by the user. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +wallDamping(=0.2) +the vertical displacements done to to control σ (CNL or CNS cases) are in fact damped, +through this wallDamping +y0(=0.0) +the height of the upper plate at the very first time step : the engine finds its value [m]. Has +to be saved, but not to be changed by the user. +class yade.wrapper.KinemCTDEngine(inherits KinemSimpleShearBox → BoundaryController → +GlobalEngine → Engine → Serializable) +To compress a simple shear sample by moving the upper box in a vertical way only, so that the +tangential displacement (defined by the horizontal gap between the upper and lower boxes) remains +constant (thus, the CTD = Constant Tangential Displacement). The lateral boxes move also to +keep always contact. All that until this box is submitted to a given stress (targetSigma). Moreover +saves are executed at each value of stresses stored in the vector sigma_save, and at targetSigma +Key(=””) +string to add at the names of the saved files +LOG(=false) +boolean controling the output of messages on the screen +alpha(=Mathr::PI/2.0) +the angle from the lower box to the left box (trigo wise). Measured by this Engine. Has to +be saved, but not to be changed by the user. +compSpeed(=0.0) +(vertical) speed of the upper box : >0 for real compression, <0 for unloading [m/s] +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +f0(=0.0) +the (vertical) force acting on the upper plate on the very first time step (determined by the +Engine). Controls of the loadings in case of KinemCNSEngine or KinemCNLEngine will be +done according to this initial value [N]. Has to be saved, but not to be changed by the user. +firstRun(=true) +boolean set to false as soon as the engine has done its job one time : useful to know if initial +height of, and normal force sustained by, the upper box are known or not (and thus if they +have to be initialized). Has to be saved, but not to be changed by the user. +232 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +id_boxback(=4) +the id of the wall at the back of the sample +id_boxbas(=1) +the id of the lower wall +id_boxfront(=5) +the id of the wall in front of the sample +id_boxleft(=0) +the id of the left wall +id_boxright(=2) +the id of the right wall +id_topbox(=3) +the id of the upper wall +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +max_vel(=1.0) +to limit the speed of the vertical displacements done to control σ (CNL or CNS cases) [m/s] +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +sigma_save(=uninitalized) +vector with the values of sigma at which a save of the simulation should be performed [kPa] +targetSigma(=0.0) +the value of sigma at which the compression should stop [kPa] +temoin_save(=uninitalized) +vector (same length as ‘gamma_save’ for ex), with 0 or 1 depending whether the save for the +corresponding value of gamma has been done (1) or not (0). Has to be saved, but not to be +changed by the user. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +wallDamping(=0.2) +the vertical displacements done to to control σ (CNL or CNS cases) are in fact damped, +through this wallDamping +y0(=0.0) +the height of the upper plate at the very first time step : the engine finds its value [m]. Has +to be saved, but not to be changed by the user. +class yade.wrapper.KinemSimpleShearBox(inherits BoundaryController → GlobalEngine → +Engine → Serializable) +This class is supposed to be a mother class for all Engines performing loadings on the simple shear +box of SimpleShear. It is not intended to be used by itself, but its declaration and implentation +will thus contain all what is useful for all these Engines. The script simpleShear.py illustrates the +use of the various corresponding Engines. +2.3. +Yade wrapper class reference +233 + +Yade Documentation, Release 3rd ed. +Key(=””) +string to add at the names of the saved files +LOG(=false) +boolean controling the output of messages on the screen +alpha(=Mathr::PI/2.0) +the angle from the lower box to the left box (trigo wise). Measured by this Engine. Has to +be saved, but not to be changed by the user. +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +f0(=0.0) +the (vertical) force acting on the upper plate on the very first time step (determined by the +Engine). Controls of the loadings in case of KinemCNSEngine or KinemCNLEngine will be +done according to this initial value [N]. Has to be saved, but not to be changed by the user. +firstRun(=true) +boolean set to false as soon as the engine has done its job one time : useful to know if initial +height of, and normal force sustained by, the upper box are known or not (and thus if they +have to be initialized). Has to be saved, but not to be changed by the user. +id_boxback(=4) +the id of the wall at the back of the sample +id_boxbas(=1) +the id of the lower wall +id_boxfront(=5) +the id of the wall in front of the sample +id_boxleft(=0) +the id of the left wall +id_boxright(=2) +the id of the right wall +id_topbox(=3) +the id of the upper wall +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +max_vel(=1.0) +to limit the speed of the vertical displacements done to control σ (CNL or CNS cases) [m/s] +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +234 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +temoin_save(=uninitalized) +vector (same length as ‘gamma_save’ for ex), with 0 or 1 depending whether the save for the +corresponding value of gamma has been done (1) or not (0). Has to be saved, but not to be +changed by the user. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +wallDamping(=0.2) +the vertical displacements done to to control σ (CNL or CNS cases) are in fact damped, +through this wallDamping +y0(=0.0) +the height of the upper plate at the very first time step : the engine finds its value [m]. Has +to be saved, but not to be changed by the user. +class yade.wrapper.Peri3dController(inherits BoundaryController → GlobalEngine → En- +gine → Serializable) +Class for controlling independently all 6 components of “engineering” stress and strain of periodic +Cell. goal are the goal values, while stressMask determines which components prescribe stress and +which prescribe strain. +If the strain is prescribed, appropriate strain rate is directly applied. If the stress is prescribed, +the strain predictor is used: from stress values in two previous steps the value of strain rate is +prescribed so as the value of stress in the next step is as close as possible to the ideal one. Current +algorithm is extremly simple and probably will be changed in future, but is roboust enough and +mostly works fine. +Stress error (difference between actual and ideal stress) is evaluated in current and previous steps +(dσi, dσi−1). Linear extrapolation is used to estimate error in the next step +dσi+1 = 2dσi − dσi−1 +According to this error, the strain rate is modified by mod parameter +dσi+1 +� > 0 → ˙εi+1 = ˙εi − max(abs(˙εi)) · mod +< 0 → ˙εi+1 = ˙εi + max(abs(˙εi)) · mod +According to this fact, the prescribed stress will (almost) never have exact prescribed value, but the +difference would be very small (and decreasing for increasing nSteps. This approach works good if +one of the dominant strain rates is prescribed. If all stresses are prescribed or if all goal strains is +prescribed as zero, a good estimation is needed for the first step, therefore the compliance matrix +is estimated (from user defined estimations of macroscopic material parameters youngEstimation +and poissonEstimation) and respective strain rates is computed form prescribed stress rates and +compliance matrix (the estimation of compliance matrix could be computed autamatically avoiding +user inputs of this kind). +The simulation on rotated periodic cell is also supported. +Firstly, the polar decomposition is +performed on cell’s transformation matrix trsf T = UP, where U is orthogonal (unitary) matrix +representing rotation and P is a positive semi-definite Hermitian matrix representing strain. A +logarithm of P should be used to obtain realistic values at higher strain values (not implemented +yet). A prescribed strain increment in global coordinates dt · ˙ε is properly rotated to cell’s local +coordinates and added to P +Pi+1 = P + UTdt · ˙εU +The new value of trsf is computed at T i+1 = UPi+1. From current and next trsf the cell’s velocity +gradient velGrad is computed (according to its definition) as +V = (T i+1T −1 − I)/dt +2.3. +Yade wrapper class reference +235 + +Yade Documentation, Release 3rd ed. +Current implementation allow user to define independent loading “path” for each prescribed com- +ponent. i.e. define the prescribed value as a function of time (or progress or steps). See Paths. +Examples examples/test/peri3dController_example1.py and examples/test/peri3dController_- +triaxialCompression.py +explain +usage +and +inputs +of +Peri3dController, +exam- +ples/test/peri3dController_shear.py is an example of using shear components and also simulation +on rotated cell. +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +doneHook(=uninitalized) +Python command (as string) to run when nSteps is achieved. If empty, the engine will be set +dead. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +goal(=Vector6r::Zero()) +Goal state; only the upper triangular matrix is considered; each component is either prescribed +stress or strain, depending on stressMask. +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +lenPe(=0) +Peri3dController internal variable +lenPs(=0) +Peri3dController internal variable +maxStrain(=1e6) +Maximal asolute value of strain allowed in the simulation. If reached, the simulation is con- +sidered as finished +maxStrainRate(=1e3) +Maximal absolute value of strain rate (both normal and shear components of strain) +mod(=.1) +Predictor modificator, by trail-and-error analysis the value 0.1 was found as the best. +nSteps(=1000) +Number of steps of the simulation. +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +pathSizes(=Vector6i::Zero()) +Peri3dController internal variable +pathsCounter(=Vector6i::Zero()) +Peri3dController internal variable +236 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +pe(=Vector6i::Zero()) +Peri3dController internal variable +poissonEstimation(=.25) +Estimation of macroscopic Poisson’s ratio, used used for the first simulation step +progress(=0.) +Actual progress of the simulation with Controller. +ps(=Vector6i::Zero()) +Peri3dController internal variable +strain(=Vector6r::Zero()) +Current strain (deformation) vector (εx,εy,εz,γyz,γzx,γxy) (auto-updated). +strainGoal(=Vector6r::Zero()) +Peri3dController internal variable +strainRate(=Vector6r::Zero()) +Current strain rate vector. +stress(=Vector6r::Zero()) +Current stress vector (σx,σy,σz,τyz,τzx,τxy)|yupdate|. +stressGoal(=Vector6r::Zero()) +Peri3dController internal variable +stressIdeal(=Vector6r::Zero()) +Ideal stress vector at current time step. +stressMask(=0, all strains) +mask determining whether components of goal are strain (0) or stress (1). +The order is +00,11,22,12,02,01 from the least significant bit. (e.g. 0b000011 is stress 00 and stress 11). +stressRate(=Vector6r::Zero()) +Current stress rate vector (that is prescribed, the actual one slightly differ). +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +xxPath +“Time function” (piecewise linear) for xx direction. Sequence of couples of numbers. First +number is time, second number desired value of respective quantity (stress or strain). The +last couple is considered as final state (equal to (nSteps, goal)), other values are relative to +this state. +Example: nSteps=1000, goal[0]=300, xxPath=((2,3),(4,1),(5,2)) +at step 400 (=5*1000/2) the value is 450 (=3*300/2), +at step 800 (=4*1000/5) the value is 150 (=1*300/2), +at step 1000 (=5*1000/5=nSteps) the value is 300 (=2*300/2=goal[0]). +See example scripts/test/peri3dController_example1 for illusration. +xyPath(=vector(1, Vector2r::Ones())) +Time function for xy direction, see xxPath +youngEstimation(=1e20) +Estimation of macroscopic Young’s modulus, used for the first simulation step +yyPath(=vector(1, Vector2r::Ones())) +Time function for yy direction, see xxPath +2.3. +Yade wrapper class reference +237 + +Yade Documentation, Release 3rd ed. +yzPath(=vector(1, Vector2r::Ones())) +Time function for yz direction, see xxPath +zxPath(=vector(1, Vector2r::Ones())) +Time function for zx direction, see xxPath +zzPath(=vector(1, Vector2r::Ones())) +Time function for zz direction, see xxPath +class yade.wrapper.PeriIsoCompressor(inherits BoundaryController → GlobalEngine → En- +gine → Serializable) +Compress/decompress cloud of spheres by controlling periodic cell size until it reaches prescribed +average stress, then moving to next stress value in given stress series. +charLen(=-1.) +Characteristic length, should be something like mean particle diameter (default -1=invalid +value)) +currUnbalanced +Current value of unbalanced force +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +doneHook(=””) +Python command to be run when reaching the last specified stress +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +globalUpdateInt(=20) +how often to recompute average stress, stiffness and unbalanced force +keepProportions(=true) +Exactly keep proportions of the cell (stress is controlled based on average, not its components +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +maxSpan(=-1.) +Maximum body span in terms of bbox, to prevent periodic cell getting too small. (auto- +computed) +maxUnbalanced(=1e-4) +if actual unbalanced force is smaller than this number, the packing is considered stable, +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +sigma +Current stress value +238 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +state(=0) +Where are we at in the stress series +stresses(=uninitalized) +Stresses that should be reached, one after another +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.PeriTriaxController(inherits BoundaryController → GlobalEngine → +Engine → Serializable) +Engine for independently controlling stress or strain in periodic simulations. +PeriTriaxController.goal contains absolute values for the controlled quantity, and Peri- +TriaxController.stressMask determines meaning of those values (0 for strain, 1 for stress): +e.g. ( 1<<0 | 1<<2 ) = 1 | 4 = 5 means that goal[0] and goal[2] are stress values, +and goal[1] is strain. +See scripts/test/periodic-triax.py for a simple example. +absStressTol(=1e3) +Absolute stress tolerance +currUnbalanced(=NaN) +current unbalanced force (updated every globUpdate) (auto-updated) +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +doneHook(=uninitalized) +python command to be run when the desired state is reached +dynCell(=false) +Imposed stress can be controlled using the packing stiffness or by applying the laws of dynamic +(dynCell=true). Don’t forget to assign a mass to the cell. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +externalWork(=0) +Work input from boundary controller. +globUpdate(=5) +How often to recompute average stress, stiffness and unbalaced force. +goal +Desired +stress +or +strain +values +(depending +on +stressMask), +strains +defined +as +strain(i)=log(Fii). +Warning: +Strains are relative to the O.cell.refSize (reference cell size), not the current +one (e.g. at the moment when the new strain value is set). +2.3. +Yade wrapper class reference +239 + +Yade Documentation, Release 3rd ed. +growDamping(=.25) +Damping of cell resizing (0=perfect control, 1=no control at all); see also wallDamping in +TriaxialStressController. +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +mass(=NaN) +mass of the cell (user set); if not set and dynCell is used, it will be computed as sum of masses +of all particles. +maxBodySpan(=Vector3r::Zero()) +maximum body dimension (auto-computed) +maxStrainRate(=Vector3r(1, 1, 1)) +Maximum strain rate of the periodic cell. +maxUnbalanced(=1e-4) +maximum unbalanced force. +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +prevGrow(=Vector3r::Zero()) +previous cell grow +relStressTol(=3e-5) +Relative stress tolerance +stiff(=Vector3r::Zero()) +average stiffness (only every globUpdate steps recomputed from interactions) (auto-updated) +strain(=Vector3r::Zero()) +cell strain (auto-updated) +strainRate(=Vector3r::Zero()) +cell strain rate (auto-updated) +stress(=Vector3r::Zero()) +diagonal terms of the stress tensor +stressMask(=0, all strains) +mask determining strain/stress (0/1) meaning for goal components +stressTensor(=Matrix3r::Zero()) +average stresses, updated at every step (only every globUpdate steps recomputed from inter- +actions if !dynCell) +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.ThreeDTriaxialEngine(inherits +TriaxialStressController +→ +Bound- +aryController +→ +GlobalEngine +→ +Engine +→ +Serializable) +The engine perform a triaxial compression with a control in direction ‘i’ in stress (if stressControl_i) +else in strain. +240 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +For a stress control the imposed stress is specified by ‘sigma_i’ with a ‘max_veli’ depending on +‘strainRatei’. To obtain the same strain rate in stress control than in strain control you need to +set ‘wallDamping = 0.8’. For a strain control the imposed strain is specified by ‘strainRatei’. With +this engine you can also perform internal compaction by growing the size of particles by using +TriaxialStressController::controlInternalStress. For that, just switch on ‘internalCom- +paction=1’ and fix sigma_iso=value of mean pressure that you want at the end of the internal +compaction. +Warning: +This engine is deprecated, please switch to TriaxialStressController if you expect +long term support. +Key(=””) +A string appended at the end of all files, use it to name simulations. +UnbalancedForce(=1) +mean resultant forces divided by mean contact force +boxVolume +Total packing volume. +computeStressStrainInterval(=10) +currentStrainRate1(=0) +current strain rate in direction 1 - converging to ThreeDTriaxialEngine::strainRate1 (./s) +currentStrainRate2(=0) +current strain rate in direction 2 - converging to ThreeDTriaxialEngine::strainRate2 (./s) +currentStrainRate3(=0) +current strain rate in direction 3 - converging to ThreeDTriaxialEngine::strainRate3 (./s) +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +depth(=0) +size of the box (2-axis) (auto-updated) +depth0(=0) +Reference size for strain definition. See TriaxialStressController::depth +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +externalWork(=0) +Mechanical work associated to the boundary conditions, i.e. +� +∂Ω T · uds with T the surface +traction and u the displacement at the boundary. (auto-updated) +finalMaxMultiplier(=1.00001) +max multiplier of diameters during internal compaction (secondary precise adjustment - Tri- +axialStressController::maxMultiplier is used in the initial stage) +frictionAngleDegree(=-1) +Value of friction used in the simulation if (updateFrictionAngle) +goal1(=0) +prescribed stress/strain rate on axis 1, as defined by TriaxialStressController::stressMask +2.3. +Yade wrapper class reference +241 + +Yade Documentation, Release 3rd ed. +goal2(=0) +prescribed stress/strain rate on axis 2, as defined by TriaxialStressController::stressMask +goal3(=0) +prescribed stress/strain rate on axis 3, as defined by TriaxialStressController::stressMask +height(=0) +size of the box (1-axis) (auto-updated) +height0(=0) +Reference size for strain definition. See TriaxialStressController::height +internalCompaction(=true) +Switch between ‘external’ (walls) and ‘internal’ (growth of particles) compaction. +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +maxMultiplier(=1.001) +max multiplier of diameters during internal compaction (initial fast increase - TriaxialStress- +Controller::finalMaxMultiplier is used in a second stage) +max_vel(=1) +Maximum allowed walls velocity [m/s]. This value superseeds the one assigned by the stress +controller if the later is higher. max_vel can be set to infinity in many cases, but sometimes +helps stabilizing packings. Based on this value, different maxima are computed for each axis +based on the dimensions of the sample, so that if each boundary moves at its maximum +velocity, the strain rate will be isotropic (see e.g. TriaxialStressController::max_vel1). +max_vel1 +see TriaxialStressController::max_vel (auto-computed) +max_vel2 +see TriaxialStressController::max_vel (auto-computed) +max_vel3 +see TriaxialStressController::max_vel (auto-computed) +meanStress(=0) +Mean stress in the packing. (auto-updated) +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +particlesVolume +Total volume of particles (clumps and dynamic spheres). (auto-computed) +porosity +Porosity of the packing, computed from particlesVolume and boxVolume. (auto-updated) +previousMultiplier(=1) +(auto-updated) +previousStress(=0) +(auto-updated) +radiusControlInterval(=10) +setContactProperties((ThreeDTriaxialEngine)arg1, (float)arg2) → None : +Assign a new friction angle (degrees) to dynamic bodies and relative interactions +242 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +spheresVolume +Shorthand for TriaxialStressController::particlesVolume +stiffnessUpdateInterval(=10) +iteration period for measuring the resultant packing-boundaries stiffnesses, for stress servo- +control +strain +Current strain in a vector (exx,eyy,ezz). The values reflect true (logarithmic) strain. +strainDamping(=0.9997) +factor used for smoothing changes in effective strain rate. +If target rate is TR, then (1- +damping)*(TR-currentRate) will be added at each iteration. With damping=0, rate=target +all the time. With damping=1, it doesn’t change. +strainRate +Current strain rate in a vector d/dt(exx,eyy,ezz). +strainRate1(=0) +target strain rate in direction 1 (./s, >0 for compression) +strainRate2(=0) +target strain rate in direction 2 (./s, >0 for compression) +strainRate3(=0) +target strain rate in direction 3 (./s, >0 for compression) +stress((TriaxialStressController)arg1, (int)id) → Vector3 : +Returns the average stress on boundary ‘id’. Here, ‘id’ refers to the internal numbering of +boundaries, between 0 and 5. +stressControl_1(=true) +Switch to choose a stress or a strain control in directions 1 +stressControl_2(=true) +Switch to choose a stress or a strain control in directions 2 +stressControl_3(=true) +Switch to choose a stress or a strain control in directions 3 +stressDamping(=0.25) +wall damping coefficient for the stress control - wallDamping=0 implies a (theoretical) perfect +control, wallDamping=1 means no movement +stressMask(=7) +Bitmask determining wether the imposed goal values are stresses (0 for none, 7 for all, 1 for +direction 1, 5 for directions 1 and 3, etc.) or strain rates +thickness(=-1) +thickness of boxes (needed by some functions) +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +updateFrictionAngle(=false) +Switch to activate the update of the intergranular frictionto the value ThreeDTriaxi- +alEngine::frictionAngleDegree. +updatePorosity(=false) +If true, solid volume will be updated once (will automatically reset to false after one calculation +step) e.g. for porosity calculation purpose. Can be used when volume of particles changes +during the simulation (e.g. when particles are erased or when clumps are created). +2.3. +Yade wrapper class reference +243 + +Yade Documentation, Release 3rd ed. +volumetricStrain(=0) +Volumetric strain (see TriaxialStressController::strain). (auto-updated) +wall_back_activated(=true) +if true, this wall moves according to the target value (stress or strain rate). +wall_back_id(=4) +id of boundary ; coordinate 2- (default value is ok if aabbWalls are appended BEFORE +spheres.) +wall_bottom_activated(=true) +if true, this wall moves according to the target value (stress or strain rate). +wall_bottom_id(=2) +id of boundary ; coordinate 1- (default value is ok if aabbWalls are appended BEFORE +spheres.) +wall_front_activated(=true) +if true, this wall moves according to the target value (stress or strain rate). +wall_front_id(=5) +id of boundary ; coordinate 2+ (default value is ok if aabbWalls are appended BEFORE +spheres.) +wall_left_activated(=true) +if true, this wall moves according to the target value (stress or strain rate). +wall_left_id(=0) +id of boundary ; coordinate 0- (default value is ok if aabbWalls are appended BEFORE +spheres.) +wall_right_activated(=true) +if true, this wall moves according to the target value (stress or strain rate). +wall_right_id(=1) +id of boundary ; coordinate 0+ (default value is ok if aabbWalls are appended BEFORE +spheres.) +wall_top_activated(=true) +if true, this wall moves according to the target value (stress or strain rate). +wall_top_id(=3) +id of boundary ; coordinate 1+ (default value is ok if aabbWalls are appended BEFORE +spheres.) +width(=0) +size of the box (0-axis) (auto-updated) +width0(=0) +Reference size for strain definition. See TriaxialStressController::width +class yade.wrapper.TriaxialCompressionEngine(inherits TriaxialStressController → Bound- +aryController → GlobalEngine → Engine → +Serializable) +The engine is a state machine with the following states; transitions my be automatic, see below. +1. STATE_ISO_COMPACTION: isotropic compaction (compression) until the prescribed mean +pressue sigmaIsoCompaction is reached and the packing is stable. The compaction happens +either by straining the walls (!internalCompaction) or by growing size of grains (internalCom- +paction). +2. STATE_ISO_UNLOADING: isotropic unloading from the previously reached state, until the +mean pressure sigmaLateralConfinement is reached (and stabilizes). +244 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +Note: +this state will be skipped if sigmaLateralConfinement == sigmaIsoCom- +paction. +3. STATE_TRIAX_LOADING: confined uniaxial compression: constant sigmaLateralConfine- +ment is kept at lateral walls (left, right, front, back), while top and bottom walls load the +packing in their axis (by straining), until the value of epsilonMax (deformation along the +loading axis) is reached. At this point, the simulation is stopped. +4. STATE_FIXED_POROSITY_COMPACTION: isotropic compaction (compression) until a +chosen porosity value (parameter:fixedPorosity). The six walls move with a chosen translation +speed (parameter StrainRate). +5. STATE_TRIAX_LIMBO: currently unused, since simulation is hard-stopped in the previous +state. +Transition from COMPACTION to UNLOADING is done automatically if autoUnload==true; +Transition from (UNLOADING to LOADING) or from (COMPACTION to LOADING: +if UNLOADING is skipped) is done automatically if autoCompressionActivation=true; +Both autoUnload and autoCompressionActivation are true by default. +Note: +Most of the algorithms used have been developed initialy for simulations reported in +[Chareyre2002a] and [Chareyre2005]. They have been ported to Yade in a second step and used in +e.g. [Kozicki2008],[Scholtes2009b]_,[Jerier2010b]. +Warning: +This engine is deprecated, please switch to TriaxialStressController if you expect +long term support. +Key(=””) +A string appended at the end of all files, use it to name simulations. +StabilityCriterion(=0.001) +tolerance in terms of TriaxialCompressionEngine::UnbalancedForce to consider the packing is +stable +UnbalancedForce(=1) +mean resultant forces divided by mean contact force +autoCompressionActivation(=true) +Auto-switch +from +isotropic +compaction +(or +unloading +state +if +sigmaLateralConfine- +ment0 for compression) +stress((TriaxialStressController)arg1, (int)id) → Vector3 : +Returns the average stress on boundary ‘id’. Here, ‘id’ refers to the internal numbering of +boundaries, between 0 and 5. +stressDamping(=0.25) +wall damping coefficient for the stress control - wallDamping=0 implies a (theoretical) perfect +control, wallDamping=1 means no movement +stressMask(=7) +Bitmask determining wether the imposed goal values are stresses (0 for none, 7 for all, 1 for +direction 1, 5 for directions 1 and 3, etc.) or strain rates +testEquilibriumInterval(=20) +interval of checks for transition between phases, higher than 1 saves computation time. +thickness(=-1) +thickness of boxes (needed by some functions) +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +translationAxis(=TriaxialStressController::normal[wall_bottom]) +compression axis +uniaxialEpsilonCurr(=1) +Current value of axial deformation during confined loading (is reference to strain[1]) +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +updatePorosity(=false) +If true, solid volume will be updated once (will automatically reset to false after one calculation +step) e.g. for porosity calculation purpose. Can be used when volume of particles changes +during the simulation (e.g. when particles are erased or when clumps are created). +volumetricStrain(=0) +Volumetric strain (see TriaxialStressController::strain). (auto-updated) +wall_back_activated(=true) +if true, this wall moves according to the target value (stress or strain rate). +wall_back_id(=4) +id of boundary ; coordinate 2- (default value is ok if aabbWalls are appended BEFORE +spheres.) +wall_bottom_activated(=true) +if true, this wall moves according to the target value (stress or strain rate). +248 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +wall_bottom_id(=2) +id of boundary ; coordinate 1- (default value is ok if aabbWalls are appended BEFORE +spheres.) +wall_front_activated(=true) +if true, this wall moves according to the target value (stress or strain rate). +wall_front_id(=5) +id of boundary ; coordinate 2+ (default value is ok if aabbWalls are appended BEFORE +spheres.) +wall_left_activated(=true) +if true, this wall moves according to the target value (stress or strain rate). +wall_left_id(=0) +id of boundary ; coordinate 0- (default value is ok if aabbWalls are appended BEFORE +spheres.) +wall_right_activated(=true) +if true, this wall moves according to the target value (stress or strain rate). +wall_right_id(=1) +id of boundary ; coordinate 0+ (default value is ok if aabbWalls are appended BEFORE +spheres.) +wall_top_activated(=true) +if true, this wall moves according to the target value (stress or strain rate). +wall_top_id(=3) +id of boundary ; coordinate 1+ (default value is ok if aabbWalls are appended BEFORE +spheres.) +warn(=0) +counter used for sending a deprecation warning once +width(=0) +size of the box (0-axis) (auto-updated) +width0(=0) +Reference size for strain definition. See TriaxialStressController::width +class yade.wrapper.TriaxialStressController(inherits BoundaryController → GlobalEngine +→ Engine → Serializable) +An engine maintaining constant stresses or constant strain rates on some boundaries of a par- +allepipedic packing. +The stress/strain control is defined for each axis using TriaxialStressCon- +troller::stressMask (a bitMask) and target values are defined by goal1,goal2, and goal3. The sign +conventions of continuum mechanics are used for strains and stresses (positive traction). +Note: +The algorithms used have been developed initialy for simulations reported in +[Chareyre2002a] and [Chareyre2005]. They have been ported to Yade in a second step and used in +e.g. [Kozicki2008],[Scholtes2009b]_,[Jerier2010b]. +boxVolume +Total packing volume. +computeStressStrainInterval(=10) +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +depth(=0) +size of the box (2-axis) (auto-updated) +2.3. +Yade wrapper class reference +249 + +Yade Documentation, Release 3rd ed. +depth0(=0) +Reference size for strain definition. See TriaxialStressController::depth +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +externalWork(=0) +Mechanical work associated to the boundary conditions, i.e. +� +∂Ω T · uds with T the surface +traction and u the displacement at the boundary. (auto-updated) +finalMaxMultiplier(=1.00001) +max multiplier of diameters during internal compaction (secondary precise adjustment - Tri- +axialStressController::maxMultiplier is used in the initial stage) +goal1(=0) +prescribed stress/strain rate on axis 1, as defined by TriaxialStressController::stressMask +goal2(=0) +prescribed stress/strain rate on axis 2, as defined by TriaxialStressController::stressMask +goal3(=0) +prescribed stress/strain rate on axis 3, as defined by TriaxialStressController::stressMask +height(=0) +size of the box (1-axis) (auto-updated) +height0(=0) +Reference size for strain definition. See TriaxialStressController::height +internalCompaction(=true) +Switch between ‘external’ (walls) and ‘internal’ (growth of particles) compaction. +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +maxMultiplier(=1.001) +max multiplier of diameters during internal compaction (initial fast increase - TriaxialStress- +Controller::finalMaxMultiplier is used in a second stage) +max_vel(=1) +Maximum allowed walls velocity [m/s]. This value superseeds the one assigned by the stress +controller if the later is higher. max_vel can be set to infinity in many cases, but sometimes +helps stabilizing packings. Based on this value, different maxima are computed for each axis +based on the dimensions of the sample, so that if each boundary moves at its maximum +velocity, the strain rate will be isotropic (see e.g. TriaxialStressController::max_vel1). +max_vel1 +see TriaxialStressController::max_vel (auto-computed) +max_vel2 +see TriaxialStressController::max_vel (auto-computed) +max_vel3 +see TriaxialStressController::max_vel (auto-computed) +meanStress(=0) +Mean stress in the packing. (auto-updated) +250 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +particlesVolume +Total volume of particles (clumps and dynamic spheres). (auto-computed) +porosity +Porosity of the packing, computed from particlesVolume and boxVolume. (auto-updated) +previousMultiplier(=1) +(auto-updated) +previousStress(=0) +(auto-updated) +radiusControlInterval(=10) +spheresVolume +Shorthand for TriaxialStressController::particlesVolume +stiffnessUpdateInterval(=10) +iteration period for measuring the resultant packing-boundaries stiffnesses, for stress servo- +control +strain +Current strain in a vector (exx,eyy,ezz). The values reflect true (logarithmic) strain. +strainDamping(=0.99) +coefficient used for smoother transitions in the strain rate. The rate reaches the target value +like dn reaches 0, where d is the damping coefficient and n is the number of steps +strainRate +Current strain rate in a vector d/dt(exx,eyy,ezz). +stress((TriaxialStressController)arg1, (int)id) → Vector3 : +Returns the average stress on boundary ‘id’. Here, ‘id’ refers to the internal numbering of +boundaries, between 0 and 5. +stressDamping(=0.25) +wall damping coefficient for the stress control - wallDamping=0 implies a (theoretical) perfect +control, wallDamping=1 means no movement +stressMask(=7) +Bitmask determining wether the imposed goal values are stresses (0 for none, 7 for all, 1 for +direction 1, 5 for directions 1 and 3, etc.) or strain rates +thickness(=-1) +thickness of boxes (needed by some functions) +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +updatePorosity(=false) +If true, solid volume will be updated once (will automatically reset to false after one calculation +step) e.g. for porosity calculation purpose. Can be used when volume of particles changes +during the simulation (e.g. when particles are erased or when clumps are created). +2.3. +Yade wrapper class reference +251 + +Yade Documentation, Release 3rd ed. +volumetricStrain(=0) +Volumetric strain (see TriaxialStressController::strain). (auto-updated) +wall_back_activated(=true) +if true, this wall moves according to the target value (stress or strain rate). +wall_back_id(=4) +id of boundary ; coordinate 2- (default value is ok if aabbWalls are appended BEFORE +spheres.) +wall_bottom_activated(=true) +if true, this wall moves according to the target value (stress or strain rate). +wall_bottom_id(=2) +id of boundary ; coordinate 1- (default value is ok if aabbWalls are appended BEFORE +spheres.) +wall_front_activated(=true) +if true, this wall moves according to the target value (stress or strain rate). +wall_front_id(=5) +id of boundary ; coordinate 2+ (default value is ok if aabbWalls are appended BEFORE +spheres.) +wall_left_activated(=true) +if true, this wall moves according to the target value (stress or strain rate). +wall_left_id(=0) +id of boundary ; coordinate 0- (default value is ok if aabbWalls are appended BEFORE +spheres.) +wall_right_activated(=true) +if true, this wall moves according to the target value (stress or strain rate). +wall_right_id(=1) +id of boundary ; coordinate 0+ (default value is ok if aabbWalls are appended BEFORE +spheres.) +wall_top_activated(=true) +if true, this wall moves according to the target value (stress or strain rate). +wall_top_id(=3) +id of boundary ; coordinate 1+ (default value is ok if aabbWalls are appended BEFORE +spheres.) +width(=0) +size of the box (0-axis) (auto-updated) +width0(=0) +Reference size for strain definition. See TriaxialStressController::width +class yade.wrapper.UniaxialStrainer(inherits BoundaryController → GlobalEngine → En- +gine → Serializable) +Axial displacing two groups of bodies in the opposite direction with given strain rate. +absSpeed(=NaN) +alternatively, absolute speed of boundary motion can be specified; this is effective only at the +beginning and if strainRate is not set; changing absSpeed directly during simulation wil have +no effect. [ms￿1] +active(=true) +Whether this engine is activated +asymmetry(=0, symmetric) +If 0, straining is symmetric for negIds and posIds; for 1 (or -1), only posIds are strained and +negIds don’t move (or vice versa) +252 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +avgStress(=0) +Current average stress (auto-updated) [Pa] +axis(=2) +The axis which is strained (0,1,2 for x,y,z) +blockDisplacements(=false) +Whether displacement of boundary bodies perpendicular to the strained axis are blocked or +are free +blockRotations(=false) +Whether rotations of boundary bodies are blocked. +crossSectionArea(=NaN) +crossSection perpendicular to he strained axis; must be given explicitly [m2] +currentStrainRate(=NaN) +Current strain rate (update automatically). (auto-updated) +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +idleIterations(=0) +Number of iterations that will pass without straining activity after stopStrain has been reached +initAccelTime(=-200) +Time for strain reaching the requested value (linear interpolation). If negative, the time is +dt*(-initAccelTime), where dt is the timestep at the first iteration. [s] +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +limitStrain(=0, disabled) +Invert the sense of straining (sharply, without transition) one this value of strain is reached. +Not effective if 0. +negIds(=uninitalized) +Bodies on which strain will be applied (on the negative end along the axis) +notYetReversed(=true) +Flag whether the sense of straining has already been reversed (only used internally). +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +originalLength(=NaN) +Distance of reference bodies in the direction of axis before straining started (computed auto- +matically) [m] +2.3. +Yade wrapper class reference +253 + +Yade Documentation, Release 3rd ed. +posIds(=uninitalized) +Bodies on which strain will be applied (on the positive end along the axis) +setSpeeds(=false) +should we set speeds at the beginning directly, instead of increasing strain rate progressively? +stopStrain(=NaN) +Strain at which we will pause simulation; inactive (nan) by default; must be reached from +below (in absolute value) +strain(=0) +Current strain value, elongation/originalLength (auto-updated) [-] +strainRate(=NaN) +Rate of strain, starting at 0, linearly raising to strainRate. [-] +stressUpdateInterval(=10) +How often to recompute stress on supports. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +Collider +Collider +GeneralIntegratorInsertionSortCollider +InsertionSortCollider +SpatialQuickSortCollider +FlatGridCollider +Fig. 28: Inheritance graph of Collider. See also: FlatGridCollider, GeneralIntegratorInsertionSortCol- +lider, InsertionSortCollider, SpatialQuickSortCollider. +class yade.wrapper.Collider(inherits GlobalEngine → Engine → Serializable) +Abstract class for finding spatial collisions between bodies. +Special constructor +Derived colliders (unless they override pyHandleCustomCtorArgs) can be given list of BoundFunc- +tors which is used to initialize the internal boundDispatcher instance. +avoidSelfInteractionMask(=0) +This mask is used to avoid the interactions inside a group of particles. To do so, the particles +must have the exact same mask and that mask should have one bit in common with this +avoidSelfInteractionMask as for their binary representations. +boundDispatcher(=new BoundDispatcher) +BoundDispatcher object that is used for creating bounds on collider’s request as necessary. +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +254 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.FlatGridCollider(inherits Collider → GlobalEngine → Engine → Serial- +izable) +Non-optimized grid collider, storing grid as dense flat array. Each body is assigned to (possibly +multiple) cells, which are arranged in regular grid between aabbMin and aabbMax, with cell size +step (same in all directions). Bodies outsize (aabbMin, aabbMax) are handled gracefully, assigned +to closest cells (this will create spurious potential interactions). verletDist determines how much is +each body enlarged to avoid collision detection at every step. +Note: +This collider keeps all cells in linear memory array, therefore will be memory-inefficient for +sparse simulations. +Warning: +objects Body::bound are not used, BoundFunctors are not used either: assigning +cells to bodies is hard-coded internally. Currently handles Shapes are: Sphere. +Note: +Periodic boundary is not handled (yet). +aabbMax(=Vector3r::Zero()) +Upper corner of grid (approximate, might be rouded up to minStep. +aabbMin(=Vector3r::Zero()) +Lower corner of grid. +avoidSelfInteractionMask(=0) +This mask is used to avoid the interactions inside a group of particles. To do so, the particles +must have the exact same mask and that mask should have one bit in common with this +avoidSelfInteractionMask as for their binary representations. +boundDispatcher(=new BoundDispatcher) +BoundDispatcher object that is used for creating bounds on collider’s request as necessary. +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +2.3. +Yade wrapper class reference +255 + +Yade Documentation, Release 3rd ed. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +step(=0) +Step in the grid (cell size) +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +verletDist(=0) +Length by which enlarge space occupied by each particle; avoids running collision detection +at every step. +class yade.wrapper.GeneralIntegratorInsertionSortCollider(inherits +InsertionSort- +Collider +→ +Collider +→ +GlobalEngine → Engine → +Serializable) +This class is the adaptive version of the InsertionSortCollider and changes the NewtonIntegrator +dependency of the collider algorithms to the Integrator interface which is more general. +allowBiggerThanPeriod +If true, tests on bodies sizes will be disabled, and the simulation will run normaly even if +bodies larger than period are found. It can be useful when the periodic problem include e.g. +a floor modelized with wall/box/facet. Be sure you know what you are doing if you touch this +flag. The result is undefined if one large body moves out of the (0,0,0) period. +avoidSelfInteractionMask(=0) +This mask is used to avoid the interactions inside a group of particles. To do so, the particles +must have the exact same mask and that mask should have one bit in common with this +avoidSelfInteractionMask as for their binary representations. +boundDispatcher(=new BoundDispatcher) +BoundDispatcher object that is used for creating bounds on collider’s request as necessary. +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +doSort(=false) +Do forced resorting of interactions. +256 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +dumpBounds((InsertionSortCollider)arg1) → tuple : +Return representation of the internal sort data. The format is ([...],[...],[...]) for 3 +axes, where each ... is a list of entries (bounds). The entry is a tuple with the fllowing items: +• coordinate (float) +• body id (int), but negated for negative bounds +• period numer (int), if the collider is in the periodic regime. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +fastestBodyMaxDist(=0) +Normalized maximum displacement of the fastest body since last run; if >= 1, we could get +out of bboxes and will trigger full run. (auto-updated) +isActivated((InsertionSortCollider)arg1) → bool : +Return true if collider needs execution at next iteration. +keepListsShort(=false) +if true remove bounds of non-existent or unbounded bodies from the lists (auto-updated); +turned true automatically in MPI mode and if bodies are erased with BodyCon- +tainer.enableRedirection‘=True. :ydefault:‘false +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +minSweepDistFactor(=0.1) +Minimal distance by which enlarge all bounding boxes; superseeds computed value of verlet- +Dist when lower that (minSweepDistFactor x verletDist). +newton(=shared_ptr()) +reference to active Newton integrator. (auto-updated) +numAction(=0) +Cummulative number of collision detection. +numReinit(=0) +Cummulative number of bound array re-initialization. +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +overlapTolerance(=1e-7) +Tolerance on determining overlap. In rare cases different parts of the code can inconsistently +lead to different results in terms of overlap, with false negative by spatialOverlapPeri possibly +leading to nasty bugs in contact detection (false positive are harmless). This tolerance is to +avoid false negative, the value can be understood as relative to 1 (i.e. independent of particle +size or any other reference length). The default should be ok. +periodic +Whether the collider is in periodic mode (read-only; for debugging) (auto-updated) +smartInsertErase(=false) +Use an algorithm optimized for heavy insert/delete (avoid initSort) - experimental. +2.3. +Yade wrapper class reference +257 + +Yade Documentation, Release 3rd ed. +sortAxis(=0) +Axis for the initial contact detection. +sortThenCollide(=false) +Separate sorting and colliding phase; it is MUCH slower, but all interactions are processed +at every step; this effectively makes the collider non-persistent, not remembering last state. +(The default behavior relies on the fact that inversions during insertion sort are overlaps of +bounding boxes that just started/ceased to exist, and only processes those; this makes the +collider much more efficient.) +strideActive +Whether striding is active (read-only; for debugging). (auto-updated) +targetInterv(=100) +(experimental) Target number of iterations between bound update, used to define a smaller +sweep distance for slower grains if >0, else always use 1*verletDist. Useful in simulations with +strong velocity contrasts between slow bodies and fast bodies. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +updatingDispFactor(=-1) +(experimental) Displacement factor used to trigger bound update: the bound is updated only +if updatingDispFactor*disp>sweepDist when >0, else all bounds are updated. +verletDist(=-.5, Automatically initialized) +Length by which to enlarge particle bounds, to avoid running collider at every step. Stride +disabled if zero. Negative value will trigger automatic computation, so that the real value will +be verletDist × minimum spherical particle radius; if there are no spherical particles, it will +be disabled. The actual length added to one bound can be only a fraction of verletDist when +InsertionSortCollider::targetInterv is > 0. +class yade.wrapper.InsertionSortCollider(inherits Collider → GlobalEngine → Engine → +Serializable) +Collider with O(n log(n)) complexity, using Aabb for bounds. +At the initial step, Bodies’ bounds (along sortAxis) are first std::sort’ed along this (sortAxis) +axis, then collided. The initial sort has O(n2) complexity, see Colliders’ performance for some +information (There are scripts in examples/collider-perf for measurements). +Insertion sort is used for sorting the bound list that is already pre-sorted from last iteration, where +each inversion calls checkOverlap which then handles either overlap (by creating interaction if +necessary) or its absence (by deleting interaction if it is only potential). +Bodies without bounding volume (such as clumps) are handled gracefully and never collide. Deleted +bodies are handled gracefully as well. +This collider handles periodic boundary conditions. There are some limitations, notably: +1. No body can have Aabb larger than cell’s half size in that respective dimension. You get +exception if it does and gets in interaction. One way to explicitly by-pass this restriction is +offered by allowBiggerThanPeriod, which can be turned on to insert a floor in the form of a +very large box for instance (see examples/periodicSandPile.py). +2. No body can travel more than cell’s distance in one step; this would mean that the simulation +is numerically exploding, and it is only detected in some cases. +Stride can be used to avoid running collider at every step by enlarging the particle’s bounds, +tracking their displacements and only re-run if they might have gone out of that bounds (see Verlet +list for brief description and background) . This requires cooperation from NewtonIntegrator as +well as BoundDispatcher, which will be found among engines automatically (exception is thrown if +they are not found). +258 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +If you wish to use strides, set verletDist (length by which bounds will be enlarged in all direc- +tions) to some value, e.g. 0.05 × typical particle radius. This parameter expresses the tradeoff +between many potential interactions (running collider rarely, but with longer exact interaction res- +olution phase) and few potential interactions (running collider more frequently, but with less exact +resolutions of interactions); it depends mainly on packing density and particle radius distribution. +If targetInterv is >1, not all particles will have their bound enlarged by verletDist; instead, +they will have bounds increased by a length in order to trigger a new colliding after targetInterv +iteration, assuming they move at almost constant velocity. Ideally in this method, all particles +would reach their bounds at the sime iteration. This is of course not the case as soon as velocities +fluctuate in time. Bound::sweepLength is tuned on the basis of the displacement recorded between +the last two runs of the collider. In this situation, verletDist defines the maximum sweep length. +allowBiggerThanPeriod +If true, tests on bodies sizes will be disabled, and the simulation will run normaly even if +bodies larger than period are found. It can be useful when the periodic problem include e.g. +a floor modelized with wall/box/facet. Be sure you know what you are doing if you touch this +flag. The result is undefined if one large body moves out of the (0,0,0) period. +avoidSelfInteractionMask(=0) +This mask is used to avoid the interactions inside a group of particles. To do so, the particles +must have the exact same mask and that mask should have one bit in common with this +avoidSelfInteractionMask as for their binary representations. +boundDispatcher(=new BoundDispatcher) +BoundDispatcher object that is used for creating bounds on collider’s request as necessary. +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +doSort(=false) +Do forced resorting of interactions. +dumpBounds((InsertionSortCollider)arg1) → tuple : +Return representation of the internal sort data. The format is ([...],[...],[...]) for 3 +axes, where each ... is a list of entries (bounds). The entry is a tuple with the fllowing items: +• coordinate (float) +• body id (int), but negated for negative bounds +• period numer (int), if the collider is in the periodic regime. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +fastestBodyMaxDist(=0) +Normalized maximum displacement of the fastest body since last run; if >= 1, we could get +out of bboxes and will trigger full run. (auto-updated) +isActivated((InsertionSortCollider)arg1) → bool : +Return true if collider needs execution at next iteration. +keepListsShort(=false) +if true remove bounds of non-existent or unbounded bodies from the lists (auto-updated); +turned true automatically in MPI mode and if bodies are erased with BodyCon- +tainer.enableRedirection‘=True. :ydefault:‘false +2.3. +Yade wrapper class reference +259 + +Yade Documentation, Release 3rd ed. +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +minSweepDistFactor(=0.1) +Minimal distance by which enlarge all bounding boxes; superseeds computed value of verlet- +Dist when lower that (minSweepDistFactor x verletDist). +newton(=shared_ptr()) +reference to active Newton integrator. (auto-updated) +numAction(=0) +Cummulative number of collision detection. +numReinit(=0) +Cummulative number of bound array re-initialization. +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +overlapTolerance(=1e-7) +Tolerance on determining overlap. In rare cases different parts of the code can inconsistently +lead to different results in terms of overlap, with false negative by spatialOverlapPeri possibly +leading to nasty bugs in contact detection (false positive are harmless). This tolerance is to +avoid false negative, the value can be understood as relative to 1 (i.e. independent of particle +size or any other reference length). The default should be ok. +periodic +Whether the collider is in periodic mode (read-only; for debugging) (auto-updated) +smartInsertErase(=false) +Use an algorithm optimized for heavy insert/delete (avoid initSort) - experimental. +sortAxis(=0) +Axis for the initial contact detection. +sortThenCollide(=false) +Separate sorting and colliding phase; it is MUCH slower, but all interactions are processed +at every step; this effectively makes the collider non-persistent, not remembering last state. +(The default behavior relies on the fact that inversions during insertion sort are overlaps of +bounding boxes that just started/ceased to exist, and only processes those; this makes the +collider much more efficient.) +strideActive +Whether striding is active (read-only; for debugging). (auto-updated) +targetInterv(=100) +(experimental) Target number of iterations between bound update, used to define a smaller +sweep distance for slower grains if >0, else always use 1*verletDist. Useful in simulations with +strong velocity contrasts between slow bodies and fast bodies. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +updatingDispFactor(=-1) +(experimental) Displacement factor used to trigger bound update: the bound is updated only +if updatingDispFactor*disp>sweepDist when >0, else all bounds are updated. +260 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +verletDist(=-.5, Automatically initialized) +Length by which to enlarge particle bounds, to avoid running collider at every step. Stride +disabled if zero. Negative value will trigger automatic computation, so that the real value will +be verletDist × minimum spherical particle radius; if there are no spherical particles, it will +be disabled. The actual length added to one bound can be only a fraction of verletDist when +InsertionSortCollider::targetInterv is > 0. +class yade.wrapper.SpatialQuickSortCollider(inherits Collider → GlobalEngine → Engine +→ Serializable) +Collider using quicksort along axes at each step, using Aabb bounds. +Its performance is lower than that of InsertionSortCollider (see Colliders’ performance), but the +algorithm is simple enought to make it good for checking other collider’s correctness. +avoidSelfInteractionMask(=0) +This mask is used to avoid the interactions inside a group of particles. To do so, the particles +must have the exact same mask and that mask should have one bit in common with this +avoidSelfInteractionMask as for their binary representations. +boundDispatcher(=new BoundDispatcher) +BoundDispatcher object that is used for creating bounds on collider’s request as necessary. +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +FieldApplier +class yade.wrapper.FieldApplier(inherits GlobalEngine → Engine → Serializable) +Base for engines applying force files on particles. Not to be used directly. +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +2.3. +Yade wrapper class reference +261 + +Yade Documentation, Release 3rd ed. +FieldApplier +HdapsGravityEngine +GravityEngine +AxialGravityEngine +CentralConstantAccelerationEngine +Fig. 29: Inheritance graph of FieldApplier. See also: AxialGravityEngine, CentralConstantAcceleratio- +nEngine, GravityEngine, HdapsGravityEngine. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.AxialGravityEngine(inherits FieldApplier → GlobalEngine → Engine → +Serializable) +Apply acceleration (independent of distance) directed towards an axis. +acceleration(=0) +Acceleration magnitude [kgms￿2] +axisDirection(=Vector3r::UnitX()) +direction of the gravity axis (will be normalized automatically) +axisPoint(=Vector3r::Zero()) +Point through which the axis is passing. +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +262 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +mask(=0) +If mask defined, only bodies with corresponding groupMask will be affected by this engine. If +0, all bodies will be affected. +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.CentralConstantAccelerationEngine(inherits +FieldApplier +→ +Glob- +alEngine → Engine → Serializ- +able) +Engine applying constant acceleration to all bodies, towards a central body. Ignoring the distance +between them. +accel(=0) +Acceleration magnitude [kgms￿2] +centralBody(=Body::ID_NONE) +The body towards which all other bodies are attracted. +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +mask(=0) +If mask defined, only bodies with corresponding groupMask will be affected by this engine. If +0, all bodies will be affected. +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +2.3. +Yade wrapper class reference +263 + +Yade Documentation, Release 3rd ed. +reciprocal(=false) +If true, acceleration will be applied on the central body as well. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.GravityEngine(inherits FieldApplier → GlobalEngine → Engine → Seri- +alizable) +Engine applying constant acceleration to all bodies. DEPRECATED, use Newton::gravity unless +you need energy tracking or selective gravity application using groupMask). +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +gravity(=Vector3r::Zero()) +Acceleration [kgms￿2] +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +mask(=0) +If mask defined, only bodies with corresponding groupMask will be affected by this engine. If +0, all bodies will be affected. +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +warnOnce(=true) +For deprecation warning once. +class yade.wrapper.HdapsGravityEngine(inherits GravityEngine → FieldApplier → Glob- +alEngine → Engine → Serializable) +Read accelerometer in Thinkpad laptops (HDAPS and accordingly set gravity within the simula- +tion. This code draws from hdaps-gl . See scripts/test/hdaps.py for an example. +accel(=Vector2i::Zero()) +reading from the sysfs file +264 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +calibrate(=Vector2i::Zero()) +Zero position; if NaN, will be read from the hdapsDir / calibrate. +calibrated(=false) +Whether calibrate was already updated. Do not set to True by hand unless you also give a +meaningful value for calibrate. +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +gravity(=Vector3r::Zero()) +Acceleration [kgms￿2] +hdapsDir(=”/sys/devices/platform/hdaps”) +Hdaps directory; contains position (with accelerometer readings) and calibration (zero +acceleration). +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +mask(=0) +If mask defined, only bodies with corresponding groupMask will be affected by this engine. If +0, all bodies will be affected. +msecUpdate(=50) +How often to update the reading. +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +updateThreshold(=4) +Minimum difference of reading from the file before updating gravity, to avoid jitter. +warnOnce(=true) +For deprecation warning once. +zeroGravity(=Vector3r(0, 0, -1)) +Gravity if the accelerometer is in flat (zero) position. +2.3. +Yade wrapper class reference +265 + +Yade Documentation, Release 3rd ed. +PartialEngine +CombinedKinematicEngine +HarmonicRotationEngine +RotationEngine +KinematicEngine +InterpolatingDirectedForceEngine +ForceEngine +LawTester +LinearDragEngine +HarmonicForceEngine +HydroForceEngine +RadialForceEngine +DragEngine +TranslationEngine +InterpolatingHelixEngine +HelixEngine +TorqueEngine +BicyclePedalEngine +StepDisplacer +ServoPIDController +HarmonicMotionEngine +Fig. 30: +Inheritance graph of PartialEngine. +See also: +BicyclePedalEngine, CombinedKinemati- +cEngine, DragEngine, ForceEngine, HarmonicForceEngine, HarmonicMotionEngine, HarmonicRotatio- +nEngine, HelixEngine, HydroForceEngine, InterpolatingDirectedForceEngine, InterpolatingHelixEngine, +KinematicEngine, LawTester, LinearDragEngine, RadialForceEngine, RotationEngine, ServoPIDCon- +troller, StepDisplacer, TorqueEngine, TranslationEngine. +2.3.4 Partial engines +class yade.wrapper.PartialEngine(inherits Engine → Serializable) +Engine affecting only particular bodies in the simulation, namely those defined in ids attribute. See +also GlobalEngine. +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +ids(=uninitalized) +Ids list of bodies affected by this PartialEngine. +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +266 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +class yade.wrapper.BicyclePedalEngine(inherits KinematicEngine → PartialEngine → En- +gine → Serializable) +Engine applying the linear motion of bicycle pedal e.g. moving points around the axis without +rotation +angularVelocity(=0) +Angular velocity. [rad/s] +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +fi(=Mathr::PI/2.0) +Initial phase [radians] +ids(=uninitalized) +Ids list of bodies affected by this PartialEngine. +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +radius(=-1.0) +Rotation radius. [m] +rotationAxis(=Vector3r::UnitX()) +Axis of rotation (direction); will be normalized automatically. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.CombinedKinematicEngine(inherits PartialEngine → Engine → Serializ- +able) +Engine for applying combined displacements on pre-defined bodies. Constructed using + operator +on regular KinematicEngines. The ids operated on are those of the first engine in the combination +(assigned automatically). +comb(=uninitalized) +Kinematic engines that will be combined by this one, run in the order given. +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +2.3. +Yade wrapper class reference +267 + +Yade Documentation, Release 3rd ed. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +ids(=uninitalized) +Ids list of bodies affected by this PartialEngine. +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.DragEngine(inherits PartialEngine → Engine → Serializable) +Apply drag force on some particles at each step, decelerating them proportionally to their linear +velocities. The applied force reads +Fd = − v +|v| +1 +2ρ|v|2CdA +where ρ is the medium density (density), v is particle’s velocity, A is particle projected area (disc), +Cd is the drag coefficient (0.47 for Sphere), +Note: +Drag force is only applied to spherical particles, listed in ids. +Cd(=0.47) +Drag coefficient ‘_. +Rho(=1.225) +Density of the medium (fluid or air), by default - the density of the air. +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +268 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +ids(=uninitalized) +Ids list of bodies affected by this PartialEngine. +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.ForceEngine(inherits PartialEngine → Engine → Serializable) +Apply contact force on some particles at each step. +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +force(=Vector3r::Zero()) +Force to apply. +ids(=uninitalized) +Ids list of bodies affected by this PartialEngine. +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +2.3. +Yade wrapper class reference +269 + +Yade Documentation, Release 3rd ed. +class yade.wrapper.HarmonicForceEngine(inherits PartialEngine → Engine → Serializable) +This engine adds a harmonic (sinusoidal) force to a set of bodies. It is identical to Harmonic- +MotionEngine except a force amplitude is prescribed instead of motion, see also the dynamics of +harmonic motion +A(=Vector3r::Zero()) +Amplitude [N] +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +f(=Vector3r::Zero()) +Frequency [hertz] +fi(=Vector3r::Zero()) +Initial phase [radians]. By default, the phase is zero such that the force starts at zero. +ids(=uninitalized) +Ids list of bodies affected by this PartialEngine. +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.HarmonicMotionEngine(inherits KinematicEngine → PartialEngine → En- +gine → Serializable) +This engine implements the harmonic oscillation of bodies. See also HarmonicForceEngine that +applies a harmonic force, see also the dynamics of harmonic motion +A(=Vector3r::Zero()) +Amplitude [m] +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +270 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +f(=Vector3r::Zero()) +Frequency [hertz] +fi(=Vector3r(Mathr::PI/2.0, Mathr::PI/2.0, Mathr::PI/2.0)) +Initial phase [radians]. By default, the body oscillates around initial position. +ids(=uninitalized) +Ids list of bodies affected by this PartialEngine. +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.HarmonicRotationEngine(inherits RotationEngine → KinematicEngine → +PartialEngine → Engine → Serializable) +This engine implements the harmonic-rotation oscillation of bodies, see also the dynamics of har- +monic motion ; please, set dynamic=False for bodies, droven by this engine, otherwise amplitude +will be 2x more, than awaited. +A(=0) +Amplitude [rad] +angularVelocity(=0) +Angular velocity. [rad/s] +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +f(=0) +Frequency [hertz] +fi(=Mathr::PI/2.0) +Initial phase [radians]. By default, the body oscillates around initial position. +ids(=uninitalized) +Ids list of bodies affected by this PartialEngine. +2.3. +Yade wrapper class reference +271 + +Yade Documentation, Release 3rd ed. +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +rotateAroundZero(=false) +If True, bodies will not rotate around their centroids, but rather around zeroPoint. +rotationAxis(=Vector3r::UnitX()) +Axis of rotation (direction); will be normalized automatically. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +zeroPoint(=Vector3r::Zero()) +Point around which bodies will rotate if rotateAroundZero is True +class yade.wrapper.HelixEngine(inherits +RotationEngine +→ +KinematicEngine +→ +Par- +tialEngine → Engine → Serializable) +Engine applying both rotation and translation, along the same axis, whence the name HelixEngine +angleTurned(=0) +How much have we turned so far. (auto-updated) [rad] +angularVelocity(=0) +Angular velocity. [rad/s] +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +ids(=uninitalized) +Ids list of bodies affected by this PartialEngine. +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +linearVelocity(=0) +Linear velocity [m/s] +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +272 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +rotateAroundZero(=false) +If True, bodies will not rotate around their centroids, but rather around zeroPoint. +rotationAxis(=Vector3r::UnitX()) +Axis of rotation (direction); will be normalized automatically. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +zeroPoint(=Vector3r::Zero()) +Point around which bodies will rotate if rotateAroundZero is True +class yade.wrapper.HydroForceEngine(inherits PartialEngine → Engine → Serializable) +Engine performing a coupling of the DEM with a volume-averaged 1D fluid resolution to simulate steady uniform unidirectional fluid flow. It has been developed and used to model steady uniform gravity-driven turbulent bedload transport [Maurin2015b] [Maurin2016] [Maurin2018], but can be also used in its current state for laminar or pressure-driven configurations. The fundamentals of the model can be found in [Maurin2015b] and [Maurin2015PhD], and in more details in [Maurin2018_VANSbasis], [Maurin2018_VANSfluidResol] and [Maurin2018_VANSvalidations]. +The engine can be decomposed in three different parts: (i) It applies the fluid force on the +particles imposed by the fluid velocity profiles and fluid properties, (ii) It evaluates averaged +solid depth profiles necessary for the fluid force application and for the fluid resolution, (iii) +It solve the volume-averaged 1D fluid momentum balance. +The three different functions are detailed below: +(i) Fluid force on particles Apply to each particles, buoyancy, drag and lift force due to +a 1D fluid flow and can apply lubrication force between two particles. The applied drag +force reads +Fd = 1 +2CdAρf|vf − v|vf − v +where ρ is the fluid density (densFluid), v is particle’s velocity, vf is the velocity of the +fluid at the particle center (taken from the fluid velocity profile vxFluid), A = πd2/4 +is particle projected area (disc), Cd is the drag coefficient. +The formulation of the +drag coefficient depends on the local particle reynolds number and the solid volume +fraction. +The formulation of the drag is [Dallavalle1948] [RevilBaudard2013] with a +correction of Richardson-Zaki [Richardson1954] to take into account the hindrance ef- +fect. +This law is classical in sediment transport. +The possibly activated lubrica- +tion force (with parameter:yref:lubrication put to True) +reads: Flubrication = 6πηfvreln +δn+εr +, with ηf the fluid dynamic viscosity viscoDyn, vreln the +normal relative velocity of the two particles, δn the distance between the two particles +surface, and εr the roughness scale of the particle (roughnessPartScale). +It is possible to activate a fluctuation of the drag force for each particle which account for the turbulent fluctuation of the fluid velocity (velFluct). Three simple discrete random walk model have been implemented for the turbulent velocity fluctuation. The main one (turbulentFluctuations) takes as input the Reynolds stress tensor Rf +xz as a function of the depth, and allows to recover the main property of the fluctuations by imposing < u′ +xu′ +z > (z) =< Rf +xz > (z)/ρf. It requires as input < Rf +xz > (z) called ReynoldStresses in the code. +The formulation of the lift is taken from [Wiberg1985] and is such that : +FL = 1 +2CLAρf((vf − v)2 +top − (vf − v)2 +bottom) +Where the subscript top and bottom means evaluated at the top (respectively the bottom) +of the sphere considered. This formulation of the lift account for the difference of pressure +at the top and the bottom of the particle inside a turbulent shear flow. As this formulation +is controversial when approaching the threshold of motion [Schmeeckle2007] it is possible to +desactivate it with the variable lift. The buoyancy is taken into account through the buoyant +weight : +Fbuoyancy = −ρfVpg +, where g is the gravity vector along the vertical, and Vp is the volume of the particle. In the case +where the fluid flow is steady and uniform, the buoyancy reduces to its wall-normal component +2.3. +Yade wrapper class reference +273 + +Yade Documentation, Release 3rd ed. +(see [Maurin2018] for a full explanation), and one should put steadyFlow to true in order to kill +the streamwise component. +(ii) Averaged solid depth profiles The function averageProfile evaluates the volume av- +eraged depth profiles (1D) of particle velocity, particle solid volume fraction and par- +ticle drag force. It uses a volume-weighting average following [Maurin2015PhD]_[Mau- +rin2015b]_, i.e. the average of a variable Ap associated to particles at a given discretized +wall-normal position z is given by: +⟨A⟩s (z) = +� +p|zp∈[z−dz/2,z+dz/2] +Ap(t)Vp +z +� +p|zp∈[z−dz/2,z+dz/2] +Vp +z +Where the sums are over the particles contained inside the slice between the wall-normal +position z − dz/2 and z + dz/2, and Vp represents the part of the volume of the given +particle effectively contained inside the slice. For more details, see [Maurin2015PhD]. +(iii) 1D volume-average fluid resolution The fluid resolution is based on the resolution +of the 1D volume-averaged fluid momentum balance. +It assumes by definition (uni- +directional) that the fluid flow is steady and uniform. It is the same fluid resolution +as [RevilBaudard2013]. +Details can be found in this paper and in [Maurin2015PhD] +[Maurin2015b]. +The three different component can be used independently, e.g. applying a fluid force due +to an imposed fluid profile or solving the fluid momentum balance for a given concentra- +tion of particles. +Cl(=0.2) +Value of the lift coefficient taken from [Wiberg1985] +ReynoldStresses(=uninitalized) +Vector of size equal to nCell containing the Reynolds stresses as a function of the depth. +ReynoldStresses(z) = ρf < u′ +xu′ +z > (z)2 +averageDrag(=uninitalized) +Discretized average drag depth profile. No role in the engine, output parameter. For practical +reason, it can be evaluated directly inside the engine, calling from python the averageProfile() +method of the engine +averageDrag1(=uninitalized) +Discretized average drag depth profile of particles of type 1. Evaluated when twoSize is set to +True. +averageDrag2(=uninitalized) +Discretized average drag depth profile of particles of type 2. Evaluated when twoSize is set to +True. +averageProfile((HydroForceEngine)arg1) → None : +Compute and store the particle velocity (vxPart, vyPart, vzPart) and solid volume fraction +(phiPart) depth profile. +For each defined cell z, the k component of the average particle +velocity reads: +< vk >z= � +p Vpvp +k/ � +p Vp, +where the sum is made over the particles contained in the cell, vp +k is the k component of the +velocity associated to particle p, and Vp is the part of the volume of the particle p contained +inside the cell. This definition allows to smooth the averaging, and is equivalent to taking +into account the center of the particles only when there is a lot of particles in each cell. As +for the solid volume fraction, it is evaluated in the same way: for each defined cell z, it reads: +< φ >z= +1 +Vcell +� +p Vp, where Vcell is the volume of the cell considered, and Vp is the volume +of particle p contained in cell z. This function gives depth profiles of average velocity and solid +volume fraction, returning the average quantities in each cell of height dz, from the reference +horizontal plane at elevation zRef (input parameter) until the plane of elevation zRef plus +274 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +nCell times deltaZ (input parameters). When the option twoSize is set to True, evaluate +in addition the average drag (averageDrag1 and averageDrag2) and solid volume fraction +(phiPart1 and phiPart2) depth profiles considering only the particles of radius respectively +radiusPart1 and radiusPart2 in the averaging. +bedElevation(=0.) +Elevation of the bed above which the fluid flow is turbulent and the particles undergo turbulent +velocity fluctuation. +channelWidth(=1.) +Fluid resolution: Channel width for the evaluation of the fluid wall friction inside the fluid +resolution. +compatibilityOldVersion(=false) +Option to make HydroForceEngine compatible with former scripts. Slow down slightly the +calculation and will eventually be removed. +computeRadiusParts((HydroForceEngine)arg1) → None : +compute the different class of radius present in the simulation. +convAcc(=uninitalized) +Convective acceleration, depth dependent +convAccOption(=false) +To activate the convective acceleration option in order to account for a convective acceleration +term inside the momentum balance. +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +deltaZ(=uninitalized) +Height of the discretization cell. +densFluid(=1000) +Density of the fluid, by default - density of water +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dpdx(=0.) +pressure gradient along streamwise direction +dtFluct(=uninitalized) +Execution time step of the turbulent fluctuation model. +enableMultiClassAverage(=false) +Enables specific averaging for all the different particle size. Uses a lot of memory if using a +lots of different particle size +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +expoRZ(=3.1) +Value of the Richardson-Zaki exponent, for the drag correction due to hindrance +fluctTime(=uninitalized) +Vector containing the time of life of the fluctuations associated to each particles. +fluidFrictionCoef(=1.) +Fluid resolution: fitting coefficient for the fluid wall friction +2.3. +Yade wrapper class reference +275 + +Yade Documentation, Release 3rd ed. +fluidResolution((HydroForceEngine)arg1, (float)arg2, (float)arg3) → None : +Solve +the +1D +volume-averaged +fluid +momentum +balance +on +the +de- +fined +mesh +(nCell, +deltaZ) +from +the +volume-averaged +solid +profiles +(phiPart,:yref:vxPart,:yref:averageDrag), +which can be evaluated with the averageProfile function. +fluidWallFriction(=false) +Fluid resolution: if set to true, introduce a sink term to account for the fluid friction at the +wall, see [Maurin2015] for details. Requires to set the width of the channel. It might slow +down significantly the calculation. +gravity(=Vector3r(0, 0, -9.81)) +Gravity vector +ids(=uninitalized) +Ids list of bodies affected by this PartialEngine. +ilm(=2) +Fluid resolution: type of mixing length resolution applied: 0: classical Prandtl mixing length, +1: Prandtl mixing length with free-surface effects, 2: Damp turbulence accounting for the +presence of particles [Li1995], see [RevilBaudard2013] for more details. +initialization((HydroForceEngine)arg1) → None : +Initialize the necessary parameters to make HydroForceEngine run. +Necessary to execute +before any simulation run, otherwise it crashes +irheolf(=0) +Fluid resolution: effective fluid viscosity option: 0: pure fluid viscosity, 1: Einstein viscosity. +iturbu(=1) +Fluid resolution: activate the turbulence resolution, 1, or not, 0 +iusl(=1) +Fluid resolution: option to set the boundary condition at the top of the fluid, 0: Dirichlet, +fixed (u = uTop en z = h), 1:Neumann, free-surface (du/dz = 0 en z = h). +kappa(=0.41) +Fluid resolution: Von Karman constant. Can be tuned to account for the effect of particles +on the fluid turbulence, see e.g. [RevilBaudard2015] +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +lift(=false) +Option to activate or not the evaluation of the lift +lubrication(=false) +Condition to activate the calculation of the lubrication force. +multiDragPart(=uninitalized) +Spatial-averaged mean drag force for each class of particle. Un-used ? Or just for debug. +multiPhiPart(=uninitalized) +Spatial-averaged solid volume fraction for each class of particle. +multiVxPart(=uninitalized) +Spatial-averaged velocity in x direction for each class of particle. +multiVyPart(=uninitalized) +Spatial-averaged velocity in y direction for each class of particle. +multiVzPart(=uninitalized) +Spatial-averaged velocity in z direction for each class of particle. +nCell(=1) +Number of cell in the depth +276 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +nbAverageT(=0) +If >0, perform a time-averaging (in addition to the spatial averaging) over nbAverage steps. +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +phiBed(=0.08) +Turbulence modelling parameter. Associated with mixing length modelling ilm = 5. +phiMax(=0.64) +Fluid resolution: maximum solid volume fraction. +phiPart(=uninitalized) +Discretized solid volume fraction depth profile. Can be taken as input parameter or evaluated +directly inside the engine, calling from python the averageProfile() function +phiPart1(=uninitalized) +Discretized solid volume fraction depth profile of particles of type 1. Evaluated when twoSize +is set to True. +phiPart2(=uninitalized) +Discretized solid volume fraction depth profile of particles of type 2. Evaluated when twoSize +is set to True. +pointParticleAverage(=false) +Evaluate the averaged with a point particle method. If False, consider the particle extent and +weigth the averaged by the volume contained in each averaging cell. +radiusPart(=0.) +Reference particle radius +radiusPart1(=0.) +Radius of the particles of type 1. Useful only when twoSize is set to True. +radiusPart2(=0.) +Radius of the particles of type 2. Useful only when twoSize is set to True. +radiusParts(=uninitalized) +Variables containing the number of different radius of particles in the simulation. Allow to +perform class averaging by particle size. +roughnessPartScale(=1e-3) +Roughness length scale of the particle. In practice, the lubrication force is cut off when the +two particles are at a distance roughnessPartScale. +steadyFlow(=true) +Condition to modify the buoyancy force according to the physical difference between a fluid +at rest and a steady fluid flow. For more details see [Maurin2018] +taufsi(=uninitalized) +Fluid Resolution: Create Taufsi/rhof = dragTerm/(rhof(vf-vxp)) to transmit to the fluid code +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +turbulentFluctuation((HydroForceEngine)arg1) → None : +Apply a discrete random walk model to the evaluation of the drag force to account for the +fluid velocity turbulent fluctuations. Very simple model applying fluctuations from the values +of the Reynolds stresses in order to recover the property < u′ +xu′ +z > (z) =< Rf +xz > (z)/ρf. The +random fluctuations are modified over a time scale given by the eddy turn over time. +2.3. +Yade wrapper class reference +277 + +Yade Documentation, Release 3rd ed. +turbulentFluctuationZDep((HydroForceEngine)arg1) → None : +Apply turbulent fluctuation to the problem similarly to turbulentFluctuation but with an +update of the fluctuation depending on the particle position. +turbulentViscosity(=uninitalized) +Fluid Resolution: turbulent viscocity as a function of the depth +twoSize(=false) +Not maintained anymore. Option to activate when considering two particle size in the simu- +lation. When activated evaluate the average solid volume fraction and drag force for the two +type of particles of diameter diameterPart1 and diameterPart2 independently. +uTop(=1.) +Fluid resolution: fluid velocity at the top boundary when iusl = 0 +unCorrelatedFluctuations(=false) +Condition to generate uncorrelated fluid fluctuations. Default case represent in free-surface +flows, for which the vertical and streamwise fluid velocity fluctuations are correlated (see e.g. +reference book of Nezu & Nagakawa 1992, turbulence in open channel flows). +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +vCell(=uninitalized) +Volume of averaging cell +vFluctX(=uninitalized) +Vector associating a streamwise fluid velocity fluctuation to each particle. Fluctuation calcu- +lated in the C++ code from the discrete random walk model +vFluctY(=uninitalized) +Vector associating a spanwise fluid velocity fluctuation to each particle. Fluctuation calculated +in the C++ code from the discrete random walk model +vFluctZ(=uninitalized) +Vector associating a normal fluid velocity fluctuation to each particle. Fluctuation calculated +in the C++ code from the discrete random walk model +vPart(=uninitalized) +Discretized streamwise solid velocity depth profile, in x, y and z direction. Only the x direction +measurement is taken into account in the 1D fluid coupling resolution. The two other can be +used as output parameters. The x component can be taken as input parameter, or evaluated +directly inside the engine, calling from python the averageProfile() function +velFluct(=false) +If true, activate the determination of turbulent fluid velocity fluctuation for the next time +step only at the position of each particle, using a simple discrete random walk (DRW) model +based on the Reynolds stresses profile (ReynoldStresses) +viscoDyn(=1e-3) +Dynamic viscosity of the fluid, by default - viscosity of water +viscousSubLayer(=0) +Fluid resolution: solve the viscous sublayer close to the bottom boundary if set to 1 +vxFluid(=uninitalized) +Discretized streamwise fluid velocity depth profile at t +vxPart(=uninitalized) +Discretized streamwise solid velocity depth profile. +Can be taken as input parameter, or +evaluated directly inside the engine, calling from python the averageProfile() function +vxPart1(=uninitalized) +Discretized solid streamwise velocity depth profile of particles of type 1. +Evaluated when +twoSize is set to True. +278 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +vxPart2(=uninitalized) +Discretized solid streamwise velocity depth profile of particles of type 2. +Evaluated when +twoSize is set to True. +vyPart(=uninitalized) +Discretized spanwise solid velocity depth profile. Can be taken as input parameter, or evalu- +ated directly inside the engine, calling from python the averageProfile() function +vyPart1(=uninitalized) +Discretized solid spanwise velocity depth profile of particles of type 1. Evaluated when twoSize +is set to True. +vyPart2(=uninitalized) +Discretized solid spanwise velocity depth profile of particles of type 2. Evaluated when twoSize +is set to True. +vzPart(=uninitalized) +Discretized wall-normal solid velocity depth profile. Can be taken as input parameter, or +evaluated directly inside the engine, calling from python the averageProfile() function +vzPart1(=uninitalized) +Discretized solid wall-normal velocity depth profile of particles of type 1. Evaluated when +twoSize is set to True. +vzPart2(=uninitalized) +Discretized solid wall-normal velocity depth profile of particles of type 2. Evaluated when +twoSize is set to True. +zRef(=0.) +Position of the reference point which correspond to the first value of the fluid velocity, i.e. to +the ground. +class yade.wrapper.InterpolatingDirectedForceEngine(inherits +ForceEngine +→ +Par- +tialEngine → Engine → Serializ- +able) +Engine for applying force of varying magnitude but constant direction on subscribed bodies. times +and magnitudes must have the same length, direction (normalized automatically) gives the orien- +tation. +As usual with interpolating engines: the first magnitude is used before the first time point, last +magnitude is used after the last time point. Wrap specifies whether time wraps around the last +time point to the first time point. +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +direction(=Vector3r::UnitX()) +Contact force direction (normalized automatically) +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +force(=Vector3r::Zero()) +Force to apply. +ids(=uninitalized) +Ids list of bodies affected by this PartialEngine. +2.3. +Yade wrapper class reference +279 + +Yade Documentation, Release 3rd ed. +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +magnitudes(=uninitalized) +Force magnitudes readings [N] +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +times(=uninitalized) +Time readings [s] +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +wrap(=false) +wrap to the beginning of the sequence if beyond the last time point +class yade.wrapper.InterpolatingHelixEngine(inherits HelixEngine → RotationEngine → +KinematicEngine → PartialEngine → Engine +→ Serializable) +Engine applying spiral motion, finding current angular velocity by linearly interpolating in times +and velocities and translation by using slope parameter. +The interpolation assumes the margin value before the first time point and last value after the last +time point. If wrap is specified, time will wrap around the last times value to the first one (note +that no interpolation between last and first values is done). +angleTurned(=0) +How much have we turned so far. (auto-updated) [rad] +angularVelocities(=uninitalized) +List of angular velocities; manadatorily of same length as times. [rad/s] +angularVelocity(=0) +Angular velocity. [rad/s] +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +ids(=uninitalized) +Ids list of bodies affected by this PartialEngine. +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +280 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +linearVelocity(=0) +Linear velocity [m/s] +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +rotateAroundZero(=false) +If True, bodies will not rotate around their centroids, but rather around zeroPoint. +rotationAxis(=Vector3r::UnitX()) +Axis of rotation (direction); will be normalized automatically. +slope(=0) +Axial translation per radian turn (can be negative) [m/rad] +times(=uninitalized) +List of time points at which velocities are given; must be increasing [s] +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +wrap(=false) +Wrap t if t>times_n, i.e. t_wrapped=t-N*(times_n-times_0) +zeroPoint(=Vector3r::Zero()) +Point around which bodies will rotate if rotateAroundZero is True +class yade.wrapper.KinematicEngine(inherits PartialEngine → Engine → Serializable) +Abstract engine for applying prescribed displacement. +Note: +Derived classes should override the apply with given list of ids (not action with Par- +tialEngine.ids), so that they work when combined together; velocity and angular velocity of all +subscribed bodies is reset before the apply method is called, it should therefore only increment +those quantities. +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +ids(=uninitalized) +Ids list of bodies affected by this PartialEngine. +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +2.3. +Yade wrapper class reference +281 + +Yade Documentation, Release 3rd ed. +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.LawTester(inherits PartialEngine → Engine → Serializable) +Prescribe and apply deformations of an interaction in terms of local mutual displacements and +rotations. +The loading path is specified either using path (as sequence of 6-vectors containing +generalized displacements ux, uy, uz, φx, φy, φz) or disPath (ux, uy, uz) and rotPath (φx, φy, +φz). Time function with time values (step numbers) corresponding to points on loading path is +given by pathSteps. Loading values are linearly interpolated between given loading path points, +and starting zero-value (the initial configuration) is assumed for both path and pathSteps. hooks +can specify python code to run when respective point on the path is reached; when the path is +finished, doneHook will be run. +LawTester should be placed between InteractionLoop and NewtonIntegrator in the simulation loop, +since it controls motion via setting linear/angular velocities on particles; those velocities are inte- +grated by NewtonIntegrator to yield an actual position change, which in turn causes IGeom to be +updated (and contact law applied) when InteractionLoop is executed. Constitutive law generating +forces on particles will not affect prescribed particle motion, since both particles have all DoFs +blocked when first used with LawTester. +LawTester uses, as much as possible, IGeom to provide useful data (such as local coordinate system), +but is able to compute those independently if absent in the respective IGeom: +IGeom +#DoFs +LawTester support level +L3Geom +3 +full +L6Geom +6 +full +ScGeom +3 +emulate local coordinate system +ScGeom6D +6 +emulate local coordinate system +Depending on IGeom, 3 (ux, uy, uz) or 6 (ux, uy, uz, φx, φy, φz) degrees of freedom (DoFs) +are controlled with LawTester, by prescribing linear and angular velocities of both particles in +contact. All DoFs controlled with LawTester are orthogonal (fully decoupled) and are controlled +independently. +When 3 DoFs are controlled, rotWeight controls whether local shear is applied by moving particle +on arc around the other one, or by rotating without changing position; although such rotation +induces mutual rotation on the interaction, it is ignored with IGeom with only 3 DoFs. When 6 +DoFs are controlled, only arc-displacement is applied for shear, since otherwise mutual rotation +would occur. +idWeight distributes prescribed motion between both particles (resulting local deformation is the +same if id1 is moved towards id2 or id2 towards id1). This is true only for ux, uy, uz, φx +however ; bending rotations φy, φz are nevertheless always distributed regardless of idWeight to +both spheres in inverse proportion to their radii, so that there is no shear induced. +LawTester knows current contact deformation from 2 sources: from its own internal data (which +are used for prescribing the displacement at every step), which can be accessed in uTest, and from +IGeom itself (depending on which data it provides), which is stored in uGeom. These two values +should be identical (disregarding numerical percision), and it is a way to test whether IGeom and +related functors compute what they are supposed to compute. +282 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +LawTester-operated interactions can be rendered with GlExtra_LawTester renderer. +See scripts/test/law-test.py for an example. +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +disPath(=uninitalized) +Loading path, where each Vector3 contains desired normal displacement and two components +of the shear displacement (in local coordinate system, which is being tracked automatically. +If shorter than rotPath, the last value is repeated. +displIsRel(=true) +Whether displacement values in disPath are normalized by reference contact length (r1+r2 +for 2 spheres). +doneHook(=uninitalized) +Python command (as string) to run when end of the path is achieved. If empty, the engine +will be set dead. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +hooks(=uninitalized) +Python commands to be run when the corresponding point in path is reached, before doing +other things in that particular step. See also doneHook. +idWeight(=1) +Float, usually ￿〈0,1〉, determining on how are displacements distributed between particles +(0 for id1, 1 for id2); intermediate values will apply respective part to each of them. This +parameter is ignored with 6-DoFs IGeom. +ids(=uninitalized) +Ids list of bodies affected by this PartialEngine. +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +pathSteps(=vector(1, 1), (constant step)) +Step number for corresponding values in path; if shorter than path, distance between last 2 +values is used for the rest. +refLength(=0) +Reference contact length, for rendering only. +renderLength(=0) +Characteristic length for the purposes of rendering, set equal to the smaller radius. +2.3. +Yade wrapper class reference +283 + +Yade Documentation, Release 3rd ed. +rotPath(=uninitalized) +Rotational components of the loading path, where each item contains torsion and two bending +rotations in local coordinates. If shorter than path, the last value is repeated. +rotWeight(=1) +Float ￿〈0,1〉 determining whether shear displacement is applied as rotation or displacement on +arc (0 is displacement-only, 1 is rotation-only). Not effective when mutual rotation is specified. +step(=1) +Step number in which this engine is active; determines position in path, using pathSteps. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +trsf(=uninitalized) +Transformation matrix for the local coordinate system. (auto-updated) +uGeom(=Vector6r::Zero()) +Current generalized displacements (3 displacements, 3 rotations), as stored in the interation +itself. They should corredpond to uTest, otherwise a bug is indicated. +uTest(=Vector6r::Zero()) +Current generalized displacements (3 displacements, 3 rotations), as they should be according +to this LawTester. Should correspond to uGeom. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +uuPrev(=Vector6r::Zero()) +Generalized displacement values reached in the previous step, for knowing which increment +to apply in the current step. +class yade.wrapper.LinearDragEngine(inherits PartialEngine → Engine → Serializable) +Apply viscous resistance or linear drag on some particles at each step, decelerating them propor- +tionally to their linear velocities. The applied force reads +Fd = −bv +where b is the linear drag, v is particle’s velocity. +b = 6πνr +where ν is the medium viscosity, r is the Stokes radius of the particle (but in this case we accept +it equal to sphere radius for simplification), +Note: +linear drag is only applied to spherical particles, listed in ids. +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +ids(=uninitalized) +Ids list of bodies affected by this PartialEngine. +284 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +nu(=0.001) +Viscosity of the medium. +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.RadialForceEngine(inherits PartialEngine → Engine → Serializable) +Apply force of given magnitude directed away from spatial axis. +axisDir(=Vector3r::UnitX()) +Axis direction (normalized automatically) +axisPt(=Vector3r::Zero()) +Point on axis +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +fNorm(=0) +Applied force magnitude +ids(=uninitalized) +Ids list of bodies affected by this PartialEngine. +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +2.3. +Yade wrapper class reference +285 + +Yade Documentation, Release 3rd ed. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.RotationEngine(inherits KinematicEngine → PartialEngine → Engine → +Serializable) +Engine applying rotation (by setting angular velocity) to subscribed bodies. If rotateAroundZero +is set, then each body is also displaced around zeroPoint. +angularVelocity(=0) +Angular velocity. [rad/s] +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +ids(=uninitalized) +Ids list of bodies affected by this PartialEngine. +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +rotateAroundZero(=false) +If True, bodies will not rotate around their centroids, but rather around zeroPoint. +rotationAxis(=Vector3r::UnitX()) +Axis of rotation (direction); will be normalized automatically. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +zeroPoint(=Vector3r::Zero()) +Point around which bodies will rotate if rotateAroundZero is True +class yade.wrapper.ServoPIDController(inherits TranslationEngine → KinematicEngine → +PartialEngine → Engine → Serializable) +PIDController servo-engine for applying prescribed force on bodies. http://en.wikipedia.org/wiki/ +PID_controller +axis(=Vector3r::Zero()) +Unit vector along which apply the velocity [-] +curVel(=0.0) +Current applied velocity [m/s] +286 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +current(=Vector3r::Zero()) +Current value for the controller [N] +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +errorCur(=0.0) +Current error [N] +errorPrev(=0.0) +Previous error [N] +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +iTerm(=0.0) +Integral term [N] +ids(=uninitalized) +Ids list of bodies affected by this PartialEngine. +iterPeriod(=100.0) +Periodicity criterion of velocity correlation [-] +iterPrevStart(=-1.0) +Previous iteration of velocity correlation [-] +kD(=0.0) +Derivative gain/coefficient for the PID-controller [-] +kI(=0.0) +Integral gain/coefficient for the PID-controller [-] +kP(=0.0) +Proportional gain/coefficient for the PID-controller [-] +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +maxVelocity(=0.0) +Velocity [m/s] +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +target(=0.0) +Target value for the controller [N] +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +translationAxis(=uninitalized) +Direction of imposed translation [Vector3] +2.3. +Yade wrapper class reference +287 + +Yade Documentation, Release 3rd ed. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +velocity(=uninitalized) +Scalar value of the imposed velocity [m/s]. Imposed vector velocity is velocity * axis +class yade.wrapper.StepDisplacer(inherits PartialEngine → Engine → Serializable) +Apply generalized displacement (displacement or rotation) stepwise on subscribed bodies. Could +be used for purposes of contact law tests (by moving one sphere compared to another), but in this +case, see rather LawTester +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +ids(=uninitalized) +Ids list of bodies affected by this PartialEngine. +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +mov(=Vector3r::Zero()) +Linear displacement step to be applied per iteration, by addition to State.pos. +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +rot(=Quaternionr::Identity()) +Rotation step to be applied per iteration (via rotation composition with State.ori). +setVelocities(=false) +If false, positions and orientations are directly updated, without changing the speeds of con- +cerned bodies. If true, only velocity and angularVelocity are modified. In this second case +integrator is supposed to be used, so that, thanks to this Engine, the bodies will have the +prescribed jump over one iteration (dt). +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.TorqueEngine(inherits PartialEngine → Engine → Serializable) +Apply given torque (momentum) value at every subscribed particle, at every step. +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +288 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +ids(=uninitalized) +Ids list of bodies affected by this PartialEngine. +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +moment(=Vector3r::Zero()) +Torque value to be applied. +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.TranslationEngine(inherits KinematicEngine → PartialEngine → Engine +→ Serializable) +Engine applying translation motion (by setting linear velocity) to subscribed bodies. +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +ids(=uninitalized) +Ids list of bodies affected by this PartialEngine. +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +2.3. +Yade wrapper class reference +289 + +Yade Documentation, Release 3rd ed. +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +translationAxis(=uninitalized) +Direction of imposed translation [Vector3] +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +velocity(=uninitalized) +Scalar value of the imposed velocity [m/s]. Imposed vector velocity is velocity * axis +2.3.5 Dispatchers +Dispatcher +LawDispatcher +GlStateDispatcher +IPhysDispatcher +BoundDispatcher +GlIPhysDispatcher +IGeomDispatcher +GlShapeDispatcher +GlIGeomDispatcher +GlBoundDispatcher +Fig. 31: Inheritance graph of Dispatcher, gray dashed classes are discussed in their own sections: LawDis- +patcher, IPhysDispatcher, BoundDispatcher, IGeomDispatcher. See also: GlBoundDispatcher, GlIGe- +omDispatcher, GlIPhysDispatcher, GlShapeDispatcher, GlStateDispatcher. +class yade.wrapper.Dispatcher(inherits Engine → Serializable) +Engine dispatching control to its associated functors, based on types of argument it receives. This +abstract base class provides no functionality in itself. +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +290 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.GlBoundDispatcher(inherits Dispatcher → Engine → Serializable) +Dispatcher calling functors based on received argument type(s). +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispFunctor((GlBoundDispatcher)arg1, (Bound)arg2) → GlBoundFunctor : +Return functor that would be dispatched for given argument(s); None if no dispatch; ambigu- +ous dispatch throws. +dispMatrix((GlBoundDispatcher)arg1[, (bool)names=True]) → dict : +Return dictionary with contents of the dispatch matrix. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +functors +Functors associated with this dispatcher. +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.GlIGeomDispatcher(inherits Dispatcher → Engine → Serializable) +Dispatcher calling functors based on received argument type(s). +2.3. +Yade wrapper class reference +291 + +Yade Documentation, Release 3rd ed. +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispFunctor((GlIGeomDispatcher)arg1, (IGeom)arg2) → GlIGeomFunctor : +Return functor that would be dispatched for given argument(s); None if no dispatch; ambigu- +ous dispatch throws. +dispMatrix((GlIGeomDispatcher)arg1[, (bool)names=True]) → dict : +Return dictionary with contents of the dispatch matrix. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +functors +Functors associated with this dispatcher. +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.GlIPhysDispatcher(inherits Dispatcher → Engine → Serializable) +Dispatcher calling functors based on received argument type(s). +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispFunctor((GlIPhysDispatcher)arg1, (IPhys)arg2) → GlIPhysFunctor : +Return functor that would be dispatched for given argument(s); None if no dispatch; ambigu- +ous dispatch throws. +dispMatrix((GlIPhysDispatcher)arg1[, (bool)names=True]) → dict : +Return dictionary with contents of the dispatch matrix. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +292 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +functors +Functors associated with this dispatcher. +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.GlShapeDispatcher(inherits Dispatcher → Engine → Serializable) +Dispatcher calling functors based on received argument type(s). +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispFunctor((GlShapeDispatcher)arg1, (Shape)arg2) → GlShapeFunctor : +Return functor that would be dispatched for given argument(s); None if no dispatch; ambigu- +ous dispatch throws. +dispMatrix((GlShapeDispatcher)arg1[, (bool)names=True]) → dict : +Return dictionary with contents of the dispatch matrix. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +functors +Functors associated with this dispatcher. +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +2.3. +Yade wrapper class reference +293 + +Yade Documentation, Release 3rd ed. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.GlStateDispatcher(inherits Dispatcher → Engine → Serializable) +Dispatcher calling functors based on received argument type(s). +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispFunctor((GlStateDispatcher)arg1, (State)arg2) → GlStateFunctor : +Return functor that would be dispatched for given argument(s); None if no dispatch; ambigu- +ous dispatch throws. +dispMatrix((GlStateDispatcher)arg1[, (bool)names=True]) → dict : +Return dictionary with contents of the dispatch matrix. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +functors +Functors associated with this dispatcher. +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +2.3.6 Functors +class yade.wrapper.Functor(inherits Serializable) +Function-like object that is called by Dispatcher, if types of arguments match those the Functor +declares to accept. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +294 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +Functor +GlIGeomFunctor +IGeomFunctor +GlShapeFunctor +GlIPhysFunctor +BoundFunctor +IPhysFunctor +GlStateFunctor +LawFunctor +GlBoundFunctor +Fig. 32: Inheritance graph of Functor, gray dashed classes are discussed in their own sections: GlIGeom- +Functor, IGeomFunctor, GlShapeFunctor, GlIPhysFunctor, BoundFunctor, IPhysFunctor, GlStateFunc- +tor, LawFunctor, GlBoundFunctor. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +2.3.7 Bounding volume creation +BoundFunctor +BoundFunctor +Bo1_Wall_Aabb +Bo1_Sphere_Aabb +Bo1_PFacet_Aabb +Bo1_GridConnection_Aabb +Bo1_ChainedCylinder_Aabb +Bo1_Box_Aabb +Bo1_Tetra_Aabb +Bo1_Facet_Aabb +Bo1_Cylinder_Aabb +Fig. 33: +Inheritance graph of BoundFunctor. +See also: +Bo1_Box_Aabb, Bo1_ChainedCylinder_- +Aabb, Bo1_Cylinder_Aabb, Bo1_Facet_Aabb, Bo1_GridConnection_Aabb, Bo1_PFacet_Aabb, Bo1_- +Sphere_Aabb, Bo1_Tetra_Aabb, Bo1_Wall_Aabb. +class yade.wrapper.BoundFunctor(inherits Functor → Serializable) +Functor for creating/updating Body::bound. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +2.3. +Yade wrapper class reference +295 + +Yade Documentation, Release 3rd ed. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Bo1_Box_Aabb(inherits BoundFunctor → Functor → Serializable) +Create/update an Aabb of a Box. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Bo1_ChainedCylinder_Aabb(inherits BoundFunctor → Functor → Serial- +izable) +Functor creating Aabb from ChainedCylinder. +aabbEnlargeFactor +Relative enlargement of the bounding box; deactivated if negative. +Note: +This attribute is used to create distant interaction, but is only meaningful with +an IGeomFunctor which will not simply discard such interactions: Ig2_Cylinder_Cylinder_- +ScGeom::interactionDetectionFactor should have the same value as aabbEnlargeFactor. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Bo1_Cylinder_Aabb(inherits BoundFunctor → Functor → Serializable) +Functor creating Aabb from Cylinder. +aabbEnlargeFactor +Relative enlargement of the bounding box; deactivated if negative. +Note: +This attribute is used to create distant interaction, but is only meaningful with +296 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +an IGeomFunctor which will not simply discard such interactions: Ig2_Cylinder_Cylinder_- +ScGeom::interactionDetectionFactor should have the same value as aabbEnlargeFactor. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Bo1_Facet_Aabb(inherits BoundFunctor → Functor → Serializable) +Creates/updates an Aabb of a Facet. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Bo1_GridConnection_Aabb(inherits BoundFunctor → Functor → Serializ- +able) +Functor creating Aabb from a GridConnection. +aabbEnlargeFactor(=-1, deactivated) +Relative enlargement of the bounding box; deactivated if negative. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Bo1_PFacet_Aabb(inherits BoundFunctor → Functor → Serializable) +Functor creating Aabb from a PFacet. +2.3. +Yade wrapper class reference +297 + +Yade Documentation, Release 3rd ed. +aabbEnlargeFactor(=-1, deactivated) +Relative enlargement of the bounding box; deactivated if negative. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Bo1_Sphere_Aabb(inherits BoundFunctor → Functor → Serializable) +Functor creating Aabb from Sphere. +aabbEnlargeFactor +Relative enlargement of the bounding box; deactivated if negative. +Note: +This attribute is used to create distant interaction, but is only meaningful with +an IGeomFunctor which will not simply discard such interactions: Ig2_Sphere_Sphere_- +ScGeom::interactionDetectionFactor should have the same value as aabbEnlargeFactor. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Bo1_Tetra_Aabb(inherits BoundFunctor → Functor → Serializable) +Create/update Aabb of a Tetra +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +298 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +class yade.wrapper.Bo1_Wall_Aabb(inherits BoundFunctor → Functor → Serializable) +Creates/updates an Aabb of a Wall +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +BoundDispatcher +class yade.wrapper.BoundDispatcher(inherits Dispatcher → Engine → Serializable) +Dispatcher calling functors based on received argument type(s). +activated(=true) +Whether the engine is activated (only should be changed by the collider) +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispFunctor((BoundDispatcher)arg1, (Shape)arg2) → BoundFunctor : +Return functor that would be dispatched for given argument(s); None if no dispatch; ambigu- +ous dispatch throws. +dispMatrix((BoundDispatcher)arg1[, (bool)names=True]) → dict : +Return dictionary with contents of the dispatch matrix. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +functors +Functors associated with this dispatcher. +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +minSweepDistFactor(=0.2) +Minimal distance by which enlarge all bounding boxes; superseeds computed value of sweep- +Dist when lower that (minSweepDistFactor x sweepDist). Updated by the collider. (auto- +updated). +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +2.3. +Yade wrapper class reference +299 + +Yade Documentation, Release 3rd ed. +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +sweepDist(=0) +Distance by which enlarge all bounding boxes, to prevent collider from being run at every +step (only should be changed by the collider). +targetInterv(=-1) +see InsertionSortCollider::targetInterv (auto-updated) +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +updatingDispFactor(=-1) +see InsertionSortCollider::updatingDispFactor (auto-updated) +2.3.8 Interaction Geometry creation +IGeomFunctor +IGeomFunctor +Ig2_Facet_Sphere_L3Geom +Ig2_Sphere_Sphere_L3Geom +Ig2_Facet_Sphere_ScGeom +Ig2_Sphere_ChainedCylinder_CylScGeom6D +Ig2_Sphere_ChainedCylinder_CylScGeom +Ig2_Box_Sphere_ScGeom +Ig2_GridConnection_GridConnection_GridCoGridCoGeom +Ig2_Sphere_Sphere_L6Geom +Ig2_Sphere_Sphere_ScGeom +Ig2_GridConnection_PFacet_ScGeom +Ig2_Sphere_GridConnection_ScGridCoGeom +Ig2_Box_Sphere_ScGeom6D +Ig2_Tetra_Tetra_TTetraGeom +Ig2_Sphere_Sphere_ScGeom6D +Ig2_Facet_Sphere_ScGeom6D +Ig2_ChainedCylinder_ChainedCylinder_ScGeom6D +Ig2_Wall_Sphere_L3Geom +Ig2_Sphere_PFacet_ScGridCoGeom +Ig2_GridNode_GridNode_GridNodeGeom6D +Ig2_PFacet_PFacet_ScGeom +Ig2_Wall_Sphere_ScGeom +Ig2_Wall_PFacet_ScGeom +Fig. 34: +Inheritance graph of IGeomFunctor. +See also: +Ig2_Box_Sphere_ScGeom, Ig2_Box_- +Sphere_ScGeom6D, Ig2_ChainedCylinder_ChainedCylinder_ScGeom6D, Ig2_Facet_Sphere_L3Geom, +Ig2_Facet_Sphere_ScGeom, Ig2_Facet_Sphere_ScGeom6D, Ig2_GridConnection_GridConnection_- +GridCoGridCoGeom, +Ig2_GridConnection_PFacet_ScGeom, +Ig2_GridNode_GridNode_GridNode- +Geom6D, +Ig2_PFacet_PFacet_ScGeom, +Ig2_Sphere_ChainedCylinder_CylScGeom, +Ig2_Sphere_- +ChainedCylinder_CylScGeom6D, Ig2_Sphere_GridConnection_ScGridCoGeom, Ig2_Sphere_PFacet_- +ScGridCoGeom, Ig2_Sphere_Sphere_L3Geom, Ig2_Sphere_Sphere_L6Geom, Ig2_Sphere_Sphere_Sc- +Geom, Ig2_Sphere_Sphere_ScGeom6D, Ig2_Tetra_Tetra_TTetraGeom, Ig2_Wall_PFacet_ScGeom, +Ig2_Wall_Sphere_L3Geom, Ig2_Wall_Sphere_ScGeom. +class yade.wrapper.IGeomFunctor(inherits Functor → Serializable) +Functor for creating/updating Interaction::geom objects. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +300 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Ig2_Box_Sphere_ScGeom(inherits IGeomFunctor → Functor → Serializ- +able) +Create an interaction geometry ScGeom from Box and Sphere, representing the box with a projected +virtual sphere of same radius. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +interactionDetectionFactor +Enlarge sphere radii by this factor (if >1), to permit creation of distant interactions. +InteractionGeometry will be computed when interactionDetectionFactor*(rad) > distance. +Note: +This parameter is functionally coupled with Bo1_Sphere_Aabb::aabbEnlargeFactor, +which will create larger bounding boxes and should be of the same value. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Ig2_Box_Sphere_ScGeom6D(inherits Ig2_Box_Sphere_ScGeom → IGeom- +Functor → Functor → Serializable) +Create an interaction geometry ScGeom6D from Box and Sphere, representing the box with a +projected virtual sphere of same radius. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +interactionDetectionFactor +Enlarge sphere radii by this factor (if >1), to permit creation of distant interactions. +InteractionGeometry will be computed when interactionDetectionFactor*(rad) > distance. +Note: +This parameter is functionally coupled with Bo1_Sphere_Aabb::aabbEnlargeFactor, +which will create larger bounding boxes and should be of the same value. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +2.3. +Yade wrapper class reference +301 + +Yade Documentation, Release 3rd ed. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Ig2_ChainedCylinder_ChainedCylinder_ScGeom6D(inherits IGeomFunc- +tor → Functor → Se- +rializable) +Create/update a ScGeom instance representing connexion between chained cylinders. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +halfLengthContacts(=true) +If True, Cylinders nodes interact like spheres of radius 0.5*length, else one node has size length +while the other has size 0. The difference is mainly the locus of rotation definition. +interactionDetectionFactor(=1) +Enlarge both radii by this factor (if >1), to permit creation of distant interactions. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Ig2_Facet_Sphere_L3Geom(inherits Ig2_Sphere_Sphere_L3Geom → IGe- +omFunctor → Functor → Serializable) +Incrementally compute L3Geom for contact between Facet and Sphere. Uses attributes of Ig2_- +Sphere_Sphere_L3Geom. +approxMask +Selectively enable geometrical approximations (bitmask); add the values for approximations +to be enabled. +1 +use previous transformation to transform velocities (which are known at mid-steps), +instead of mid-step transformation computed as quaternion slerp at t=0.5. +2 +do not take average (mid-step) normal when computing relative shear displacement, +use previous value instead +4 +do not re-normalize average (mid-step) normal, if used.… +By default, the mask is zero, wherefore none of these approximations is used. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +distFactor(=1) +Create interaction if spheres are not futher than distFactor *(r1+r2). If negative, zero normal +deformation will be set to be the initial value (otherwise, the geometrical distance is the ‘’zero’’ +one). +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +302 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +noRatch(=true) +See Ig2_Sphere_Sphere_ScGeom.avoidGranularRatcheting. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +trsfRenorm(=100) +How often to renormalize trsf; if non-positive, never renormalized (simulation might be un- +stable) +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Ig2_Facet_Sphere_ScGeom(inherits IGeomFunctor → Functor → Serializ- +able) +Create/update a ScGeom instance representing intersection of Facet and Sphere. The equivalent +radius for the Facet (ScGeom.refR1) is chosen as twice the Sphere’s one. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +hertzian(=false) +The equivalent radius for the Facet (ScGeom.refR1) is chosen as 1e8 times the Sphere’s radius +(closer to Hertzian therory, where it is infinite). +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +shrinkFactor(=0, no shrinking) +The radius of the inscribed circle of the facet is decreased by the value of the sphere’s ra- +dius multiplied by shrinkFactor. From the definition of contact point on the surface made +of facets, the given surface is not continuous and becomes in effect surface covered with tri- +angular tiles, with gap between the separate tiles equal to the sphere’s radius multiplied by +2×*shrinkFactor*. If zero, no shrinking is done. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Ig2_Facet_Sphere_ScGeom6D(inherits +Ig2_Facet_Sphere_ScGeom +→ +IGeomFunctor → Functor → Serializable) +Create an interaction geometry ScGeom6D from Facet and Sphere, representing the Facet with a +projected virtual sphere of same radius. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +hertzian(=false) +The equivalent radius for the Facet (ScGeom.refR1) is chosen as 1e8 times the Sphere’s radius +(closer to Hertzian therory, where it is infinite). +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +2.3. +Yade wrapper class reference +303 + +Yade Documentation, Release 3rd ed. +shrinkFactor(=0, no shrinking) +The radius of the inscribed circle of the facet is decreased by the value of the sphere’s ra- +dius multiplied by shrinkFactor. From the definition of contact point on the surface made +of facets, the given surface is not continuous and becomes in effect surface covered with tri- +angular tiles, with gap between the separate tiles equal to the sphere’s radius multiplied by +2×*shrinkFactor*. If zero, no shrinking is done. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Ig2_GridConnection_GridConnection_GridCoGridCoGeom(inherits IGe- +omFunctor +→ Functor → +Serializable) +Create/update a GridCoGridCoGeom instance representing the geometry of a contact point be- +tween two GridConnection , including relative rotations. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Ig2_GridConnection_PFacet_ScGeom(inherits +Ig2_Sphere_GridConnec- +tion_ScGridCoGeom +→ +IGeom- +Functor → Functor → Serializable) +Create/update a ScGeom instance representing intersection of Facet and GridConnection. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +interactionDetectionFactor(=1) +Enlarge both radii by this factor (if >1), to permit creation of distant interactions. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +shrinkFactor(=0, no shrinking) +The radius of the inscribed circle of the facet is decreased by the value of the sphere’s ra- +dius multipled by shrinkFactor. From the definition of contact point on the surface made +of facets, the given surface is not continuous and becomes in effect surface covered with tri- +angular tiles, with gap between the separate tiles equal to the sphere’s radius multiplied by +2×*shrinkFactor*. If zero, no shrinking is done. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +304 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Ig2_GridNode_GridNode_GridNodeGeom6D(inherits Ig2_Sphere_Sphere_- +ScGeom → IGeomFunctor → +Functor → Serializable) +Create/update a GridNodeGeom6D instance representing the geometry of a contact point between +two GridNode, including relative rotations. +avoidGranularRatcheting +Define relative velocity so that ratcheting is avoided. It applies for sphere-sphere contacts. It +eventualy also apply for sphere-emulating interactions (i.e. convertible into the ScGeom type), +if the virtual sphere’s motion is defined correctly (see e.g. Ig2_Sphere_ChainedCylinder_- +CylScGeom). +Short explanation of what we want to avoid : +Numerical ratcheting is best understood considering a small elastic cycle at a contact between +two grains : assuming b1 is fixed, impose this displacement to b2 : +1. translation dx in the normal direction +2. rotation a +3. translation -dx (back to the initial position) +4. rotation -a (back to the initial orientation) +If the branch vector used to define the relative shear in rotation×branch is not constant +(typically if it is defined from the vector center→contactPoint), then the shear displacement +at the end of this cycle is not zero: rotations a and -a are multiplied by branches of different +lengths. +It results in a finite contact force at the end of the cycle even though the positions and +orientations are unchanged, in total contradiction with the elastic nature of the problem. It +could also be seen as an inconsistent energy creation or loss. Given that DEM simulations tend +to generate oscillations around equilibrium (damped mass-spring), it can have a significant +impact on the evolution of the packings, resulting for instance in slow creep in iterations under +constant load. +The solution adopted here to avoid ratcheting is as proposed by McNamara and co-workers. +They analyzed the ratcheting problem in detail - even though they comment on the basis +of a cycle that differs from the one shown above. One will find interesting discussions in +e.g. [McNamara2008], even though solution it suggests is not fully applied here (equations of +motion are not incorporating alpha, in contradiction with what is suggested by McNamara et +al.). +bases +Ordered list of types (as strings) this functor accepts. +creep(=false) +Substract rotational creep from relative rotation. The rotational creep ScGeom6D::twistCreep +is a quaternion and has to be updated inside a constitutive law, see for instance Law2_- +ScGeom6D_CohFrictPhys_CohesionMoment. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +interactionDetectionFactor +Enlarge both radii by this factor (if >1), to permit creation of distant interactions. +InteractionGeometry will be computed when interactionDetectionFactor*(rad1+rad2) > dis- +tance. +2.3. +Yade wrapper class reference +305 + +Yade Documentation, Release 3rd ed. +Note: +This parameter is functionally coupled with Bo1_Sphere_Aabb::aabbEnlargeFactor, +which will create larger bounding boxes and should be of the same value. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +updateRotations(=true) +Precompute relative rotations. Turning this false can speed up simulations when rotations +are not needed in constitutive laws (e.g. when spheres are compressed without cohesion and +moment in early stage of a triaxial test), but is not foolproof. Change this value only if you +know what you are doing. +class yade.wrapper.Ig2_PFacet_PFacet_ScGeom(inherits +Ig2_Sphere_PFacet_ScGridCo- +Geom → Ig2_Sphere_GridConnection_Sc- +GridCoGeom → IGeomFunctor → Functor +→ Serializable) +Create/update a ScGridCoGeom instance representing intersection of Facet and Sphere. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +interactionDetectionFactor(=1) +Enlarge both radii by this factor (if >1), to permit creation of distant interactions. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +shrinkFactor(=0, no shrinking) +The radius of the inscribed circle of the facet is decreased by the value of the sphere’s ra- +dius multipled by shrinkFactor. From the definition of contact point on the surface made +of facets, the given surface is not continuous and becomes in effect surface covered with tri- +angular tiles, with gap between the separate tiles equal to the sphere’s radius multiplied by +2×*shrinkFactor*. If zero, no shrinking is done. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Ig2_Sphere_ChainedCylinder_CylScGeom(inherits +IGeomFunctor +→ +Functor → Serializable) +Create/update a ScGeom instance representing intersection of two Spheres. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +306 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +interactionDetectionFactor(=1) +Enlarge both radii by this factor (if >1), to permit creation of distant interactions. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Ig2_Sphere_ChainedCylinder_CylScGeom6D(inherits +Ig2_Sphere_- +ChainedCylinder_CylSc- +Geom → IGeomFunctor → +Functor → Serializable) +Create/update a ScGeom6D instance representing the geometry of a contact point between two +Spheres, including relative rotations. +bases +Ordered list of types (as strings) this functor accepts. +creep(=false) +Substract rotational creep from relative rotation. The rotational creep ScGeom6D::twistCreep +is a quaternion and has to be updated inside a constitutive law, see for instance Law2_- +ScGeom6D_CohFrictPhys_CohesionMoment. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +interactionDetectionFactor(=1) +Enlarge both radii by this factor (if >1), to permit creation of distant interactions. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +updateRotations(=false) +Precompute relative rotations. Turning this false can speed up simulations when rotations +are not needed in constitutive laws (e.g. when spheres are compressed without cohesion and +moment in early stage of a triaxial test), but is not foolproof. Change this value only if you +know what you are doing. +class yade.wrapper.Ig2_Sphere_GridConnection_ScGridCoGeom(inherits +IGeomFunctor +→ +Functor → Serializable) +Create/update a ScGridCoGeom6D instance representing the geometry of a contact point between +a GricConnection and a Sphere including relative rotations. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +interactionDetectionFactor(=1) +Enlarge both radii by this factor (if >1), to permit creation of distant interactions. +2.3. +Yade wrapper class reference +307 + +Yade Documentation, Release 3rd ed. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Ig2_Sphere_PFacet_ScGridCoGeom(inherits +Ig2_Sphere_GridConnec- +tion_ScGridCoGeom → IGeomFunc- +tor → Functor → Serializable) +Create/update a ScGridCoGeom instance representing intersection of PFacet and Sphere. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +interactionDetectionFactor(=1) +Enlarge both radii by this factor (if >1), to permit creation of distant interactions. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +shrinkFactor(=0, no shrinking) +The radius of the inscribed circle of the facet is decreased by the value of the sphere’s ra- +dius multipled by shrinkFactor. From the definition of contact point on the surface made +of facets, the given surface is not continuous and becomes in effect surface covered with tri- +angular tiles, with gap between the separate tiles equal to the sphere’s radius multiplied by +2×*shrinkFactor*. If zero, no shrinking is done. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Ig2_Sphere_Sphere_L3Geom(inherits IGeomFunctor → Functor → Serial- +izable) +Functor for computing incrementally configuration of 2 Spheres stored in L3Geom; the configuration +is positioned in global space by local origin c (contact point) and rotation matrix T (orthonormal +transformation matrix), and its degrees of freedom are local displacement u (in one normal and +two shear directions); with Ig2_Sphere_Sphere_L6Geom and L6Geom, there is additionally φ. +The first row of T, i.e. local x-axis, is the contact normal noted n for brevity. Additionally, quasi- +constant values of u0 (and φ0) are stored as shifted origins of u (and φ); therefore, current value +of displacement is always u◦ − u0. +Suppose two spheres with radii ri, positions xi, velocities vi, angular velocities ωi. +When there is not yet contact, it will be created if uN = |x◦ +2 − x◦ +1| − |fd|(r1 + r2) < 0, where fd is +distFactor (sometimes also called ‘‘interaction radius’’). If fd > 0, then u0x will be initalized to +uN, otherwise to 0. In another words, contact will be created if spheres enlarged by |fd| touch, and +the ‘‘equilibrium distance’’ (where ux − u − 0x is zero) will be set to the current distance if fd is +positive, and to the geometrically-touching distance if negative. +Local axes (rows of T) are initially defined as follows: +• local x-axis is n = xl = � +x2 − x1; +• local y-axis positioned arbitrarily, but in a deterministic manner: aligned with the xz plane +(if ny < nz) or xy plane (otherwise); +308 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +• local z-axis zl = xl × yl. +If there has already been contact between the two spheres, it is updated to keep track of rigid +motion of the contact (one that does not change mutual configuration of spheres) and mutual +configuration changes. Rigid motion transforms local coordinate system and can be decomposed +in rigid translation (affecting c), and rigid rotation (affecting T), which can be split in rotation or +perpendicular to the normal and rotation ot (‘‘twist’’) parallel with the normal: +o⊖ +r = n− × n◦. +Since velocities are known at previous midstep (t − ∆t/2), we consider mid-step normal +n⊖ = n− + n◦ +2 +. +For the sake of numerical stability, n⊖ is re-normalized after being computed, unless prohibited by +approxMask. If approxMask has the appropriate bit set, the mid-normal is not compute, and we +simply use n⊖ ≈ n−. +Rigid rotation parallel with the normal is +o⊖ +t = n⊖ +� +n⊖ · ω⊖ +1 + ω⊖ +2 +2 +� +∆t. +Branch vectors b1, b2 (connecting x◦ +1, x◦ +2 with c◦ are computed depending on noRatch (see here). +b1 = +� +r1n◦ +with noRatch +c◦ − x◦ +1 +otherwise +b2 = +� +−r2n◦ +with noRatch +c◦ − x◦ +2 +otherwise +Relative velocity at c◦ can be computed as +v⊖ +r = (˜v⊖ +2 + ω2 × b2) − (v1 + ω1 × b1) +where ˜v2 is v2 without mean-field velocity gradient in periodic boundary conditions (see +Cell.homoDeform). In the numerial implementation, the normal part of incident velocity is re- +moved (since it is computed directly) with v⊖ +r2 = v⊖ +r − (n⊖ · v⊖ +r )n⊖. +Any vector a expressed in global coordinates transforms during one timestep as +a◦ = a− + v⊖ +r ∆t − a− × o⊖ +r − a− × t⊖ +r +where the increments have the meaning of relative shear, rigid rotation normal to n and rigid +rotation parallel with n. Local coordinate system orientation, rotation matrix T, is updated by +rows, i.e. +T ◦ = +� +� +n◦ +x +n◦ +y +n◦ +z +T − +1,• − T − +1,• × o⊖ +r − T − +1,• × o⊖ +t +T − +2,• − T − +2,• × o⊖ +r − T − +,• × o⊖ +t +� +� +This matrix is re-normalized (unless prevented by approxMask) and mid-step transformation is +computed using quaternion spherical interpolation as +T ⊖ = Slerp +� +T −; T ◦; t = 1/2 +� +. +Depending on approxMask, this computation can be avoided by approximating T ⊖ = T −. +Finally, current displacement is evaluated as +u◦ = u− + T ⊖v⊖ +r ∆t. +2.3. +Yade wrapper class reference +309 + +Yade Documentation, Release 3rd ed. +For the normal component, non-incremental evaluation is preferred, giving +u◦ +x = |x◦ +2 − x◦ +1| − (r1 + r2) +If this functor is called for L6Geom, local rotation is updated as +φ◦ = φ− + T ⊖∆t(ω2 − ω1) +approxMask +Selectively enable geometrical approximations (bitmask); add the values for approximations +to be enabled. +1 +use previous transformation to transform velocities (which are known at mid-steps), +instead of mid-step transformation computed as quaternion slerp at t=0.5. +2 +do not take average (mid-step) normal when computing relative shear displacement, +use previous value instead +4 +do not re-normalize average (mid-step) normal, if used.… +By default, the mask is zero, wherefore none of these approximations is used. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +distFactor(=1) +Create interaction if spheres are not futher than distFactor *(r1+r2). If negative, zero normal +deformation will be set to be the initial value (otherwise, the geometrical distance is the ‘’zero’’ +one). +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +noRatch(=true) +See Ig2_Sphere_Sphere_ScGeom.avoidGranularRatcheting. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +trsfRenorm(=100) +How often to renormalize trsf; if non-positive, never renormalized (simulation might be un- +stable) +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Ig2_Sphere_Sphere_L6Geom(inherits +Ig2_Sphere_Sphere_L3Geom +→ +IGeomFunctor → Functor → Serializable) +Incrementally compute L6Geom for contact of 2 spheres. +approxMask +Selectively enable geometrical approximations (bitmask); add the values for approximations +to be enabled. +1 +use previous transformation to transform velocities (which are known at mid-steps), +instead of mid-step transformation computed as quaternion slerp at t=0.5. +2 +do not take average (mid-step) normal when computing relative shear displacement, +use previous value instead +4 +do not re-normalize average (mid-step) normal, if used.… +310 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +By default, the mask is zero, wherefore none of these approximations is used. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +distFactor(=1) +Create interaction if spheres are not futher than distFactor *(r1+r2). If negative, zero normal +deformation will be set to be the initial value (otherwise, the geometrical distance is the ‘’zero’’ +one). +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +noRatch(=true) +See Ig2_Sphere_Sphere_ScGeom.avoidGranularRatcheting. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +trsfRenorm(=100) +How often to renormalize trsf; if non-positive, never renormalized (simulation might be un- +stable) +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Ig2_Sphere_Sphere_ScGeom(inherits IGeomFunctor → Functor → Serial- +izable) +Create/update a ScGeom instance representing the geometry of a contact point between two +Spheres s. +avoidGranularRatcheting +Define relative velocity so that ratcheting is avoided. It applies for sphere-sphere contacts. It +eventualy also apply for sphere-emulating interactions (i.e. convertible into the ScGeom type), +if the virtual sphere’s motion is defined correctly (see e.g. Ig2_Sphere_ChainedCylinder_- +CylScGeom). +Short explanation of what we want to avoid : +Numerical ratcheting is best understood considering a small elastic cycle at a contact between +two grains : assuming b1 is fixed, impose this displacement to b2 : +1. translation dx in the normal direction +2. rotation a +3. translation -dx (back to the initial position) +4. rotation -a (back to the initial orientation) +If the branch vector used to define the relative shear in rotation×branch is not constant +(typically if it is defined from the vector center→contactPoint), then the shear displacement +at the end of this cycle is not zero: rotations a and -a are multiplied by branches of different +lengths. +It results in a finite contact force at the end of the cycle even though the positions and +orientations are unchanged, in total contradiction with the elastic nature of the problem. It +could also be seen as an inconsistent energy creation or loss. Given that DEM simulations tend +to generate oscillations around equilibrium (damped mass-spring), it can have a significant +impact on the evolution of the packings, resulting for instance in slow creep in iterations under +constant load. +2.3. +Yade wrapper class reference +311 + +Yade Documentation, Release 3rd ed. +The solution adopted here to avoid ratcheting is as proposed by McNamara and co-workers. +They analyzed the ratcheting problem in detail - even though they comment on the basis +of a cycle that differs from the one shown above. One will find interesting discussions in +e.g. [McNamara2008], even though solution it suggests is not fully applied here (equations of +motion are not incorporating alpha, in contradiction with what is suggested by McNamara et +al.). +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +interactionDetectionFactor +Enlarge both radii by this factor (if >1), to permit creation of distant interactions. +InteractionGeometry will be computed when interactionDetectionFactor*(rad1+rad2) > dis- +tance. +Note: +This parameter is functionally coupled with Bo1_Sphere_Aabb::aabbEnlargeFactor, +which will create larger bounding boxes and should be of the same value. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Ig2_Sphere_Sphere_ScGeom6D(inherits Ig2_Sphere_Sphere_ScGeom → +IGeomFunctor → Functor → Serializable) +Create/update a ScGeom6D instance representing the geometry of a contact point between two +Spheres, including relative rotations. +avoidGranularRatcheting +Define relative velocity so that ratcheting is avoided. It applies for sphere-sphere contacts. It +eventualy also apply for sphere-emulating interactions (i.e. convertible into the ScGeom type), +if the virtual sphere’s motion is defined correctly (see e.g. Ig2_Sphere_ChainedCylinder_- +CylScGeom). +Short explanation of what we want to avoid : +Numerical ratcheting is best understood considering a small elastic cycle at a contact between +two grains : assuming b1 is fixed, impose this displacement to b2 : +1. translation dx in the normal direction +2. rotation a +3. translation -dx (back to the initial position) +4. rotation -a (back to the initial orientation) +If the branch vector used to define the relative shear in rotation×branch is not constant +(typically if it is defined from the vector center→contactPoint), then the shear displacement +at the end of this cycle is not zero: rotations a and -a are multiplied by branches of different +lengths. +It results in a finite contact force at the end of the cycle even though the positions and +orientations are unchanged, in total contradiction with the elastic nature of the problem. It +312 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +could also be seen as an inconsistent energy creation or loss. Given that DEM simulations tend +to generate oscillations around equilibrium (damped mass-spring), it can have a significant +impact on the evolution of the packings, resulting for instance in slow creep in iterations under +constant load. +The solution adopted here to avoid ratcheting is as proposed by McNamara and co-workers. +They analyzed the ratcheting problem in detail - even though they comment on the basis +of a cycle that differs from the one shown above. One will find interesting discussions in +e.g. [McNamara2008], even though solution it suggests is not fully applied here (equations of +motion are not incorporating alpha, in contradiction with what is suggested by McNamara et +al.). +bases +Ordered list of types (as strings) this functor accepts. +creep(=false) +Substract rotational creep from relative rotation. The rotational creep ScGeom6D::twistCreep +is a quaternion and has to be updated inside a constitutive law, see for instance Law2_- +ScGeom6D_CohFrictPhys_CohesionMoment. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +interactionDetectionFactor +Enlarge both radii by this factor (if >1), to permit creation of distant interactions. +InteractionGeometry will be computed when interactionDetectionFactor*(rad1+rad2) > dis- +tance. +Note: +This parameter is functionally coupled with Bo1_Sphere_Aabb::aabbEnlargeFactor, +which will create larger bounding boxes and should be of the same value. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +updateRotations(=true) +Precompute relative rotations. Turning this false can speed up simulations when rotations +are not needed in constitutive laws (e.g. when spheres are compressed without cohesion and +moment in early stage of a triaxial test), but is not foolproof. Change this value only if you +know what you are doing. +class yade.wrapper.Ig2_Tetra_Tetra_TTetraGeom(inherits IGeomFunctor → Functor → Se- +rializable) +Create/update geometry of collision between 2 tetrahedra (TTetraGeom instance) +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +2.3. +Yade wrapper class reference +313 + +Yade Documentation, Release 3rd ed. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Ig2_Wall_PFacet_ScGeom(inherits Ig2_Wall_Sphere_ScGeom → IGeom- +Functor → Functor → Serializable) +Create/update a ScGeom instance representing intersection of Wall and PFacet. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +hertzian(=false) +The equivalent radius for the Wall (ScGeom.refR1) is chosen as 1e8 times the Sphere’s radius +(closer to Hertzian therory, where it is infinite). +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +noRatch(=true) +Avoid granular ratcheting +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Ig2_Wall_Sphere_L3Geom(inherits Ig2_Sphere_Sphere_L3Geom → IGe- +omFunctor → Functor → Serializable) +Incrementally compute L3Geom for contact between Wall and Sphere. Uses attributes of Ig2_- +Sphere_Sphere_L3Geom. +approxMask +Selectively enable geometrical approximations (bitmask); add the values for approximations +to be enabled. +1 +use previous transformation to transform velocities (which are known at mid-steps), +instead of mid-step transformation computed as quaternion slerp at t=0.5. +2 +do not take average (mid-step) normal when computing relative shear displacement, +use previous value instead +4 +do not re-normalize average (mid-step) normal, if used.… +By default, the mask is zero, wherefore none of these approximations is used. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +distFactor(=1) +Create interaction if spheres are not futher than distFactor *(r1+r2). If negative, zero normal +deformation will be set to be the initial value (otherwise, the geometrical distance is the ‘’zero’’ +one). +314 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +noRatch(=true) +See Ig2_Sphere_Sphere_ScGeom.avoidGranularRatcheting. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +trsfRenorm(=100) +How often to renormalize trsf; if non-positive, never renormalized (simulation might be un- +stable) +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Ig2_Wall_Sphere_ScGeom(inherits IGeomFunctor → Functor → Serializ- +able) +Create/update a ScGeom instance representing intersection of Wall and Sphere. The equivalent +radius for the Wall (ScGeom.refR1) is chosen equal to the Sphere’s radius. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +hertzian(=false) +The equivalent radius for the Wall (ScGeom.refR1) is chosen as 1e8 times the Sphere’s radius +(closer to Hertzian therory, where it is infinite). +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +noRatch(=true) +Avoid granular ratcheting +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +IGeomDispatcher +class yade.wrapper.IGeomDispatcher(inherits Dispatcher → Engine → Serializable) +Dispatcher calling functors based on received argument type(s). +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispFunctor((IGeomDispatcher)arg1, (Shape)arg2, (Shape)arg3) → IGeomFunctor : +Return functor that would be dispatched for given argument(s); None if no dispatch; ambigu- +ous dispatch throws. +dispMatrix((IGeomDispatcher)arg1[, (bool)names=True]) → dict : +Return dictionary with contents of the dispatch matrix. +2.3. +Yade wrapper class reference +315 + +Yade Documentation, Release 3rd ed. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +functors +Functors associated with this dispatcher. +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +2.3.9 Interaction Physics creation +IPhysFunctor +class yade.wrapper.IPhysFunctor(inherits Functor → Serializable) +Functor for creating/updating Interaction::phys objects from bodies’ material properties. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Ip2_2xInelastCohFrictMat_InelastCohFrictPhys(inherits +IPhysFunc- +tor → Functor → Se- +rializable) +Generates cohesive-frictional interactions with moments. +Used in the contact law Law2_Sc- +Geom6D_InelastCohFrictPhys_CohesionMoment. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +316 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +IPhysFunctor +Ip2_FrictMat_CpmMat_FrictPhys +Ip2_2xInelastCohFrictMat_InelastCohFrictPhys +Ip2_FrictMat_FrictMatCDM_MindlinPhysCDM +Ip2_ElastMat_ElastMat_NormPhys +Ip2_CohFrictMat_CohFrictMat_CohFrictPhys +Ip2_FrictMat_FrictViscoMat_FrictViscoPhys +Ip2_ViscElMat_ViscElMat_ViscElPhys +Ip2_FrictMat_FrictMat_CapillaryPhys +Ip2_CpmMat_CpmMat_CpmPhys +Ip2_FrictMat_FrictMat_LubricationPhys +Ip2_BubbleMat_BubbleMat_BubblePhys +Ip2_JCFpmMat_JCFpmMat_JCFpmPhys +Ip2_LudingMat_LudingMat_LudingPhys +Ip2_FrictMat_FrictMat_MindlinCapillaryPhys +Ip2_FrictMat_FrictMat_ViscoFrictPhys +Ip2_FrictMat_FrictMat_FrictPhys +Ip2_ViscElCapMat_ViscElCapMat_ViscElCapPhys +Ip2_FrictMat_FrictMat_MindlinPhys +Ip2_ElastMat_ElastMat_NormShearPhys +Ip2_WireMat_WireMat_WirePhys +Ip2_FrictMatCDM_FrictMatCDM_MindlinPhysCDM +Ip2_FrictViscoMat_FrictViscoMat_FrictViscoPhys +Ip2_MortarMat_MortarMat_MortarPhys +Fig. +35: +Inheritance +graph +of +IPhysFunctor. +See +also: +Ip2_2xInelastCohFrictMat_Inelast- +CohFrictPhys, +Ip2_BubbleMat_BubbleMat_BubblePhys, +Ip2_CohFrictMat_CohFrictMat_CohFrict- +Phys, Ip2_CpmMat_CpmMat_CpmPhys, Ip2_ElastMat_ElastMat_NormPhys, Ip2_ElastMat_Elast- +Mat_NormShearPhys, +Ip2_FrictMatCDM_FrictMatCDM_MindlinPhysCDM, +Ip2_FrictMat_Cpm- +Mat_FrictPhys, Ip2_FrictMat_FrictMatCDM_MindlinPhysCDM, Ip2_FrictMat_FrictMat_Capillary- +Phys, Ip2_FrictMat_FrictMat_FrictPhys, Ip2_FrictMat_FrictMat_LubricationPhys, Ip2_FrictMat_- +FrictMat_MindlinCapillaryPhys, +Ip2_FrictMat_FrictMat_MindlinPhys, +Ip2_FrictMat_FrictMat_- +ViscoFrictPhys, +Ip2_FrictMat_FrictViscoMat_FrictViscoPhys, +Ip2_FrictViscoMat_FrictViscoMat_- +FrictViscoPhys, Ip2_JCFpmMat_JCFpmMat_JCFpmPhys, Ip2_LudingMat_LudingMat_LudingPhys, +Ip2_MortarMat_MortarMat_MortarPhys, Ip2_ViscElCapMat_ViscElCapMat_ViscElCapPhys, Ip2_- +ViscElMat_ViscElMat_ViscElPhys, Ip2_WireMat_WireMat_WirePhys. +2.3. +Yade wrapper class reference +317 + +Yade Documentation, Release 3rd ed. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Ip2_BubbleMat_BubbleMat_BubblePhys(inherits IPhysFunctor → Functor +→ Serializable) +Generates bubble interactions.Used in the contact law Law2_ScGeom_BubblePhys_Bubble. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Ip2_CohFrictMat_CohFrictMat_CohFrictPhys(inherits IPhysFunctor → +Functor → Serializable) +Generates cohesive-frictional interactions with moments, used in the contact law Law2_Sc- +Geom6D_CohFrictPhys_CohesionMoment. The normal/shear stiffness and friction definitions are +the same as in Ip2_FrictMat_FrictMat_FrictPhys, check the documentation there for details. +Adhesions related to the normal and the shear components are calculated from CohFrict- +Mat::normalCohesion (Cn) and CohFrictMat::shearCohesion (Cs). For particles of size R1,R2 the +adhesion will be ai = Cimin(R1, R2)2, i = n, s. +Twist and rolling stiffnesses are proportional to the shear stiffness through dimensionless factors +alphaKtw and alphaKr, such that the rotational stiffnesses are defined by ksαiR1R2, i = tw r +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +frictAngle(=uninitalized) +Instance of MatchMaker determining how to compute interaction’s friction angle. If None, +minimum value is used. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +normalCohesion(=uninitalized) +Instance of MatchMaker determining tensile strength +setCohesionNow(=false) +If true, assign cohesion to all existing contacts in current time-step. The flag is turned false +automatically, so that assignment is done in the current timestep only. +setCohesionOnNewContacts(=false) +If true, assign cohesion at all new contacts. If false, only existing contacts can be cohesive (also +318 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +see Ip2_CohFrictMat_CohFrictMat_CohFrictPhys::setCohesionNow), and new contacts are +only frictional. +shearCohesion(=uninitalized) +Instance of MatchMaker determining cohesive part of the shear strength (a frictional term +might be added depending on CohFrictPhys::cohesionDisablesFriction) +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Ip2_CpmMat_CpmMat_CpmPhys(inherits IPhysFunctor → Functor → Serial- +izable) +Convert 2 CpmMat instances to CpmPhys with corresponding parameters. Uses simple (arithmetic) +averages if material are different. Simple copy of parameters is performed if the material is shared +between both particles. See cpm-model for detals. +E(=uninitalized) +Instance of MatchMaker determining how to compute interaction’s normal modulus. If None, +average value is used. +bases +Ordered list of types (as strings) this functor accepts. +cohesiveThresholdIter(=10) +Should new contacts be cohesive? They will before this iter#, they will not be afterwards. If +0, they will never be. If negative, they will always be created as cohesive (10 by default). +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Ip2_ElastMat_ElastMat_NormPhys(inherits IPhysFunctor → Functor → +Serializable) +Create a NormPhys from two ElastMats. TODO. EXPERIMENTAL +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +2.3. +Yade wrapper class reference +319 + +Yade Documentation, Release 3rd ed. +class yade.wrapper.Ip2_ElastMat_ElastMat_NormShearPhys(inherits IPhysFunctor → Func- +tor → Serializable) +Create a NormShearPhys from two ElastMats. TODO. EXPERIMENTAL +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Ip2_FrictMatCDM_FrictMatCDM_MindlinPhysCDM(inherits +IPhysFunctor +→ Functor → Serializ- +able) +Create a MindlinPhysCDM from two FrictMatCDMsExts. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +frictAngle(=uninitalized) +Instance of MatchMaker determining how to compute interaction’s friction angle. If None, +minimum value is used. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Ip2_FrictMat_CpmMat_FrictPhys(inherits IPhysFunctor → Functor → +Serializable) +Convert CpmMat instance and FrictMat instance to FrictPhys with corresponding parameters +(young, poisson, frictionAngle). Uses simple (arithmetic) averages if material parameters are dif- +ferent. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +frictAngle(=uninitalized) +See Ip2_FrictMat_FrictMat_FrictPhys. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +320 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Ip2_FrictMat_FrictMatCDM_MindlinPhysCDM(inherits +IPhysFunctor +→ +Functor → Serializable) +Create a MindlinPhysCDM from one FrictMat and one FrictMatCDM instance. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +frictAngle(=uninitalized) +Instance of MatchMaker determining how to compute interaction’s friction angle. If None, +minimum value is used. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Ip2_FrictMat_FrictMat_CapillaryPhys(inherits IPhysFunctor → Func- +tor → Serializable) +RelationShips to use with Law2_ScGeom_CapillaryPhys_Capillarity. +In these RelationShips all the interaction attributes are computed. +Warning: +as in the others Ip2 functors, most of the attributes are computed only once, when +the interaction is new. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Ip2_FrictMat_FrictMat_FrictPhys(inherits IPhysFunctor → Functor → +Serializable) +Create a FrictPhys from two FrictMats. The compliance of one sphere under point load is defined +here as 1/(E.D), with E the stiffness of the sphere and D its diameter. The compliance of the +contact itself is taken as the sum of compliances from each sphere, i.e. 1/(E1.D1) + 1/(E2.D2) +in the general case, or 2/(E.D) in the special case of equal sizes and equal stiffness. Note that +summing compliances is equivalent to summing the harmonic average of stiffnesses. This reasoning +2.3. +Yade wrapper class reference +321 + +Yade Documentation, Release 3rd ed. +is applied in both the normal and the tangential directions (as in e.g. [Scholtes2009a]), hence the +general form of the contact stiffness: +k = E1D1∗E2D2 +E1D1+E2D2 = k1∗k2 +k1+k2 , with ki = EiDi. +In the above equation Ei is taken equal to FrictMat::young of sphere i for the normal stiffness, +and FrictMat::young × ElastMat::poisson for the shear stiffness. In the case of a contact between +a ViscElMat and a FrictMat, be sure to set FrictMat::young and FrictMat::poisson, otherwise the +default value will be used. +The contact friction is defined according to Ip2_FrictMat_FrictMat_FrictPhys::frictAngle (mini- +mum of the two materials by default). +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +frictAngle(=uninitalized) +Instance of MatchMaker determining how to compute interaction’s friction angle. If None, +minimum value is used. +kn(=uninitalized) +Instance of MatchMaker determining how to compute interaction’s normal stiffness. If None, +harmonic average is used. +ks(=uninitalized) +Instance of MatchMaker determining how to compute interaction’s shear stiffness. If None, +harmonic average is used. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Ip2_FrictMat_FrictMat_LubricationPhys(inherits +IPhysFunctor +→ +Functor → Serializable) +Ip2 creating LubricationPhys from two Material instances. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +eps(=0.001) +Roughness: fraction of radius enlargement for contact asperities +eta(=1) +Fluid viscosity [Pa.s] +keps(=1) +Dimensionless stiffness coefficient of the asperities, relative to the stiffness of the surface (the +final stiffness will be keps*kn). Only used with resolution method=0, with resolution>0 it is +always equal to 1. [-] +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +322 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Ip2_FrictMat_FrictMat_MindlinCapillaryPhys(inherits +IPhysFunctor +→ Functor → Serializ- +able) +RelationShips to use with Law2_ScGeom_CapillaryPhys_Capillarity +In these RelationShips all the interaction attributes are computed. +Warning: +as in the others Ip2 functors, most of the attributes are computed only once, when +the interaction is new. +bases +Ordered list of types (as strings) this functor accepts. +betan(=uninitalized) +Normal viscous damping ratio βn. +betas(=uninitalized) +Shear viscous damping ratio βs. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +en(=uninitalized) +Normal coefficient of restitution en. +es(=uninitalized) +Shear coefficient of restitution es. +eta(=0.0) +Coefficient to determine the plastic bending moment +gamma(=0.0) +Surface energy parameter [J/m^2] per each unit contact surface, to derive DMT formulation +from HM +krot(=0.0) +Rotational stiffness for moment contact law +ktwist(=0.0) +Torsional stiffness for moment contact law +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Ip2_FrictMat_FrictMat_MindlinPhys(inherits IPhysFunctor → Functor +→ Serializable) +Calculate some physical parameters needed to obtain the normal and shear stiffnesses according to +the Hertz-Mindlin formulation (as implemented in PFC). The viscous damping coefficients cn, cs +can be specified either using viscous damping ratios (βn, βs) or coefficients of restitution (en, es). +2.3. +Yade wrapper class reference +323 + +Yade Documentation, Release 3rd ed. +# If the viscous damping ratio βn (βs) is given, it is assigned directly to MindlinPhys.betan +(MindlinPhys.betas) and the viscous damping coefficient is calculated as cn = 2 · βn · √mbar · kn +(cs = 2 · βs · √mbar · ks), where kn (ks) the tangential normal (shear) stiffness. Replacing kn = +3/2 · kno · uN0.5 (ks = kso · uN0.5) and kno = 4/3 · E · +√ +R (kso = 2 · +√ +4 · R · G/(2 − ν)), we get +cn = 2 · βn · √mbar · +� +2 · E · +√ +R · uN0.25 (cs = 2 · βs · √mbar · +� +4 · +√ +R · G/(2 − ν) · uN0.25), +where mbar, R, E, G the effective mass and mean radius, elastic and shear moduli of the interacting +particles. +# If the coefficient of restitution en is given instead, the normal viscous damping ratio is computed +using βn = −(log en)/ +� +π2 + (log en)2. The shear coefficient of restitution is considered as es = +en and the viscous damping coefficient is calculated as cn = cs = α · √mbar · uN0.25, where +α = 2 · +� +5/6 · βn · +� +2 · E · +√ +R, i.e. cn = cs = 2 · +� +5/6 · βn · √mbar · +� +2 · E · +√ +R · uN0.25. +In both cases, the viscous forces are calculated as Fn,viscous = cn · vn (Fs,viscous = cs · vs), where +vn (vs) the normal (shear) component of the relative velocity. The following rules apply: +# If βn and βs are used, then MindlinPhys.alpha =0; if en is defined instead, then Mindlin- +Phys.betan = MindlinPhys.betan =0.0. +# It is an error (exception) to specify both en and βn (es and βs). +# If neither en nor βn is given, zero value for MindlinPhys.betan is used; there will be no viscous +effects. +# If neither es nor βs is given, the value of MindlinPhys.betan is used for MindlinPhys.betas as +well. +# To consider different viscous coefficients in the normal and shear contact directions, use βn, βs, +instead of en. +The en, βn, es, βs are MatchMaker objects; they can be constructed from float values to always +return constant values. See scripts/examples/spheresFactory.py for an example of specifying en +based on combination of parameters, for different materials in contact. +bases +Ordered list of types (as strings) this functor accepts. +betan(=uninitalized) +Normal viscous damping ratio βn. +betas(=uninitalized) +Shear viscous damping ratio βs. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +en(=uninitalized) +Normal coefficient of restitution en. +es(=uninitalized) +Shear coefficient of restitution es. +eta(=0.0) +Coefficient to determine the plastic bending moment +frictAngle(=uninitalized) +Instance of MatchMaker determining how to compute the friction angle of an interaction. If +None, minimum value is used. +gamma(=0.0) +Surface energy parameter [J/m^2] per each unit contact surface, to derive DMT formulation +from HM +krot(=0.0) +Rotational stiffness for moment contact law +324 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +ktwist(=0.0) +Torsional stiffness for moment contact law +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Ip2_FrictMat_FrictMat_ViscoFrictPhys(inherits +Ip2_FrictMat_Frict- +Mat_FrictPhys → IPhysFunc- +tor → Functor → Serializable) +Create a FrictPhys from two FrictMats. The compliance of one sphere under symetric point loads +is defined here as 1/(E.r), with E the stiffness of the sphere and r its radius, and corresponds to +a compliance 1/(2.E.r)=1/(E.D) from each contact point. The compliance of the contact itself +will be the sum of compliances from each sphere, i.e. 1/(E.D1)+1/(E.D2) in the general case, +or 1/(E.r) in the special case of equal sizes. +Note that summing compliances corresponds to +an harmonic average of stiffnesss, which is how kn is actually computed in the Ip2_FrictMat_- +FrictMat_FrictPhys functor. +The shear stiffness ks of one sphere is defined via the material parameter ElastMat::poisson, as +ks=poisson*kn, and the resulting shear stiffness of the interaction will be also an harmonic average. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +frictAngle(=uninitalized) +Instance of MatchMaker determining how to compute interaction’s friction angle. If None, +minimum value is used. +kn(=uninitalized) +Instance of MatchMaker determining how to compute interaction’s normal stiffness. If None, +harmonic average is used. +ks(=uninitalized) +Instance of MatchMaker determining how to compute interaction’s shear stiffness. If None, +harmonic average is used. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Ip2_FrictMat_FrictViscoMat_FrictViscoPhys(inherits IPhysFunctor → +Functor → Serializable) +Converts a FrictMat and FrictViscoMat instance to FrictViscoPhys with corresponding parameters. +Basically this functor corresponds to Ip2_FrictMat_FrictMat_FrictPhys with the only difference +that damping in normal direction can be considered. +bases +Ordered list of types (as strings) this functor accepts. +2.3. +Yade wrapper class reference +325 + +Yade Documentation, Release 3rd ed. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +frictAngle(=uninitalized) +Instance of MatchMaker determining how to compute interaction’s friction angle. If None, +minimum value is used. +kRatio(=uninitalized) +Instance of MatchMaker determining how to compute interaction’s shear contact stiffnesses. +If this value is not given the elastic properties (i.e. poisson) of the two colliding materials are +used to calculate the stiffness. +kn(=uninitalized) +Instance of MatchMaker determining how to compute interaction’s normal contact stiffnesses. +If this value is not given the elastic properties (i.e. young) of the two colliding materials are +used to calculate the stiffness. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Ip2_FrictViscoMat_FrictViscoMat_FrictViscoPhys(inherits +IPhys- +Functor → Func- +tor +→ +Serializ- +able) +Converts 2 FrictViscoMat instances to FrictViscoPhys with corresponding parameters. Basically +this functor corresponds to Ip2_FrictMat_FrictMat_FrictPhys with the only difference that damp- +ing in normal direction can be considered. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +frictAngle(=uninitalized) +Instance of MatchMaker determining how to compute interaction’s friction angle. If None, +minimum value is used. +kRatio(=uninitalized) +Instance of MatchMaker determining how to compute interaction’s shear contact stiffnesses. +If this value is not given the elastic properties (i.e. poisson) of the two colliding materials are +used to calculate the stiffness. +kn(=uninitalized) +Instance of MatchMaker determining how to compute interaction’s normal contact stiffnesses. +If this value is not given the elastic properties (i.e. young) of the two colliding materials are +used to calculate the stiffness. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +326 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +class yade.wrapper.Ip2_JCFpmMat_JCFpmMat_JCFpmPhys(inherits IPhysFunctor → Functor → +Serializable) +Converts 2 JCFpmMat instances to one JCFpmPhys instance, with corresponding parameters. See +JCFpmMat and [Duriez2016] for details +bases +Ordered list of types (as strings) this functor accepts. +cohesiveTresholdIteration(=1) +should new contacts be cohesive? If strictly negativ, they will in any case. If positiv, they +will before this iter, they won’t afterward. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +weibullCutOffMax(=10) +Factor that cuts off the largest values of the weibull distributed interaction areas. +weibullCutOffMin(=0.) +Factor that cuts off the smallest values of the weibull distributed interaction areas. +xSectionWeibullScaleParameter(=1) +Scale parameter used to generate interaction radii for the crosssectional areas (changing +strength criteria only) according to Weibull distribution. Activated for any value other than +0. Needs to be combined with a shape parameter +xSectionWeibullShapeParameter(=0) +Shape parameter used to generate interaction radii for the crossSectional areas (changing +strength criteria only) according to Weibull distribution. Activated for any value other than +0. Needs to be combined with a scale parameter) +class yade.wrapper.Ip2_LudingMat_LudingMat_LudingPhys(inherits IPhysFunctor → Functor +→ Serializable) +Convert 2 instances of LudingMat to LudingPhys using the rule of consecutive connection. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Ip2_MortarMat_MortarMat_MortarPhys(inherits IPhysFunctor → Functor +→ Serializable) +Ip2 creating MortarPhys from two MortarMat instances. +2.3. +Yade wrapper class reference +327 + +Yade Documentation, Release 3rd ed. +bases +Ordered list of types (as strings) this functor accepts. +cohesiveThresholdIter(=2) +Should new contacts be cohesive? They will before this iter#, they will not be afterwards. If +<=0, they will never be. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Ip2_ViscElCapMat_ViscElCapMat_ViscElCapPhys(inherits +Ip2_ViscEl- +Mat_ViscElMat_Vis- +cElPhys +→ +IPhys- +Functor +→ +Functor +→ Serializable) +Convert 2 instances of ViscElCapMat to ViscElCapPhys using the rule of consecutive connection. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +en(=uninitalized) +Instance of MatchMaker determining restitution coefficient in normal direction +et(=uninitalized) +Instance of MatchMaker determining restitution coefficient in tangential direction +frictAngle(=uninitalized) +Instance of MatchMaker determining how to compute interaction’s friction angle. If None, +minimum value is used. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +tc(=uninitalized) +Instance of MatchMaker determining contact time +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Ip2_ViscElMat_ViscElMat_ViscElPhys(inherits IPhysFunctor → Functor +→ Serializable) +Convert 2 instances of ViscElMat to ViscElPhys using the rule of consecutive connection. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +328 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +en(=uninitalized) +Instance of MatchMaker determining restitution coefficient in normal direction +et(=uninitalized) +Instance of MatchMaker determining restitution coefficient in tangential direction +frictAngle(=uninitalized) +Instance of MatchMaker determining how to compute interaction’s friction angle. If None, +minimum value is used. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +tc(=uninitalized) +Instance of MatchMaker determining contact time +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Ip2_WireMat_WireMat_WirePhys(inherits IPhysFunctor → Functor → Se- +rializable) +Converts 2 WireMat instances to WirePhys with corresponding parameters. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +linkThresholdIteration(=1) +Iteration to create the link. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +IPhysDispatcher +class yade.wrapper.IPhysDispatcher(inherits Dispatcher → Engine → Serializable) +Dispatcher calling functors based on received argument type(s). +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispFunctor((IPhysDispatcher)arg1, (Material)arg2, (Material)arg3) → IPhysFunctor : +Return functor that would be dispatched for given argument(s); None if no dispatch; ambigu- +ous dispatch throws. +dispMatrix((IPhysDispatcher)arg1[, (bool)names=True]) → dict : +Return dictionary with contents of the dispatch matrix. +2.3. +Yade wrapper class reference +329 + +Yade Documentation, Release 3rd ed. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +functors +Functors associated with this dispatcher. +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +2.3.10 Constitutive laws +LawFunctor +class yade.wrapper.LawFunctor(inherits Functor → Serializable) +Functor for applying constitutive laws on interactions. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Law2_ChCylGeom6D_CohFrictPhys_CohesionMoment(inherits LawFunctor +→ Functor → Serial- +izable) +Law for linear compression, and Mohr-Coulomb plasticity surface without cohesion. This law imple- +ments the classical linear elastic-plastic law from [CundallStrack1979] (see also [Pfc3dManual30]). +The normal force is (with the convention of positive tensile forces) Fn = min(knun, 0). +The +shear force is Fs = ksus, the plasticity condition defines the maximum value of the shear force : +Fmax +s += Fn tan(φ), with φ the friction angle. +330 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +LawFunctor +Law2_ScGridCoGeom_FrictPhys_CundallStrack +Law2_ScGeom_ViscoFrictPhys_CundallStrack +Law2_ScGeom_FrictPhys_CundallStrack +Law2_ScGeom_VirtualLubricationPhys +Law2_CylScGeom_FrictPhys_CundallStrack +Law2_ScGeom_BubblePhys_Bubble +Law2_ScGeom_ImplicitLubricationPhys +Law2_ScGeom_LudingPhys_Basic +Law2_ScGeom_ViscElCapPhys_Basic +Law2_ScGeom_MindlinPhys_MindlinDeresiewitz +Law2_ScGeom_MindlinPhys_HertzWithLinearShear +Law2_ScGeom_MindlinPhys_Mindlin +Law2_ScGeom_CpmPhys_Cpm +Law2_ScGeom_PotentialLubricationPhys +Law2_ScGeom6D_InelastCohFrictPhys_CohesionMoment +Law2_ScGeom_JCFpmPhys_JointedCohesiveFrictionalPM +Law2_L3Geom_FrictPhys_ElPerfPl +Law2_ScGeom6D_CohFrictPhys_CohesionMoment +Law2_ScGeom_MortarPhys_Lourenco +Law2_L6Geom_FrictPhys_Linear +Law2_ScGeom_FrictViscoPhys_CundallStrackVisco +Law2_ScGeom_ViscElPhys_Basic +Law2_GridCoGridCoGeom_FrictPhys_CundallStrack +Law2_ScGridCoGeom_CohFrictPhys_CundallStrack +Law2_ChCylGeom6D_CohFrictPhys_CohesionMoment +Law2_CylScGeom6D_CohFrictPhys_CohesionMoment +Law2_ScGeom_WirePhys_WirePM +Law2_ScGeom_MindlinPhysCDM_HertzMindlinCDM +Fig. 36: Inheritance graph of LawFunctor. See also: Law2_ChCylGeom6D_CohFrictPhys_CohesionMo- +ment, Law2_CylScGeom6D_CohFrictPhys_CohesionMoment, Law2_CylScGeom_FrictPhys_Cundall- +Strack, +Law2_GridCoGridCoGeom_FrictPhys_CundallStrack, +Law2_L3Geom_FrictPhys_ElPerfPl, +Law2_L6Geom_FrictPhys_Linear, +Law2_ScGeom6D_CohFrictPhys_CohesionMoment, +Law2_Sc- +Geom6D_InelastCohFrictPhys_CohesionMoment, +Law2_ScGeom_BubblePhys_Bubble, +Law2_Sc- +Geom_CpmPhys_Cpm, Law2_ScGeom_FrictPhys_CundallStrack, Law2_ScGeom_FrictViscoPhys_- +CundallStrackVisco, +Law2_ScGeom_ImplicitLubricationPhys, +Law2_ScGeom_JCFpmPhys_Jointed- +CohesiveFrictionalPM, Law2_ScGeom_LudingPhys_Basic, Law2_ScGeom_MindlinPhysCDM_Hertz- +MindlinCDM, Law2_ScGeom_MindlinPhys_HertzWithLinearShear, Law2_ScGeom_MindlinPhys_- +Mindlin, +Law2_ScGeom_MindlinPhys_MindlinDeresiewitz, +Law2_ScGeom_MortarPhys_Lourenco, +Law2_ScGeom_PotentialLubricationPhys, Law2_ScGeom_VirtualLubricationPhys, Law2_ScGeom_- +ViscElCapPhys_Basic, Law2_ScGeom_ViscElPhys_Basic, Law2_ScGeom_ViscoFrictPhys_Cundall- +Strack, +Law2_ScGeom_WirePhys_WirePM, +Law2_ScGridCoGeom_CohFrictPhys_CundallStrack, +Law2_ScGridCoGeom_FrictPhys_CundallStrack. +2.3. +Yade wrapper class reference +331 + +Yade Documentation, Release 3rd ed. +Note: +This law is well tested in the context of triaxial simulation, and has been used for a +number of published results (see e.g. [Scholtes2009b] and other papers from the same authors). +It is generalised by Law2_ScGeom6D_CohFrictPhys_CohesionMoment, which adds cohesion and +moments at contact. +always_use_moment_law(=false) +If true, use bending/twisting moments at all contacts. If false, compute moments only for +cohesive contacts. +bases +Ordered list of types (as strings) this functor accepts. +creep_viscosity(=1) +creep viscosity [Pa.s/m]. probably should be moved to Ip2_CohFrictMat_CohFrictMat_- +CohFrictPhys… +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +neverErase(=false) +Keep interactions even if particles go away from each other (only in case another constitutive +law is in the scene, e.g. Law2_ScGeom_CapillaryPhys_Capillarity) +shear_creep(=false) +activate creep on the shear force, using CohesiveFrictionalContactLaw::creep_viscosity. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +twist_creep(=false) +activate creep on the twisting moment, using CohesiveFrictionalContactLaw::creep_viscosity. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +useIncrementalForm(=false) +use the incremental formulation to compute bending and twisting moments. Creep on the +twisting moment is not included in such a case. +class yade.wrapper.Law2_CylScGeom6D_CohFrictPhys_CohesionMoment(inherits LawFunctor +→ Functor → Serial- +izable) +This law generalises Law2_CylScGeom_FrictPhys_CundallStrack by adding cohesion and mo- +ments at contact. +always_use_moment_law(=false) +If true, use bending/twisting moments at all contacts. If false, compute moments only for +cohesive contacts. +bases +Ordered list of types (as strings) this functor accepts. +creep_viscosity(=1) +creep viscosity [Pa.s/m]. probably should be moved to Ip2_CohFrictMat_CohFrictMat_- +CohFrictPhys… +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +332 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +neverErase(=false) +Keep interactions even if particles go away from each other (only in case another constitutive +law is in the scene, e.g. Law2_ScGeom_CapillaryPhys_Capillarity) +shear_creep(=false) +activate creep on the shear force, using CohesiveFrictionalContactLaw::creep_viscosity. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +twist_creep(=false) +activate creep on the twisting moment, using CohesiveFrictionalContactLaw::creep_viscosity. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +useIncrementalForm(=false) +use the incremental formulation to compute bending and twisting moments. Creep on the +twisting moment is not included in such a case. +class yade.wrapper.Law2_CylScGeom_FrictPhys_CundallStrack(inherits +LawFunctor +→ +Functor → Serializable) +Law for linear compression, and Mohr-Coulomb plasticity surface without cohesion. This law imple- +ments the classical linear elastic-plastic law from [CundallStrack1979] (see also [Pfc3dManual30]). +The normal force is (with the convention of positive tensile forces) Fn = min(knun, 0). +The +shear force is Fs = ksus, the plasticity condition defines the maximum value of the shear force : +Fmax +s += Fn tan(φ), with φ the friction angle. +Note: +This law uses ScGeom. +Note: +This law is well tested in the context of triaxial simulation, and has been used for a +number of published results (see e.g. [Scholtes2009b] and other papers from the same authors). +It is generalised by Law2_ScGeom6D_CohFrictPhys_CohesionMoment, which adds cohesion and +moments at contact. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +neverErase(=false) +Keep interactions even if particles go away from each other (only in case another constitutive +law is in the scene, e.g. Law2_ScGeom_CapillaryPhys_Capillarity) +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +2.3. +Yade wrapper class reference +333 + +Yade Documentation, Release 3rd ed. +class yade.wrapper.Law2_GridCoGridCoGeom_FrictPhys_CundallStrack(inherits +Law2_Sc- +Geom_FrictPhys_- +CundallStrack +→ +LawFunctor +→ +Functor → Serializ- +able) +Frictional elastic contact law between two gridConnection . See Law2_ScGeom_FrictPhys_Cun- +dallStrack for more details. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +elasticEnergy((Law2_ScGeom_FrictPhys_CundallStrack)arg1) → float : +Compute and return the total elastic energy in all “FrictPhys” contacts +initPlasticDissipation((Law2_ScGeom_FrictPhys_CundallStrack)arg1, (float)arg2) → +None : +Initialize cummulated plastic dissipation to a value (0 by default). +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +neverErase(=false) +Keep interactions even if particles go away from each other (only in case another constitutive +law is in the scene, e.g. Law2_ScGeom_CapillaryPhys_Capillarity) +plasticDissipation((Law2_ScGeom_FrictPhys_CundallStrack)arg1) → float : +Total energy dissipated in plastic slips at all FrictPhys contacts. Computed only if Law2_- +ScGeom_FrictPhys_CundallStrack::traceEnergy is true. +sphericalBodies(=true) +If true, compute branch vectors from radii (faster), else use contactPoint-position. Turning +this flag true is safe for sphere-sphere contacts and a few other specific cases. It will give +wrong values of torques on facets or boxes. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +traceEnergy(=false) +Define the total energy dissipated in plastic slips at all contacts. This will trace only plastic +energy in this law, see O.trackEnergy for a more complete energies tracing +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Law2_L3Geom_FrictPhys_ElPerfPl(inherits LawFunctor → Functor → Se- +rializable) +Basic law for testing L3Geom; it bears no cohesion (unless noBreak is True), and plastic slip obeys +the Mohr-Coulomb criterion (unless noSlip is True). +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +noBreak(=false) +Do not break contacts when particles separate. +334 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +noSlip(=false) +No plastic slipping. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Law2_L6Geom_FrictPhys_Linear(inherits +Law2_L3Geom_FrictPhys_- +ElPerfPl → LawFunctor → Functor → +Serializable) +Basic law for testing L6Geom – linear in both normal and shear sense, without slip or breakage. +bases +Ordered list of types (as strings) this functor accepts. +charLen(=1) +Characteristic length with the meaning of the stiffness ratios bending/shear and tor- +sion/normal. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +noBreak(=false) +Do not break contacts when particles separate. +noSlip(=false) +No plastic slipping. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Law2_ScGeom6D_CohFrictPhys_CohesionMoment(inherits LawFunctor → +Functor → Serializable) +Law for linear traction-compression-bending-twisting, with cohesion+friction and Mohr-Coulomb +plasticity surface. This law adds adhesion and moments to Law2_ScGeom_FrictPhys_Cundall- +Strack. +The normal force is (with the convention of positive tensile forces) Fn = min(kn ∗ (un − up +n), an), +with an the normal adhesion and up +n the plastic part of normal displacement. The shear force is +Fs = ks ∗ us, the plasticity condition defines the maximum value of the shear force, by default +Fmax +s += Fn ∗ tan(φ) + as, with φ the friction angle and as the shear adhesion. +If CohFrict- +Phys::cohesionDisablesFriction is True, friction is ignored as long as adhesion is active, and the +maximum shear force is only Fmax +s += as. +If the maximum tensile or maximum shear force is reached and CohFrictPhys::fragile =True (de- +fault), the cohesive link is broken, and an, as are set back to zero. If a tensile force is present, the +contact is lost, else the shear strength is Fmax +s += Fn ∗ tan(φ). If CohFrictPhys::fragile =False, the +behaviour is perfectly plastic, and the shear strength is kept constant. +If Law2_ScGeom6D_CohFrictPhys_CohesionMoment::momentRotationLaw =True, bending and +twisting moments are computed using a linear law with moduli respectively kt and kr, so that +the moments are : Mb = kb ∗ Θb and Mt = kt ∗ Θt, with Θb,t the relative rotations between +interacting bodies (details can be found in [Bourrier2013]). The maximum value of moments can +be defined and takes the form of rolling friction. Cohesive -type moment may also be included in +the future. +2.3. +Yade wrapper class reference +335 + +Yade Documentation, Release 3rd ed. +Creep at contact is implemented in this law, as defined in [Hassan2010]. If activated, there is a +viscous behaviour of the shear and twisting components, and the evolution of the elastic parts of +shear displacement and relative twist is given by dus,e/dt = −Fs/νs and dΘt,e/dt = −Mt/νt. +always_use_moment_law(=false) +If true, use bending/twisting moments at all contacts. If false, compute moments only for +cohesive contacts. Both cases also require CohFrictPhys::momentRotationLaw to be true. +bases +Ordered list of types (as strings) this functor accepts. +bendingElastEnergy((Law2_ScGeom6D_CohFrictPhys_CohesionMoment)arg1) → float : +Compute bending elastic energy. +creep_viscosity(=1) +creep viscosity [Pa.s/m]. probably should be moved to Ip2_CohFrictMat_CohFrictMat_- +CohFrictPhys. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +elasticEnergy((Law2_ScGeom6D_CohFrictPhys_CohesionMoment)arg1) → float : +Compute total elastic energy. +initPlasticDissipation((Law2_ScGeom6D_CohFrictPhys_CohesionMoment)arg1, +(float)arg2) → None : +Initialize cummulated plastic dissipation to a value (0 by default). +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +neverErase(=false) +Keep interactions even if particles go away from each other (only in case another constitutive +law is in the scene, e.g. Law2_ScGeom_CapillaryPhys_Capillarity) +normElastEnergy((Law2_ScGeom6D_CohFrictPhys_CohesionMoment)arg1) → float : +Compute normal elastic energy. +plasticDissipation((Law2_ScGeom6D_CohFrictPhys_CohesionMoment)arg1) → float : +Total energy dissipated in plastic slips at all CohFrictPhys contacts. Computed only if Law2_- +ScGeom_FrictPhys_CundallStrack::traceEnergy is true. +shearElastEnergy((Law2_ScGeom6D_CohFrictPhys_CohesionMoment)arg1) → float : +Compute shear elastic energy. +shear_creep(=false) +activate creep on the shear force, using CohesiveFrictionalContactLaw::creep_viscosity. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +traceEnergy(=false) +Define the total energy dissipated in plastic slips at contacts. Note that it will not reflect +any energy associated to de-bonding, as it may occur for fragile contacts, nor does it include +plastic dissipation in traction. +twistElastEnergy((Law2_ScGeom6D_CohFrictPhys_CohesionMoment)arg1) → float : +Compute twist elastic energy. +twist_creep(=false) +activate creep on the twisting moment, using CohesiveFrictionalContactLaw::creep_viscosity. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +336 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +useIncrementalForm(=false) +use the incremental formulation to compute bending and twisting moments. Creep on the +twisting moment is not included in such a case. +class yade.wrapper.Law2_ScGeom6D_InelastCohFrictPhys_CohesionMoment(inherits +Law- +Functor +→ +Functor +→ +Serializable) +This law is currently under developpement. Final version and documentation will come before the +end of 2014. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +normElastEnergy((Law2_ScGeom6D_InelastCohFrictPhys_CohesionMoment)arg1) +→ +float : +Compute normal elastic energy. +shearElastEnergy((Law2_ScGeom6D_InelastCohFrictPhys_CohesionMoment)arg1) +→ +float : +Compute shear elastic energy. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Law2_ScGeom_BubblePhys_Bubble(inherits LawFunctor → Functor → Se- +rializable) +Constitutive law for Bubble model. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +pctMaxForce(=0.1) +Chan[2011] states the contact law is valid only for small interferences; therefore an exponential +force-displacement curve models the contact stiffness outside that regime (large penetration). +This artificial stiffening ensures that bubbles will not pass through eachother or completely +overlap during the simulation. The maximum force is Fmax = (2*pi*surfaceTension*rAvg). +pctMaxForce is the percentage of the maximum force dictates the separation threshold, Dmax, +for each contact. Penetrations less than Dmax calculate the reaction force from the derived +contact law, while penetrations equal to or greater than Dmax calculate the reaction force +from the artificial exponential curve. +surfaceTension(=0.07197) +The surface tension in the liquid surrounding the bubbles. The default value is that of water +at 25 degrees Celcius. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +2.3. +Yade wrapper class reference +337 + +Yade Documentation, Release 3rd ed. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Law2_ScGeom_CpmPhys_Cpm(inherits LawFunctor → Functor → Serializ- +able) +Constitutive law for the cpm-model. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +elasticEnergy((Law2_ScGeom_CpmPhys_Cpm)arg1) → float : +Compute and return the total elastic energy in all “CpmPhys” contacts +epsSoft(=1., approximates confinement (for -3e-3) -20MPa precisely, -100MPa a little over, +-200 and -400 are OK (secant)) +Strain at which softening in compression starts (non-negative to deactivate). +The default +value is such that plasticity does not occur +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +omegaThreshold(=1., >=1. to deactivate, i.e. never delete any contacts) +damage after which the contact disappears (<1), since omega reaches 1 only for strain →+∞ +relKnSoft(=.3) +Relative rigidity of the softening branch in compression (0=perfect elastic-plastic, <0 soften- +ing, >0 hardening) +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +yieldEllipseShift(=NaN) +horizontal scaling of the ellipse (shifts on the +x axis as interactions with +y are given) +yieldLogSpeed(=.1) +scaling in the logarithmic yield surface (should be <1 for realistic results; >=0 for meaningful +results) +yieldSigmaTMagnitude((Law2_ScGeom_CpmPhys_Cpm)arg1, (float)sigmaN, (float)omega, +(float)undamagedCohesion, (float)tanFrictionAngle) → float : +Return radius of yield surface for given material and state parameters; uses attributes of the +current instance (yieldSurfType etc), change them before calling if you need that. +yieldSurfType(=2) +yield function: 0: mohr-coulomb (original); 1: parabolic; 2: logarithmic, 3: log+lin_tension, +4: elliptic, 5: elliptic+log +class yade.wrapper.Law2_ScGeom_FrictPhys_CundallStrack(inherits LawFunctor → Functor +→ Serializable) +Law for linear compression, and Mohr-Coulomb plasticity surface without cohesion. This law imple- +ments the classical linear elastic-plastic law from [CundallStrack1979] (see also [Pfc3dManual30]). +The normal force is (with the convention of positive tensile forces) Fn = min(knun, 0). +The +shear force is Fs = ksus, the plasticity condition defines the maximum value of the shear force : +Fmax +s += Fn tan(φ), with φ the friction angle. +This law is well tested in the context of triaxial simulation, and has been used for a number of +published results (see e.g. [Scholtes2009b] and other papers from the same authors). It is gener- +alised by Law2_ScGeom6D_CohFrictPhys_CohesionMoment, which adds cohesion and moments +at contact. +338 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +elasticEnergy((Law2_ScGeom_FrictPhys_CundallStrack)arg1) → float : +Compute and return the total elastic energy in all “FrictPhys” contacts +initPlasticDissipation((Law2_ScGeom_FrictPhys_CundallStrack)arg1, (float)arg2) → +None : +Initialize cummulated plastic dissipation to a value (0 by default). +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +neverErase(=false) +Keep interactions even if particles go away from each other (only in case another constitutive +law is in the scene, e.g. Law2_ScGeom_CapillaryPhys_Capillarity) +plasticDissipation((Law2_ScGeom_FrictPhys_CundallStrack)arg1) → float : +Total energy dissipated in plastic slips at all FrictPhys contacts. Computed only if Law2_- +ScGeom_FrictPhys_CundallStrack::traceEnergy is true. +sphericalBodies(=true) +If true, compute branch vectors from radii (faster), else use contactPoint-position. Turning +this flag true is safe for sphere-sphere contacts and a few other specific cases. It will give +wrong values of torques on facets or boxes. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +traceEnergy(=false) +Define the total energy dissipated in plastic slips at all contacts. This will trace only plastic +energy in this law, see O.trackEnergy for a more complete energies tracing +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Law2_ScGeom_FrictViscoPhys_CundallStrackVisco(inherits +LawFunc- +tor → Functor → +Serializable) +Constitutive law for the FrictViscoPM. Corresponds to Law2_ScGeom_FrictPhys_CundallStrack +with the only difference that viscous damping in normal direction can be considered. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +elasticEnergy((Law2_ScGeom_FrictViscoPhys_CundallStrackVisco)arg1) → float : +Compute and return the total elastic energy in all “FrictViscoPhys” contacts +initPlasticDissipation((Law2_ScGeom_FrictViscoPhys_CundallStrackVisco)arg1, +(float)arg2) → None : +Initialize cummulated plastic dissipation to a value (0 by default). +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +neverErase(=false) +Keep interactions even if particles go away from each other (only in case another constitutive +law is in the scene, e.g. Law2_ScGeom_CapillaryPhys_Capillarity) +2.3. +Yade wrapper class reference +339 + +Yade Documentation, Release 3rd ed. +plasticDissipation((Law2_ScGeom_FrictViscoPhys_CundallStrackVisco)arg1) → float : +Total energy dissipated in plastic slips at all FrictPhys contacts. +Computed only if +:yref:Law2_ScGeom_FrictViscoPhys_CundallStrackVisco::traceEnergy‘ is true. +sphericalBodies(=true) +If true, compute branch vectors from radii (faster), else use contactPoint-position. Turning +this flag true is safe for sphere-sphere contacts and a few other specific cases. It will give +wrong values of torques on facets or boxes. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +traceEnergy(=false) +Define the total energy dissipated in plastic slips at all contacts. This will trace only plastic +energy in this law, see O.trackEnergy for a more complete energies tracing +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Law2_ScGeom_ImplicitLubricationPhys(inherits Law2_ScGeom_Virtu- +alLubricationPhys → LawFunc- +tor → Functor → Serializable) +Material law for lubrication and contact between two spheres, solved using implicit method. The +full description of this contact law is available in [Chevremont2020] . Several resolution methods +are available. Iterative exact, solving the 2nd order polynomia. Other resolutions methods are nu- +merical (Newton-Rafson and Dichotomy) with a variable change δ = log(u), solved in dimentionless +coordinates. +MaxDist(=2.) +Maximum distance (d/a) for the interaction +MaxIter(=30) +Maximum iterations for numerical resolution (Dichotomy and Newton-Rafson) +SolutionTol(=1.e-8) +Tolerance for numerical resolution (Dichotomy and Newton-Rafson) +activateRollLubrication(=true) +Activate roll lubrication (default: true) +activateTangencialLubrication(=true) +Activate tangencial lubrication (default: true) +activateTwistLubrication(=true) +Activate twist lubrication (default: true) +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +static getStressForEachBody() → tuple : +Get stresses tensors for each bodies: normal contact stress, shear contact stress, normal +lubrication stress, shear lubrication stress, stress from additionnal potential forces. +static getTotalStresses() → tuple : +Get total stresses tensors: normal contact stress, shear contact stress, normal lubrication +stress, shear lubrication stress, stress from additionnal potential forces. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +340 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +maxSubSteps(=4) +max recursion depth of adaptative timestepping in the theta-method, the minimal time in- +terval is thus Omega::dt/2depth. If still not converged the integrator will switch to backward +Euler. +resolution(=0) +Change normal component resolution method, 0: Iterative exact resolution with substepping +(theta method, linear contact), 1: Newton-Rafson dimensionless resolution (theta method, +linear contact), 2: (default) Dichotomy dimensionless resolution (theta method, linear con- +tact), 3: Exact dimensionless solution with contact prediction (theta method, linear contact). +Method 3 is better if the volumic fraction is not too high. Use 2 otherwise. +theta(=0.55) +parameter of the ‘theta’-method, 1: backward Euler, 0.5: trapezoidal rule, 0: not used, 0.55: +suggested optimum) +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Law2_ScGeom_JCFpmPhys_JointedCohesiveFrictionalPM(inherits +Law- +Functor +→ +Functor +→ +Serializable) +Interaction law for cohesive frictional material, e.g. rock, possibly presenting joint surfaces, that +can be mechanically described with a smooth contact logic [Ivars2011] (implemented in Yade in +[Scholtes2012]). +See examples/jointedCohesiveFrictionalPM for script examples. +Joint surface +definitions (through stl meshes or direct definition with gts module) are illustrated there. +Key(=””) +string specifying the name of saved file ‘cracks___.txt’, when recordCracks is true. +bases +Ordered list of types (as strings) this functor accepts. +clusterMoments(=true) +computer clustered moments? (on by default +computedCentroid(=false) +computer clustered moments? +cracksFileExist(=false) +if true (and if recordCracks), data are appended to an existing ‘cracksKey’ text file; otherwise +its content is reset. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +eventNumber(=0) +cluster event number (used for clustering and paraview visualization of groups). +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +momentFudgeFactor(=1.) +Fudge factor used by Hazzard and Damjanac 2013 to improve moment size accuracy (set to +1 for no impact by default) +momentRadiusFactor(=5.) +Average particle diameter multiplier for moment magnitude calculation +2.3. +Yade wrapper class reference +341 + +Yade Documentation, Release 3rd ed. +momentsFileExist(=false) +if true (and if recordCracks), data are appended to an existing ‘momentsKey’ text file; other- +wise its content is reset. +nbShearCracks(=0) +number of shear microcracks. +nbTensCracks(=0) +number of tensile microcracks. +neverErase(=false) +Keep interactions even if particles go away from each other (only in case another constitutive +law is in the scene +recordCracks(=false) +if true, data about interactions that lose their cohesive feature are stored in the text file +cracksKey.txt (see Key and cracksFileExist). It contains 9 columns: the break iteration, the +3 coordinates of the contact point, the type (1 means shear break, while 0 corresponds to +tensile break), the ‘’cross section’’ (mean radius of the 2 spheres) and the 3 coordinates of the +contact normal. +recordMoments(=false) +Combines with :yref: Key +to compute acoustic emissions according to clustered broken bond method? (off by default) +smoothJoint(=false) +if true, interactions of particles belonging to joint surface (JCFpmPhys.isOnJoint) are handled +according to a smooth contact logic [Ivars2011], [Scholtes2012]. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +totalCracksSurface(=0.) +calculate the total cracked surface. +totalShearCracksE(=0.) +calculate the overall energy dissipated by interparticle microcracking in shear. +totalTensCracksE(=0.) +calculate the overall energy dissipated by interparticle microcracking in tension. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +useStrainEnergy(=true) +use strain energy for moment magnitude estimation (if false, use kinetic energy) +class yade.wrapper.Law2_ScGeom_LudingPhys_Basic(inherits LawFunctor → Functor → Seri- +alizable) +Linear viscoelastic model operating on ScGeom and LudingPhys. See [Luding2008] ,[Singh2013]_- +for more details. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +342 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Law2_ScGeom_MindlinPhysCDM_HertzMindlinCDM(inherits LawFunctor → +Functor +→ +Serializ- +able) +Hertz-Mindlin model extended: Normal direction: conical damage model from Harkness et al. +2016./ Suhr & Six 2017. Tangential direction: stress dependent interparticle friction coefficient, +Suhr & Six 2016. Both models can be switched on/off separately. In this version there is NO +damping (neither viscous nor linear), NO adhesion and NO calc_energy, NO includeMoment, NO +preventGranularRatcheting. NOT tested for periodic simulations. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +neverErase(=false) +Keep interactions even if particles go away from each other (only in case another constitutive +law is in the scene, e.g. Law2_ScGeom_CapillaryPhys_Capillarity) +ratioSlidingContacts((Law2_ScGeom_MindlinPhysCDM_HertzMindlinCDM)arg1) +→ +float : +Return the ratio between the number of contacts sliding to the total number at a given time. +ratioYieldingContacts((Law2_ScGeom_MindlinPhysCDM_HertzMindlinCDM)arg1) → +float : +Return the ratio between the number of contacts yielding to the total number at a given time. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Law2_ScGeom_MindlinPhys_HertzWithLinearShear(inherits LawFunctor +→ Functor → Serial- +izable) +Constitutive law for the Hertz formulation (using MindlinPhys.kno) and linear behavior in shear +(using MindlinPhys.kso for stiffness and FrictPhys.tangensOfFrictionAngle). +Note: No viscosity or damping. If you need those, look at Law2_ScGeom_MindlinPhys_Mindlin, +which also includes non-linear Mindlin shear. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +neverErase(=false) +Keep interactions even if particles go away from each other (only in case another constitutive +law is in the scene, e.g. Law2_ScGeom_CapillaryPhys_Capillarity) +2.3. +Yade wrapper class reference +343 + +Yade Documentation, Release 3rd ed. +nonLin(=0) +Shear force nonlinearity (the value determines how many features of the non-linearity are +taken in account). 1: ks as in HM 2: shearElastic increment computed as in HM 3. granular +ratcheting disabled. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Law2_ScGeom_MindlinPhys_Mindlin(inherits LawFunctor → Functor → +Serializable) +Constitutive law for the Hertz-Mindlin formulation. It includes non linear elasticity in the normal +direction as predicted by Hertz for two non-conforming elastic contact bodies. In the shear direc- +tion, instead, it reseambles the simplified case without slip discussed in Mindlin’s paper, where a +linear relationship between shear force and tangential displacement is provided. Finally, the Mohr- +Coulomb criterion is employed to established the maximum friction force which can be developed +at the contact. Moreover, it is also possible to include the effect of linear viscous damping through +the definition of the parameters βn and βs. +bases +Ordered list of types (as strings) this functor accepts. +calcEnergy(=false) +bool to calculate energy terms (shear potential energy, dissipation of energy due to friction +and dissipation of energy due to normal and tangential damping) +contactsAdhesive((Law2_ScGeom_MindlinPhys_Mindlin)arg1) → float : +Compute total number of adhesive contacts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +frictionDissipation(=uninitalized) +Energy dissipation due to sliding +includeAdhesion(=false) +bool to include the adhesion force following the DMT formulation. If true, also the normal +elastic energy takes into account the adhesion effect. +includeMoment(=false) +bool to consider rolling resistance (if Ip2_FrictMat_FrictMat_MindlinPhys::eta is 0.0, no +plastic condition is applied.) +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +neverErase(=false) +Keep interactions even if particles go away from each other (only in case another constitutive +law is in the scene, e.g. Law2_ScGeom_CapillaryPhys_Capillarity) +normDampDissip(=uninitalized) +Energy dissipated by normal damping +normElastEnergy((Law2_ScGeom_MindlinPhys_Mindlin)arg1) → float : +Compute normal elastic potential energy. It handles the DMT formulation if Law2_ScGeom_- +MindlinPhys_Mindlin::includeAdhesion is set to true. +preventGranularRatcheting(=true) +bool to avoid granular ratcheting +ratioSlidingContacts((Law2_ScGeom_MindlinPhys_Mindlin)arg1) → float : +Return the ratio between the number of contacts sliding to the total number at a given time. +344 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +shearDampDissip(=uninitalized) +Energy dissipated by tangential damping +shearEnergy(=uninitalized) +Shear elastic potential energy +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Law2_ScGeom_MindlinPhys_MindlinDeresiewitz(inherits LawFunctor → +Functor +→ +Serializ- +able) +Hertz-Mindlin contact law with partial slip solution, as described in [Thornton1991]. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +neverErase(=false) +Keep interactions even if particles go away from each other (only in case another constitutive +law is in the scene, e.g. Law2_ScGeom_CapillaryPhys_Capillarity) +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Law2_ScGeom_MortarPhys_Lourenco(inherits LawFunctor → Functor → +Serializable) +Material law for mortar layer according to [Lourenco1994]. The contact behaves elastic until brittle +failure when reaching strength envelope. The envelope has three parts. +Tensile with condition σN − ft. +Shear part with Mohr-Coulomb condition |σT| + σN tan φ − c. +Compressive part with condition σ2 +N + A2σ2 +T − f2 +c +The main idea is to begin simulation with this model and when the contact is broken, to use +standard non-cohesive Law2_PolyhedraGeom_PolyhedraPhys_Volumetric. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +2.3. +Yade wrapper class reference +345 + +Yade Documentation, Release 3rd ed. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Law2_ScGeom_PotentialLubricationPhys(inherits +Law2_ScGeom_- +ImplicitLubricationPhys +→ +Law2_ScGeom_VirtualLubri- +cationPhys → LawFunctor → +Functor → Serializable) +Material law for lubrication + potential between two spheres. The potential model include contact. +This material law will solve the system with lubrication and the provided potential. +MaxDist(=2.) +Maximum distance (d/a) for the interaction +MaxIter(=30) +Maximum iterations for numerical resolution (Dichotomy and Newton-Rafson) +SolutionTol(=1.e-8) +Tolerance for numerical resolution (Dichotomy and Newton-Rafson) +activateRollLubrication(=true) +Activate roll lubrication (default: true) +activateTangencialLubrication(=true) +Activate tangencial lubrication (default: true) +activateTwistLubrication(=true) +Activate twist lubrication (default: true) +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +static getStressForEachBody() → tuple : +Get stresses tensors for each bodies: normal contact stress, shear contact stress, normal +lubrication stress, shear lubrication stress, stress from additionnal potential forces. +static getTotalStresses() → tuple : +Get total stresses tensors: normal contact stress, shear contact stress, normal lubrication +stress, shear lubrication stress, stress from additionnal potential forces. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +maxSubSteps(=4) +max recursion depth of adaptative timestepping in the theta-method, the minimal time in- +terval is thus Omega::dt/2depth. If still not converged the integrator will switch to backward +Euler. +potential(=new GenericPotential()) +Physical potential force between spheres. +resolution(=0) +Change normal component resolution method, 0: Iterative exact resolution with substepping +(theta method, linear contact), 1: Newton-Rafson dimensionless resolution (theta method, +linear contact), 2: (default) Dichotomy dimensionless resolution (theta method, linear con- +tact), 3: Exact dimensionless solution with contact prediction (theta method, linear contact). +Method 3 is better if the volumic fraction is not too high. Use 2 otherwise. +theta(=0.55) +parameter of the ‘theta’-method, 1: backward Euler, 0.5: trapezoidal rule, 0: not used, 0.55: +suggested optimum) +346 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Law2_ScGeom_VirtualLubricationPhys(inherits LawFunctor → Functor +→ Serializable) +Virtual class for sheared lubrication functions. This don’t do any computation and shouldn’t be +used directly! +MaxDist(=2.) +Maximum distance (d/a) for the interaction +activateRollLubrication(=true) +Activate roll lubrication (default: true) +activateTangencialLubrication(=true) +Activate tangencial lubrication (default: true) +activateTwistLubrication(=true) +Activate twist lubrication (default: true) +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +static getStressForEachBody() → tuple : +Get stresses tensors for each bodies: normal contact stress, shear contact stress, normal +lubrication stress, shear lubrication stress, stress from additionnal potential forces. +static getTotalStresses() → tuple : +Get total stresses tensors: normal contact stress, shear contact stress, normal lubrication +stress, shear lubrication stress, stress from additionnal potential forces. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Law2_ScGeom_ViscElCapPhys_Basic(inherits LawFunctor → Functor → +Serializable) +Extended version of Linear viscoelastic model with capillary parameters. +NLiqBridg(=uninitalized) +The total number of liquid bridges +VLiqBridg(=uninitalized) +The total volume of liquid bridges +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +2.3. +Yade wrapper class reference +347 + +Yade Documentation, Release 3rd ed. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Law2_ScGeom_ViscElPhys_Basic(inherits LawFunctor → Functor → Seri- +alizable) +Linear viscoelastic model operating on ScGeom and ViscElPhys. The contact law is visco-elastic +in the normal direction, and visco-elastic frictional in the tangential direction. The normal contact +is modelled as a spring of equivalent stiffness kn, placed in parallel with a viscous damper of +equivalent viscosity cn. As for the tangential contact, it is made of a spring-dashpot system (in +parallel with equivalent stiffness ks and viscosity cs) in serie with a slider of friction coefficient +µ = tan φ. +The friction coefficient µ = tan φ is always evaluated as tan(min(φ1, φ2)), where φ1 and φ2 +are respectively the friction angle of particle 1 and 2. For the other parameters, depending on the +material input, the equivalent parameters of the contact (Kn,Cn,Ks,Cs,φ) are evaluated differently. +In the following, the quantities in parenthesis are the material constant which are precised for each +particle. They are then associated to particle 1 and 2 (e.g. kn1,kn2,cn1…), and should not be +confused with the equivalent parameters of the contact (Kn,Cn,Ks,Cs,φ). +• If contact time (tc), normal and tangential restitution coefficient (en,et) are precised, the +equivalent parameters are evaluated following the formulation of Pournin [Pournin2001]. +• If normal and tangential stiffnesses (kn, ks) and damping constant (cn,cs) of each particle +are precised, the equivalent stiffnesses and damping constants of each contact made of two +particles 1 and 2 is made A = 2 a1a2 +a1+a2 , where A is Kn, Ks, Cn and Cs, and 1 and 2 refer to +the value associated to particle 1 and 2. +• Alternatively it is possible to precise the Young’s modulus (young) and Poisson’s ratio (pois- +son) instead of the normal and spring constant (kn and ks). +In this case, the equivalent +parameters are evaluated the same way as the previous case with knx = Exdx, ksx = vxknx, +where Ex, vx and dx are Young’s modulus, Poisson’s ratio and diameter of particle x. +• If Young’s modulus (young), Poisson’s ratio (poisson), normal and tangential restitution co- +efficient (en,et)are precised, the equivalent stiffnesses are evaluated as previously: Kn = +2 kn1kn2 +kn1+kn2 , knx = Exdx, Ks = 2(ks1ks2)/(ks1 + ks2), ksx = vknx. +The damping con- +stant is computed at each contact in order to fulfill the normal restitution coefficient +en = (en1 + en2)/2. This is achieved resolving numerically equation 21 of [Schwager2007] +(There is in fact a mistake in the article from equation 18 to 19, so that there is a change +in sign). Be careful in this configuration the tangential restitution coefficient is set to 1 (no +tangential damping). This formulation imposes directly the normal restitution coefficient of +the collisions instead of the damping constant. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +348 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +class yade.wrapper.Law2_ScGeom_ViscoFrictPhys_CundallStrack(inherits Law2_ScGeom_- +FrictPhys_CundallStrack +→ LawFunctor → Functor +→ Serializable) +Law similar to Law2_ScGeom_FrictPhys_CundallStrack with the addition of shear creep at con- +tacts. +bases +Ordered list of types (as strings) this functor accepts. +creepStiffness(=1) +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +elasticEnergy((Law2_ScGeom_FrictPhys_CundallStrack)arg1) → float : +Compute and return the total elastic energy in all “FrictPhys” contacts +initPlasticDissipation((Law2_ScGeom_FrictPhys_CundallStrack)arg1, (float)arg2) → +None : +Initialize cummulated plastic dissipation to a value (0 by default). +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +neverErase(=false) +Keep interactions even if particles go away from each other (only in case another constitutive +law is in the scene, e.g. Law2_ScGeom_CapillaryPhys_Capillarity) +plasticDissipation((Law2_ScGeom_FrictPhys_CundallStrack)arg1) → float : +Total energy dissipated in plastic slips at all FrictPhys contacts. Computed only if Law2_- +ScGeom_FrictPhys_CundallStrack::traceEnergy is true. +shearCreep(=false) +sphericalBodies(=true) +If true, compute branch vectors from radii (faster), else use contactPoint-position. Turning +this flag true is safe for sphere-sphere contacts and a few other specific cases. It will give +wrong values of torques on facets or boxes. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +traceEnergy(=false) +Define the total energy dissipated in plastic slips at all contacts. This will trace only plastic +energy in this law, see O.trackEnergy for a more complete energies tracing +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +viscosity(=1) +class yade.wrapper.Law2_ScGeom_WirePhys_WirePM(inherits LawFunctor → Functor → Seri- +alizable) +Constitutive law for the wire model. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +2.3. +Yade wrapper class reference +349 + +Yade Documentation, Release 3rd ed. +linkThresholdIteration(=1) +Iteration to create the link. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Law2_ScGridCoGeom_CohFrictPhys_CundallStrack(inherits LawFunctor +→ Functor → Serial- +izable) +Law between a cohesive frictional GridConnection and a cohesive frictional Sphere. Almost the +same than Law2_ScGeom6D_CohFrictPhys_CohesionMoment, but THE ROTATIONAL MO- +MENTS ARE NOT COMPUTED. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +neverErase(=false) +Keep interactions even if particles go away from each other (only in case another constitutive +law is in the scene, e.g. Law2_ScGeom_CapillaryPhys_Capillarity) +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Law2_ScGridCoGeom_FrictPhys_CundallStrack(inherits LawFunctor → +Functor → Serializable) +Law between a frictional GridConnection and a frictional Sphere. Almost the same than Law2_- +ScGeom_FrictPhys_CundallStrack, but the force is divided and applied on the two GridNodes +only. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +neverErase(=false) +Keep interactions even if particles go away from each other (only in case another constitutive +law is in the scene, e.g. Law2_ScGeom_CapillaryPhys_Capillarity) +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +350 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +LawDispatcher +class yade.wrapper.LawDispatcher(inherits Dispatcher → Engine → Serializable) +Dispatcher calling functors based on received argument type(s). +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispFunctor((LawDispatcher)arg1, (IGeom)arg2, (IPhys)arg3) → LawFunctor : +Return functor that would be dispatched for given argument(s); None if no dispatch; ambigu- +ous dispatch throws. +dispMatrix((LawDispatcher)arg1[, (bool)names=True]) → dict : +Return dictionary with contents of the dispatch matrix. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +functors +Functors associated with this dispatcher. +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +2.3.11 Internal forces +InternalForceFunctor +InternalForceDispatcher +2.3.12 Callbacks +IntrCallback +SumIntrForcesCb +Fig. 37: Inheritance graph of IntrCallback. See also: SumIntrForcesCb. +2.3. +Yade wrapper class reference +351 + +Yade Documentation, Release 3rd ed. +class yade.wrapper.IntrCallback(inherits Serializable) +Abstract callback object which will be called for every (real) Interaction after the interaction has +been processed by InteractionLoop. +At the beginning of the interaction loop, stepInit is called, initializing the object; it returns either +NULL (to deactivate the callback during this time step) or pointer to function, which will then be +passed (1) pointer to the callback object itself and (2) pointer to Interaction. +Note: +(NOT YET DONE) This functionality is accessible from python by passing 4th argument +to InteractionLoop constructor, or by appending the callback object to InteractionLoop::callbacks. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.SumIntrForcesCb(inherits IntrCallback → Serializable) +Callback summing magnitudes of forces over all interactions. IPhys of interactions must derive +from NormShearPhys (responsability fo the user). +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +2.3.13 Preprocessors +FileGenerator +SimpleShear +TriaxialTest +Fig. 38: Inheritance graph of FileGenerator. See also: SimpleShear, TriaxialTest. +class yade.wrapper.FileGenerator(inherits Serializable) +Base class for scene generators, preprocessors. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +generate((FileGenerator)arg1, (str)out) → None : +Generate scene, save to given file +load((FileGenerator)arg1) → None : +Generate scene, save to temporary file and load immediately +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.SimpleShear(inherits FileGenerator → Serializable) +Preprocessor for a simple shear box model. The packing initially conforms a gas-like, very loose, +state (see utils.makeCloud function), but importing some existing packing from a text file can be +also performed after little change in the source code. In its current state, the preprocessor carries out +an oedometric compression, until a value of normal stress equal to 2 MPa (and a stable mechanical +state). Others Engines such as KinemCNDEngine, KinemCNSEngine and KinemCNLEngine, could +be used to apply resp. constant normal displacement, constant normal rigidity and constant normal +stress paths using such a simple shear box. +352 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +density(=2600) +density of the spheres [kg/m3] +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +generate((FileGenerator)arg1, (str)out) → None : +Generate scene, save to given file +gravApplied(=false) +depending on this, GravityEngine is added or not to the scene to take into account the weight +of particles +gravity(=Vector3r(0, -9.81, 0)) +vector corresponding to used gravity (if gravApplied) [m/s2] +height(=0.02) +initial height (along y-axis) of the shear box [m] +length(=0.1) +initial length (along x-axis) of the shear box [m] +load((FileGenerator)arg1) → None : +Generate scene, save to temporary file and load immediately +matFrictionDeg(=37) +value of FrictMat.frictionAngle within the packing and for the two horizontal boundaries +(friction is zero along other boundaries) [◦] (the necessary conversion in [rad] is done auto- +matically) +matPoissonRatio(=0.04) +value of FrictMat.poisson for the bodies [-] +matYoungModulus(=4.0e9) +value of FrictMat.young for the bodies [Pa] +thickness(=0.001) +thickness of the boxes constituting the shear box [m] +timeStepUpdateInterval(=50) +value of TimeStepper::timeStepUpdateInterval for the TimeStepper used here +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +width(=0.04) +initial width (along z-axis) of the shear box [m] +class yade.wrapper.TriaxialTest(inherits FileGenerator → Serializable) +Create a scene for triaxal test. +Introduction Yade includes tools to simulate triaxial tests on particles assemblies. +This pre- +processor (and variants like e.g. CapillaryTriaxialTest) illustrate how to use them. It generates +a scene which will - by default - go through the following steps : +• generate random loose packings in a parallelepiped. +• compress the packing isotropicaly, either squeezing the packing between moving rigid +boxes or expanding the particles while boxes are fixed (depending on flag internalCom- +paction). The confining pressure in this stage is defined via sigmaIsoCompaction. +• when the packing is dense and stable, simulate a loading path and get the mechanical +response as a result. +The default loading path corresponds to a constant lateral stress (sigmaLateralConfinement) +in 2 directions and constant strain rate on the third direction. This default loading path is +performed when the flag autoCompressionActivation it True, otherwise the simulation stops +after isotropic compression. +2.3. +Yade wrapper class reference +353 + +Yade Documentation, Release 3rd ed. +Different loading paths might be performed. In order to define them, the user can modify +the flags found in engine TriaxialStressController at any point in the simulation (in c++). +If TriaxialStressController.wall_X_activated is true boundary X is moved automati- +cally to maintain the defined stress level sigmaN (see axis conventions below). If false the +boundary is not controlled by the engine at all. In that case the user is free to prescribe fixed +position, constant velocity, or more complex conditions. +Note: +Axis conventions. Boundaries perpendicular to the x axis are called “left” and “right”, +y corresponds to “top” and “bottom”, and axis z to “front” and “back”. In the default loading +path, strain rate is assigned along y, and constant stresses are assigned on x and z. +Essential engines +1. The TriaxialCompressionEngine is used for controlling the state of the sample and simu- +lating loading paths. TriaxialCompressionEngine inherits from TriaxialStressController, +which computes stress- and strain-like quantities in the packing and maintain a constant +level of stress at each boundary. TriaxialCompressionEngine has few more members in +order to impose constant strain rate and control the transition between isotropic com- +pression and triaxial test. Transitions are defined by changing some flags of the Triaxial- +StressController, switching from/to imposed strain rate to/from imposed stress. +2. The class TriaxialStateRecorder is used to write to a file the history of stresses and strains. +3. TriaxialTest is using GlobalStiffnessTimeStepper to compute an appropriate ∆t for the +numerical scheme. +Note: +TriaxialStressController::ComputeUnbalancedForce returns a value that can +be useful for evaluating the stability of the packing. It is defined as (mean force on parti- +cles)/(mean contact force), so that it tends to 0 in a stable packing. This parameter is checked +by TriaxialCompressionEngine to switch from one stage of the simulation to the next one (e.g. +stop isotropic confinment and start axial loading) +Frequently Asked Questions +1. How is generated the packing? How to change particles sizes distribution? Why do I have a message “Exceeded 3000 tries to insert non-overlapping sphere? +The initial positioning of spheres is done by generating random (x,y,z) in a box and +checking if a sphere of radius R (R also randomly generated with respect to a uniform +distribution between mean*(1-std_dev) and mean*(1+std_dev) can be inserted at this +location without overlaping with others. +If the sphere overlaps, new (x,y,z)’s are generated until a free position for the new sphere is +found. This explains the message you have: after 3000 trial-and-error, the sphere couldn’t +be placed, and the algorithm stops. +You get the message above if you try to generate an initialy dense packing, which is not +possible with this algorithm. It can only generate clouds. You should keep the default +value of porosity (n~0.7), or even increase if it is still to low in some cases. The dense +state will be obtained in the second step (compaction, see below). +2. How is the compaction done, what are the parameters maxWallVelocity and finalMaxMultiplier? +Compaction is done +1. by moving rigid boxes or +2. by increasing the sizes of the particles (decided using the option internalCompaction +￿ size increase). +354 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +Both algorithm needs numerical parameters to prevent instabilities. For instance, with +the method (1) maxWallVelocity is the maximum wall velocity, with method (2) final- +MaxMultiplier is the max value of the multiplier applied on sizes at each iteration (always +something like 1.00001). +3. During the simulation of triaxial compression test, the wall in one direction moves with an increment of strain while the stresses in other two directions are adjusted to sigma_iso. How the stresses in other directions are maintained constant to sigma_iso? What is the mechanism? Where is it implemented in Yade? +The control of stress on a boundary is based on the total stiffness K of all contacts +between the packing and this boundary. In short, at each step, displacement=stress_- +error/K. This algorithm is implemented in TriaxialStressController, and the control +itself is in TriaxialStressController::ControlExternalStress. +The control can +be turned off independently for each boundary, using the flags wall_XXX_activated, +with XXX￿{top, bottom, left, right, back, front}. The imposed sress is a unique value +(sigma_iso) for all directions if TriaxialStressController.isAxisymetric, or 3 independent +values sigma1, sigma2, sigma3. +4. Which value of friction angle do you use during the compaction phase of the Triaxial Test? +The friction during the compaction (whether you are using the expansion method or +the compression one for the specimen generation) can be anything between 0 and the +final value used during the Triaxial phase. Note that higher friction than the final one +would result in volumetric collapse at the beginning of the test. The purpose of using a +different value of friction during this phase is related to the fact that the final porosity +you get at the end of the sample generation essentially depends on it as well as on the +assumed Particle Size Distribution. Changing the initial value of friction will get to a +different value of the final porosity. +5. Which is the aim of the bool isRadiusControlIteration? This internal variable (up- +dated automatically) is true each N timesteps (with N=radiusControlInterval). For other +timesteps, there is no expansion. Cycling without expanding is just a way to speed up the +simulation, based on the idea that 1% increase each 10 iterations needs less operations +than 0.1% at each iteration, but will give similar results. +6. How comes the unbalanced force reaches a low value only after many timesteps in the compaction phase? +The value of unbalanced force (dimensionless) is expected to reach low value (i.e. identi- +fying a static-equilibrium condition for the specimen) only at the end of the compaction +phase. The code is not aiming at simulating a quasistatic isotropic compaction process, +it is only giving a stable packing at the end of it. +Key(=””) +A code that is added to output filenames. +StabilityCriterion(=0.01) +Value of unbalanced force for which the system is considered stable. Used in conditionals to +switch between loading stages. +WallStressRecordFile(=”./WallStresses”+Key) +autoCompressionActivation(=true) +Do we just want to generate a stable packing under isotropic pressure (false) or do we want +the triaxial loading to start automatically right after compaction stage (true)? +autoStopSimulation(=false) +freeze the simulation when conditions are reached (don’t activate this if you want to be able +to run/stop from Qt GUI) +autoUnload(=true) +auto adjust the isotropic stress state from TriaxialTest::sigmaIsoCompaction to Triaxial- +Test::sigmaLateralConfinement if they have different values. See docs for TriaxialCompres- +sionEngine::autoUnload +boxFrictionDeg(=0.0) +Friction angle [°] of boundaries contacts. +2.3. +Yade wrapper class reference +355 + +Yade Documentation, Release 3rd ed. +boxKsDivKn(=0.5) +Ratio of shear vs. normal contact stiffness for boxes. +boxYoungModulus(=15000000.0) +Stiffness of boxes. +compactionFrictionDeg(=sphereFrictionDeg) +Friction angle [°] of spheres during compaction (different values result in different porosities)]. +This value is overridden by TriaxialTest::sphereFrictionDeg before triaxial testing. +dampingForce(=0.2) +Coefficient of Cundal-Non-Viscous damping (applied on on the 3 components of forces) +dampingMomentum(=0.2) +Coefficient of Cundal-Non-Viscous damping (applied on on the 3 components of torques) +defaultDt(=-1) +Max time-step. Used as initial value if defined. Latter adjusted by the time stepper. +density(=2600) +density of spheres +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +facetWalls(=false) +Use facets for boundaries (not tested) +finalMaxMultiplier(=1.001) +max multiplier of diameters during internal compaction (secondary precise adjustment) +fixedBoxDims(=””) +string that contains some subset (max. 2) of {‘x’,’y’,’z’} ; contains axes will have box dimension +hardcoded, even if box is scaled as mean_radius is prescribed: scaling will be applied on the +rest. +generate((FileGenerator)arg1, (str)out) → None : +Generate scene, save to given file +importFilename(=””) +File with positions and sizes of spheres. +internalCompaction(=false) +flag for choosing between moving boundaries or increasing particles sizes during the com- +paction stage. +load((FileGenerator)arg1) → None : +Generate scene, save to temporary file and load immediately +lowerCorner(=Vector3r(0, 0, 0)) +Lower corner of the box. +maxMultiplier(=1.01) +max multiplier of diameters during internal compaction (initial fast increase) +maxWallVelocity(=10) +max velocity of boundaries. Usually useless, but can help stabilizing the system in some cases. +noFiles(=false) +Do not create any files during run (.xml, .spheres, wall stress records) +numberOfGrains(=400) +Number of generated spheres. +radiusControlInterval(=10) +interval between size changes when growing spheres. +356 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +radiusMean(=-1) +Mean radius. If negative (default), autocomputed to as a function of box size and Triaxial- +Test::numberOfGrains +radiusStdDev(=0.3) +Normalized standard deviation of generated sizes. +recordIntervalIter(=20) +interval between file outputs +seed(=0) +Seed used for the call to makeCloud +sigmaIsoCompaction(=-50000) +Confining stress during isotropic compaction (< 0 for real - compressive - compaction). +sigmaLateralConfinement(=-50000) +Lateral stress during triaxial loading (< 0 for classical compressive cases). An isotropic un- +loading is performed if the value is not equal to TriaxialTest::sigmaIsoCompaction. +sphereFrictionDeg(=18.0) +Friction angle [°] of spheres assigned just before triaxial testing. +sphereKsDivKn(=0.5) +Ratio of shear vs. normal contact stiffness for spheres. +sphereYoungModulus(=15000000.0) +Stiffness of spheres. +strainRate(=0.1) +Strain rate in triaxial loading. +thickness(=0.001) +thickness of boundaries. It is arbitrary and should have no effect +timeStepUpdateInterval(=50) +interval for GlobalStiffnessTimeStepper +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +upperCorner(=Vector3r(1, 1, 1)) +Upper corner of the box. +wallOversizeFactor(=1.3) +Make boundaries larger than the packing to make sure spheres don’t go out during deforma- +tion. +wallStiffnessUpdateInterval(=10) +interval for updating the stiffness of sample/boundaries contacts +wallWalls(=false) +Use walls for boundaries (not tested) +2.3.14 Rendering +OpenGLRenderer +class yade.wrapper.OpenGLRenderer(inherits Serializable) +Class responsible for rendering scene on OpenGL devices. +bgColor(=Vector3r(.2, .2, .2)) +Color of the background canvas (RGB) +blinkHighlight(=BlinkHighlight::NORMAL) +Adjust blinking of the body selected in the ‘Simulation Inspection’ window. +2.3. +Yade wrapper class reference +357 + +Yade Documentation, Release 3rd ed. +bound(=false) +Render body Bound +cellColor(=Vector3r(1, 1, 0)) +Color of the periodic cell (RGB). +clipPlaneActive(=vector(numClipPlanes, false)) +Activate/deactivate respective clipping planes +clipPlaneSe3(=vector(numClipPlanes, +Se3r(Vector3r::Zero(), +Quater- +nionr::Identity()))) +Position and orientation of clipping planes +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +dispScale(=Vector3r::Ones(), disable scaling) +Artificially enlarge (scale) dispalcements from bodies’ reference positions by this relative +amount, so that they become better visible (independently in 3 dimensions). Disbled if (1,1,1). +dof(=false) +Show which degrees of freedom are blocked for each body +extraDrawers(=uninitalized) +Additional rendering components (GlExtraDrawer). +ghosts(=true) +Render objects crossing periodic cell edges by cloning them in multiple places (periodic sim- +ulations only). +hideBody((OpenGLRenderer)arg1, (int)id) → None : +Hide body from id (see OpenGLRenderer::showBody) +id(=false) +Show body id’s +intrAllWire(=false) +Draw wire for all interactions, blue for potential and green for real ones (mostly for debugging) +intrGeom(=false) +Render Interaction::geom objects. +intrPhys(=false) +Render Interaction::phys objects +intrWire(=false) +If rendering interactions, use only wires to represent them. +light1(=true) +Turn light 1 on. +light2(=true) +Turn light 2 on. +light2Color(=Vector3r(0.5, 0.5, 0.1)) +Per-color intensity of secondary light (RGB). +light2Pos(=Vector3r(-130, 75, 30)) +Position of secondary OpenGL light source in the scene. +lightColor(=Vector3r(0.6, 0.6, 0.6)) +Per-color intensity of primary light (RGB). +lightPos(=Vector3r(75, 130, 0)) +Position of OpenGL light source in the scene. +mask(=~0, draw everything) +Bitmask for showing only bodies where ((mask & Body::mask)!=0) +358 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +render((OpenGLRenderer)arg1) → None : +Render the scene in the current OpenGL context. +rotScale(=1., disable scaling) +Artificially enlarge (scale) rotations of bodies relative to their reference orientation, so the +they are better visible. +selId(=Body::ID_NONE) +Id of particle that was selected by the user. +setRefSe3((OpenGLRenderer)arg1) → None : +Make current positions and orientation reference for scaleDisplacements and scaleRotations. +shape(=true) +Render body Shape +showBody((OpenGLRenderer)arg1, (int)id) → None : +Make body visible (see OpenGLRenderer::hideBody) +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +wire(=false) +Render all bodies with wire only (faster) +GlShapeFunctor +GlShapeFunctor +Gl1_Sphere +Gl1_PFacet +Gl1_ChainedCylinder +Gl1_Cylinder +Gl1_Facet +Gl1_Box +Gl1_Tetra +Gl1_Wall +Gl1_GridConnection +Fig. 39: Inheritance graph of GlShapeFunctor. See also: Gl1_Box, Gl1_ChainedCylinder, Gl1_Cylin- +der, Gl1_Facet, Gl1_GridConnection, Gl1_PFacet, Gl1_Sphere, Gl1_Tetra, Gl1_Wall. +class yade.wrapper.GlShapeFunctor(inherits Functor → Serializable) +Abstract functor for rendering Shape objects. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +2.3. +Yade wrapper class reference +359 + +Yade Documentation, Release 3rd ed. +class yade.wrapper.Gl1_Box(inherits GlShapeFunctor → Functor → Serializable) +Renders Box object +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Gl1_ChainedCylinder(inherits Gl1_Cylinder → GlShapeFunctor → Func- +tor → Serializable) +Renders ChainedCylinder object including a shift for compensating flexion. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +glutNormalize = True +glutSlices = 8 +glutStacks = 4 +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +wire = False +class yade.wrapper.Gl1_Cylinder(inherits GlShapeFunctor → Functor → Serializable) +Renders Cylinder object +wire(=false) [static] +Only show wireframe (controlled by glutSlices and glutStacks. +glutNormalize(=true) [static] +Fix normals for non-wire rendering +glutSlices(=8) [static] +Number of sphere slices. +glutStacks(=4) [static] +Number of sphere stacks. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +360 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +glutNormalize = True +glutSlices = 8 +glutStacks = 4 +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +wire = False +class yade.wrapper.Gl1_Facet(inherits GlShapeFunctor → Functor → Serializable) +Renders Facet object +normals(=false) [static] +In wire mode, render normals of facets and edges; facet’s colors are disregarded in that case. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +normals = False +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Gl1_GridConnection(inherits GlShapeFunctor → Functor → Serializable) +Renders Cylinder object +wire(=false) [static] +Only show wireframe (controlled by glutSlices and glutStacks. +glutNormalize(=true) [static] +Fix normals for non-wire rendering +glutSlices(=8) [static] +Number of cylinder slices. +glutStacks(=4) [static] +Number of cylinder stacks. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +glutNormalize = True +glutSlices = 8 +glutStacks = 4 +2.3. +Yade wrapper class reference +361 + +Yade Documentation, Release 3rd ed. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +wire = False +class yade.wrapper.Gl1_PFacet(inherits GlShapeFunctor → Functor → Serializable) +Renders Facet object +wire(=false) [static] +Only show wireframe (controlled by glutSlices and glutStacks. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +wire = False +class yade.wrapper.Gl1_Sphere(inherits GlShapeFunctor → Functor → Serializable) +Renders Sphere object +quality(=1.0) [static] +Change discretization level of spheres. quality>1 for better image quality, at the price of more +cpu/gpu usage, 0::infinity()) [static] +Reference (minimum) particle radius; used only if maxRadius is negative. This value will be +decreased (but not increased ) automatically. (auto-updated) +maxRadius(=-1) [static] +Cylinder radius corresponding to the maximum normal force. If negative, auto-updated re- +fRadius will be used instead. +slices(=6) [static] +Number of sphere slices; (see glutCylinder reference) +stacks(=1) [static] +Number of sphere stacks; (see glutCylinder reference) +maxWeakFn(=NaN) [static] +Value that divides contacts by their normal force into the ‘weak fabric’ and ‘strong fabric’. +This value is set as side-effect by utils.fabricTensor. +weakFilter(=0) [static] +If non-zero, only display contacts belonging to the ‘weak’ (-1) or ‘strong’ (+1) fabric. +368 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +weakScale(=1.) [static] +If maxWeakFn is set, scale radius of the weak fabric by this amount (usually smaller than 1). +If zero, 1 pixel line is displayed. Colors are not affected by this value. +bases +Ordered list of types (as strings) this functor accepts. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +label(=uninitalized) +Textual label for this object; must be a valid python identifier, you can refer to it directly +from python. +maxFn = 0.0 +maxRadius = -1.0 +maxWeakFn = nan +refRadius = inf +signFilter = 0 +slices = 6 +stacks = 1 +timingDeltas +Detailed information about timing inside the Dispatcher itself. Empty unless enabled in the +source code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +weakFilter = 0 +weakScale = 1.0 +2.3.15 Simulation data +Omega +class yade.wrapper.Omega +addScene((Omega)arg1) → int : +Add new scene to Omega, returns its number +bodies +Bodies in the current simulation (container supporting index access by id and iteration) +cell +Periodic cell of the current scene (None if the scene is aperiodic). +childClassesNonrecursive((Omega)arg1, (str)arg2) → list : +Return list of all classes deriving from given class, as registered in the class factory +disableGdb((Omega)arg1) → None : +Revert SEGV and ABRT handlers to system defaults. +dt +Current timestep (∆t) value. See dynDt for enabling/disabling automatic ∆t updates through +a TimeStepper. +dynDt +Whether a TimeStepper (when present in O.engines) is used for dynamic ∆t control. +2.3. +Yade wrapper class reference +369 + +Yade Documentation, Release 3rd ed. +dynDtAvailable +Whether a TimeStepper is amongst O.engines, activated or not. +energy +EnergyTracker of the current simulation. (meaningful only with O.trackEnergy) +engines +List of engines in the simulation (corresponds to Scene::engines in C++ source code). +exitNoBacktrace((Omega)arg1[, (int)status=0]) → None : +Disable SEGV handler and exit, optionally with given status number. +filename +Filename under which the current simulation was saved (None if never saved). +forceSyncCount +Counter for number of syncs in ForceContainer, for profiling purposes. +forces +ForceContainer (forces, torques, displacements) in the current simulation. +interactions +Access to interactions of simulation, by using +1. id’s of both Bodies of the interactions, e.g. O.interactions[23,65] +2. iteraction over the whole container: +for i in O.interactions: print i.id1,i.id2 +Note: +Iteration silently skips interactions that are not real. +isChildClassOf((Omega)arg1, (str)arg2, (str)arg3) → bool : +Tells whether the first class derives from the second one (both given as strings). +iter +Get current step number +labeledEngine((Omega)arg1, (str)arg2) → object : +Return instance of engine/functor with the given label. This function shouldn’t be called +by the user directly; every ehange in O.engines will assign respective global python variables +according to labels. +For example: +O.engines=[InsertionSortCollider(label=’collider’)] +collider.nBins=5 # collider has become a variable after assignment to O.engines +automatically +load((Omega)arg1, (str)file[, (bool)quiet=False]) → None : +Load simulation from file. The file should have been saved in the same version of Yade built +or compiled with the same features, otherwise compatibility is not guaranteed. Compatibility +may also be affected by different versions of external libraries such as Boost +loadTmp((Omega)arg1[, (str)mark=”[, (bool)quiet=False]]) → None : +Load simulation previously stored in memory by saveTmp. +mark optionally distinguishes +multiple saved simulations +lsTmp((Omega)arg1) → list : +Return list of all memory-saved simulations. +materials +Shared materials; they can be accessed by id or by label +370 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +miscParams +MiscParams in the simulation (Scene::mistParams), usually used to save serializables that +don’t fit anywhere else, like GL functors +numThreads +Get maximum number of threads openMP can use. +pause((Omega)arg1) → None : +Stop simulation execution. (May be called from within the loop, and it will stop after the +current step). +periodic +Get/set whether the scene is periodic or not (True/False). +plugins((Omega)arg1) → list : +Return list of all plugins registered in the class factory. +realtime +Return clock (human world) time the simulation has been running. +reload((Omega)arg1[, (bool)quiet=False]) → None : +Reload current simulation +reset((Omega)arg1) → None : +Reset simulations completely (including another scenes!). +resetAllScenes((Omega)arg1) → None : +Reset all scenes. +resetCurrentScene((Omega)arg1) → None : +Reset current scene. +resetThisScene((Omega)arg1) → None : +Reset current scene. +resetTime((Omega)arg1) → None : +Reset simulation time: step number, virtual and real time. (Doesn’t touch anything else, +including timings). +run((Omega)arg1[, (int)nSteps=-1[, (bool)wait=False]]) → None : +Run the simulation. nSteps how many steps to run, then stop (if positive); wait will cause +not returning to python until simulation will have stopped. +runEngine((Omega)arg1, (Engine)arg2) → None : +Run given engine exactly once; simulation time, step number etc. will not be incremented +(use only if you know what you do). +running +Whether background thread is currently running a simulation. +save((Omega)arg1, (str)file[, (bool)quiet=False]) → None : +Save current simulation to file (should be .xml or .xml.bz2 or .yade or .yade.gz). .xml files are +bigger than .yade, but can be more or less easily (due to their size) opened and edited, e.g. +with text editors. .bz2 and .gz correspond both to compressed versions. There are software +requirements for successful reloads, see O.load. +saveTmp((Omega)arg1[, (str)mark=”[, (bool)quiet=False]]) → None : +Save simulation to memory (disappears at shutdown), can be loaded later with loadTmp. +mark optionally distinguishes different memory-saved simulations. +sceneToString((Omega)arg1) → object : +Return the entire scene as a string. Equivalent to using O.save(…) except that the scene goes +to a string instead of a file. (see also stringToScene()) +speed +Return current calculation speed [iter/sec]. +2.3. +Yade wrapper class reference +371 + +Yade Documentation, Release 3rd ed. +step((Omega)arg1) → None : +Advance the simulation by one step. Returns after the step will have finished. +stopAtIter +Get/set number of iteration after which the simulation will stop. +stopAtTime +Get/set time after which the simulation will stop. +stringToScene((Omega)arg1, (str)arg2[, (str)mark=”]) → None : +Load simulation from a string passed as argument (see also sceneToString). +subStep +Get the current subStep number (only meaningful if O.subStepping==True); -1 when out- +side the loop, otherwise either 0 (O.subStepping==False) or number of engine to be run +(O.subStepping==True) +subStepping +Get/set whether subStepping is active. +switchScene((Omega)arg1) → None : +Switch to alternative simulation (while keeping the old one). +Calling the function again +switches back to the first one. Note that most variables from the first simulation will still +refer to the first simulation even after the switch (e.g. b=O.bodies[4]; O.switchScene(); [b still +refers to the body in the first simulation here]) +switchToScene((Omega)arg1, (int)arg2) → None : +Switch to defined scene. Default scene has number 0, other scenes have to be created by +addScene method. +tags +Tags (string=string dictionary) of the current simulation (container supporting string-index +access/assignment) +thisScene +Return current scene’s id. +time +Return virtual (model world) time of the simulation. +timingEnabled +Globally enable/disable timing services (see documentation of the timing module). +tmpFilename((Omega)arg1) → str : +Return unique name of file in temporary directory which will be deleted when yade exits. +tmpToFile((Omega)arg1, (str)fileName[, (str)mark=”]) → None : +Save XML of saveTmp’d simulation into fileName. +tmpToString((Omega)arg1[, (str)mark=”]) → str : +Return XML of saveTmp’d simulation as string. +trackEnergy +When energy tracking is enabled or disabled in this simulation. +wait((Omega)arg1) → None : +Don’t return until the simulation will have been paused. (Returns immediately if not running). +BodyContainer +class yade.wrapper.BodyContainer +__init__((object)arg1, (BodyContainer)arg2) → None +372 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +addToClump((BodyContainer)arg1, (object)arg2, (int)arg3[, (int)discretization=0]) → None +: +Add body b (or a list of bodies) to an existing clump c. c must be clump and b may not be +a clump member of c. Clump masses and inertia are adapted automatically (for details see +clump()). +See examples/clumps/addToClump-example.py for an example script. +Note: +If b is a clump itself, then all members will be added to c and b will be deleted. If +b is a clump member of clump d, then all members from d will be added to c and d will be +deleted. If you need to add just clump member b, release this member from d first. +append((BodyContainer)arg1, (Body)arg2) → int : +Append one Body instance, return its id. +append( (BodyContainer)arg1, (object)arg2) -> object : Append list of Body in- +stance, return list of ids +appendClumped((BodyContainer)arg1, (object)arg2[, (int)discretization=0]) → tuple : +Append given list of bodies as a clump (rigid aggregate); returns a tuple of (clumpId, +[memberId1,memberId2,...]). Clump masses and inertia are computed automatically de- +pending upon discretization (for details see clump()). +clear((BodyContainer)arg1) → None : +Remove all bodies (interactions not checked) +clump((BodyContainer)arg1, (object)arg2[, (int)discretization=0]) → int : +Clump given bodies together (creating a rigid aggregate); returns clumpId. A precise defini- +tion of clump masses and inertia when clump members overlap requires discretization>0 and +is achieved in this case by integration/summation over mass points using a regular grid of +cells (grid cells length is defined as Lmin/discretization, where Lmin is the minimum length +of an Axis-Aligned Bounding Box. If *discretization*<=0 sum of inertias from members is +simply used, which is faster but accurate only for non-overlapping members). +deleteClumpBody((BodyContainer)arg1, (Body)arg2) → None : +Erase clump member. +deleteClumpMember((BodyContainer)arg1, (Body)arg2, (Body)arg3) → None : +Erase clump member. +enableRedirection +let collider switch to optimized algorithm with body redirection when bodies are erased - true +by default +erase((BodyContainer)arg1, (int)arg2[, (bool)eraseClumpMembers=0]) → bool : +Erase body with the given id; all interaction will be deleted by InteractionLoop in the next +step. +If a clump is erased use O.bodies.erase(clumpId,True) to erase the clump AND its +members. +getRoundness((BodyContainer)arg1[, (list)excludeList=[]]) → float : +Returns roundness coefficient RC = R2/R1. R1 is the equivalent sphere radius of a clump. +R2 is the minimum radius of a sphere, that imbeds the clump. If just spheres are present +RC = 1. If clumps are present 0 < RC < 1. Bodies can be excluded from the calculation by +giving a list of ids: O.bodies.getRoundness([ids]). +See examples/clumps/replaceByClumps-example.py for an example script. +insertAtId((BodyContainer)arg1, (Body)arg2, (int)insertatid) → int : +Insert a body at theid, (no body should exist in this id) +releaseFromClump((BodyContainer)arg1, (int)arg2, (int)arg3[, (int)discretization=0]) → +None : +Release body b from clump c. b must be a clump member of c. Clump masses and inertia +2.3. +Yade wrapper class reference +373 + +Yade Documentation, Release 3rd ed. +are adapted automatically (for details see clump()). +See examples/clumps/releaseFromClump-example.py for an example script. +Note: +If c contains only 2 members b will not be released and a warning will appear. In +this case clump c should be erased. +replace((BodyContainer)arg1, (object)arg2) → object +replaceByClumps((BodyContainer)arg1, (list)arg2, (object)arg3[, (int)discretization=0]) → +list : +Replace spheres by clumps using a list of clump templates and a list of amounts; returns a list +of tuples: [(clumpId1,[memberId1,memberId2,...]),(clumpId2,[memberId1,memberId2, +...]),...]. A new clump will have the same volume as the sphere, that was replaced. Clump +masses and inertia are adapted automatically (for details see clump()). +O.bodies.replaceByClumps( [utils.clumpTemplate([1,1],[.5,.5])] , [.9] ) #will replace +90 % of all standalone spheres by ‘dyads’ +See examples/clumps/replaceByClumps-example.py for an example script. +updateClumpProperties((BodyContainer)arg1[, (list)excludeList=[][, (int)discretization=5 +]]) → None : +Manually force Yade to update clump properties mass, volume and inertia (for details of +‘discretization’ value see clump()). Can be used, when clumps are modified or erased dur- +ing a simulation. +Clumps can be excluded from the calculation by giving a list of ids: +O.bodies.updateProperties([ids]). +useRedirection +true if the scene uses up-to-date lists for boundedBodies and realBodies; turned true auto- +matically 1/ after removal of bodies if enableRedirection=True , and 2/ in MPI execution. +(auto-updated) +InteractionContainer +class yade.wrapper.InteractionContainer +Access to interactions of simulation, by using +1. id’s of both Bodies of the interactions, e.g. O.interactions[23,65] +2. iteraction over the whole container: +for i in O.interactions: print i.id1,i.id2 +Note: +Iteration silently skips interactions that are virtual i.e. not real. +__init__((object)arg1, (InteractionContainer)arg2) → None +all((InteractionContainer)arg1[, (bool)onlyReal=False]) → list : +Return list of all interactions. +Virtual interaction are filtered out if onlyReal=True, else +(default) it dumps the full content. +clear((InteractionContainer)arg1) → None : +Remove all interactions, and invalidate persistent collider data (if the collider supports it). +countReal((InteractionContainer)arg1) → int : +Return number of interactions that are real. +erase((InteractionContainer)arg1, (int)arg2, (int)arg3) → None : +Erase one interaction, given by id1, id2 (internally, requestErase is called – the interaction +might still exist as potential, if the Collider decides so). +374 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +eraseNonReal((InteractionContainer)arg1) → None : +Erase all interactions that are not real . +has((InteractionContainer)arg1, (int)id1, (int)id2[, (bool)onlyReal=False]) → bool : +Tell if a pair of ids id1, id2 corresponds to an existing interaction (real or not depending on +onlyReal) +nth((InteractionContainer)arg1, (int)arg2) → Interaction : +Return n-th interaction from the container (usable for picking random interaction). +The +virtual interactions are not reached. +serializeSorted +withBody((InteractionContainer)arg1, (int)arg2) → list : +Return list of real interactions of given body. +withBodyAll((InteractionContainer)arg1, (int)arg2) → list : +Return list of all (real as well as non-real) interactions of given body. +ForceContainer +class yade.wrapper.ForceContainer +__init__((object)arg1, (ForceContainer)arg2) → None +addF((ForceContainer)arg1, (int)id, (Vector3)f[, (bool)permanent=False]) → None : +Apply force on body (accumulates). The force applies for one iteration, then it is reset by +ForceResetter. # permanent parameter is deprecated, instead of addF(…,permanent=True) +use setPermF(…). +addT((ForceContainer)arg1, (int)id, (Vector3)t[, (bool)permanent=False]) → None : +Apply torque on body (accumulates). The torque applies for one iteration, then it is reset by +ForceResetter. # permanent parameter is deprecated, instead of addT(…,permanent=True) +use setPermT(…). +f((ForceContainer)arg1, (int)id[, (bool)sync=False]) → Vector3 : +Resultant force on body, excluding gravity. For clumps in openMP, synchronize the force +container with sync=True, else the value will be wrong. +getPermForceUsed((ForceContainer)arg1) → bool : +Check wether permanent forces are present. +m((ForceContainer)arg1, (int)id[, (bool)sync=False]) → Vector3 : +Deprecated alias for t (torque). +permF((ForceContainer)arg1, (int)id) → Vector3 : +read the value of permanent force on body (set with setPermF()). +permT((ForceContainer)arg1, (int)id) → Vector3 : +read the value of permanent torque on body (set with setPermT()). +reset((ForceContainer)arg1[, (bool)resetAll=True]) → None : +Reset the force container, including user defined permanent forces/torques. resetAll=False +will keep permanent forces/torques unchanged. +setPermF((ForceContainer)arg1, (int)arg2, (Vector3)arg3) → None : +set the value of permanent force on body. +setPermT((ForceContainer)arg1, (int)arg2, (Vector3)arg3) → None : +set the value of permanent torque on body. +syncCount +Number of synchronizations of ForceContainer (cummulative); if significantly higher than +number of steps, there might be unnecessary syncs hurting performance. +2.3. +Yade wrapper class reference +375 + +Yade Documentation, Release 3rd ed. +t((ForceContainer)arg1, (int)id[, (bool)sync=False]) → Vector3 : +Torque applied on body. +For clumps in openMP, synchronize the force container with +sync=True, else the value will be wrong. +MaterialContainer +class yade.wrapper.MaterialContainer +Container for Materials. A material can be accessed using +1. numerical index in range(0,len(cont)), like cont[2]; +2. textual label that was given to the material, like cont[‘steel’]. +This entails traversing all +materials and should not be used frequently. +__init__((object)arg1, (MaterialContainer)arg2) → None +append((MaterialContainer)arg1, (Material)arg2) → int : +Add new shared Material; changes its id and return it. +append( (MaterialContainer)arg1, (object)arg2) -> object : Append list of Material +instances, return list of ids. +index((MaterialContainer)arg1, (str)arg2) → int : +Return id of material, given its label. +Scene +class yade.wrapper.Scene(inherits Serializable) +Object comprising the whole simulation. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +doSort(=false) +Used, when new body is added to the scene. +dt(=1e-8) +Current timestep for integration. +isPeriodic(=false) +Whether periodic boundary conditions are active. +iter(=0) +Current iteration (computational step) number +selectedBody(=-1) +Id of body that is selected by the user +speed(=0) +Current calculation speed [iter/s] +stopAtIter(=0) +Iteration after which to stop the simulation. +stopAtTime(=0) +Time after which to stop the simulation +subStep(=-1) +Number of sub-step; not to be changed directly. -1 means to run loop prologue (cell integra- +tion), 0…n-1 runs respective engines (n is number of engines), n runs epilogue (increment step +number and time. +subStepping(=false) +Whether we currently advance by one engine in every step (rather than by single run through +all engines). +376 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +tags(=uninitalized) +Arbitrary key=value associations (tags like mp3 tags: author, date, version, description etc.) +time(=0) +Simulation time (virtual time) [s] +trackEnergy(=false) +Whether energies are being traced. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +Cell +class yade.wrapper.Cell(inherits Serializable) +Parameters of periodic boundary conditions. Only applies if O.isPeriodic==True. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +getDefGrad((Cell)arg1) → Matrix3 : +Returns deformation gradient tensor F of the cell deformation (http://en.wikipedia.org/wiki/ +Finite_strain_theory) +getEulerianAlmansiStrain((Cell)arg1) → Matrix3 : +Returns Eulerian-Almansi strain tensor e = 1 +2(I − b−1) = 1 +2(I − (FFT)−1) of the cell (http: +//en.wikipedia.org/wiki/Finite_strain_theory) +getLCauchyGreenDef((Cell)arg1) → Matrix3 : +Returns left Cauchy-Green deformation tensor b = FFT of the cell (http://en.wikipedia.org/ +wiki/Finite_strain_theory) +getLagrangianStrain((Cell)arg1) → Matrix3 : +Returns Lagrangian strain tensor E = +1 +2(C − I) = +1 +2(FTF − I) = +1 +2(U2 − I) of the cell +(http://en.wikipedia.org/wiki/Finite_strain_theory) +getLeftStretch((Cell)arg1) → Matrix3 : +Returns left (spatial) stretch tensor of the cell (matrix U from polar decomposition F = RU ) +getPolarDecOfDefGrad((Cell)arg1) → tuple : +Returns orthogonal matrix R and symmetric positive semi-definite matrix U as polar decom- +position of deformation gradient F of the cell ( F = RU ) +getRCauchyGreenDef((Cell)arg1) → Matrix3 : +Returns right Cauchy-Green deformation tensor C = FTF of the cell (http://en.wikipedia.org/ +wiki/Finite_strain_theory) +getRightStretch((Cell)arg1) → Matrix3 : +Returns right (material) stretch tensor of the cell (matrix V from polar decomposition F = +RU = VR → V = FRT ) +getRotation((Cell)arg1) → Matrix3 : +Returns rotation of the cell (orthogonal matrix R from polar decomposition F = RU ) +getSmallStrain((Cell)arg1) → Matrix3 : +Returns small strain tensor ε = 1 +2(F+FT)−I of the cell (http://en.wikipedia.org/wiki/Finite_ +strain_theory) +getSpin((Cell)arg1) → Vector3 : +Returns the spin defined by the skew symmetric part of velGrad +hSize +Base cell vectors (columns of the matrix), updated at every step from velGrad (trsf accumu- +lates applied velGrad transformations). Setting hSize during a simulation is not supported +2.3. +Yade wrapper class reference +377 + +Yade Documentation, Release 3rd ed. +by most contact laws, it is only meant to be used at iteration 0 before any interactions have +been created. +hSize0 +Value of untransformed hSize, with respect to current trsf (computed as trsf ￿1 × hSize. +homoDeform(=2) +If >0, deform (velGrad) the cell homothetically by adjusting positions and velocities of bodies. +The velocity change is obtained by deriving the expression v=￿v.x, where ￿v is the macroscopic +velocity gradient, giving in an incremental form: ∆v=∆ ￿v x + ￿v ∆x. As a result, velocities +are modified as soon as velGrad changes, according to the first term: ∆v(t)=∆ ￿v x(t), while +the 2nd term reflects a convective term: ∆v’= ￿v v(t-dt/2). The second term is neglected if +homoDeform=1. All terms are included if homoDeform=2 (default) +nextVelGrad(=Matrix3r::Zero()) +see Cell.velGrad. +prevHSize(=Matrix3r::Identity()) +hSize from the previous step, used in the definition of relative velocity across periods. +prevVelGrad(=Matrix3r::Zero()) +Velocity gradient in the previous step. +refHSize(=Matrix3r::Identity()) +Reference cell configuration, only used with OpenGLRenderer.dispScale. Updated automati- +cally when hSize or trsf is assigned directly; also modified by utils.setRefSe3 (called e.g. by +the Reference button in the UI). +refSize +Reference size of the cell (lengths of initial cell vectors, i.e. column norms of hSize). +Note: +Modifying this value is deprecated, use setBox instead. +setBox((Cell)arg1, (Vector3)arg2) → None : +Set Cell shape to be rectangular, with dimensions along axes specified by given argument. +Shorthand for assigning diagonal matrix with respective entries to hSize. +setBox( (Cell)arg1, (float)arg2, (float)arg3, (float)arg4) -> None : Set Cell shape +to be rectangular, with dimensions along x, y, z specified by arguments. +Shorthand +for assigning diagonal matrix with the respective entries to hSize. +shearPt((Cell)arg1, (Vector3)arg2) → Vector3 : +Apply shear (cell skew+rot) on the point +shearTrsf +Current skew+rot transformation (no resize) +size +Current size of the cell, i.e. lengths of the 3 cell lateral vectors contained in Cell.hSize columns. +Updated automatically at every step. +trsf +Current transformation matrix of the cell, obtained from time integration of Cell.velGrad. +unshearPt((Cell)arg1, (Vector3)arg2) → Vector3 : +Apply inverse shear on the point (removes skew+rot of the cell) +unshearTrsf +Inverse of the current skew+rot transformation (no resize) +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +378 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +velGrad +Velocity gradient of the transformation; used in NewtonIntegrator. Values of velGrad accu- +mulate in trsf at every step. +note: changing velGrad at the beginning of a timestep would lead to inaccurate +integration for that step, as it should normally be changed after the contact laws +(but before Newton). To avoid this problem, assignment is deferred automatically. +The assigned value is internaly stored in Cell.nextVelGrad and will be applied right +in time by Newton integrator. +Warning: +Assigning individual components as in O.cell.velGrad[0,0]=1 is not possible +(it will not return any error but it will have no effect). Instead, the whole matrix should +be assigned, as in O.cell.velGrad=Matrix3(…). Alternatively nextVelGrad can be assigned +directly (both per-component or as a whole) and the effect should be the same. +velGradChanged(=false) +true when velGrad has been changed manually (see also Cell.nextVelGrad) +volume +Current volume of the cell. +wrap((Cell)arg1, (Vector3)arg2) → Vector3 : +Transform an arbitrary point into a point in the reference cell +wrapPt((Cell)arg1, (Vector3)arg2) → Vector3 : +Wrap point inside the reference cell, assuming the cell has no skew+rot. +2.3.16 Other classes +class yade.wrapper.TimingDeltas +data +Get timing data as list of tuples (label, execTime[nsec], execCount) (one tuple per checkpoint) +reset((TimingDeltas)arg1) → None : +Reset timing information +class yade.wrapper.Serializable +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.GlExtra_LawTester(inherits GlExtraDrawer → Serializable) +Find an instance of LawTester and show visually its data. +dead(=false) +Deactivate the object (on error/exception). +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +tester(=uninitalized) +Associated LawTester object. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +2.3. +Yade wrapper class reference +379 + +Yade Documentation, Release 3rd ed. +class yade.wrapper.MatchMaker(inherits Serializable) +Class matching pair of ids to return pre-defined (for a pair of ids defined in matches) or derived +value (computed using algo) of a scalar parameter. It can be called (id1, id2, val1=NaN, val2=NaN) +in both python and c++. +Note: +There is a converter from python number defined for this class, which creates a new +MatchMaker returning the value of that number; instead of giving the object instance therefore, +you can only pass the number value and it will be converted automatically. +algo +Algorithm used to compute value when no match for ids is found. Possible values are +• ‘avg’ (arithmetic average) +• ‘min’ (minimum value) +• ‘max’ (maximum value) +• ‘harmAvg’ (harmonic average) +The following algo algorithms do not require meaningful input values in order to work: +• ‘val’ (return value specified by val) +• ‘zero’ (always return 0.) +computeFallback((MatchMaker)arg1, (float)val1, (float)val2) → float : +Compute algo value for val1 and val2, using algorithm specified by algo. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +matches(=uninitalized) +Array of (id1,id2,value) items; queries matching id1 + id2 or id2 + id1 will return value +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +val(=NaN) +Constant value returned if there is no match and algo is val +class yade.wrapper.Engine(inherits Serializable) +Basic execution unit of simulation, called from the simulation loop (O.engines) +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +380 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.EnergyTracker(inherits Serializable) +Storage for tracing energies. Only to be used if O.trackEnergy is True. +clear((EnergyTracker)arg1) → None : +Clear all stored values. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +energies(=uninitalized) +Energy values, in linear array +items((EnergyTracker)arg1) → list : +Return contents as list of (name,value) tuples. +keys((EnergyTracker)arg1) → list : +Return defined energies. +total((EnergyTracker)arg1) → float : +Return sum of all energies. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.LinExponentialPotential(inherits CundallStrackPotential → GenericPo- +tential → Serializable) +LinExponential Potential with only Cundall-and-Strack-like contact. The LinExponential potential +formula is F(u) = k∗(xe−x0) +xe +(u/a − x0) exp +� +−(u/a) +xe−x0 +� +. Where k is the slope at the origin, x0 is the +position where the potential cross 0 and xe is the position of the extremum. +F0(=1) +Force at contact. Force when F0 = F(u = 0) (LinExponential) +Fe(=1) +Extremum force. Value of force at extremum. (LinExponential) +alpha(=1) +Bulk-to-roughness stiffness ratio +computeParametersFromF0((LinExponentialPotential)arg1, (float)F0, (float)xe, (float)k) → +None : +Set parameters of the potential, with k computed from F0 +computeParametersFromF0Fe((LinExponentialPotential)arg1, (float)xe, (float)Fe, (float)F0) +→ None : +Set parameters of the potential, with k and x0 computed from F0 and Fe +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +k(=1) +Slope at the origin (stiffness). (LinExponential) +2.3. +Yade wrapper class reference +381 + +Yade Documentation, Release 3rd ed. +potential((LinExponentialPotential)arg1, (float)u) → float : +Get potential value at any point. +setParameters((LinExponentialPotential)arg1, (float)x0, (float)xe, (float)k) → None : +Set parameters of the potential +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +x0(=0) +Equilibrium distance. Potential force is 0 at x0 (LinExponential) +xe(=1) +Extremum position. Position of local max/min of force. (LinExponential) +class yade.wrapper.CundallStrackPotential(inherits GenericPotential → Serializable) +Potential with only Cundall-and-Strack-like contact. +alpha(=1) +Bulk-to-roughness stiffness ratio +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.GlExtra_OctreeCubes(inherits GlExtraDrawer → Serializable) +Render boxed read from file +boxesFile(=uninitalized) +File to read boxes from; ascii files with x0 y0 z0 x1 y1 z1 c records, where c is an integer +specifying fill (0 for wire, 1 for filled). +dead(=false) +Deactivate the object (on error/exception). +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +fillRangeDraw(=Vector2i(-2, 2)) +Range of fill indices that will be rendered. +fillRangeFill(=Vector2i(2, 2)) +Range of fill indices that will be filled. +levelRangeDraw(=Vector2i(-2, 2)) +Range of levels that will be rendered. +noFillZero(=true) +Do not fill 0-fill boxed (those that are further subdivided) +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.ParallelEngine(inherits Engine → Serializable) +Engine for running other Engine in parallel. +__init__((object)arg1) → None +object __init__(tuple args, dict kwds) +__init__( (object)arg1, (list)arg2) -> object : Construct from (possibly nested) list +of slaves. +dead(=false) +If true, this engine will not run at all; can be used for making an engine temporarily deactivated +and only resurrect it at a later point. +382 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +execCount +Cumulative count this engine was run (only used if O.timingEnabled==True). +execTime +Cumulative +time +in +nanoseconds +this +Engine +took +to +run +(only +used +if +O.timingEnabled==True). +label(=uninitalized) +Textual label for this object; must be valid python identifier, you can refer to it directly from +python. +ompThreads(=-1) +Number of threads to be used in the engine. If ompThreads<0 (default), the number will be +typically OMP_NUM_THREADS or the number N defined by ‘yade -jN’ (this behavior can +depend on the engine though). This attribute will only affect engines whose code includes +openMP parallel regions (e.g. InteractionLoop). This attribute is mostly useful for experi- +ments or when combining ParallelEngine with engines that run parallel regions, resulting in +nested OMP loops with different number of threads at each level. +slaves +List of lists of Engines; each top-level group will be run in parallel with other groups, while +Engines inside each group will be run sequentially, in given order. +timingDeltas +Detailed information about timing inside the Engine itself. Empty unless enabled in the source +code and O.timingEnabled==True. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.Cell(inherits Serializable) +Parameters of periodic boundary conditions. Only applies if O.isPeriodic==True. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +getDefGrad((Cell)arg1) → Matrix3 : +Returns deformation gradient tensor F of the cell deformation (http://en.wikipedia.org/wiki/ +Finite_strain_theory) +getEulerianAlmansiStrain((Cell)arg1) → Matrix3 : +Returns Eulerian-Almansi strain tensor e = 1 +2(I − b−1) = 1 +2(I − (FFT)−1) of the cell (http: +//en.wikipedia.org/wiki/Finite_strain_theory) +getLCauchyGreenDef((Cell)arg1) → Matrix3 : +Returns left Cauchy-Green deformation tensor b = FFT of the cell (http://en.wikipedia.org/ +wiki/Finite_strain_theory) +getLagrangianStrain((Cell)arg1) → Matrix3 : +Returns Lagrangian strain tensor E = +1 +2(C − I) = +1 +2(FTF − I) = +1 +2(U2 − I) of the cell +(http://en.wikipedia.org/wiki/Finite_strain_theory) +getLeftStretch((Cell)arg1) → Matrix3 : +Returns left (spatial) stretch tensor of the cell (matrix U from polar decomposition F = RU ) +getPolarDecOfDefGrad((Cell)arg1) → tuple : +Returns orthogonal matrix R and symmetric positive semi-definite matrix U as polar decom- +position of deformation gradient F of the cell ( F = RU ) +getRCauchyGreenDef((Cell)arg1) → Matrix3 : +Returns right Cauchy-Green deformation tensor C = FTF of the cell (http://en.wikipedia.org/ +wiki/Finite_strain_theory) +2.3. +Yade wrapper class reference +383 + +Yade Documentation, Release 3rd ed. +getRightStretch((Cell)arg1) → Matrix3 : +Returns right (material) stretch tensor of the cell (matrix V from polar decomposition F = +RU = VR → V = FRT ) +getRotation((Cell)arg1) → Matrix3 : +Returns rotation of the cell (orthogonal matrix R from polar decomposition F = RU ) +getSmallStrain((Cell)arg1) → Matrix3 : +Returns small strain tensor ε = 1 +2(F+FT)−I of the cell (http://en.wikipedia.org/wiki/Finite_ +strain_theory) +getSpin((Cell)arg1) → Vector3 : +Returns the spin defined by the skew symmetric part of velGrad +hSize +Base cell vectors (columns of the matrix), updated at every step from velGrad (trsf accumu- +lates applied velGrad transformations). Setting hSize during a simulation is not supported +by most contact laws, it is only meant to be used at iteration 0 before any interactions have +been created. +hSize0 +Value of untransformed hSize, with respect to current trsf (computed as trsf ￿1 × hSize. +homoDeform(=2) +If >0, deform (velGrad) the cell homothetically by adjusting positions and velocities of bodies. +The velocity change is obtained by deriving the expression v=￿v.x, where ￿v is the macroscopic +velocity gradient, giving in an incremental form: ∆v=∆ ￿v x + ￿v ∆x. As a result, velocities +are modified as soon as velGrad changes, according to the first term: ∆v(t)=∆ ￿v x(t), while +the 2nd term reflects a convective term: ∆v’= ￿v v(t-dt/2). The second term is neglected if +homoDeform=1. All terms are included if homoDeform=2 (default) +nextVelGrad(=Matrix3r::Zero()) +see Cell.velGrad. +prevHSize(=Matrix3r::Identity()) +hSize from the previous step, used in the definition of relative velocity across periods. +prevVelGrad(=Matrix3r::Zero()) +Velocity gradient in the previous step. +refHSize(=Matrix3r::Identity()) +Reference cell configuration, only used with OpenGLRenderer.dispScale. Updated automati- +cally when hSize or trsf is assigned directly; also modified by utils.setRefSe3 (called e.g. by +the Reference button in the UI). +refSize +Reference size of the cell (lengths of initial cell vectors, i.e. column norms of hSize). +Note: +Modifying this value is deprecated, use setBox instead. +setBox((Cell)arg1, (Vector3)arg2) → None : +Set Cell shape to be rectangular, with dimensions along axes specified by given argument. +Shorthand for assigning diagonal matrix with respective entries to hSize. +setBox( (Cell)arg1, (float)arg2, (float)arg3, (float)arg4) -> None : Set Cell shape +to be rectangular, with dimensions along x, y, z specified by arguments. +Shorthand +for assigning diagonal matrix with the respective entries to hSize. +shearPt((Cell)arg1, (Vector3)arg2) → Vector3 : +Apply shear (cell skew+rot) on the point +shearTrsf +Current skew+rot transformation (no resize) +384 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +size +Current size of the cell, i.e. lengths of the 3 cell lateral vectors contained in Cell.hSize columns. +Updated automatically at every step. +trsf +Current transformation matrix of the cell, obtained from time integration of Cell.velGrad. +unshearPt((Cell)arg1, (Vector3)arg2) → Vector3 : +Apply inverse shear on the point (removes skew+rot of the cell) +unshearTrsf +Inverse of the current skew+rot transformation (no resize) +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +velGrad +Velocity gradient of the transformation; used in NewtonIntegrator. Values of velGrad accu- +mulate in trsf at every step. +note: changing velGrad at the beginning of a timestep would lead to inaccurate +integration for that step, as it should normally be changed after the contact laws +(but before Newton). To avoid this problem, assignment is deferred automatically. +The assigned value is internaly stored in Cell.nextVelGrad and will be applied right +in time by Newton integrator. +Warning: +Assigning individual components as in O.cell.velGrad[0,0]=1 is not possible +(it will not return any error but it will have no effect). Instead, the whole matrix should +be assigned, as in O.cell.velGrad=Matrix3(…). Alternatively nextVelGrad can be assigned +directly (both per-component or as a whole) and the effect should be the same. +velGradChanged(=false) +true when velGrad has been changed manually (see also Cell.nextVelGrad) +volume +Current volume of the cell. +wrap((Cell)arg1, (Vector3)arg2) → Vector3 : +Transform an arbitrary point into a point in the reference cell +wrapPt((Cell)arg1, (Vector3)arg2) → Vector3 : +Wrap point inside the reference cell, assuming the cell has no skew+rot. +class yade.wrapper.CundallStrackAdhesivePotential(inherits +CundallStrackPotential +→ +GenericPotential → Serializable) +CundallStrack model with adhesive part. Contact is created when u/a − ε < 0 and released when +u/a − ε > ladh, where ladh = fadh/kn. This lead to an hysteretic attractive part. +alpha(=1) +Bulk-to-roughness stiffness ratio +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +fadh(=0) +Adhesion force. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.GenericPotential(inherits Serializable) +Generic class for potential representation in PotentialLubrication law. Don’t do anything. If set +as potential, the result will be a lubrication-only simulation. +2.3. +Yade wrapper class reference +385 + +Yade Documentation, Release 3rd ed. +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +class yade.wrapper.GlExtraDrawer(inherits Serializable) +Performing arbitrary OpenGL drawing commands; called from OpenGLRenderer (see OpenGLRen- +derer.extraDrawers) once regular rendering routines will have finished. +This class itself does not render anything, derived classes should override the render method. +dead(=false) +Deactivate the object (on error/exception). +dict((Serializable)arg1) → dict : +Return dictionary of attributes. +updateAttrs((Serializable)arg1, (dict)arg2) → None : +Update object attributes from given dictionary +2.4 Yade modules reference +2.4.1 yade.bodiesHandling module +Miscellaneous functions, which are useful for handling bodies. +yade.bodiesHandling.facetsDimensions(idFacets=[], mask=-1) +The function accepts the list of facet id’s or list of facets and calculates max and min dimensions, +geometrical center. +Parameters +• idFacets (list) – list of spheres +• mask (int) – Body.mask for the checked bodies +Returns dictionary with keys min (minimal dimension, Vector3), max (maximal dimen- +sion, Vector3), minId (minimal dimension facet Id, Vector3), maxId (maximal dimen- +sion facet Id, Vector3), center (central point of bounding box, Vector3), extends +(sizes of bounding box, Vector3), number (number of facets, int), +yade.bodiesHandling.sphereDuplicate(idSphere) +The functions makes a copy of sphere +yade.bodiesHandling.spheresModify(idSpheres=[], mask=-1, shift=Vector3(0, 0, 0), scale=1.0, +orientation=Quaternion((1, 0, 0), 0), copy=False) +The function accepts the list of spheres id’s or list of bodies and modifies them: rotating, scaling, +shifting. if copy=True copies bodies and modifies them. Also the mask can be given. If idSpheres +not empty, the function affects only bodies, where the mask passes. If idSpheres is empty, the +function search for bodies, where the mask passes. +Parameters +• shift (Vector3) – Vector3(X,Y,Z) parameter moves spheres. +• scale (float) – factor scales given spheres. +• orientation (Quaternion) – orientation of spheres +• mask (int) – Body.mask for the checked bodies +Returns list of bodies if copy=True, and Boolean value if copy=False +386 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +yade.bodiesHandling.spheresPackDimensions(idSpheres=[], mask=-1) +The function accepts the list of spheres id’s or list of bodies and calculates max and min dimensions, +geometrical center. +Parameters +• idSpheres (list) – list of spheres +• mask (int) – Body.mask for the checked bodies +Returns dictionary with keys min (minimal dimension, Vector3), max (maximal di- +mension, Vector3), minId (minimal dimension sphere Id, Vector3), maxId (maximal +dimension sphere Id, Vector3), center (central point of bounding box, Vector3), +extends (sizes of bounding box, Vector3), volume (volume of spheres, Real), mass +(mass of spheres, Real), number (number of spheres, int), +2.4.2 yade.export module +Export (not only) geometry to various formats. +class yade.export.VTKExporter(inherits object) +Class for exporting data to VTK Simple Legacy File (for example if, for some reason, you are not +able to use VTKRecorder). Supported export of: +• spheres +• facets +• polyhedra +• PotentialBlocks +• interactions +• contact points +• periodic cell +Usage: +• create object vtkExporter = VTKExporter('baseFileName'), +• add to O.engines a PyRunner with command='vtkExporter.exportSomething(...)' +• alternatively, just use vtkExporter.exportSomething(...) at the end of the script for in- +stance +Example: +examples/test/vtk-exporter/vtkExporter.py, +examples/test/unv- +read/unvReadVTKExport.py. +Parameters +• baseName (string) – name of the exported files. The files would be named, e.g., +baseName-spheres-snapNb.vtk or baseName-facets-snapNb.vtk +• startSnap (int) – the numbering of files will start form startSnap +exportContactPoints(ids=’all’, +what={}, +useRef={}, +comment=’comment’, +numLa- +bel=None) +exports contact points (CPs) and defined properties. +Parameters +• ids ([(int,int)]) – see exportInteractions() +• what (dictionary) – see exportInteractions() +• useRef (bool) – see exportInteractions() +• comment (string) – comment to add to vtk file +2.4. +Yade modules reference +387 + +Yade Documentation, Release 3rd ed. +• numLabel (int) – number of file (e.g. time step), if unspecified, the last used +value + 1 will be used +exportFacets(ids=’all’, what={}, comment=’comment’, numLabel=None) +exports facets (positions) and defined properties. Facets are exported with multiplicated nodes +Parameters +• ids ([int]|"all") – if “all”, then export all facets, otherwise only facets from +integer list +• what (dictionary) – see exportSpheres() +• comment (string) – comment to add to vtk file +• numLabel (int) – number of file (e.g. time step), if unspecified, the last used +value + 1 will be used +exportFacetsAsMesh(ids=’all’, connectivityTable=None, what={}, comment=’comment’, +numLabel=None) +exports facets (positions) and defined properties. +Facets are exported as mesh (not with +multiplicated nodes). Therefore additional parameters connectivityTable is needed +Parameters +• ids ([int]|"all") – if “all”, then export all facets, otherwise only facets from +integer list +• what (dictionary) – see exportSpheres() +• comment (string) – comment to add to vtk file +• numLabel (int) – number of file (e.g. time step), if unspecified, the last used +value + 1 will be used +• nodes ([(float,float,float)|Vector3]) – list of coordinates of nodes +• connectivityTable ([(int,int,int)]) – list of node ids of individual ele- +ments (facets) +exportInteractions(ids=’all’, what={}, verticesWhat={}, comment=’comment’, numLa- +bel=None, useRef=False) +exports interactions and defined properties. +Parameters +• ids ([(int,int)]|"all") – if “all”, then export all interactions, otherwise +only interactions from (int,int) list +• what (dictionary) – what to export. parameter is a name->command dic- +tionary. Name is string under which it is saved to vtk, command is string to +evaluate. Note that the interactions are labeled as i in this function. Scalar, +vector and tensor variables are supported. For example, to export the stiff- +ness difference (named as dStiff) from a certain value (1e9) you should write: +what=dict(dStiff='i.phys.kn-1e9', ... ) +• verticesWhat (dictionary) – what to export on connected bodies. Bodies +are labeled as b (or b1 and b2 if you need to treat both bodies differently) +• comment (string) – comment to add to vtk file +• numLabel (int) – number of file (e.g. time step), if unspecified, the last used +value + 1 will be used +• useRef (bool) – if False (default), use current position of the bodies for export, +use reference position otherwise +exportPeriodicCell(comment=’comment’, numLabel=None) +exports the Cell geometry for periodic simulations. +388 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +Parameters +• comment (string) – comment to add to vtk file +• numLabel (int) – number of file (e.g. time step), if unspecified, the last used +value + 1 will be used +exportPolyhedra(ids=’all’, +what={}, +comment=’comment’, +numLabel=None, +useRef=False) +Exports polyhedrons and defined properties. +Parameters +• ids ([int] | "all") – if “all”, then export all polyhedrons, otherwise only +polyhedrons from integer list +• what (dictionary) – which additional quantities (in addition to the posi- +tions) to export. parameter is name->command dictionary. Name is string +under which it is saved to vtk, command is string to evaluate. +Note that +the bodies are labeled as b in this function. +Scalar, vector and tensor +variables are supported. +For example, to export velocity (named as parti- +cleVelocity) and the distance from point (0,0,0) (named as dist) you should +write: what=dict(particleVelocity='b.state.vel',dist='b.state.pos. +norm()', ... ) +• comment (string) – comment to add to vtk file +• numLabel (int) – number of file (e.g. time step), if unspecified, the last used +value + 1 will be used +exportPotentialBlocks(ids=’all’, +what={}, +comment=’comment’, +numLabel=None, +useRef=False) +Exports Potential Blocks and defined properties. +Parameters +• ids ([int] | "all") – if “all”, then export all Potential Blocks, otherwise +only Potential Blocks from integer list +• what (dictionary) – which additional quantities (in addition to the posi- +tions) to export. parameter is name->command dictionary. Name is string +under which it is saved to vtk, command is string to evaluate. +Note that +the bodies are labeled as b in this function. +Scalar, vector and tensor +variables are supported. +For example, to export velocity (named as parti- +cleVelocity) and the distance from point (0,0,0) (named as dist) you should +write: what=dict(particleVelocity='b.state.vel',dist='b.state.pos. +norm()', ... ) +• comment (string) – comment to add to vtk file +• numLabel (int) – number of file (e.g. time step), if unspecified, the last used +value + 1 will be used +exportSpheres(ids=’all’, what={}, comment=’comment’, numLabel=None, useRef=False) +exports spheres (positions and radius) and defined properties. +Parameters +• ids ([int]|"all") – if “all”, then export all spheres, otherwise only spheres +from integer list +• what (dictionary) – which additional quantities (other than the position and +the radius) to export. +parameter is name->command dictionary. +Name is +string under which it is save to vtk, command is string to evaluate. +Note +that the bodies are labeled as b in this function. Scalar, vector and tensor +variables are supported. For example, to export velocity (with name parti- +cleVelocity) and the distance form point (0,0,0) (named as dist) you should +2.4. +Yade modules reference +389 + +Yade Documentation, Release 3rd ed. +write: what=dict(particleVelocity='b.state.vel',dist='b.state.pos. +norm()', ... ) +• comment (string) – comment to add to vtk file +• numLabel (int) – number of file (e.g. time step), if unspecified, the last used +value + 1 will be used +• useRef (bool) – if False (default), use current position of the spheres for export, +use reference position otherwise +class yade.export.VTKWriter(inherits object) +USAGE: create object vtk_writer = VTKWriter(‘base_file_name’), add to engines PyRunner with +command=’vtk_writer.snapshot()’ +snapshot() +yade.export.gmshGeo(filename, comment=”, mask=-1, accuracy=-1) +Save spheres in geo-file for the following using in GMSH (http://www.geuz.org/gmsh/doc/texinfo/) +program. The spheres can be there meshed. +Parameters +• filename (string) – the name of the file, where sphere coordinates will be +exported. +• mask (int) – export only spheres with the corresponding mask export only +spheres with the corresponding mask +• accuracy (float) – the accuracy parameter, which will be set for the poinst in +geo-file. By default: 1./10. of the minimal sphere diameter. +Returns number of spheres which were exported. +Return type int +yade.export.text(filename, mask=-1) +Save sphere coordinates into a text file; the format of the line is: x y z r. Non-spherical bodies are +silently skipped. Example added to examples/regular-sphere-pack/regular-sphere-pack.py +Parameters +• filename (string) – the name of the file, where sphere coordinates will be +exported. +• mask (int) – export only spheres with the corresponding mask +Returns number of spheres which were written. +Return type int +yade.export.text2vtk(inFileName, outFileName, comment=’comment’) +Converts text file (created by export.textExt function) into vtk file. See examples/test/paraview- +spheres-solid-section/export_text.py example +Parameters +• inFileName (str) – name of input text file +• outFileName (str) – name of output vtk file +• comment (str) – optional comment in vtk file +yade.export.text2vtkSection(inFileName, outFileName, point, normal=(1, 0, 0)) +Converts section through spheres from text file (created by export.textExt function) into vtk file. +See examples/test/paraview-spheres-solid-section/export_text.py example +Parameters +• inFileName (str) – name of input text file +• outFileName (str) – name of output vtk file +390 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +• point (Vector3|(float,float,float)) – coordinates of a point lying on the +section plane +• normal (Vector3|(float,float,float)) – normal vector of the section plane +yade.export.textClumps(filename, format=’x_y_z_r_clumpId’, comment=”, mask=-1) +Save clumps-members into a text file. Non-clumps members are bodies are silently skipped. +Parameters +• filename (string) – the name of the file, where sphere coordinates will be +exported. +• comment (string) – the text, which will be added as a comment at the top of +file. If you want to create several lines of text, please use ‘\n#’ for next lines. +• mask (int) – export only spheres with the corresponding mask export only +spheres with the corresponding mask +Returns number of clumps, number of spheres which were written. +Return type int +yade.export.textExt(filename, format=’x_y_z_r’, comment=”, mask=-1, attrs=[]) +Save sphere coordinates and other parameters into a text file in specific format. Non-spherical +bodies are silently skipped. Users can add here their own specific format, giving meaningful names. +The first file row will contain the format name. Be sure to add the same format specification in +ymport.textExt. +Parameters +• filename (string) – the name of the file, where sphere coordinates will be +exported. +• format (string) – the name of output format. Supported ‘x_y_z_r’(default), +‘x_y_z_r_matId’, ‘x_y_z_r_attrs’ (use proper comment) +• comment (string) – the text, which will be added as a comment at the top of +file. If you want to create several lines of text, please use ‘\n#’ for next lines. +With ‘x_y_z_r_attrs’ format, the last (or only) line should consist of column +headers of quantities passed as attrs (1 comment word for scalars, 3 comment +words for vectors and 9 comment words for matrices) +• mask (int) – export only spheres with the corresponding mask export only +spheres with the corresponding mask +• attrs ([str]) – attributes to be exported with ‘x_y_z_r_attrs’ format. +Each str in the list is evaluated for every body exported with body=b (i.e. +‘b.state.pos.norm()’ would stand for distance of body from coordinate system +origin) +Returns number of spheres which were written. +Return type int +yade.export.textPolyhedra(fileName, comment=”, mask=-1, explanationComment=True, at- +trs=[]) +Save polyhedra into a text file. Non-polyhedra bodies are silently skipped. +Parameters +• filename (string) – the name of the output file +• comment (string) – the text, which will be added as a comment at the top of +file. If you want to create several lines of text, please use ‘\n#’ for next lines. +• mask (int) – export only polyhedra with the corresponding mask +• explanationComment (str) – inclde explanation of format to the beginning of +file +2.4. +Yade modules reference +391 + +Yade Documentation, Release 3rd ed. +Returns number of polyhedra which were written. +Return type int +2.4.3 yade.geom module +Creates geometry objects from facets. +yade.geom.facetBox(center, extents, orientation=Quaternion((1, 0, 0), 0), wallMask=63, **kw) +Create arbitrarily-aligned box composed of facets, with given center, extents and orientation. If +any of the box dimensions is zero, corresponding facets will not be created. The facets are oriented +outwards from the box. +Parameters +• center (Vector3) – center of the box +• extents (Vector3) – half lengths of the box sides +• orientation (Quaternion) – orientation of the box +• wallMask (bitmask) – determines which walls will be created, in the order -x +(1), +x (2), -y (4), +y (8), -z (16), +z (32). The numbers are ANDed; the +default 63 means to create all walls +• **kw – (unused keyword arguments) passed to utils.facet +Returns list of facets forming the box +yade.geom.facetBunker(center, dBunker, dOutput, hBunker, hOutput, hPipe=0.0, orienta- +tion=Quaternion((1, 0, 0), 0), segmentsNumber=10, wallMask=4, an- +gleRange=None, closeGap=False, **kw) +Create arbitrarily-aligned bunker, composed of facets, with given center, radii, heights and orien- +tation. Return List of facets forming the bunker; +dBunker +______________ +| +| +| +| +| +| hBunker +| +| +| +| +| +| +|____________| +\ +/ +\ +/ +\ +/ +hOutput +\ +/ +\____/ +| +| +|____| +hPipe +dOutput +Parameters +• center (Vector3) – center of the created bunker +• dBunker (float) – bunker diameter, top +• dOutput (float) – bunker output diameter +• hBunker (float) – bunker height +• hOutput (float) – bunker output height +• hPipe (float) – bunker pipe height +392 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +• orientation (Quaternion) – orientation of the bunker; the reference orientation +has axis along the +x axis. +• segmentsNumber (int) – number of edges on the bunker surface (>=5) +• wallMask (bitmask) – determines which walls will be created, in the order up +(1), down (2), side (4). The numbers are ANDed; the default 7 means to create +all walls +• angleRange ((ϑmin,Θmax)) – allows one to create only part of bunker by spec- +ifying range of angles; if None, (0,2*pi) is assumed. +• closeGap (bool) – close range skipped in angleRange with triangular facets at +cylinder bases. +• **kw – (unused keyword arguments) passed to utils.facet; +yade.geom.facetCone(center, +radiusTop, +radiusBottom, +height, +orientation=Quaternion((1, +0, +0), +0), +segmentsNumber=10, +wallMask=7, +angleRange=None, +closeGap=False, radiusTopInner=-1, radiusBottomInner=-1, **kw) +Create arbitrarily-aligned cone composed of facets, with given center, radius, height and orientation. +Return List of facets forming the cone; +Parameters +• center (Vector3) – center of the created cylinder +• radiusTop (float) – cone top radius +• radiusBottom (float) – cone bottom radius +• radiusTopInner (float) – inner radius of cones top, -1 by default +• radiusBottomInner (float) – inner radius of cones bottom, -1 by default +• height (float) – cone height +• orientation (Quaternion) – orientation of the cone; the reference orientation +has axis along the +x axis. +• segmentsNumber (int) – number of edges on the cone surface (>=5) +• wallMask (bitmask) – determines which walls will be created, in the order up +(1), down (2), side (4). The numbers are ANDed; the default 7 means to create +all walls +• angleRange ((ϑmin,Θmax)) – allows one to create only part of cone by specifying +range of angles; if None, (0,2*pi) is assumed. +• closeGap (bool) – close range skipped in angleRange with triangular facets at +cylinder bases. +• **kw – (unused keyword arguments) passed to utils.facet; +yade.geom.facetCylinder(center, radius, height, orientation=Quaternion((1, 0, 0), 0), seg- +mentsNumber=10, wallMask=7, angleRange=None, closeGap=False, +radiusTopInner=-1, radiusBottomInner=-1, **kw) +Create arbitrarily-aligned cylinder composed of facets, with given center, radius, height and orien- +tation. Return List of facets forming the cylinder; +Parameters +• center (Vector3) – center of the created cylinder +• radius (float) – cylinder radius +• height (float) – cylinder height +• radiusTopInner (float) – inner radius of cylinders top, -1 by default +• radiusBottomInner (float) – inner radius of cylinders bottom, -1 by default +2.4. +Yade modules reference +393 + +Yade Documentation, Release 3rd ed. +• orientation (Quaternion) – orientation of the cylinder; the reference orienta- +tion has axis along the +x axis. +• segmentsNumber (int) – number of edges on the cylinder surface (>=5) +• wallMask (bitmask) – determines which walls will be created, in the order up +(1), down (2), side (4). The numbers are ANDed; the default 7 means to create +all walls +• angleRange ((ϑmin,Θmax)) – allows one to create only part of bunker by spec- +ifying range of angles; if None, (0,2*pi) is assumed. +• closeGap (bool) – close range skipped in angleRange with triangular facets at +cylinder bases. +• **kw – (unused keyword arguments) passed to utils.facet; +yade.geom.facetCylinderConeGenerator(center, radiusTop, height, orientation=Quaternion((1, +0, 0), 0), segmentsNumber=10, wallMask=7, an- +gleRange=None, closeGap=False, radiusBottom=-1, +radiusTopInner=-1, radiusBottomInner=-1, **kw) +Please, do not use this function directly! Use geom.facetCylinder and geom.facetCone instead. +This is the base function for generating cylinders and cones from facets. +Parameters +• radiusTop (float) – top radius +• radiusBottom (float) – bottom radius +• **kw – (unused keyword arguments) passed to utils.facet; +yade.geom.facetHelix(center, radiusOuter, pitch, orientation=Quaternion((1, 0, 0), 0), seg- +mentsNumber=10, angleRange=None, radiusInner=0, **kw) +Create arbitrarily-aligned helix composed of facets, with given center, radius (outer and inner), +pitch and orientation. Return List of facets forming the helix; +Parameters +• center (Vector3) – center of the created cylinder +• radiusOuter (float) – outer radius +• radiusInner (float) – inner height (can be 0) +• orientation (Quaternion) – orientation of the helix; the reference orientation +has axis along the +x axis. +• segmentsNumber (int) – number of edges on the helix surface (>=3) +• angleRange ((ϑmin,Θmax)) – range of angles; if None, (0,2*pi) is assumed. +• **kw – (unused keyword arguments) passed to utils.facet; +yade.geom.facetParallelepiped(center, extents, height, orientation=Quaternion((1, 0, 0), 0), +wallMask=63, **kw) +Create arbitrarily-aligned Parallelepiped composed of facets, with given center, extents, height and +orientation. If any of the parallelepiped dimensions is zero, corresponding facets will not be created. +The facets are oriented outwards from the parallelepiped. +Parameters +• center (Vector3) – center of the parallelepiped +• extents (Vector3) – half lengths of the parallelepiped sides +• height (Real) – height of the parallelepiped (along axis z) +• orientation (Quaternion) – orientation of the parallelepiped +394 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +• wallMask (bitmask) – determines which walls will be created, in the order -x +(1), +x (2), -y (4), +y (8), -z (16), +z (32). The numbers are ANDed; the +default 63 means to create all walls +• **kw – (unused keyword arguments) passed to utils.facet +Returns list of facets forming the parallelepiped +yade.geom.facetPolygon(center, +radiusOuter, +orientation=Quaternion((1, +0, +0), +0), +seg- +mentsNumber=10, angleRange=None, radiusInner=0, **kw) +Create arbitrarily-aligned polygon composed of facets, with given center, radius (outer and inner) +and orientation. Return List of facets forming the polygon; +Parameters +• center (Vector3) – center of the created cylinder +• radiusOuter (float) – outer radius +• radiusInner (float) – inner height (can be 0) +• orientation (Quaternion) – orientation of the polygon; the reference orienta- +tion has axis along the +x axis. +• segmentsNumber (int) – number of edges on the polygon surface (>=3) +• angleRange ((ϑmin,Θmax)) – allows one to create only part of polygon by +specifying range of angles; if None, (0,2*pi) is assumed. +• **kw – (unused keyword arguments) passed to utils.facet; +yade.geom.facetPolygonHelixGenerator(center, +radiusOuter, +pitch=0, +orienta- +tion=Quaternion((1, 0, 0), 0), segmentsNumber=10, +angleRange=None, radiusInner=0, **kw) +Please, do not use this function directly! Use geom.facetPloygon and geom.facetHelix instead. This +is the base function for generating polygons and helixes from facets. +yade.geom.facetSphere(center, +radius, +thetaResolution=8, +phiResolution=8, +returnEle- +mentMap=False, **kw) +Create arbitrarily-aligned sphere composed of facets, with given center, radius and orientation. +Return List of facets forming the sphere. Parameters inspired by ParaView sphere glyph +Parameters +• center (Vector3) – center of the created sphere +• radius (float) – sphere radius +• thetaResolution (int) – number of facets around “equator” +• phiResolution (int) – number of facets between “poles” + 1 +• returnElementMap (bool) – returns also tuple of nodes ((x1,y1,z1),(x2,y2,z2),…) +and elements ((id01,id02,id03),(id11,id12,id13),…) if true, only facets otherwise +• **kw – (unused keyword arguments) passed to utils.facet; +2.4.4 yade.gridpfacet module +Helper functions for creating cylinders, grids and membranes. For more details on this type of elements +see [Effeindzourou2016], [Effeindzourou2015a], [Bourrier2013],. +For examples using GridConnections, see +• examples/grids/CohesiveGridConnectionSphere.py +• examples/grids/GridConnection_Spring.py +• examples/grids/Simple_Grid_Falling.py +2.4. +Yade modules reference +395 + +Yade Documentation, Release 3rd ed. +• examples/grids/Simple_GridConnection_Falling.py +For examples using PFacets, see +• examples/pfacet/gts-pfacet.py +• examples/pfacet/mesh-pfacet.py +• examples/pfacet/pfacetcreators.py +yade.gridpfacet.chainedCylinder(begin=Vector3(0, 0, 0), end=Vector3(1, 0, 0), radius=0.2, +dynamic=None, fixed=False, wire=False, color=None, high- +light=False, material=-1, mask=1) +Create and connect a chainedCylinder with given parameters. The shape generated by repeted +calls of this function is the Minkowski sum of polyline and sphere. +Parameters +• radius (Real) – radius of sphere in the Minkowski sum. +• begin (Vector3) – first point positioning the line in the Minkowski sum +• last (Vector3) – last point positioning the line in the Minkowski sum +In order to build a correct chain, last point of element of rank N must correspond to first point of +element of rank N+1 in the same chain (with some tolerance, since bounding boxes will be used to +create connections. +Returns Body object with the ChainedCylinder shape. +Note: +ChainedCylinder is deprecated and will be removed in the future, use GridConnection +instead. See gridpfacet.cylinder and gridpfacet.cylinderConnection. +yade.gridpfacet.cylinder(begin=Vector3(0, +0, +0), +end=Vector3(1, +0, +0), +radius=0.2, +nodesIds=[], cylIds=[], dynamic=None, fixed=False, wire=False, +color=None, +highlight=False, +intMaterial=-1, +extMaterial=-1, +mask=1) +Create a cylinder with given parameters. The shape corresponds to the Minkowski sum of line- +segment and sphere, hence, the cylinder has rounded vertices. The cylinder (GridConnection) and +its corresponding nodes (yref:GridNodes) are automatically added to the simulation. +The lists with nodes and cylinder ids will be updated automatically. +Parameters +• begin (Vector3) – first point of the Minkowski sum in the global coordinate +system. +• end (Vector3) – last point of the Minkowski sum in the global coordinate system. +• radius (Real) – radius of sphere in the Minkowski sum. +• nodesIds (list) – list with ids of already existing GridNodes. New ids will be +added. +• cylIds (list) – list with ids of already existing GridConnections. New id will +be added. +• intMaterial – Body.material used to create the interaction physics between the +two GridNodes +• extMaterial – Body.material used to create the interaction physics between the +Cylinder (GridConnection) and other bodies (e.g., spheres interaction with the +cylinder) +See utils.sphere’s documentation for meaning of other parameters. +396 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +yade.gridpfacet.cylinderConnection(vertices, +radius=0.2, +nodesIds=[], +cylIds=[], +dy- +namic=None, +fixed=False, +wire=False, +color=None, +highlight=False, +intMaterial=-1, +extMaterial=-1, +mask=1) +Create a chain of cylinders with given parameters. The cylinders (GridConnection) and its cor- +responding nodes (yref:GridNodes) are automatically added to the simulation. The +lists with nodes and cylinder ids will be updated automatically. +Parameters vertices ([Vector3]) – coordinates of vertices to connect in the global +coordinate system. +See gridpfacet.cylinder documentation for meaning of other parameters. +yade.gridpfacet.gmshPFacet(meshfile=’file.mesh’, +shift=Vector3(0, +0, +0), +scale=1.0, +ori- +entation=Quaternion((1, +0, +0), +0), +radius=1.0, +wire=True, +fixed=True, materialNodes=-1, material=-1, color=None) +Imports mesh geometry from .mesh file and automatically creates connected PFacet elements. For +an example see examples/pfacet/mesh-pfacet.py. +Parameters +• filename (string) – .gts file to read. +• shift ([float,float,float]) – [X,Y,Z] parameter shifts the mesh. +• scale (float) – factor scales the mesh. +• orientation (quaternion) – orientation of the imported geometry. +• radius (float) – radius used to create the PFacets. +• materialNodes – specify Body.material of GridNodes. This material is used to +make the internal connections. +• material – specify Body.material of PFacets. This material is used for interac- +tions with external bodies. +See documentation of utils.sphere for meaning of other parameters. +Returns lists of GridNode ids nodesIds, GridConnection ids cylIds, and PFacet ids pfIds +mesh files can easily be created with GMSH. +Additional examples of mesh-files can be downloaded from http://www-roc.inria.fr/gamma/ +download/download.php +yade.gridpfacet.gridConnection(id1, id2, radius, wire=False, color=None, highlight=False, +material=-1, mask=1, cellDist=None) +Create a GridConnection by connecting two GridNodes. +Parameters +• id1,id2 – the two GridNodes forming the cylinder. +• radius (float) – radius of the cylinder. Note that the radius needs to be the +same as the one for the GridNodes. +• cellDist (Vector3) – for periodic boundary conditions, see Interaction.cellDist. +Note: +periodic boundary conditions for gridConnections are not yet imple- +mented! +See documentation of utils.sphere for meaning of other parameters. +Returns Body object with the GridConnection shape. +Note: +The material of the GridNodes will be used to set the constitutive behaviour of the internal +connection, i.e., the constitutive behaviour of the cylinder. The material of the GridConnection is +used for interactions with other (external) bodies. +2.4. +Yade modules reference +397 + +Yade Documentation, Release 3rd ed. +yade.gridpfacet.gridNode(center, +radius, +dynamic=None, +fixed=False, +wire=False, +color=None, highlight=False, material=-1) +Create a GridNode which is needed to set up GridConnections. +See documentation of utils.sphere for meaning of parameters. +Returns Body object with the gridNode shape. +yade.gridpfacet.gtsPFacet(meshfile, shift=Vector3(0, 0, 0), scale=1.0, radius=1, wire=True, +fixed=True, materialNodes=-1, material=-1, color=None) +Imports mesh geometry from .gts file and automatically creates connected PFacet3 elements. For +an example see examples/pfacet/gts-pfacet.py. +Parameters +• filename (string) – .gts file to read. +• shift ([float,float,float]) – [X,Y,Z] parameter shifts the mesh. +• scale (float) – factor scales the mesh. +• radius (float) – radius used to create the PFacets. +• materialNodes – specify Body.material of GridNodes. This material is used to +make the internal connections. +• material – specify Body.material of PFacets. This material is used for interac- +tions with external bodies. +See documentation of utils.sphere for meaning of other parameters. +Returns lists of GridNode ids nodesIds, GridConnection ids cylIds, and PFacet ids pfIds +yade.gridpfacet.pfacet(id1, id2, id3, wire=True, color=None, highlight=False, material=-1, +mask=1, cellDist=None) +Create a PFacet element from 3 GridNodes which are already connected via 3 GridConnections: +Parameters +• id1,id2,id3 – already with GridConnections connected GridNodes +• wire (bool) – if True, top and bottom facet are shown as skeleton; otherwise +facets are filled. +• color (Vector3-or-None) – color of the PFacet; random color will be assigned +if None. +• cellDist (Vector3) – for periodic boundary conditions, see Interaction.cellDist. +Note: periodic boundary conditions are not yet implemented for PFacets! +See documentation of utils.sphere for meaning of other parameters. +Returns Body object with the PFacet shape. +Note: +GridNodes and GridConnections need to have the same radius. This is also the radius +used to create the PFacet +yade.gridpfacet.pfacetCreator1(vertices, radius, nodesIds=[], cylIds=[], pfIds=[], wire=False, +fixed=True, materialNodes=-1, material=-1, color=None) +Create a PFacet element from 3 vertices and automatically append to simulation. The function +uses the vertices to create GridNodes and automatically checks for existing nodes. +Parameters +• vertices ([Vector3,Vector3,Vector3]) – coordinates of vertices in the global +coordinate system. +• radius (float) – radius used to create the PFacets. +398 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +• nodesIds (list) – list with ids of already existing GridNodes. New ids will be +added. +• cylIds (list) – list with ids of already existing GridConnections. New ids will +be added. +• pfIds (list) – list with ids of already existing PFacets. New ids will be added. +• materialNodes – specify Body.material of GridNodes. This material is used to +make the internal connections. +• material – specify Body.material of PFacets. This material is used for interac- +tions with external bodies. +See documentation of utils.sphere for meaning of other parameters. +yade.gridpfacet.pfacetCreator2(id1, +id2, +vertex, +radius, +nodesIds=[], +wire=True, +materialNodes=-1, material=-1, color=None, fixed=True) +Create a PFacet element from 2 already existing and connected GridNodes and one vertex. The +element is automatically appended to the simulation. +Parameters +• id1,id2 (int) – ids of already with GridConnection connected GridNodes. +• vertex (Vector3) – coordinates of the vertex in the global coordinate system. +See documentation of gridpfacet.pfacetCreator1 for meaning of other parameters. +yade.gridpfacet.pfacetCreator3(id1, id2, id3, cylIds=[], pfIds=[], wire=True, material=-1, +color=None, fixed=True, mask=-1) +Create a PFacet element from 3 already existing GridNodes which are not yet connected. The +element is automatically appended to the simulation. +Parameters id1,id2,id3 (int) – id of the 3 GridNodes forming the PFacet. +See documentation of gridpfacet.pfacetCreator1 for meaning of other parameters. +yade.gridpfacet.pfacetCreator4(id1, id2, id3, pfIds=[], wire=True, material=-1, color=None, +fixed=True, mask=-1) +Create a PFacet element from 3 already existing GridConnections. The element is automatically +appended to the simulation. +Parameters id1,id2,id3 (int) – id of the 3 GridConnections forming the PFacet. +See documentation of gridpfacet.pfacetCreator1 for meaning of other parameters. +2.4.5 yade.libVersions module +The yade.libVersions module tracks versions of all libraries it was compiled with. Example usage is +as follows: +from yade.libVersions import * +if(getVersion('cgal') > (4,9,0)): +… +else: +… +To obtain a list of all libraries use the function libVersions.printAllVersions. +All libraries listed in prerequisites are detected by this module. +Note: +If we need a version of some library not listed in prerequisites, then it must also be added to +that list. +When adding a new version please have a look at these three files: +2.4. +Yade modules reference +399 + +Yade Documentation, Release 3rd ed. +1. py/_libVersions.cpp: detection of versions from #include files by C++. +2. py/libVersions.py.in: python module which is constructed by cmake during compilation. All *.in +files are processed by cmake. +3. cMake/FindMissingVersions.cmake: forced detection of library with undetectable version. +Hint: +The safest way to compare versions is to use builtin python tuple comparison e.g. if(cgalVer +> (4,9,0) and cgalVer < (5,1,1)):. +yade.libVersions.getAllVersions(rstFormat=False) +Returns str - this function returns the result of printAllVersions(rstFormat) call inside +a string variable. +yade.libVersions.getAllVersionsCmake() +This function returns library versions as provided by cmake during compilation. +Returns dictionary +in +following +format: +{ "libName" : [ (major, minor, +patchlevel) , "versionString" ] } +As an example the dict below reflects what libraries this documentation was compiled with (here +are only those detected by CMAKE): +Yade [1]: from yade.libVersions import * +Yade [2]: getAllVersionsCmake() +Out[2]: +{'cmake': [(3, 16, 3), '3.16.3'], +'compiler': [(9, 4, 0), '/usr/bin/c++ 9.4.0'], +'boost': [(1, 71, 0), '107100'], +'freeglut': [(2, 8, 1), '2.8.1'], +'python': [(3, 8, 10), '3.8.10'], +'eigen': [(3, 3, 7), '3.3.7'], +'ipython': [(7, 13, 0), '7.13.0'], +'sphinx': [(1, 8, 5), '1.8.5-final-0'], +'mpi4py': [(3, 0, 3), '3.0.3'], +'mpmath': [(1, 1, 0), '1.1.0']} +Note: +Please add here detection of other libraries when yade starts using them or if you discover +how to extract from cmake a version which I didn’t add here. +yade.libVersions.getArchitecture() +Returns string containing processor architecture name, as reported by uname -m call +or from CMAKE_HOST_SYSTEM_PROCESSOR cmake variable. +yade.libVersions.getLinuxVersion() +Returns string containing linux release and version, preferably the value of PRETTY_- +NAME from file /etc/os-release. +yade.libVersions.getVersion(libName) +This function returns the tuple (major, minor, patchlevel) with library version number. The +yade --test in file py/tests/libVersions.py tests that this version is the same as detected by cmake +and C++. If only one of those could detect the library version, then this number is used. +Parameters libName (string) – the name of the library +Returns tuple in format (major, minor, patchlevel) if libName exists. Otherwise +it returns None. +400 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +Note: +library openblas has no properly defined version in header files, this function will return +(0,0,0) for openblas. Parsing the version string would be unreliable. The mpi version detected +by cmake sometimes is different than version detected by C++, this needs further investigation. +yade.libVersions.printAllVersions(rstFormat=False) +This function prints a nicely formatted table with library versions. +Parameters rstFormat (bool) – whether to print table using the reStructuredText +formatting. Defaults to False and prints using Gitlab markdown rules so that it is +easy to paste into gitlab discussions. +As an example the table below actually reflects with what libraries this documentation was com- +piled: +Yade [1]: printAllVersions() +``` +Yade version +: +2021-11-19.git-639e121 +Yade features +: +QT5 +Yade config dir: +~/.yadeflip +Yade precision : +53 bits, 15 decimal places, without mpmath, PrecisionDouble +``` +Libraries used : +| library +| cmake +| C++ +| +| ------------- | -------------------- | ----------- | +| boost +| 107100 +| 1.71.0 +| +| cmake +| 3.16.3 +| +| +| compiler +| /usr/bin/c++ 9.4.0 +| gcc 9.4.0 +| +| eigen +| 3.3.7 +| 3.3.7 +| +| freeglut +| 2.8.1 +| +| +| gl +| +| 20190805 +| +| ipython +| 7.13.0 +| +| +| mpi4py +| 3.0.3 +| +| +| mpmath +| 1.1.0 +| +| +| python +| 3.8.10 +| 3.8.10 +| +| qglviewer +| +| 2.6.3 +| +| qt +| +| 5.12.8 +| +| sphinx +| 1.8.5-final-0 +| +| +| sqlite +| +| 3.31.1 +| +``` +Linux version +: +Ubuntu 20.04.4 LTS +Architecture +: +amd64 +Little endian +: +True +``` +Note: +For convenience at startup from yade.libVersions import printAllVersions is exe- +cuted, so that this function is readily accessible. +yade._libVersions.getAllVersionsCpp() → dict +This function returns library versions as discovered by C++ during compilation from all the +#include headers. This can be useful in debugging to detect some library .so conflicts. +Returns dictionary in folowing format: { "libName" : [ (major, minor, patch) , +"versionString" ] } +As an example the dict below reflects what libraries this documentation was compiled with (here +are only those detected by C++): +2.4. +Yade modules reference +401 + +Yade Documentation, Release 3rd ed. +Yade [1]: from yade.libVersions import * +Yade [2]: getAllVersionsCpp() +Out[2]: +{'compiler': [(9, 4, 0), 'gcc 9.4.0'], +'boost': [(1, 71, 0), '1.71.0'], +'qt': [(5, 12, 8), '5.12.8'], +'gl': [(2019, 8, 5), '20190805'], +'qglviewer': [(2, 6, 3), '2.6.3'], +'python': [(3, 8, 10), '3.8.10'], +'eigen': [(3, 3, 7), '3.3.7'], +'sqlite': [(3, 31, 1), '3.31.1'], +'vtk': [], +'cgal': [], +'suitesparse': [], +'openblas': [], +'metis': [], +'mpi': [], +'clp': [], +'coinutils': [], +'mpfr': [], +'mpc': []} +Note: +Please add here C++ detection of other libraries when yade starts using them. +2.4.6 yade.linterpolation module +Module for rudimentary support of manipulation with piecewise-linear functions (which are usually +interpolations of higher-order functions, whence the module name). Interpolation is always given as two +lists of the same length, where the x-list must be increasing. +Periodicity is supported by supposing that the interpolation can wrap from the last x-value to the first +x-value (which should be 0 for meaningful results). +Non-periodic interpolation can be converted to periodic one by padding the interpolation with constant +head and tail using the sanitizeInterpolation function. +There +is +a +c++ +template +function +for +interpolating +on +such +sequences +in +pkg/common/Engine/PartialEngine/LinearInterpolate.hpp +(stateful, +therefore +fast +for +sequential +reads). +TODO: Interpolating from within python is not (yet) supported. +yade.linterpolation.integral(x, y) +Return integral of piecewise-linear function given by points x0,x1,… and y0,y1,… +yade.linterpolation.revIntegrateLinear(I, x0, y0, x1, y1) +Helper function, returns value of integral variable x for linear function f passing through +(x0,y0),(x1,y1) such that 1. x￿[x0,x1] 2. ￿_x0^x f dx=I and raise exception if such number doesn’t +exist or the solution is not unique (possible?) +yade.linterpolation.sanitizeInterpolation(x, y, x0, x1) +Extends piecewise-linear function in such way that it spans at least the x0…x1 interval, by adding +constant padding at the beginning (using y0) and/or at the end (using y1) or not at all. +yade.linterpolation.xFractionalFromIntegral(integral, x, y) +Return x within range x0…xn such that ￿_x0^x f dx==integral. Raises error if the integral value +is not reached within the x-range. +402 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +yade.linterpolation.xFromIntegral(integralValue, x, y) +Return x such that ￿_x0^x f dx==integral. x wraps around at xn. For meaningful results, therefore, +x0 should == 0 +2.4.7 yade.log module +The yade.log module serves as an interface to yade logging framework implemented on top of boost::log. +For full documentation see debugging section. Example usage in python is as follows: +import yade.log +yade.log.setLevel('PeriTriaxController',yade.log.TRACE) +Example usage in C++ is as follows: +LOG_WARN("Something: "< higher precision math functions can be accessed in python by using the .HPn module +scope. For example: +import yade.math as mth +mth.HP2.sqrt(2) # produces square root of 2 using RealHP<2> precision +mth.sqrt(2) +# without using HPn module scope it defaults to RealHP<1> +yade.math.Real(arg) +This function is for compatibility of calls like: +g = yade.math.toHP1("-9.81"). +If yade is +compiled with default Real precision set as double, then python won’t accept string argu- +ments as numbers. +However when using higher precisions only calls yade.math.toHP1("1. +234567890123456789012345678901234567890") do not cut to the first 15 decimal places. The +calls such as yade.math.toHP1(1.234567890123456789012345678901234567890) will use default +python ￿ double conversion and will cut the number to its first 15 digits. +If you are debugging a high precision python script, and have difficulty finding places where such +cuts have happened you should use yade.math.toHP1(string) for declaring all python floating +point numbers which are physically important in the simulation. This function will throw exception +if bad conversion is about to take place. +Also see example high precision check checkGravityRungeKuttaCashKarp54.py. +yade.math.Real1(arg) +This function is for compatibility of calls like: +g = yade.math.toHP1("-9.81"). +If yade is +compiled with default Real precision set as double, then python won’t accept string argu- +ments as numbers. +However when using higher precisions only calls yade.math.toHP1("1. +234567890123456789012345678901234567890") do not cut to the first 15 decimal places. The +calls such as yade.math.toHP1(1.234567890123456789012345678901234567890) will use default +python ￿ double conversion and will cut the number to its first 15 digits. +If you are debugging a high precision python script, and have difficulty finding places where such +cuts have happened you should use yade.math.toHP1(string) for declaring all python floating +point numbers which are physically important in the simulation. This function will throw exception +if bad conversion is about to take place. +Also see example high precision check checkGravityRungeKuttaCashKarp54.py. +yade.math.degrees(arg) +Returns arg in radians converted to degrees, using yade.math.Real precision. +yade.math.degreesHP1(arg) +Returns arg in radians converted to degrees, using yade.math.Real precision. +yade.math.getRealHPCppDigits10() +Returns tuple containing amount of decimal digits supported on C++ side by Eigen +and CGAL. +yade.math.getRealHPPythonDigits10() +Returns tuple containing amount of decimal digits supported on python side by +yade.minieigenHP. +yade.math.linspace(a, b, num) +This function calls numpy.linspace(…) or mpmath.linspace(…), because numpy.linspace func- +tion does not work with mpmath. +2.4. +Yade modules reference +405 + +Yade Documentation, Release 3rd ed. +yade.math.needsMpmathAtN(N) +Parameters N – The int N value of RealHP in question. Must be N >= 1. +Returns True or False with information if using mpmath is necessary to avoid losing +precision when working with RealHP. +yade.math.radians(arg) +The default python function import math ; math.radians(arg) only works on 15 digit double +precision. If you want to carry on calculations in higher precision it is advisable to use this function +yade.math.radiansHP1(arg) instead. It uses full yade Real precision numbers. +NOTE: in the future this function may replace radians(…) function which is called in yade in +many scripts, and which in fact is a call to native python math.radians. We only need to find +the best backward compatible approach for this. The function yade.math.radiansHP1(arg) will +remain as the function which uses native yade Real precision. +yade.math.radiansHP1(arg) +The default python function import math ; math.radians(arg) only works on 15 digit double +precision. If you want to carry on calculations in higher precision it is advisable to use this function +yade.math.radiansHP1(arg) instead. It uses full yade Real precision numbers. +NOTE: in the future this function may replace radians(…) function which is called in yade in +many scripts, and which in fact is a call to native python math.radians. We only need to find +the best backward compatible approach for this. The function yade.math.radiansHP1(arg) will +remain as the function which uses native yade Real precision. +yade.math.toHP1(arg) +This function is for compatibility of calls like: +g = yade.math.toHP1("-9.81"). +If yade is +compiled with default Real precision set as double, then python won’t accept string argu- +ments as numbers. +However when using higher precisions only calls yade.math.toHP1("1. +234567890123456789012345678901234567890") do not cut to the first 15 decimal places. The +calls such as yade.math.toHP1(1.234567890123456789012345678901234567890) will use default +python ￿ double conversion and will cut the number to its first 15 digits. +If you are debugging a high precision python script, and have difficulty finding places where such +cuts have happened you should use yade.math.toHP1(string) for declaring all python floating +point numbers which are physically important in the simulation. This function will throw exception +if bad conversion is about to take place. +Also see example high precision check checkGravityRungeKuttaCashKarp54.py. +yade.math.usesHP() +Returns True if yade is using default Real precision higher than 15 digit (53 bits) +double type. +yade._math.Catalan([(int)Precision=53]) → float +Returns Real The catalan constant, exposed to python for testing of eigen numerical +traits. +yade._math.Euler([(int)Precision=53]) → float +Returns Real The Euler–Mascheroni constant, exposed to python for testing of eigen +numerical traits. +class yade._math.HP1 +AddCost = 1 +Catalan([(int)Precision=53]) → float : +Returns Real The catalan constant, exposed to python for testing of eigen numer- +ical traits. +ComplexAddCost = 2 +406 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +ComplexMulCost = 6 +ComplexReadCost = 2 +Euler([(int)Precision=53]) → float : +Returns Real The Euler–Mascheroni constant, exposed to python for testing +of +eigen numerical traits. +IsComplex = 0 +IsInteger = 0 +IsSigned = 1 +Log2([(int)Precision=53]) → float : +Returns Real natural logarithm of 2, exposed to python for testing +of eigen nu- +merical traits. +MulCost = 1 +Pi([(int)Precision=53]) → float : +Returns Real The π constant, exposed to python for testing +of eigen numerical +traits. +ReadCost = 1 +RequireInitialization = 0 +class Var +The Var class is used to test to/from python converters for arbitrary precision Real +cpl +one Complex variable to test reading from and writing to it. +val +one Real variable for testing. +abs((complex)x) → float : +Returns the Real absolute value of the Complex argument. Depending on compila- +tion options wraps ::boost::multiprecision::abs(…) or std::abs(…) function. +abs( (float)x) -> float : +return the Real absolute value of the Real argument. Depending on compilation +options wraps ::boost::multiprecision::abs(…) or std::abs(…) function. +acos((complex)x) → complex : +Returns Complex the arc-cosine of the Complex argument in radians. Depending on +compilation options wraps ::boost::multiprecision::acos(…) or std::acos(…) +function. +acos( (float)x) -> float : +return Real the arcus cosine of the argument. Depending on compilation options +wraps ::boost::multiprecision::acos(…) or std::acos(…) function. +acosh((complex)x) → complex : +Returns Complex the arc-hyperbolic cosine of the Complex argument in radians. +Depending on compilation options wraps ::boost::multiprecision::acosh(…) +or std::acosh(…) function. +acosh( (float)x) -> float : +2.4. +Yade modules reference +407 + +Yade Documentation, Release 3rd ed. +return Real the hyperbolic arcus cosine of the argument. Depending on compi- +lation options wraps ::boost::multiprecision::acosh(…) or std::acosh(…) +function. +arg((complex)x) → float : +Returns Real +the +arg +(Phase +angle +of +complex +in +radians) +of +the +Complex argument in radians. +Depending on compilation options wraps +::boost::multiprecision::arg(…) or std::arg(…) function. +asin((complex)x) → complex : +Returns Complex the arc-sine of the Complex argument in radians. Depending on +compilation options wraps ::boost::multiprecision::asin(…) or std::asin(…) +function. +asin( (float)x) -> float : +return Real the arcus sine of the argument. Depending on compilation options +wraps ::boost::multiprecision::asin(…) or std::asin(…) function. +asinh((complex)x) → complex : +Returns Complex the arc-hyperbolic sine of the Complex argument in radians. De- +pending on compilation options wraps ::boost::multiprecision::asinh(…) or +std::asinh(…) function. +asinh( (float)x) -> float : +return Real the hyperbolic arcus sine of the argument. Depending on compi- +lation options wraps ::boost::multiprecision::asinh(…) or std::asinh(…) +function. +atan((complex)x) → complex : +Returns Complex the arc-tangent of the Complex argument in radians. +De- +pending on compilation options wraps ::boost::multiprecision::atan(…) or +std::atan(…) function. +atan( (float)x) -> float : +return Real the arcus tangent of the argument. Depending on compilation op- +tions wraps ::boost::multiprecision::atan(…) or std::atan(…) function. +atan2((float)x, (float)y) → float : +Returns Real the arc tangent of y/x using the signs of the arguments x and y +to determine the correct quadrant. +Depending on compilation options wraps +::boost::multiprecision::atan2(…) or std::atan2(…) function. +atanh((complex)x) → complex : +Returns Complex the arc-hyperbolic tangent of the Complex argument in radians. +Depending on compilation options wraps ::boost::multiprecision::atanh(…) +or std::atanh(…) function. +atanh( (float)x) -> float : +return Real the hyperbolic arcus tangent of the argument. Depending on compi- +lation options wraps ::boost::multiprecision::atanh(…) or std::atanh(…) +function. +cbrt((float)x) → float : +408 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +Returns Real cubic root of the argument. Depending on compilation options wraps +::boost::multiprecision::cbrt(…) or std::cbrt(…) function. +ceil((float)x) → float : +Returns Real Computes the smallest integer value not less than arg. Depending on +compilation options wraps ::boost::multiprecision::ceil(…) or std::ceil(…) +function. +conj((complex)x) → complex : +Returns the complex conjugation a Complex argument. Depending on compilation +options wraps ::boost::multiprecision::conj(…) or std::conj(…) function. +cos((complex)x) → complex : +Returns Complex the cosine of the Complex argument in radians. Depending on +compilation options wraps ::boost::multiprecision::cos(…) or std::cos(…) +function. +cos( (float)x) -> float : +return Real the cosine of the Real argument in radians. Depending on compi- +lation options wraps ::boost::multiprecision::cos(…) or std::cos(…) func- +tion. +cosh((complex)x) → complex : +Returns Complex the hyperbolic cosine of the Complex argument in radians. De- +pending on compilation options wraps ::boost::multiprecision::cosh(…) or +std::cosh(…) function. +cosh( (float)x) -> float : +return Real the hyperbolic cosine of the Real argument in radians. Depend- +ing on compilation options wraps ::boost::multiprecision::cosh(…) or +std::cosh(…) function. +cylBesselJ((int)k, (float)x) → float : +Returns Real the Bessel Functions of the First Kind of the order k and the Real ar- +gument. See: ‘__ +defprec = 53 +dummy_precision() → float : +Returns similar to the function epsilon, but assumes that last 10% of bits con- +tain the numerical error only. +This is sometimes used by Eigen when calling +isEqualFuzzy to determine if values differ a lot or if they are vaguely close to +each other. +epsilon([(int)Precision=53]) → float : +Returns Real returns the difference between 1.0 and the next representable value +of the Real type. Wraps std::numeric_limits::epsilon() function. +epsilon( (float)x) -> float : +return Real returns the difference between 1.0 and the next representable value +of the Real type. Wraps std::numeric_limits::epsilon() function. +erf((float)x) → float : +Returns Real Computes the error function of argument. Depending on compilation +options wraps ::boost::multiprecision::erf(…) or std::erf(…) function. +2.4. +Yade modules reference +409 + +Yade Documentation, Release 3rd ed. +erfc((float)x) → float : +Returns Real +Computes +the +complementary +error +function +of +argu- +ment, +that is 1.0-erf(arg). +Depending on compilation options wraps +::boost::multiprecision::erfc(…) or std::erfc(…) function. +exp((complex)x) → complex : +Returns the base e exponential of a Complex argument. Depending on compilation +options wraps ::boost::multiprecision::exp(…) or std::exp(…) function. +exp( (float)x) -> float : +return the base e exponential of a Real argument. Depending on compilation +options wraps ::boost::multiprecision::exp(…) or std::exp(…) function. +exp2((float)x) → float : +Returns the base 2 exponential of a Real argument. Depending on compilation +options wraps ::boost::multiprecision::exp2(…) or std::exp2(…) function. +expm1((float)x) → float : +Returns the base e exponential of a Real argument minus 1.0. Depending on com- +pilation options wraps ::boost::multiprecision::expm1(…) or std::expm1(…) +function. +fabs((float)x) → float : +Returns the Real absolute value of the argument. Depending on compilation op- +tions wraps ::boost::multiprecision::abs(…) or std::abs(…) function. +factorial((int)x) → float : +Returns Real the factorial of the Real argument. See: ‘__ +floor((float)x) → float : +Returns Real Computes the largest integer value not greater than arg. +De- +pending on compilation options wraps ::boost::multiprecision::floor(…) +or std::floor(…) function. +fma((float)x, (float)y, (float)z) → float : +Returns Real - computes (x*y) + z as if to infinite precision and rounded +only once to fit the result type. +Depending on compilation options wraps +::boost::multiprecision::fma(…) or std::fma(…) function. +fmod((float)x, (float)y) → float : +Returns Real +the +floating-point +remainder +of +the +division +operation +x/y +of the arguments x and y. +Depending on compilation options wraps +::boost::multiprecision::fmod(…) or std::fmod(…) function. +frexp((float)x) → tuple : +Returns tuple of (Real,int), decomposes given floating point Real argument into +a normalized fraction and an integral power of two. Depending on compilation +options wraps ::boost::multiprecision::frexp(…) or std::frexp(…) function. +fromBits((str)bits[, (int)exp=0[, (int)sign=1]]) → float : +Parameters +• bits – str - a string containing ‘0’, ‘1’ characters. +• exp – int - the binary exponent which shifts the bits. +410 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +• sign – int - the sign, should be -1 or +1, but it is not checked. It multiplies +the result when construction from bits is finished. +Returns RealHP constructed from string containing ‘0’, ‘1’ bits. +This is for +debugging purposes, rather slow. +getDecomposedReal((float)x) → dict : +Returns dict - the dictionary with the debug information how the DecomposedReal +class sees this type. This is for debugging purposes, rather slow. Includes result +from fpclassify function call, a binary representation and other useful info. See +also fromBits. +getDemangledName() → str : +Returns string - the demangled C++ typnename of RealHP. +getDemangledNameComplex() → str : +Returns string - the demangled C++ typnename of ComplexHP. +getFloatDistanceULP((float)arg1, (float)arg2) → float : +Returns an integer value stored in RealHP, the ULP distance calculated by +boost::math::float_distance, also see Floating-point Comparison and Prof. Ka- +han paper about this topic. +Warning: +The returned value is the directed distance between two arguments, this +means that it can be negative. +getRawBits((float)x) → str : +Returns string - the raw bits in memory representing this type. Be careful: it +only checks the system endianness and either prints bytes in reverse order or +not. Does not make any attempts to further interpret the bits of: sign, exponent +or significand (on a typical x86 processor they are printed in that order), and +different processors might store them differently. It is not useful for types which +internally use a pointer because for them this function prints not the floating +point number but a pointer. This is for debugging purposes. +hasInfinityNan = True +highest([(int)Precision=53]) → float : +Returns Real +returns the +largest +finite +value +of +the +Real +type. +Wraps +std::numeric_limits::max() function. +hypot((float)x, (float)y) → float : +Returns Real the square root of the sum of the squares of x and y, without +undue overflow or underflow at intermediate stages of the computation. +De- +pending on compilation options wraps ::boost::multiprecision::hypot(…) +or std::hypot(…) function. +ilogb((float)x) → float : +Returns Real extracts the value of the unbiased exponent from the floating-point +argument arg, and returns it as a signed integer value. Depending on compilation +options wraps ::boost::multiprecision::ilogb(…) or std::ilogb(…) function. +imag((complex)x) → float : +Returns the imag part of a Complex argument. Depending on compilation options +wraps ::boost::multiprecision::imag(…) or std::imag(…) function. +isApprox((float)a, (float)b, (float)eps) → bool : +2.4. +Yade modules reference +411 + +Yade Documentation, Release 3rd ed. +Returns bool, True if a is approximately equal b with provided eps, see also here +isApproxOrLessThan((float)a, (float)b, (float)eps) → bool : +Returns bool, True if a is approximately less than or equal b with provided eps, +see also here +isEqualFuzzy((float)arg1, (float)arg2, (float)arg3) → bool : +Returns bool, True if the absolute difference between two numbers is smaller than +std::numeric_limits::epsilon() +isMuchSmallerThan((float)a, (float)b, (float)eps) → bool : +Returns bool, True if a is less than b with provided eps, see also here +isfinite((float)x) → bool : +Returns bool indicating if the Real argument is Inf. Depending on compilation op- +tions wraps ::boost::multiprecision::isfinite(…) or std::isfinite(…) func- +tion. +isinf((float)x) → bool : +Returns bool indicating if the Real argument is Inf. Depending on compilation +options wraps ::boost::multiprecision::isinf(…) or std::isinf(…) function. +isnan((float)x) → bool : +Returns bool indicating if the Real argument is NaN. Depending on compilation +options wraps ::boost::multiprecision::isnan(…) or std::isnan(…) function. +laguerre((int)n, (int)m, (float)x) → float : +Returns Real the Laguerre polynomial of the orders n, m and the Real argu- +ment. +See: +‘__ +ldexp((float)x, (int)y) → float : +Returns Multiplies a floating point value x by the number 2 raised to the exp power. +Depending on compilation options wraps ::boost::multiprecision::ldexp(…) +or std::ldexp(…) function. +lgamma((float)x) → float : +Returns Real +Computes +the +natural +logarithm +of +the +absolute +value +of +the gamma function of arg. +Depending on compilation options wraps +::boost::multiprecision::lgamma(…) or std::lgamma(…) function. +log((complex)x) → complex : +Returns the Complex natural (base e) logarithm of a complex value z with a +branch cut along the negative real axis. Depending on compilation options wraps +::boost::multiprecision::log(…) or std::log(…) function. +log( (float)x) -> float : +return the Real natural (base e) logarithm of a real value. Depending on compi- +lation options wraps ::boost::multiprecision::log(…) or std::log(…) func- +tion. +log10((complex)x) → complex : +Returns the Complex (base 10) logarithm of a complex value z with a branch +cut along the negative real axis. +Depending on compilation options wraps +::boost::multiprecision::log10(…) or std::log10(…) function. +log10( (float)x) -> float : +412 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +return the Real decimal (base 10) logarithm of a real value. Depending on com- +pilation options wraps ::boost::multiprecision::log10(…) or std::log10(…) +function. +log1p((float)x) → float : +Returns the Real natural (base e) logarithm of 1+argument. Depending on com- +pilation options wraps ::boost::multiprecision::log1p(…) or std::log1p(…) +function. +log2((float)x) → float : +Returns the Real binary (base 2) logarithm of a real value. Depending on compi- +lation options wraps ::boost::multiprecision::log2(…) or std::log2(…) func- +tion. +logb((float)x) → float : +Returns Extracts the value of the unbiased radix-independent exponent from the +floating-point argument arg, and returns it as a floating-point value. +De- +pending on compilation options wraps ::boost::multiprecision::logb(…) or +std::logb(…) function. +lowest([(int)Precision=53]) → float : +Returns Real returns the lowest (negative) finite value of the Real type. Wraps +std::numeric_limits::lowest() function. +max((float)x, (float)y) → float : +Returns Real larger of the two arguments. Depending on compilation options wraps +::boost::multiprecision::max(…) or std::max(…) function. +max_exp2 = 1024 +min((float)x, (float)y) → float : +Returns Real smaller of the two arguments. +Depending on compilation options +wraps ::boost::multiprecision::min(…) or std::min(…) function. +modf((float)x) → tuple : +Returns tuple of (Real,Real), decomposes given floating point Real into integral +and fractional parts, each having the same type and sign as x. Depending on com- +pilation options wraps ::boost::multiprecision::modf(…) or std::modf(…) +function. +polar((float)x, (float)y) → complex : +Returns Complex the polar (Complex from polar components) of the Real rho +(length), Real theta (angle) arguments in radians. Depending on compilation +options wraps ::boost::multiprecision::polar(…) or std::polar(…) function. +pow((complex)x, (complex)pow) → complex : +Returns the Complex complex arg1 raised to the Complex power arg2. Depending on +compilation options wraps ::boost::multiprecision::pow(…) or std::pow(…) +function. +pow( (float)x, (float)y) -> float : +return Real the value of base raised to the power exp. Depending on compila- +tion options wraps ::boost::multiprecision::pow(…) or std::pow(…) func- +tion. +proj((complex)x) → complex : +2.4. +Yade modules reference +413 + +Yade Documentation, Release 3rd ed. +Returns Complex the proj (projection of the complex number onto the Riemann +sphere) of the Complex argument in radians. Depending on compilation options +wraps ::boost::multiprecision::proj(…) or std::proj(…) function. +random() → float : +Returns Real a symmetric random number in interval (-1,1). Used by Eigen. +random( (float)a, (float)b) -> float : +return Real a random number in interval (a,b). Used by Eigen. +real((complex)x) → float : +Returns the real part of a Complex argument. Depending on compilation options +wraps ::boost::multiprecision::real(…) or std::real(…) function. +remainder((float)x, (float)y) → float : +Returns Real +the +IEEE +remainder +of +the +floating +point +divi- +sion +operation +x/y. +Depending +on +compilation +options +wraps +::boost::multiprecision::remainder(…) or std::remainder(…) function. +remquo((float)x, (float)y) → tuple : +Returns tuple of (Real,long), the floating-point remainder of the division oper- +ation x/y as the std::remainder() function does. Additionally, the sign and at +least the three of the last bits of x/y are returned, sufficient to determine the +octant of the result within a period. Depending on compilation options wraps +::boost::multiprecision::remquo(…) or std::remquo(…) function. +rint((float)x) → float : +Returns Rounds the floating-point argument arg to an integer value (in floating- +point format), using the current rounding mode. Depending on compilation op- +tions wraps ::boost::multiprecision::rint(…) or std::rint(…) function. +round((float)x) → float : +Returns Real the nearest integer value to arg (in floating-point format), rounding +halfway cases away from zero, regardless of the current rounding mode.. +De- +pending on compilation options wraps ::boost::multiprecision::round(…) +or std::round(…) function. +roundTrip((float)x) → float : +Returns Real returns the argument x. +Can be used to convert type to native +RealHP accuracy. +sgn((float)x) → int : +Returns int the sign of the argument: -1, 0 or 1. +sign((float)x) → int : +Returns int the sign of the argument: -1, 0 or 1. +sin((complex)x) → complex : +Returns Complex the sine of the Complex argument in radians. Depending on com- +pilation options wraps ::boost::multiprecision::sin(…) or std::sin(…) func- +tion. +sin( (float)x) -> float : +return Real the sine of the Real argument in radians. Depending on compilation +options wraps ::boost::multiprecision::sin(…) or std::sin(…) function. +sinh((complex)x) → complex : +414 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +Returns Complex the hyperbolic sine of the Complex argument in radians. +De- +pending on compilation options wraps ::boost::multiprecision::sinh(…) or +std::sinh(…) function. +sinh( (float)x) -> float : +return Real the hyperbolic sine of the Real argument in radians. +Depend- +ing on compilation options wraps ::boost::multiprecision::sinh(…) or +std::sinh(…) function. +smallest_positive() → float : +Returns Real the smallest number greater than zero. +Wraps std::numeric_lim- +its::min() +sphericalHarmonic((int)l, (int)m, (float)theta, (float)phi) → complex : +Returns Real +the +spherical +harmonic +polynomial +of +the +orders +l +(un- +signed +int), +m +(signed int) +and +the +Real +arguments +theta +and +phi. +See: +‘__ +sqrt((complex)x) → complex : +Returns the Complex square root of Complex argument. Depending on compilation +options wraps ::boost::multiprecision::sqrt(…) or std::sqrt(…) function. +sqrt( (float)x) -> float : +return Real square root of the argument. Depending on compilation options +wraps ::boost::multiprecision::sqrt(…) or std::sqrt(…) function. +squaredNorm((complex)x) → float : +Returns Real the norm (squared magnitude) of the Complex argument in radians. +Depending on compilation options wraps ::boost::multiprecision::norm(…) +or std::norm(…) function. +tan((complex)x) → complex : +Returns Complex the tangent of the Complex argument in radians. Depending on +compilation options wraps ::boost::multiprecision::tan(…) or std::tan(…) +function. +tan( (float)x) -> float : +return Real the tangent of the Real argument in radians. Depending on compi- +lation options wraps ::boost::multiprecision::tan(…) or std::tan(…) func- +tion. +tanh((complex)x) → complex : +Returns Complex the hyperbolic tangent of the Complex argument in radians. De- +pending on compilation options wraps ::boost::multiprecision::tanh(…) or +std::tanh(…) function. +tanh( (float)x) -> float : +return Real the hyperbolic tangent of the Real argument in radians. +De- +pending on compilation options wraps ::boost::multiprecision::tanh(…) +or std::tanh(…) function. +testArray() → None : +This function tests call to std::vector::data(…) function in order to extract the array. +2.4. +Yade modules reference +415 + +Yade Documentation, Release 3rd ed. +testCgalNumTraits = False +testConstants() → None : +This function tests lib/high-precision/Constants.hpp, the yade::math::ConstantsHP, for- +mer yade::Mathr constants. +tgamma((float)x) → float : +Returns Real Computes the gamma function of arg. Depending on compilation +options wraps ::boost::multiprecision::tgamma(…) or std::tgamma(…) func- +tion. +toDouble((float)x) → float : +Returns float converts Real type to double and returns a native python float. +toHP1((float)x) → float : +Returns RealHP<1> converted from argument RealHP<1> as a result of static_- +cast>(arg). +toInt((float)x) → int : +Returns int converts Real type to int and returns a native python int. +toLong((float)x) → int : +Returns int converts Real type to long int and returns a native python int. +toLongDouble((float)x) → float : +Returns float converts Real type to long double and returns a native python +float. +trunc((float)x) → float : +Returns Real the nearest integer not greater in magnitude than arg. +Depend- +ing on compilation options wraps ::boost::multiprecision::trunc(…) or +std::trunc(…) function. +yade._math.Log2([(int)Precision=53]) → float +Returns Real natural logarithm of 2, exposed to python for testing of eigen numerical +traits. +yade._math.Pi([(int)Precision=53]) → float +Returns Real The π constant, exposed to python for testing of eigen numerical traits. +class yade._math.RealHPConfig +RealHPConfig class provides information about RealHP type. +Variables +• extraStringDigits10 – this static variable allows to control how many extra +digits to use when converting to decimal strings. Assign a different value to it to +affect the string conversion done in C++ ￿ python conversions as well as in all +other conversions. Be careful, because values smaller than 3 can fail the round +trip conversion test. +• isFloat128Broken – provides runtime information if Yade was compiled with +g++ version < 9.2.1 and thus boost::multiprecision::float128 cannot work. +• isEnabledRealHP – provides runtime information RealHP is available for N +higher than 1. +• workaroundSlowBoostBinFloat – boost::multiprecision::cpp_bin_float +has some problem that importing it in python is very slow when these functions +are exported: erf, erfc, lgamma, tgamma. In such case the python import yade. +math can take more than minute. The workaround is to make them unavailable +416 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +in python for higher N values. See invocation of IfConstexprForSlowFunctions +in _math.cpp. This variable contains the highest N in which these functions are +available. It equals to highest N when boost::multiprecision::cpp_bin_- +float is not used. +extraStringDigits10 = 4 +getDigits10((int)N) → int : +This is a yade.math.RealHPConfig diagnostic function. +Parameters N – int - the value of N in RealHP. +Returns the int representing std::numeric_limits>::digits10 +getDigits2((int)N) → int : +This is a yade.math.RealHPConfig diagnostic function. +Parameters N – int - the value of N in RealHP. +Returns the int representing std::numeric_limits>::digits, which +corresponds to the number of significand bits used by this type. +getSupportedByEigenCgal() → tuple : +Returns the tuple containing N from RealHP precisions supported by Eigen and +CGAL +getSupportedByMinieigen() → tuple : +Returns the +tuple +containing +N +from +RealHP +precisions +supported +by +minieigenHP +isEnabledRealHP = False +isFloat128Broken = False +isFloat128Present = False +workaroundSlowBoostBinFloat = 1 +class yade._math.Var +The Var class is used to test to/from python converters for arbitrary precision Real +cpl +one Complex variable to test reading from and writing to it. +val +one Real variable for testing. +yade._math.abs((complex)x) → float +return the Real absolute value of the Complex argument. Depending on com- +pilation options wraps ::boost::multiprecision::abs(…) or std::abs(…) +function. +abs( (float)x) → float : +return the Real absolute value of the Real argument. Depending on compilation +options wraps ::boost::multiprecision::abs(…) or std::abs(…) function. +yade._math.acos((complex)x) → complex +return Complex the arc-cosine of the Complex argument in radians. Depend- +ing on compilation options wraps ::boost::multiprecision::acos(…) or +std::acos(…) function. +acos( (float)x) → float : +return Real the arcus cosine of the argument. Depending on compilation options +wraps ::boost::multiprecision::acos(…) or std::acos(…) function. +2.4. +Yade modules reference +417 + +Yade Documentation, Release 3rd ed. +yade._math.acosh((complex)x) → complex +return Complex +the +arc-hyperbolic +cosine +of +the +Complex +argu- +ment +in +radians. +Depending +on +compilation +options +wraps +::boost::multiprecision::acosh(…) or std::acosh(…) function. +acosh( (float)x) → float : +return Real the hyperbolic arcus cosine of the argument. Depending on compilation +options wraps ::boost::multiprecision::acosh(…) or std::acosh(…) function. +yade._math.arg((complex)x) → float +Returns Real +the +arg +(Phase +angle +of +complex +in +radians) +of +the +Complex +argument +in +radians. +Depending +on +compilation +options +wraps +::boost::multiprecision::arg(…) or std::arg(…) function. +yade._math.asin((complex)x) → complex +return Complex the arc-sine of the Complex argument in radians. +Depend- +ing on compilation options wraps ::boost::multiprecision::asin(…) or +std::asin(…) function. +asin( (float)x) → float : +return Real the arcus sine of the argument. +Depending on compilation options +wraps ::boost::multiprecision::asin(…) or std::asin(…) function. +yade._math.asinh((complex)x) → complex +return Complex +the +arc-hyperbolic +sine +of +the +Complex +argu- +ment +in +radians. +Depending +on +compilation +options +wraps +::boost::multiprecision::asinh(…) or std::asinh(…) function. +asinh( (float)x) → float : +return Real the hyperbolic arcus sine of the argument. Depending on compilation +options wraps ::boost::multiprecision::asinh(…) or std::asinh(…) function. +yade._math.atan((complex)x) → complex +return Complex the arc-tangent of the Complex argument in radians. Depend- +ing on compilation options wraps ::boost::multiprecision::atan(…) or +std::atan(…) function. +atan( (float)x) → float : +return Real the arcus tangent of the argument. Depending on compilation options +wraps ::boost::multiprecision::atan(…) or std::atan(…) function. +yade._math.atan2((float)x, (float)y) → float +Returns Real the arc tangent of y/x using the signs of the arguments x and y +to determine the correct quadrant. +Depending on compilation options wraps +::boost::multiprecision::atan2(…) or std::atan2(…) function. +yade._math.atanh((complex)x) → complex +return Complex +the +arc-hyperbolic +tangent +of +the +Complex +argu- +ment +in +radians. +Depending +on +compilation +options +wraps +::boost::multiprecision::atanh(…) or std::atanh(…) function. +atanh( (float)x) → float : +418 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +return Real the hyperbolic arcus tangent of the argument. +Depending on com- +pilation options wraps ::boost::multiprecision::atanh(…) or std::atanh(…) +function. +yade._math.cbrt((float)x) → float +Returns Real cubic root of the argument. Depending on compilation options wraps +::boost::multiprecision::cbrt(…) or std::cbrt(…) function. +yade._math.ceil((float)x) → float +Returns Real Computes the smallest integer value not less than arg. +Depending +on compilation options wraps ::boost::multiprecision::ceil(…) or std::ceil(…) +function. +yade._math.conj((complex)x) → complex +Returns the complex conjugation a Complex argument. +Depending on compilation +options wraps ::boost::multiprecision::conj(…) or std::conj(…) function. +yade._math.cos((complex)x) → complex +return Complex the cosine of the Complex argument in radians. +Depend- +ing on compilation options wraps ::boost::multiprecision::cos(…) or +std::cos(…) function. +cos( (float)x) → float : +return Real the cosine of the Real argument in radians. Depending on compilation +options wraps ::boost::multiprecision::cos(…) or std::cos(…) function. +yade._math.cosh((complex)x) → complex +return Complex +the +hyperbolic +cosine +of +the +Complex +argu- +ment +in +radians. +Depending +on +compilation +options +wraps +::boost::multiprecision::cosh(…) or std::cosh(…) function. +cosh( (float)x) → float : +return Real the hyperbolic cosine of the Real argument in radians. Depending on +compilation options wraps ::boost::multiprecision::cosh(…) or std::cosh(…) +function. +yade._math.cylBesselJ((int)k, (float)x) → float +Returns Real the Bessel Functions of the First Kind of the order k and the Real +argument. +See: ‘__ +yade._math.dummy_precision() → float +Returns similar to the function epsilon, but assumes that last 10% of bits contain the +numerical error only. This is sometimes used by Eigen when calling isEqualFuzzy +to determine if values differ a lot or if they are vaguely close to each other. +yade._math.epsilon([(int)Precision=53]) → float +return Real returns the difference between 1.0 and the next representable +value of the Real type. Wraps std::numeric_limits::epsilon() func- +tion. +epsilon( (float)x) → float : +return Real returns the difference between 1.0 and the next representable value of +the Real type. Wraps std::numeric_limits::epsilon() function. +2.4. +Yade modules reference +419 + +Yade Documentation, Release 3rd ed. +yade._math.erf((float)x) → float +Returns Real Computes the error function of argument. Depending on compilation +options wraps ::boost::multiprecision::erf(…) or std::erf(…) function. +yade._math.erfc((float)x) → float +Returns Real +Computes +the +complementary +error +function +of +argument, +that +is +1.0-erf(arg). +Depending +on +compilation +options +wraps +::boost::multiprecision::erfc(…) or std::erfc(…) function. +yade._math.exp((complex)x) → complex +return the base e exponential of a Complex argument. +Depending on com- +pilation options wraps ::boost::multiprecision::exp(…) or std::exp(…) +function. +exp( (float)x) → float : +return the base e exponential of a Real argument. Depending on compilation op- +tions wraps ::boost::multiprecision::exp(…) or std::exp(…) function. +yade._math.exp2((float)x) → float +Returns the base 2 exponential of a Real argument. Depending on compilation options +wraps ::boost::multiprecision::exp2(…) or std::exp2(…) function. +yade._math.expm1((float)x) → float +Returns the base e exponential of a Real argument minus 1.0. Depending on compi- +lation options wraps ::boost::multiprecision::expm1(…) or std::expm1(…) func- +tion. +yade._math.fabs((float)x) → float +Returns the Real absolute value of the argument. Depending on compilation options +wraps ::boost::multiprecision::abs(…) or std::abs(…) function. +yade._math.factorial((int)x) → float +Returns Real the factorial of the Real argument. See: ‘__ +yade._math.floor((float)x) → float +Returns Real Computes the largest integer value not greater than arg. Depending on +compilation options wraps ::boost::multiprecision::floor(…) or std::floor(…) +function. +yade._math.fma((float)x, (float)y, (float)z) → float +Returns Real +- +computes +(x*y) + z +as +if +to +infinite +precision +and +rounded +only once to fit the result type. +Depending on compilation options wraps +::boost::multiprecision::fma(…) or std::fma(…) function. +yade._math.fmod((float)x, (float)y) → float +Returns Real +the +floating-point +remainder +of +the +division +operation +x/y +of +the +arguments +x +and +y. +Depending +on +compilation +options +wraps +::boost::multiprecision::fmod(…) or std::fmod(…) function. +yade._math.frexp((float)x) → tuple +Returns tuple of (Real,int), decomposes given floating point Real argument into a +normalized fraction and an integral power of two. Depending on compilation options +wraps ::boost::multiprecision::frexp(…) or std::frexp(…) function. +yade._math.fromBits((str)bits[, (int)exp=0[, (int)sign=1]]) → float +420 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +Parameters +• bits – str - a string containing ‘0’, ‘1’ characters. +• exp – int - the binary exponent which shifts the bits. +• sign – int - the sign, should be -1 or +1, but it is not checked. It multiplies the +result when construction from bits is finished. +Returns RealHP constructed from string containing ‘0’, ‘1’ bits. This is for debug- +ging purposes, rather slow. +yade._math.getDecomposedReal((float)x) → dict +Returns dict - the dictionary with the debug information how the DecomposedReal +class sees this type. This is for debugging purposes, rather slow. Includes result +from fpclassify function call, a binary representation and other useful info. See also +fromBits. +yade._math.getDemangledName() → str +Returns string - the demangled C++ typnename of RealHP. +yade._math.getDemangledNameComplex() → str +Returns string - the demangled C++ typnename of ComplexHP. +yade._math.getEigenFlags() → dict +Returns A python dictionary listing flags for all types, see: https://eigen.tuxfamily. +org/dox/group__flags.html +yade._math.getEigenStorageOrders() → dict +Returns A python dictionary listing options for all types, see: https://eigen.tuxfamily. +org/dox/group__TopicStorageOrders.html +yade._math.getFloatDistanceULP((float)arg1, (float)arg2) → float +Returns an integer value stored in RealHP, the ULP distance calculated by +boost::math::float_distance, also see Floating-point Comparison and Prof. Kahan +paper about this topic. +The returned value is the directed distance between two arguments, this means that it can be +negative. +yade._math.getRawBits((float)x) → str +Returns string - the raw bits in memory representing this type. Be careful: it only +checks the system endianness and either prints bytes in reverse order or not. Does +not make any attempts to further interpret the bits of: sign, exponent or significand +(on a typical x86 processor they are printed in that order), and different processors +might store them differently. It is not useful for types which internally use a pointer +because for them this function prints not the floating point number but a pointer. +This is for debugging purposes. +yade._math.getRealHPErrors((list)testLevelsHP[, +(int)testCount=10[, +(float)minX=- +10.0[, +(float)maxX=10.0[, +(bool)useRandomArgs=False[, +(int)printEveryNth=1000[, +(bool)collectArgs=False[, +(bool)extraChecks=False]]]]]]]) → dict +Tests mathematical functions against the highest precision in argument testLevelsHP and returns +the largest ULP distance found with getFloatDistanceULP. A testCount randomized tries with +function arguments in range minX ... maxX are performed on the RealHP types where N is +from the list provided in testLevelsHP argument. +Parameters +2.4. +Yade modules reference +421 + +Yade Documentation, Release 3rd ed. +• testLevelsHP – a list of int values consisting of high precision levels N (in +RealHP) for which the tests should be done. Must consist at least of two ele- +ments so that there is a higher precision type available against which to perform +the tests. +• testCount – int - specifies how many randomized tests of each function to +perform. +• minX – Real - start of the range in which the random arguments are generated. +• maxX – Real - end of that range. +• useRandomArgs – If False (default) then minX ... maxX is divided into +testCount equidistant points. If True then each call is a random number. This +applies only to the first argument of a function, if a function takes more than one +argument, then remaining arguments are random - 2D scans are not performed. +• printEveryNth – will print using LOG_INFO the progress information every Nth +step in the testCount loop. To see it e.g. start using yade -f6, also see logger +documentation. +• collectArgs – if True then in returned results will be a longer list of arguments +that produce incorrect results. +• extraChecks – will perform extra checks while executing this funcion. Useful +only for debugging of getRealHPErrors. +Returns A python dictionary with the largest ULP distance to the correct function +value. For each function name there is a dictionary consisting of: how many binary +digits (bits) are in the tested RealHP type, the worst arguments for this function, +and the ULP distance to the reference value. +The returned ULP error is an absolute value, as opposed to getFloatDistanceULP which is signed. +yade._math.highest([(int)Precision=53]) → float +Returns Real returns the largest finite value of the Real type. Wraps std::numeric_- +limits::max() function. +yade._math.hypot((float)x, (float)y) → float +Returns Real the square root of the sum of the squares of x and y, without undue +overflow or underflow at intermediate stages of the computation. +Depending on +compilation options wraps ::boost::multiprecision::hypot(…) or std::hypot(…) +function. +yade._math.ilogb((float)x) → float +Returns Real extracts the value of the unbiased exponent from the floating-point ar- +gument arg, and returns it as a signed integer value. Depending on compilation +options wraps ::boost::multiprecision::ilogb(…) or std::ilogb(…) function. +yade._math.imag((complex)x) → float +Returns the imag part of a Complex argument. +Depending on compilation options +wraps ::boost::multiprecision::imag(…) or std::imag(…) function. +yade._math.isApprox((float)a, (float)b, (float)eps) → bool +Returns bool, True if a is approximately equal b with provided eps, see also here +yade._math.isApproxOrLessThan((float)a, (float)b, (float)eps) → bool +Returns bool, True if a is approximately less than or equal b with provided eps, see +also here +yade._math.isEqualFuzzy((float)arg1, (float)arg2, (float)arg3) → bool +422 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +Returns bool, True if the absolute difference between two numbers is smaller than +std::numeric_limits::epsilon() +yade._math.isMuchSmallerThan((float)a, (float)b, (float)eps) → bool +Returns bool, True if a is less than b with provided eps, see also here +yade._math.isThisSystemLittleEndian() → bool +Returns True if this system uses little endian architecture, False otherwise. +yade._math.isfinite((float)x) → bool +Returns bool indicating if the Real argument is Inf. Depending on compilation options +wraps ::boost::multiprecision::isfinite(…) or std::isfinite(…) function. +yade._math.isinf((float)x) → bool +Returns bool indicating if the Real argument is Inf. Depending on compilation options +wraps ::boost::multiprecision::isinf(…) or std::isinf(…) function. +yade._math.isnan((float)x) → bool +Returns bool indicating if the Real argument is NaN. Depending on compilation op- +tions wraps ::boost::multiprecision::isnan(…) or std::isnan(…) function. +yade._math.laguerre((int)n, (int)m, (float)x) → float +Returns Real the Laguerre polynomial of the orders n, +m and the Real ar- +gument. +See: +‘__ +yade._math.ldexp((float)x, (int)y) → float +Returns Multiplies a floating point value x by the number 2 raised to the exp power. +Depending on compilation options wraps ::boost::multiprecision::ldexp(…) or +std::ldexp(…) function. +yade._math.lgamma((float)x) → float +Returns Real +Computes +the +natural +logarithm +of +the +absolute +value +of +the +gamma +function +of +arg. +Depending +on +compilation +options +wraps +::boost::multiprecision::lgamma(…) or std::lgamma(…) function. +yade._math.log((complex)x) → complex +return the Complex natural (base e) logarithm of a complex value z with a +branch cut along the negative real axis. Depending on compilation options +wraps ::boost::multiprecision::log(…) or std::log(…) function. +log( (float)x) → float : +return the Real natural (base e) logarithm of a real value. Depending on compila- +tion options wraps ::boost::multiprecision::log(…) or std::log(…) function. +yade._math.log10((complex)x) → complex +return the Complex (base 10) logarithm of a complex value z with a branch +cut along the negative real axis. Depending on compilation options wraps +::boost::multiprecision::log10(…) or std::log10(…) function. +log10( (float)x) → float : +return the Real decimal (base 10) logarithm of a real value. Depending on com- +pilation options wraps ::boost::multiprecision::log10(…) or std::log10(…) +function. +yade._math.log1p((float)x) → float +2.4. +Yade modules reference +423 + +Yade Documentation, Release 3rd ed. +Returns the Real natural (base e) logarithm of 1+argument. Depending on compilation +options wraps ::boost::multiprecision::log1p(…) or std::log1p(…) function. +yade._math.log2((float)x) → float +Returns the Real binary (base 2) logarithm of a real value. Depending on compilation +options wraps ::boost::multiprecision::log2(…) or std::log2(…) function. +yade._math.logb((float)x) → float +Returns Extracts the value of the unbiased radix-independent exponent from the +floating-point argument arg, and returns it as a floating-point value. +Depending +on compilation options wraps ::boost::multiprecision::logb(…) or std::logb(…) +function. +yade._math.lowest([(int)Precision=53]) → float +Returns Real returns the lowest (negative) finite value of the Real type. +Wraps +std::numeric_limits::lowest() function. +yade._math.max((float)x, (float)y) → float +Returns Real larger of the two arguments. Depending on compilation options wraps +::boost::multiprecision::max(…) or std::max(…) function. +yade._math.min((float)x, (float)y) → float +Returns Real smaller of the two arguments. Depending on compilation options wraps +::boost::multiprecision::min(…) or std::min(…) function. +yade._math.modf((float)x) → tuple +Returns tuple of (Real,Real), decomposes given floating point Real into integral and +fractional parts, each having the same type and sign as x. Depending on compilation +options wraps ::boost::multiprecision::modf(…) or std::modf(…) function. +yade._math.polar((float)x, (float)y) → complex +Returns Complex the polar (Complex from polar components) of the Real rho (length), +Real theta (angle) arguments in radians. Depending on compilation options wraps +::boost::multiprecision::polar(…) or std::polar(…) function. +yade._math.pow((complex)x, (complex)pow) → complex +return the Complex complex arg1 raised to the Complex power arg2. Depend- +ing on compilation options wraps ::boost::multiprecision::pow(…) or +std::pow(…) function. +pow( (float)x, (float)y) → float : +return Real the value of base raised to the power exp. Depending on compilation +options wraps ::boost::multiprecision::pow(…) or std::pow(…) function. +yade._math.proj((complex)x) → complex +Returns Complex the proj (projection of the complex number onto the Riemann sphere) +of the Complex argument in radians. +Depending on compilation options wraps +::boost::multiprecision::proj(…) or std::proj(…) function. +yade._math.random() → float +return Real a symmetric random number in interval (-1,1). Used by Eigen. +random( (float)a, (float)b) → float : +return Real a random number in interval (a,b). Used by Eigen. +yade._math.real((complex)x) → float +424 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +Returns the real part of a Complex argument. Depending on compilation options wraps +::boost::multiprecision::real(…) or std::real(…) function. +yade._math.remainder((float)x, (float)y) → float +Returns Real the IEEE remainder of the floating point division operation x/y. De- +pending on compilation options wraps ::boost::multiprecision::remainder(…) +or std::remainder(…) function. +yade._math.remquo((float)x, (float)y) → tuple +Returns tuple of (Real,long), the floating-point remainder of the division opera- +tion x/y as the std::remainder() function does. +Additionally, the sign and at +least the three of the last bits of x/y are returned, sufficient to determine the +octant of the result within a period. +Depending on compilation options wraps +::boost::multiprecision::remquo(…) or std::remquo(…) function. +yade._math.rint((float)x) → float +Returns Rounds the floating-point argument arg to an integer value (in floating-point +format), using the current rounding mode. Depending on compilation options wraps +::boost::multiprecision::rint(…) or std::rint(…) function. +yade._math.round((float)x) → float +Returns Real the nearest integer value to arg (in floating-point format), rounding +halfway cases away from zero, regardless of the current rounding mode.. +De- +pending on compilation options wraps ::boost::multiprecision::round(…) or +std::round(…) function. +yade._math.roundTrip((float)x) → float +Returns Real returns the argument x. +Can be used to convert type to native Re- +alHP accuracy. +yade._math.sgn((float)x) → int +Returns int the sign of the argument: -1, 0 or 1. +yade._math.sign((float)x) → int +Returns int the sign of the argument: -1, 0 or 1. +yade._math.sin((complex)x) → complex +return Complex the sine of the Complex argument in radians. +Depend- +ing on compilation options wraps ::boost::multiprecision::sin(…) or +std::sin(…) function. +sin( (float)x) → float : +return Real the sine of the Real argument in radians. Depending on compilation +options wraps ::boost::multiprecision::sin(…) or std::sin(…) function. +yade._math.sinh((complex)x) → complex +return Complex +the +hyperbolic +sine +of +the +Complex +argu- +ment +in +radians. +Depending +on +compilation +options +wraps +::boost::multiprecision::sinh(…) or std::sinh(…) function. +sinh( (float)x) → float : +return Real the hyperbolic sine of the Real argument in radians. Depending on +compilation options wraps ::boost::multiprecision::sinh(…) or std::sinh(…) +function. +yade._math.smallest_positive() → float +2.4. +Yade modules reference +425 + +Yade Documentation, Release 3rd ed. +Returns Real the smallest number greater than zero. +Wraps std::numeric_lim- +its::min() +yade._math.sphericalHarmonic((int)l, (int)m, (float)theta, (float)phi) → complex +Returns Real +the +spherical +harmonic +polynomial +of +the +orders +l +(unsigned +int), +m +(signed int) +and +the +Real +arguments +theta +and +phi. +See: +‘__ +yade._math.sqrt((complex)x) → complex +return the Complex square root of Complex argument. Depending on compi- +lation options wraps ::boost::multiprecision::sqrt(…) or std::sqrt(…) +function. +sqrt( (float)x) → float : +return Real square root of the argument. Depending on compilation options wraps +::boost::multiprecision::sqrt(…) or std::sqrt(…) function. +yade._math.squaredNorm((complex)x) → float +Returns Real the norm (squared magnitude) of the Complex argument in radians. +Depending on compilation options wraps ::boost::multiprecision::norm(…) or +std::norm(…) function. +yade._math.tan((complex)x) → complex +return Complex the tangent of the Complex argument in radians. +Depend- +ing on compilation options wraps ::boost::multiprecision::tan(…) or +std::tan(…) function. +tan( (float)x) → float : +return Real the tangent of the Real argument in radians. Depending on compilation +options wraps ::boost::multiprecision::tan(…) or std::tan(…) function. +yade._math.tanh((complex)x) → complex +return Complex +the +hyperbolic +tangent +of +the +Complex +argu- +ment +in +radians. +Depending +on +compilation +options +wraps +::boost::multiprecision::tanh(…) or std::tanh(…) function. +tanh( (float)x) → float : +return Real the hyperbolic tangent of the Real argument in radians. Depending on +compilation options wraps ::boost::multiprecision::tanh(…) or std::tanh(…) +function. +yade._math.testArray() → None +This function tests call to std::vector::data(…) function in order to extract the array. +yade._math.testConstants() → None +This function tests lib/high-precision/Constants.hpp, the yade::math::ConstantsHP, former +yade::Mathr constants. +yade._math.testLoopRealHP() → None +This function tests lib/high-precision/Constants.hpp, but the C++ side: all precisions, even those +inaccessible from python +yade._math.tgamma((float)x) → float +Returns Real Computes the gamma function of arg. Depending on compilation options +wraps ::boost::multiprecision::tgamma(…) or std::tgamma(…) function. +426 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +yade._math.toDouble((float)x) → float +Returns float converts Real type to double and returns a native python float. +yade._math.toHP1((float)x) → float +Returns RealHP<1> converted from argument RealHP<1> as a result of static_- +cast>(arg). +yade._math.toInt((float)x) → int +Returns int converts Real type to int and returns a native python int. +yade._math.toLong((float)x) → int +Returns int converts Real type to long int and returns a native python int. +yade._math.toLongDouble((float)x) → float +Returns float converts Real type to long double and returns a native python float. +yade._math.trunc((float)x) → float +Returns Real the nearest integer not greater in magnitude than arg. Depending on +compilation options wraps ::boost::multiprecision::trunc(…) or std::trunc(…) +function. +2.4.9 yade.minieigenHP module +When yade uses high-precision number as Real type the usual (old): +from minieigen import * +has to be replaced with: +from yade.minieigenHP import * +This command ensures backward compatibility between both. It is then guaranteed that python uses +the same number of decimal places as yade is using everywhere else. +Please note that used precision can be very arbitrary, because cpp_bin_float or mpfr take it as a +compile-time argument. Hence such yade.minieigenHP cannot be separately precompiled as a package. +Though it could be precompiled for some special types such as boost::multiprecision::float128. +The RealHP higher precision vectors and matrices can be accessed in python by using the .HPn module +scope. For example: +import yade.minieigenHP as mne +mne.HP2.Vector3(1,2,3) # produces Vector3 using RealHP<2> precision +mne.Vector3(1,2,3) +# without using HPn module scope it defaults to RealHP<1> +miniEigen is wrapper for a small part of the Eigen library. Refer to its documentation for details. All +classes in this module support pickling. +class yade._minieigenHP.AlignedBox2 +Axis-aligned box object in 2d, defined by its minimum and maximum corners +__init__((object)arg1) → None +__init__( (object)arg1, (AlignedBox2)other) -> None +__init__( (object)arg1, (Vector2)min, (Vector2)max) -> None +center((AlignedBox2)arg1) → Vector2 +clamp((AlignedBox2)arg1, (AlignedBox2)arg2) → None +contains((AlignedBox2)arg1, (Vector2)arg2) → bool +contains( (AlignedBox2)arg1, (AlignedBox2)arg2) -> bool +2.4. +Yade modules reference +427 + +Yade Documentation, Release 3rd ed. +empty((AlignedBox2)arg1) → bool +extend((AlignedBox2)arg1, (Vector2)arg2) → None +extend( (AlignedBox2)arg1, (AlignedBox2)arg2) -> None +intersection((AlignedBox2)arg1, (AlignedBox2)arg2) → AlignedBox2 +max +merged((AlignedBox2)arg1, (AlignedBox2)arg2) → AlignedBox2 +min +sizes((AlignedBox2)arg1) → Vector2 +volume((AlignedBox2)arg1) → float +class yade._minieigenHP.AlignedBox3 +Axis-aligned box object, defined by its minimum and maximum corners +__init__((object)arg1) → None +__init__( (object)arg1, (AlignedBox3)other) -> None +__init__( (object)arg1, (Vector3)min, (Vector3)max) -> None +center((AlignedBox3)arg1) → Vector3 +clamp((AlignedBox3)arg1, (AlignedBox3)arg2) → None +contains((AlignedBox3)arg1, (Vector3)arg2) → bool +contains( (AlignedBox3)arg1, (AlignedBox3)arg2) -> bool +empty((AlignedBox3)arg1) → bool +extend((AlignedBox3)arg1, (Vector3)arg2) → None +extend( (AlignedBox3)arg1, (AlignedBox3)arg2) -> None +intersection((AlignedBox3)arg1, (AlignedBox3)arg2) → AlignedBox3 +max +merged((AlignedBox3)arg1, (AlignedBox3)arg2) → AlignedBox3 +min +sizes((AlignedBox3)arg1) → Vector3 +volume((AlignedBox3)arg1) → float +class yade._minieigenHP.HP1 +class AlignedBox2 +Axis-aligned box object in 2d, defined by its minimum and maximum corners +__init__((object)arg1) → None +__init__( (object)arg1, (AlignedBox2)other) -> None +__init__( (object)arg1, (Vector2)min, (Vector2)max) -> None +center((AlignedBox2)arg1) → Vector2 +clamp((AlignedBox2)arg1, (AlignedBox2)arg2) → None +contains((AlignedBox2)arg1, (Vector2)arg2) → bool +contains( (AlignedBox2)arg1, (AlignedBox2)arg2) -> bool +empty((AlignedBox2)arg1) → bool +extend((AlignedBox2)arg1, (Vector2)arg2) → None +extend( (AlignedBox2)arg1, (AlignedBox2)arg2) -> None +intersection((AlignedBox2)arg1, (AlignedBox2)arg2) → AlignedBox2 +428 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +max +merged((AlignedBox2)arg1, (AlignedBox2)arg2) → AlignedBox2 +min +sizes((AlignedBox2)arg1) → Vector2 +volume((AlignedBox2)arg1) → float +class AlignedBox3 +Axis-aligned box object, defined by its minimum and maximum corners +__init__((object)arg1) → None +__init__( (object)arg1, (AlignedBox3)other) -> None +__init__( (object)arg1, (Vector3)min, (Vector3)max) -> None +center((AlignedBox3)arg1) → Vector3 +clamp((AlignedBox3)arg1, (AlignedBox3)arg2) → None +contains((AlignedBox3)arg1, (Vector3)arg2) → bool +contains( (AlignedBox3)arg1, (AlignedBox3)arg2) -> bool +empty((AlignedBox3)arg1) → bool +extend((AlignedBox3)arg1, (Vector3)arg2) → None +extend( (AlignedBox3)arg1, (AlignedBox3)arg2) -> None +intersection((AlignedBox3)arg1, (AlignedBox3)arg2) → AlignedBox3 +max +merged((AlignedBox3)arg1, (AlignedBox3)arg2) → AlignedBox3 +min +sizes((AlignedBox3)arg1) → Vector3 +volume((AlignedBox3)arg1) → float +class Matrix3 +3x3 float matrix. +Supported operations (m is a Matrix3, f if a float/int, v is a Vector3): -m, m+m, m+=m, m-m, +m-=m, m*f, f*m, m*=f, m/f, m/=f, m*m, m*=m, m*v, v*m, m==m, m!=m. +Static attributes: Zero, Ones, Identity. +Identity = Matrix3(1,0,0, 0,1,0, 0,0,1) +Ones = Matrix3(1,1,1, 1,1,1, 1,1,1) +static Random() → Matrix3 : +Return an object where all elements are randomly set to values between 0 and 1. +Zero = Matrix3(0,0,0, 0,0,0, 0,0,0) +__init__((object)arg1) → None +__init__( (object)arg1, (Quaternion)q) -> None +__init__( (object)arg1, (Matrix3)other) -> None +__init__( (object)arg1, (Vector3)diag) -> object +__init__( (object)arg1, (float)m00, (float)m01, (float)m02, (float)m10, (float)m11, +(float)m12, (float)m20, (float)m21, (float)m22) -> object +__init__( (object)arg1, (Vector3)r0, (Vector3)r1, (Vector3)r2 [, (bool)cols=False]) -> +object +col((Matrix3)arg1, (int)col) → Vector3 : +Return column as vector. +2.4. +Yade modules reference +429 + +Yade Documentation, Release 3rd ed. +cols((Matrix3)arg1) → int : +Number of columns. +computeUnitaryPositive((Matrix3)arg1) → tuple : +Compute polar decomposition (unitary matrix U and positive semi-definite symmetric +matrix P such that self=U*P). +determinant((Matrix3)arg1) → float : +Return matrix determinant. +diagonal((Matrix3)arg1) → Vector3 : +Return diagonal as vector. +inverse((Matrix3)arg1) → Matrix3 : +Return inverted matrix. +isApprox((Matrix3)arg1, (Matrix3)other[, (float)prec=1e-12]) → bool : +Approximate comparison with precision prec. +jacobiSVD((Matrix3)arg1) → tuple : +Compute +SVD +decomposition +of +square +matrix, +retuns +(U,S,V) +such +that +self=U*S*V.transpose() +maxAbsCoeff((Matrix3)arg1) → float : +Maximum absolute value over all elements. +maxCoeff((Matrix3)arg1) → float : +Maximum value over all elements. +mean((Matrix3)arg1) → float : +Mean value over all elements. +minCoeff((Matrix3)arg1) → float : +Minimum value over all elements. +norm((Matrix3)arg1) → float : +Euclidean norm. +normalize((Matrix3)arg1) → None : +Normalize this object in-place. +normalized((Matrix3)arg1) → Matrix3 : +Return normalized copy of this object +polarDecomposition((Matrix3)arg1) → tuple : +Alias for computeUnitaryPositive. +prod((Matrix3)arg1) → float : +Product of all elements. +pruned((Matrix3)arg1[, (float)absTol=1e-06]) → Matrix3 : +Zero all elements which are greater than absTol. Negative zeros are not pruned. +row((Matrix3)arg1, (int)row) → Vector3 : +Return row as vector. +rows((Matrix3)arg1) → int : +Number of rows. +selfAdjointEigenDecomposition((Matrix3)arg1) → tuple : +Compute eigen (spectral) decomposition of symmetric matrix, returns (eigVecs,eigVals). +eigVecs is orthogonal Matrix3 with columns ar normalized eigenvectors, eigVals is Vector3 +with corresponding eigenvalues. self=eigVecs*diag(eigVals)*eigVecs.transpose(). +spectralDecomposition((Matrix3)arg1) → tuple : +Alias for selfAdjointEigenDecomposition. +squaredNorm((Matrix3)arg1) → float : +Square of the Euclidean norm. +430 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +sum((Matrix3)arg1) → float : +Sum of all elements. +svd((Matrix3)arg1) → tuple : +Alias for jacobiSVD. +trace((Matrix3)arg1) → float : +Return sum of diagonal elements. +transpose((Matrix3)arg1) → Matrix3 : +Return transposed matrix. +class Matrix3c +/TODO/ +Identity = Matrix3c(1,0,0, 0,1,0, 0,0,1) +Ones = Matrix3c(1,1,1, 1,1,1, 1,1,1) +static Random() → Matrix3c : +Return an object where all elements are randomly set to values between 0 and 1. +Zero = Matrix3c(0,0,0, 0,0,0, 0,0,0) +__init__((object)arg1) → None +__init__( (object)arg1, (Matrix3c)other) -> None +__init__( (object)arg1, (Vector3c)diag) -> object +__init__( (object)arg1, (complex)m00, (complex)m01, (complex)m02, (complex)m10, +(complex)m11, (complex)m12, (complex)m20, (complex)m21, (complex)m22) -> object +__init__( (object)arg1, (Vector3c)r0, (Vector3c)r1, (Vector3c)r2 [, (bool)cols=False]) +-> object +col((Matrix3c)arg1, (int)col) → Vector3c : +Return column as vector. +cols((Matrix3c)arg1) → int : +Number of columns. +determinant((Matrix3c)arg1) → complex : +Return matrix determinant. +diagonal((Matrix3c)arg1) → Vector3c : +Return diagonal as vector. +inverse((Matrix3c)arg1) → Matrix3c : +Return inverted matrix. +isApprox((Matrix3c)arg1, (Matrix3c)other[, (float)prec=1e-12]) → bool : +Approximate comparison with precision prec. +maxAbsCoeff((Matrix3c)arg1) → float : +Maximum absolute value over all elements. +mean((Matrix3c)arg1) → complex : +Mean value over all elements. +norm((Matrix3c)arg1) → float : +Euclidean norm. +normalize((Matrix3c)arg1) → None : +Normalize this object in-place. +normalized((Matrix3c)arg1) → Matrix3c : +Return normalized copy of this object +prod((Matrix3c)arg1) → complex : +Product of all elements. +2.4. +Yade modules reference +431 + +Yade Documentation, Release 3rd ed. +pruned((Matrix3c)arg1[, (float)absTol=1e-06]) → Matrix3c : +Zero all elements which are greater than absTol. Negative zeros are not pruned. +row((Matrix3c)arg1, (int)row) → Vector3c : +Return row as vector. +rows((Matrix3c)arg1) → int : +Number of rows. +squaredNorm((Matrix3c)arg1) → float : +Square of the Euclidean norm. +sum((Matrix3c)arg1) → complex : +Sum of all elements. +trace((Matrix3c)arg1) → complex : +Return sum of diagonal elements. +transpose((Matrix3c)arg1) → Matrix3c : +Return transposed matrix. +class Matrix6 +6x6 float matrix. Constructed from 4 3x3 sub-matrices, from 6xVector6 (rows). +Supported operations (m is a Matrix6, f if a float/int, v is a Vector6): -m, m+m, m+=m, m-m, +m-=m, m*f, f*m, m*=f, m/f, m/=f, m*m, m*=m, m*v, v*m, m==m, m!=m. +Static attributes: Zero, Ones, Identity. +Identity = Matrix6( (1,0,0,0,0,0), (0,1,0,0,0,0), (0,0,1,0,0,0), (0,0,0,1,0,0), (0,0,0,0,1,0), (0,0,0,0,0,1) ) +Ones = Matrix6( (1,1,1,1,1,1), (1,1,1,1,1,1), (1,1,1,1,1,1), (1,1,1,1,1,1), (1,1,1,1,1,1), (1,1,1,1,1,1) ) +static Random() → Matrix6 : +Return an object where all elements are randomly set to values between 0 and 1. +Zero = Matrix6( (0,0,0,0,0,0), (0,0,0,0,0,0), (0,0,0,0,0,0), (0,0,0,0,0,0), (0,0,0,0,0,0), (0,0,0,0,0,0) ) +__init__((object)arg1) → None +__init__( (object)arg1, (Matrix6)other) -> None +__init__( (object)arg1, (Vector6)diag) -> object +__init__( (object)arg1, (Matrix3)ul, (Matrix3)ur, (Matrix3)ll, (Matrix3)lr) -> object +__init__( (object)arg1, (Vector6)l0, (Vector6)l1, (Vector6)l2, (Vector6)l3, (Vector6)l4, +(Vector6)l5 [, (bool)cols=False]) -> object +col((Matrix6)arg1, (int)col) → Vector6 : +Return column as vector. +cols((Matrix6)arg1) → int : +Number of columns. +computeUnitaryPositive((Matrix6)arg1) → tuple : +Compute polar decomposition (unitary matrix U and positive semi-definite symmetric +matrix P such that self=U*P). +determinant((Matrix6)arg1) → float : +Return matrix determinant. +diagonal((Matrix6)arg1) → Vector6 : +Return diagonal as vector. +inverse((Matrix6)arg1) → Matrix6 : +Return inverted matrix. +isApprox((Matrix6)arg1, (Matrix6)other[, (float)prec=1e-12]) → bool : +Approximate comparison with precision prec. +432 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +jacobiSVD((Matrix6)arg1) → tuple : +Compute +SVD +decomposition +of +square +matrix, +retuns +(U,S,V) +such +that +self=U*S*V.transpose() +ll((Matrix6)arg1) → Matrix3 : +Return lower-left 3x3 block +lr((Matrix6)arg1) → Matrix3 : +Return lower-right 3x3 block +maxAbsCoeff((Matrix6)arg1) → float : +Maximum absolute value over all elements. +maxCoeff((Matrix6)arg1) → float : +Maximum value over all elements. +mean((Matrix6)arg1) → float : +Mean value over all elements. +minCoeff((Matrix6)arg1) → float : +Minimum value over all elements. +norm((Matrix6)arg1) → float : +Euclidean norm. +normalize((Matrix6)arg1) → None : +Normalize this object in-place. +normalized((Matrix6)arg1) → Matrix6 : +Return normalized copy of this object +polarDecomposition((Matrix6)arg1) → tuple : +Alias for computeUnitaryPositive. +prod((Matrix6)arg1) → float : +Product of all elements. +pruned((Matrix6)arg1[, (float)absTol=1e-06]) → Matrix6 : +Zero all elements which are greater than absTol. Negative zeros are not pruned. +row((Matrix6)arg1, (int)row) → Vector6 : +Return row as vector. +rows((Matrix6)arg1) → int : +Number of rows. +selfAdjointEigenDecomposition((Matrix6)arg1) → tuple : +Compute eigen (spectral) decomposition of symmetric matrix, returns (eigVecs,eigVals). +eigVecs is orthogonal Matrix3 with columns ar normalized eigenvectors, eigVals is Vector3 +with corresponding eigenvalues. self=eigVecs*diag(eigVals)*eigVecs.transpose(). +spectralDecomposition((Matrix6)arg1) → tuple : +Alias for selfAdjointEigenDecomposition. +squaredNorm((Matrix6)arg1) → float : +Square of the Euclidean norm. +sum((Matrix6)arg1) → float : +Sum of all elements. +svd((Matrix6)arg1) → tuple : +Alias for jacobiSVD. +trace((Matrix6)arg1) → float : +Return sum of diagonal elements. +transpose((Matrix6)arg1) → Matrix6 : +Return transposed matrix. +2.4. +Yade modules reference +433 + +Yade Documentation, Release 3rd ed. +ul((Matrix6)arg1) → Matrix3 : +Return upper-left 3x3 block +ur((Matrix6)arg1) → Matrix3 : +Return upper-right 3x3 block +class Matrix6c +/TODO/ +Identity = Matrix6c( (1,0,0,0,0,0), (0,1,0,0,0,0), (0,0,1,0,0,0), (0,0,0,1,0,0), (0,0,0,0,1,0), (0,0,0,0,0,1) ) +Ones = Matrix6c( (1,1,1,1,1,1), (1,1,1,1,1,1), (1,1,1,1,1,1), (1,1,1,1,1,1), (1,1,1,1,1,1), (1,1,1,1,1,1) ) +static Random() → Matrix6c : +Return an object where all elements are randomly set to values between 0 and 1. +Zero = Matrix6c( (0,0,0,0,0,0), (0,0,0,0,0,0), (0,0,0,0,0,0), (0,0,0,0,0,0), (0,0,0,0,0,0), (0,0,0,0,0,0) ) +__init__((object)arg1) → None +__init__( (object)arg1, (Matrix6c)other) -> None +__init__( (object)arg1, (Vector6c)diag) -> object +__init__( (object)arg1, (Matrix3c)ul, (Matrix3c)ur, (Matrix3c)ll, (Matrix3c)lr) -> ob- +ject +__init__( (object)arg1, (Vector6c)l0, (Vector6c)l1, (Vector6c)l2, (Vector6c)l3, (Vec- +tor6c)l4, (Vector6c)l5 [, (bool)cols=False]) -> object +col((Matrix6c)arg1, (int)col) → Vector6c : +Return column as vector. +cols((Matrix6c)arg1) → int : +Number of columns. +determinant((Matrix6c)arg1) → complex : +Return matrix determinant. +diagonal((Matrix6c)arg1) → Vector6c : +Return diagonal as vector. +inverse((Matrix6c)arg1) → Matrix6c : +Return inverted matrix. +isApprox((Matrix6c)arg1, (Matrix6c)other[, (float)prec=1e-12]) → bool : +Approximate comparison with precision prec. +ll((Matrix6c)arg1) → Matrix3c : +Return lower-left 3x3 block +lr((Matrix6c)arg1) → Matrix3c : +Return lower-right 3x3 block +maxAbsCoeff((Matrix6c)arg1) → float : +Maximum absolute value over all elements. +mean((Matrix6c)arg1) → complex : +Mean value over all elements. +norm((Matrix6c)arg1) → float : +Euclidean norm. +normalize((Matrix6c)arg1) → None : +Normalize this object in-place. +normalized((Matrix6c)arg1) → Matrix6c : +Return normalized copy of this object +prod((Matrix6c)arg1) → complex : +Product of all elements. +434 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +pruned((Matrix6c)arg1[, (float)absTol=1e-06]) → Matrix6c : +Zero all elements which are greater than absTol. Negative zeros are not pruned. +row((Matrix6c)arg1, (int)row) → Vector6c : +Return row as vector. +rows((Matrix6c)arg1) → int : +Number of rows. +squaredNorm((Matrix6c)arg1) → float : +Square of the Euclidean norm. +sum((Matrix6c)arg1) → complex : +Sum of all elements. +trace((Matrix6c)arg1) → complex : +Return sum of diagonal elements. +transpose((Matrix6c)arg1) → Matrix6c : +Return transposed matrix. +ul((Matrix6c)arg1) → Matrix3c : +Return upper-left 3x3 block +ur((Matrix6c)arg1) → Matrix3c : +Return upper-right 3x3 block +class MatrixX +XxX (dynamic-sized) float matrix. Constructed from list of rows (as VectorX). +Supported operations (m is a MatrixX, f if a float/int, v is a VectorX): -m, m+m, m+=m, m-m, +m-=m, m*f, f*m, m*=f, m/f, m/=f, m*m, m*=m, m*v, v*m, m==m, m!=m. +static Identity((int)arg1, (int)rank) → MatrixX : +Create identity matrix with given rank (square). +static Ones((int)rows, (int)cols) → MatrixX : +Create matrix of given dimensions where all elements are set to 1. +static Random((int)rows, (int)cols) → MatrixX : +Create matrix with given dimensions where all elements are set to number between 0 and +1 (uniformly-distributed). +static Zero((int)rows, (int)cols) → MatrixX : +Create zero matrix of given dimensions +__init__((object)arg1) → None +__init__( (object)arg1, (MatrixX)other) -> None +__init__( (object)arg1, (VectorX)diag) -> object +__init__( (object)arg1 [, (VectorX)r0=VectorX() [, (VectorX)r1=VectorX() [, (Vec- +torX)r2=VectorX() +[, +(VectorX)r3=VectorX() +[, +(VectorX)r4=VectorX() +[, +(Vec- +torX)r5=VectorX() +[, +(VectorX)r6=VectorX() +[, +(VectorX)r7=VectorX() +[, +(Vec- +torX)r8=VectorX() [, (VectorX)r9=VectorX() [, (bool)cols=False]]]]]]]]]]]) -> object +__init__( (object)arg1, (object)rows [, (bool)cols=False]) -> object +col((MatrixX)arg1, (int)col) → VectorX : +Return column as vector. +cols((MatrixX)arg1) → int : +Number of columns. +computeUnitaryPositive((MatrixX)arg1) → tuple : +Compute polar decomposition (unitary matrix U and positive semi-definite symmetric +matrix P such that self=U*P). +2.4. +Yade modules reference +435 + +Yade Documentation, Release 3rd ed. +determinant((MatrixX)arg1) → float : +Return matrix determinant. +diagonal((MatrixX)arg1) → VectorX : +Return diagonal as vector. +inverse((MatrixX)arg1) → MatrixX : +Return inverted matrix. +isApprox((MatrixX)arg1, (MatrixX)other[, (float)prec=1e-12]) → bool : +Approximate comparison with precision prec. +jacobiSVD((MatrixX)arg1) → tuple : +Compute +SVD +decomposition +of +square +matrix, +retuns +(U,S,V) +such +that +self=U*S*V.transpose() +maxAbsCoeff((MatrixX)arg1) → float : +Maximum absolute value over all elements. +maxCoeff((MatrixX)arg1) → float : +Maximum value over all elements. +mean((MatrixX)arg1) → float : +Mean value over all elements. +minCoeff((MatrixX)arg1) → float : +Minimum value over all elements. +norm((MatrixX)arg1) → float : +Euclidean norm. +normalize((MatrixX)arg1) → None : +Normalize this object in-place. +normalized((MatrixX)arg1) → MatrixX : +Return normalized copy of this object +polarDecomposition((MatrixX)arg1) → tuple : +Alias for computeUnitaryPositive. +prod((MatrixX)arg1) → float : +Product of all elements. +pruned((MatrixX)arg1[, (float)absTol=1e-06]) → MatrixX : +Zero all elements which are greater than absTol. Negative zeros are not pruned. +resize((MatrixX)arg1, (int)rows, (int)cols) → None : +Change size of the matrix, keep values of elements which exist in the new matrix +row((MatrixX)arg1, (int)row) → VectorX : +Return row as vector. +rows((MatrixX)arg1) → int : +Number of rows. +selfAdjointEigenDecomposition((MatrixX)arg1) → tuple : +Compute eigen (spectral) decomposition of symmetric matrix, returns (eigVecs,eigVals). +eigVecs is orthogonal Matrix3 with columns ar normalized eigenvectors, eigVals is Vector3 +with corresponding eigenvalues. self=eigVecs*diag(eigVals)*eigVecs.transpose(). +spectralDecomposition((MatrixX)arg1) → tuple : +Alias for selfAdjointEigenDecomposition. +squaredNorm((MatrixX)arg1) → float : +Square of the Euclidean norm. +sum((MatrixX)arg1) → float : +Sum of all elements. +436 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +svd((MatrixX)arg1) → tuple : +Alias for jacobiSVD. +trace((MatrixX)arg1) → float : +Return sum of diagonal elements. +transpose((MatrixX)arg1) → MatrixX : +Return transposed matrix. +class MatrixXc +/TODO/ +static Identity((int)arg1, (int)rank) → MatrixXc : +Create identity matrix with given rank (square). +static Ones((int)rows, (int)cols) → MatrixXc : +Create matrix of given dimensions where all elements are set to 1. +static Random((int)rows, (int)cols) → MatrixXc : +Create matrix with given dimensions where all elements are set to number between 0 and +1 (uniformly-distributed). +static Zero((int)rows, (int)cols) → MatrixXc : +Create zero matrix of given dimensions +__init__((object)arg1) → None +__init__( (object)arg1, (MatrixXc)other) -> None +__init__( (object)arg1, (VectorXc)diag) -> object +__init__( (object)arg1 [, (VectorXc)r0=VectorXc() [, (VectorXc)r1=VectorXc() [, (Vec- +torXc)r2=VectorXc() [, (VectorXc)r3=VectorXc() [, (VectorXc)r4=VectorXc() [, (Vec- +torXc)r5=VectorXc() [, (VectorXc)r6=VectorXc() [, (VectorXc)r7=VectorXc() [, (Vec- +torXc)r8=VectorXc() [, (VectorXc)r9=VectorXc() [, (bool)cols=False]]]]]]]]]]]) -> object +__init__( (object)arg1, (object)rows [, (bool)cols=False]) -> object +col((MatrixXc)arg1, (int)col) → VectorXc : +Return column as vector. +cols((MatrixXc)arg1) → int : +Number of columns. +determinant((MatrixXc)arg1) → complex : +Return matrix determinant. +diagonal((MatrixXc)arg1) → VectorXc : +Return diagonal as vector. +inverse((MatrixXc)arg1) → MatrixXc : +Return inverted matrix. +isApprox((MatrixXc)arg1, (MatrixXc)other[, (float)prec=1e-12]) → bool : +Approximate comparison with precision prec. +maxAbsCoeff((MatrixXc)arg1) → float : +Maximum absolute value over all elements. +mean((MatrixXc)arg1) → complex : +Mean value over all elements. +norm((MatrixXc)arg1) → float : +Euclidean norm. +normalize((MatrixXc)arg1) → None : +Normalize this object in-place. +normalized((MatrixXc)arg1) → MatrixXc : +Return normalized copy of this object +2.4. +Yade modules reference +437 + +Yade Documentation, Release 3rd ed. +prod((MatrixXc)arg1) → complex : +Product of all elements. +pruned((MatrixXc)arg1[, (float)absTol=1e-06]) → MatrixXc : +Zero all elements which are greater than absTol. Negative zeros are not pruned. +resize((MatrixXc)arg1, (int)rows, (int)cols) → None : +Change size of the matrix, keep values of elements which exist in the new matrix +row((MatrixXc)arg1, (int)row) → VectorXc : +Return row as vector. +rows((MatrixXc)arg1) → int : +Number of rows. +squaredNorm((MatrixXc)arg1) → float : +Square of the Euclidean norm. +sum((MatrixXc)arg1) → complex : +Sum of all elements. +trace((MatrixXc)arg1) → complex : +Return sum of diagonal elements. +transpose((MatrixXc)arg1) → MatrixXc : +Return transposed matrix. +class Quaternion +Quaternion representing rotation. +Supported operations (q is a Quaternion, v is a Vector3): q*q (rotation composition), q*=q, +q*v (rotating v by q), q==q, q!=q. +Static attributes: Identity. +Note: +Quaternion is represented as axis-angle when printed (e.g. +Identity is +Quaternion((1,0,0),0), and can also be constructed from the axis-angle representation. +This is however different from the data stored inside, which can be accessed by indices +[0] (x), [1] (y), [2] (z), [3] (w). To obtain axis-angle programatically, use Quaternion. +toAxisAngle which returns the tuple. +Identity = Quaternion((1,0,0),0) +Rotate((Quaternion)arg1, (Vector3)v) → Vector3 +__init__((object)arg1) → None +__init__( (object)arg1, (Vector3)axis, (float)angle) -> object +__init__( (object)arg1, (float)angle, (Vector3)axis) -> object +__init__( (object)arg1, (Vector3)u, (Vector3)v) -> object +__init__( (object)arg1, (float)w, (float)x, (float)y, (float)z) -> None : +Initialize from coefficients. +Note: The order of coefficients is w, x, y, z. The [] operator numbers them differently, +0…4 for x y z w! +__init__( (object)arg1, (Matrix3)rotMatrix) -> None +__init__( (object)arg1, (Quaternion)other) -> None +angularDistance((Quaternion)arg1, (Quaternion)arg2) → float +conjugate((Quaternion)arg1) → Quaternion +inverse((Quaternion)arg1) → Quaternion +438 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +norm((Quaternion)arg1) → float +normalize((Quaternion)arg1) → None +normalized((Quaternion)arg1) → Quaternion +setFromTwoVectors((Quaternion)arg1, (Vector3)u, (Vector3)v) → None +slerp((Quaternion)arg1, (float)t, (Quaternion)other) → Quaternion +toAngleAxis((Quaternion)arg1) → tuple +toAxisAngle((Quaternion)arg1) → tuple +toRotationMatrix((Quaternion)arg1) → Matrix3 +toRotationVector((Quaternion)arg1) → Vector3 +class Vector2 +3-dimensional float vector. +Supported operations (f if a float/int, v is a Vector3): -v, v+v, v+=v, v-v, v-=v, v*f, f*v, +v*=f, v/f, v/=f, v==v, v!=v. +Implicit conversion from sequence (list, tuple, …) of 2 floats. +Static attributes: Zero, Ones, UnitX, UnitY. +Identity = Vector2(1,0) +Ones = Vector2(1,1) +static Random() → Vector2 : +Return an object where all elements are randomly set to values between 0 and 1. +static Unit((int)arg1) → Vector2 +UnitX = Vector2(1,0) +UnitY = Vector2(0,1) +Zero = Vector2(0,0) +__init__((object)arg1) → None +__init__( (object)arg1, (Vector2)other) -> None +__init__( (object)arg1, (float)x, (float)y) -> None +asDiagonal((Vector2)arg1) → object : +Return diagonal matrix with this vector on the diagonal. +cols((Vector2)arg1) → int : +Number of columns. +dot((Vector2)arg1, (Vector2)other) → float : +Dot product with other. +isApprox((Vector2)arg1, (Vector2)other[, (float)prec=1e-12]) → bool : +Approximate comparison with precision prec. +maxAbsCoeff((Vector2)arg1) → float : +Maximum absolute value over all elements. +maxCoeff((Vector2)arg1) → float : +Maximum value over all elements. +mean((Vector2)arg1) → float : +Mean value over all elements. +minCoeff((Vector2)arg1) → float : +Minimum value over all elements. +2.4. +Yade modules reference +439 + +Yade Documentation, Release 3rd ed. +norm((Vector2)arg1) → float : +Euclidean norm. +normalize((Vector2)arg1) → None : +Normalize this object in-place. +normalized((Vector2)arg1) → Vector2 : +Return normalized copy of this object +outer((Vector2)arg1, (Vector2)other) → object : +Outer product with other. +prod((Vector2)arg1) → float : +Product of all elements. +pruned((Vector2)arg1[, (float)absTol=1e-06]) → Vector2 : +Zero all elements which are greater than absTol. Negative zeros are not pruned. +rows((Vector2)arg1) → int : +Number of rows. +squaredNorm((Vector2)arg1) → float : +Square of the Euclidean norm. +sum((Vector2)arg1) → float : +Sum of all elements. +class Vector2c +/TODO/ +Identity = Vector2c(1,0) +Ones = Vector2c(1,1) +static Random() → Vector2c : +Return an object where all elements are randomly set to values between 0 and 1. +static Unit((int)arg1) → Vector2c +UnitX = Vector2c(1,0) +UnitY = Vector2c(0,1) +Zero = Vector2c(0,0) +__init__((object)arg1) → None +__init__( (object)arg1, (Vector2c)other) -> None +__init__( (object)arg1, (complex)x, (complex)y) -> None +asDiagonal((Vector2c)arg1) → object : +Return diagonal matrix with this vector on the diagonal. +cols((Vector2c)arg1) → int : +Number of columns. +dot((Vector2c)arg1, (Vector2c)other) → complex : +Dot product with other. +isApprox((Vector2c)arg1, (Vector2c)other[, (float)prec=1e-12]) → bool : +Approximate comparison with precision prec. +maxAbsCoeff((Vector2c)arg1) → float : +Maximum absolute value over all elements. +mean((Vector2c)arg1) → complex : +Mean value over all elements. +norm((Vector2c)arg1) → float : +Euclidean norm. +440 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +normalize((Vector2c)arg1) → None : +Normalize this object in-place. +normalized((Vector2c)arg1) → Vector2c : +Return normalized copy of this object +outer((Vector2c)arg1, (Vector2c)other) → object : +Outer product with other. +prod((Vector2c)arg1) → complex : +Product of all elements. +pruned((Vector2c)arg1[, (float)absTol=1e-06]) → Vector2c : +Zero all elements which are greater than absTol. Negative zeros are not pruned. +rows((Vector2c)arg1) → int : +Number of rows. +squaredNorm((Vector2c)arg1) → float : +Square of the Euclidean norm. +sum((Vector2c)arg1) → complex : +Sum of all elements. +class Vector2i +2-dimensional integer vector. +Supported operations (i if an int, v is a Vector2i): -v, v+v, v+=v, v-v, v-=v, v*i, i*v, v*=i, +v==v, v!=v. +Implicit conversion from sequence (list, tuple, …) of 2 integers. +Static attributes: Zero, Ones, UnitX, UnitY. +Identity = Vector2i(1,0) +Ones = Vector2i(1,1) +static Random() → Vector2i : +Return an object where all elements are randomly set to values between 0 and 1. +static Unit((int)arg1) → Vector2i +UnitX = Vector2i(1,0) +UnitY = Vector2i(0,1) +Zero = Vector2i(0,0) +__init__((object)arg1) → None +__init__( (object)arg1, (Vector2i)other) -> None +__init__( (object)arg1, (int)x, (int)y) -> None +asDiagonal((Vector2i)arg1) → object : +Return diagonal matrix with this vector on the diagonal. +cols((Vector2i)arg1) → int : +Number of columns. +dot((Vector2i)arg1, (Vector2i)other) → int : +Dot product with other. +isApprox((Vector2i)arg1, (Vector2i)other[, (int)prec=0]) → bool : +Approximate comparison with precision prec. +maxAbsCoeff((Vector2i)arg1) → int : +Maximum absolute value over all elements. +maxCoeff((Vector2i)arg1) → int : +Maximum value over all elements. +2.4. +Yade modules reference +441 + +Yade Documentation, Release 3rd ed. +mean((Vector2i)arg1) → int : +Mean value over all elements. +minCoeff((Vector2i)arg1) → int : +Minimum value over all elements. +outer((Vector2i)arg1, (Vector2i)other) → object : +Outer product with other. +prod((Vector2i)arg1) → int : +Product of all elements. +rows((Vector2i)arg1) → int : +Number of rows. +sum((Vector2i)arg1) → int : +Sum of all elements. +class Vector3 +3-dimensional float vector. +Supported operations (f if a float/int, v is a Vector3): -v, v+v, v+=v, v-v, v-=v, v*f, f*v, +v*=f, v/f, v/=f, v==v, v!=v, plus operations with Matrix3 and Quaternion. +Implicit conversion from sequence (list, tuple, …) of 3 floats. +Static attributes: Zero, Ones, UnitX, UnitY, UnitZ. +Identity = Vector3(1,0,0) +Ones = Vector3(1,1,1) +static Random() → Vector3 : +Return an object where all elements are randomly set to values between 0 and 1. +static Unit((int)arg1) → Vector3 +UnitX = Vector3(1,0,0) +UnitY = Vector3(0,1,0) +UnitZ = Vector3(0,0,1) +Zero = Vector3(0,0,0) +__init__((object)arg1) → None +__init__( (object)arg1, (Vector3)other) -> None +__init__( (object)arg1 [, (float)x=0.0 [, (float)y=0.0 [, (float)z=0.0]]]) -> None +asDiagonal((Vector3)arg1) → Matrix3 : +Return diagonal matrix with this vector on the diagonal. +cols((Vector3)arg1) → int : +Number of columns. +cross((Vector3)arg1, (Vector3)arg2) → Vector3 +dot((Vector3)arg1, (Vector3)other) → float : +Dot product with other. +isApprox((Vector3)arg1, (Vector3)other[, (float)prec=1e-12]) → bool : +Approximate comparison with precision prec. +maxAbsCoeff((Vector3)arg1) → float : +Maximum absolute value over all elements. +maxCoeff((Vector3)arg1) → float : +Maximum value over all elements. +442 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +mean((Vector3)arg1) → float : +Mean value over all elements. +minCoeff((Vector3)arg1) → float : +Minimum value over all elements. +norm((Vector3)arg1) → float : +Euclidean norm. +normalize((Vector3)arg1) → None : +Normalize this object in-place. +normalized((Vector3)arg1) → Vector3 : +Return normalized copy of this object +outer((Vector3)arg1, (Vector3)other) → Matrix3 : +Outer product with other. +prod((Vector3)arg1) → float : +Product of all elements. +pruned((Vector3)arg1[, (float)absTol=1e-06]) → Vector3 : +Zero all elements which are greater than absTol. Negative zeros are not pruned. +rows((Vector3)arg1) → int : +Number of rows. +squaredNorm((Vector3)arg1) → float : +Square of the Euclidean norm. +sum((Vector3)arg1) → float : +Sum of all elements. +xy((Vector3)arg1) → Vector2 +xz((Vector3)arg1) → Vector2 +yx((Vector3)arg1) → Vector2 +yz((Vector3)arg1) → Vector2 +zx((Vector3)arg1) → Vector2 +zy((Vector3)arg1) → Vector2 +class Vector3c +/TODO/ +Identity = Vector3c(1,0,0) +Ones = Vector3c(1,1,1) +static Random() → Vector3c : +Return an object where all elements are randomly set to values between 0 and 1. +static Unit((int)arg1) → Vector3c +UnitX = Vector3c(1,0,0) +UnitY = Vector3c(0,1,0) +UnitZ = Vector3c(0,0,1) +Zero = Vector3c(0,0,0) +__init__((object)arg1) → None +__init__( (object)arg1, (Vector3c)other) -> None +__init__( (object)arg1 [, (complex)x=0j [, (complex)y=0j [, (complex)z=0j]]]) -> None +asDiagonal((Vector3c)arg1) → Matrix3c : +Return diagonal matrix with this vector on the diagonal. +2.4. +Yade modules reference +443 + +Yade Documentation, Release 3rd ed. +cols((Vector3c)arg1) → int : +Number of columns. +cross((Vector3c)arg1, (Vector3c)arg2) → Vector3c +dot((Vector3c)arg1, (Vector3c)other) → complex : +Dot product with other. +isApprox((Vector3c)arg1, (Vector3c)other[, (float)prec=1e-12]) → bool : +Approximate comparison with precision prec. +maxAbsCoeff((Vector3c)arg1) → float : +Maximum absolute value over all elements. +mean((Vector3c)arg1) → complex : +Mean value over all elements. +norm((Vector3c)arg1) → float : +Euclidean norm. +normalize((Vector3c)arg1) → None : +Normalize this object in-place. +normalized((Vector3c)arg1) → Vector3c : +Return normalized copy of this object +outer((Vector3c)arg1, (Vector3c)other) → Matrix3c : +Outer product with other. +prod((Vector3c)arg1) → complex : +Product of all elements. +pruned((Vector3c)arg1[, (float)absTol=1e-06]) → Vector3c : +Zero all elements which are greater than absTol. Negative zeros are not pruned. +rows((Vector3c)arg1) → int : +Number of rows. +squaredNorm((Vector3c)arg1) → float : +Square of the Euclidean norm. +sum((Vector3c)arg1) → complex : +Sum of all elements. +xy((Vector3c)arg1) → Vector2c +xz((Vector3c)arg1) → Vector2c +yx((Vector3c)arg1) → Vector2c +yz((Vector3c)arg1) → Vector2c +zx((Vector3c)arg1) → Vector2c +zy((Vector3c)arg1) → Vector2c +class Vector3i +3-dimensional integer vector. +Supported operations (i if an int, v is a Vector3i): -v, v+v, v+=v, v-v, v-=v, v*i, i*v, v*=i, +v==v, v!=v. +Implicit conversion from sequence (list, tuple, …) of 3 integers. +Static attributes: Zero, Ones, UnitX, UnitY, UnitZ. +Identity = Vector3i(1,0,0) +Ones = Vector3i(1,1,1) +444 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +static Random() → Vector3i : +Return an object where all elements are randomly set to values between 0 and 1. +static Unit((int)arg1) → Vector3i +UnitX = Vector3i(1,0,0) +UnitY = Vector3i(0,1,0) +UnitZ = Vector3i(0,0,1) +Zero = Vector3i(0,0,0) +__init__((object)arg1) → None +__init__( (object)arg1, (Vector3i)other) -> None +__init__( (object)arg1 [, (int)x=0 [, (int)y=0 [, (int)z=0]]]) -> None +asDiagonal((Vector3i)arg1) → object : +Return diagonal matrix with this vector on the diagonal. +cols((Vector3i)arg1) → int : +Number of columns. +cross((Vector3i)arg1, (Vector3i)arg2) → Vector3i +dot((Vector3i)arg1, (Vector3i)other) → int : +Dot product with other. +isApprox((Vector3i)arg1, (Vector3i)other[, (int)prec=0]) → bool : +Approximate comparison with precision prec. +maxAbsCoeff((Vector3i)arg1) → int : +Maximum absolute value over all elements. +maxCoeff((Vector3i)arg1) → int : +Maximum value over all elements. +mean((Vector3i)arg1) → int : +Mean value over all elements. +minCoeff((Vector3i)arg1) → int : +Minimum value over all elements. +outer((Vector3i)arg1, (Vector3i)other) → object : +Outer product with other. +prod((Vector3i)arg1) → int : +Product of all elements. +rows((Vector3i)arg1) → int : +Number of rows. +sum((Vector3i)arg1) → int : +Sum of all elements. +xy((Vector3i)arg1) → Vector2i +xz((Vector3i)arg1) → Vector2i +yx((Vector3i)arg1) → Vector2i +yz((Vector3i)arg1) → Vector2i +zx((Vector3i)arg1) → Vector2i +zy((Vector3i)arg1) → Vector2i +class Vector4 +4-dimensional float vector. +2.4. +Yade modules reference +445 + +Yade Documentation, Release 3rd ed. +Supported operations (f if a float/int, v is a Vector3): -v, v+v, v+=v, v-v, v-=v, v*f, f*v, +v*=f, v/f, v/=f, v==v, v!=v. +Implicit conversion from sequence (list, tuple, …) of 4 floats. +Static attributes: Zero, Ones. +Identity = Vector4(1,0,0, 0) +Ones = Vector4(1,1,1, 1) +static Random() → Vector4 : +Return an object where all elements are randomly set to values between 0 and 1. +static Unit((int)arg1) → Vector4 +Zero = Vector4(0,0,0, 0) +__init__((object)arg1) → None +__init__( (object)arg1, (Vector4)other) -> None +__init__( (object)arg1, (float)v0, (float)v1, (float)v2, (float)v3) -> None +asDiagonal((Vector4)arg1) → object : +Return diagonal matrix with this vector on the diagonal. +cols((Vector4)arg1) → int : +Number of columns. +dot((Vector4)arg1, (Vector4)other) → float : +Dot product with other. +isApprox((Vector4)arg1, (Vector4)other[, (float)prec=1e-12]) → bool : +Approximate comparison with precision prec. +maxAbsCoeff((Vector4)arg1) → float : +Maximum absolute value over all elements. +maxCoeff((Vector4)arg1) → float : +Maximum value over all elements. +mean((Vector4)arg1) → float : +Mean value over all elements. +minCoeff((Vector4)arg1) → float : +Minimum value over all elements. +norm((Vector4)arg1) → float : +Euclidean norm. +normalize((Vector4)arg1) → None : +Normalize this object in-place. +normalized((Vector4)arg1) → Vector4 : +Return normalized copy of this object +outer((Vector4)arg1, (Vector4)other) → object : +Outer product with other. +prod((Vector4)arg1) → float : +Product of all elements. +pruned((Vector4)arg1[, (float)absTol=1e-06]) → Vector4 : +Zero all elements which are greater than absTol. Negative zeros are not pruned. +rows((Vector4)arg1) → int : +Number of rows. +squaredNorm((Vector4)arg1) → float : +Square of the Euclidean norm. +446 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +sum((Vector4)arg1) → float : +Sum of all elements. +class Vector6 +6-dimensional float vector. +Supported operations (f if a float/int, v is a Vector6): -v, v+v, v+=v, v-v, v-=v, v*f, f*v, +v*=f, v/f, v/=f, v==v, v!=v. +Implicit conversion from sequence (list, tuple, …) of 6 floats. +Static attributes: Zero, Ones. +Identity = Vector6(1,0,0, 0,0,0) +Ones = Vector6(1,1,1, 1,1,1) +static Random() → Vector6 : +Return an object where all elements are randomly set to values between 0 and 1. +static Unit((int)arg1) → Vector6 +Zero = Vector6(0,0,0, 0,0,0) +__init__((object)arg1) → None +__init__( (object)arg1, (Vector6)other) -> None +__init__( (object)arg1, (float)v0, (float)v1, (float)v2, (float)v3, (float)v4, (float)v5) -> +object +__init__( (object)arg1, (Vector3)head, (Vector3)tail) -> object +asDiagonal((Vector6)arg1) → Matrix6 : +Return diagonal matrix with this vector on the diagonal. +cols((Vector6)arg1) → int : +Number of columns. +dot((Vector6)arg1, (Vector6)other) → float : +Dot product with other. +head((Vector6)arg1) → Vector3 +isApprox((Vector6)arg1, (Vector6)other[, (float)prec=1e-12]) → bool : +Approximate comparison with precision prec. +maxAbsCoeff((Vector6)arg1) → float : +Maximum absolute value over all elements. +maxCoeff((Vector6)arg1) → float : +Maximum value over all elements. +mean((Vector6)arg1) → float : +Mean value over all elements. +minCoeff((Vector6)arg1) → float : +Minimum value over all elements. +norm((Vector6)arg1) → float : +Euclidean norm. +normalize((Vector6)arg1) → None : +Normalize this object in-place. +normalized((Vector6)arg1) → Vector6 : +Return normalized copy of this object +outer((Vector6)arg1, (Vector6)other) → Matrix6 : +Outer product with other. +2.4. +Yade modules reference +447 + +Yade Documentation, Release 3rd ed. +prod((Vector6)arg1) → float : +Product of all elements. +pruned((Vector6)arg1[, (float)absTol=1e-06]) → Vector6 : +Zero all elements which are greater than absTol. Negative zeros are not pruned. +rows((Vector6)arg1) → int : +Number of rows. +squaredNorm((Vector6)arg1) → float : +Square of the Euclidean norm. +sum((Vector6)arg1) → float : +Sum of all elements. +tail((Vector6)arg1) → Vector3 +class Vector6c +/TODO/ +Identity = Vector6c(1,0,0, 0,0,0) +Ones = Vector6c(1,1,1, 1,1,1) +static Random() → Vector6c : +Return an object where all elements are randomly set to values between 0 and 1. +static Unit((int)arg1) → Vector6c +Zero = Vector6c(0,0,0, 0,0,0) +__init__((object)arg1) → None +__init__( (object)arg1, (Vector6c)other) -> None +__init__( (object)arg1, (complex)v0, (complex)v1, (complex)v2, (complex)v3, (com- +plex)v4, (complex)v5) -> object +__init__( (object)arg1, (Vector3c)head, (Vector3c)tail) -> object +asDiagonal((Vector6c)arg1) → Matrix6c : +Return diagonal matrix with this vector on the diagonal. +cols((Vector6c)arg1) → int : +Number of columns. +dot((Vector6c)arg1, (Vector6c)other) → complex : +Dot product with other. +head((Vector6c)arg1) → Vector3c +isApprox((Vector6c)arg1, (Vector6c)other[, (float)prec=1e-12]) → bool : +Approximate comparison with precision prec. +maxAbsCoeff((Vector6c)arg1) → float : +Maximum absolute value over all elements. +mean((Vector6c)arg1) → complex : +Mean value over all elements. +norm((Vector6c)arg1) → float : +Euclidean norm. +normalize((Vector6c)arg1) → None : +Normalize this object in-place. +normalized((Vector6c)arg1) → Vector6c : +Return normalized copy of this object +outer((Vector6c)arg1, (Vector6c)other) → Matrix6c : +Outer product with other. +448 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +prod((Vector6c)arg1) → complex : +Product of all elements. +pruned((Vector6c)arg1[, (float)absTol=1e-06]) → Vector6c : +Zero all elements which are greater than absTol. Negative zeros are not pruned. +rows((Vector6c)arg1) → int : +Number of rows. +squaredNorm((Vector6c)arg1) → float : +Square of the Euclidean norm. +sum((Vector6c)arg1) → complex : +Sum of all elements. +tail((Vector6c)arg1) → Vector3c +class Vector6i +6-dimensional float vector. +Supported operations (f if a float/int, v is a Vector6): -v, v+v, v+=v, v-v, v-=v, v*f, f*v, +v*=f, v/f, v/=f, v==v, v!=v. +Implicit conversion from sequence (list, tuple, …) of 6 ints. +Static attributes: Zero, Ones. +Identity = Vector6i(1,0,0, 0,0,0) +Ones = Vector6i(1,1,1, 1,1,1) +static Random() → Vector6i : +Return an object where all elements are randomly set to values between 0 and 1. +static Unit((int)arg1) → Vector6i +Zero = Vector6i(0,0,0, 0,0,0) +__init__((object)arg1) → None +__init__( (object)arg1, (Vector6i)other) -> None +__init__( (object)arg1, (int)v0, (int)v1, (int)v2, (int)v3, (int)v4, (int)v5) -> object +__init__( (object)arg1, (Vector3i)head, (Vector3i)tail) -> object +asDiagonal((Vector6i)arg1) → object : +Return diagonal matrix with this vector on the diagonal. +cols((Vector6i)arg1) → int : +Number of columns. +dot((Vector6i)arg1, (Vector6i)other) → int : +Dot product with other. +head((Vector6i)arg1) → Vector3i +isApprox((Vector6i)arg1, (Vector6i)other[, (int)prec=0]) → bool : +Approximate comparison with precision prec. +maxAbsCoeff((Vector6i)arg1) → int : +Maximum absolute value over all elements. +maxCoeff((Vector6i)arg1) → int : +Maximum value over all elements. +mean((Vector6i)arg1) → int : +Mean value over all elements. +minCoeff((Vector6i)arg1) → int : +Minimum value over all elements. +2.4. +Yade modules reference +449 + +Yade Documentation, Release 3rd ed. +outer((Vector6i)arg1, (Vector6i)other) → object : +Outer product with other. +prod((Vector6i)arg1) → int : +Product of all elements. +rows((Vector6i)arg1) → int : +Number of rows. +sum((Vector6i)arg1) → int : +Sum of all elements. +tail((Vector6i)arg1) → Vector3i +class VectorX +Dynamic-sized float vector. +Supported operations (f if a float/int, v is a VectorX): -v, v+v, v+=v, v-v, v-=v, v*f, f*v, +v*=f, v/f, v/=f, v==v, v!=v. +Implicit conversion from sequence (list, tuple, …) of X floats. +static Ones((int)arg1) → VectorX +static Random((int)len) → VectorX : +Return vector of given length with all elements set to values between 0 and 1 randomly. +static Unit((int)arg1, (int)arg2) → VectorX +static Zero((int)arg1) → VectorX +__init__((object)arg1) → None +__init__( (object)arg1, (VectorX)other) -> None +__init__( (object)arg1, (object)vv) -> object +asDiagonal((VectorX)arg1) → MatrixX : +Return diagonal matrix with this vector on the diagonal. +cols((VectorX)arg1) → int : +Number of columns. +dot((VectorX)arg1, (VectorX)other) → float : +Dot product with other. +isApprox((VectorX)arg1, (VectorX)other[, (float)prec=1e-12]) → bool : +Approximate comparison with precision prec. +maxAbsCoeff((VectorX)arg1) → float : +Maximum absolute value over all elements. +maxCoeff((VectorX)arg1) → float : +Maximum value over all elements. +mean((VectorX)arg1) → float : +Mean value over all elements. +minCoeff((VectorX)arg1) → float : +Minimum value over all elements. +norm((VectorX)arg1) → float : +Euclidean norm. +normalize((VectorX)arg1) → None : +Normalize this object in-place. +normalized((VectorX)arg1) → VectorX : +Return normalized copy of this object +450 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +outer((VectorX)arg1, (VectorX)other) → MatrixX : +Outer product with other. +prod((VectorX)arg1) → float : +Product of all elements. +pruned((VectorX)arg1[, (float)absTol=1e-06]) → VectorX : +Zero all elements which are greater than absTol. Negative zeros are not pruned. +resize((VectorX)arg1, (int)arg2) → None +rows((VectorX)arg1) → int : +Number of rows. +squaredNorm((VectorX)arg1) → float : +Square of the Euclidean norm. +sum((VectorX)arg1) → float : +Sum of all elements. +class VectorXc +/TODO/ +static Ones((int)arg1) → VectorXc +static Random((int)len) → VectorXc : +Return vector of given length with all elements set to values between 0 and 1 randomly. +static Unit((int)arg1, (int)arg2) → VectorXc +static Zero((int)arg1) → VectorXc +__init__((object)arg1) → None +__init__( (object)arg1, (VectorXc)other) -> None +__init__( (object)arg1, (object)vv) -> object +asDiagonal((VectorXc)arg1) → MatrixXc : +Return diagonal matrix with this vector on the diagonal. +cols((VectorXc)arg1) → int : +Number of columns. +dot((VectorXc)arg1, (VectorXc)other) → complex : +Dot product with other. +isApprox((VectorXc)arg1, (VectorXc)other[, (float)prec=1e-12]) → bool : +Approximate comparison with precision prec. +maxAbsCoeff((VectorXc)arg1) → float : +Maximum absolute value over all elements. +mean((VectorXc)arg1) → complex : +Mean value over all elements. +norm((VectorXc)arg1) → float : +Euclidean norm. +normalize((VectorXc)arg1) → None : +Normalize this object in-place. +normalized((VectorXc)arg1) → VectorXc : +Return normalized copy of this object +outer((VectorXc)arg1, (VectorXc)other) → MatrixXc : +Outer product with other. +prod((VectorXc)arg1) → complex : +Product of all elements. +2.4. +Yade modules reference +451 + +Yade Documentation, Release 3rd ed. +pruned((VectorXc)arg1[, (float)absTol=1e-06]) → VectorXc : +Zero all elements which are greater than absTol. Negative zeros are not pruned. +resize((VectorXc)arg1, (int)arg2) → None +rows((VectorXc)arg1) → int : +Number of rows. +squaredNorm((VectorXc)arg1) → float : +Square of the Euclidean norm. +sum((VectorXc)arg1) → complex : +Sum of all elements. +vectorize = False +class yade._minieigenHP.Matrix3 +3x3 float matrix. +Supported operations (m is a Matrix3, f if a float/int, v is a Vector3): -m, m+m, m+=m, m-m, m-=m, +m*f, f*m, m*=f, m/f, m/=f, m*m, m*=m, m*v, v*m, m==m, m!=m. +Static attributes: Zero, Ones, Identity. +Identity = Matrix3(1,0,0, 0,1,0, 0,0,1) +Ones = Matrix3(1,1,1, 1,1,1, 1,1,1) +static Random() → Matrix3 : +Return an object where all elements are randomly set to values between 0 and 1. +Zero = Matrix3(0,0,0, 0,0,0, 0,0,0) +__init__((object)arg1) → None +__init__( (object)arg1, (Quaternion)q) -> None +__init__( (object)arg1, (Matrix3)other) -> None +__init__( (object)arg1, (Vector3)diag) -> object +__init__( +(object)arg1, +(float)m00, +(float)m01, +(float)m02, +(float)m10, +(float)m11, +(float)m12, (float)m20, (float)m21, (float)m22) -> object +__init__( (object)arg1, (Vector3)r0, (Vector3)r1, (Vector3)r2 [, (bool)cols=False]) -> object +col((Matrix3)arg1, (int)col) → Vector3 : +Return column as vector. +cols((Matrix3)arg1) → int : +Number of columns. +computeUnitaryPositive((Matrix3)arg1) → tuple : +Compute polar decomposition (unitary matrix U and positive semi-definite symmetric matrix +P such that self=U*P). +determinant((Matrix3)arg1) → float : +Return matrix determinant. +diagonal((Matrix3)arg1) → Vector3 : +Return diagonal as vector. +inverse((Matrix3)arg1) → Matrix3 : +Return inverted matrix. +isApprox((Matrix3)arg1, (Matrix3)other[, (float)prec=1e-12]) → bool : +Approximate comparison with precision prec. +jacobiSVD((Matrix3)arg1) → tuple : +Compute +SVD +decomposition +of +square +matrix, +retuns +(U,S,V) +such +that +self=U*S*V.transpose() +452 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +maxAbsCoeff((Matrix3)arg1) → float : +Maximum absolute value over all elements. +maxCoeff((Matrix3)arg1) → float : +Maximum value over all elements. +mean((Matrix3)arg1) → float : +Mean value over all elements. +minCoeff((Matrix3)arg1) → float : +Minimum value over all elements. +norm((Matrix3)arg1) → float : +Euclidean norm. +normalize((Matrix3)arg1) → None : +Normalize this object in-place. +normalized((Matrix3)arg1) → Matrix3 : +Return normalized copy of this object +polarDecomposition((Matrix3)arg1) → tuple : +Alias for computeUnitaryPositive. +prod((Matrix3)arg1) → float : +Product of all elements. +pruned((Matrix3)arg1[, (float)absTol=1e-06]) → Matrix3 : +Zero all elements which are greater than absTol. Negative zeros are not pruned. +row((Matrix3)arg1, (int)row) → Vector3 : +Return row as vector. +rows((Matrix3)arg1) → int : +Number of rows. +selfAdjointEigenDecomposition((Matrix3)arg1) → tuple : +Compute eigen (spectral) decomposition of symmetric matrix, returns (eigVecs,eigVals). +eigVecs is orthogonal Matrix3 with columns ar normalized eigenvectors, eigVals is Vector3 +with corresponding eigenvalues. self=eigVecs*diag(eigVals)*eigVecs.transpose(). +spectralDecomposition((Matrix3)arg1) → tuple : +Alias for selfAdjointEigenDecomposition. +squaredNorm((Matrix3)arg1) → float : +Square of the Euclidean norm. +sum((Matrix3)arg1) → float : +Sum of all elements. +svd((Matrix3)arg1) → tuple : +Alias for jacobiSVD. +trace((Matrix3)arg1) → float : +Return sum of diagonal elements. +transpose((Matrix3)arg1) → Matrix3 : +Return transposed matrix. +class yade._minieigenHP.Matrix3c +/TODO/ +Identity = Matrix3c(1,0,0, 0,1,0, 0,0,1) +Ones = Matrix3c(1,1,1, 1,1,1, 1,1,1) +static Random() → Matrix3c : +Return an object where all elements are randomly set to values between 0 and 1. +Zero = Matrix3c(0,0,0, 0,0,0, 0,0,0) +2.4. +Yade modules reference +453 + +Yade Documentation, Release 3rd ed. +__init__((object)arg1) → None +__init__( (object)arg1, (Matrix3c)other) -> None +__init__( (object)arg1, (Vector3c)diag) -> object +__init__( (object)arg1, (complex)m00, (complex)m01, (complex)m02, (complex)m10, (com- +plex)m11, (complex)m12, (complex)m20, (complex)m21, (complex)m22) -> object +__init__( (object)arg1, (Vector3c)r0, (Vector3c)r1, (Vector3c)r2 [, (bool)cols=False]) -> +object +col((Matrix3c)arg1, (int)col) → Vector3c : +Return column as vector. +cols((Matrix3c)arg1) → int : +Number of columns. +determinant((Matrix3c)arg1) → complex : +Return matrix determinant. +diagonal((Matrix3c)arg1) → Vector3c : +Return diagonal as vector. +inverse((Matrix3c)arg1) → Matrix3c : +Return inverted matrix. +isApprox((Matrix3c)arg1, (Matrix3c)other[, (float)prec=1e-12]) → bool : +Approximate comparison with precision prec. +maxAbsCoeff((Matrix3c)arg1) → float : +Maximum absolute value over all elements. +mean((Matrix3c)arg1) → complex : +Mean value over all elements. +norm((Matrix3c)arg1) → float : +Euclidean norm. +normalize((Matrix3c)arg1) → None : +Normalize this object in-place. +normalized((Matrix3c)arg1) → Matrix3c : +Return normalized copy of this object +prod((Matrix3c)arg1) → complex : +Product of all elements. +pruned((Matrix3c)arg1[, (float)absTol=1e-06]) → Matrix3c : +Zero all elements which are greater than absTol. Negative zeros are not pruned. +row((Matrix3c)arg1, (int)row) → Vector3c : +Return row as vector. +rows((Matrix3c)arg1) → int : +Number of rows. +squaredNorm((Matrix3c)arg1) → float : +Square of the Euclidean norm. +sum((Matrix3c)arg1) → complex : +Sum of all elements. +trace((Matrix3c)arg1) → complex : +Return sum of diagonal elements. +transpose((Matrix3c)arg1) → Matrix3c : +Return transposed matrix. +454 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +class yade._minieigenHP.Matrix6 +6x6 float matrix. Constructed from 4 3x3 sub-matrices, from 6xVector6 (rows). +Supported operations (m is a Matrix6, f if a float/int, v is a Vector6): -m, m+m, m+=m, m-m, m-=m, +m*f, f*m, m*=f, m/f, m/=f, m*m, m*=m, m*v, v*m, m==m, m!=m. +Static attributes: Zero, Ones, Identity. +Identity = Matrix6( (1,0,0,0,0,0), (0,1,0,0,0,0), (0,0,1,0,0,0), (0,0,0,1,0,0), (0,0,0,0,1,0), (0,0,0,0,0,1) ) +Ones = Matrix6( (1,1,1,1,1,1), (1,1,1,1,1,1), (1,1,1,1,1,1), (1,1,1,1,1,1), (1,1,1,1,1,1), (1,1,1,1,1,1) ) +static Random() → Matrix6 : +Return an object where all elements are randomly set to values between 0 and 1. +Zero = Matrix6( (0,0,0,0,0,0), (0,0,0,0,0,0), (0,0,0,0,0,0), (0,0,0,0,0,0), (0,0,0,0,0,0), (0,0,0,0,0,0) ) +__init__((object)arg1) → None +__init__( (object)arg1, (Matrix6)other) -> None +__init__( (object)arg1, (Vector6)diag) -> object +__init__( (object)arg1, (Matrix3)ul, (Matrix3)ur, (Matrix3)ll, (Matrix3)lr) -> object +__init__( (object)arg1, (Vector6)l0, (Vector6)l1, (Vector6)l2, (Vector6)l3, (Vector6)l4, (Vec- +tor6)l5 [, (bool)cols=False]) -> object +col((Matrix6)arg1, (int)col) → Vector6 : +Return column as vector. +cols((Matrix6)arg1) → int : +Number of columns. +computeUnitaryPositive((Matrix6)arg1) → tuple : +Compute polar decomposition (unitary matrix U and positive semi-definite symmetric matrix +P such that self=U*P). +determinant((Matrix6)arg1) → float : +Return matrix determinant. +diagonal((Matrix6)arg1) → Vector6 : +Return diagonal as vector. +inverse((Matrix6)arg1) → Matrix6 : +Return inverted matrix. +isApprox((Matrix6)arg1, (Matrix6)other[, (float)prec=1e-12]) → bool : +Approximate comparison with precision prec. +jacobiSVD((Matrix6)arg1) → tuple : +Compute +SVD +decomposition +of +square +matrix, +retuns +(U,S,V) +such +that +self=U*S*V.transpose() +ll((Matrix6)arg1) → Matrix3 : +Return lower-left 3x3 block +lr((Matrix6)arg1) → Matrix3 : +Return lower-right 3x3 block +maxAbsCoeff((Matrix6)arg1) → float : +Maximum absolute value over all elements. +maxCoeff((Matrix6)arg1) → float : +Maximum value over all elements. +mean((Matrix6)arg1) → float : +Mean value over all elements. +minCoeff((Matrix6)arg1) → float : +Minimum value over all elements. +2.4. +Yade modules reference +455 + +Yade Documentation, Release 3rd ed. +norm((Matrix6)arg1) → float : +Euclidean norm. +normalize((Matrix6)arg1) → None : +Normalize this object in-place. +normalized((Matrix6)arg1) → Matrix6 : +Return normalized copy of this object +polarDecomposition((Matrix6)arg1) → tuple : +Alias for computeUnitaryPositive. +prod((Matrix6)arg1) → float : +Product of all elements. +pruned((Matrix6)arg1[, (float)absTol=1e-06]) → Matrix6 : +Zero all elements which are greater than absTol. Negative zeros are not pruned. +row((Matrix6)arg1, (int)row) → Vector6 : +Return row as vector. +rows((Matrix6)arg1) → int : +Number of rows. +selfAdjointEigenDecomposition((Matrix6)arg1) → tuple : +Compute eigen (spectral) decomposition of symmetric matrix, returns (eigVecs,eigVals). +eigVecs is orthogonal Matrix3 with columns ar normalized eigenvectors, eigVals is Vector3 +with corresponding eigenvalues. self=eigVecs*diag(eigVals)*eigVecs.transpose(). +spectralDecomposition((Matrix6)arg1) → tuple : +Alias for selfAdjointEigenDecomposition. +squaredNorm((Matrix6)arg1) → float : +Square of the Euclidean norm. +sum((Matrix6)arg1) → float : +Sum of all elements. +svd((Matrix6)arg1) → tuple : +Alias for jacobiSVD. +trace((Matrix6)arg1) → float : +Return sum of diagonal elements. +transpose((Matrix6)arg1) → Matrix6 : +Return transposed matrix. +ul((Matrix6)arg1) → Matrix3 : +Return upper-left 3x3 block +ur((Matrix6)arg1) → Matrix3 : +Return upper-right 3x3 block +class yade._minieigenHP.Matrix6c +/TODO/ +Identity = Matrix6c( (1,0,0,0,0,0), (0,1,0,0,0,0), (0,0,1,0,0,0), (0,0,0,1,0,0), (0,0,0,0,1,0), (0,0,0,0,0,1) ) +Ones = Matrix6c( (1,1,1,1,1,1), (1,1,1,1,1,1), (1,1,1,1,1,1), (1,1,1,1,1,1), (1,1,1,1,1,1), (1,1,1,1,1,1) ) +static Random() → Matrix6c : +Return an object where all elements are randomly set to values between 0 and 1. +Zero = Matrix6c( (0,0,0,0,0,0), (0,0,0,0,0,0), (0,0,0,0,0,0), (0,0,0,0,0,0), (0,0,0,0,0,0), (0,0,0,0,0,0) ) +__init__((object)arg1) → None +__init__( (object)arg1, (Matrix6c)other) -> None +__init__( (object)arg1, (Vector6c)diag) -> object +456 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +__init__( (object)arg1, (Matrix3c)ul, (Matrix3c)ur, (Matrix3c)ll, (Matrix3c)lr) -> object +__init__( (object)arg1, (Vector6c)l0, (Vector6c)l1, (Vector6c)l2, (Vector6c)l3, (Vector6c)l4, +(Vector6c)l5 [, (bool)cols=False]) -> object +col((Matrix6c)arg1, (int)col) → Vector6c : +Return column as vector. +cols((Matrix6c)arg1) → int : +Number of columns. +determinant((Matrix6c)arg1) → complex : +Return matrix determinant. +diagonal((Matrix6c)arg1) → Vector6c : +Return diagonal as vector. +inverse((Matrix6c)arg1) → Matrix6c : +Return inverted matrix. +isApprox((Matrix6c)arg1, (Matrix6c)other[, (float)prec=1e-12]) → bool : +Approximate comparison with precision prec. +ll((Matrix6c)arg1) → Matrix3c : +Return lower-left 3x3 block +lr((Matrix6c)arg1) → Matrix3c : +Return lower-right 3x3 block +maxAbsCoeff((Matrix6c)arg1) → float : +Maximum absolute value over all elements. +mean((Matrix6c)arg1) → complex : +Mean value over all elements. +norm((Matrix6c)arg1) → float : +Euclidean norm. +normalize((Matrix6c)arg1) → None : +Normalize this object in-place. +normalized((Matrix6c)arg1) → Matrix6c : +Return normalized copy of this object +prod((Matrix6c)arg1) → complex : +Product of all elements. +pruned((Matrix6c)arg1[, (float)absTol=1e-06]) → Matrix6c : +Zero all elements which are greater than absTol. Negative zeros are not pruned. +row((Matrix6c)arg1, (int)row) → Vector6c : +Return row as vector. +rows((Matrix6c)arg1) → int : +Number of rows. +squaredNorm((Matrix6c)arg1) → float : +Square of the Euclidean norm. +sum((Matrix6c)arg1) → complex : +Sum of all elements. +trace((Matrix6c)arg1) → complex : +Return sum of diagonal elements. +transpose((Matrix6c)arg1) → Matrix6c : +Return transposed matrix. +ul((Matrix6c)arg1) → Matrix3c : +Return upper-left 3x3 block +2.4. +Yade modules reference +457 + +Yade Documentation, Release 3rd ed. +ur((Matrix6c)arg1) → Matrix3c : +Return upper-right 3x3 block +class yade._minieigenHP.MatrixX +XxX (dynamic-sized) float matrix. Constructed from list of rows (as VectorX). +Supported operations (m is a MatrixX, f if a float/int, v is a VectorX): -m, m+m, m+=m, m-m, m-=m, +m*f, f*m, m*=f, m/f, m/=f, m*m, m*=m, m*v, v*m, m==m, m!=m. +static Identity((int)arg1, (int)rank) → MatrixX : +Create identity matrix with given rank (square). +static Ones((int)rows, (int)cols) → MatrixX : +Create matrix of given dimensions where all elements are set to 1. +static Random((int)rows, (int)cols) → MatrixX : +Create matrix with given dimensions where all elements are set to number between 0 and 1 +(uniformly-distributed). +static Zero((int)rows, (int)cols) → MatrixX : +Create zero matrix of given dimensions +__init__((object)arg1) → None +__init__( (object)arg1, (MatrixX)other) -> None +__init__( (object)arg1, (VectorX)diag) -> object +__init__( (object)arg1 [, +(VectorX)r0=VectorX() [, +(VectorX)r1=VectorX() [, +(Vec- +torX)r2=VectorX() +[, +(VectorX)r3=VectorX() +[, +(VectorX)r4=VectorX() +[, +(Vec- +torX)r5=VectorX() +[, +(VectorX)r6=VectorX() +[, +(VectorX)r7=VectorX() +[, +(Vec- +torX)r8=VectorX() [, (VectorX)r9=VectorX() [, (bool)cols=False]]]]]]]]]]]) -> object +__init__( (object)arg1, (object)rows [, (bool)cols=False]) -> object +col((MatrixX)arg1, (int)col) → VectorX : +Return column as vector. +cols((MatrixX)arg1) → int : +Number of columns. +computeUnitaryPositive((MatrixX)arg1) → tuple : +Compute polar decomposition (unitary matrix U and positive semi-definite symmetric matrix +P such that self=U*P). +determinant((MatrixX)arg1) → float : +Return matrix determinant. +diagonal((MatrixX)arg1) → VectorX : +Return diagonal as vector. +inverse((MatrixX)arg1) → MatrixX : +Return inverted matrix. +isApprox((MatrixX)arg1, (MatrixX)other[, (float)prec=1e-12]) → bool : +Approximate comparison with precision prec. +jacobiSVD((MatrixX)arg1) → tuple : +Compute +SVD +decomposition +of +square +matrix, +retuns +(U,S,V) +such +that +self=U*S*V.transpose() +maxAbsCoeff((MatrixX)arg1) → float : +Maximum absolute value over all elements. +maxCoeff((MatrixX)arg1) → float : +Maximum value over all elements. +mean((MatrixX)arg1) → float : +Mean value over all elements. +458 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +minCoeff((MatrixX)arg1) → float : +Minimum value over all elements. +norm((MatrixX)arg1) → float : +Euclidean norm. +normalize((MatrixX)arg1) → None : +Normalize this object in-place. +normalized((MatrixX)arg1) → MatrixX : +Return normalized copy of this object +polarDecomposition((MatrixX)arg1) → tuple : +Alias for computeUnitaryPositive. +prod((MatrixX)arg1) → float : +Product of all elements. +pruned((MatrixX)arg1[, (float)absTol=1e-06]) → MatrixX : +Zero all elements which are greater than absTol. Negative zeros are not pruned. +resize((MatrixX)arg1, (int)rows, (int)cols) → None : +Change size of the matrix, keep values of elements which exist in the new matrix +row((MatrixX)arg1, (int)row) → VectorX : +Return row as vector. +rows((MatrixX)arg1) → int : +Number of rows. +selfAdjointEigenDecomposition((MatrixX)arg1) → tuple : +Compute eigen (spectral) decomposition of symmetric matrix, returns (eigVecs,eigVals). +eigVecs is orthogonal Matrix3 with columns ar normalized eigenvectors, eigVals is Vector3 +with corresponding eigenvalues. self=eigVecs*diag(eigVals)*eigVecs.transpose(). +spectralDecomposition((MatrixX)arg1) → tuple : +Alias for selfAdjointEigenDecomposition. +squaredNorm((MatrixX)arg1) → float : +Square of the Euclidean norm. +sum((MatrixX)arg1) → float : +Sum of all elements. +svd((MatrixX)arg1) → tuple : +Alias for jacobiSVD. +trace((MatrixX)arg1) → float : +Return sum of diagonal elements. +transpose((MatrixX)arg1) → MatrixX : +Return transposed matrix. +class yade._minieigenHP.MatrixXc +/TODO/ +static Identity((int)arg1, (int)rank) → MatrixXc : +Create identity matrix with given rank (square). +static Ones((int)rows, (int)cols) → MatrixXc : +Create matrix of given dimensions where all elements are set to 1. +static Random((int)rows, (int)cols) → MatrixXc : +Create matrix with given dimensions where all elements are set to number between 0 and 1 +(uniformly-distributed). +static Zero((int)rows, (int)cols) → MatrixXc : +Create zero matrix of given dimensions +2.4. +Yade modules reference +459 + +Yade Documentation, Release 3rd ed. +__init__((object)arg1) → None +__init__( (object)arg1, (MatrixXc)other) -> None +__init__( (object)arg1, (VectorXc)diag) -> object +__init__( (object)arg1 [, (VectorXc)r0=VectorXc() [, (VectorXc)r1=VectorXc() [, (Vec- +torXc)r2=VectorXc() [, +(VectorXc)r3=VectorXc() [, +(VectorXc)r4=VectorXc() [, +(Vec- +torXc)r5=VectorXc() [, +(VectorXc)r6=VectorXc() [, +(VectorXc)r7=VectorXc() [, +(Vec- +torXc)r8=VectorXc() [, (VectorXc)r9=VectorXc() [, (bool)cols=False]]]]]]]]]]]) -> object +__init__( (object)arg1, (object)rows [, (bool)cols=False]) -> object +col((MatrixXc)arg1, (int)col) → VectorXc : +Return column as vector. +cols((MatrixXc)arg1) → int : +Number of columns. +determinant((MatrixXc)arg1) → complex : +Return matrix determinant. +diagonal((MatrixXc)arg1) → VectorXc : +Return diagonal as vector. +inverse((MatrixXc)arg1) → MatrixXc : +Return inverted matrix. +isApprox((MatrixXc)arg1, (MatrixXc)other[, (float)prec=1e-12]) → bool : +Approximate comparison with precision prec. +maxAbsCoeff((MatrixXc)arg1) → float : +Maximum absolute value over all elements. +mean((MatrixXc)arg1) → complex : +Mean value over all elements. +norm((MatrixXc)arg1) → float : +Euclidean norm. +normalize((MatrixXc)arg1) → None : +Normalize this object in-place. +normalized((MatrixXc)arg1) → MatrixXc : +Return normalized copy of this object +prod((MatrixXc)arg1) → complex : +Product of all elements. +pruned((MatrixXc)arg1[, (float)absTol=1e-06]) → MatrixXc : +Zero all elements which are greater than absTol. Negative zeros are not pruned. +resize((MatrixXc)arg1, (int)rows, (int)cols) → None : +Change size of the matrix, keep values of elements which exist in the new matrix +row((MatrixXc)arg1, (int)row) → VectorXc : +Return row as vector. +rows((MatrixXc)arg1) → int : +Number of rows. +squaredNorm((MatrixXc)arg1) → float : +Square of the Euclidean norm. +sum((MatrixXc)arg1) → complex : +Sum of all elements. +trace((MatrixXc)arg1) → complex : +Return sum of diagonal elements. +460 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +transpose((MatrixXc)arg1) → MatrixXc : +Return transposed matrix. +class yade._minieigenHP.Quaternion +Quaternion representing rotation. +Supported operations (q is a Quaternion, v is a Vector3): q*q (rotation composition), q*=q, q*v +(rotating v by q), q==q, q!=q. +Static attributes: Identity. +Note: +Quaternion is represented as axis-angle when printed (e.g. Identity is Quaternion((1, +0,0),0), and can also be constructed from the axis-angle representation. This is however different +from the data stored inside, which can be accessed by indices [0] (x), [1] (y), [2] (z), [3] (w). +To obtain axis-angle programatically, use Quaternion.toAxisAngle which returns the tuple. +Identity = Quaternion((1,0,0),0) +Rotate((Quaternion)arg1, (Vector3)v) → Vector3 +__init__((object)arg1) → None +__init__( (object)arg1, (Vector3)axis, (float)angle) -> object +__init__( (object)arg1, (float)angle, (Vector3)axis) -> object +__init__( (object)arg1, (Vector3)u, (Vector3)v) -> object +__init__( (object)arg1, (float)w, (float)x, (float)y, (float)z) -> None : Initialize +from coefficients. +Note: +The order of coefficients is w, x, y, z. The [] operator numbers them differently, +0…4 for x y z w! +__init__( (object)arg1, (Matrix3)rotMatrix) -> None +__init__( (object)arg1, (Quaternion)other) -> None +angularDistance((Quaternion)arg1, (Quaternion)arg2) → float +conjugate((Quaternion)arg1) → Quaternion +inverse((Quaternion)arg1) → Quaternion +norm((Quaternion)arg1) → float +normalize((Quaternion)arg1) → None +normalized((Quaternion)arg1) → Quaternion +setFromTwoVectors((Quaternion)arg1, (Vector3)u, (Vector3)v) → None +slerp((Quaternion)arg1, (float)t, (Quaternion)other) → Quaternion +toAngleAxis((Quaternion)arg1) → tuple +toAxisAngle((Quaternion)arg1) → tuple +toRotationMatrix((Quaternion)arg1) → Matrix3 +toRotationVector((Quaternion)arg1) → Vector3 +class yade._minieigenHP.Vector2 +3-dimensional float vector. +Supported operations (f if a float/int, v is a Vector3): -v, v+v, v+=v, v-v, v-=v, v*f, f*v, v*=f, +v/f, v/=f, v==v, v!=v. +Implicit conversion from sequence (list, tuple, …) of 2 floats. +2.4. +Yade modules reference +461 + +Yade Documentation, Release 3rd ed. +Static attributes: Zero, Ones, UnitX, UnitY. +Identity = Vector2(1,0) +Ones = Vector2(1,1) +static Random() → Vector2 : +Return an object where all elements are randomly set to values between 0 and 1. +static Unit((int)arg1) → Vector2 +UnitX = Vector2(1,0) +UnitY = Vector2(0,1) +Zero = Vector2(0,0) +__init__((object)arg1) → None +__init__( (object)arg1, (Vector2)other) -> None +__init__( (object)arg1, (float)x, (float)y) -> None +asDiagonal((Vector2)arg1) → object : +Return diagonal matrix with this vector on the diagonal. +cols((Vector2)arg1) → int : +Number of columns. +dot((Vector2)arg1, (Vector2)other) → float : +Dot product with other. +isApprox((Vector2)arg1, (Vector2)other[, (float)prec=1e-12]) → bool : +Approximate comparison with precision prec. +maxAbsCoeff((Vector2)arg1) → float : +Maximum absolute value over all elements. +maxCoeff((Vector2)arg1) → float : +Maximum value over all elements. +mean((Vector2)arg1) → float : +Mean value over all elements. +minCoeff((Vector2)arg1) → float : +Minimum value over all elements. +norm((Vector2)arg1) → float : +Euclidean norm. +normalize((Vector2)arg1) → None : +Normalize this object in-place. +normalized((Vector2)arg1) → Vector2 : +Return normalized copy of this object +outer((Vector2)arg1, (Vector2)other) → object : +Outer product with other. +prod((Vector2)arg1) → float : +Product of all elements. +pruned((Vector2)arg1[, (float)absTol=1e-06]) → Vector2 : +Zero all elements which are greater than absTol. Negative zeros are not pruned. +rows((Vector2)arg1) → int : +Number of rows. +squaredNorm((Vector2)arg1) → float : +Square of the Euclidean norm. +462 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +sum((Vector2)arg1) → float : +Sum of all elements. +class yade._minieigenHP.Vector2c +/TODO/ +Identity = Vector2c(1,0) +Ones = Vector2c(1,1) +static Random() → Vector2c : +Return an object where all elements are randomly set to values between 0 and 1. +static Unit((int)arg1) → Vector2c +UnitX = Vector2c(1,0) +UnitY = Vector2c(0,1) +Zero = Vector2c(0,0) +__init__((object)arg1) → None +__init__( (object)arg1, (Vector2c)other) -> None +__init__( (object)arg1, (complex)x, (complex)y) -> None +asDiagonal((Vector2c)arg1) → object : +Return diagonal matrix with this vector on the diagonal. +cols((Vector2c)arg1) → int : +Number of columns. +dot((Vector2c)arg1, (Vector2c)other) → complex : +Dot product with other. +isApprox((Vector2c)arg1, (Vector2c)other[, (float)prec=1e-12]) → bool : +Approximate comparison with precision prec. +maxAbsCoeff((Vector2c)arg1) → float : +Maximum absolute value over all elements. +mean((Vector2c)arg1) → complex : +Mean value over all elements. +norm((Vector2c)arg1) → float : +Euclidean norm. +normalize((Vector2c)arg1) → None : +Normalize this object in-place. +normalized((Vector2c)arg1) → Vector2c : +Return normalized copy of this object +outer((Vector2c)arg1, (Vector2c)other) → object : +Outer product with other. +prod((Vector2c)arg1) → complex : +Product of all elements. +pruned((Vector2c)arg1[, (float)absTol=1e-06]) → Vector2c : +Zero all elements which are greater than absTol. Negative zeros are not pruned. +rows((Vector2c)arg1) → int : +Number of rows. +squaredNorm((Vector2c)arg1) → float : +Square of the Euclidean norm. +sum((Vector2c)arg1) → complex : +Sum of all elements. +2.4. +Yade modules reference +463 + +Yade Documentation, Release 3rd ed. +class yade._minieigenHP.Vector2i +2-dimensional integer vector. +Supported operations (i if an int, v is a Vector2i): -v, v+v, v+=v, v-v, v-=v, v*i, i*v, v*=i, v==v, +v!=v. +Implicit conversion from sequence (list, tuple, …) of 2 integers. +Static attributes: Zero, Ones, UnitX, UnitY. +Identity = Vector2i(1,0) +Ones = Vector2i(1,1) +static Random() → Vector2i : +Return an object where all elements are randomly set to values between 0 and 1. +static Unit((int)arg1) → Vector2i +UnitX = Vector2i(1,0) +UnitY = Vector2i(0,1) +Zero = Vector2i(0,0) +__init__((object)arg1) → None +__init__( (object)arg1, (Vector2i)other) -> None +__init__( (object)arg1, (int)x, (int)y) -> None +asDiagonal((Vector2i)arg1) → object : +Return diagonal matrix with this vector on the diagonal. +cols((Vector2i)arg1) → int : +Number of columns. +dot((Vector2i)arg1, (Vector2i)other) → int : +Dot product with other. +isApprox((Vector2i)arg1, (Vector2i)other[, (int)prec=0]) → bool : +Approximate comparison with precision prec. +maxAbsCoeff((Vector2i)arg1) → int : +Maximum absolute value over all elements. +maxCoeff((Vector2i)arg1) → int : +Maximum value over all elements. +mean((Vector2i)arg1) → int : +Mean value over all elements. +minCoeff((Vector2i)arg1) → int : +Minimum value over all elements. +outer((Vector2i)arg1, (Vector2i)other) → object : +Outer product with other. +prod((Vector2i)arg1) → int : +Product of all elements. +rows((Vector2i)arg1) → int : +Number of rows. +sum((Vector2i)arg1) → int : +Sum of all elements. +class yade._minieigenHP.Vector3 +3-dimensional float vector. +Supported operations (f if a float/int, v is a Vector3): -v, v+v, v+=v, v-v, v-=v, v*f, f*v, v*=f, +v/f, v/=f, v==v, v!=v, plus operations with Matrix3 and Quaternion. +464 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +Implicit conversion from sequence (list, tuple, …) of 3 floats. +Static attributes: Zero, Ones, UnitX, UnitY, UnitZ. +Identity = Vector3(1,0,0) +Ones = Vector3(1,1,1) +static Random() → Vector3 : +Return an object where all elements are randomly set to values between 0 and 1. +static Unit((int)arg1) → Vector3 +UnitX = Vector3(1,0,0) +UnitY = Vector3(0,1,0) +UnitZ = Vector3(0,0,1) +Zero = Vector3(0,0,0) +__init__((object)arg1) → None +__init__( (object)arg1, (Vector3)other) -> None +__init__( (object)arg1 [, (float)x=0.0 [, (float)y=0.0 [, (float)z=0.0]]]) -> None +asDiagonal((Vector3)arg1) → Matrix3 : +Return diagonal matrix with this vector on the diagonal. +cols((Vector3)arg1) → int : +Number of columns. +cross((Vector3)arg1, (Vector3)arg2) → Vector3 +dot((Vector3)arg1, (Vector3)other) → float : +Dot product with other. +isApprox((Vector3)arg1, (Vector3)other[, (float)prec=1e-12]) → bool : +Approximate comparison with precision prec. +maxAbsCoeff((Vector3)arg1) → float : +Maximum absolute value over all elements. +maxCoeff((Vector3)arg1) → float : +Maximum value over all elements. +mean((Vector3)arg1) → float : +Mean value over all elements. +minCoeff((Vector3)arg1) → float : +Minimum value over all elements. +norm((Vector3)arg1) → float : +Euclidean norm. +normalize((Vector3)arg1) → None : +Normalize this object in-place. +normalized((Vector3)arg1) → Vector3 : +Return normalized copy of this object +outer((Vector3)arg1, (Vector3)other) → Matrix3 : +Outer product with other. +prod((Vector3)arg1) → float : +Product of all elements. +pruned((Vector3)arg1[, (float)absTol=1e-06]) → Vector3 : +Zero all elements which are greater than absTol. Negative zeros are not pruned. +2.4. +Yade modules reference +465 + +Yade Documentation, Release 3rd ed. +rows((Vector3)arg1) → int : +Number of rows. +squaredNorm((Vector3)arg1) → float : +Square of the Euclidean norm. +sum((Vector3)arg1) → float : +Sum of all elements. +xy((Vector3)arg1) → Vector2 +xz((Vector3)arg1) → Vector2 +yx((Vector3)arg1) → Vector2 +yz((Vector3)arg1) → Vector2 +zx((Vector3)arg1) → Vector2 +zy((Vector3)arg1) → Vector2 +class yade._minieigenHP.Vector3c +/TODO/ +Identity = Vector3c(1,0,0) +Ones = Vector3c(1,1,1) +static Random() → Vector3c : +Return an object where all elements are randomly set to values between 0 and 1. +static Unit((int)arg1) → Vector3c +UnitX = Vector3c(1,0,0) +UnitY = Vector3c(0,1,0) +UnitZ = Vector3c(0,0,1) +Zero = Vector3c(0,0,0) +__init__((object)arg1) → None +__init__( (object)arg1, (Vector3c)other) -> None +__init__( (object)arg1 [, (complex)x=0j [, (complex)y=0j [, (complex)z=0j]]]) -> None +asDiagonal((Vector3c)arg1) → Matrix3c : +Return diagonal matrix with this vector on the diagonal. +cols((Vector3c)arg1) → int : +Number of columns. +cross((Vector3c)arg1, (Vector3c)arg2) → Vector3c +dot((Vector3c)arg1, (Vector3c)other) → complex : +Dot product with other. +isApprox((Vector3c)arg1, (Vector3c)other[, (float)prec=1e-12]) → bool : +Approximate comparison with precision prec. +maxAbsCoeff((Vector3c)arg1) → float : +Maximum absolute value over all elements. +mean((Vector3c)arg1) → complex : +Mean value over all elements. +norm((Vector3c)arg1) → float : +Euclidean norm. +normalize((Vector3c)arg1) → None : +Normalize this object in-place. +466 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +normalized((Vector3c)arg1) → Vector3c : +Return normalized copy of this object +outer((Vector3c)arg1, (Vector3c)other) → Matrix3c : +Outer product with other. +prod((Vector3c)arg1) → complex : +Product of all elements. +pruned((Vector3c)arg1[, (float)absTol=1e-06]) → Vector3c : +Zero all elements which are greater than absTol. Negative zeros are not pruned. +rows((Vector3c)arg1) → int : +Number of rows. +squaredNorm((Vector3c)arg1) → float : +Square of the Euclidean norm. +sum((Vector3c)arg1) → complex : +Sum of all elements. +xy((Vector3c)arg1) → Vector2c +xz((Vector3c)arg1) → Vector2c +yx((Vector3c)arg1) → Vector2c +yz((Vector3c)arg1) → Vector2c +zx((Vector3c)arg1) → Vector2c +zy((Vector3c)arg1) → Vector2c +class yade._minieigenHP.Vector3i +3-dimensional integer vector. +Supported operations (i if an int, v is a Vector3i): -v, v+v, v+=v, v-v, v-=v, v*i, i*v, v*=i, v==v, +v!=v. +Implicit conversion from sequence (list, tuple, …) of 3 integers. +Static attributes: Zero, Ones, UnitX, UnitY, UnitZ. +Identity = Vector3i(1,0,0) +Ones = Vector3i(1,1,1) +static Random() → Vector3i : +Return an object where all elements are randomly set to values between 0 and 1. +static Unit((int)arg1) → Vector3i +UnitX = Vector3i(1,0,0) +UnitY = Vector3i(0,1,0) +UnitZ = Vector3i(0,0,1) +Zero = Vector3i(0,0,0) +__init__((object)arg1) → None +__init__( (object)arg1, (Vector3i)other) -> None +__init__( (object)arg1 [, (int)x=0 [, (int)y=0 [, (int)z=0]]]) -> None +asDiagonal((Vector3i)arg1) → object : +Return diagonal matrix with this vector on the diagonal. +cols((Vector3i)arg1) → int : +Number of columns. +cross((Vector3i)arg1, (Vector3i)arg2) → Vector3i +2.4. +Yade modules reference +467 + +Yade Documentation, Release 3rd ed. +dot((Vector3i)arg1, (Vector3i)other) → int : +Dot product with other. +isApprox((Vector3i)arg1, (Vector3i)other[, (int)prec=0]) → bool : +Approximate comparison with precision prec. +maxAbsCoeff((Vector3i)arg1) → int : +Maximum absolute value over all elements. +maxCoeff((Vector3i)arg1) → int : +Maximum value over all elements. +mean((Vector3i)arg1) → int : +Mean value over all elements. +minCoeff((Vector3i)arg1) → int : +Minimum value over all elements. +outer((Vector3i)arg1, (Vector3i)other) → object : +Outer product with other. +prod((Vector3i)arg1) → int : +Product of all elements. +rows((Vector3i)arg1) → int : +Number of rows. +sum((Vector3i)arg1) → int : +Sum of all elements. +xy((Vector3i)arg1) → Vector2i +xz((Vector3i)arg1) → Vector2i +yx((Vector3i)arg1) → Vector2i +yz((Vector3i)arg1) → Vector2i +zx((Vector3i)arg1) → Vector2i +zy((Vector3i)arg1) → Vector2i +class yade._minieigenHP.Vector4 +4-dimensional float vector. +Supported operations (f if a float/int, v is a Vector3): -v, v+v, v+=v, v-v, v-=v, v*f, f*v, v*=f, +v/f, v/=f, v==v, v!=v. +Implicit conversion from sequence (list, tuple, …) of 4 floats. +Static attributes: Zero, Ones. +Identity = Vector4(1,0,0, 0) +Ones = Vector4(1,1,1, 1) +static Random() → Vector4 : +Return an object where all elements are randomly set to values between 0 and 1. +static Unit((int)arg1) → Vector4 +Zero = Vector4(0,0,0, 0) +__init__((object)arg1) → None +__init__( (object)arg1, (Vector4)other) -> None +__init__( (object)arg1, (float)v0, (float)v1, (float)v2, (float)v3) -> None +asDiagonal((Vector4)arg1) → object : +Return diagonal matrix with this vector on the diagonal. +468 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +cols((Vector4)arg1) → int : +Number of columns. +dot((Vector4)arg1, (Vector4)other) → float : +Dot product with other. +isApprox((Vector4)arg1, (Vector4)other[, (float)prec=1e-12]) → bool : +Approximate comparison with precision prec. +maxAbsCoeff((Vector4)arg1) → float : +Maximum absolute value over all elements. +maxCoeff((Vector4)arg1) → float : +Maximum value over all elements. +mean((Vector4)arg1) → float : +Mean value over all elements. +minCoeff((Vector4)arg1) → float : +Minimum value over all elements. +norm((Vector4)arg1) → float : +Euclidean norm. +normalize((Vector4)arg1) → None : +Normalize this object in-place. +normalized((Vector4)arg1) → Vector4 : +Return normalized copy of this object +outer((Vector4)arg1, (Vector4)other) → object : +Outer product with other. +prod((Vector4)arg1) → float : +Product of all elements. +pruned((Vector4)arg1[, (float)absTol=1e-06]) → Vector4 : +Zero all elements which are greater than absTol. Negative zeros are not pruned. +rows((Vector4)arg1) → int : +Number of rows. +squaredNorm((Vector4)arg1) → float : +Square of the Euclidean norm. +sum((Vector4)arg1) → float : +Sum of all elements. +class yade._minieigenHP.Vector6 +6-dimensional float vector. +Supported operations (f if a float/int, v is a Vector6): -v, v+v, v+=v, v-v, v-=v, v*f, f*v, v*=f, +v/f, v/=f, v==v, v!=v. +Implicit conversion from sequence (list, tuple, …) of 6 floats. +Static attributes: Zero, Ones. +Identity = Vector6(1,0,0, 0,0,0) +Ones = Vector6(1,1,1, 1,1,1) +static Random() → Vector6 : +Return an object where all elements are randomly set to values between 0 and 1. +static Unit((int)arg1) → Vector6 +Zero = Vector6(0,0,0, 0,0,0) +2.4. +Yade modules reference +469 + +Yade Documentation, Release 3rd ed. +__init__((object)arg1) → None +__init__( (object)arg1, (Vector6)other) -> None +__init__( (object)arg1, (float)v0, (float)v1, (float)v2, (float)v3, (float)v4, (float)v5) -> ob- +ject +__init__( (object)arg1, (Vector3)head, (Vector3)tail) -> object +asDiagonal((Vector6)arg1) → Matrix6 : +Return diagonal matrix with this vector on the diagonal. +cols((Vector6)arg1) → int : +Number of columns. +dot((Vector6)arg1, (Vector6)other) → float : +Dot product with other. +head((Vector6)arg1) → Vector3 +isApprox((Vector6)arg1, (Vector6)other[, (float)prec=1e-12]) → bool : +Approximate comparison with precision prec. +maxAbsCoeff((Vector6)arg1) → float : +Maximum absolute value over all elements. +maxCoeff((Vector6)arg1) → float : +Maximum value over all elements. +mean((Vector6)arg1) → float : +Mean value over all elements. +minCoeff((Vector6)arg1) → float : +Minimum value over all elements. +norm((Vector6)arg1) → float : +Euclidean norm. +normalize((Vector6)arg1) → None : +Normalize this object in-place. +normalized((Vector6)arg1) → Vector6 : +Return normalized copy of this object +outer((Vector6)arg1, (Vector6)other) → Matrix6 : +Outer product with other. +prod((Vector6)arg1) → float : +Product of all elements. +pruned((Vector6)arg1[, (float)absTol=1e-06]) → Vector6 : +Zero all elements which are greater than absTol. Negative zeros are not pruned. +rows((Vector6)arg1) → int : +Number of rows. +squaredNorm((Vector6)arg1) → float : +Square of the Euclidean norm. +sum((Vector6)arg1) → float : +Sum of all elements. +tail((Vector6)arg1) → Vector3 +class yade._minieigenHP.Vector6c +/TODO/ +Identity = Vector6c(1,0,0, 0,0,0) +Ones = Vector6c(1,1,1, 1,1,1) +470 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +static Random() → Vector6c : +Return an object where all elements are randomly set to values between 0 and 1. +static Unit((int)arg1) → Vector6c +Zero = Vector6c(0,0,0, 0,0,0) +__init__((object)arg1) → None +__init__( (object)arg1, (Vector6c)other) -> None +__init__( (object)arg1, (complex)v0, (complex)v1, (complex)v2, (complex)v3, (complex)v4, +(complex)v5) -> object +__init__( (object)arg1, (Vector3c)head, (Vector3c)tail) -> object +asDiagonal((Vector6c)arg1) → Matrix6c : +Return diagonal matrix with this vector on the diagonal. +cols((Vector6c)arg1) → int : +Number of columns. +dot((Vector6c)arg1, (Vector6c)other) → complex : +Dot product with other. +head((Vector6c)arg1) → Vector3c +isApprox((Vector6c)arg1, (Vector6c)other[, (float)prec=1e-12]) → bool : +Approximate comparison with precision prec. +maxAbsCoeff((Vector6c)arg1) → float : +Maximum absolute value over all elements. +mean((Vector6c)arg1) → complex : +Mean value over all elements. +norm((Vector6c)arg1) → float : +Euclidean norm. +normalize((Vector6c)arg1) → None : +Normalize this object in-place. +normalized((Vector6c)arg1) → Vector6c : +Return normalized copy of this object +outer((Vector6c)arg1, (Vector6c)other) → Matrix6c : +Outer product with other. +prod((Vector6c)arg1) → complex : +Product of all elements. +pruned((Vector6c)arg1[, (float)absTol=1e-06]) → Vector6c : +Zero all elements which are greater than absTol. Negative zeros are not pruned. +rows((Vector6c)arg1) → int : +Number of rows. +squaredNorm((Vector6c)arg1) → float : +Square of the Euclidean norm. +sum((Vector6c)arg1) → complex : +Sum of all elements. +tail((Vector6c)arg1) → Vector3c +class yade._minieigenHP.Vector6i +6-dimensional float vector. +Supported operations (f if a float/int, v is a Vector6): -v, v+v, v+=v, v-v, v-=v, v*f, f*v, v*=f, +v/f, v/=f, v==v, v!=v. +2.4. +Yade modules reference +471 + +Yade Documentation, Release 3rd ed. +Implicit conversion from sequence (list, tuple, …) of 6 ints. +Static attributes: Zero, Ones. +Identity = Vector6i(1,0,0, 0,0,0) +Ones = Vector6i(1,1,1, 1,1,1) +static Random() → Vector6i : +Return an object where all elements are randomly set to values between 0 and 1. +static Unit((int)arg1) → Vector6i +Zero = Vector6i(0,0,0, 0,0,0) +__init__((object)arg1) → None +__init__( (object)arg1, (Vector6i)other) -> None +__init__( (object)arg1, (int)v0, (int)v1, (int)v2, (int)v3, (int)v4, (int)v5) -> object +__init__( (object)arg1, (Vector3i)head, (Vector3i)tail) -> object +asDiagonal((Vector6i)arg1) → object : +Return diagonal matrix with this vector on the diagonal. +cols((Vector6i)arg1) → int : +Number of columns. +dot((Vector6i)arg1, (Vector6i)other) → int : +Dot product with other. +head((Vector6i)arg1) → Vector3i +isApprox((Vector6i)arg1, (Vector6i)other[, (int)prec=0]) → bool : +Approximate comparison with precision prec. +maxAbsCoeff((Vector6i)arg1) → int : +Maximum absolute value over all elements. +maxCoeff((Vector6i)arg1) → int : +Maximum value over all elements. +mean((Vector6i)arg1) → int : +Mean value over all elements. +minCoeff((Vector6i)arg1) → int : +Minimum value over all elements. +outer((Vector6i)arg1, (Vector6i)other) → object : +Outer product with other. +prod((Vector6i)arg1) → int : +Product of all elements. +rows((Vector6i)arg1) → int : +Number of rows. +sum((Vector6i)arg1) → int : +Sum of all elements. +tail((Vector6i)arg1) → Vector3i +class yade._minieigenHP.VectorX +Dynamic-sized float vector. +Supported operations (f if a float/int, v is a VectorX): -v, v+v, v+=v, v-v, v-=v, v*f, f*v, v*=f, +v/f, v/=f, v==v, v!=v. +Implicit conversion from sequence (list, tuple, …) of X floats. +static Ones((int)arg1) → VectorX +472 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +static Random((int)len) → VectorX : +Return vector of given length with all elements set to values between 0 and 1 randomly. +static Unit((int)arg1, (int)arg2) → VectorX +static Zero((int)arg1) → VectorX +__init__((object)arg1) → None +__init__( (object)arg1, (VectorX)other) -> None +__init__( (object)arg1, (object)vv) -> object +asDiagonal((VectorX)arg1) → MatrixX : +Return diagonal matrix with this vector on the diagonal. +cols((VectorX)arg1) → int : +Number of columns. +dot((VectorX)arg1, (VectorX)other) → float : +Dot product with other. +isApprox((VectorX)arg1, (VectorX)other[, (float)prec=1e-12]) → bool : +Approximate comparison with precision prec. +maxAbsCoeff((VectorX)arg1) → float : +Maximum absolute value over all elements. +maxCoeff((VectorX)arg1) → float : +Maximum value over all elements. +mean((VectorX)arg1) → float : +Mean value over all elements. +minCoeff((VectorX)arg1) → float : +Minimum value over all elements. +norm((VectorX)arg1) → float : +Euclidean norm. +normalize((VectorX)arg1) → None : +Normalize this object in-place. +normalized((VectorX)arg1) → VectorX : +Return normalized copy of this object +outer((VectorX)arg1, (VectorX)other) → MatrixX : +Outer product with other. +prod((VectorX)arg1) → float : +Product of all elements. +pruned((VectorX)arg1[, (float)absTol=1e-06]) → VectorX : +Zero all elements which are greater than absTol. Negative zeros are not pruned. +resize((VectorX)arg1, (int)arg2) → None +rows((VectorX)arg1) → int : +Number of rows. +squaredNorm((VectorX)arg1) → float : +Square of the Euclidean norm. +sum((VectorX)arg1) → float : +Sum of all elements. +class yade._minieigenHP.VectorXc +/TODO/ +static Ones((int)arg1) → VectorXc +2.4. +Yade modules reference +473 + +Yade Documentation, Release 3rd ed. +static Random((int)len) → VectorXc : +Return vector of given length with all elements set to values between 0 and 1 randomly. +static Unit((int)arg1, (int)arg2) → VectorXc +static Zero((int)arg1) → VectorXc +__init__((object)arg1) → None +__init__( (object)arg1, (VectorXc)other) -> None +__init__( (object)arg1, (object)vv) -> object +asDiagonal((VectorXc)arg1) → MatrixXc : +Return diagonal matrix with this vector on the diagonal. +cols((VectorXc)arg1) → int : +Number of columns. +dot((VectorXc)arg1, (VectorXc)other) → complex : +Dot product with other. +isApprox((VectorXc)arg1, (VectorXc)other[, (float)prec=1e-12]) → bool : +Approximate comparison with precision prec. +maxAbsCoeff((VectorXc)arg1) → float : +Maximum absolute value over all elements. +mean((VectorXc)arg1) → complex : +Mean value over all elements. +norm((VectorXc)arg1) → float : +Euclidean norm. +normalize((VectorXc)arg1) → None : +Normalize this object in-place. +normalized((VectorXc)arg1) → VectorXc : +Return normalized copy of this object +outer((VectorXc)arg1, (VectorXc)other) → MatrixXc : +Outer product with other. +prod((VectorXc)arg1) → complex : +Product of all elements. +pruned((VectorXc)arg1[, (float)absTol=1e-06]) → VectorXc : +Zero all elements which are greater than absTol. Negative zeros are not pruned. +resize((VectorXc)arg1, (int)arg2) → None +rows((VectorXc)arg1) → int : +Number of rows. +squaredNorm((VectorXc)arg1) → float : +Square of the Euclidean norm. +sum((VectorXc)arg1) → complex : +Sum of all elements. +2.4.10 yade.mpy module +This module defines mpirun(), a parallel implementation of run() using a distributed memory approach. +Message passing is done with mpi4py mainly, however some messages are also handled in c++ (with +openmpi). +474 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +Note: +Many internals of the mpy module listed on this page are not helpful to the user. Instead, please +find introductory material on mpy module in user manual. +Logic: +The logic for an initially centralized scene is as follows: +1. Instanciate a complete, ordinary, yade scene +2. Insert subdomains as special yade bodies. This is somehow similar to adding a clump body on the +top of clump members +3. Broadcast this scene to all workers. In the initialization phase the workers will: +• define the bounding box of their assigned bodies and return it to other workers +• detect which assigned bodies are virtually in interaction with other domains (based on their +bounding boxes) and communicate the lists to the relevant workers +• erase the bodies which are neither assigned nor virtually interacting with the subdomain +4. Run a number of ‘regular’ iterations without re-running collision detection (verlet dist mechanism). +In each regular iteration the workers will: +• calculate internal and cross-domains interactions +• execute Newton on assigned bodies (modified Newton skips other domains) +• send updated positions to other workers and partial force on floor to master +5. When one worker triggers collision detection all workers will follow. It will result in updating the +intersections between subdomains. +6. If enabled, bodies may be re-allocated to different domains just after a collision detection, based +on a filter. Custom filters are possible. One is predidefined here (medianFilter) +Rules: +#- intersections[0] has 0-bodies (to which we need to send force) #- intersections[thisDomain] +has ids of the other domains overlapping the current ones #- intersections[otherDomain] has +ids of bodies in _current_ domain which are overlapping with other domain (for which we +need to send updated pos/vel) +Hints: +#- +handle +subD.intersections +with +care +(same +for +mirrorIntersections). +subD.intersections.append() will not reach the c++ object. +subD.intersections can +only be assigned (a list of list of int) +yade.mpy.MAX_RANK_OUTPUT = 5 +larger ranks will be skipped in mprint +yade.mpy.REALLOCATE_FILTER(i, j, giveAway) +Returns bodies in “i” to be assigned to “j” based on median split between the center points of +subdomain’s AABBs If giveAway!=0, positive or negative, “i” will give/acquire this number to “j” +with nothing in return (for load balancing purposes) +class yade.mpy.Timing_comm(inherits object) +Allgather(timing_name, *args, **kwargs) +2.4. +Yade modules reference +475 + +Yade Documentation, Release 3rd ed. +Gather(timing_name, *args, **kwargs) +Gatherv(timing_name, *args, **kwargs) +allreduce(timing_name, *args, **kwargs) +bcast(timing_name, *args, **kwargs) +clear() +enable_timing() +mpiSendStates(timing_name, *args, **kwargs) +mpiWait(timing_name, *args, **kwargs) +mpiWaitReceived(timing_name, *args, **kwargs) +print_all() +recv(timing_name, *args, **kwargs) +send(timing_name, *args, **kwargs) +yade.mpy.bodyErase(ids) +The parallel version of O.bodies.erase(id), should be called collectively else the distributed scenes +become inconsistent with each other (even the subdomains which don’t have ‘id’ can call safely). +For performance, better call on a list: bodyErase([i,j,k]). +yade.mpy.checkAndCollide() +return true if collision detection needs activation in at least one SD, else false. If COPY_MIR- +ROR_BODIES_WHEN_COLLIDE run collider when needed, and in that case return False. +yade.mpy.colorDomains() +Apply color to body to reflect their subdomain idx +yade.mpy.configure() +Import MPI and define context, configure will no spawn workers by itself, that is done by initialize() +openmpi environment variables needs to be set before calling configure() +yade.mpy.declareMasterInteractive() +This is to signal that we are in interactive session, so TIMEOUT will be reset to 0 (ignored) +yade.mpy.disconnect() +Kill all mpi processes, leaving python interpreter to rank 0 as in single-threaded execution. The +scenes in workers are lost since further reconnexion to mpi will just spawn new processes. The +scene in master thread is left unchanged. +yade.mpy.eraseRemote() +yade.mpy.genLocalIntersections(subdomains) +Defines sets of bodies within current domain overlapping with other domains. +The structure of the data for domain ‘k’ is: [[id1, id2, …], <———– intersections[0] = ids of bodies +in domain k interacting with master domain (subdomain k itself excluded) [id3, id4, …], <—— +—– intersections[1] = ids of bodies in domain k interacting with domain rank=1 (subdomain k +itself excluded) … [domain1, domain2, domain3, …], <———- intersections[k] = ranks (not ids!) of +external domains interacting with domain k … ] +yade.mpy.genUpdatedStates(b_ids) +return list of [id,state] (or [id,state,shape] conditionnaly) to be sent to other workers +yade.mpy.initialize(np) +yade.mpy.isendRecvForces() +Communicate forces from subdomain to master Warning: the sending sides (everyone but master) +must wait() the returned list of requests +yade.mpy.makeColorScale(n=None) +yade.mpy.makeMpiArgv() +476 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +yade.mpy.maskedConnection(b, boolArray) +List bodies within a facet selectively, the ones marked ‘True’ in boolArray (i.e. already selected +from another facet) are discarded +yade.mpy.maskedPFacet(b, boolArray) +List bodies within a facet selectively, the ones marked ‘True’ in boolArray (i.e. already selected +from another facet) are discarded +yade.mpy.medianFilter(i, j, giveAway) +Returns bodies in “i” to be assigned to “j” based on median split between the center points of +subdomain’s AABBs If giveAway!=0, positive or negative, “i” will give/acquire this number to “j” +with nothing in return (for load balancing purposes) +yade.mpy.mergeScene() +yade.mpy.migrateBodies(ids, origin, destination) +Reassign bodies from origin to destination. The function has to be called by both origin (send) +and destination (recv). Note: subD.completeSendBodies() will have to be called after a series of +reassignement since subD.sendBodies() is non-blocking +yade.mpy.mpiStats() +yade.mpy.mpirun(nSteps, np=None, withMerge=False) +Parallel version of O.run() using MPI domain decomposition. +Parameters +nSteps : The numer of steps to compute np : number of mpi workers (master+subdomains), if=1 +the function fallback to O.run() withMerge : wether subdomains should be merged into master +at the end of the run (default False). If True the scene in the master process is exactly in the +same state as after O.run(nSteps,True). The merge can be time consumming, it is recommended +to activate only if post-processing or other similar tasks require it. +yade.mpy.mprint(*args, force=False) +Print with rank-reflecting color regardless of mpy.VERBOSE_OUTPUT, still limited to +rank<=mpy.MAX_RANK_OUTPUT +yade.mpy.pairOp(talkTo) +yade.mpy.parallelCollide() +yade.mpy.probeRecvMessage(source, tag) +yade.mpy.projectedBounds(i, j) +Returns sorted list of projections of bounds on a given axis, with bounds taken in i->j and j->i +intersections +yade.mpy.reallocateBodiesPairWiseBlocking(_filter, otherDomain) +Re-assign bodies from/to otherDomain based on ‘_filter’ argument. Requirement: ‘_filter’ is a +function taking ranks of origin and destination and returning the list of bodies (by index) to be +moved. That’s where the decomposition strategy is defined. See example medianFilter (used by +default). +yade.mpy.reallocateBodiesToSubdomains(_filter=, blocking=True) +Re-assign bodies to subdomains based on ‘_filter’ argument. Requirement: ‘_filter’ is a function +taking ranks of origin and destination and returning the list of bodies (by index) to be moved. +That’s where the decomposition strategy is defined. See example medianFilter (used by default). +This function must be called in parallel, hence if ran interactively the command needs to be sent +explicitely: mp.sendCommand(“all”,”reallocateBodiesToSubdomains(medianFilter)”,True) +yade.mpy.reboundRemoteBodies(ids) +update states of bodies handled by other workers, argument ‘states’ is a list of [id,state] (or +[id,state,shape] conditionnaly) +yade.mpy.receiveForces(subdomains) +Accumulate forces from subdomains (only executed by master process), should happen after +2.4. +Yade modules reference +477 + +Yade Documentation, Release 3rd ed. +ForceResetter but before Newton and before any other force-dependent engine (e.g. StressCon- +troller), could be inserted via yade’s pyRunner. +yade.mpy.recordMpiTiming(name, val) +append val to a list of values defined by ‘name’ in the dictionnary timing.mpi +yade.mpy.runOnSynchronouslPairs(workers, command) +Locally (from one worker POV), this function runs interactive mpi tasks defined by ‘command’ on a +list of other workers (typically the list of interacting subdomains). Overall, peer-to-peer connexions +are established so so that ‘command’ is executed symmetrically and simultaneously on both sides +of each worker pair. I.e. if worker “i” executes “command” with argument “j” (index of another +worker), then by design “j” will execute the same thing with argument “i” simultaneously. +In many cases a similar series of data exchanges can be obtained more simply (and fastly) with +asynchronous irecv+send like below. +for w in workers: m=comm.irecv(w) comm.send(data,dest=w) +The above only works if the messages are all known in advance locally, before any communication. +If the interaction with workers[1] depends on the result of a previous interaction with workers[0] +OTOH, it needs synchronous execution, hence this function. Synchronicity is also required if more +than one blocking call is present in ‘command’, else an obvious deadlock as if ‘irecv’ was replaced +by ‘recv’ in that naive loop. Both cases occur with the ‘medianFilter’ algorithm, hence why we +need this synchronous method. +In this function pair connexions are established by the workers in a non-supervized and non- +deterministic manner. Each time an interactive communication (i,j) is established ‘command’ is +executed simultaneously by i and j. It is guaranted that all possible pairs are visited. +The function can be used for all-to-all operations (N^2 pairs), but more interestingly it works +with workers=intersections[rank] (O(N) pairs). It can be tested with the dummy funtion ‘pairOp’: +runOnSynchronouslPairs(range(numThreads),pairOp) +command: a function taking index of another worker as argument, can include blocking com- +munications with the other worker since runOnSynchronouslPairs guarantee that the other +worker will be running the command symmetrically. +yade.mpy.sendCommand(executors, command, wait=True, workerToWorker=False) +Send a command to a worker (or list of) from master or from another worker. Accepted executors +are “i”, “[i,j,k]”, “slaves”, “all” (then even master will execute the command). +yade.mpy.sendRecvStates() +yade.mpy.shrinkIntersections() +Reduce intersections and mirrorIntersections to bodies effectively interacting with another statefull +body form current subdomain This will reduce the number of updates in sendRecvStates Initial +lists are backed-up and need to be restored (and all states updated) before collision detection (see +checkAndCollide()) +yade.mpy.spawnedProcessWaitCommand() +yade.mpy.splitScene() +Split a monolithic scene into distributed scenes on threads. +Precondition: the bodies have subdomain no. set in user script +yade.mpy.unboundRemoteBodies() +Turn bounding boxes on/off depending on rank +yade.mpy.updateAllIntersections() +yade.mpy.updateDomainBounds(subdomains) +Update bounds of current subdomain, broadcast, and receive updated bounds from other subdo- +mains Precondition: collider.boundDispatcher.__call__() +yade.mpy.updateMirrorOwners() +478 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +yade.mpy.updateRemoteStates(states, setBounded=False) +update states of bodies handled by other workers, argument ‘states’ is a list of [id,state] (or +[id,state,shape] conditionnaly) +yade.mpy.waitForces() +wait until all forces are sent to master. O.freqs is empty for master, and for all threads if not +ACCUMULATE_FORCES +yade.mpy.wprint(*args) +Print with rank-reflecting color, only if mpy.VERBOSE_OUTPUT=True (else see mpy.mprint), +limited to rank<=mpy.MAX_RANK_OUTPUT +2.4.11 yade.pack module +Creating packings and filling volumes defined by boundary representation or constructive solid geometry. +For examples, see +• examples/gts-horse/gts-operators.py +• examples/gts-horse/gts-random-pack-obb.py +• examples/gts-horse/gts-random-pack.py +• examples/test/pack-cloud.py +• examples/test/pack-predicates.py +• examples/packs/packs.py +• examples/gts-horse/gts-horse.py +• examples/WireMatPM/wirepackings.py +yade.pack.SpherePack_toSimulation(self, rot=Matrix3(1, 0, 0, 0, 1, 0, 0, 0, 1), **kw) +Append spheres directly to the simulation. In addition calling O.bodies.append, this method also +appropriately sets periodic cell information of the simulation. +>>> from yade import pack; from math import * +>>> sp=pack.SpherePack() +Create random periodic packing with 20 spheres: +>>> sp.makeCloud((0,0,0),(5,5,5),rMean=.5,rRelFuzz=.5,periodic=True,num=20) +20 +Virgin simulation is aperiodic: +>>> O.reset() +>>> O.periodic +False +Add generated packing to the simulation, rotated by 45° along +z +>>> sp.toSimulation(rot=Quaternion((0,0,1),pi/4),color=(0,0,1)) +[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19] +Periodic properties are transferred to the simulation correctly, including rotation (this could be +avoided by explicitly passing “hSize=O.cell.hSize” as an argument): +>>> O.periodic +True +>>> O.cell.refSize +Vector3(5,5,5) +(continues on next page) +2.4. +Yade modules reference +479 + +Yade Documentation, Release 3rd ed. +(continued from previous page) +>>> O.cell.hSize # doctest: +SKIP +Matrix3(3.53553,-3.53553,0, 3.53553,3.53553,0, 0,0,5) +The current state (even if rotated) is taken as mechanically undeformed, i.e. with identity trans- +formation: +>>> O.cell.trsf +Matrix3(1,0,0, 0,1,0, 0,0,1) +Parameters +• rot (Quaternion/Matrix3) – rotation of the packing, which will be applied on +spheres and will be used to set Cell.trsf as well. +• **kw – passed to utils.sphere +Returns list of body ids added (like O.bodies.append) +yade.pack.filterSpherePack(predicate, spherePack, returnSpherePack=None, **kw) +Using given SpherePack instance, return spheres that satisfy predicate. +It returns either a +pack.SpherePack (if returnSpherePack) or a list. +The packing will be recentered to match the +predicate and warning is given if the predicate is larger than the packing. +yade.pack.gtsSurface2Facets(surf, **kw) +Construct facets from given GTS surface. **kw is passed to utils.facet. +yade.pack.gtsSurfaceBestFitOBB(surf) +Return (Vector3 center, Vector3 halfSize, Quaternion orientation) describing best-fit oriented +bounding box (OBB) for the given surface. See cloudBestFitOBB for details. +yade.pack.hexaNet(radius, +cornerCoord=[0, +0, +0], +xLength=1.0, +yLength=0.5, +mos=0.08, +a=0.04, b=0.04, startAtCorner=True, isSymmetric=False, **kw) +Definition of the particles for a hexagonal wire net in the x-y-plane for the WireMatPM. +Parameters +• radius – radius of the particle +• cornerCoord – coordinates of the lower left corner of the net +• xLenght – net length in x-direction +• yLenght – net length in y-direction +• mos – mesh opening size (horizontal distance between the double twists) +• a – length of double-twist +• b – height of single wire section +• startAtCorner – if true the generation starts with a double-twist at the lower +left corner +• isSymmetric – defines if the net is symmetric with respect to the y-axis +Returns set of spheres which defines the net (net) and exact dimensions of the net +(lx,ly). +Note: +This packing works for the WireMatPM only. The particles at the corner are always +generated first. For examples on how to use this packing see examples/WireMatPM. In order to +create the proper interactions for the net the interaction radius has to be adapted in the simulation. +class yade.pack.inConvexPolyhedron(inherits Predicate) +480 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +aabb((Predicate)arg1) → tuple +aabb( (Predicate)arg1) -> None +center((Predicate)arg1) → Vector3 +dim((Predicate)arg1) → Vector3 +class yade.pack.inGtsSurface_py(inherits Predicate) +This class was re-implemented in c++, but should stay here to serve as reference for implementing +Predicates in pure python code. C++ allows us to play dirty tricks in GTS which are not accessible +through pygts itself; the performance penalty of pygts comes from fact that if constructs and +destructs bb tree for the surface at every invocation of gts.Point().is_inside(). That is cached in +the c++ code, provided that the surface is not manipulated with during lifetime of the object +(user’s responsibility). +— +Predicate for GTS surfaces. Constructed using an already existing surfaces, which must be closed. +import gts surf=gts.read(open(‘horse.gts’)) inGtsSurface(surf) +Note: +Padding is optionally supported by testing 6 points along the axes in the pad distance. +This must be enabled in the ctor by saying doSlowPad=True. If it is not enabled and pad is not +zero, warning is issued. +aabb((Predicate)arg1) → tuple +aabb( (Predicate)arg1) -> None +center((Predicate)arg1) → Vector3 +dim((Predicate)arg1) → Vector3 +class yade.pack.inHalfSpace(inherits Predicate) +Predicate returning True any points, with infinite bounding box. +aabb((Predicate)arg1) → tuple +aabb( (Predicate)arg1) -> None +center((Predicate)arg1) → Vector3 +dim((Predicate)arg1) → Vector3 +class yade.pack.inSpace(inherits Predicate) +Predicate returning True for any points, with infinite bounding box. +aabb((Predicate)arg1) → tuple +aabb( (Predicate)arg1) -> None +center((Predicate)arg1) → Vector3 +dim((Predicate)arg1) → Vector3 +yade.pack.randomDensePack(predicate, radius, material=-1, dim=None, cropLayers=0, rRel- +Fuzz=0.0, spheresInCell=0, memoizeDb=None, useOBB=False, +memoDbg=False, color=None, returnSpherePack=None, seed=-1) +Generator of random dense packing with given geometry properties, using TriaxialTest (aperiodic) +or PeriIsoCompressor (periodic). The periodicity depens on whether the spheresInCell parameter +is given. +O.switchScene() magic is used to have clean simulation for TriaxialTest without deleting the original +simulation. This function therefore should never run in parallel with some code accessing your +simulation. +Parameters +• predicate – solid-defining predicate for which we generate packing +2.4. +Yade modules reference +481 + +Yade Documentation, Release 3rd ed. +• spheresInCell – if given, the packing will be periodic, with given number of +spheres in the periodic cell. +• radius – mean radius of spheres +• rRelFuzz – relative fuzz of the radius – e.g. +radius=10, rRelFuzz=.2, then +spheres will have radii 10 ± (10*.2)), with an uniform distribution. 0 by default, +meaning all spheres will have exactly the same radius. +• cropLayers – (aperiodic only) how many layers of spheres will be added to +the computed dimension of the box so that there no (or not so much, at least) +boundary effects at the boundaries of the predicate. +• dim – dimension of the packing, to override dimensions of the predicate (if it is +infinite, for instance) +• memoizeDb – name of sqlite database (existent or nonexistent) to find an already +generated packing or to store the packing that will be generated, if not found (the +technique of caching results of expensive computations is known as memoization). +Fuzzy matching is used to select suitable candidate – packing will be scaled, +rRelFuzz and dimensions compared. Packing that are too small are dictarded. +From the remaining candidate, the one with the least number spheres will be +loaded and returned. +• useOBB – effective only if a inGtsSurface predicate is given. If true (not default), +oriented bounding box will be computed first; it can reduce substantially num- +ber of spheres for the triaxial compression (like 10× depending on how much +asymmetric the body is), see examples/gts-horse/gts-random-pack-obb.py +• memoDbg – show packings that are considered and reasons why they are re- +jected/accepted +• returnSpherePack – see the corresponding argument in pack.filterSpherePack +Returns SpherePack object with spheres, filtered by the predicate. +yade.pack.randomPeriPack(radius, initSize, rRelFuzz=0.0, memoizeDb=None, noPrint=False, +seed=-1) +Generate periodic dense packing. +A cell of initSize is stuffed with as many spheres as possible, then we run periodic compression +with PeriIsoCompressor, just like with randomDensePack. +Parameters +• radius – mean sphere radius +• rRelFuzz – relative fuzz of sphere radius (equal distribution); see the same param +for randomDensePack. +• initSize – initial size of the periodic cell. +Returns SpherePack object, which also contains periodicity information. +yade.pack.regularHexa(predicate, radius, gap, **kw) +Return set of spheres in regular hexagonal grid, clipped inside solid given by predicate. Created +spheres will have given radius and will be separated by gap space. +yade.pack.regularOrtho(predicate, radius, gap, **kw) +Return set of spheres in regular orthogonal grid, clipped inside solid given by predicate. Created +spheres will have given radius and will be separated by gap space. +yade.pack.revolutionSurfaceMeridians(sects, +angles, +origin=Vector3(0, +0, +0), +orienta- +tion=Quaternion((1, 0, 0), 0)) +Revolution surface given sequences of 2d points and sequence of corresponding angles, returning +sequences of 3d points representing meridian sections of the revolution surface. The 2d sections +are turned around z-axis, but they can be transformed using the origin and orientation arguments +to give arbitrary orientation. +482 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +yade.pack.sweptPolylines2gtsSurface(pts, threshold=0, capStart=False, capEnd=False) +Create swept suface (as GTS triangulation) given same-length sequences of points (as 3-tuples). +If threshold is given (>0), then +• degenerate faces (with edges shorter than threshold) will not be created +• gts.Surface().cleanup(threshold) will be called before returning, which merges vertices mutu- +ally closer than threshold. In case your pts are closed (last point concident with the first +one) this will the surface strip of triangles. If you additionally have capStart==True and +capEnd==True, the surface will be closed. +Note: +capStart and capEnd make the most naive polygon triangulation (diagonals) and will +perhaps fail for non-convex sections. +Warning: +the algorithm connects points sequentially; if two polylines are mutually rotated or +have inverse sense, the algorithm will not detect it and connect them regardless in their given +order. +Creation, manipulation, IO for generic sphere packings. +class yade._packSpheres.SpherePack +Set of spheres represented as centers and radii. This class is returned by pack.randomDensePack, +pack.randomPeriPack and others. The object supports iteration over spheres, as in +>>> sp=SpherePack() +>>> for center,radius in sp: print center,radius +>>> for sphere in sp: print sphere[0],sphere[1] +## same, but without unpacking the␣ +�→tuple automatically +>>> for i in range(0,len(sp)): print sp[i][0], sp[i][1] +## same, but accessing spheres␣ +�→by index +Special constructors +Construct from list of [(c1,r1),(c2,r2),…]. To convert two same-length lists of centers and +radii, construct with zip(centers,radii). +__init__((object)arg1[, (list)list]) → None : +Empty constructor, optionally taking list [ ((cx,cy,cz),r), … ] for initial data. +aabb((SpherePack)arg1) → tuple : +Get axis-aligned bounding box coordinates, as 2 3-tuples. +add((SpherePack)arg1, (Vector3)arg2, (float)arg3) → None : +Add single sphere to packing, given center as 3-tuple and radius +appliedPsdScaling +A factor between 0 and 1, uniformly applied on all sizes of of the PSD. +cellFill((SpherePack)arg1, (Vector3)arg2) → None : +Repeat the packing (if periodic) so that the results has dim() >= given size. The packing +retains periodicity, but changes cellSize. Raises exception for non-periodic packing. +cellRepeat((SpherePack)arg1, (Vector3i)arg2) → None : +Repeat the packing given number of times in each dimension. Periodicity is retained, cellSize +changes. Raises exception for non-periodic packing. +2.4. +Yade modules reference +483 + +Yade Documentation, Release 3rd ed. +cellSize +Size of periodic cell; is Vector3(0,0,0) if not periodic. (Change this property only if you know +what you’re doing). +center((SpherePack)arg1) → Vector3 : +Return coordinates of the bounding box center. +dim((SpherePack)arg1) → Vector3 : +Return dimensions of the packing in terms of aabb(), as a 3-tuple. +fromList((SpherePack)arg1, (list)arg2) → None : +Make packing from given list, same format as for constructor. Discards current data. +fromList( (SpherePack)arg1, (object)centers, (object)radii) -> None : Make pack- +ing from given list, same format as for constructor. Discards current data. +fromSimulation((SpherePack)arg1) → None : +Make packing corresponding to the current simulation. Discards current data. +getClumps((SpherePack)arg1) → tuple : +Return lists of sphere ids sorted by clumps they belong to. +The return value is (stan- +dalones,[clump1,clump2,…]), where each item is list of id’s of spheres. +hasClumps((SpherePack)arg1) → bool : +Whether this object contains clumps. +isPeriodic +was the packing generated in periodic boundaries? +load((SpherePack)arg1, (str)fileName) → None : +Load packing from external text file (current data will be discarded). +makeCloud((SpherePack)arg1[, +(Vector3)minCorner=Vector3(0, +0, +0)[, +(Vec- +tor3)maxCorner=Vector3(0, +0, +0)[, +(float)rMean=-1[, +(float)rRelFuzz=0[, +(int)num=-1[, (bool)periodic=False[, (float)porosity=0.65[, (object)psdSizes=[][, +(object)psdCumm=[][, +(bool)distributeMass=False[, +(int)seed=-1[, +(Ma- +trix3)hSize=Matrix3(0, 0, 0, 0, 0, 0, 0, 0, 0)]]]]]]]]]]]]) → int : +Create a random cloud of particles enclosed in a parallelepiped. The resulting packing is a +gas-like state with no contacts between particles initially. Usually used as a first step before +reaching a dense packing. +Parameters +• minCorner (Vector3) – lower corner of an axis-aligned box +• maxCorner (Vector3) – upper corner of an axis-aligned box +• hSize (Matrix3) – base vectors of a generalized box (arbitrary parallelepiped, +typically Cell::hSize), superseeds minCorner and maxCorner if defined. For +periodic boundaries only. +• rMean (float) – mean radius or spheres +• rRelFuzz (float) – dispersion of radius relative to rMean +• num (int) – number of spheres to be generated. If negative (default), generate +as many as possible with stochastic sizes, ending after a fixed number of tries to +place the sphere in space, else generate exactly num spheres with deterministic +size distribution. +• periodic (bool) – whether the packing to be generated should be periodic +• porosity (float) – initial guess for the iterative generation procedure (if +num>1). The algorithm will be retrying until the number of generated spheres +is num. The first iteration tries with the provided porosity, but next iterations +484 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +increase it if necessary (hence an initialy high porosity can speed-up the al- +gorithm). If psdSizes is not defined, rRelFuzz (z) and num (N) are used so +that the porosity given (ρ) is approximately achieved at the end of generation, +rm = +3� +V(1−ρ) +4 +3 π(1+z2)N. +The default is ρ=0.5. +The optimal value depends on +rRelFuzz or psdSizes. +• psdSizes – sieve sizes (particle diameters) when particle size distribution +(PSD) is specified. +• psdCumm – cummulative fractions of particle sizes given by psdSizes; must be +the same length as psdSizes and should be non-decreasing. +• distributeMass (bool) – if True, given distribution will be used to distribute +sphere’s mass rather than radius of them. +• seed – number used to initialize the random number generator. +Returns number of created spheres, which can be lower than num depending on the +method used. +Note: +• Works in 2D if minCorner[k]=maxCorner[k] for one coordinate. +• If num is defined, then sizes generation is deterministic, giving the best fit of target +distribution. It enables spheres placement in descending size order, thus giving lower +porosity than the random generation. +• By default (with distributeMass==False), the distribution is applied to particle count +(i.e. particle count percent passing). The typical geomechanics sense of “particle size +distribution” is the distribution of mass fraction (i.e. mass percent passing); this can be +achieved with distributeMass=True. +• Sphere radius distribution can be specified using one of the following ways: +1. rMean, rRelFuzz and num gives uniform radius distribution in rMean×(1±rRelFuzz). +Less than num spheres can be generated if it is too high. +2. rRelFuzz, num and (optional) porosity, which estimates mean radius so that +porosity is attained at the end. rMean must be less than 0 (default). porosity +is only an initial guess for the generation algorithm, which will retry with higher +porosity until the prescibed num is obtained. +3. psdSizes and psdCumm, two arrays specifying points of the particle size distribution +function. As many spheres as possible are generated. +4. psdSizes, psdCumm, num, and (optional) porosity, like above but if num is not ob- +tained, psdSizes will be scaled down uniformly, until num is obtained (see appliedPs- +dScaling). +makeClumpCloud((SpherePack)arg1, +(Vector3)minCorner, +(Vector3)maxCorner, +(ob- +ject)clumps[, (bool)periodic=False[, (int)num=-1[, (int)seed=-1]]]) → int +: +Create a random loose packing of clumps the same way makeCloud does with spheres. The pa- +rameters minCorner, maxCorner, periodic, num and seed are the same as in makeCloud. The +parameter clumps is a list containing all the different clumps to be appended as SpherePack +objects. Here is an exemple that shows how to create a cloud made of 10 identical clumps : +clp = SpherePack([((0,0,0), 1e-2), ((1e-2,0,0), 1e-2)]) # The clump we want a cloud␣ +�→of +sp = SpherePack() +sp.makeClumpCloud((0,0,0), (1,1,1), [clp], num=10, seed=42) +sp.toSimulation() # All the particles in the cloud are now appended to O.bodies +2.4. +Yade modules reference +485 + +Yade Documentation, Release 3rd ed. +psd((SpherePack)arg1[, (int)bins=50[, (bool)mass=True]]) → tuple : +Return particle size distribution of the packing. +Parameters +• bins (int) – number of bins between minimum and maximum diameter +• mass – Compute relative mass rather than relative particle count for each bin. +Corresponds to distributeMass parameter for makeCloud. +Returns tuple of (cumm,edges), where cumm are cummulative fractions for respec- +tive diameters and edges are those diameter values. Dimension of both arrays is +equal to bins+1. +relDensity((SpherePack)arg1) → float : +Relative packing density, measured as sum of spheres’ volumes / aabb volume. +(Sphere +overlaps are ignored.) +rotate((SpherePack)arg1, (Vector3)axis, (float)angle) → None : +Rotate all spheres around packing center (in terms of aabb()), given axis and angle of the +rotation. +save((SpherePack)arg1, (str)fileName) → None : +Save packing to external text file (will be overwritten). +scale((SpherePack)arg1, (float)arg2) → None : +Scale the packing around its center (in terms of aabb()) by given factor (may be negative). +toList((SpherePack)arg1) → list : +Return packing data as python list. +toSimulation(rot=Matrix3(1, 0, 0, 0, 1, 0, 0, 0, 1), **kw) +Append spheres directly to the simulation. In addition calling O.bodies.append, this method +also appropriately sets periodic cell information of the simulation. +>>> from yade import pack; from math import * +>>> sp=pack.SpherePack() +Create random periodic packing with 20 spheres: +>>> sp.makeCloud((0,0,0),(5,5,5),rMean=.5,rRelFuzz=.5,periodic=True,num=20) +20 +Virgin simulation is aperiodic: +>>> O.reset() +>>> O.periodic +False +Add generated packing to the simulation, rotated by 45° along +z +>>> sp.toSimulation(rot=Quaternion((0,0,1),pi/4),color=(0,0,1)) +[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19] +Periodic properties are transferred to the simulation correctly, including rotation (this could +be avoided by explicitly passing “hSize=O.cell.hSize” as an argument): +>>> O.periodic +True +>>> O.cell.refSize +Vector3(5,5,5) +>>> O.cell.hSize # doctest: +SKIP +Matrix3(3.53553,-3.53553,0, 3.53553,3.53553,0, 0,0,5) +486 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +The current state (even if rotated) is taken as mechanically undeformed, i.e. with identity +transformation: +>>> O.cell.trsf +Matrix3(1,0,0, 0,1,0, 0,0,1) +Parameters +• rot (Quaternion/Matrix3) – rotation of the packing, which will be applied on +spheres and will be used to set Cell.trsf as well. +• **kw – passed to utils.sphere +Returns list of body ids added (like O.bodies.append) +translate((SpherePack)arg1, (Vector3)arg2) → None : +Translate all spheres by given vector. +class yade._packSpheres.SpherePackIterator +__init__((object)arg1, (SpherePackIterator)arg2) → None +next() +__next__( (SpherePackIterator)arg1) -> tuple +Spatial predicates for volumes (defined analytically or by triangulation). +class yade._packPredicates.Predicate +aabb((Predicate)arg1) → tuple +aabb( (Predicate)arg1) -> None +center((Predicate)arg1) → Vector3 +dim((Predicate)arg1) → Vector3 +class yade._packPredicates.PredicateBoolean(inherits Predicate) +Boolean operation on 2 predicates (abstract class) +A +B +__init__() +Raises an exception This class cannot be instantiated from Python +aabb((Predicate)arg1) → tuple +aabb( (Predicate)arg1) -> None +center((Predicate)arg1) → Vector3 +dim((Predicate)arg1) → Vector3 +class yade._packPredicates.PredicateDifference(inherits PredicateBoolean → Predicate) +Difference (conjunction with negative predicate) of 2 predicates. A point has to be inside the first +and outside the second predicate. Can be constructed using the - operator on predicates: pred1 +- pred2. +A +B +__init__((object)arg1, (object)arg2, (object)arg3) → None +aabb((Predicate)arg1) → tuple +aabb( (Predicate)arg1) -> None +center((Predicate)arg1) → Vector3 +2.4. +Yade modules reference +487 + +Yade Documentation, Release 3rd ed. +dim((Predicate)arg1) → Vector3 +class yade._packPredicates.PredicateIntersection(inherits PredicateBoolean → Predicate) +Intersection (conjunction) of 2 predicates. +A point has to be inside both predicates. +Can be +constructed using the & operator on predicates: pred1 & pred2. +A +B +__init__((object)arg1, (object)arg2, (object)arg3) → None +aabb((Predicate)arg1) → tuple +aabb( (Predicate)arg1) -> None +center((Predicate)arg1) → Vector3 +dim((Predicate)arg1) → Vector3 +class yade._packPredicates.PredicateSymmetricDifference(inherits +PredicateBoolean +→ +Predicate) +SymmetricDifference (exclusive disjunction) of 2 predicates. +A point has to be in exactly one +predicate of the two. Can be constructed using the ^ operator on predicates: pred1 ^ pred2. +A +B +__init__((object)arg1, (object)arg2, (object)arg3) → None +aabb((Predicate)arg1) → tuple +aabb( (Predicate)arg1) -> None +center((Predicate)arg1) → Vector3 +dim((Predicate)arg1) → Vector3 +class yade._packPredicates.PredicateUnion(inherits PredicateBoolean → Predicate) +Union (non-exclusive disjunction) of 2 predicates. A point has to be inside any of the two predicates +to be inside. Can be constructed using the | operator on predicates: pred1 | pred2. +A +B +__init__((object)arg1, (object)arg2, (object)arg3) → None +aabb((Predicate)arg1) → tuple +aabb( (Predicate)arg1) -> None +center((Predicate)arg1) → Vector3 +dim((Predicate)arg1) → Vector3 +class yade._packPredicates.inAlignedBox(inherits Predicate) +Axis-aligned box predicate +__init__((object)arg1, (Vector3)minAABB, (Vector3)maxAABB) → None : +Ctor taking minumum and maximum points of the box (as 3-tuples). +aabb((Predicate)arg1) → tuple +aabb( (Predicate)arg1) -> None +center((Predicate)arg1) → Vector3 +dim((Predicate)arg1) → Vector3 +class yade._packPredicates.inCylinder(inherits Predicate) +Cylinder predicate +__init__((object)arg1, (Vector3)centerBottom, (Vector3)centerTop, (float)radius) → None : +Ctor taking centers of the lateral walls (as 3-tuples) and radius. +488 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +aabb((Predicate)arg1) → tuple +aabb( (Predicate)arg1) -> None +center((Predicate)arg1) → Vector3 +dim((Predicate)arg1) → Vector3 +class yade._packPredicates.inEllipsoid(inherits Predicate) +Ellipsoid predicate +__init__((object)arg1, (Vector3)centerPoint, (Vector3)abc) → None : +Ctor taking center of the ellipsoid (3-tuple) and its 3 radii (3-tuple). +aabb((Predicate)arg1) → tuple +aabb( (Predicate)arg1) -> None +center((Predicate)arg1) → Vector3 +dim((Predicate)arg1) → Vector3 +class yade._packPredicates.inHyperboloid(inherits Predicate) +Hyperboloid predicate +__init__((object)arg1, +(Vector3)centerBottom, +(Vector3)centerTop, +(float)radius, +(float)skirt) → None : +Ctor taking centers of the lateral walls (as 3-tuples), radius at bases and skirt (middle radius). +aabb((Predicate)arg1) → tuple +aabb( (Predicate)arg1) -> None +center((Predicate)arg1) → Vector3 +dim((Predicate)arg1) → Vector3 +class yade._packPredicates.inParallelepiped(inherits Predicate) +Parallelepiped predicate +__init__((object)arg1, (Vector3)o, (Vector3)a, (Vector3)b, (Vector3)c) → None : +Ctor taking four points: o (for origin) and then a, b, c which define endpoints of 3 respective +edges from o. +aabb((Predicate)arg1) → tuple +aabb( (Predicate)arg1) -> None +center((Predicate)arg1) → Vector3 +dim((Predicate)arg1) → Vector3 +class yade._packPredicates.inSphere(inherits Predicate) +Sphere predicate. +__init__((object)arg1, (Vector3)center, (float)radius) → None : +Ctor taking center (as a 3-tuple) and radius +aabb((Predicate)arg1) → tuple +aabb( (Predicate)arg1) -> None +center((Predicate)arg1) → Vector3 +dim((Predicate)arg1) → Vector3 +class yade._packPredicates.notInNotch(inherits Predicate) +Outside of infinite, rectangle-shaped notch predicate +__init__((object)arg1, +(Vector3)centerPoint, +(Vector3)edge, +(Vector3)normal, +(float)aperture) → None : +Ctor taking point in the symmetry plane, vector pointing along the edge, plane normal and +aperture size. The side inside the notch is edge×normal. Normal is made perpendicular to +the edge. All vectors are normalized at construction time. +2.4. +Yade modules reference +489 + +Yade Documentation, Release 3rd ed. +aabb((Predicate)arg1) → tuple +aabb( (Predicate)arg1) -> None +center((Predicate)arg1) → Vector3 +dim((Predicate)arg1) → Vector3 +Computation of oriented bounding box for cloud of points. +yade._packObb.cloudBestFitOBB((tuple)arg1) → tuple +Return (Vector3 center, Vector3 halfSize, Quaternion orientation) of best-fit oriented bounding-box +for given tuple of points (uses brute-force velome minimization, do not use for very large clouds). +2.4.12 yade.plot module +Module containing utility functions for plotting inside yade. See examples/simple-scene/simple-scene- +plot.py or examples/concrete/uniax.py for example of usage. +yade.plot.data = {'eps': [0.0001, 0.001, nan], 'force': [nan, nan, 1000.0], 'sigma': [12, nan, nan]} +Global dictionary containing all data values, common for all plots, in the form {‘name’:[value,…],…}. +Data should be added using plot.addData function. All [value,…] columns have the same length, +they are padded with NaN if unspecified. +yade.plot.plots = {'i': ('t',), 'i ': ('z1', 'v1')} +dictionary x-name -> (yspec,…), where yspec is either y-name or (y-name,’line-specification’). If +(yspec,...) is None, then the plot has meaning of image, which will be taken from respective field +of plot.imgData. +yade.plot.labels = {} +Dictionary converting names in data to human-readable names (TeX names, for instance); if a +variable is not specified, it is left untranslated. +yade.plot.live = True +Enable/disable live plot updating. +yade.plot.liveInterval = 1 +Interval for the live plot updating, in seconds. +yade.plot.setLiveForceAlwaysUpdate(forceLiveUpdate) +The plot.liveInterval and plot.live control live refreshing of the plot during calculations. The re- +freshing is done in a separate thread, so that it does not interfere with calculations. +Drawing +the data will not work when at exactly the same time it is being updated in other thread. Use +yade.plot.setLiveForceAlwaysUpdate(True) if you want calculations to PAUSE during the +plot updates. This function returns current bool value of forced updates if the call was a success, +otherwise it returns a str with explanation why it failed. It is guaranteed to work if simulation +was paused with O.pause() call. +yade.plot.autozoom = True +Enable/disable automatic plot rezooming after data update. Sometimes rezooming must be skipped +unless a call to plot.setLiveForceAlwaysUpdate forces it to work. +yade.plot.plot(noShow=False, subPlots=True) +Do the actual plot, which is either shown on screen (and nothing is returned: if noShow is False +- note that your yade compilation should present qt4 feature so that figures can be displayed) or, +if noShow is True, returned as matplotlib’s Figure object or list of them. +You can use +>>> from yade import plot +>>> plot.resetData() +>>> plot.plots={'foo':('bar',)} +>>> plot.plot(noShow=True).savefig('someFile.pdf') +>>> import os +(continues on next page) +490 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +(continued from previous page) +>>> os.path.exists('someFile.pdf') +True +>>> os.remove('someFile.pdf') +to save the figure to file automatically. +Note: +For backwards compatibility reasons, noShow option will return list of figures for multiple +figures but a single figure (rather than list with 1 element) if there is only 1 figure. +yade.plot.reset() +Reset all plot-related variables (data, plots, labels) +yade.plot.resetData() +Reset all plot data; keep plots and labels intact. +yade.plot.splitData() +Make all plots discontinuous at this point (adds nan’s to all data fields) +yade.plot.reverseData() +Reverse yade.plot.data order. +Useful for tension-compression test, where the initial (zero) state is loaded and, to make data +continuous, last part must end in the zero state. +yade.plot.addData(*d_in, **kw) +Add +data +from +arguments +name1=value1,name2=value2 +to +yade.plot.data. +(the +old +{‘name1’:value1,’name2’:value2} is deprecated, but still supported) +New data will be padded with nan’s, unspecified data will be nan (nan’s don’t appear in graphs). +This way, equal length of all data is assured so that they can be plotted one against any other. +>>> from yade import plot +>>> from pprint import pprint +>>> plot.resetData() +>>> plot.addData(a=1) +>>> plot.addData(b=2) +>>> plot.addData(a=3,b=4) +>>> pprint(plot.data) +{'a': [1, nan, 3], 'b': [nan, 2, 4]} +Some sequence types can be given to addData; they will be saved in synthesized columns for +individual components. +>>> plot.resetData() +>>> plot.addData(c=Vector3(5,6,7),d=Matrix3(8,9,10, 11,12,13, 14,15,16)) +>>> pprint(plot.data) +{'c_x': [5.0], +'c_y': [6.0], +'c_z': [7.0], +'d_xx': [8.0], +'d_xy': [9.0], +'d_xz': [10.0], +'d_yx': [11.0], +'d_yy': [12.0], +'d_yz': [13.0], +'d_zx': [14.0], +'d_zy': [15.0], +'d_zz': [16.0]} +yade.plot.addAutoData() +2.4. +Yade modules reference +491 + +Yade Documentation, Release 3rd ed. +Add data by evaluating contents of plot.plots. Expressions rasing exceptions will be handled grace- +fully, but warning is printed for each. +>>> from yade import plot +>>> from pprint import pprint +>>> O.reset() +>>> plot.resetData() +>>> plot.plots={'O.iter':('O.time',None,'numParticles=len(O.bodies)')} +>>> plot.addAutoData() +>>> pprint(plot.data) +{'O.iter': [0], 'O.time': [0.0], 'numParticles': [0]} +Note that each item in plot.plots can be +• an expression to be evaluated (using the eval builtin); +• name=expression string, where name will appear as label in plots, and expression will be +evaluated each time; +• a dictionary-like object – current keys are labels of plots and current values are added to +plot.data. The contents of the dictionary can change over time, in which case new lines will +be created as necessary. +A simple simulation with plot can be written in the following way; note how the energy plot is +specified. +>>> from yade import plot, utils +>>> plot.plots={'i=O.iter':(O.energy,None,'total energy=O.energy.total()')} +>>> # we create a simple simulation with one ball falling down +>>> plot.resetData() +>>> O.bodies.append(utils.sphere((0,0,0),1)) +0 +>>> O.dt=utils.PWaveTimeStep() +>>> O.engines=[ +... +ForceResetter(), +... +GravityEngine(gravity=(0,0,-10),warnOnce=False), +... +NewtonIntegrator(damping=.4,kinSplit=True), +... +# get data required by plots at every step +... +PyRunner(command='yade.plot.addAutoData()',iterPeriod=1,initRun=True) +... ] +>>> O.trackEnergy=True +>>> O.run(2,True) +>>> pprint(plot.data) +#doctest: +ELLIPSIS +{'gravWork': [0.0, -25.13274...], +'i': [0, 1], +'kinRot': [0.0, 0.0], +'kinTrans': [0.0, 7.5398...], +'nonviscDamp': [0.0, 10.0530...], +'total energy': [0.0, -7.5398...]} +yade.plot.saveGnuplot(baseName, +term=’wxt’, +extension=None, +timestamp=False, +com- +ment=None, title=None, varData=False) +Save data added with plot.addData into (compressed) file and create .gnuplot file that attempts to +mimick plots specified with plot.plots. +Parameters +• baseName – used for creating baseName.gnuplot (command file for gnuplot), +associated baseName.data.bz2 (data) and output files (if applicable) in the form +baseName.[plot number].extension +• term – specify the gnuplot terminal; defaults to x11, in which case gnuplot will +draw persistent windows to screen and terminate; other useful terminals are png, +cairopdf and so on +492 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +• extension – extension for baseName defaults to terminal name; fine for png for +example; if you use cairopdf, you should also say extension='pdf' however +• timestamp (bool) – append numeric time to the basename +• varData (bool) – whether file to plot will be declared as variable or be in-place +in the plot expression +• comment – a user comment (may be multiline) that will be embedded in the +control file +Returns name of the gnuplot file created. +yade.plot.saveDataTxt(fileName, vars=None, headers=None) +Save plot data into a (optionally compressed) text file. The first line contains a comment (starting +with #) giving variable name for each of the columns. This format is suitable for being loaded for +further processing (outside yade) with numpy.genfromtxt function, which recognizes those variable +names (creating numpy array with named entries) and handles decompression transparently. +>>> from yade import plot +>>> from pprint import pprint +>>> plot.reset() +>>> plot.addData(a=1,b=11,c=21,d=31) +# add some data here +>>> plot.addData(a=2,b=12,c=22,d=32) +>>> pprint(plot.data) +{'a': [1, 2], 'b': [11, 12], 'c': [21, 22], 'd': [31, 32]} +>>> plot.saveDataTxt('/tmp/dataFile.txt.tar.gz',vars=('a','b','c')) +>>> import numpy +>>> d=numpy.genfromtxt('/tmp/dataFile.txt.tar.gz',dtype=None,names=True) +>>> d['a'] +array([1, 2]) +>>> d['b'] +array([11, 12]) +>>> import os # cleanup +>>> os.remove('/tmp/dataFile.txt.tar.gz') +Parameters +• fileName – file to save data to; if it ends with .bz2 / .gz, the file will be +compressed using bzip2 / gzip. +• vars – Sequence (tuple/list/set) of variable names to be saved. If None (default), +all variables in plot.plot are saved. +• headers – Set of parameters to write on header +yade.plot.savePlotSequence(fileBase, stride=1, imgRatio=(5, 7), title=None, titleFrames=20, +lastFrames=30) +Save sequence of plots, each plot corresponding to one line in history. It is especially meant to be +used for utils.makeVideo. +Parameters +• stride – only consider every stride-th line of history (default creates one frame +per each line) +• title – Create title frame, where lines of title are separated with newlines (\n) +and optional subtitle is separated from title by double newline. +• titleFrames (int) – Create this number of frames with title (by repeating its +filename), determines how long the title will stand in the movie. +• lastFrames (int) – Repeat the last frame this number of times, so that the +movie does not end abruptly. +Returns List of filenames with consecutive frames. +2.4. +Yade modules reference +493 + +Yade Documentation, Release 3rd ed. +2.4.13 yade.polyhedra_utils module +2.4.14 yade.post2d module +Module for 2d postprocessing, containing classes to project points from 3d to 2d in various ways, providing +basic but flexible framework for extracting arbitrary scalar values from bodies/interactions and plotting +the results. There are 2 basic components: flatteners and extractors. +The algorithms operate on bodies (default) or interactions, depending on the intr parameter of +post2d.data. +Flatteners +Instance of classes that convert 3d (model) coordinates to 2d (plot) coordinates. Their interface is defined +by the post2d.Flatten class (__call__, planar, normal). +Extractors +Callable objects returning scalar or vector value, given a body/interaction object. +If a 3d vector is +returned, Flattener.planar is called, which should return only in-plane components of the vector. +Example +This example can be found in examples/concrete/uniax-post.py +from yade import post2d +import pylab # the matlab-like interface of matplotlib +O.load('/tmp/uniax-tension.xml.bz2') +# flattener that project to the xz plane +flattener=post2d.AxisFlatten(useRef=False,axis=1) +# return scalar given a Body instance +extractDmg=lambda b: b.state.normDmg +# will call flattener.planar implicitly +# the same as: extractVelocity=lambda b: flattener.planar(b,b.state.vel) +extractVelocity=lambda b: b.state.vel +# create new figure +pylab.figure() +# plot raw damage +post2d.plot(post2d.data(extractDmg,flattener)) +# plot smooth damage into new figure +pylab.figure(); ax,map=post2d.plot(post2d.data(extractDmg,flattener,stDev=2e-3)) +# show color scale +pylab.colorbar(map,orientation='horizontal') +# raw velocity (vector field) plot +pylab.figure(); post2d.plot(post2d.data(extractVelocity,flattener)) +# smooth velocity plot; data are sampled at regular grid +pylab.figure(); ax,map=post2d.plot(post2d.data(extractVelocity,flattener,stDev=1e-3)) +# save last (current) figure to file +pylab.gcf().savefig('/tmp/foo.png') +# show the figures +pylab.show() +494 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +class yade.post2d.AxisFlatten(inherits Flatten → object) +__init__(useRef=False, axis=2) +Parameters +• useRef (bool) – use reference positions rather than actual positions (only +meaningful when operating on Bodies) +• axis ({0,1,2}) – axis normal to the plane; the return value will be simply +position with this component dropped. +normal(pos, vec) +Given position and vector value, return lenght of the vector normal to the flat plane. +planar(pos, vec) +Given position and vector value, project the vector value to the flat plane and return its 2 +in-plane components. +class yade.post2d.CylinderFlatten(inherits Flatten → object) +Class for converting 3d point to 2d based on projection onto plane from circle. The y-axis in the +projection corresponds to the rotation axis; the x-axis is distance form the axis. +__init__(useRef, axis=2) +Parameters +• useRef – (bool) use reference positions rather than actual positions +• axis – axis of the cylinder, ￿{0,1,2} +normal(b, vec) +Given position and vector value, return lenght of the vector normal to the flat plane. +planar(b, vec) +Given position and vector value, project the vector value to the flat plane and return its 2 +in-plane components. +class yade.post2d.Flatten(inherits object) +Abstract class for converting 3d point into 2d. Used by post2d.data2d. +normal(pos, vec) +Given position and vector value, return lenght of the vector normal to the flat plane. +planar(pos, vec) +Given position and vector value, project the vector value to the flat plane and return its 2 +in-plane components. +class yade.post2d.HelixFlatten(inherits Flatten → object) +Class converting 3d point to 2d based on projection from helix. +The y-axis in the projection +corresponds to the rotation axis +__init__(useRef, thetaRange, dH_dTheta, axis=2, periodStart=0) +Parameters +• useRef (bool) – use reference positions rather than actual positions +• thetaRange ((ϑmin,ϑmax)) – bodies outside this range will be discarded +• dH_dTheta (float) – inclination of the spiral (per radian) +• axis ({0,1,2}) – axis of rotation of the spiral +• periodStart (float) – height of the spiral for zero angle +normal(pos, vec) +Given position and vector value, return lenght of the vector normal to the flat plane. +2.4. +Yade modules reference +495 + +Yade Documentation, Release 3rd ed. +planar(b, vec) +Given position and vector value, project the vector value to the flat plane and return its 2 +in-plane components. +yade.post2d.data(extractor, flattener, intr=False, onlyDynamic=True, stDev=None, relThresh- +old=3.0, perArea=0, div=(50, 50), margin=(0, 0), radius=1) +Filter all bodies/interactions, project them to 2d and extract required scalar value; return either +discrete array of positions and values, or smoothed data, depending on whether the stDev value is +specified. +The intr parameter determines whether we operate on bodies or interactions; the extractor pro- +vided should expect to receive body/interaction. +Parameters +• extractor (callable) – receives Body (or Interaction, if intr is True) in- +stance, should return scalar, a 2-tuple (vector fields) or None (to skip that +body/interaction) +• flattener (callable) – post2d.Flatten instance, receiving body/interaction, re- +turns its 2d coordinates or None (to skip that body/interaction) +• intr (bool) – operate on interactions rather than bodies +• onlyDynamic (bool) – skip all non-dynamic bodies +• stDev (float/None) – standard deviation for averaging, enables smoothing; None +(default) means raw mode, where discrete points are returned +• relThreshold (float) – threshold for the gaussian weight function relative to +stDev (smooth mode only) +• perArea (int) – if 1, compute weightedSum/weightedArea rather than weighted +average (weightedSum/sumWeights); the first is useful to compute average stress; +if 2, compute averages on subdivision elements, not using weight function +• div ((int,int)) – number of cells for the gaussian grid (smooth mode only) +• margin ((float,float)) – x,y margins around bounding box for data (smooth +mode only) +• radius (float/callable) – Fallback value for radius (for raw plotting) for non- +spherical bodies or interactions; if a callable, receives body/interaction and re- +turns radius +Returns dictionary +Returned +dictionary +always +containing +keys +‘type’ +(one +of +‘rawScalar’,’rawVector’,’smoothScalar’,’smoothVector’, depending on value of smooth and on +return value from extractor), ‘x’, ‘y’, ‘bbox’. +Raw data further contains ‘radii’. +Scalar fields contain ‘val’ (value from extractor), vector fields have ‘valX’ and ‘valY’ (2 components +returned by the extractor). +yade.post2d.plot(data, axes=None, alpha=0.5, clabel=True, cbar=False, aspect=’equal’, **kw) +Given output from post2d.data, plot the scalar as discrete or smooth plot. +For raw discrete data, plot filled circles with radii of particles, colored by the scalar value. +For smooth discrete data, plot image with optional contours and contour labels. +For vector data (raw or smooth), plot quiver (vector field), with arrows colored by the magnitude. +Parameters +• axes – matplotlib.axesinstance where the figure will be plotted; if None, will be +created from scratch. +496 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +• data – value returned by post2d.data +• clabel (bool) – show contour labels (smooth mode only), or annotate cells with +numbers inside (with perArea==2) +• cbar (bool) – show colorbar (equivalent to calling pylab.colorbar(mappable) on +the returned mappable) +Returns tuple of (axes,mappable); mappable can be used in further calls to py- +lab.colorbar. +2.4.15 yade.qt module +Common initialization core for yade. +This file is executed when anything is imported from yade for the first time. It loads yade plugins and +injects c++ class constructors to the __builtins__ (that might change in the future, though) namespace, +making them available everywhere. +class yade.qt._GLViewer.GLViewer +__init__() +Raises an exception This class cannot be instantiated from Python +axes +Show arrows for axes. +center((GLViewer)arg1[, (bool)median=True[, (float)suggestedRadius=-1.0]]) → None : +Center view. View is centered either so that all bodies fit inside (median = False), or so that +75% of bodies fit inside (median = True). If radius cannot be determined automatically then +suggestedRadius is used. +close((GLViewer)arg1) → None +eyePosition +Camera position. +fitAABB((GLViewer)arg1, (Vector3)mn, (Vector3)mx) → None : +Adjust scene bounds so that Axis-aligned bounding box given by its lower and upper corners +mn, mx fits in. +fitSphere((GLViewer)arg1, (Vector3)center, (float)radius) → None : +Adjust scene bounds so that sphere given by center and radius fits in. +fps +Show frames per second indicator. +grid +Display square grid in zero planes, as 3-tuple of bools for yz, xz, xy planes. +loadState((GLViewer)arg1[, (str)stateFilename=’.qglviewer.xml’]) → None : +Load display parameters from file saved previously into. +lookAt +Point at which camera is directed. +ortho +Whether orthographic projection is used; if false, use perspective projection. +saveSnapshot((GLViewer)arg1, (str)filename) → None : +Save the current view to image file +saveState((GLViewer)arg1[, (str)stateFilename=’.qglviewer.xml’]) → None : +Save display parameters into a file. Saves state for both GLViewer and associated OpenGLRen- +derer. +2.4. +Yade modules reference +497 + +Yade Documentation, Release 3rd ed. +scale +Scale of the view (?) +sceneRadius +Visible scene radius. +screenSize +Size of the viewer’s window, in screen pixels +selection +showEntireScene((GLViewer)arg1) → None +timeDisp +Time displayed on in the vindow; is a string composed of characters r, v, i standing respectively +for real time, virtual time, iteration number. +upVector +Vector that will be shown oriented up on the screen. +viewDir +Camera orientation (as vector). +yade.qt._GLViewer.Renderer() → OpenGLRenderer +Return the active OpenGLRenderer object. +yade.qt._GLViewer.View([(float)timeout=5.0]) → GLViewer +Create a new 3d view. +yade.qt._GLViewer.center([(float)suggestedRadius=-1.0[, +(Vector3)gridOrigin=Vector3(0, +0, +0)[, +(Vector3)suggestedCenter=Vector3(0, +0, +0)[, +(int)gridDecimalPlaces=4]]]]) → None +Center all views. +Parameters +• suggestedRadius – optional parameter, if provided and positive then it will be +used instead of automatic radius detection. This parameter affects the (1) size +of grid being drawn (2) the Z-clipping distance in OpenGL, it means that if +clipping is too large and some of your scene is not being drawn but is “cut” or +“sliced” then this parameter needs to be bigger. +• gridOrigin – optional parameter, if provided it will be used as the origin for +drawing the grid. Meaning the intersection of all three grids will not be at 0,0,0; +but at the provided coordinate rounded to the nearest gridStep. +• suggestedCenter – optional parameter, if provided other than (0,0,0) then it +will be used instead of automatic calculation of scene center using bounding +boxes. This parameter affects the drawn rotation-center. If you try to rotate the +view, and the rotation is around some strange point, then this parameter needs +to be changed. +• gridDecimalPlaces – default value=4, determines the number of decimal places +to be shown on grid labels using stringstream (extra zeros are not shown). +Note: +You can get the current values of all these four arguments by invoking command: +qt.centerValues() +yade.qt._GLViewer.centerValues() → dict +Returns a dictionary with all parameters currently used by yade.qt.center(…), see +qt.center or type yade.qt.center? for details. Returns zeros if view is closed. +yade.qt._GLViewer.views() → list +498 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +Returns a list of all open qt.GLViewer objects +If one needs to exactly copy camera position and settings between two different yade sessions, the +following commands can be used: +v=yade.qt.views()[0] +## to obtain a handle of currently opened␣ +�→view. +v.lookAt, v.viewDir, v.eyePosition, v.upVector ## to print the current camera parameters␣ +�→of the view. +## Then copy the output of this command into the second yade session to reposition the␣ +�→camera. +v.lookAt, v.viewDir, v.eyePosition, v.upVector = (Vector3(-0.5,1.6,0.47),Vector3(-0.5,0.6, +�→0.4),Vector3(0.015,0.98,-0.012),Vector3(0.84,0.46,0.27)) +## Since these parameters depend on each other it might be necessary to execute this␣ +�→command twice. +Also one can call qt.centerValues() to obtain current settings of axis and scene radius (if defaults +are not used) and apply them via call to qt.center in the second yade session. +This cumbersome method above may be improved in the future. +2.4.16 yade.timing module +Functions for accessing timing information stored in engines and functors. +See Timing section of the programmer’s manual, wiki page for some examples. +yade.timing.reset() +Zero all timing data. +yade.timing.runtime() +Return total running time (same as last line in the output of stats()) in nanoseconds +yade.timing.stats() +Print summary table of timing information from engines and functors. Absolute times as well as +percentages are given. Sample output: +Name +Count +Time +␣ +�→ +Rel. time +------------------------------------------------------------------------------------------ +�→------------- +ForceResetter +102 +2150us +␣ +�→ +0.02% +"collider" +5 +64200us +␣ +�→ +0.60% +InteractionLoop +102 +10571887us +␣ +�→ 98.49% +"combEngine" +102 +8362us +␣ +�→ +0.08% +"newton" +102 +73166us +␣ +�→ +0.68% +"cpmStateUpdater" +1 +9605us +␣ +�→ +0.09% +PyRunner +1 +136us +␣ +�→ +0.00% +"plotDataCollector" +1 +291us +␣ +�→ +0.00% +TOTAL +10733564us +␣ +�→100.00% +sample output (compiled with -DENABLE_PROFILING=1 option): +2.4. +Yade modules reference +499 + +Yade Documentation, Release 3rd ed. +Name +Count +Time +␣ +�→ +Rel. time +------------------------------------------------------------------------------------------ +�→------------- +ForceResetter +102 +2150us +␣ +�→ +0.02% +"collider" +5 +64200us +␣ +�→ +0.60% +InteractionLoop +102 +10571887us +␣ +�→ 98.49% +Ig2_Sphere_Sphere_ScGeom +1222186 +1723168us +␣ +�→ +16.30% +Ig2_Sphere_Sphere_ScGeom +1222186 +1723168us +␣ +�→ +100.00% +Ig2_Facet_Sphere_ScGeom +753 +1157us +␣ +�→ +0.01% +Ig2_Facet_Sphere_ScGeom +753 +1157us +␣ +�→ +100.00% +Ip2_CpmMat_CpmMat_CpmPhys +11712 +26015us +␣ +�→ +0.25% +end of Ip2_CpmPhys +11712 +26015us +␣ +�→ +100.00% +Ip2_FrictMat_CpmMat_FrictPhys +0 +0us +␣ +�→ +0.00% +Law2_ScGeom_CpmPhys_Cpm +3583872 +4819289us +␣ +�→ +45.59% +GO A +1194624 +1423738us +␣ +�→ +29.54% +GO B +1194624 +1801250us +␣ +�→ +37.38% +rest +1194624 +1594300us +␣ +�→ +33.08% +TOTAL +3583872 +4819289us +␣ +�→ +100.00% +Law2_ScGeom_FrictPhys_CundallStrack +0 +0us +␣ +�→ +0.00% +"combEngine" +102 +8362us +␣ +�→ +0.08% +"newton" +102 +73166us +␣ +�→ +0.68% +"cpmStateUpdater" +1 +9605us +␣ +�→ +0.09% +PyRunner +1 +136us +␣ +�→ +0.00% +"plotDataCollector" +1 +291us +␣ +�→ +0.00% +TOTAL +10733564us +␣ +�→100.00% +2.4.17 yade.utils module +Heap of functions that don’t (yet) fit anywhere else. +Devs: please DO NOT ADD more functions here, it is getting too crowded! +yade.utils.NormalRestitution2DampingRate(en) +Compute the normal damping rate as a function of the normal coefficient of restitution en. For +en ∈ ⟨0, 1⟩ damping rate equals +− +log en +� +e2n + π2 +500 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +yade.utils.SpherePWaveTimeStep(radius, density, young) +Compute P-wave critical timestep for a single (presumably representative) sphere, using formula +for P-Wave propagation speed ∆tc = +r +√ +E/ρ. If you want to compute minimum critical timestep +for all spheres in the simulation, use utils.PWaveTimeStep instead. +>>> SpherePWaveTimeStep(1e-3,2400,30e9) +2.8284271247461903e-07 +class yade.utils.TableParamReader(inherits object) +Class for reading simulation parameters from text file. +Each parameter is represented by one column, each parameter set by one line. Colums are separated +by blanks (no quoting). +First non-empty line contains column titles (without quotes). You may use special column named +‘description’ to describe this parameter set; if such colum is absent, description will be built by +concatenating column names and corresponding values (param1=34,param2=12.22,param4=foo) +• from columns ending in ! (the ! is not included in the column name) +• from all columns, if no columns end in !. +Empty lines within the file are ignored (although counted); # starts comment till the end of line. +Number of blank-separated columns must be the same for all non-empty lines. +A special value = can be used instead of parameter value; value from the previous non-empty line +will be used instead (works recursively). +This class is used by utils.readParamsFromTable. +__init__(file) +Setup the reader class, read data into memory. +paramDict() +Return dictionary containing data from file given to constructor. Keys are line numbers (which +might be non-contiguous and refer to real line numbers that one can see in text editors), values +are dictionaries mapping parameter names to their values given in the file. The special value +‘=’ has already been interpreted, ! (bangs) (if any) were already removed from column titles, +description column has already been added (if absent). +yade.utils.aabbDim(cutoff=0.0, centers=False) +Return dimensions of the axis-aligned bounding box, optionally with relative part cutoff cut away. +yade.utils.aabbExtrema2d(pts) +Return 2d bounding box for a sequence of 2-tuples. +yade.utils.aabbWalls(extrema=None, thickness=0, oversizeFactor=1.5, **kw) +Return 6 boxes that will wrap existing packing as walls from all sides. +Parameters +• extrema – extremal points of the Aabb of the packing, as a list of two Vector3, +or any equivalent type (will be calculated if not specified) +• thickness (float) – is wall thickness (will be 1/10 of the X-dimension if not +specified) +• oversizeFactor (float) – factor to enlarge walls in their plane. +Returns a +list +of +6 +wall +Bodies +enclosing +the +packing, +in +the +order +minX,maxX,minY,maxY,minZ,maxZ. +yade.utils.avgNumInteractions(cutoff=0.0, skipFree=False, considerClumps=False) +Return average numer of interactions per particle, also known as coordination number Z. This +number is defined as +Z = 2C/N +2.4. +Yade modules reference +501 + +Yade Documentation, Release 3rd ed. +where C is number of contacts and N is number of particles. When clumps are present, number of +particles is the sum of standalone spheres plus the sum of clumps. Clumps are considered in the +calculation if cutoff != 0 or skipFree = True. If cutoff=0 (default) and skipFree=False (default) +one needs to set considerClumps=True to consider clumps in the calculation. +With skipFree, particles not contributing to stable state of the packing are skipped, following +equation (8) given in [Thornton2000]: +Zm = +2C − N1 +N − N0 − N1 +Parameters +• cutoff – cut some relative part of the sample’s bounding box away. +• skipFree – see above. +• considerClumps – also consider clumps if cutoff=0 and skipFree=False; for fur- +ther explanation see above. +yade.utils.box(center, +extents, +orientation=Quaternion((1, +0, +0), +0), +dynamic=None, +fixed=False, wire=False, color=None, highlight=False, material=-1, mask=1) +Create box (cuboid) with given parameters. +Parameters +• extents (Vector3) – half-sizes along x,y,z axes. Use can be made of orientation +parameter in case those box-related axes do not conform the simulation axes +• orientation (Quaternion) – assigned to the body’s orientation, which corre- +sponds to rotating the extents axes +See utils.sphere’s documentation for meaning of other parameters. +class yade.utils.clumpTemplate(inherits object) +Create a clump template by a list of relative radii and a list of relative positions. Both lists must +have the same length. +Parameters +• relRadii ([float,float,..]) – list of relative radii (minimum length = 2) +• relPositions ([Vector3,Vector3,..]) – list of relative positions (minimum +length = 2) +yade.utils.defaultMaterial() +Return default material, when creating bodies with utils.sphere and friends, material is unspecified +and there is no shared material defined yet. By default, this function returns +FrictMat(density=1e3,young=1e7,poisson=.3,frictionAngle=.5,label='defaultMat') +yade.utils.facet(vertices, +dynamic=None, +fixed=True, +wire=True, +color=None, +high- +light=False, noBound=False, material=-1, mask=1, chain=-1) +Create facet with given parameters. +Parameters +• vertices ([Vector3,Vector3,Vector3]) – coordinates of vertices in the global +coordinate system. +• wire (bool) – if True, facets are shown as skeleton; otherwise facets are filled +• noBound (bool) – set Body.bounded +• color (Vector3-or-None) – color of the facet; random color will be assigned if +None. +See utils.sphere’s documentation for meaning of other parameters. +502 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +yade.utils.fractionalBox(fraction=1.0, minMax=None) +Return (min,max) that is the original minMax box (or aabb of the whole simulation if not specified) +linearly scaled around its center to the fraction factor +yade.utils.levelSetBody(shape=”, center=Vector3(0, 0, 0), radius=0, extents=Vector3(0, 0, +0), epsilons=Vector2(0, 0), clump=None, spacing=0.1, grid=None, +distField=[], nSurfNodes=27, nodesPath=2, nodesTol=50, orienta- +tion=Quaternion((1, 0, 0), 0), dynamic=True, material=-1) +Creates a LevelSet shaped body through various workflows: one can choose among pre-defined +shapes (through shape and related attributes), or to mimick a Clump instance (clump attribute, +for comparison purposes), or directly assign the discrete distance field on some grid (distField and +grid attributes) :param string shape: use this argument to enjoy predefined shapes among ‘sphere’, +‘box’ (for a rectangular parallelepiped), ‘disk’ (for a 2D analysis in (x,y) plane), or ‘superellipsoid’; +in conjunction with extents or radius attributes. Superellipsoid surfaces are defined in local axes +(inertial frame) by the following equation: f(x, y, z) = (|x/rx|2/εe +|y/ry|2/εe)εe/εn +|z/rz|2/εn = 1 +and their distance field is obtained thanks to a Fast Marching Method. :param Vector3 center: +(initial) position of that body :param Clump clump: pass here a multi-sphere instance to mimick, +if desired :param Real radius: imposed radius in case shape = ‘sphere’ or ‘disk’ :param Vector3 +extents: half extents along the local axes in case shape = ‘box’ or ‘superellipsoid’ (rx, ry, rz for +the latter) :param Vector2 epsilons: in case shape = ‘superellipsoid’, the (εe, εn) exponents :param +Real spacing: spatial increment of the level set grid, if you picked a pre-defined shape or a clump +:param list distField: the discrete distance field on grid (if given) as a list (of list of list; use +.tolist() if working initially with 3D numpy arrays), where distField[i][j][k] is the distance value +at grid.gridPoint(i,j,k) :param RegularGrid grid: the grid carrying the distance field, when the +latter is directly assigned through distField :param int nSurfNodes: number of boundary nodes, +passed to LevelSet.nSurfNodes :param int nodesPath: path for the boundary nodes, passed to +LevelSet.nodesPath :param Real nodesTol: tolerance while ray tracing boundary nodes, passed to +LevelSet.nodesTol :param Quaternion orientation: the initial orientation of the body :param bool +dynamic: passed to Body.dynamic :param Material material: passed to Body.material :return: a +corresponding body instance +yade.utils.loadVars(mark=None) +Load variables from utils.saveVars, which are saved inside the simulation. If mark==None, all save +variables are loaded. Otherwise only those with the mark passed. +yade.utils.makeVideo(frameSpec, +out, +renameNotOverwrite=True, +fps=24, +kbps=6000, +bps=None) +Create a video from external image files using mencoder. Two-pass encoding using the default +mencoder codec (mpeg4) is performed, running multi-threaded with number of threads equal to +number of OpenMP threads allocated for Yade. +Parameters +• frameSpec – wildcard | sequence of filenames. If list or tuple, filenames to be +encoded in given order; otherwise wildcard understood by mencoder’s mf:// URI +option (shell wildcards such as /tmp/snap-*.png or and printf-style pattern like +/tmp/snap-%05d.png) +• out (str) – file to save video into +• renameNotOverwrite (bool) – if True, existing same-named video file will have +-number appended; will be overwritten otherwise. +• fps (int) – Frames per second (-mf fps=…) +• kbps (int) – Bitrate (-lavcopts vbitrate=…) in kb/s +yade.utils.perpendicularArea(axis) +Return area perpendicular to given axis (0=x,1=y,2=z) generated by bodies for which the function +consider returns True (defaults to returning True always) and which is of the type Sphere. +yade.utils.phiIniPy(ioPyFn, grid) +Returns a 3D discrete field appropriate to serve as FastMarchingMethod.phiIni (LS_DEM feature +required), applying a user-made Python function ioPyFn +2.4. +Yade modules reference +503 + +Yade Documentation, Release 3rd ed. +Parameters +• ioPyFn – an existing inside-outside Python function that takes three numbers +(cartesian coordinates) as arguments +• grid (RegularGrid) – the RegularGrid instance to operate on +Return list an appropriate 3D discrete field to pass at FastMarchingMethod.phiIni +yade.utils.plotDirections(aabb=(), mask=0, bins=20, numHist=True, noShow=False, sph- +Sph=False) +Plot 3 histograms for distribution of interaction directions, in yz,xz and xy planes and (optional but +default) histogram of number of interactions per body. If sphSph only sphere-sphere interactions +are considered for the 3 directions histograms. +Returns If noShow is False, displays the figure and returns nothing. If noShow, the +figure object is returned without being displayed (works the same way as plot.plot). +yade.utils.plotNumInteractionsHistogram(cutoff=0.0) +Plot histogram with number of interactions per body, optionally cutting away cutoff relative axis- +aligned box from specimen margin. +yade.utils.polyhedron(vertices, +fixed=False, +wire=True, +color=None, +highlight=False, +noBound=False, material=-1, mask=1, chain=-1) +Create polyhedron with given parameters. +Parameters vertices ([Vector3]) – coordinates of vertices in the global coordinate +system. +See utils.sphere’s documentation for meaning of other parameters. +yade.utils.psd(bins=5, mass=True, mask=-1) +Calculates particle size distribution. +Parameters +• bins (int) – number of bins +• mass (bool) – if true, the mass-PSD will be calculated +• mask (int) – Body.mask for the body +Returns +• binsSizes: list of bin’s sizes +• binsProc: how much material (in percents) are in the bin, cumulative +• binsSumCum: how much material (in units) are in the bin, cumulative +binsSizes, binsProc, binsSumCum +yade.utils.randomColor(seed=None) +Return random Vector3 with each component in interval 0…1 (uniform distribution) +yade.utils.randomOrientation() +Returns +(uniformly +distributed) +random +orientation. +Taken +from +Eigen::Quaternion::UnitRandom() +source +code. +Uses +standard +Python +random.random() +function(s), you can random.seed() it +yade.utils.randomizeColors(onlyDynamic=False) +Assign random colors to Shape::color. +If onlyDynamic is true, only dynamic bodies will have the color changed. +yade.utils.readParamsFromTable(tableFileLine=None, noTableOk=True, unknownOk=False, +**kw) +Read parameters from a file and assign them to __builtin__ variables. +The format of the file is as follows (commens starting with # and empty lines allowed): +504 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +# commented lines allowed anywhere +name1 name2 … # first non-blank line are column headings +# empty line is OK, with or without comment +val1 +val2 +… # 1st parameter set +val2 +val2 +… # 2nd +… +Assigned tags (the description column is synthesized if absent,see utils.TableParamReader); +O.tags[‘description’]=… +# +assigns +the +description +column; +might +be +synthe- +sized O.tags[‘params’]=”name1=val1,name2=val2,…” # all explicitly assigned pa- +rameters O.tags[‘defaultParams’]=”unassignedName1=defaultValue1,…” # parameters +that were left at their defaults O.tags[‘d.id’]=O.tags[‘id’]+’.’+O.tags[‘description’] +O.tags[‘id.d’]=O.tags[‘description’]+’.’+O.tags[‘id’] +All parameters (default as well as settable) are saved using utils.saveVars('table'). +Parameters +• tableFileLine – string attribute to define which line number (as seen in a +text editor) from wich text file (with one value per blank-separated columns) +to get the values from. A ‘:’ should appear between the two informations, e.g. +‘file.table:4’ to read the 4th line from file.table file +• noTableOk (bool) – if False, raise exception if the file cannot be open; use default +values otherwise +• unknownOk (bool) – do not raise exception if unknown column name is found in +the file, and assign it as well +Returns number of assigned parameters +yade.utils.replaceCollider(colliderEngine) +Replaces collider (Collider) engine with the engine supplied. Raises error if no collider is in engines. +yade.utils.runningInBatch() +Tell whether we are running inside the batch or separately. +yade.utils.saveVars(mark=”, loadNow=True, **kw) +Save passed variables into the simulation so that it can be recovered when the simulation is loaded +again. +For example, variables a, b and c are defined. To save them, use: +>>> saveVars('something',a=1,b=2,c=3) +>>> from yade.params.something import * +>>> a,b,c +(1, 2, 3) +those variables will be save in the .xml file, when the simulation itself is saved. To recover those +variables once the .xml is loaded again, use loadVars('something') and they will be defined in the +yade.params.mark module. The loadNow parameter calls utils.loadVars after saving automatically. +If ‘something’ already exists, given variables will be inserted. +yade.utils.sphere(center, radius, dynamic=None, fixed=False, wire=False, color=None, high- +light=False, material=-1, mask=1) +Create sphere with given parameters; mass and inertia computed automatically. +Last assigned material is used by default (material = -1), and utils.defaultMaterial() will be used +if no material is defined at all. +Parameters +• center (Vector3) – center +• radius (float) – radius +2.4. +Yade modules reference +505 + +Yade Documentation, Release 3rd ed. +• dynamic (float) – deprecated, see “fixed” +• fixed (float) – generate the body with all DOFs blocked? +• material – +specify Body.material; different types are accepted: +– int: O.materials[material] will be used; as a special case, if material==- +1 and there is no shared materials defined, utils.defaultMaterial() will be +assigned to O.materials[0] +– string: label of an existing material that will be used +– Material instance: this instance will be used +– callable: will be called without arguments; returned Material value will be +used (Material factory object, if you like) +• mask (int) – Body.mask for the body +• wire – display as wire sphere? +• highlight – highlight this body in the viewer? +• Vector3-or-None – body’s color, as normalized RGB; random color will be +assigned if None. +Returns A Body instance with desired characteristics. +Creating default shared material if none exists neither is given: +>>> O.reset() +>>> from yade import utils +>>> len(O.materials) +0 +>>> s0=utils.sphere([2,0,0],1) +>>> len(O.materials) +1 +Instance of material can be given: +>>> s1=utils.sphere([0,0,0],1,wire=False,color=(0,1,0),material=ElastMat(young=30e9, +�→density=2e3)) +>>> s1.shape.wire +False +>>> s1.shape.color +Vector3(0,1,0) +>>> s1.mat.density +2000.0 +Material can be given by label: +>>> O.materials.append(FrictMat(young=10e9,poisson=.11,label='myMaterial')) +1 +>>> s2=utils.sphere([0,0,2],1,material='myMaterial') +>>> s2.mat.label +'myMaterial' +>>> s2.mat.poisson +0.11 +Finally, material can be a callable object (taking no arguments), which returns a Material instance. +Use this if you don’t call this function directly (for instance, through yade.pack.randomDensePack), +passing only 1 material parameter, but you don’t want material to be shared. +For instance, randomized material properties can be created like this: +506 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +>>> import random +>>> def matFactory(): return ElastMat(young=1e10*random.random(),density=1e3+1e3*random. +�→random()) +... +>>> s3=utils.sphere([0,2,0],1,material=matFactory) +>>> s4=utils.sphere([1,2,0],1,material=matFactory) +yade.utils.tetra(vertices, +strictCheck=True, +fixed=False, +wire=True, +color=None, +high- +light=False, noBound=False, material=-1, mask=1, chain=-1) +Create tetrahedron with given parameters. +Parameters +• vertices ([Vector3,Vector3,Vector3,Vector3]) – coordinates of vertices in +the global coordinate system. +• strictCheck (bool) – checks vertices order, raise RuntimeError for negative +volume +See utils.sphere’s documentation for meaning of other parameters. +yade.utils.tetraPoly(vertices, +fixed=False, +wire=True, +color=None, +highlight=False, +noBound=False, material=-1, mask=1, chain=-1) +Create tetrahedron (actually simple Polyhedra) with given parameters. +Parameters vertices ([Vector3,Vector3,Vector3,Vector3]) – coordinates of ver- +tices in the global coordinate system. +See utils.sphere’s documentation for meaning of other parameters. +yade.utils.trackPerfomance(updateTime=5) +Track perfomance of a simulation. (Experimental) Will create new thread to produce some plots. +Useful for track perfomance of long run simulations (in bath mode for example). +yade.utils.typedEngine(name) +Return first engine from current O.engines, identified by its type (as string). For example: +>>> from yade import utils +>>> O.engines=[InsertionSortCollider(),NewtonIntegrator(),GravityEngine()] +>>> utils.typedEngine("NewtonIntegrator") == O.engines[1] +True +yade.utils.uniaxialTestFeatures(filename=None, areaSections=10, axis=-1, distFactor=2.2, +**kw) +Get some data about the current packing useful for uniaxial test: +1. Find the dimensions that is the longest (uniaxial loading axis) +2. Find the minimum cross-section area of the specimen by examining several (areaSections) +sections perpendicular to axis, computing area of the convex hull for each one. This will work +also for non-prismatic specimen. +3. Find the bodies that are on the negative/positive boundary, to which the straining condition +should be applied. +Parameters +• filename – if given, spheres will be loaded from this file (ASCII format); if not, +current simulation will be used. +• areaSection (float) – number of section that will be used to estimate cross- +section +• axis (￿{0,1,2}) – if given, force strained axis, rather than computing it from +predominant length +Returns dictionary with keys negIds, posIds, axis, area. +2.4. +Yade modules reference +507 + +Yade Documentation, Release 3rd ed. +Warning: +The function utils.approxSectionArea uses convex hull algorithm to find the area, +but the implementation is reported to be buggy (bot works in some cases). Always check this +number, or fix the convex hull algorithm (it is documented in the source, see py/_utils.cpp). +yade.utils.vmData() +Return memory usage data from Linux’s /proc/[pid]/status, line VmData. +yade.utils.voxelPorosityTriaxial(triax, resolution=200, offset=0) +Calculate the porosity of a sample, given the TriaxialCompressionEngine. +A function utils.voxelPorosity is invoked, with the volume of a box enclosed by TriaxialCompres- +sionEngine walls. The additional parameter offset allows using a smaller volume inside the box, +where each side of the volume is at offset distance from the walls. By this way it is possible to find +a more precise porosity of the sample, since at walls’ contact the porosity is usually reduced. +A recommended value of offset is bigger or equal to the average radius of spheres inside. +The value of resolution depends on size of spheres used. It can be calibrated by invoking voxel- +PorosityTriaxial with offset=0 and comparing the result with TriaxialCompressionEngine.porosity. +After calibration, the offset can be set to radius, or a bigger value, to get the result. +Parameters +• triax – the TriaxialCompressionEngine handle +• resolution – voxel grid resolution +• offset – offset distance +Returns the porosity of the sample inside given volume +Example invocation: +from yade import utils +rAvg=0.03 +TriaxialTest(numberOfGrains=200,radiusMean=rAvg).load() +O.dt=-1 +O.run(1000) +O.engines[4].porosity +0.44007807740143889 +utils.voxelPorosityTriaxial(O.engines[4],200,0) +0.44055412500000002 +utils.voxelPorosityTriaxial(O.engines[4],200,rAvg) +0.36798199999999998 +yade.utils.waitIfBatch() +Block the simulation if running inside a batch. Typically used at the end of script so that it does +not finish prematurely in batch mode (the execution would be ended in such a case). +yade.utils.wall(position, axis, sense=0, color=None, material=-1, mask=1) +Return ready-made wall body. +Parameters +• position (float-or-Vector3) – center of the wall. If float, it is the position +along given axis, the other 2 components being zero +• axis (￿{0,1,2}) – orientation of the wall normal (0,1,2) for x,y,z (sc. planes yz, +xz, xy) +• sense (￿{-1,0,1}) – sense in which to interact (0: both, -1: negative, +1: +positive; see Wall) +See utils.sphere’s documentation for meaning of other parameters. +508 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +yade.utils.xMirror(half) +Mirror a sequence of 2d points around the x axis (changing sign on the y coord). The sequence +should start up and then it will wrap from y downwards (or vice versa). If the last point’s x coord +is zero, it will not be duplicated. +yade._utils.PWaveTimeStep() → float +Get timestep accoring to the velocity of P-Wave propagation; computed from sphere radii, rigidities +and masses. +yade._utils.RayleighWaveTimeStep() → float +Determination of time step according to Rayleigh wave speed of force propagation. +yade._utils.TetrahedronCentralInertiaTensor((object)arg1) → Matrix3 +TODO +yade._utils.TetrahedronInertiaTensor((object)arg1) → Matrix3 +TODO +yade._utils.TetrahedronSignedVolume((object)arg1) → float +TODO +yade._utils.TetrahedronVolume((object)arg1) → float +TODO +yade._utils.TetrahedronWithLocalAxesPrincipal((Body)arg1) → Quaternion +TODO +yade._utils.aabbExtrema([(float)cutoff=0.0[, (bool)centers=False]]) → tuple +Return coordinates of box enclosing all spherical bodies +Parameters +• centers (bool) – do not take sphere radii in account, only their centroids +• cutoff (float￿〈0…1〉) – relative dimension by which the box will be cut away +at its boundaries. +Returns [lower corner, upper corner] as [Vector3,Vector3] +yade._utils.angularMomentum([(Vector3)origin=Vector3(0, 0, 0)]) → Vector3 +TODO +yade._utils.approxSectionArea((float)arg1, (int)arg2) → float +Compute area of convex hull when when taking (swept) spheres crossing the plane at coord, per- +pendicular to axis. +yade._utils.bodyNumInteractionsHistogram((tuple)aabb) → tuple +yade._utils.bodyStressTensors() → list +Compute and return a table with per-particle stress tensors. Each tensor represents the average +stress in one particle, obtained from the contour integral of applied load as detailed below. This +definition is considering each sphere as a continuum. It can be considered exact in the context of +spheres at static equilibrium, interacting at contact points with negligible volume changes of the +solid phase (this last assumption is not restricting possible deformations and volume changes at +the packing scale). +Proof: +First, we remark the identity: σij = δikσkj = xi,kσkj = (xiσkj),k − xiσkj,k. +At equilibrium, the divergence of stress is null: σkj,k = 0. Consequently, after divergence theorem: +1 +V +� +V σijdV = 1 +V +� +V(xiσkj),kdV = 1 +V +� +∂V xiσkjnkdS = 1 +V +� +b xb +i fb +j . +The last equality is implicitely based on the representation of external loads as Dirac distributions +whose zeros are the so-called contact points: 0-sized surfaces on which the contact forces are applied, +located at xi in the deformed configuration. +2.4. +Yade modules reference +509 + +Yade Documentation, Release 3rd ed. +A weighted average of per-body stresses will give the average stress inside the solid phase. There is +a simple relation between the stress inside the solid phase and the stress in an equivalent continuum +in the absence of fluid pressure. For porosity n, the relation reads: σequ. +ij += (1 − n)σsolid +ij +. +This last relation may not be very useful if porosity is not homogeneous. If it happens, one can +define the equivalent bulk stress a the particles scale by assigning a volume to each particle. This +volume can be obtained from TesselationWrapper (see e.g. [Catalano2014a]) +yade._utils.calm([(int)mask=-1]) → None +Set translational and rotational velocities of bodies to zero. Applied to all bodies by default. To +calm only some bodies, use mask parameter, it will calm only bodies with groupMask compatible +to given value +yade._utils.coordsAndDisplacements((int)axis[, (tuple)Aabb=()]) → tuple +Return tuple of 2 same-length lists for coordinates and displacements (coordinate minus reference +coordinate) along given axis (1st arg); if the Aabb=((x_min,y_min,z_min),(x_max,y_max,z_- +max)) box is given, only bodies within this box will be considered. +yade._utils.createInteraction((int)id1, (int)id2[, (bool)virtualI=False]) → Interaction +Create interaction between given bodies by hand. +If virtualI=False, current engines are searched for IGeomDispatcher and IPhysDispatcher (might +be both hidden in InteractionLoop). Geometry is created using force parameter of the geometry +dispatcher, wherefore the interaction will exist even if bodies do not spatially overlap and the +functor would return false under normal circumstances. +If virtualI=True the interaction is left in a virtual state. +Warning: +This function will very likely behave incorrectly for periodic simulations (though +it could be extended it to handle it farily easily). +yade._utils.fabricTensor([(float)cutoff=0.0[, +(bool)splitTensor=False[, +(float)thresholdForce=nan]]]) → tuple +Computes the fabric tensor Fij = +1 +nc +� +c ninj [Satake1982], for all interactions c. +Parameters +• cutoff (Real) – intended to disregard boundary effects: to define in [0;1] to +focus on the interactions located in the centered inner (1-cutoff)^3*V part of the +spherical packing V. +• splitTensor (bool) – split the fabric tensor into two parts related to the strong +(greatest compressive normal forces) and weak contact forces respectively. +• thresholdForce (Real) – if the fabric tensor is split into two parts, a threshold +value can be specified otherwise the mean contact force is considered by default. +Use negative signed values for compressive states. To note that this value could +be set to zero if one wanted to make distinction between compressive and tensile +forces. +yade._utils.flipCell([(Matrix3)flip=Matrix3(0, 0, 0, 0, 0, 0, 0, 0, 0)]) → Matrix3 +Flip periodic cell so that angles between R3 axes and transformed axes are as small as possible, +using the two following facts:1. repeating in R3 space the corners of a periodic cell defines a regular +grid; 2. two cells leading through this process to a unique grid are equivalent and can be flipped one +over another. Flipping necessitates adjustment of Interaction.cellDist for interactions that cross +the boundary and didn’t before (or vice versa), and re-initialization of collider. The flip argument +can be used to specify desired flip: integers, each column for one axis; if zero matrix, best fit +(minimizing the angles) is computed automatically. +In c++, this function is accessible as Shop::flipCell. +yade._utils.forcesOnCoordPlane((float)arg1, (int)arg2) → Vector3 +510 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +yade._utils.forcesOnPlane((Vector3)planePt, (Vector3)normal) → Vector3 +Find all interactions deriving from NormShearPhys that cross given plane and sum forces (both +normal and shear) on them. +Parameters +• planePt (Vector3) – a point on the plane +• normal (Vector3) – plane normal (will be normalized). +yade._utils.getBodyIdsContacts([(int)bodyID=0]) → list +Get a list of body-ids, which contacts the given body. +yade._utils.getCapillaryStress([(float)volume=0[, (bool)mindlin=False]]) → Matrix3 +Compute and return Love-Weber capillary stress tensor: +σcap +ij += 1 +V +� +b lb +i fcap,b +j +, where the sum is over all interactions, with l the branch vector +(joining centers of the bodies) and fcap is the capillary force. +V can be passed to +the function. If it is not, it will be equal to one in non-periodic cases, or equal to the +volume of the cell in periodic cases. Only the CapillaryPhys interaction type is supported +presently. Using this function with physics MindlinCapillaryPhys needs to pass True as +second argument. +yade._utils.getDepthProfiles((float)volume, +(int)nCell, +(float)dz, +(float)zRef, +(bool)activateCond, +(float)radiusPy, +(int)direction) +→ +tu- +ple +Compute and return the particle velocity and solid volume fraction (porosity) depth profile along +the direction specified (default is z; 0=>x,1=>y,2=>z). For each defined cell z, the k component +of the average particle velocity reads: +< vk >z= � +p Vpvp +k/ � +p Vp, +where the sum is made over the particles contained in the cell, vp +k is the k component of the velocity +associated to particle p, and Vp is the part of the volume of the particle p contained inside the cell. +This definition allows to smooth the averaging, and is equivalent to taking into account the center +of the particles only when there is a lot of particles in each cell. As for the solid volume fraction, +it is evaluated in the same way: for each defined cell z, it reads: +< φ >z= +1 +Vcell +� +p Vp, where Vcell is the volume of the cell considered, and Vp is the volume of particle p contained in cell z. +This function gives depth profiles of average velocity and solid volume fraction, returning the +average quantities in each cell of height dz, from the reference horizontal plane at elevation +zRef (input parameter) until the plane of elevation zRef+nCell*dz (input parameters). If +the argument activateCond is set to true, do the average only on particles of radius equal to +radiusPy (input parameter) +yade._utils.getDepthProfiles_center((float)volume, +(int)nCell, +(float)dz, +(float)zRef, +(bool)activateCond, (float)radiusPy) → tuple +Same as getDepthProfiles but taking into account particles as points located at the particle center. +yade._utils.getDynamicStress() → list +Compute the dynamic stress tensor for each body: σp +D = − 1 +Vp mpu′p ⊗ u′p +yade._utils.getSpheresMass([(int)mask=-1]) → float +Compute the total mass of spheres in the simulation, mask parameter is considered +yade._utils.getSpheresVolume([(int)mask=-1]) → float +Compute the total volume of spheres in the simulation, mask parameter is considered +yade._utils.getSpheresVolume2D([(int)mask=-1]) → float +Compute the total volume of discs in the simulation, mask parameter is considered +yade._utils.getStress([(float)volume=0]) → Matrix3 +Compute and return Love-Weber stress tensor: +σij = 1 +V +� +b fb +i lb +j , where the sum is over all interactions, with f the contact force and l +the branch vector (joining centers of the bodies). Stress is negativ for repulsive contact +2.4. +Yade modules reference +511 + +Yade Documentation, Release 3rd ed. +forces, i.e. compression. V can be passed to the function. If it is not, it will be equal to +the volume of the cell in periodic cases, or to the one deduced from utils.aabbDim() in +non-periodic cases. +yade._utils.getStressAndTangent([(float)volume=0[, (bool)symmetry=True]]) → tuple +Compute overall stress of periodic cell using the same equation as function getStress. In addition, +the tangent operator is calculated using the equation published in [Kruyt and Rothenburg1998]_: +Sijkl = 1 +V +� +c +(knniljnkll + kttiljtkll) +Parameters +• volume (float) – same as in function getStress +• symmetry (bool) – make the tensors symmetric. +Returns macroscopic stress tensor and tangent operator as py::tuple +yade._utils.getStressProfile((float)volume, +(int)nCell, +(float)dz, +(float)zRef, +(ob- +ject)vPartAverageX, +(object)vPartAverageY, +(ob- +ject)vPartAverageZ) → tuple +Compute and return the stress tensor depth profile, including the contribution from Love-Weber +stress tensor and the dynamic stress tensor taking into account the effect of particles inertia. For +each defined cell z, the stress tensor reads: +σz +ij = 1 +V +� +c fc +i lc,z +j +− 1 +V +� +p mpu′p +i u′p +j , +where the first sum is made over the contacts which are contained or cross the cell z, f^c is the +contact force from particle 1 to particle 2, and l^{c,z} is the part of the branch vector from particle +2 to particle 1, contained in the cell. The second sum is made over the particles, and u’^p is the +velocity fluctuations of the particle p with respect to the spatial averaged particle velocity at this +point (given as input parameters). The expression of the stress tensor is the same as the one given in +getStress plus the inertial contribution. Apart from that, the main difference with getStress stands +in the fact that it gives a depth profile of stress tensor, i.e. from the reference horizontal plane at +elevation zRef (input parameter) until the plane of elevation zRef+nCell*dz (input parameters), it +is computing the stress tensor for each cell of height dz. For the love-Weber stress contribution, the +branch vector taken into account in the calculations is only the part of the branch vector contained +in the cell considered. To validate the formulation, it has been checked that activating only the +Love-Weber stress tensor, and suming all the contributions at the different altitude, we recover the +same stress tensor as when using getStress. For my own use, I have troubles with strong overlap +between fixed object, so that I made a condition to exclude the contribution to the stress tensor +of the fixed objects, this can be desactivated easily if needed (and should be desactivated for the +comparison with getStress). +yade._utils.getStressProfile_contact((float)volume, (int)nCell, (float)dz, (float)zRef) → tu- +ple +same as getStressProfile, only contact contribution. +yade._utils.getTotalDynamicStress([(float)volume=0]) → Matrix3 +Compute the total dynamic stress tensor : σD = − 1 +V +� +p mpu′p ⊗ u′p. The volume have to be +provided for non-periodic simulations. It is computed from cell volume for periodic simulations. +yade._utils.getViscoelasticFromSpheresInteraction((float)tc, (float)en, (float)es) → dict +Attention! The function is deprecated! Compute viscoelastic interaction parameters from analytical +512 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +solution of a pair spheres collision problem: +kn = m +t2c +� +π2 + (ln en)2� +cn = −2m +tc +ln en +kt = 2 +7 +m +t2c +� +π2 + (ln et)2� +ct = −2 +7 +m +tc +ln et +where kn, cn are normal elastic and viscous coefficients and kt, ct shear elastic and viscous coeffi- +cients. For details see [Pournin2001]. +Parameters +• m (float) – sphere mass m +• tc (float) – collision time tc +• en (float) – normal restitution coefficient en +• es (float) – tangential restitution coefficient es +Returns dictionary with keys kn (the value of kn), cn (cn), kt (kt), ct (ct). +yade._utils.growParticle((int)bodyID, (float)multiplier[, (bool)updateMass=True]) → None +Change the size of a single sphere (to be implemented: single clump). If updateMass=True, then +the mass is updated. +yade._utils.growParticles((float)multiplier[, +(bool)updateMass=True[, +(bool)dynamicOnly=True]]) → None +Change the size of spheres and clumps of spheres by the multiplier. If updateMass=True, then the +mass and inertia are updated. dynamicOnly=True will select dynamic bodies. +yade._utils.highlightNone() → None +Reset highlight on all bodies. +yade._utils.initMPI() → None +Initialize MPI communicator, for Foam Coupling +yade._utils.inscribedCircleCenter((Vector3)v1, (Vector3)v2, (Vector3)v3) → Vector3 +Return center of inscribed circle for triangle given by its vertices v1, v2, v3. +yade._utils.interactionAnglesHistogram((int)axis[, +(int)mask=0[, +(int)bins=20[, +(tuple)aabb=()[, +(bool)sphSph=0[, +(float)minProjLen=1e-06]]]]]) → tuple +yade._utils.intrsOfEachBody() → list +returns list of lists of interactions of each body +yade._utils.kineticEnergy([(bool)findMaxId=False]) → object +Compute overall kinetic energy of the simulation as +� 1 +2 +� +miv2 +i + ω(IωT) +� +. +For aspherical bodies, the inertia tensor I is transformed to global frame, before multiplied by ω, +therefore the value should be accurate. +yade._utils.maxOverlapRatio() → float +Return maximum overlap ration in interactions (with ScGeom) of two spheres. The ratio is com- +puted as +uN +2(r1r2)/r1+r2 , where uN is the current overlap distance and r1, r2 are radii of the two +spheres in contact. +2.4. +Yade modules reference +513 + +Yade Documentation, Release 3rd ed. +yade._utils.momentum() → Vector3 +TODO +yade._utils.negPosExtremeIds((int)axis, (float)distFactor) → tuple +Return list of ids for spheres (only) that are on extremal ends of the specimen along given axis; +distFactor multiplies their radius so that sphere that do not touch the boundary coordinate can +also be returned. +yade._utils.normalShearStressTensors([(bool)compressionPositive=False[, +(bool)splitNormalTensor=False[, +(float)thresholdForce=nan]]]) → tuple +Compute overall stress tensor of the periodic cell decomposed in 2 parts, one contributed by normal +forces, the other by shear forces. The formulation can be found in [Thornton2000], eq. (3): +σij = 2 +V +� +RNninj + 2 +V +� +RTnitj +where V is the cell volume, R is “contact radius” (in our implementation, current distance between +particle centroids), n is the normal vector, t is a vector perpendicular to n, N and T are norms of +normal and shear forces. +Parameters +• splitNormalTensor (bool) – if true the function returns normal stress tensor +split into two parts according to the two subnetworks of strong an weak forces. +• thresholdForce (Real) – threshold value according to which the normal stress +tensor can be split (e.g. a zero value would make distinction between tensile and +compressive forces). +yade._utils.numIntrsOfEachBody() → list +returns list of number of interactions of each body +yade._utils.pointInsidePolygon((tuple)arg1, (object)arg2) → bool +yade._utils.porosity([(float)volume=-1]) → float +Compute packing porosity V−Vs +V +where V is overall volume and Vs is volume of spheres. +Parameters volume (float) – overall volume V. +For periodic simulations, current +volume of the Cell is used. +For aperiodic simulations, the value deduced from +utils.aabbDim() is used. For compatibility reasons, positive values passed by the +user are also accepted in this case. +yade._utils.ptInAABB((Vector3)arg1, (Vector3)arg2, (Vector3)arg3) → bool +Return True/False whether the point p is within box given by its min and max corners +yade._utils.scalarOnColorScale((float)x[, (float)xmin=0[, (float)xmax=1]]) → Vector3 +Map scalar variable to color scale. +Parameters +• x (float) – scalar value which the function applies to. +• xmin (float) – minimum value for the color scale, with a return value of (0,0,1) +for x ≤ xmin, i.e. blue color in RGB. +• xmax (float) – maximum value, with a return value of (1,0,0) for x ≥ xmax, i.e. +red color in RGB. +Returns a Vector3 depending on the relative position of x on a [xmin;*xmax*] scale. +yade._utils.setBodyAngularVelocity((int)id, (Vector3)angVel) → None +Set a body angular velocity from its id and a new Vector3r. +Parameters +• id (int) – the body id. +514 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +• angVel (Vector3) – the desired updated angular velocity. +yade._utils.setBodyColor((int)id, (Vector3)color) → None +Set a body color from its id and a new Vector3r. +Parameters +• id (int) – the body id. +• color (Vector3) – the desired updated color. +yade._utils.setBodyOrientation((int)id, (Quaternion)ori) → None +Set a body orientation from its id and a new Quaternionr. +Parameters +• id (int) – the body id. +• ori (Quaternion) – the desired updated orientation. +yade._utils.setBodyPosition((int)id, (Vector3)pos[, (str)axis=’xyz’]) → None +Set a body position from its id and a new vector3r. +Parameters +• id (int) – the body id. +• pos (Vector3) – the desired updated position. +• axis (str) – the axis along which the position has to be updated (ex: +if +axis==”xy” and pos==Vector3r(r0,r1,r2), r2 will be ignored and the position +along z will not be updated). +yade._utils.setBodyVelocity((int)id, (Vector3)vel[, (str)axis=’xyz’]) → None +Set a body velocity from its id and a new vector3r. +Parameters +• id (int) – the body id. +• vel (Vector3) – the desired updated velocity. +• axis (str) – the axis along which the velocity has to be updated (ex: +if +axis==”xy” and vel==Vector3r(r0,r1,r2), r2 will be ignored and the velocity +along z will not be updated). +yade._utils.setContactFriction((float)angleRad) → None +Modify the friction angle (in radians) inside the material classes and existing contacts. The friction +for non-dynamic bodies is not modified. +yade._utils.setNewVerticesOfFacet((Body)b, (Vector3)v1, (Vector3)v2, (Vector3)v3) → None +Sets new vertices (in global coordinates) to given facet. +yade._utils.setRefSe3() → None +Set reference positions and orientations of all bodies equal to their current positions and orienta- +tions. +yade._utils.shiftBodies((list)ids, (Vector3)shift) → float +Shifts bodies listed in ids without updating their velocities. +yade._utils.spiralProject((Vector3)pt, +(float)dH_dTheta[, +(int)axis=2[, +(float)periodStart=nan[, (float)theta0=0]]]) → tuple +yade._utils.sumFacetNormalForces((object)ids[, (int)axis=-1]) → float +Sum force magnitudes on given bodies (must have shape of the Facet type), considering only part +of forces perpendicular to each facet’s face; if axis has positive value, then the specified axis (0=x, +1=y, 2=z) will be used instead of facet’s normals. +2.4. +Yade modules reference +515 + +Yade Documentation, Release 3rd ed. +yade._utils.sumForces((list)ids, (Vector3)direction) → float +Return summary force on bodies with given ids, projected on the direction vector. +yade._utils.sumTorques((list)ids, (Vector3)axis, (Vector3)axisPt) → float +Sum forces and torques on bodies given in ids with respect to axis specified by a point axisPt and +its direction axis. +yade._utils.totalForceInVolume() → tuple +Return summed forces on all interactions and average isotropic stiffness, as tuple (Vector3,float) +yade._utils.unbalancedForce([(bool)useMaxForce=False]) → float +Compute the ratio of mean (or maximum, if useMaxForce) summary force on bodies and mean force +magnitude on interactions. For perfectly static equilibrium, summary force on all bodies is zero +(since forces from interactions cancel out and induce no acceleration of particles); this ratio will tend +to zero as simulation stabilizes, though zero is never reached because of finite precision computation. +Sufficiently small value can be e.g. 1e-2 or smaller, depending on how much equilibrium it should +be. +yade._utils.voidratio2D([(float)zlen=1]) → float +Compute 2D packing void ratio V−Vs +Vs +where V is overall volume and Vs is volume of disks. +Parameters zlen (float) – length in the third direction. +yade._utils.voxelPorosity([(int)resolution=200[, +(Vector3)start=Vector3(0, +0, +0)[, +(Vec- +tor3)end=Vector3(0, 0, 0)]]]) → float +Compute packing porosity V−Vv +V +where V is a specified volume (from start to end) and Vv is volume +of voxels that fall inside any sphere. The calculation method is to divide whole volume into a dense +grid of voxels (at given resolution), and count the voxels that fall inside any of the spheres. This +method allows one to calculate porosity in any given sub-volume of a whole sample. It is properly +excluding part of a sphere that does not fall inside a specified volume. +Parameters +• resolution (int) – voxel grid resolution, values bigger than resolution=1600 +require a 64 bit operating system, because more than 4GB of RAM is used, a +resolution=800 will use 500MB of RAM. +• start (Vector3) – start corner of the volume. +• end (Vector3) – end corner of the volume. +yade._utils.wireAll() → None +Set Shape::wire on all bodies to True, rendering them with wireframe only. +yade._utils.wireNoSpheres() → None +Set Shape::wire to True on non-spherical bodies (Facets, Walls). +yade._utils.wireNone() → None +Set Shape::wire on all bodies to False, rendering them as solids. +2.4.18 yade.ymport module +Import geometry from various formats (‘import’ is python keyword, hence the name ‘ymport’). +yade.ymport.ele(nodeFileName, eleFileName, shift=(0, 0, 0), scale=1.0, **kw) +Import tetrahedral mesh from .ele file, return list of created tetrahedrons. +Parameters +• nodeFileName (string) – name of .node file +• eleFileName (string) – name of .ele file +• shift ((float,float,float)|Vector3) – (X,Y,Z) parameter moves the speci- +men. +516 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +• scale (float) – factor scales the given data. +• **kw – (unused keyword arguments) is passed to utils.polyhedron +yade.ymport.gengeo(mntable, shift=Vector3(0, 0, 0), scale=1.0, **kw) +Imports geometry from LSMGenGeo library and creates spheres. +Since 2012 the package is +available in Debian/Ubuntu and known as python-demgengeo http://packages.qa.debian.org/p/ +python-demgengeo.html +Parameters +mntable: mntable object, which creates by LSMGenGeo library, see example +shift: [float,float,float] [X,Y,Z] parameter moves the specimen. +scale: float factor scales the given data. +**kw: (unused keyword arguments) is passed to utils.sphere +LSMGenGeo library allows one to create pack of spheres with given [Rmin:Rmax] with null stress +inside the specimen. Can be useful for Mining Rock simulation. +Example: +examples/packs/packs.py, +usage +of +LSMGenGeo +library +in +exam- +ples/test/genCylLSM.py. +• https://answers.launchpad.net/esys-particle/+faq/877 +• http://www.access.edu.au/lsmgengeo_python_doc/current/pythonapi/html/ +GenGeo-module.html +• https://svn.esscc.uq.edu.au/svn/esys3/lsm/contrib/LSMGenGeo/ +yade.ymport.gengeoFile(fileName=’file.geo’, +shift=Vector3(0, +0, +0), +scale=1.0, +orienta- +tion=Quaternion((1, 0, 0), 0), **kw) +Imports geometry from LSMGenGeo .geo file and creates spheres. +Since 2012 the package is +available in Debian/Ubuntu and known as python-demgengeo http://packages.qa.debian.org/p/ +python-demgengeo.html +Parameters +filename: string file which has 4 colums [x, y, z, radius]. +shift: Vector3 Vector3(X,Y,Z) parameter moves the specimen. +scale: float factor scales the given data. +orientation: quaternion orientation of the imported geometry +**kw: (unused keyword arguments) is passed to utils.sphere +Returns list of spheres. +LSMGenGeo library allows one to create pack of spheres with given [Rmin:Rmax] with null stress +inside the specimen. Can be useful for Mining Rock simulation. +Example: +examples/packs/packs.py, +usage +of +LSMGenGeo +library +in +exam- +ples/test/genCylLSM.py. +• https://answers.launchpad.net/esys-particle/+faq/877 +• http://www.access.edu.au/lsmgengeo_python_doc/current/pythonapi/html/ +GenGeo-module.html +• https://svn.esscc.uq.edu.au/svn/esys3/lsm/contrib/LSMGenGeo/ +yade.ymport.gmsh(meshfile=’file.mesh’, +shift=Vector3(0, +0, +0), +scale=1.0, +orienta- +tion=Quaternion((1, 0, 0), 0), **kw) +Imports geometry from .mesh file and creates facets. +Parameters +shift: [float,float,float] [X,Y,Z] parameter moves the specimen. +2.4. +Yade modules reference +517 + +Yade Documentation, Release 3rd ed. +scale: float factor scales the given data. +orientation: quaternion orientation of the imported mesh +**kw: (unused keyword arguments) is passed to utils.facet +Returns list of facets forming the specimen. +mesh files can easily be created with GMSH. Example added to examples/packs/packs.py +Additional examples of mesh-files can be downloaded from http://www-roc.inria.fr/gamma/ +download/download.php +yade.ymport.gts(meshfile, shift=Vector3(0, 0, 0), scale=1.0, **kw) +Read given meshfile in gts format. +Parameters +meshfile: string name of the input file. +shift: [float,float,float] [X,Y,Z] parameter moves the specimen. +scale: float factor scales the given data. +**kw: (unused keyword arguments) is passed to utils.facet +Returns list of facets. +yade.ymport.iges(fileName, shift=(0, 0, 0), scale=1.0, returnConnectivityTable=False, **kw) +Import triangular mesh from .igs file, return list of created facets. +Parameters +• fileName (string) – name of iges file +• shift ((float,float,float)|Vector3) – (X,Y,Z) parameter moves the speci- +men. +• scale (float) – factor scales the given data. +• **kw – (unused keyword arguments) is passed to utils.facet +• returnConnectivityTable (bool) – if True, apart from facets returns also nodes +(list of (x,y,z) nodes coordinates) and elements (list of (id1,id2,id3) element nodes +ids). If False (default), returns only facets +yade.ymport.stl(file, dynamic=None, fixed=True, wire=True, color=None, highlight=False, +noBound=False, material=-1, scale=1.0, shift=Vector3(0, 0, 0)) +Import a .stl geometry in the form of a set of Facet-shaped bodies. +Parameters +• file (string) – the .stl file serving as geometry input +• dynamic (bool) – controls Body.dynamic +• fixed +(bool) +– +controls +Body.dynamic +(with +fixed += +True +imposing +Body.dynamic = False) if dynamic attribute is not given +• wire (bool) – rendering option, passed to Facet.wire +• color – rendering option, passed to Facet.color +• highlight (bool) – rendering option, passed to Facet.highlight +• noBound (bool) – sets Body.bounded to False if True, preventing collision detec- +tion (and vice-versa) +• material – defines material properties, see Defining materials for usage +• scale (float) – scaling factor to e.g. dilate the geometry if > 1 +• shift (Vector3) – for translating the geometry +518 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +Returns a corresponding list of Facet-shaped bodies +yade.ymport.text(fileName, shift=Vector3(0, 0, 0), scale=1.0, **kw) +Load sphere coordinates from file, returns a list of corresponding bodies; that may be inserted to +the simulation with O.bodies.append(). +Parameters +• filename (string) – file which has 4 colums [x, y, z, radius]. +• shift ([float,float,float]) – [X,Y,Z] parameter moves the specimen. +• scale (float) – factor scales the given data. +• **kw – (unused keyword arguments) is passed to utils.sphere +Returns list of spheres. +Lines starting with # are skipped +yade.ymport.textClumps(fileName, +shift=Vector3(0, +0, +0), +discretization=0, +orienta- +tion=Quaternion((1, 0, 0), 0), scale=1.0, **kw) +Load clumps-members from file in a format selected by the format argument, insert them to the +simulation. +Parameters +• filename (str) – file name +• format (str) – selected input format. +Supported 'x_y_z_r'``(default), +``'x_y_z_r_clumpId' +• shift ([float,float,float]) – [X,Y,Z] parameter moves the specimen. +• scale (float) – factor scales the given data. +• **kw – (unused keyword arguments) is passed to utils.sphere +Returns list of spheres. +Lines starting with # are skipped +yade.ymport.textExt(fileName, format=’x_y_z_r’, shift=Vector3(0, 0, 0), scale=1.0, attrs=[], +**kw) +Load sphere coordinates from file in a format selected by the format argument, returns a list of +corresponding bodies; that may be inserted to the simulation with O.bodies.append(). +Parameters +• filename (str) – file name +• format (str) – selected input format. +Supported 'x_y_z_r'``(default), +``'x_y_z_r_matId', 'x_y_z_r_attrs' +• shift ([float,float,float]) – [X,Y,Z] parameter moves the specimen. +• scale (float) – factor scales the given data. +• attrs (list) – attrs read from file if export.textExt(format=’x_y_z_r_attrs’) +were used (‘passed by reference’ style) +• **kw – (unused keyword arguments) is passed to utils.sphere +Returns list of spheres. +Lines starting with # are skipped +yade.ymport.textFacets(fileName, +format=’x1_y1_z1_x2_y2_z2_x3_y3_z3’, +shift=Vector3(0, 0, 0), scale=1.0, attrs=[], **kw) +Load facet coordinates from file in a format selected by the format argument, returns a list of +corresponding bodies; that may be inserted to the simulation with O.bodies.append(). +Parameters +2.4. +Yade modules reference +519 + +Yade Documentation, Release 3rd ed. +• filename (str) – file name +• format (str) – selected input format. +Supported 'x1_y1_z1_x2_y2_- +z2_x3_y3_z3'``(default), ``'x1_y1_z1_x2_y2_z2_x3_y3_z3_matId', +'id_x1_y1_z1_x2_y2_z2_x3_y3_z3_matId' or 'x1_y1_z1_x2_y2_z2_x3_y3_- +z3_attrs' +• shift ([float,float,float]) – [X,Y,Z] parameter moves the specimen. +• scale (float) – factor scales the given data. +• attrs (list) – attrs read from file (‘passed by reference’ style) +• **kw – (unused keyword arguments) is passed to utils.facet +Returns list of facets. +Lines starting with # are skipped +yade.ymport.textPolyhedra(fileName, material, shift=Vector3(0, 0, 0), scale=1.0, orienta- +tion=Quaternion((1, 0, 0), 0), **kw) +Load polyhedra from a text file. +Parameters +• filename (str) – file name. +Expected file format is the one output by ex- +port.textPolyhedra. +• shift ([float,float,float]) – [X,Y,Z] parameter moves the specimen. +• scale (float) – factor scales the given data. +• orientation (quaternion) – orientation of the imported polyhedra +• **kw – (unused keyword arguments) is passed to polyhedra_utils.polyhedra +Returns list of polyhedras. +Lines starting with # are skipped +yade.ymport.unv(fileName, shift=(0, 0, 0), scale=1.0, returnConnectivityTable=False, **kw) +Import geometry from unv file, return list of created facets. +param string fileName name of unv file +param (float,float,float)|Vector3 shift (X,Y,Z) parameter moves the spec- +imen. +param float scale factor scales the given data. +param **kw (unused keyword arguments) is passed to utils.facet +param bool returnConnectivityTable if True, apart from facets returns +also nodes (list of (x,y,z) nodes coordinates) and elements (list of (id1,id2,id3) +element nodes ids). If False (default), returns only facets +unv files are mainly used for FEM analyses (are used by OOFEM and Abaqus), but triangular +elements can be imported as facets. These files cen be created e.g. with open-source free software +Salome. +Example: examples/test/unv-read/unvRead.py. +2.5 Installation +• Linux systems: Yade can be installed from packages (pre-compiled binaries) or source code. The +choice depends on what you need: if you don’t plan to modify Yade itself, package installation is +easier. In the contrary case, you must download and install the source code. +• Other Operating Systems: Jump to the last section of this page. +520 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +• 64 bit Operating Systems required; no support for 32 bit (i386). +2.5.1 Packages +Stable packages +Since 2011, all Ubuntu (starting from 11.10, Oneiric) and Debian (starting from Wheezy) versions have +Yade in their main repositories. There are only stable releases in place. To install Yade, run the following: +sudo apt-get install yade +After that you can normally start Yade using the command yade or yade-batch. +This image shows versions and up to date status of Yade in some repositories. +Daily packages +Pre-built packages updated more frequently than the stable versions are provided for all currently sup- +ported Debian and Ubuntu versions and available on yade-dem.org/packages . +These are “daily” versions of the packages which are being updated regularly and, hence, include all the +newly added features. +To install the daily-version you need to add the repository to your /etc/apt/sources.list. +• Debian 9 stretch: +sudo bash -c 'echo "deb http://www.yade-dem.org/packages/ stretch main" >> /etc/apt/ +�→sources.list' +• Debian 10 buster: +sudo bash -c 'echo "deb http://www.yade-dem.org/packages/ buster main" >> /etc/apt/ +�→sources.list' +• Debian 11 bullseye: +sudo bash -c 'echo "deb http://www.yade-dem.org/packages/ bullseye main" >> /etc/apt/ +�→sources.list' +• Debian 12 bookworm: +sudo bash -c 'echo "deb http://www.yade-dem.org/packages/ bookworm main" >> /etc/apt/ +�→sources.list' +• Ubuntu 16.04 xenial: +sudo bash -c 'echo "deb http://www.yade-dem.org/packages/ xenial main" >> /etc/apt/ +�→sources.list' +• Ubuntu 18.04 bionic: +sudo bash -c 'echo "deb http://www.yade-dem.org/packages/ bionic main" >> /etc/apt/ +�→sources.list' +• Ubuntu 20.04 focal: +sudo bash -c 'echo "deb http://www.yade-dem.org/packages/ focal main" >> /etc/apt/sources. +�→list' +• Ubuntu 22.04 jammy: +sudo bash -c 'echo "deb http://www.yade-dem.org/packages/ jammy main" >> /etc/apt/sources. +�→list' +2.5. +Installation +521 + +Yade Documentation, Release 3rd ed. +Add the PGP-key AA915EEB as trusted and install yadedaily: +wget -O - http://www.yade-dem.org/packages/yadedev_pub.gpg | sudo apt-key add - +sudo apt-get update +sudo apt-get install yadedaily +After that you can normally start Yade using the command yadedaily or yadedaily-batch. yadedaily +on older distributions can have some disabled features due to older library versions, shipped with par- +ticular distribution. +The Git-repository for packaging stuff is available on GitLab. +If +you +do +not +need +yadedaily-package +anymore, +just +remove +the +corresponding +line +in +/etc/apt/sources.list and the package itself: +sudo apt-get remove yadedaily +To remove our key from keyring, execute the following command: +sudo apt-key remove AA915EEB +Daily and stable Yade versions can coexist without any conflicts, i.e., you can use yade and yadedaily +at the same time. +2.5.2 Docker +Yade can be installed using docker images, which are daily built. Images contain both stable and dialy +versions of packages. Docker images are based on supported distributions: +• Debian 9 stretch: +docker run -it registry.gitlab.com/yade-dev/docker-prod:debian-stretch +• Debian 10 buster: +docker run -it registry.gitlab.com/yade-dev/docker-prod:debian-buster +• Debian 11 bullseye: +docker run -it registry.gitlab.com/yade-dev/docker-prod:debian-bullseye +• Debian 12 bookworm: +docker run -it registry.gitlab.com/yade-dev/docker-prod:debian-bookworm +• Ubuntu 16.04 xenial: +docker run -it registry.gitlab.com/yade-dev/docker-prod:ubuntu16.04 +• Ubuntu 18.04 bionic: +docker run -it registry.gitlab.com/yade-dev/docker-prod:ubuntu18.04 +• Ubuntu 20.04 focal: +docker run -it registry.gitlab.com/yade-dev/docker-prod:ubuntu20.04 +• Ubuntu 22.04 jammy: +docker run -it registry.gitlab.com/yade-dev/docker-prod:ubuntu22.04 +After the container is pulled and is running, Yade functionality can be checked: +522 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +yade --test +yade --check +yadedaily --test +yadedaily --check +2.5.3 Source code +Installation from source code is reasonable, when you want to add or modify constitutive laws, engines, +functions etc. Installing the latest trunk version allows one to use newly added features, which are not +yet available in packaged versions. +Download +If you want to install from source, you can install either a release (numbered version, which is frozen) +or the current development version (updated by the developers frequently). You should download the +development version (called trunk) if you want to modify the source code, as you might encounter +problems that will be fixed by the developers. Release versions will not be updated (except for updates +due to critical and easy-to-fix bugs), but generally they are more stable than the trunk. +1. Releases can be downloaded from the download page, as compressed archive. Uncompressing the +archive gives you a directory with the sources. +2. The development version (trunk) can be obtained from the code repository at GitLab. +We use GIT (the git command) for code management (install the git package on your system and +create a GitLab account): +git clone git@gitlab.com:yade-dev/trunk.git +will download the whole code repository of the trunk. Check out Yade on GitLab for more details on +how to collaborate using git. +Alternatively, a read-only checkout is possible via https without a GitLab account (easier if you don’t +want to modify the trunk version): +git clone https://gitlab.com/yade-dev/trunk.git +For those behind a firewall, you can download the sources from our GitLab repository as compressed +archive. +Release and trunk sources are compiled in exactly the same way. +Prerequisites +Yade relies on a number of external software to run; they are checked before the compilation starts. +Some of them are only optional. +• cmake build system +• gcc compiler (g++); other compilers will not work; you need g++>=4.2 for openMP support +• boost 1.47 or later +• Qt library +• freeglut3 +• libQGLViewer +• python, numpy, ipython, sphinx, mpi4py +• matplotlib +2.5. +Installation +523 + +Yade Documentation, Release 3rd ed. +• eigen algebra library (minimal required version 3.2.1) +• gdb debugger +• sqlite3 database engine +• VTK library (optional but recommended) +• CGAL library (optional) +• SuiteSparse sparse algebra library (fluid coupling, optional, requires eigen>=3.1) +• OpenBLAS optimized and parallelized alternative to the standard blas+lapack (fluid coupling +FlowEngine, optional) +• Metis matrix preconditioning (fluid coupling, optional) +• OpenMPI library for parallel distributed computing (For MPI and OpenFOAM coupling, optional) +• python3-mpi4py MPI for Python (For MPI, optional) +• coin-or COIN-OR Linear Programming Solver (For PotentialBlock, optional) +• mpfr in C++ and mpmath in python for high precision Real or for CGAL exact predicates (optional) +• mpc is an MPFR extension to complex numbers. It is used explicitly together with MPFR. +Most of the list above is very likely already packaged for your distribution. In case you are confronted +with some errors concerning not available packages (e.g., package libmetis-dev is not available) it may +be necessary to add yade external ppa from https://launchpad.net/~yade-users/+archive/external (see +below) as well as http://www.yade-dem.org/packages (see the top of this page): +sudo add-apt-repository ppa:yade-users/external +sudo apt-get update +The following commands have to be executed in the command line of your corresponding distribution. +Just copy&paste to the terminal. Note, to execute these commands you need root privileges. +• Ubuntu 20.04, 18.04, Debian 9, 10, 11 and their derivatives: +sudo apt install cmake git freeglut3-dev libloki-dev libboost-all-dev fakeroot \ +dpkg-dev build-essential g++ python3-dev python3-ipython python3-matplotlib \ +libsqlite3-dev python3-numpy python3-tk gnuplot libgts-dev python3-pygraphviz \ +libvtk6-dev libeigen3-dev python3-xlib python3-pyqt5 pyqt5-dev-tools python3-mpi4py \ +python3-pyqt5.qtwebkit gtk2-engines-pixbuf python3-pyqt5.qtsvg libqglviewer-dev-qt5 \ +python3-pil libjs-jquery python3-sphinx python3-git libxmu-dev libxi-dev libcgal-dev \ +help2man libbz2-dev zlib1g-dev libopenblas-dev libsuitesparse-dev \ +libmetis-dev python3-bibtexparser python3-future coinor-clp coinor-libclp-dev \ +python3-mpmath libmpfr-dev libmpfrc++-dev libmpc-dev +• For Ubuntu +16.04 libqglviewer-dev-qt5 is to be replaced by libqglviewer-dev and +python3-ipython by ipython3. +• The packages python3-mpmath libmpfr-dev libmpfrc++-dev in above list are required only if +one wants to use high precision calculations. The latter two only if mpfr will be used. See high +precision documentation for more details. +• For building documentation (the make doc invocation explained below) additional package +texlive-xetex is required. +On some multi-language systems an error Building format(s) +--all. This may take some time... fmtutil failed. may occur, in that case a package +locales-all is required. +Some of the packages (for example, cmake, eigen3) are mandatory, some of them are optional. Watch +for notes and warnings/errors, which are shown by cmake during the configuration step. If the miss- +ing package is optional, some of Yade features will be disabled (see the messages at the end of the +configuration). +524 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +Some packages listed here are relatively new and they can be absent in your distribution (for example, +libmetis-dev). +They can be installed from yade-dem.org/packages or from our external PPA. If not +installed the related features will be disabled automatically. +If you are using other distributions than Debian or its derivatives you should install the software packages +listed above. Their names in other distributions can differ from the names of the Debian-packages. +Warning: +If you have Ubuntu 14.04 Trusty, you need to add -DCMAKE_CXX_FLAGS=- +frounding-math during the configuration step of compilation (see below) or to install libcgal-dev +from our external PPA. Otherwise the following error occurs on AMD64 architectures: +terminate called after throwing an instance of 'CGAL::Assertion_exception' +what(): +CGAL ERROR: assertion violation! +Expr: -CGAL_IA_MUL(-1.1, 10.1) != CGAL_IA_MUL(1.1, 10.1) +File: /usr/include/CGAL/Interval_nt.h +Line: 209 +Explanation: Wrong rounding: did you forget the +-frounding-math +option if you use GCC (or␣ +�→ -fp-model strict +for Intel)? +Aborted +Compilation +You should create a separate build-place-folder, where Yade will be configured and where the source code +will be compiled. Here is an example for a folder structure: +myYade/ +## base directory +trunk/ +## folder for source code in which you use git +build/ +## folder in which the sources will be compiled; build-directory; use␣ +�→cmake here +install/ +## install folder; contains the executables +Then, inside this build-directory you should call cmake to configure the compilation process: +cmake -DCMAKE_INSTALL_PREFIX=/path/to/installfolder /path/to/sources +For the folder structure given above call the following command in the folder “build”: +cmake -DCMAKE_INSTALL_PREFIX=../install ../trunk +Additional options can be configured in the same line with the following syntax: +cmake -DOPTION1=VALUE1 -DOPTION2=VALUE2 +For example: +cmake -DENABLE_POTENTIAL_BLOCKS=ON +The following cmake options are available: (see the source code for a most up-to-date list) +• CMAKE_INSTALL_PREFIX: path where Yade should be installed (/usr/local by default) +• LIBRARY_OUTPUT_PATH: path to install libraries (lib by default) +• DEBUG: compile in debug-mode (OFF by default) +• MAX_LOG_LEVEL: set maximum level for LOG_* macros compiled with ENABLE_LOGGER, +(default is 5) +• CMAKE_VERBOSE_MAKEFILE: output additional information during compiling (OFF by de- +fault) +• SUFFIX: suffix, added after binary-names (version number by default) +2.5. +Installation +525 + +Yade Documentation, Release 3rd ed. +• NOSUFFIX: do not add a suffix after binary-name (OFF by default) +• YADE_VERSION: explicitly set version number (is defined from git-directory by default) +• ENABLE_ASAN: AddressSanitizer allows detection of memory errors, memory leaks, heap cor- +ruption errors and out-of-bounds accesses (but it is slow) +• ENABLE_CGAL: enable CGAL option (ON by default) +• ENABLE_COMPLEX_MP: use boost multiprecision complex for ComplexHP, otherwise use +std::complex>. +See high precision documentation for additional details. +(ON by +default if possible: requires boost >= 1.71) +• ENABLE_DEFORM: enable constant volume deformation engine (OFF by default) +• ENABLE_FAST_NATIVE: use maximum optimization compiler flags including -Ofast and +-mtune=native. Note: native means that code will only run on the same processor type on +which it was compiled. Observed speedup was 2% (below standard deviation measurement error) +and above 5% if clang compiler was used. (OFF by default) +• ENABLE_FEMLIKE: enable meshed solids, FEM-like (ON by default) +• ENABLE_GL2PS: enable GL2PS-option (ON by default) +• ENABLE_GTS: enable GTS-option (ON by default) +• ENABLE_GUI: enable GUI option (ON by default) +• ENABLE_LBMFLOW: enable LBMFLOW-option, LBM_ENGINE (ON by default) +• ENABLE_LS_DEM: enable a LevelSet shape description (ON by default) +• ENABLE_LINSOLV: enable LINSOLV-option (ON by default) +• ENABLE_LIQMIGRATION: enable LIQMIGRATION-option, see [Mani2013] for details (OFF +by default) +• ENABLE_LOGGER: use boost::log library for logging separately for each class (ON by default) +• ENABLE_MASK_ARBITRARY: enable MASK_ARBITRARY option (OFF by default) +• ENABLE_MPFR: use mpfr in C++ and mpmath in python. It can be used for higher precision +Real or for CGAL exact predicates (OFF by default) +• ENABLE_MPI: Enable MPI enviroment and communication, required distributed memory and +for Yade-OpenFOAM coupling (ON by default) +• ENABLE_OAR: generate a script for oar-based task scheduler (OFF by default) +• ENABLE_OPENMP: enable OpenMP-parallelizing option (ON by default) +• ENABLE_PARTIALSAT : enable the partially saturated clay engine, under construction (ON by +default) +• ENABLE_PFVFLOW: enable PFVFLOW-option, FlowEngine (ON by default) +• ENABLE_POTENTIAL_BLOCKS: enable potential blocks option (ON by default) +• ENABLE_POTENTIAL_PARTICLES: enable potential particles option (ON by default) +• ENABLE_PROFILING: enable profiling, e.g., shows some more metrics, which can define bottle- +necks of the code (OFF by default) +• ENABLE_REAL_HP: allow using twice, quadruple or higher precisions of Real as RealHP<2>, +RealHP<4> or RealHP in computationally demanding sections of C++ code. See high precision +documentation for additional details (ON by default). +• ENABLE_SPH: enable SPH-option, Smoothed Particle Hydrodynamics (OFF by default) +• ENABLE_THERMAL : enable thermal engine (ON by default, experimental)” +• ENABLE_TWOPHASEFLOW: enable TWOPHASEFLOW-option, TwoPhaseFlowEngine (ON +by default) +526 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +• ENABLE_USEFUL_ERRORS: enable useful compiler errors which help a lot in error-free devel- +opment (ON by default) +• ENABLE_VTK: enable VTK-export option (ON by default) +• REAL_PRECISION_BITS, REAL_DECIMAL_PLACES: specify either of them to use a custom +calculation precision of Real type. By default double (64 bits, 15 decimal places) precision is used +as the Real type. See high precision documentation for additional details. +• runtimePREFIX: used for packaging, when install directory is not the same as runtime directory +(/usr/local by default) +• VECTORIZE: enables vectorization and alignment in Eigen3 library, experimental (OFF by de- +fault) +• USE_QT5: use QT5 for GUI (ON by default) +• CHOLMOD_GPU link Yade to custom SuiteSparse installation and activate GPU accelerated +PFV (OFF by default) +• SUITESPARSEPATH: define this variable with the path to a custom suitesparse install +• PYTHON_VERSION: force Python version to the given one, e.g. -DPYTHON_VERSION=3.5. Set +to -1 to automatically use the last version on the system (-1 by default) +It is possible to disable all options to create the slim build: +cmake -DDISABLE_ALL=ON +In this case all available options will be switched off. In this case some required options can be enabled +explicitely: +cmake -DDISABLE_ALL=ON -DENABLE_VTK=ON +For using more extended parameters of cmake, please follow the corresponding documentation on +https://cmake.org/documentation. +Warning: +Only Qt5 is supported. On Debian/Ubuntu operating systems libQGLViewer of version +2.6.3 and higher are compiled against Qt5 (for other operating systems refer to the package archive +of your distribution). If you mix Qt-versions a Segmentation fault will appear just after Yade is +started. To provide necessary build dependencies for Qt5, install python-pyqt5 pyqt5-dev-tools. +If cmake finishes without errors, you will see all enabled and disabled options at the end. Then start the +actual compilation process with: +make +The compilation process can take a considerable amount of time, be patient. If you are using a multi-core +systems you can use the parameter -j to speed-up the compilation and split the compilation onto many +cores. For example, on 4-core machines it would be reasonable to set the parameter -j4. Note, Yade +requires approximately 3GB RAM per core for compilation, otherwise the swap-file will be used and +compilation time dramatically increases. +The installation is performed with the following command: +make install +The install command will in fact also recompile if source files have been modified. Hence there is no +absolute need to type the two commands separately. You may receive make errors if you don’t have +permission to write into the target folder. These errors are not critical but without writing permissions +Yade won’t be installed in /usr/local/bin/. +After +the +compilation +finished +successfully, +the +new +built +can +be +started +by +navigating +to +/path/to/installfolder/bin and calling yade via (based on version yade-2014-02-20.git-a7048f4): +2.5. +Installation +527 + +Yade Documentation, Release 3rd ed. +cd /path/to/installfolder/bin +./yade-2014-02-20.git-a7048f4 +For building the documentation you should at first execute the command make install and +then make doc to build it. +The generated files will be stored in your current install directory +/path/to/installfolder/share/doc/yade-your-version. Once again writing permissions are necessary for +installing into /usr/local/share/doc/. To open your local documentation go into the folder html and +open the file index.html with a browser. +make manpage command generates and moves manpages in a standard place. make check command +executes standard test to check the functionality of the compiled program. +Yade can be compiled not only by GCC-compiler, but also by CLANG front-end for the LLVM compiler. +For that you set the environment variables CC and CXX upon detecting the C and C++ compiler to +use: +export CC=/usr/bin/clang +export CXX=/usr/bin/clang++ +cmake -DOPTION1=VALUE1 -DOPTION2=VALUE2 +Clang does not support OpenMP-parallelizing for the moment, that is why the feature will be disabled. +Supported linux releases +Currently supported1 linux releases and their respective docker files are: +• Ubuntu 16.04 xenial +• Ubuntu 18.04 bionic +• Debian 9 stretch +• Debian 10 buster +• openSUSE 15 +These are the bash commands used to prepare the linux distribution and environment for installing and +testing yade. These instructions are automatically performed using the gitlab continuous integration +service after each merge to master. This makes sure that yade always works correctly on these linux +distributions. In fact yade can be installed manually by following step by step these instructions in +following order: +1. Bash commands in the respective Dockerfile to install necessary packages, +2. do git clone https://gitlab.com/yade-dev/trunk.git, +3. then the cmake_* commands in the .gitlab-ci.yml file for respective distribution, +4. then the make_* commands to compile yade, +5. and finally the --check and --test commands. +6. Optionally documentation can be built with make doc command, however currently it is not guar- +anteed to work on all linux distributions due to frequent interface changes in sphinx. +These instructions use ccache and ld.gold to speed-up compilation as described below. +Python 2 backward compatibility +Following the end of Python 2 support (beginning of 2020), Yade compilation on a Python 2 ecosystem +is no longer garanteed since the 6e097e95 trunk version. +Python 2-compilation of the latter is still +possible using the above PYTHON_VERSION cmake option, requiring Python 2 version of prerequisites +1 To see details of the latest build log click on the master branch. +528 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +packages whose list can be found in the corresponding paragraph (Python 2 backward compatibility) of +the historical doc. +Ongoing development of Yade now assumes a Python 3 environment, and you may refer to some notes +about converting Python 2 scripts into Python 3 if needed. +2.5.4 Speed-up compilation +Compile with ccache +Caching previous compilations with ccache can significantly speed up re-compilation: +cmake -DCMAKE_CXX_COMPILER_LAUNCHER=ccache [options as usual] +Additionally one can check current ccache status with command ccache --show-stats (ccache -s for +short) or change the default cache size stored in file ~/.ccache/ccache.conf. +Compile with distcc +When spliting the compilation on many cores (make -jN), N is limited by the available cores and memory. +It is possible to use more cores if remote computers are available, distributing the compilation with distcc +(see distcc documentation for configuring slaves and master): +export CC="distcc gcc" +export CXX="distcc g++" +cmake [options as usual] +make -jN +The two tools can be combined, adding to the above exports: +export CCACHE_PREFIX="distcc" +Compile with cmake UNITY_BUILD +This option concatenates source files in batches containing several *.cpp each, in order to share the +overhead of include directives (since most source files include the same boost headers, typically). It +accelerates full compilation from scratch (quite significantly). It is activated by adding the following to +cmake command, CMAKE_UNITY_BUILD_BATCH_SIZE defines the maximum number of files to be concate- +nated together (the higher the better, main limitation might be available RAM): +-DCMAKE_UNITY_BUILD=ON -DCMAKE_UNITY_BUILD_BATCH_SIZE=18 +This method is helpless for incremental re-compilation and might even be detrimental since a full batch +has to be recompiled each time a single file is modified. If it is anticipated that specific files will need +incremental compilation they can be excluded from the unity build by assigning their full path to cmake +flag NO_UNITY (a single file or a comma-separated list): +-DCMAKE_UNITY_BUILD=ON -DCMAKE_UNITY_BUILD_BATCH_SIZE=18 -DNO_UNITY=../trunk/pkg/dem/ +�→CohesiveFrictionalContactLaw.cpp +Link time +The link time can be reduced by changing the default linker from ld to ld.gold. They are both in the +same package binutils (on opensuse15 it is package binutils-gold). To perform the switch execute +these commands as root: +2.5. +Installation +529 + +Yade Documentation, Release 3rd ed. +ld --version +update-alternatives --install "/usr/bin/ld" "ld" "/usr/bin/ld.gold" 20 +update-alternatives --install "/usr/bin/ld" "ld" "/usr/bin/ld.bfd" 10 +ld --version +To switch back run the commands above with reversed priorities 10 ￿ 20. Alternatively a manual selection +can be performed by command: update-alternatives --config ld. +Note: ld.gold is incompatible with the compiler wrapper mpicxx in some distributions, which is mani- +fested as an error in the cmake stage. We do not use mpicxx for our gitlab builds currently. If you want +to use it then disable ld.gold. Cmake MPI-related failures have also been reported without the mpicxx +compiler, if it happens then the only solution is to disable either ld.gold or the MPI feature. +2.5.5 Cloud Computing +It is possible to exploit cloud computing services to run Yade. The combo Yade/Amazon Web Service +has been found to work well, namely. Detailed instructions for migrating to amazon can be found in the +section Using YADE with cloud computing on Amazon EC2. +2.5.6 GPU Acceleration +The FlowEngine can be accelerated with CHOLMOD’s GPU accelerated solver. The specific hardware +and software requirements are outlined in the section Accelerating Yade’s FlowEngine with GPU. +2.5.7 Special builds +The software can be compiled by a special way to find some specific bugs and problems in it: memory +corruptions, data races, undefined behaviour etc. +The listed sanitizers are runtime-detectors. They can only find the problems in the code, if the particular +part of the code is executed. If you have written a new C++ class (constitutive law, engine etc.) try to +run your Python script with the sanitized software to check, whether the problem in your code exist. +AddressSanitizer +AddressSanitizer is a memory error detector, which helps to find heap corruptions, out-of-bounds errors +and many other memory errors, leading to crashes and even wrong results. +To compile Yade with this type of sanitizer, use ENABLE_ASAN option: +cmake -DENABLE_ASAN=1 +The compilation time, memory consumption during build and the size of build-files are much higher +than during the normall build. Monitor RAM and disk usage during compilation to prevent out-of-RAM +problems. +To find the proper libasan library in your particular distribution, use locate or find /usr -iname +"libasan*so" command. Then, launch your yade executable in connection with that libasan library, +e.g.: +LD_PRELOAD=/some/path/to/libasan.so yade +By default the leak detector is enabled in the asan build. Yade is producing a lot of leak warnings at the +moment. To mute those warnings and concentrate on other memory errors, one can use detect_leaks=0 +option. Accounting for the latter, the full command to run tests with the AddressSanitized-Yade on +Debian 10 Buster is: +530 +Chapter 2. +Yade for users + +Yade Documentation, Release 3rd ed. +ASAN_OPTIONS=detect_leaks=0:verify_asan_link_order=false yade --test +If you add a new check script, it is being run automatically through the AddressSanitizer in the CI- +pipeline. +2.5.8 Yubuntu +If you are not running a Linux system there is a way to create an Ubuntu live-usb on any usb mass- +storage device (minimum size 10GB). It is a way to boot the computer on a linux system with Yadedaily +pre-installed without affecting the original system. More informations about this alternative are available +here (see the README file first). +Alternatively, images of a linux virtual machine can be downloaded, here again, and they should run on +any system with a virtualization software (tested with VirtualBox and VMWare). +2.6 Acknowledging Yade +We kindly ask Yade users to cite this documentation as a whole in scientific publications as a way to +assess Yade’s contribution to their field. It can be done using the following reference: +• V. +Šmilauer +et +al. +(2021), +Yade +Documentation +3rd +ed. +The +Yade +Project. +DOI:10.5281/zenodo.5705394 (http://yade-dem.org/doc/) +Beyond acknowledging the work of the developpers, it helps finding new use cases or new users by +tracking the citations on Yade’s Scholar profile. +2.6. +Acknowledging Yade +531 + +Yade Documentation, Release 3rd ed. +532 +Chapter 2. +Yade for users + +Chapter 3 +Yade for programmers +3.1 Programmer’s manual +3.1.1 Build system +Yade uses cmake the cross-platform, open-source build system for managing the build process. It takes +care of configuration, compilation and installation. CMake is used to control the software compilation +process using simple platform and compiler independent configuration files. CMake generates native +makefiles and workspaces that can be used in the compiler environment of your choice. +Building +The structure of Yade source tree is presented below. We shall call each top-level component module (ex- +cluding, doc, examples and scripts which don’t participate in the build process). Some subdirectories +of modules are skipped for brevity, see README.rst files therein for more information: +cMake/ +## cmake files used to detect compilation requirements +core/ +## core simulation building blocks +data/ +## data files used by yade, packaged separately +doc/ +## this documentation +examples/ +## examples directory +gui/ +## user interfaces +qt5/ +## same, but for qt5 +lib/ +## support libraries, not specific to simulations +preprocessing/ +## files associated with creation or generation of the simulation +dem/ +## creating a DEM simulation +potential/ +## creating a PotentialBlocks or PotentialParticles simulation +README.rst +## more information about this directory +pkg/ +## simulation-specific files +common/ +## generally useful classes +dem/ +## classes for Discrete Element Method +README.rst +## more information about this directory +postprocessing/ +## files associated with extracting results for postprocessing +dem/ +## general data extraction from DEM, no particular data target +image/ +## creating images from simulation +vtk/ +## extracting data for VTK +README.rst +## more information about this directory +py/ +## python modules +scripts/ +## helper scripts including packaging and checks-and-tests +533 + +Yade Documentation, Release 3rd ed. +Header installation +CMAKE uses the original source layout and it is advised to use #include style of +inclusion rather than #include "Class.hpp" even if you are in the same directory. The following table +gives a few examples: +Original header location +Included as +core/Scene.hpp +#include +lib/base/Logging.hpp +#include +lib/serialization/Serializable.hpp +#include +pkg/dem/SpherePack.hpp +#include +Automatic compilation +In the pkg/ directory, situation is different. In order to maximally ease addition of modules to yade, all +*.cpp files are automatically scanned recursively by CMAKE and considered for compilation. +To enable/disable some component use the cmake flags ENABLE_FEATURE, which are listed in: +1. compilation instructions. +2. CMakeLists.txt. +When some component is enabled an extra #define flag YADE_FEATURE is passed from cmake to the +compiler. Then inside the code both the .cpp and .hpp files which contain the FEATURE feature should +have an #ifdef YADE_FEATURE guard at the beginning. +Linking +The order in which modules might depend on each other is given as follows: +mod- +ule +resulting shared library +dependencies +lib +libyade-support.so +can depend on external libraries, may not depend on any other +part of Yade. +core +libcore.so +yade-support; may depend on external libraries. +pkg +libplugins.so +core, yade-support +gui +libQtGUI.so, +libPythonUI.so +lib, core, pkg +py +(many files) +lib, core, pkg, external +3.1.2 Development tools +Integrated Development Environment and other tools +A frequently used IDE is Kdevelop. We recommend using this software for navigating in the sources, +compiling and debugging. Other useful tools for debugging and profiling are Valgrind and KCachegrind. +A series of wiki pages is dedicated to these tools in the development section of the wiki. +Hosting and versioning +The Yade project is kindly hosted at Launchpad and GitLab: +• source code on gitlab +• issue and bug tracking on gitlab +534 +Chapter 3. +Yade for programmers + +Yade Documentation, Release 3rd ed. +• release downloads on launchpad +• yade-dev mailing list on launchpad: yade-dev@lists.launchpad.net +• yade-users mailing list on launchpad: yade-users@lists.launchpad.net +• questions and answers on launchpad +The versioning software used is GIT, for which a short tutorial can be found in Yade on GitLab. GIT is +a distributed revision control system. It is available packaged for all major linux distributions. +The source code is periodically imported to Launchpad for building PPA-packages. The repository can +be http-browsed. +Build robot +A build robot hosted at UMS Gricad is tracking source code changes via gitlab pipeline mechanism. +Each time a change in the source code is committed to the main development branch via GIT, or a +Merge Request (MR) is submitted the “buildbot” downloads and compiles the new version, and then +starts a series of tests. +If a compilation error has been introduced, it will be notified to the yade-dev mailing list and to the +committer, thus helping to fix problems quickly. If the compilation is successful, the buildbot starts +unit regression tests and “check tests” (see below) and report the results. If all tests are passed, a new +version of the documentation is generated and uploaded to the website in html and pdf formats. As a +consequence, those two links always point to the documentation (the one you are reading now) of the last +successful build, and the delay between commits and documentation updates are very short (minutes). +The buildbot activity and logs can be browsed online. +The output of each particular build is directly accessible by clicking the green “Passed” button, and then +clicking “Browse” in the “Job Artifacts” on the right. +3.1.3 Debugging +For yade debugging two tools are available: +1. Use the debug build so that the stack trace provides complete information about potential crash. +This can be achieved in two ways: +a) Compiling yade with cmake option -DDEBUG=ON, +b) Installing yade-dbgsym debian/ubuntu package (this option will be available after this task +is completed). +2. Use Logging framework described below. +These tools can be used in conjunction with other software. A detailed discussion of these is on yade +wiki. These tools include: kdevelop, valgrind, alleyoop, kcachegrind, ddd, gdb, kompare, kdiff3, meld. +Note: +On some linux systems stack trace will not be shown and a message ptrace: Operation +not permitted will appear instead. +To enable stack trace issue command: sudo echo 0 > /proc/ +sys/kernel/yama/ptrace_scope. To disable stack trace issue command sudo echo 1 > /proc/sys/ +kernel/yama/ptrace_scope. +Hint: +When debugging make sure there is enough free space in /tmp. +3.1. +Programmer’s manual +535 + +Yade Documentation, Release 3rd ed. +Logging +Yade uses boost::log library for flexible logging levels and per-class debugging. See also description of +log module. A cmake compilation option -DENABLE_LOGGER=ON must be supplied during compilation1. +Figure imgLogging shows example use of logging framework. +Usually a ClassName appears in place +of _log.cpp shown on the screenshot. It is there because the yade.log module uses CREATE_CPP_- +LOCAL_LOGGER macro instead of the regular DECLARE_LOGGER and CREATE_LOGGER, which are discussed +below. +Note: +Default format of log message is: + ClassName:LineNumber FunctionName: Log Message +special macro LOG_NOFILTER is printed without ClassName because it lacks one. +Config files can be saved and loaded via readConfigFile and saveConfigFile. The defaultConfigFileName +is read upon startup if it exists. The filter level setting -f supplied from command line will override the +setting in config file. +Log levels +Following debug levels are supported: +1 Without -DENABLE_LOGGER=ON cmake option the debug macros in /lib/base/Logging.hpp use regular std::cerr for +output, per-class logging and log levels do not work. +536 +Chapter 3. +Yade for programmers + +1og +In [2l: yade.log.setLevel("_log-cpp",5) +IFO> + _log.cpp:1o1 yoid setLeyel(std::-_cxx11::string. int): filter log level for _log.cpp has been set to 5 +In [3l: log.setLevel("NewtonIntegrator",4) + _log.cpp:101 void setLevei(std::-_cxx11: :string, int): filter log level for NewtonIntegrator has been set to 4 +In [4]: log-getUsedLevels() +Out[4]:{'Default':3."NewtonIntegrator':4."_log-cpp':5} +In [5l: yade.log.testAllLevels() +:54voidtestAllLevels():Testloglevel:LOG_o_NOFILTER,testint:0 +_log.cpp:55 void testAllLevels(): Test log level: LOG_1_FATAL, test int: 1 +KDEBUG> + :62 yoid testAllLevels(): Below 6 yariables are printed at filter level TRACE, then macro TRACE: is usedYade Documentation, Release 3rd ed. +Table 1: Yade logging verbosity levels. +macro name +filter name +option +explanation +LOG_NOFILTER +log.NOFILTER +-f0 +Will print only the unfiltered messages. +The LOG_- +NOFILTER macro is for developer use only, so basically +-f0 means that nothing will be printed. This log level is +not useful unless a very silent mode is necessary. +LOG_FATAL +log.FATAL +-f1 +Will print only critical errors. +Even a throw to yade +python interface will not recover from this situation. This +is usually followed by yade exiting to shell. +LOG_ERROR +log.ERROR +-f2 +Will also print errors which do not require to throw to +yade python interface. +Calculations will continue, but +very likely the results will be all wrong. +LOG_WARN +log.WARN +-f3 +Will also print warnings about recoverable problems that +you should be notified about (e.g., invalid value in a con- +figuration file, so yade fell back to the default value). +LOG_INFO +log.INFO +-f4 +Will also print all informational messages (e.g. +some- +thing was loaded, something was called, etc.). +LOG_DEBUG +log.DEBUG +-f5 +Will also print debug messages. A yade developer puts +them everywhere, and yade user enables them on per- +class basis to provide some extra debug info. +LOG_TRACE +log.TRACE +-f6 +Trace messages, they capture every possible detail about +yade behavior. +Yade default log level is yade.log.WARN which is the same as invoking yade -f3. +Setting a filter level +Warning: +The messages (such as a << b << " message.") given as arguments to LOG_* macros +are used only if the message passes the filter level. Do not use such messages to perform mission +critical calculations. +There are two settings for the filter level, the Default level used when no ClassName (or "filename. +cpp") specific filter is set and a filter level set for specific ClassName (or "filename.cpp"). They can +be set with following means: +1. When starting yade with yade -fN command, where N sets the Default filter level. The default +value is yade.log.WARN (3). +2. To change Default filter level during runtime invoke command log.setLevel("Default",value) +or log.setDefaultLogLevel(value): +Yade [1]: import log +Yade [2]: log.setLevel("Default",log.WARN) +Yade [3]: log.setLevel("Default",3) +Yade [4]: log.setDefaultLogLevel(log.WARN) +Yade [5]: log.setDefaultLogLevel(3) +3. To change filter level for SomeClass invoke command: +Yade [6]: import log +(continues on next page) +3.1. +Programmer’s manual +537 + +Yade Documentation, Release 3rd ed. +(continued from previous page) +Yade [7]: log.setLevel("NewtonIntegrator",log.TRACE) +Yade [8]: log.setLevel("NewtonIntegrator",6) +4. To change the filter level for "filename.cpp" use the name specified when creating it. For example +manipulating filter log level of "_log.cpp" might look like following: +Yade [9]: import log +Yade [10]: log.getUsedLevels() +Out[10]: {} +Yade [11]: log.setLevel("_log.cpp",log.WARN) +Yade [12]: log.getUsedLevels() +Out[12]: {} +Yade [13]: log.getAllLevels()["_log.cpp"] +--------------------------------------------------------------------------- +KeyError +Traceback (most recent call last) +~/yade/lib/x86_64-linux-gnu/yadeflip/py/yade/__init__.py in +----> 1 log.getAllLevels()["_log.cpp"] +KeyError: '_log.cpp' +Debug macros +To enable debugging for particular class the DECLARE_LOGGER; macro should be put in class definition +inside header to create a separate named logger for that class. Then the CREATE_LOGGER(ClassName); +macro must be used in the class implementation .cpp file to create the static variable. Sometimes a logger +is necessary outside the class, such named logger can be created inside a .cpp file and by convention its +name should correspond to the name of the file, use the macro CREATE_CPP_LOCAL_LOGGER("filename. +cpp"); for this. On rare occasions logging is necessary inside .hpp file outside of a class (where the local +class named logger is unavailable), then the solution is to use LOG_NOFILTER(…) macro, because it is the +only one that can work without a named logger. If the need arises this solution can be improved, see +Logging.cpp for details. +All debug macros (LOG_TRACE, LOG_DEBUG, LOG_INFO, LOG_WARN, LOG_ERROR, LOG_FATAL, LOG_NOFILTER) +listed in section above accept the std::ostream syntax inside the brackets, such as LOG_TRACE( a << +b << " text" ). The LOG_NOFILTER is special because it is always printed regardless of debug level, +hence it should be used only in development branches. +Additionally seven macros for printing variables at LOG_TRACE level are available: TRVAR1, TRVAR2, +TRVAR3, TRVAR4, TRVAR5, TRVAR6 and TRVARn. They print the variables, e.g.: TRVAR3(testInt,testStr, +testReal); or TRVARn((testInt)(testStr)(testReal)). See function testAllLevels for example use. +The macro TRACE; prints a "Been here" message at TRACE log filter level, and can be used for quick +debugging. +Utility debug macros +The LOG_TIMED_* family of macros: +In some situations it is useful to debug variables inside a very fast, or maybe a multithreaded, loop. +In such situations it would be useful to: +1. Avoid spamming console with very fast printed messages and add some print timeout to them, +preferably specified with units of seconds or milliseconds. +538 +Chapter 3. +Yade for programmers + +Yade Documentation, Release 3rd ed. +2. Make sure that each separate thread has opportunity to print message, without interleaving such +messages with other threads. +To use above functionality one must #include in the .cpp file which +provides the LOG_TIMED_* and TIMED_TRVAR* macro family. Example usage can be found in function +testTimedLevels. +To satisfy the first requirement all LOG_TIMED_* macros accept two arguments, where the first argument +is the wait timeout, using standard C++14 / C++20 time units, example use is LOG_TIMED_INFO( 2s +, "test int: " << testInt++); to print every 2 seconds. +But only seconds and milliseconds are +accepted (this can be changed if necessary). +To satisfy the second requirement a thread_local static Timer variable is used. This way each thread in +a parallel loop can print a message every 500ms or 10s e.g. in this parallel loop. The time of last print to +console is stored independently for each thread and an extra code block which checks time is added. It +means that a bit more checks are done than typical LOG_* which only perform an integer comparison to +check filter level. Therefore suggested use is only during heavy debugging. When debugging is finished +then better to remove them. +Note: +The *_TRACE family of macros are removed by compiler during the release builds, because the +default -DMAX_LOG_LEVEL is 5. So those are very safe to use, but to have them working locally make +sure to compile yade with cmake -DMAX_LOG_LEVEL=6 option. +The LOG_ONCE_* family of macros: +In a similar manner a LOG_ONCE_* and ONCE_TRVAR* family of macros is provided inside file Log- +gingUtils.hpp. Then the message is printed only once. +All debug macros are summarized in the table below: +3.1. +Programmer’s manual +539 + +Yade Documentation, Release 3rd ed. +Table 2: Yade debug macros. +macro name +explanation +DECLARE_LOGGER; +Declares logger variable inside class definition in +.hpp file. +CREATE_LOGGER(ClassName); +Creates +logger +static +variable +(with +name +"ClassName") inside class implementation in . +cpp file. +TEMPLATE_CREATE_- +LOGGER(ClassName); +Creates +logger +static +variable +(with +name +"ClassName") inside class imple- +mentation in a .cpp file. Use this for templated +classes. +CREATE_CPP_LOCAL_LOGGER("filename.cpp"); +Creates logger static variable outside of any +class (with name "filename.cpp") inside the +filename.cpp file. +LOG_TRACE, LOG_TIMED_TRACE, LOG_ONCE_TRACE, +LOG_DEBUG, LOG_TIMED_DEBUG, LOG_ONCE_DEBUG, +LOG_INFO, LOG_TIMED_INFO, LOG_ONCE_INFO, +LOG_WARN, LOG_TIMED_WARN, LOG_ONCE_WARN, +LOG_ERROR, LOG_TIMED_ERROR, LOG_ONCE_ERROR, +LOG_FATAL, LOG_TIMED_FATAL, LOG_ONCE_FATAL, +LOG_NOFILTER, LOG_TIMED_NOFILTER, +LOG_ONCE_NOFILTER +Prints message using std::ostream syntax, like: +LOG_TRACE( a << b << " text" ) +LOG_TIMED_TRACE( 5s , a << b << " text" +); , prints every 5 seconds +LOG_TIMED_DEBUG( 500ms , a );, prints every +500 milliseconds +LOG_ONCE_TRACE( a << b << " text" ); , +prints just once +LOG_ONCE_DEBUG( a );, prints only once +TRVAR1, TIMED_TRVAR1, ONCE_TRVAR1, +TRVAR2, TIMED_TRVAR2, ONCE_TRVAR2, +TRVAR3, TIMED_TRVAR3, ONCE_TRVAR3, +TRVAR4, TIMED_TRVAR4, ONCE_TRVAR4, +TRVAR5, TIMED_TRVAR5, ONCE_TRVAR5, +TRVAR6, TIMED_TRVAR6, ONCE_TRVAR6, +TRVARn, TIMED_TRVARn, ONCE_TRVARn +Prints provided variables like: +TRVAR3(testInt,testStr,testReal); +TRVARn((testInt)(testStr)(testReal)); +TIMED_TRVAR3( 10s , testInt , testStr , +testReal); +ONCE_TRVARn( +(testInt)(testStr)(testReal)); +See file py/_log.cpp for example use. +TRACE; +Prints a "Been here" message at TRACE log filter +level. +LOG_TIMED_6, LOG_6_TRACE, LOG_ONCE_6, +LOG_TIMED_5, LOG_5_DEBUG, LOG_ONCE_5, +LOG_TIMED_4, LOG_4_INFO, LOG_ONCE_4, +LOG_TIMED_3, LOG_3_WARN, LOG_ONCE_3, +LOG_TIMED_2, LOG_2_ERROR, LOG_ONCE_2, +LOG_TIMED_1, LOG_1_FATAL, LOG_ONCE_1, +LOG_TIMED_0, LOG_0_NOFILTER, LOG_ONCE_0, +LOG_TIMED_6_TRACE, LOG_6, LOG_ONCE_6_TRACE, +LOG_TIMED_5_DEBUG, LOG_5, LOG_ONCE_5_DEBUG, +LOG_TIMED_4_INFO, LOG_4, LOG_ONCE_4_INFO, +LOG_TIMED_3_WARN, LOG_3, LOG_ONCE_3_WARN, +LOG_TIMED_2_ERROR, LOG_2, LOG_ONCE_2_ERROR, +LOG_TIMED_1_FATAL, LOG_1, LOG_ONCE_1_FATAL, +LOG_TIMED_0_NOFILTER, LOG_0 +LOG_ONCE_0_NOFILTER, +Additional macro aliases for easier use in editors +with tab completion. They have have a filter level +number in their name. +540 +Chapter 3. +Yade for programmers + +Yade Documentation, Release 3rd ed. +Maximum log level +Using boost::log for log filtering means that each call to LOG_* macro must perform a single integer +comparison to determine if the message passes current filter level. +For production use calculations +should be as fast as possible and this filtering is not optimal, because the macros are not optimized +out, as they can be re-enabled with a simple call to log.setLevel("Default",log.TRACE) or log. +setLevel("Default",6). The remedy is to use the cmake compilation option MAX_LOG_LEVEL=4 (or 3) +which will remove macros higher than the specified level during compilation. The code will run slightly +faster and the command log.setLevel("Default",6) will only print a warning that such high log level +(which can be checked with log.getMaxLevel() call) is impossible to obtain with current build. +Note: +At the time when logging was introduced into yade the speed-up gain was so small, that +it turned out to be impossible to measure with yade -f0 --stdperformance command. Hence this +option MAX_LOG_LEVEL was introduced only on principle. +The upside of this approach is that yade can be compiled in a non-debug build, and the log filtering +framework can be still used. +3.1.4 Regression tests +Yade contains two types of regression tests, some are unit tests while others are testing more complex +simulations. Although both types can be considered regression tests, the usage is that we name the first +simply “regression tests”, while the latest are called “check tests”. Both series of tests can be ran at yade +startup by passing the options “test” or “checkall” +yade --test +yade --checkall +The yade --checkall is a complete check. To skip checks lasting more than 30 seconds one can use +this command +yade --check +Unit regression tests +Unit regression tests are testing the output of individual functors and engines in well defined conditions. +They are defined in the folder py/tests/. The purpose of unit testing is to make sure that the behaviour +of the most important classes remains correct during code development. Since they test classes one by +one, unit tests can’t detect problems coming from the interaction between different engines in a typical +simulation. That is why check tests have been introduced. +To add a new test, the following steps must be performed: +1. Place a new file such as py/tests/dummyTest.py. +2. Add the file name such as dummyTest to the py/tests/__init__.py file. +3. If necessary modify the import and allModules lines in py/tests/__init__.py. +4. According to instructions in python unittest documentation use commands such as self. +assertTrue(…), self.assertFalse(…) or self.assertRaises(…,…) to report possible errors. +Note: +It is important that all variables used in the test are stored inside the class (using the self. +accessor), and that all preparations are done inside the function setUp(). +3.1. +Programmer’s manual +541 + +Yade Documentation, Release 3rd ed. +Check tests +Check tests (also see README) perform comparisons of simulation results between different versions of +yade, as discussed here. They differ with regression tests in the sense that they simulate more complex +situations and combinations of different engines, and usually don’t have a mathematical proof (though +there is no restriction on the latest). They compare the values obtained in version N with values obtained +in a previous version or any other “expected” results. The reference values must be hardcoded in the +script itself or in data files provided with the script. Check tests are based on regular yade scripts, so +that users can easily commit their own scripts to trunk in order to get some automatized testing after +commits from other developers. +When +check +fails +the +script +should +return +an +error +message +via +python +command +raise +YadeCheckError(messageString) telling what went wrong. If the script itself fails for some reason +and can’t generate an output, the log will contain only “scriptName failure”. If the script defines differ- +ences on obtained and awaited data, it should print some useful information about the problem. After +this occurs, the automatic test will stop the execution with error message. +An example dummy check test scripts/checks-and-tests/checks/checkTestDummy.py demonstrates a +minimal empty test. A little more functional example check test can be found in scripts/checks-and- +tests/checks/checkTestTriax.py. It shows results comparison, output, and how to define the path to +data files using checksPath. Users are encouraged to add their own scripts into the scripts/checks-and- +tests/checks/ folder. Discussion of some specific checktests design in questions and answers is welcome. +Note that re-compiling is required before the newly added scripts can be launched by yade --check (or +direct changes have to be performed in “lib” subfolders). A check test should never need more than a few +seconds to run. If your typical script needs more, try to reduce the number of elements or the number +of steps. +To add a new check, the following steps must be performed: +1. Place a new file such as scripts/checks-and-tests/checks/checkTestDummy.py, +2. Inside the new script use checksPath when it is necessary to load some data file, like scripts/checks- +and-tests/checks/data/100spheres +3. When error occurs raise exception with command raise YadeCheckError(messageString) +GUI Tests +In order to add a new GUI test one needs to add a file to scripts/checks-and-tests/gui directory. File +must be named according to the following convention: testGuiName.py with an appropriate test Name +in place (the testGui.sh script is searching for files matching this pattern). The scripts/checks-and- +tests/gui/testGuiBilliard.py may serve as a boilerplate example. The important “extra” parts of the +code (taken from e.g. example directory) are: +1. from testGuiHelper import TestGUIHelper +2. scr = TestGUIHelper("Billiard"), make sure to put the chosen test Name in place of Billiard. +3. Establish a reasonable value of guiIterPeriod which makes the test finish in less than 30 seconds. +4. Inside +O.engines +there +has +to +be +a +call +at +the +end +of +the +loop +to +PyRunner(iterPeriod=guiIterPeriod, command='scr.screenshotEngine()'). +5. The last command in the script should be O.run(guiIterPeriod * scr.getTestNum() + 1) to +start the test process. +6. Make sure to push to yade-data repository the reference screenshots (for dealing with ./data dir +see Yade on GitLab). These screenshots can be also obtained from artifacts by clicking “Download” +button in the gitlab pipeline, next to the “Browse” button in the right pane. +These tests can be run locally, after adjusting the paths at the start of testGui.sh script. Two modes of +operation are possible: +542 +Chapter 3. +Yade for programmers + +Yade Documentation, Release 3rd ed. +1. Launch on the local desktop via command: scripts/checks-and-tests/gui/testGui.sh, in this +case the screenshots will be different from those used during the test. +2. Or launch inside a virtual xserver via command: xvfb-run -a -s "-screen 0 1600x1200x24" +scripts/checks-and-tests/gui/testGui.sh, then the screenshots will be similar to those used +in the test, but still there may be some differences in the font size. In such case it is recommended +to use the reference screenshots downloaded from the artifacts in the gitlab pipeline (see point 6. +above). +Care should be taken to not use random colors of bodies used in the test. Also no windows such as 3d +View or Inspector view should be opened in the script testGuiName.py, because they are opened during +the test by the TestGUIHelper class. +Note: +It is not possible to call GUI tests from a call such as yade --test because of the necessity to +launch YADE inside a virtual xserver. +3.1.5 Conventions +The following coding rules should be respected; documentation is treated separately. +• general +– C++ source files have .hpp and .cpp extensions (for headers and implementation, respec- +tively). In rare cases .ipp is used for pure template code. +– All header files should have the #pragma once multiple-inclusion guard. +– Do not type using namespace … in header files, this can lead to obscure bugs due to names- +pace pollution. +– Avoid using std::something in .hpp files. Feel free to use them as much as you like inside +.cpp files. But remember that the usual problems with this practice still apply: wrong type +or function might be used instead of the one that you would expect. But since it’s limited to a +single .cpp file, it will be easier to debug and the convenience might outweight the associated +dangers. +– Use tabs for indentation. +While this is merely visual in C++, it has semantic meaning in +python; inadvertently mixing tabs and spaces can result in syntax errors. +• capitalization style +– Types should be always capitalized. +Use CamelCase for composed class and typenames +(GlobalEngine). Underscores should be used only in special cases, such as functor names. +– Class data members and methods must not be capitalized, composed names should use low- +ercase camelCase (glutSlices). The same applies for functions in python modules. +– Preprocessor macros are uppercase, separated by underscores; those that are used outside the +core take (with exceptions) the form YADE_*, such as YADE_CLASS_BASE_DOC_* macro +family. +• programming style +– Be defensive, if it has no significant performance impact. Use assertions abundantly: they +don’t affect performance (in the optimized build) and make spotting error conditions much +easier. +– Use YADE_CAST and YADE_PTR_CAST where you want type-check during debug builds, but fast +casting in optimized build. +– Initialize all class variables in the default constructor. This avoids bugs that may manifest +randomly and are difficult to fix. Initializing with NaN’s will help you find otherwise unitialized +variable. (This is taken care of by YADE_CLASS_BASE_DOC_* macro family macros for +user classes) +3.1. +Programmer’s manual +543 + +Yade Documentation, Release 3rd ed. +Using clang-format +The file .clang-format contains the config which should produce always the same results. It works with +clang-format --version >= 10. The aim is to eliminate commits that change formatting. The script +scripts/clang-formatter.sh can be invoked on either file or a directory and will do the reformatting. +Usually this can be integrated with the editor, see clang-format documentation (except that for vim +py3f command has to be used), and in kdevelop it is added as a custom formatter. +The script scripts/python-formatter.sh applies our coding conventions to formatting of python scripts. +It should be used before committing changes to python scripts. +For more help see: +1. clang-format documentation +2. yapf3 documentation +Sometimes it is useful to disable formatting in a small section of the file. In order to do so, put the +guards around this section: +1. In C++ use: +// clang-format off +…… +// clang-format on +2. In Python use: +# yapf: disable +…… +# yapf: enable +Class naming +Although for historical reasons the naming scheme is not completely consistent, these rules should be +obeyed especially when adding a new class. +GlobalEngines and PartialEngines GlobalEngines should be named in a way suggesting that it is +a performer of certain action (like ForceResetter, InsertionSortCollider, Recorder); if this is not +appropriate, append the Engine to the characteristics name (e.g. GravityEngine). PartialEngines +have no special naming convention different from GlobalEngines. +Dispatchers Names of all dispatchers end in Dispatcher. The name is composed of type it creates or, +in case it doesn’t create any objects, its main characteristics. Currently, the following dispatchers2 +are defined: +dispatcher +arity +dispatch +types +created +type +functor type +functor pre- +fix +BoundDis- +patcher +1 +Shape +Bound +BoundFunc- +tor +Bo1 +IGeomDis- +patcher +2 (symetric) +2 × Shape +IGeom +IGeomFunc- +tor +Ig2 +IPhysDis- +patcher +2 (symetric) +2 × Mate- +rial +IPhys +IPhysFunc- +tor +Ip2 +LawDispatcher +2 +(asymet- +ric) +IGeom +IPhys +(none) +LawFunctor +Law2 +Respective abstract functors for each dispatchers are BoundFunctor, IGeomFunctor, IPhys- +Functor and LawFunctor. +2 Not considering OpenGL dispatchers, which might be replaced by regular virtual functions in the future. +544 +Chapter 3. +Yade for programmers + +Yade Documentation, Release 3rd ed. +Functors Functor name is composed of 3 parts, separated by underscore. +1. prefix, composed of abbreviated functor type and arity (see table above) +2. Types entering the dispatcher logic (1 for unary and 2 for binary functors) +3. Return type for functors that create instances, simple characteristics for functors that don’t +create instances. +To give a few examples: +• Bo1_Sphere_Aabb is a BoundFunctor which is called for Sphere, creating an instance of Aabb. +• Ig2_Facet_Sphere_ScGeom is binary functor called for Facet and Sphere, creating and instace +of ScGeom. +• Law2_ScGeom_CpmPhys_Cpm is binary functor (LawFunctor) called for types ScGeom +(Geom) and CpmPhys. +Documentation +Documenting code properly is one of the most important aspects of sustained development. +Read it again. +Most code in research software like Yade is not only used, but also read, by developers or even by regular +users. Therefore, when adding new class, always mention the following in the documentation: +• purpose +• details of the functionality, unless obvious (algorithms, internal logic) +• limitations (by design, by implementation), bugs +• bibliographical reference, if using non-trivial published algorithms (see below) +• references to other related classes +• hyperlinks to bugs, blueprints, wiki or mailing list about this particular feature. +As much as it is meaningful, you should also +• update any other documentation affected +• provide a simple python script demonstrating the new functionality in scripts/test. +Sphinx documentation +Most c++ classes are wrapped in Python, which provides good introspection and interactive documen- +tation (try writing Material? in the ipython prompt; or help(CpmState)). +Syntax of documentation is ReST (reStructuredText, see reStructuredText Primer). It is the same for +c++ and python code. +• Documentation of c++ classes exposed to python is given as 3rd argument to YADE_CLASS_- +BASE_DOC_* macro family introduced below. +• Python classes/functions are documented using regular python docstrings. +Besides explaining +functionality, meaning and types of all arguments should also be documented. Short pieces of code +might be very helpful. See the utils module for an example. +Note: +Use C++ string literal when writing docstrings in C++. By convention the R"""(raw text)""" +is used. For example see here and here. +3.1. +Programmer’s manual +545 + +Yade Documentation, Release 3rd ed. +Note: +Remember that inside C++ docstrings it is possible to invoke python commands which are +executed by yade when documentation is being compiled. For example compare this source docstring +with the final effect. +In addition to standard ReST syntax, yade provides several shorthand macros: +:yref: creates hyperlink to referenced term, for instance: +:yref:`CpmMat` +becomes CpmMat; link name and target can be different: +:yref:`Material used in the CPM model` +yielding Material used in the CPM model. +:ysrc: creates hyperlink to file within the source tree (to its latest version in the repository), for instance +core/Cell.hpp. Just like with :yref:, alternate text can be used with +:ysrc:`Link text` +like this. This cannot be used to link to a specified line number, since changing the file will cause +the line numbers to become outdated. To link to a line number use :ysrccommit: described below. +:ysrccommit: creates hyperlink to file within the source tree at the specified commit hash. This allows +to link to the line numbers using for example #L121 at the end of the link. Use it just like the +:ysrc: except that commit hash must be provided at the beginning: +:ysrccommit:`Link text` +:ysrccommit:`default engines<775ae7436/py/__init__.py.in#L112>` +becomes default engines. +Linking to inheritanceGraph* To link to an inheritance graph of some base class a global an- +chor is created with name inheritanceGraph* added in front of the class name, for example +:ref:`Shape` yields link to inheritance graph of Shape. +|ycomp| is used in attribute description for those that should not be provided by the user, but are +auto-computed instead; |ycomp| expands to (auto-computed). +|yupdate| marks attributes that are periodically updated, being subset of the previous. |yupdate| +expands to (auto-updated). +$...$ delimits inline math expressions; they will be replaced by: +:math:`...` +and rendered via LaTeX. To write a single dollar sign, escape it with backslash \$. +Displayed mathematics (standalone equations) can be inserted as explained in Math support for +HTML outputs in Sphinx. +As a reminder in the standard ReST syntax the references are: +:ref: is the the standard restructured text reference to an anchor placed elsewere in the text. For +instance an anchor .. _NumericalDamping: is placed in formulation.rst then it is linked to with +:ref:`NumericalDamping` inside the source code. +.. _anchor-name: is used to place anchors in the text, to be referenced from elsewhere in the text. +Symbol _ is forbidden in the anchor name, because it has a special meaning: _anchor specifies +anchor, while anchor_ links to it, see below. +546 +Chapter 3. +Yade for programmers + +Yade Documentation, Release 3rd ed. +anchor-name_ is used to place a link to anchor within the same file. It is a shorter form compared to +the one which works between different files: :ref:. For example usage on anchor imgQtGui see +here and here. +Note: +The command :scale: NN % (with percent) does not work well with .html + .pdf output, +better to specify :width: NN cm. Then it is the same size in .html and .pdf.. For example see here +which becomes this picture. But bear in mind that maximum picture width in .pdf is 16.2 cm. +Bibliographical references +As in any scientific documentation, references to publications are very important. To cite an article, first +add it in BibTeX format to files doc/references.bib or doc/yade-*.bib depending whether that reference +used Yade (the latter cases) or not (the former). Please adhere to the following conventions: +1. Keep entries in the form Author2008 (Author is the first author), Author2008b etc if multiple +articles from one author; +2. Try to fill mandatory fields for given type of citation; +3. Do not use \'{i} funny escapes for accents, since they will not work with the HTML output; put +everything in straight utf-8. +In your docstring, the Author2008 article can be then cited by [Author2008]_; for example: +According to [Allen1989]_, the integration scheme … +will be rendered as +According to [Allen1989], the integration scheme … +Separate class/function documentation +Some c++ might have long or content-rich documentation, which is rather inconvenient to type in the +c++ source itself as string literals. Yade provides a way to write documentation separately in py/_- +extraDocs.py file: it is executed after loading c++ plugins and can set __doc__ attribute of any object +directly, overwriting docstring from c++. In such (exceptional) cases: +1. Provide at least a brief description of the class in the c++ code nevertheless, for people only reading +the code. +2. Add notice saying “This class is documented in detail in the py/_extraDocs.py file”. +3. Add documentation to py/_extraDocs.py in this way: +module.YourClass.__doc__=''' +This is the docstring for YourClass. +Class, methods and functions can be documented this way. +.. note:: It can use any syntax features you like. +''' +Note: +Boost::python embeds function signatures in the docstring (before the one provided by the user). +Therefore, before creating separate documentation of your function, have a look at its __doc__ attribute +and copy the first line (and the blank line afterwards) in the separate docstring. The first line is then +used to create the function signature (arguments and return value). +3.1. +Programmer’s manual +547 + +Yade Documentation, Release 3rd ed. +Internal c++ documentation +doxygen was used for automatic generation of c++ code. Since user-visible classes are defined with +sphinx now, it is not meaningful to use doxygen to generate overall documentation. +However, take +care to document well internal parts of code using regular comments, including public and private data +members. +3.1.6 Support framework +Besides the framework provided by the c++ standard library (including STL), boost and other depen- +dencies, Yade provides its own specific services. +Pointers +Shared pointers +Yade makes extensive use of shared pointers shared_ptr.3 Although it probably has some performance +impacts, it greatly simplifies memory management, ownership management of c++ objects in python +and so forth. To obtain raw pointer from a shared_ptr, use its get() method; raw pointers should be +used in case the object will be used only for short time (during a function call, for instance) and not +stored anywhere. +Python defines thin wrappers for most c++ Yade classes (for all those registered with YADE_CLASS_- +BASE_DOC_* macro family and several others), which can be constructed from shared_ptr; in this +way, Python reference counting blends with the shared_ptr reference counting model, preventing crashes +due to python objects pointing to c++ objects that were destructed in the meantime. +Typecasting +Frequently, pointers have to be typecast; there is choice between static and dynamic casting. +• dynamic_cast (dynamic_pointer_cast for a shared_ptr) assures cast admissibility by checking +runtime type of its argument and returns NULL if the cast is invalid; such check obviously costs +time. Invalid cast is easily caught by checking whether the pointer is NULL or not; even if such +check (e.g. assert) is absent, dereferencing NULL pointer is easily spotted from the stacktrace +(debugger output) after crash. Moreover, shared_ptr checks that the pointer is non-NULL before +dereferencing in debug build and aborts with “Assertion ‘px!=0’ failed.” if the check fails. +• static_cast is fast but potentially dangerous (static_pointer_cast for shared_ptr). Static +cast will return non-NULL pointer even if types don’t allow the cast (such as casting from State* +to Material*); the consequence of such cast is interpreting garbage data as instance of the class +cast to, leading very likely to invalid memory access (segmentation fault, “crash” for short). +To have both speed and safety, Yade provides 2 macros: +YADE_CAST expands to static_cast in optimized builds and to dynamic_cast in debug builds. +YADE_PTR_CAST expands to static_pointer_cast in optimized builds and to dynamic_pointer_cast +in debug builds. +Basic numerics +The floating point type to use in Yade is Real, which is by default typedef for double (64 bits, 15 decimal +places).4 +3 Either boost::shared_ptr or tr1::shared_ptr is used, but it is always imported with the using statement so that +unqualified shared_ptr can be used. +4 See high precision documentation for additional details. +548 +Chapter 3. +Yade for programmers + +Yade Documentation, Release 3rd ed. +Yade uses the Eigen library for computations. It provides classes for 2d and 3d vectors, quaternions and +3x3 matrices templated by number type; their specialization for the Real type are typedef’ed with the +“r” suffix, and occasionally useful integer types with the “i” suffix: +• Vector2r, Vector2i +• Vector3r, Vector3i +• Quaternionr +• Matrix3r +Yade +additionally +defines +a +class +named +Se3r, +which +contains +spatial +position +(Vector3r +Se3r::position) and orientation (Quaternionr Se3r::orientation), since they are frequently used +one with another, and it is convenient to pass them as single parameter to functions. +Eigen provides full rich linear algebra functionality. Some code further uses the [cgal] library for com- +putational geometry. +In Python, basic numeric types are wrapped and imported from the yade.minieigenHP module; the +types drop the r type qualifier at the end, the syntax is otherwise similar. Se3r is not wrapped at all, +only converted automatically, rarely as it is needed, from/to a (Vector3,Quaternion) tuple/list. See +high precision section for more details. +# cross product +Yade [14]: Vector3(1,2,3).cross(Vector3(0,0,1)) +Out[14]: Vector3(2,-1,0) +# construct quaternion from axis and angle +Yade [15]: Quaternion(Vector3(0,0,1),pi/2) +Out[15]: Quaternion((0,0,1),1.570796326794896558) +Note: +Quaternions are internally stored as 4 numbers. Their usual human-readable representation +is, however, (normalized) axis and angle of rotation around that axis, and it is also how they are +input/output in Python. Raw internal values can be accessed using the [0] … [3] element access (or +.W(), .X(), .Y() and .Z() methods), in both c++ and Python. +Run-time type identification (RTTI) +Since serialization and dispatchers need extended type and inheritance information, which is not suffi- +ciently provided by standard RTTI. Each yade class is therefore derived from Factorable and it must +use macro to override its virtual functions providing this extended RTTI: +YADE_CLASS_BASE_DOC(Foo,Bar Baz,"Docstring") creates the following virtual methods (mediated +via the REGISTER_CLASS_AND_BASE macro, which is not user-visible and should not be used directly): +• std::string getClassName() +returning +class +name +(Foo) +as +string. +(There +is +the +typeid(instanceOrType).name() standard c++ construct, but the name returned is compiler- +dependent.) +• unsigned getBaseClassNumber() returning number of base classes (in this case, 2). +• std::string getBaseClassName(unsigned i=0) returning name of i-th base class (here, Bar for +i=0 and Baz for i=1). +Warning: RTTI relies on virtual functions; in order for virtual functions to work, at least one virtual +method must be present in the implementation (.cpp) file. Otherwise, virtual method table (vtable) +will not be generated for this class by the compiler, preventing virtual methods from functioning +properly. +3.1. +Programmer’s manual +549 + +Yade Documentation, Release 3rd ed. +Some RTTI information can be accessed from python: +Yade [16]: yade.system.childClasses('Shape') +Out[16]: +{'Box', +'ChainedCylinder', +'Clump', +'Cylinder', +'Facet', +'GridConnection', +'GridNode', +'PFacet', +'Sphere', +'Tetra', +'Wall'} +Yade [17]: Sphere().__class__.__name__ +## getClassName() +Out[17]: 'Sphere' +Serialization +Serialization serves to save simulation to file and restore it later. This process has several necessary +conditions: +• classes know which attributes (data members) they have and what are their names (as strings); +• creating class instances based solely on its name; +• knowing what classes are defined inside a particular shared library (plugin). +This functionality is provided by 3 macros and 4 optional methods; details are provided below. +Serializable::preLoad, Serializable::preSave, Serializable::postLoad, Serializable::postSave +Prepare attributes before serialization (saving) or deserialization (loading) or process them after +serialization or deserialization. +See Attribute registration. +YADE_CLASS_BASE_DOC_* Inside the class declaration (i.e. in the .hpp file within the class Foo { /* +… */}; block). See Attribute registration. +Enumerate class attributes that should be saved and loaded; associate each attribute with its literal +name, which can be used to retrieve it. See YADE_CLASS_BASE_DOC_* macro family. +Additionally documents the class in python, adds methods for attribute access from python, and +documents each attribute. +REGISTER_SERIALIZABLE In header file, but after the class declaration block. See Class factory. +Associate literal name of the class with functions that will create its new instance (ClassFactory). +Must be declared inside namespace yade. +YADE_PLUGIN In the implementation .cpp file. See Plugin registration. +Declare what classes are declared inside a particular plugin at time the plugin is being loaded (yade +startup). +Must be declared inside namespace yade. +Attribute registration +All (serializable) types in Yade are one of the following: +550 +Chapter 3. +Yade for programmers + +Yade Documentation, Release 3rd ed. +• Type deriving from Serializable, which provide information on how to serialize themselves via over- +riding the Serializable::registerAttributes method; it declares data members that should +be serialzed along with their literal names, by which they are identified. This method then invokes +registerAttributes of its base class (until Serializable itself is reached); in this way, derived +classes properly serialize data of their base classes. +This funcionality is hidden behind the macro YADE_CLASS_BASE_DOC_* macro family used +in class declaration body (header file), which takes base class and list of attributes: +YADE_CLASS_BASE_DOC_ATTRS(ThisClass,BaseClass,"class documentation",((type1,attribute1, +�→initValue1,,"Documentation for attribute 1"))((type2,attribute2,initValue2,, +�→"Documentation for attribute 2"))); +Note that attributes are encoded in double parentheses, not separated by commas. +Empty +attribute list can be given simply by YADE_CLASS_BASE_DOC_ATTRS(ThisClass,BaseClass, +"documentation",) (the last comma is mandatory), or by omiting ATTRS from macro name and +last parameter altogether. +• Fundamental type: strings, various number types, booleans, Vector3r and others. Their “handlers” +(serializers and deserializers) are defined in lib/serialization. +• Standard container of any serializable objects. +• Shared pointer to serializable object. +Yade uses the excellent boost::serialization library internally for serialization of data. +Note: +YADE_CLASS_BASE_DOC_ATTRS also generates code for attribute access from python; this will +be discussed later. Since this macro serves both purposes, the consequence is that attributes that are +serialized can always be accessed from python. +Yade +also +provides +callback +for +before/after +(de) +serialization, +virtual +functions +Serial- +izable::preProcessAttributes +and +Serializable::postProcessAttributes, +which +receive +one +bool +deserializing argument (true when deserializing, false when serializing). +Their default im- +plementation in Serializable doesn’t do anything, but their typical use is: +• converting some non-serializable internal data structure of the class (such as multi-dimensional +array, hash table, array of pointers) into a serializable one (pre-processing) and fill this non- +serializable structure back after deserialization (post-processing); for instance, InteractionCon- +tainer uses these hooks to ask its concrete implementation to store its contents to a unified storage +(vector >) before serialization and to restore from it after deserial- +ization. +• precomputing non-serialized attributes from the serialized values; e.g. Facet computes its (local) +edge normals and edge lengths from vertices’ coordinates. +Class factory +Each serializable class must use REGISTER_SERIALIZABLE, which defines function to create that class by +ClassFactory. ClassFactory is able to instantiate a class given its name (as string), which is necessary +for deserialization. +Although mostly used internally by the serialization framework, programmer can ask for a class instanti- +ation using shared_ptr f=ClassFactory::instance().createShared("ClassName");, +casting the returned shared_ptr to desired type afterwards. Serializable itself derives +from Factorable, i.e. all serializable types are also factorable. +Note: +Both macros REGISTER_SERIALIZABLE and YADE_PLUGIN have to be declared inside yade names- +pace. +3.1. +Programmer’s manual +551 + +Yade Documentation, Release 3rd ed. +Plugin registration +Yade loads dynamic libraries containing all its functionality at startup. ClassFactory must be taught +about classes each particular file provides. YADE_PLUGIN serves this purpose and, contrary to YADE_- +CLASS_BASE_DOC_* macro family, must be placed in the implementation (.cpp) file, inside yade +namespace. It simply enumerates classes that are provided by this file: +YADE_PLUGIN((ClassFoo)(ClassBar)); +Note: +You must use parentheses around the class name even if there is only one class (preprocessor +limitation): YADE_PLUGIN((classFoo));. If there is no class in this file, do not use this macro at all. +Internally, +this macro creates function registerThisPluginClasses_ declared specially as __- +attribute__((constructor)) (see GCC Function Attributes); this attributes makes the function being +executed when the plugin is loaded via dlopen from ClassFactory::load(...). It registers all fac- +torable classes from that file in the Class factory. +Note: +Classes that do not derive from Factorable, such as Shop or SpherePack, are not declared with +YADE_PLUGIN. +This is an example of a serializable class header: +namespace yade { +/*! Homogeneous gravity field; applies gravity×mass force on all bodies. */ +class GravityEngine: public GlobalEngine{ +public: +virtual void action(); +// registering class and its base for the RTTI system +YADE_CLASS_BASE_DOC_ATTRS(GravityEngine,GlobalEngine, +// documentation visible from python and generated reference documentation +"Homogeneous gravity field; applies gravity×mass force on all bodies.", +// enumerating attributes here, include documentation +((Vector3r,gravity,Vector3r::ZERO,"acceleration, zero by default [kgms￿2]")) +); +}; +// registration function for ClassFactory +REGISTER_SERIALIZABLE(GravityEngine); +} // namespace yade +and this is the implementation: +#include +#include +namespace yade { +// registering the plugin +YADE_PLUGIN((GravityEngine)); +void GravityEngine::action(){ +/* do the work here */ +} +} // namespace yade +We can create a mini-simulation (with only one GravityEngine): +552 +Chapter 3. +Yade for programmers + +Yade Documentation, Release 3rd ed. +Yade [18]: O.engines=[GravityEngine(gravity=Vector3(0,0,-9.81))] +Yade [19]: O.save('abc.xml') +and the XML save looks like this: + + + + + + +
1.00000000000000002e-08
+0 +0 +-1 + +0.00000000000000000e+00 +0 +0.00000000000000000e+00 +0 +0 +0 +1 +-1 + +5 +0 +author=bchareyre~(bchareyre@HP-ZBook-15-G3) +isoTime=20220726T141500 +id=20220726T141500p61547 +d.id=20220726T141500p61547 +id.d=20220726T141500p61547 + + +1 +1 + + + + + + +0 +-1 + + + + + +0.00000000000000000e+00 +0.00000000000000000e+00 +-9.81000000000000050e+00 + +0 +1 +(continues on next page) +3.1. +Programmer’s manual +553 + +Yade Documentation, Release 3rd ed. +(continued from previous page) + + + +<_nextEngines> +0 +1 + + + + + +0 +1 + + +0 +0 + + +0 +0 + + +0 +0 + +0 +1 + + + + + + +0 +1 + +0 +1 + + + + + + +0 + + +0 +0 + + +0 + + + + +0 +1 + + +(continues on next page) +554 +Chapter 3. +Yade for programmers + +Yade Documentation, Release 3rd ed. +(continued from previous page) + + + + + + +1.00000000000000000e+00 +0.00000000000000000e+00 +0.00000000000000000e+00 +0.00000000000000000e+00 +1.00000000000000000e+00 +0.00000000000000000e+00 +0.00000000000000000e+00 +0.00000000000000000e+00 +1.00000000000000000e+00 + + +1.00000000000000000e+00 +0.00000000000000000e+00 +0.00000000000000000e+00 +0.00000000000000000e+00 +1.00000000000000000e+00 +0.00000000000000000e+00 +0.00000000000000000e+00 +0.00000000000000000e+00 +1.00000000000000000e+00 + + +1.00000000000000000e+00 +0.00000000000000000e+00 +0.00000000000000000e+00 +0.00000000000000000e+00 +1.00000000000000000e+00 +0.00000000000000000e+00 +0.00000000000000000e+00 +0.00000000000000000e+00 +1.00000000000000000e+00 + + +1.00000000000000000e+00 +0.00000000000000000e+00 +0.00000000000000000e+00 +0.00000000000000000e+00 +1.00000000000000000e+00 +0.00000000000000000e+00 +0.00000000000000000e+00 +0.00000000000000000e+00 +1.00000000000000000e+00 + + +0.00000000000000000e+00 +0.00000000000000000e+00 +0.00000000000000000e+00 +0.00000000000000000e+00 +0.00000000000000000e+00 +0.00000000000000000e+00 +0.00000000000000000e+00 +0.00000000000000000e+00 +0.00000000000000000e+00 + + +(continues on next page) +3.1. +Programmer’s manual +555 + +Yade Documentation, Release 3rd ed. +(continued from previous page) +0.00000000000000000e+00 +0.00000000000000000e+00 +0.00000000000000000e+00 +0.00000000000000000e+00 +0.00000000000000000e+00 +0.00000000000000000e+00 +0.00000000000000000e+00 +0.00000000000000000e+00 +0.00000000000000000e+00 + + +0.00000000000000000e+00 +0.00000000000000000e+00 +0.00000000000000000e+00 +0.00000000000000000e+00 +0.00000000000000000e+00 +0.00000000000000000e+00 +0.00000000000000000e+00 +0.00000000000000000e+00 +0.00000000000000000e+00 + +2 +0 + + + +0 +1 + + +0 +1 + + + + +Warning: Since XML files closely reflect implementation details of Yade, they will not be compatible +between different versions. Use them only for short-term saving of scenes. Python is the high-level +description Yade uses. +Python attribute access +The macro YADE_CLASS_BASE_DOC_* macro family introduced above is (behind the scenes) also +used to create functions for accessing attributes from Python. As already noted, set of serialized at- +tributes and set of attributes accessible from Python are identical. +Besides attribute access, these +wrapper classes imitate also some functionality of regular python dictionaries: +Yade [20]: s=Sphere() +Yade [21]: s.radius +## read-access +Out[21]: nan +Yade [22]: s.radius=4. +## write access +Yade [23]: s.dict().keys() +## show all available keys +Out[23]: dict_keys(['radius', 'color', 'wire', 'highlight']) +(continues on next page) +556 +Chapter 3. +Yade for programmers + +Yade Documentation, Release 3rd ed. +(continued from previous page) +Yade [24]: for k in s.dict().keys(): print(s.dict()[k]) +## iterate over keys, print their␣ +�→values +....: +4.0 +Vector3(1,1,1) +False +False +Yade [25]: s.dict()['radius'] +## same as: 'radius' in s.keys() +Out[25]: 4.0 +Yade [26]: s.dict() +## show dictionary of both attributes and values +Out[26]: {'radius': 4.0, 'color': Vector3(1,1,1), 'wire': False, 'highlight': False} +YADE_CLASS_BASE_DOC_* macro family +There is several macros that hide behind them the functionality of Sphinx documentation, Run-time type +identification (RTTI), Attribute registration, Python attribute access, plus automatic attribute initializa- +tion and documentation. They are all defined as shorthands for base macro YADE_CLASS_BASE_DOC_- +ATTRS_INIT_CTOR_PY with some arguments left out. They must be placed in class declaration’s body +(.hpp file): +#define YADE_CLASS_BASE_DOC(klass,base,doc) \ +YADE_CLASS_BASE_DOC_ATTRS(klass,base,doc,) +#define YADE_CLASS_BASE_DOC_ATTRS(klass,base,doc,attrs) \ +YADE_CLASS_BASE_DOC_ATTRS_CTOR(klass,base,doc,attrs,) +#define YADE_CLASS_BASE_DOC_ATTRS_CTOR(klass,base,doc,attrs,ctor) \ +YADE_CLASS_BASE_DOC_ATTRS_CTOR_PY(klass,base,doc,attrs,ctor,) +#define YADE_CLASS_BASE_DOC_ATTRS_CTOR_PY(klass,base,doc,attrs,ctor,py) \ +YADE_CLASS_BASE_DOC_ATTRS_INIT_CTOR_PY(klass,base,doc,attrs,,ctor,py) +#define YADE_CLASS_BASE_DOC_ATTRS_INIT_CTOR_PY(klass,base,doc,attrs,init,ctor,py) \ +YADE_CLASS_BASE_DOC_ATTRS_INIT_CTOR_PY(klass,base,doc,attrs,inits,ctor,py) +Expected parameters are indicated by macro name components separated with underscores. Their mean- +ing is as follows: +klass (unquoted) name of this class (used for RTTI and python) +base (unquoted) name of the base class (used for RTTI and python) +doc docstring of this class, written in the ReST syntax. This docstring will appear in generated docu- +mentation (such as CpmMat). It can be as long as necessary, use string literal to avoid sequences +interpreted by c++ compiler (so that some backslashes don’t have to be doubled, like in σ = εE) +instead of writing this: +":math:`\\sigma=\\epsilon E" +Write following: R"""(:math:`\sigma=\epsilon E`)""". When the R"""(raw text)""" is used +the escaped characters \n and \t do not have to be written. Newlines and tabs can be used instead. +For example see here and here. Hyperlink the documentation abundantly with yref (all references +to other classes should be hyperlinks). See previous section about syntax on using references and +anchors. +attrs Attribute must be written in the form of parethesized list: +((type1,attr1,initValue1,attrFlags,"Attribute 1 documentation")) +((type2,attr2,,,"Attribute 2 documentation")) +// initValue and attrFlags unspecified +This will expand to +3.1. +Programmer’s manual +557 + +Yade Documentation, Release 3rd ed. +1. data members declaration in c++ (note that all attributes are public): +public: type1 attr1; +type2 attr2; +2. Initializers of the default (argument-less) constructor, for attributes that have non-empty +initValue: +Klass(): attr1(initValue1), attr2() { /* constructor body */ } +No initial value will be assigned for attribute of which initial value is left empty (as +is for attr2 in the above example). Note that you still have to write the commas. +3. Registration of the attribute in the serialization system (unless disabled by attrFlags – see +below) +4. Registration of the attribute in python (unless disabled by attrFlags), so that it can be accessed as klass().name1. +The attribute is read-write by default, see attrFlags to change that. +This attribute will carry the docstring provided, along with knowledge of the initial value. +You can add text description to the default value using the comma operator of c++ and +casting the char* to (void): +((Real,dmgTau,((void)"deactivated if negative",-1),,"Characteristic time for␣ +�→normal viscosity. [s]")) +leading to CpmMat::dmgTau. +The attribute is registered via boost::python::add_property specifying return_by_- +value policy rather than return_internal_reference, which is the default when using +def_readwrite. The reason is that we need to honor custom converters for those values; +see note in Custom converters for details. +Attribute flags +By default, an attribute will be serialized and will be read-write from python. There is a +number of flags that can be passed as the 4th argument (empty by default) to change that: +• Attr::noSave avoids serialization of the attribute (while still keeping its accessibility +from Python) +• Attr::readonly makes the attribute read-only from Python +• Attr::triggerPostLoad will trigger call to postLoad function to handle attribute +change after its value is set from Python; this is to ensure consistency of other pre- +computed data which depend on this value (such as Cell.trsf and such) +• Attr::hidden will not expose the attribute to Python at all +• Attr::noResize will not permit changing size of the array from Python [not yet used] +Flags can be combined as usual using bitwise disjunction | (such as Attr::noSave | +Attr::readonly), though in such case the value should be parenthesized to avoid a warning +with some compilers (g++ specifically), i.e. (Attr::noSave | Attr::readonly). +Currently, the flags logic handled at runtime; that means that even for attributes with +Attr::noSave, their serialization template must be defined (although it will never be used). +In the future, the implementation might be template-based, avoiding this necessity. +deprec List of deprecated attribute names. The syntax is +((oldName1,newName1,"Explanation why renamed etc.")) +((oldName2,newName2,"! Explanation why removed and what to do instead.")) +558 +Chapter 3. +Yade for programmers + +Yade Documentation, Release 3rd ed. +This will make accessing oldName1 attribute from Python return value of newName, but displaying +warning message about the attribute name change, displaying provided explanation. This happens +whether the access is read or write. +If the explanation’s first character is ! (bang), the message will be displayed upon attribute access, +but exception will be thrown immediately. Use this in cases where attribute is no longer meaningful +or was not straightforwardsly replaced by another, but more complex adaptation of user’s script is +needed. You still have to give newName2, although its value will never be used – you can use any +variable you like, but something must be given for syntax reasons). +Warning: Due to compiler limitations, this feature only works if Yade is compiled with gcc >= +4.4. In the contrary case, deprecated attribute functionality is disabled, even if such attributes +are declared. +init Parethesized list of the form: +((attr3,value3)) ((attr4,value4)) +which will be expanded to initializers in the default ctor: +Klass(): /* attributes declared with the attrs argument */ attr4(value4), attr5(value5) {␣ +�→/* constructor body */ } +The purpose of this argument is to make it possible to initialize constants and references (which +are not declared as attributes using this macro themselves, but separately), as that cannot be done +in constructor body. This argument is rarely used, though. +ctor will be put directly into the generated constructor’s body. Mostly used for calling createIndex(); +in the constructor. +Note: +The code must not contain commas outside parentheses (since preprocessor uses commas +to separate macro arguments). If you need complex things at construction time, create a separate +init() function and call it from the constructor instead. +py will be appended directly after generated python code that registers the class and all its attributes. +You can use it to access class methods from python, for instance, to override an existing attribute +with the same name etc: +.def_readonly("omega",&CpmPhys::omega,"Damage internal variable") +.def_readonly("Fn",&CpmPhys::Fn,"Magnitude of normal force.") +def_readonly will not work for custom types (such as std::vector), as it bypasses conversion +registry; see Custom converters for details. +Exposing function-attributes to GUI +Usually to expose a more complex data a getter and setter functions are used, for example Body::mask. +They are accessible from python. To make them visible in GUI without a corresponding variable at all a +function virtual ::boost::python::dict pyDictCustom() const { …… }; must be overridden. For +example see Interaction.hpp where a special attribute isReal is exposed to GUI. To mark such attribute +as readonly an extra information has to be added to its docstring: :yattrflags:`2`. Normally it is +put there by the class attribute registration macros. But since it is not a variable, such attribute has to +be added manually. +3.1. +Programmer’s manual +559 + +Yade Documentation, Release 3rd ed. +Special python constructors +The Python wrapper automatically creates constructor that takes keyword (named) arguments corre- +sponding to instance attributes; those attributes are set to values provided in the constructor. In some +cases, more flexibility is desired (such as InteractionLoop, which takes 3 lists of functors). For such cases, +you can override the function Serializable::pyHandleCustomCtorArgs, which can arbitrarily modify +the new (already existing) instance. It should modify in-place arguments given to it, as they will be +passed further down to the routine which sets attribute values. In such cases, you should document the +constructor: +.. admonition:: Special constructor +Constructs from lists of … +which then appears in the documentation similar to InteractionLoop. +Enums +It is possible to expose enum and enum class (the enum class is the preferred one because it has +stronger type safety to protect programmer from mistakes) in GUI in a dropdown menu. +This +approach is backward compatible, an assignment of int value in an old python script will work +the same as before. +Additionally it will be possible to assign the string type values to an +enum. +To enable the dropdown menu one must put a macro YADE_ENUM( Scope , EnumName , +(ValueName1)(ValueName2)(ValueName3)(ValueName4) ) in a .cpp file. Where each macro argument +means: +1. Scope is the full scope name in which the enum resides. +For example the scope of +yade::OpenGLRenderer::BlinkHighlight is yade::OpenGLRenderer. +2. EnumName is the name of the enum to be registered +3. ValueName are all enum values that are to be exposed to python. They have to be updated if the +C++ enum declaration in .hpp file changes. +After it is registered, like for example in OpenGLRenderer.cpp it is available for use. Additionally the +registered enum class type definitions are exposed in yade.EnumClass_* scope, for example one can +check the names and values dictionaries: +Yade [27]: yade.EnumClass_BlinkHighlight.names +Out[27]: +{'NEVER': yade.EnumClass_BlinkHighlight.NEVER, +'NORMAL': yade.EnumClass_BlinkHighlight.NORMAL, +'WEAK': yade.EnumClass_BlinkHighlight.WEAK} +Yade [28]: yade.EnumClass_BlinkHighlight.values +Out[28]: +{0: yade.EnumClass_BlinkHighlight.NEVER, +1: yade.EnumClass_BlinkHighlight.NORMAL, +2: yade.EnumClass_BlinkHighlight.WEAK} +Keep in mind that these are not the variable instances hence trying to assign something to them will +not change the blinkHighlight setting in GUI. To change enum value from python the respective variable +must be assigned to, such as yade.qt.Renderer().blinkHighlight. Trying to assign an incorrect value will +throw an exception. For example: +Yade [29]: r = yade.FlowEngine() # this is only a test of enum, not of FlowEngine +--------------------------------------------------------------------------- +AttributeError +Traceback (most recent call last) +~/yade/lib/x86_64-linux-gnu/yadeflip/py/yade/__init__.py in +----> 1 r = yade.FlowEngine() # this is only a test of enum, not of FlowEngine +(continues on next page) +560 +Chapter 3. +Yade for programmers + +Yade Documentation, Release 3rd ed. +(continued from previous page) +AttributeError: module 'yade' has no attribute 'FlowEngine' +Yade [30]: r.useSolver +--------------------------------------------------------------------------- +NameError +Traceback (most recent call last) +~/yade/lib/x86_64-linux-gnu/yadeflip/py/yade/__init__.py in +----> 1 r.useSolver +NameError: name 'r' is not defined +Yade [31]: r.useSolver = 'GaussSeidel' +--------------------------------------------------------------------------- +NameError +Traceback (most recent call last) +~/yade/lib/x86_64-linux-gnu/yadeflip/py/yade/__init__.py in +----> 1 r.useSolver = 'GaussSeidel' +NameError: name 'r' is not defined +Yade [32]: try: +....: +r.useSolver = 20 +# assigning incorrect value has no effect +....: except: +....: +print("Error, value is still equal to:",r.useSolver) +....: +--------------------------------------------------------------------------- +NameError +Traceback (most recent call last) +~/yade/lib/x86_64-linux-gnu/yadeflip/py/yade/__init__.py in +1 try: +----> 2 +r.useSolver = 20 +# assigning incorrect value has no effect +3 except: +NameError: name 'r' is not defined +During handling of the above exception, another exception occurred: +NameError +Traceback (most recent call last) +~/yade/lib/x86_64-linux-gnu/yadeflip/py/yade/__init__.py in +2 +r.useSolver = 20 +# assigning incorrect value has no effect +3 except: +----> 4 +print("Error, value is still equal to:",r.useSolver) +NameError: name 'r' is not defined +Yade [33]: r.useSolver +--------------------------------------------------------------------------- +NameError +Traceback (most recent call last) +~/yade/lib/x86_64-linux-gnu/yadeflip/py/yade/__init__.py in +----> 1 r.useSolver +NameError: name 'r' is not defined +Alternatively the dropdown menu in GUI can be used for the same effect. +Static attributes +Some classes (such as OpenGL functors) are instantiated automatically; since we want their attributes +to be persistent throughout the session, they are static. To expose class with static attributes, use the +YADE_CLASS_BASE_DOC_STATICATTRS macro. Attribute syntax is the same as for YADE_CLASS_BASE_- +DOC_ATTRS: +3.1. +Programmer’s manual +561 + +Yade Documentation, Release 3rd ed. +class SomeClass: public BaseClass{ +YADE_CLASS_BASE_DOC_STATICATTRS(SomeClass,BaseClass,"Documentation of SomeClass", +((Type1,attr1,default1,"doc for attr1")) +((Type2,attr2,default2,"doc for attr2")) +); +}; +additionally, you have to allocate memory for static data members in the .cpp file (otherwise, error +about undefined symbol will appear when the plugin is loaded): +There is no way to expose class that has both static and non-static attributes using YADE_CLASS_BASE_* +macros. You have to expose non-static attributes normally and wrap static attributes separately in the +py parameter. +Returning attribute by value or by reference +When attribute is passed from c++ to python, it can be passed either as +• value: new python object representing the original c++ object is constructed, but not bound to it; +changing the python object doesn’t modify the c++ object, unless explicitly assigned back to it, +where inverse conversion takes place and the c++ object is replaced. +• reference: only reference to the underlying c++ object is given back to python; modifying python +object will make the c++ object modified automatically. +The way of passing attributes given to YADE_CLASS_BASE_DOC_ATTRS in the attrs parameter is deter- +mined automatically in the following manner: +• Vector3, Vector3i, Vector2, Vector2i, Matrix3 and Quaternion objects are passed by reference. For instance:: +O.bodies[0].state.pos[0]=1.33 +will assign correct value to x component of position, without changing the other ones. +• Yade classes (all that use shared_ptr when declared in python: all classes deriving from Serializable declared with YADE_CLASS_BASE_DOC_*, and some others) are passed as references (technically speaking, they are passed by value of the shared_ptr, but by virtue of its sharedness, they appear as references). For instance:: +O.engines[4].damping=.3 +will change damping parameter on the original engine object, not on its copy. +• All other types are passed by value. This includes, most importantly, sequence types declared in Custom converters, such as std::vector >. For this reason, :: +O.engines[4]=NewtonIntegrator() +will not work as expected; it will replace 5th element of a copy of the sequence, and this change +will not propagate back to c++. +Multiple dispatch +Multiple dispatch is generalization of virtual methods: a Dispatcher decides based on type(s) of its +argument(s) which of its Functors to call. Number of arguments (currently 1 or 2) determines arity of +the dispatcher (and of the functor): unary or binary. For example: +InsertionSortCollider([Bo1_Sphere_Aabb(),Bo1_Facet_Aabb()]) +creates InsertionSortCollider, which internally contains Collider.boundDispatcher, a BoundDispatcher (a +Dispatcher), with 2 functors; they receive Sphere or Facet instances and create Aabb. This code would +look like this in c++: +shared_ptr collider=(new InsertionSortCollider); +collider->boundDispatcher->add(new Bo1_Sphere_Aabb()); +collider->boundDispatcher->add(new Bo1_Facet_Aabb()); +There are currenly 4 predefined dispatchers (see dispatcher-names) and corresponding functor types. +They are inherited from template instantiations of Dispatcher1D or Dispatcher2D (for functors, +562 +Chapter 3. +Yade for programmers + +Yade Documentation, Release 3rd ed. +Functor1D or Functor2D). These templates themselves derive from DynlibDispatcher (for dispatch- +ers) and FunctorWrapper (for functors). +Example: IGeomDispatcher +Let’s take (the most complicated perhaps) IGeomDispatcher. IGeomFunctor, which is dispatched based +on types of 2 Shape instances (a Functor), takes a number of arguments and returns bool. The functor +“call” is always provided by its overridden Functor::go method; it always receives the dispatched +instances as first argument(s) (2 × const shared_ptr&) and a number of other arguments it +needs: +class IGeomFunctor: public Functor2D< +bool, +//return type +TYPELIST_7(const shared_ptr&, +// 1st class for dispatch +const shared_ptr&, +// 2nd class for dispatch +const State&, +// other arguments passed to ::go +const State&, +// … +const Vector3r&, +// … +const bool&, +// … +const shared_ptr& +// … +) +> +The dispatcher is declared as follows: +class IGeomDispatcher: public Dispatcher2D< +Shape, +// 1st class for dispatch +Shape, +// 2nd class for dispatch +IGeomFunctor, +// functor type +bool, +// return type of the functor +// follow argument types for functor call +// they must be exactly the same as types +// given to the IGeomFunctor above. +TYPELIST_7(const shared_ptr&, +const shared_ptr&, +const State&, +const State&, +const Vector3r&, +const bool &, +const shared_ptr& +), +// handle symetry automatically +// (if the dispatcher receives Sphere+Facet, +// the dispatcher might call functor for Facet+Sphere, +// reversing the arguments) +false +> +{ /* … */ } +Functor derived from IGeomFunctor must then +• override the ::go method with appropriate arguments (they must match exactly types given to +TYPELIST_* macro); +• declare what types they should be dispatched for, and in what order if they are not the same. +class Ig2_Facet_Sphere_ScGeom: public IGeomFunctor{ +public: +(continues on next page) +3.1. +Programmer’s manual +563 + +Yade Documentation, Release 3rd ed. +(continued from previous page) +// override the IGeomFunctor::go +// +(it is really inherited from FunctorWrapper template, +// +therefore not declare explicitly in the +// +IGeomFunctor declaration as such) +// since dispatcher dispatches only for declared types +// +(or types derived from them), we can do +// +static_cast(shape1) and static_cast(shape2) +// +in the ::go body, without worrying about types being wrong. +virtual bool go( +// objects for dispatch +const shared_ptr& shape1, const shared_ptr& shape2, +// other arguments +const State& state1, const State& state2, const Vector3r& shift2, +const bool& force, const shared_ptr& c +); +/* … */ +// this declares the type we want to be dispatched for, matching +// +first 2 arguments to ::go and first 2 classes in TYPELIST_7 above +// +shape1 is a Facet and shape2 is a Sphere +// +(or vice versa, see lines below) +FUNCTOR2D(Facet,Sphere); +// declare how to swap the arguments +// +so that we can receive those as well +DEFINE_FUNCTOR_ORDER_2D(Facet,Sphere); +/* … */ +}; +Dispatch resolution +The dispatcher doesn’t always have functors that exactly match the actual types it receives. In the same +way as virtual methods, it tries to find the closest match in such way that: +1. the actual instances are derived types of those the functor accepts, or exactly the accepted types; +2. sum of distances from actual to accepted types is sharp-minimized (each step up in the class +hierarchy counts as 1) +If no functor is able to accept given types (first condition violated) or multiple functors have the same +distance (in condition 2), an exception is thrown. +This resolution mechanism makes it possible, for instance, to have a hierarchy of ScGeom classes (for +different combination of shapes), but only provide a LawFunctor accepting ScGeom, rather than having +different laws for each shape combination. +Note: +Performance implications of dispatch resolution are relatively low. The dispatcher lookup is only +done once, and uses fast lookup matrix (1D or 2D); then, the functor found for this type(s) is cached +within the Interaction (or Body) instance. Thus, regular functor call costs the same as dereferencing +pointer and calling virtual method. There is blueprint to avoid virtual function call as well. +Note: +At the beginning, the dispatch matrix contains just entries exactly matching given functors. +Only when necessary (by passing other types), appropriate entries are filled in as well. +564 +Chapter 3. +Yade for programmers + +Yade Documentation, Release 3rd ed. +Indexing dispatch types +Classes entering the dispatch mechanism must provide for fast identification of themselves and of their +parent class.5 +This is called class indexing and all such classes derive from Indexable. +There are +top-level Indexables (types that the dispatchers accept) and each derived class registers its index +related to this top-level Indexable. Currently, there are: +Top-level Indexable +used by +Shape +BoundFunctor, IGeomDispatcher +Material +IPhysDispatcher +IPhys +LawDispatcher +IGeom +LawDispatcher +The top-level Indexable must use the REGISTER_INDEX_COUNTER macro, which sets up the machinery +for identifying types of derived classes; they must then use the REGISTER_CLASS_INDEX macro and call +createIndex() in their constructor. For instance, taking the Shape class (which is a top-level Indexable): +// derive from Indexable +class Shape: public Serializable, public Indexable { +// never call createIndex() in the top-level Indexable ctor! +/* … */ +// allow index registration for classes deriving from ``Shape`` +REGISTER_INDEX_COUNTER(Shape); +}; +Now, all derived classes (such as Sphere or Facet) use this: +class Sphere: public Shape{ +/* … */ +YADE_CLASS_BASE_DOC_ATTRS_CTOR(Sphere,Shape,"docstring", +((Type1,attr1,default1,"docstring1")) +/* … */, +// this is the CTOR argument +// important; assigns index to the class at runtime +createIndex(); +); +// register index for this class, and give name of the immediate parent class +// +(i.e. if there were a class deriving from Sphere, it would use +// +REGISTER_CLASS_INDEX(SpecialSphere,Sphere), +// +not REGISTER_CLASS_INDEX(SpecialSphere,Shape)!) +REGISTER_CLASS_INDEX(Sphere,Shape); +}; +At runtime, each class within the top-level Indexable hierarchy has its own unique numerical index. +These indices serve to build the dispatch matrix for each dispatcher. +Inspecting dispatch in python +If there is a need to debug/study multiple dispatch, python provides convenient interface for this low-level +functionality. +We can inspect indices with the dispIndex property (note that the top-level indexable Shape has negative +(invalid) class index; we purposively didn’t call createIndex in its constructor): +5 The functionality described in Run-time type identification (RTTI) serves a different purpose (serialization) and would +hurt the performance here. For this reason, classes provide numbers (indices) in addition to strings. +3.1. +Programmer’s manual +565 + +Yade Documentation, Release 3rd ed. +Yade [34]: Sphere().dispIndex, Facet().dispIndex, Wall().dispIndex +Out[34]: (1, 5, 10) +Yade [35]: Shape().dispIndex +# top-level indexable +Out[35]: -1 +Dispatch hierarchy for a particular class can be shown with the dispHierarchy() function, returning +list of class names: 0th element is the instance itself, last element is the top-level indexable (again, with +invalid index); for instance: +Yade [36]: ScGeom().dispHierarchy() +# parent class of all other ScGeom_ classes +Out[36]: ['ScGeom', 'GenericSpheresContact', 'IGeom'] +Yade [37]: ScGridCoGeom().dispHierarchy(), ScGeom6D().dispHierarchy(), CylScGeom(). +�→dispHierarchy() +Out[37]: +(['ScGridCoGeom', 'ScGeom6D', 'ScGeom', 'GenericSpheresContact', 'IGeom'], +['ScGeom6D', 'ScGeom', 'GenericSpheresContact', 'IGeom'], +['CylScGeom', 'ScGeom', 'GenericSpheresContact', 'IGeom']) +Yade [38]: CylScGeom().dispHierarchy(names=False) +# show numeric indices instead +Out[38]: [4, 1, 0, -1] +Dispatchers can also be inspected, using the .dispMatrix() method: +Yade [39]: ig=IGeomDispatcher([ +....: +Ig2_Sphere_Sphere_ScGeom(), +....: +Ig2_Facet_Sphere_ScGeom(), +....: +Ig2_Wall_Sphere_ScGeom() +....: ]) +....: +Yade [40]: ig.dispMatrix() +Out[40]: +{('Sphere', 'Sphere'): 'Ig2_Sphere_Sphere_ScGeom', +('Sphere', 'Facet'): 'Ig2_Facet_Sphere_ScGeom', +('Sphere', 'Wall'): 'Ig2_Wall_Sphere_ScGeom', +('Facet', 'Sphere'): 'Ig2_Facet_Sphere_ScGeom', +('Wall', 'Sphere'): 'Ig2_Wall_Sphere_ScGeom'} +Yade [41]: ig.dispMatrix(False) +# don't convert to class names +Out[41]: +{(1, 1): 'Ig2_Sphere_Sphere_ScGeom', +(1, 5): 'Ig2_Facet_Sphere_ScGeom', +(1, 10): 'Ig2_Wall_Sphere_ScGeom', +(5, 1): 'Ig2_Facet_Sphere_ScGeom', +(10, 1): 'Ig2_Wall_Sphere_ScGeom'} +We can see that functors make use of symmetry (i.e. that Sphere+Wall are dispatched to the same +functor as Wall+Sphere). +Finally, dispatcher can be asked to return functor suitable for given argument(s): +Yade [42]: ld=LawDispatcher([Law2_ScGeom_CpmPhys_Cpm()]) +Yade [43]: ld.dispMatrix() +Out[43]: {('GenericSpheresContact', 'CpmPhys'): 'Law2_ScGeom_CpmPhys_Cpm'} +# see how the entry for ScGridCoGeom will be filled after this request +Yade [44]: ld.dispFunctor(ScGridCoGeom(),CpmPhys()) +Out[44]: +(continues on next page) +566 +Chapter 3. +Yade for programmers + +Yade Documentation, Release 3rd ed. +(continued from previous page) +Yade [45]: ld.dispMatrix() +Out[45]: +{('GenericSpheresContact', 'CpmPhys'): 'Law2_ScGeom_CpmPhys_Cpm', +('ScGridCoGeom', 'CpmPhys'): 'Law2_ScGeom_CpmPhys_Cpm'} +OpenGL functors +OpenGL rendering is being done also by 1D functors (dispatched for the type to be rendered). Since it +is sufficient to have exactly one class for each rendered type, the functors are found automatically. Their +base functor types are GlShapeFunctor, GlBoundFunctor, GlIGeomFunctor and so on. These classes +register the type they render using the RENDERS macro: +namespace yade { // Cannot have #include directive inside. +class Gl1_Sphere: public GlShapeFunctor { +public : +virtual void go(const shared_ptr&, +const shared_ptr&, +bool wire, +const GLViewInfo& +); +RENDERS(Sphere); +YADE_CLASS_BASE_DOC_STATICATTRS(Gl1_Sphere,GlShapeFunctor,"docstring", +((Type1,staticAttr1,informativeDefault,"docstring")) +/* … */ +); +}; +REGISTER_SERIALIZABLE(Gl1_Sphere); +} // namespace yade +You can list available functors of a particular type by querying child classes of the base functor: +Yade [46]: yade.system.childClasses('GlShapeFunctor') +Out[46]: +{'Gl1_Box', +'Gl1_ChainedCylinder', +'Gl1_Cylinder', +'Gl1_Facet', +'Gl1_GridConnection', +'Gl1_PFacet', +'Gl1_Sphere', +'Gl1_Tetra', +'Gl1_Wall'} +Note: +OpenGL functors may disappear in the future, being replaced by virtual functions of each class +that can be rendered. +Parallel execution +Yade was originally not designed with parallel computation in mind, but rather with maximum flexibility +(for good or for bad). Parallel execution was added later; in order to not have to rewrite whole Yade +from scratch, relatively non-instrusive way of parallelizing was used: OpenMP. OpenMP is standartized +shared-memory parallel execution environment, where parallel sections are marked by special #pragma +in the code (which means that they can compile with compiler that doesn’t support OpenMP) and a few +functions to query/manipulate OpenMP runtime if necessary. +There is parallelism at 3 levels: +3.1. +Programmer’s manual +567 + +Yade Documentation, Release 3rd ed. +• Computation, interaction (python, GUI) and rendering threads are separate. +This is done via +regular threads (boost::threads) and is not related to OpenMP. +• ParallelEngine can run multiple engine groups (which are themselves run serially) in parallel; it +rarely finds use in regular simulations, but it could be used for example when coupling with an +independent expensive computation: +ParallelEngine([ +[Engine1(),Engine2()], +# Engine1 will run before Engine2 +[Engine3()] +# Engine3() will run in parallel with the group␣ +�→[Engine1(),Engine2()] +# arbitrary number of groups can be used +]) +Engine2 will be run after Engine1, but in parallel with Engine3. +Warning: +It is your reponsibility to avoid concurrent access to data when using +ParallelEngine. Make sure you understand very well what the engines run in parallel +do. +• Parallelism inside Engines. Some loops over bodies or interactions are parallelized (notably Inter- +actionLoop and NewtonIntegrator, which are treated in detail later (FIXME: link)): +#pragma omp parallel for +for(long id=0; id& b(scene->bodies[id]); +/* … */ +} +Note: +OpenMP requires loops over contiguous range of integers (OpenMP 3 also +accepts containers with random-access iterators). +If you consider running parallelized loop in your engine, always evalue its benefits. +OpenMP has some overhead fo creating threads and distributing workload, which is +proportionally more expensive if the loop body execution is fast. The results are highly +hardware-dependent (CPU caches, RAM controller). +Maximum number of OpenMP threads is determined by the OMP_NUM_THREADS environment variable +and is constant throughout the program run. Yade main program also sets this variable (before loading +OpenMP libraries) if you use the -j/--threads option. It can be queried at runtime with the omp_- +get_max_threads function. +At places which are susceptible of being accessed concurrently from multiple threads, Yade provides some +mutual exclusion mechanisms, discussed elsewhere (FIXME): +• simultaneously writeable container for ForceContainer, +• mutex for Body::state. +Timing +Yade provides 2 services for measuring time spent in different parts of the code. One has the granularity +of engine and can be enabled at runtime. The other one is finer, but requires adjusting and recompiling +the code being measured. +568 +Chapter 3. +Yade for programmers + +Yade Documentation, Release 3rd ed. +Per-engine timing +The coarser timing works by merely accumulating number of invocations and time (with the precision +of the clock_gettime function) spent in each engine, which can be then post-processed by associated +Python module yade.timing. There is a static bool variable controlling whether such measurements +take place (disabled by default), which you can change +TimingInfo::enabled=True; +// in c++ +O.timingEnabled=True +## in python +After running the simulation, yade.timing.stats() function will show table with the results and per- +centages: +Yade [47]: TriaxialTest(numberOfGrains=100).load() +Yade [48]: O.engines[0].label='firstEngine' +## labeled engines will show by labels in the␣ +�→stats table +Yade [49]: import yade.timing; +Yade [50]: O.timingEnabled=True +Yade [51]: yade.timing.reset() +## not necessary if used for the first time +Yade [52]: O.run(50); O.wait() +Yade [53]: yade.timing.stats() +Name +Count +Time +␣ +�→Rel. time +----------------------------------------------------------------------------------------------- +�→-------- +"firstEngine" +50 +15.278us +0. +�→26% +InsertionSortCollider +25 +1866.469us +32. +�→16% +InteractionLoop +50 +2849.897us +49. +�→11% +GlobalStiffnessTimeStepper +2 +15.417us +0. +�→27% +TriaxialCompressionEngine +50 +262.545us +4. +�→52% +TriaxialStateRecorder +3 +145.883us +2. +�→51% +NewtonIntegrator +50 +647.859us +11. +�→16% +forces sync +50 +7.259us +␣ +�→1.12% +motion integration +50 +624.789us +␣ +�→96.44% +sync max vel +50 +3.264us +␣ +�→0.50% +terminate +50 +2.337us +␣ +�→0.36% +TOTAL +200 +637.649us +␣ +�→98.42% +TOTAL +5803.348us +100. +�→00% +Exec count and time can be accessed and manipulated through Engine::timingInfo from c++ or +Engine().execCount and Engine().execTime properties in Python. +3.1. +Programmer’s manual +569 + +Yade Documentation, Release 3rd ed. +In-engine and in-functor timing +Timing within engines (and functors) is based on TimingDeltas class which is by default instantiated +in engines and functors as Engine::timingDeltas and Functor::timingDeltas (Engine.timingDeltas and +Functor.timingDeltas in Python). +It is made for timing loops (functors’ loop is in their respective +dispatcher) and stores cummulatively time differences between checkpoints. +Note: Fine timing with TimingDeltas will only work if timing is enabled globally (see previous section). +The code would still run, but giving zero times and exec counts. +1. Preferably define the timingDeltas attributes in the constructor: +// header file +class Law2_ScGeom_CpmPhys_Cpm: public LawFunctor { +/* … */ +YADE_CLASS_BASE_DOC_ATTRS_CTOR(Law2_ScGeom_CpmPhys_Cpm,LawFunctor,"docstring", +/* attrs */, +/* constructor */ +timingDeltas=shared_ptr(new TimingDeltas); // timingDeltas␣ +�→object is automatically initialized when using -DENABLE_PROFILING=1 cmake␣ +�→option +); +// ... +}; +2. Inside the loop, start the timing by calling timingDeltas->start(); +3. At places of interest, call timingDeltas->checkpoint("label"). The label is used only for post- +processing, data are stored based on the checkpoint position, not the label. +Warning: +Checkpoints must be always reached in the same order, otherwise the +timing data will be garbage. Your code can still branch, but you have to put check- +points to places which are in common. +void Law2_ScGeom_CpmPhys_Cpm::go(shared_ptr& _geom, +shared_ptr& _phys, +Interaction* I, +Scene* scene) +{ +timingDeltas->start(); +// the point at which the first␣ +�→timing starts +// prepare some variables etc here +timingDeltas->checkpoint("setup"); +// find geometrical data (deformations) here +timingDeltas->checkpoint("geom"); +// compute forces here +timingDeltas->checkpoint("material"); +// apply forces, cleanup here +timingDeltas->checkpoint("rest"); +} +4. Alternatively, you can compile Yade using -DENABLE_PROFILING=1 cmake option and use predefined macros TIMING_DELTAS_START and TIMING_DELTAS_CHECKPOINT. Without -DENABLE_PROFILING options, those macros are empty and do nothing. +void Law2_ScGeom_CpmPhys_Cpm::go(shared_ptr& _geom, +shared_ptr& _phys, +Interaction* I, +Scene* scene) +(continues on next page) +570 +Chapter 3. +Yade for programmers + +Yade Documentation, Release 3rd ed. +(continued from previous page) +{ +TIMING_DELTAS_START(); +// prepare some variables etc here +TIMING_DELTAS_CHECKPOINT("setup") +// find geometrical data (deformations) here +TIMING_DELTAS_CHECKPOINT("geom") +// compute forces here +TIMING_DELTAS_CHECKPOINT("material") +// apply forces, cleanup here +TIMING_DELTAS_CHECKPOINT("rest") +} +The output might look like this (note that functors are nested inside dispatchers and TimingDeltas +inside their engine/functor): +Name +Count +Time +Rel. time +------------------------------------------------------------------------------------- +ForceReseter +400 +9449µs +0.01% +BoundDispatcher +400 +1171770µs +1.15% +InsertionSortCollider +400 +9433093µs +9.24% +IGeomDispatcher +400 +15177607µs +14.87% +IPhysDispatcher +400 +9518738µs +9.33% +LawDispatcher +400 +64810867µs +63.49% +Law2_ScGeom_CpmPhys_Cpm +setup +4926145 +7649131µs +15.25% +geom +4926145 +23216292µs +46.28% +material +4926145 +8595686µs +17.14% +rest +4926145 +10700007µs +21.33% +TOTAL +50161117µs +100.00% +NewtonIntegrator +400 +1866816µs +1.83% +"strainer" +400 +21589µs +0.02% +"plotDataCollector" +160 +64284µs +0.06% +"damageChecker" +9 +3272µs +0.00% +TOTAL +102077490µs +100.00% +Warning: +Do not use TimingDeltas in parallel sections, results might not be meaningful. +In +particular, avoid timing functors inside InteractionLoop when running with multiple OpenMP threads. +TimingDeltas data are accessible from Python as list of (label,*time*,*count*) tuples, one tuple repre- +senting each checkpoint: +deltas=someEngineOrFunctor.timingDeltas.data() +deltas[0][0] # 0th checkpoint label +deltas[0][1] # 0th checkpoint time in nanoseconds +deltas[0][2] # 0th checkpoint execution count +deltas[1][0] # 1st checkpoint label +# … +deltas.reset() +Timing overhead +The overhead of the coarser, per-engine timing, is very small. +For simulations with at least several +hundreds of elements, they are below the usual time variance (a few percent). +The finer TimingDeltas timing can have major performance impact and should be only used during +debugging and performance-tuning phase. +The parts that are file-timed will take disproportionally +longer time that the rest of engine; in the output presented above, LawDispatcher takes almost ￿ of total +3.1. +Programmer’s manual +571 + +Yade Documentation, Release 3rd ed. +simulation time in average, but the number would be twice of thrice lower typically (note that each +checkpoint was timed almost 5 million times in this particular case). +OpenGL Rendering +Yade provides 3d rendering based on QGLViewer. It is not meant to be full-featured rendering and +post-processing, but rather a way to quickly check that scene is as intended or that simulation behaves +sanely. +Note: +Although 3d rendering runs in a separate thread, it has performance impact on the computa- +tion itself, since interaction container requires mutual exclusion for interaction creation/deletion. The +InteractionContainer::drawloopmutex is either held by the renderer (OpenGLRenderingEngine) or +by the insertion/deletion routine. +Warning: +There are 2 possible causes of crash, which are not prevented because of serious perfor- +mance penalty that would result: +1. access to BodyContainer, in particular deleting bodies from simulation; this is a rare operation, +though. +2. deleting Interaction::phys or Interaction::geom. +Renderable entities (Shape, State, Bound, IGeom, IPhys) have their associated OpenGL functors. An +entity is rendered if +1. Rendering such entities is enabled by appropriate attribute in OpenGLRenderingEngine +2. Functor for that particular entity type is found via the dispatch mechanism. +Gl1_* functors operating on Body’s attributes (Shape, State, Bound) are called with the OpenGL con- +text translated and rotated according to State::pos and State::ori. Interaction functors work in global +coordinates. +3.1.7 Simulation framework +Besides the support framework mentioned in the previous section, some functionality pertaining to +simulation itself is also provided. +There are special containers for storing bodies, interactions and (generalized) forces. Their internal func- +tioning is normally opaque to the programmer, but should be understood as it can influence performance. +Scene +Scene is the object containing the whole simulation. Although multiple scenes can be present in the +memory, only one of them is active. Saving and loading (serializing and deserializing) the Scene object +should make the simulation run from the point where it left off. +Note: +All Engines and functors have interally a Scene* scene pointer which is updated regularly by +engine/functor callers; this ensures that the current scene can be accessed from within user code. +For outside functions (such as those called from python, or static functions in Shop), you can use +Omega::instance().getScene() to retrieve a shared_ptr of the current scene. +572 +Chapter 3. +Yade for programmers + +Yade Documentation, Release 3rd ed. +Body container +Body container is linear storage of bodies. Each body in the simulation has its unique id, under which it +must be found in the BodyContainer. Body that is not yet part of the simulation typically has id equal +to invalid value Body::ID_NONE, and will have its id assigned upon insertion into the container. The +requirements on BodyContainer are +• O(1) access to elements, +• linear-addressability (0…n indexability), +• store shared_ptr, not objects themselves, +• no mutual exclusion for insertion/removal (this must be assured by the called, if desired), +• intelligent allocation of id for new bodies (tracking removed bodies), +• easy iteration over all bodies. +Note: +Currently, there is “abstract” class BodyContainer, from which derive concrete implementations; +the initial idea was the ability to select at runtime which implementation to use (to find one that performs +the best for given simulation). This incurs the penalty of many virtual function calls, and will probably +change in the future. All implementations of BodyContainer were removed in the meantime, except +BodyVector (internally a vector > plus a few methods around), which is the fastest. +Insertion/deletion +Body insertion is typically used in FileGenerator’s: +shared_ptr body(new Body); +// … (body setup) +scene->bodies->insert(body); // assigns the id +Bodies are deleted only rarely: +scene->bodies->erase(id); +Warning: +Since mutual exclusion is not assured, never insert/erase bodies from parallel sections, +unless you explicitly assure there will be no concurrent access. +Iteration +The container can be iterated over using for(const auto& …… : …… ) C++ syntax: +for(const auto& b : *scene->bodies){ +if(!b) continue; +// skip deleted bodies, nullptr-check +/* do something here */ +} +The same loop can be also written by using the type const shared_ptr& explicitly: +for(const shared_ptr& b : *scene->bodies){ +if(!b) continue; +// skip deleted bodies, nullptr-check +/* do something here */ +} +3.1. +Programmer’s manual +573 + +Yade Documentation, Release 3rd ed. +Warning: +The previously used macro FOREACH is now deprecated. +Note a few important things: +1. Always use const shared_ptr& (const reference); that avoids incrementing and decrement- +ing the reference count on each shared_ptr. +2. Take care to skip NULL bodies (if(!b) continue): deleted bodies are deallocated from the +container, but since body id’s must be persistent, their place is simply held by an empty shared_- +ptr() object, which is implicitly convertible to false. +In python, the BodyContainer wrapper also has iteration capabilities; for convenience (which is different +from the c++ iterator), NULL bodies as silently skipped: +Yade [54]: O.bodies.append([Body(),Body(),Body()]) +Out[54]: [0, 1, 2] +Yade [55]: O.bodies.erase(1) +Out[55]: True +Yade [56]: [b.id for b in O.bodies] +Out[56]: [0, 2] +In loops parallelized using OpenMP, the loop must traverse integer interval (rather than using iterators): +const long size=(long)bodies.size(); +// store this value, since it doesn't change during␣ +�→the loop +#pragma omp parallel for +for(long _id=0; _id& b(bodies[_id]); +if(!b) continue; +/* … */ +} +InteractionContainer +Interactions are stored in special container, and each interaction must be uniquely identified by pair of +ids (id1,id2). +• O(1) access to elements, +• linear-addressability (0…n indexability), +• store shared_ptr, not objects themselves, +• mutual exclusion for insertion/removal, +• easy iteration over all interactions, +• addressing symmetry, i.e. interaction(id1,id2)￿interaction(id2,id1) +Note: +As with BodyContainer, there is “abstract” class InteractionContainer, and then its concrete +implementations. +Currently, only InteractionVecMap implementation is used and all the other were +removed. +Therefore, the abstract InteractionContainer class may disappear in the future, to avoid +unnecessary virtual calls. +Further, there is a blueprint for storing interactions inside bodies, as that would give extra advantage of +quickly getting all interactions of one particular body (currently, this necessitates loop over all interac- +tions); in that case, InteractionContainer would disappear. +574 +Chapter 3. +Yade for programmers + +Yade Documentation, Release 3rd ed. +Insert/erase +Creating new interactions and deleting them is delicate topic, since many eleents of simulation must be +synchronized; the exact workflow is described in Handling interactions. You will almost certainly never +need to insert/delete an interaction manually from the container; if you do, consider designing your code +differently. +// both insertion and erase are internally protected by a mutex, +// and can be done from parallel sections safely +scene->interactions->insert(shared_ptr(new Interactions(id1,id2))); +scene->interactions->erase(id1,id2); +Iteration +As with BodyContainer, iteration over interactions should use the for(const auto& …… : …… ) C++ +syntax, also const shared_ptr& can be used instead of auto&: +for(const shared_ptr& i : *scene->interactions){ +if(!i->isReal()) continue; +/* … */ +} +Warning: +The previously used macro FOREACH is now deprecated. +Again, note the usage const reference for i. +The check if(!i->isReal()) filters away interactions +that exist only potentially, i.e. there is only Bound overlap of the two bodies, but not (yet) overlap of +bodies themselves. The i->isReal() function is equivalent to i->geom && i->phys. Details are again +explained in Handling interactions. +In some cases, such as OpenMP-loops requiring integral index (OpenMP >= 3.0 allows parallelization +using random-access iterator as well), you need to iterate over interaction indices instead: +int nIntr=(int)scene->interactions->size(); // hoist container size +#pragma omp parallel for +for(int j=0; j& i=(*scene->interactions)[j]; +if(!i->isReal()) continue; +/* … */ +} +ForceContainer +ForceContainer holds “generalized forces”, i.e. forces, torques, (explicit) dispalcements and rotations for +each body. +During each computation step, there are typically 3 phases pertaining to forces: +1. Resetting forces to zero (usually done by the ForceResetter engine) +2. Incrementing forces from parallel sections (solving interactions – from LawFunctor) +3. Reading absolute force values sequentially for each body: forces applied from different interactions +are summed together to give overall force applied on that body (NewtonIntegrator, but also various +other engine that read forces) +This scenario leads to special design, which allows fast parallel write access: +3.1. +Programmer’s manual +575 + +Yade Documentation, Release 3rd ed. +• each thread has its own storage (zeroed upon request), and only writes to its own storage; this +avoids concurrency issues. Each thread identifies itself by the omp_get_thread_num() function +provided by the OpenMP runtime. +• before reading absolute values, the container must be synchronized, i.e. values from all threads +are summed up and stored separately. This is a relatively slow operation and we provide Force- +Container::syncCount that you might check to find cummulative number of synchronizations and +compare it against number of steps. Ideally, ForceContainer is only synchronized once at each +step. +• the container is resized whenever an element outside the current range is read/written to (the +read returns zero in that case); this avoids the necessity of tracking number of bodies, but also is +potential danger (such as scene->forces.getForce(1000000000), which will probably exhaust +your RAM). Unlike c++, Python does check given id against number of bodies. +// resetting forces (inside ForceResetter) +scene->forces.reset() +// in a parallel section +scene->forces.addForce(id,force); // add force +// container is not synced after we wrote to it, sync before reading +scene->forces.sync(); +const Vector3r& f=scene->forces.getForce(id); +Synchronization is handled automatically if values are read from python: +Yade [57]: O.bodies.append(Body()) +Out[57]: 3 +Yade [58]: O.forces.addF(0,Vector3(1,2,3)) +Yade [59]: O.forces.f(0) +Out[59]: Vector3(1,2,3) +Yade [60]: O.forces.f(100) +--------------------------------------------------------------------------- +IndexError +Traceback (most recent call last) +~/yade/lib/x86_64-linux-gnu/yadeflip/py/yade/__init__.py in +----> 1 O.forces.f(100) +IndexError: Body id out of range. +Handling interactions +Creating and removing interactions is a rather delicate topic and number of components must cooperate +so that the whole behaves as expected. +Terminologically, we distinguish +potential interactions, having neither geometry nor physics. Interaction.isReal can be used to query +the status (Interaction::isReal() in c++). +real interactions, having both geometry and physics. Below, we shall discuss the possibility of inter- +actions that only have geometry but no physics. +During each step in the simulation, the following operations are performed on interactions in a typical +simulation: +1. Collider creates potential interactions based on spatial proximity. +Not all pairs of bodies are +susceptible of entering interaction; the decision is done in Collider::mayCollide: +• clumps may not enter interactions (only their members can) +576 +Chapter 3. +Yade for programmers + +Yade Documentation, Release 3rd ed. +• clump members may not interact if they belong to the same clump +• bitwise AND on both bodies’ masks must be non-zero (i.e. there must be at least one bit set +in common) +2. Collider erases interactions that were requested for being erased (see below). +3. InteractionLoop (via IGeomDispatcher) calls appropriate IGeomFunctor based on Shape combina- +tion of both bodies, if such functor exists. For real interactions, the functor updates associated +IGeom. For potential interactions, the functor returns +false if there is no geometrical overlap, and the interaction will stillremain potential- +only +true if there is geometrical overlap; the functor will have created an IGeom in such case. +Note: +For real interactions, the functor must return true, even if there is no more +spatial overlap between bodies. If you wish to delete an interaction without geometrical +overlap, you have to do this in the LawFunctor. +This behavior is deliberate, since different laws have different requirements, though ide- +ally using relatively small number of generally useful geometry functors. +Note: +If there is no functor suitable to handle given combination of shapes, the inter- +action will be left in potential state, without raising any error. +4. For real interactions (already existing or just created in last step), InteractionLoop (via IPhys- +Dispatcher) calls appropriate IPhysFunctor based on Material combination of both bodies. The +functor must update (or create, if it doesn’t exist yet) associated IPhys instance. It is an error if +no suitable functor is found, and an exception will be thrown. +5. For real interactions, InteractionLoop (via LawDispatcher) calls appropriate LawFunctor based on +combination of IGeom and IPhys of the interaction. Again, it is an error if no functor capable of +handling it is found. +6. LawDispatcher takes care of erasing those interactions that are no longer active (such as if bodies +get too far apart for non-cohesive laws; or in case of complete damage for damage models). This +is triggered by the LawFunctor returning false. For this reason it is of upmost importance for the +LawFunctor to return consistently. +Such interaction will not be deleted immediately, but will be reset to potential state. +At the next +execution of the collider InteractionContainer::conditionalyEraseNonReal will be called, which +will completely erase interactions only if the bounding boxes ceased to overlap; the rest will be kept in +potential state. +Creating interactions explicitly +Interactions may still be created explicitly with utils.createInteraction, without any spatial requirements. +This function searches current engines for dispatchers and uses them. IGeomFunctor is called with the +force parameter, obliging it to return true even if there is no spatial overlap. +Associating Material and State types +Some models keep extra state information in the Body.state object, therefore requiring strict association +of a Material with a certain State (for instance, CpmMat is associated to CpmState and this combination +is supposed by engines such as CpmStateUpdater). +If a Material has such a requirement, it must override 2 virtual methods: +3.1. +Programmer’s manual +577 + +Yade Documentation, Release 3rd ed. +1. Material.newAssocState, which returns a new State object of the corresponding type. The default +implementation returns State itself. +2. Material.stateTypeOk, which checks whether a given State object is of the corresponding type (this +check is run at the beginning of the simulation for all particles). +In c++, the code looks like this (for CpmMat): +class CpmMat: public FrictMat { +public: +virtual shared_ptr newAssocState() const { return shared_ptr(new CpmState); +�→ } +virtual bool stateTypeOk(State* s) const { return (bool)dynamic_cast(s); } +/* ... */ +}; +This allows one to construct Body objects from functions such as utils.sphere only by knowing the requires +Material type, enforcing the expectation of the model implementor. +3.1.8 Runtime structure +Startup sequence +Yade’s main program is python script in core/main/main.py.in; the build system replaces a few +${variables} in that file before copying it to its install location. It does the following: +1. Process command-line options, set environment variables based on those options. +2. Import main yade module (import yade), residing in py/__init__.py.in. This module locates +plugins (recursive search for files lib*.so in the lib installation directory). yade.boot module is +used to setup temporary directory, … and, most importantly, loads plugins. +3. Manage further actions, such as running scripts given at command line, opening qt.Controller (if +desired), launching the ipython prompt. +Singletons +There are several “global variables” that are always accessible from c++ code; properly speaking, they +are Singletons, classes of which exactly one instance always exists. The interest is to have some general +functionality acessible from anywhere in the code, without the necessity of passing pointers to such objects +everywhere. The instance is created at startup and can be always retrieved (as non-const reference) using +the instance() static method (e.g. Omega::instance().getScene()). +There are 3 singletons: +ClassFactory Registers classes from plugins and able to factor instance of a class given its name as +string (the class must derive from Factorable). Not exposed to python. +Omega Access to simulation(s); deserves separate section due to its importance. +Logging Handles logging filters for all named loggers, see logging verbosity. +Omega +The Omega class handles all simulation-related functionality: loading/saving, running, pausing. +In python, the wrapper class to the singleton is instantiated6 as global variable O. For convenience, +Omega is used as proxy for scene’s attribute: although multiple Scene objects may be instantiated in +c++, it is always the current scene that Omega represents. +6 It is understood that instantiating Omega() in python only instantiates the wrapper class, not the singleton itself. +578 +Chapter 3. +Yade for programmers + +Yade Documentation, Release 3rd ed. +The correspondence of data is literal: Omega.materials corresponds to Scene::materials of the current +scene; likewise for materials, bodies, interactions, tags, cell, engines, initializers, miscParams. +To give an overview of (some) variables: +Python +c++ +Omega.iter +Scene::iter +Omega.dt +Scene::dt +Omega.time +Scene::time +Omega.realtime +Omega::getRealTime() +Omega.stopAtIter +Scene::stopAtIter +Omega in c++ contains pointer to the current scene (Omega::scene, retrieved by Omega::instance(). +getScene()). Using Omega.switchScene, it is possible to swap this pointer with Omega::sceneAnother, a +completely independent simulation. This can be useful for example (and this motivated this functionality) +if while constructing simulation, another simulation has to be run to dynamically generate (i.e. +by +running simulation) packing of spheres. +Engine loop +Running simulation consists in looping over Engines and calling them in sequence. This loop is defined +in Scene::moveToNextTimeStep function in core/Scene.cpp. Before the loop starts, O.initializers are +called; they are only run once. The engine loop does the following in each iteration over O.engines: +1. set Engine::scene pointer to point to the current Scene. +2. Call Engine::isActivated(); if it returns false, the engine is skipped. +3. Call Engine::action() +4. If O.timingEnabled, increment Engine::execTime by the difference from the last time reading (either +after the previous engine was run, or immediately before the loop started, if this engine comes first). +Increment Engine::execCount by 1. +After engines are processed, virtual time is incremented by timestep and iteration number is incremented +by 1. +Background execution +The engine loop is (normally) executed in background thread (handled by SimulationFlow class), leaving +foreground thread free to manage user interaction or running python script. The background thread is +managed by O.run() and O.pause() commands. Foreground thread can be blocked until the loop finishes +using O.wait(). +Single iteration can be run without spawning additional thread using O.step(). +3.1.9 Python framework +Wrapping c++ classes +Each class deriving from Serializable is automatically exposed to python, with access to its (registered) +attributes. This is achieved via YADE_CLASS_BASE_DOC_* macro family. All classes registered +in class factory are default-constructed in Omega::buildDynlibDatabase. Then, each serializable class +calls Serializable::pyRegisterClass virtual method, which injects the class wrapper into (initially +empty) yade.wrapper module. pyRegisterClass is defined by YADE_CLASS_BASE_DOC and knows about +class, base class, docstring, attributes, which subsequently all appear in boost::python class definition. +Wrapped classes define special constructor taking keyword arguments corresponding to class attributes; +therefore, it is the same to write: +3.1. +Programmer’s manual +579 + +Yade Documentation, Release 3rd ed. +Yade [61]: f1=ForceEngine() +Yade [62]: f1.ids=[0,4,5] +Yade [63]: f1.force=Vector3(0,-1,-2) +and +Yade [64]: f2=ForceEngine(ids=[0,4,5],force=Vector3(0,-1,-2)) +Yade [65]: print(f1.dict()) +{'force': Vector3(0,-1,-2), 'ids': [0, 4, 5], 'dead': False, 'ompThreads': -1, 'label': ''} +Yade [66]: print(f2.dict()) +{'force': Vector3(0,-1,-2), 'ids': [0, 4, 5], 'dead': False, 'ompThreads': -1, 'label': ''} +Wrapped classes also inherit from Serializable several special virtual methods: dict() returning all reg- +istered class attributes as dictionary (shown above), clone() returning copy of instance (by copying +attribute values), updateAttrs() and updateExistingAttrs() assigning attributes from given dictionary +(the former thrown for unknown attribute, the latter doesn’t). And pyDictCustom() explained also in +preceeding section. +Read-only property name wraps c++ method getClassName() returning class name as string. (Since +c++ class and the wrapper class always have the same name, getting python type using __class__ and +its property __name__ will give the same value). +Yade [67]: s=Sphere() +Yade [68]: s.__class__.__name__ +Out[68]: 'Sphere' +Subclassing c++ types in python +In some (rare) cases, it can be useful to derive new class from wrapped c++ type in pure python. This is +done in the yade.pack module module: Predicate is c++ base class; from this class, several c++ classes are +derived (such as inGtsSurface), but also python classes (such as the trivial inSpace predicate). inSpace +derives from python class Predicate; it is, however, not direct wrapper of the c++ Predicate class, +since virtual methods would not work. +boost::python provides special boost::python::wrapper template for such cases, where each overrid- +able virtual method has to be declared explicitly, requesting python override of that method, if present. +See Overridable virtual functions for more details. +When python code is called from C++, the calling thread must hold the python “Global Interpreter +Lock” (GIL). When initalizing the script as well as running one iteration calling O.step(), the running +thread is the same as python, and no additional code is required. On the other hand, calling python +code inside the simulation loop using O.run() needs the lock to be acquired by the thread, or a segfault +error will occurs. See implementation of pyGenericPotential () for a complete exemple. +Reference counting +Python internally uses reference counting on all its objects, which is not visible to casual user. It has to +be handled explicitly if using pure Python/C API with Py_INCREF and similar functions. +boost::python used in Yade fortunately handles reference counting internally. Additionally, it auto- +matically integrates reference counting for shared_ptr and python objects, if class A is wrapped as +boost::python::class_>. Since all Yade classes wrapped using YADE_CLASS_- +BASE_DOC_* macro family are wrapped in this way, returning shared_ptr<…> objects from is the +preferred way of passing objects from c++ to python. +580 +Chapter 3. +Yade for programmers + +Yade Documentation, Release 3rd ed. +Returning shared_ptr is much more efficient, since only one pointer is returned and reference count +internally incremented. Modifying the object from python will modify the (same) object in c++ and +vice versa. It also makes sure that the c++ object will not be deleted as long as it is used somewhere in +python, preventing (important) source of crashes. +Custom converters +When an object is passed from c++ to python or vice versa, then either +1. the type is basic type which is transparently passed between c++ and python (int, bool, std::string +etc) +2. the type is wrapped by boost::python (such as Yade classes, Vector3 and so on), in which case +wrapped object is returned;7 +Other classes, including template containers such as std::vector must have their custom converters +written separately. Some of them are provided in py/wrapper/customConverters.cpp, notably converters +between python (homogeneous, i.e. with all elements of the same type) sequences and c++ std::vector +of corresponding type; look in that source file to add your own converter or for inspiration. +When an object is crossing c++/python boundary, boost::python’s global “converters registry” is +searched for class that can perform conversion between corresponding c++ and python types. +The +“converters registry” is common for the whole program instance: there is no need to register convert- +ers in each script (by importing _customConverters, for instance), as that is done by yade at startup +already. +Note: +Custom converters only work for value that are passed by value to python (not “by reference”): +some attributes defined using YADE_CLASS_BASE_DOC_* macro family are passed by value, but if +you define your own, make sure that you read and understand Why is my automatic to-python conversion +not being found?. +In short, the default for def_readwrite and def_readonly is to return references to underlying c++ +objects, which avoids performing conversion on them. For that reason, return value policy must be set +to return_by_value explicitly, using slighly more complicated add_property syntax, as explained at +the page referenced. +This deficiency is addressed presently in the file lib/serialization/PyClassCustom.hpp for the .def_- +readonly(…) function. It can be improved later if the need arises. +3.1.10 Adding a new python/C++ module +Modules are placed in py/ directory, the C++ parts of the modules begin their name with an underscore +_. The procedure to add a new module is following: +1. Create your new files: +1. The yourNewModule.py file like this. +2. The _yourNewModule.cpp file like this, if part of your module will be written in C++. +2. Update the module redirection map in these two places: +1. mods in doc/sphinx/yadeSphinx.py. +2. moduleMap in doc/sphinx/conf.py, if the new module has a C++ part (this duplication of data +will hopefully be soon removed). +7 Wrapped classes are automatically registered when the class wrapper is created. +If wrapped class derives from +another wrapped class (and if this dependency is declared with the boost::python::bases template, which Yade’s +classes do automatically), parent class must be registered before derived class, however. +(This is handled via loop in +Omega::buildDynlibDatabase, which reiterates over classes, skipping failures, until they all successfully register) Math +classes (Vector3, Matrix3, Quaternion) are wrapped in minieigenHP. See high precision documentation for more details. +3.1. +Programmer’s manual +581 + +Yade Documentation, Release 3rd ed. +3. Add the C++ file into py/CMakeLists.txt like this. +4. Modify the CMakeLists.txt but only if the file will depend on cmake compilation variables, eg. +like this. The file then needs an additional extension .in and be put in two places: +1. The cmake command to generate the file from .in input: like this. +2. The cmake command to install it: like this. +Hint: +The last step regarding yourNewModule.py.in (or _yourNewModule.cpp.in) is needed only on +very rare occasions, and is included here only for the sake of completeness. +Debugging boundary between python and C++ +During normal use all C++ exceptions are propagated back to python interface with full information +associated with them. +The only situation where this might not be the case is during execution of +command import module inside a python script. It might happen that when importing a new module +some cryptic errors occur like: initialization of module raised unreported exception. +These +unreported exceptions happen in the situation when the C++ code executed a python code inside +it (this is called embedding) and this python code threw an exception. The proper way to deal with +this situation is to wrap entire module declaration inside a try {} catch(...) {} block. It might be +possible to deal with specific exceptions also (see here for other example catch blocks), however the +general solution is to properly inform python that importing this module did not work. In this catch +block it is possible to execute PyErr_Print(); command to see what the problem was and propagate +the exception back to python, however during import module command only the SystemError python +exception can get through. Hence the catch(...) block after BOOST_PYTHON_MODULE(_yourNewModule) +should look like this: +#include +CREATE_CPP_LOCAL_LOGGER("_yourNewModule.cpp"); +BOOST_PYTHON_MODULE(_yourNewModule) +try { +py::def("foo", foo, R"""( +The description of function foo(). +:param arg1: description of first argument +:param arg2: description of second argument +:type arg1: type description +:type arg2: type description +:return: return description +:rtype: the return type description +Example usage of foo: +.. ipython:: +In [1]: from yade.yourNewModule import * +In [1]: foo() +.. note:: Notes, hints and warnings about how to use foo(). +)"""); +} catch (...) { +LOG_FATAL("Importing this module caused an exception and this module is in an␣ +�→inconsistent state now."); +(continues on next page) +582 +Chapter 3. +Yade for programmers + +Yade Documentation, Release 3rd ed. +(continued from previous page) +PyErr_Print(); +PyErr_SetString(PyExc_SystemError, __FILE__); +boost::python::handle_exception(); +throw; +} +Note: +Pay attention to the _yourNewModule inside BOOST_PYTHON_MODULE(…), it has to match the file +name of the .cpp file. +Further reading, about how to work with python exceptions: +1. Example in boost::python reference manual. +2. Example in boost::python tutorial. +3. When PyErr_Print(); is not enough. +3.1.11 Maintaining compatibility +In Yade development, we identified compatibility to be very strong desire of users. Compatibility concerns +python scripts, not simulations saved in XML or old c++ code. +Renaming class +Script scripts/rename-class.py should be used to rename class in c++ code. It takes 2 parameters (old +name and new name) and must be run from top-level source directory: +$ scripts/rename-class.py OldClassName NewClassName +Replaced 4 occurences, moved 0 files and 0 directories +Update python scripts (if wanted) by running: perl -pi -e 's/\bOldClassName\b/NewClassName/g'␣ +�→`ls **/*.py |grep -v py/system.py` +This has the following effects: +1. If file or directory has basename OldClassName (plus extension), it will be renamed using bzr. +2. All occurences of whole word OldClassName will be replaced by NewClassName in c++ sources. +3. An entry is added to py/system.py, which contains map of deprecated class names. At yade startup, +proxy class with OldClassName will be created, which issues a DeprecationWarning when being +instantiated, informing you of the new name you should use; it creates an instance of NewClassName, +hence not disruting your script’s functioning: +Yade [3]: SimpleViscoelasticMat() +/usr/local/lib/yade-trunk/py/yade/__init__.py:1: DeprecationWarning: Class␣ +�→`SimpleViscoelasticMat' was renamed to (or replaced by) `ViscElMat', update your code!␣ +�→(you can run 'yade --update script.py' to do that automatically) +-> +[3]: +As you have just been informed, you can run yade --update to all old names with their new names in +scripts you provide: +$ yade-trunk --update script1.py some/where/script2.py +This gives you enough freedom to make your class name descriptive and intuitive. +3.1. +Programmer’s manual +583 + +Yade Documentation, Release 3rd ed. +Renaming class attribute +Renaming class attribute is handled from c++ code. You have the choice of merely warning at accessing +old attribute (giving the new name), or of throwing exception in addition, both with provided explanation. +See deprec parameter to YADE_CLASS_BASE_DOC_* macro family for details. +3.2 Yade on GitLab +3.2.1 Fast checkout (read-only) +Getting the source code without registering on GitLab can be done via a single command. It will not +allow interactions with the remote repository, which you access the read-only way: +git clone --recurse-submodules https://gitlab.com/yade-dev/trunk.git +3.2.2 Branches on GitLab +Most useful commands are listed in the sections below. For more details, see these git guides: +1. ProGit online Book, +2. Guide on setting up git, +3. Git “choose your own adventure”, +4. Guide on fixing the conflicts. +Setup +1. Register on gitlab.com +2. Add your SSH key to GitLab +3. Set your username and email through terminal +git config --global user.name "Firstname Lastname" +git config --global user.email "your_email@youremail.com" +You can check these settings with git config --list. +4. To fork the repository (optional), click the “Fork” button on the gitlab page, and also fork the +YADE data files. +Note: +By default gitlab will try and compile the forked repository, and it will fail if you don’t +have runners attached to your account. To avoid receiving failure notifications go to repository +settings (bottom of left panel->general->permissions) to turn of pipelines. +5. Set Up Your Local Repo through terminal. The argument --recurse-submodules is to make sure +that ./data directory is filled with the recent data from yade-data (the path is relative to your +gitlab profile): +git clone --recurse-submodules git@gitlab.com:username/trunk.git +This creates a new folder, named trunk, that contains the whole code (make sure username is +replaced by your GitLab name). If you already have a cloned yade repository with ./data directory +in it, then you can populate your existing repository using command: +584 +Chapter 3. +Yade for programmers + +Yade Documentation, Release 3rd ed. +git submodule update --init --recursive +6. Configure remotes +cd to/newly/created/folder +git remote add upstream git@gitlab.com:yade-dev/trunk.git +git remote update +Now, your “trunk” folder is linked with two remote repositories both hosted on gitlab.com, the +original trunk from yade-dev (called “upstream” after the last command) and the fork which resides +in your personal account (called “origin” and always configured by default). Through appropriate +commands explained below, you will be able to update your code to include changes commited by +others, or to commit yourself changes that others can get. +Holding a fork under personnal account is in fact not strictly necessary. It is recommended, however, +and in what follows it is assumed that the above steps have been followed. +Older versions +In case you want to work with, or compile, an older version of Yade which is not tagged, you can create +your own (local) branch of the corresponding daily build. Look here for details. +Committing and updating +Inspecting changes +After changing the source code in the local repository you may start by inspecting them with a few +commands. For the “diff” command, it is convenient to copy from the output of “status” instead of +typing the path to modified files. +git status +git diff path/to/modified/file.cpp +Pushing changes +Depending on the remote repository you want to push to, follow one of the methods below. +1. Push to yade-dev +Merging changes into yade-dev’s master branch cannot be done directly with a push, only by merge +request (see below). It is possible however to push changes to a new branch of yade-dev repository +for members of that group. It is currently the only way to have merge requests tested by the gitlab +CI pipeline before being effectively merged. To push to a new yade-dev/branch: +git branch localBranch +git checkout localBranch +git add path/to/new/file.cpp +#Version a newly created file +git commit path/to/new_or_modified/file.cpp -m 'Commit message' +#stage (register) change␣ +�→in the local repository +git pull --rebase upstream master #get updated version of sources from yade-dev repo and␣ +�→apply your commits on the top of them +git push upstream localBranch:newlyCreatedBranch #Push all commits to a new remote branch. +The first two lines are optional, if ignored the commits will go the to the default branch, called +“master”. In the last command localBranch is the local branch name on which you were working +(possibly master) and newlyCreatedBranch will be the name of that branch on the remote. Please +choose a descriptive name as much as you can (e.g. “fixBug457895”). +3.2. +Yade on GitLab +585 + +Yade Documentation, Release 3rd ed. +Note: +If you run into any problems with command git pull --rebase upstream master, you always +can revert or even better fix the conflicts. +2. Push to personnal repository +After previous steps proceed to commit through terminal, “localBranch” should be replaced by a +relevant name: +git branch localBranch +git checkout localBranch +git add path/to/new/file.cpp +#Version a newly created file +git commit path/to/new_or_modified/file.cpp -m 'Commit message' +#stage (register) change␣ +�→in the local repository +git push +#Push all commits to the remote branch +The changes will be pushed to your personal fork. +Updating +You may want to get changes done by others to keep your local and remote repositories synced with the +upstream: +git pull --rebase upstream master #Pull new updates from the upstream to your branch. Eq. of +�→"bzr update", updating the local branch from the upstream yade-dev/trunk/master +git push +#Merge changes from upstream into your gitlab repo (origin) +If you have local uncommited changes this will return an error. A workaround to update while preserving +them is to “stash”: +git stash #backup and hide changes +git pull --rebase upstream master +git push +git stash pop #restore backed up changes +Auto rebase +We promote “rebasing” to avoid confusing logs after each commit/pull/push cycle. It can be convenient +to setup automatic rebase, so it does not have to be added everytime in the above commands: +git config --global branch.autosetuprebase always +Now your file ~/.gitconfig should include: +[branch] +autosetuprebase = always +Check also .git/config file in your local trunk folder (rebase = true): +[remote "origin"] +url = git@gitlab.com:yade-dev/trunk.git +fetch = +refs/heads/*:refs/remotes/origin/* +[branch "master"] +remote = origin +merge = refs/heads/master +rebase = true +586 +Chapter 3. +Yade for programmers + +Yade Documentation, Release 3rd ed. +Pulling a rebased branch +If someone else rebased on the gitlab server the branch on which you are working on locally, the command +git pull may complain that the branches have diverged, and refuse to perform operation, in that case +this command: +git pull --rebase upstream branchName +Will match your local branch history with the one present on the gitlab server. +If you are afraid of messing up your local branch you can always make a copy of this branch with +command: +git branch backupCopyName +If you forgot to make that backup-copy and want to go back, then make a copy anyway and go back +with this command: +git reset --merge ORIG_HEAD +The ORIG_HEAD backs up the position of HEAD before a potentially dangerous operation (merge, rebase, +etc.). +A tutorial on fixing the conflicts is a recommended read. +Note: +If you are lost about how to fix your git problems try a git choose your own adventure. +3.2.3 Merge requests +Members of yade-dev +If you have tested your changes and you are ready to merge them into yade-dev’s master branch, you’ll +have to make a “merge request” (MR) from the gitlab.com interface (see the “+” button at the top of the +repository webpage). Set source branch and target branch, from yade-dev/trunk/newlyCreatedBranch +to yade-dev/trunk/master. The MR will trigger a pipeline which includes compiling, running regression +tests, and generating the documentation (the newly built documentation is accessible via settings->pages +or by clicking on the “Browse” button in the “Job artifacts” (in the right pane) in the doc_18_04 build +from the pipeline; then navigating to path Artifacts/install/share/doc). If the full pipeline succeeds +the merge request can be merged into the master branch. +Note: +In case of MR to yade-dev’s master from another branch of yade-dev, the pipeline will use group +runners attached to yade-dev (the group runners are kindly provided by 3SR, UMS Gricad and Gdańsk +University of Technology). +New developers +Welcome! At start it is very convenient to work on a local fork of YADE in your own gitlab profile. +When you are confident that your changes are ready to be merged into official YADE release, please +open a Merge Request (MR) in the following way: +1. Make sure that your work is in a separate branch, not in the master branch. You can “copy” your +branch into another branch with command git checkout -b myNewFeature. Please make sure +that the amount of changes as compared to the master branch is not large. In case of larger code +improvements it is better to split it into several smaller merge requests. This way it will be faster +for us to check it and merge. +3.2. +Yade on GitLab +587 + +Yade Documentation, Release 3rd ed. +2. Push your branch to the repository on your gitlab profile with command such as: +git push --set-upstream origin myNewFeature +3. You should see something like: +remote: +remote: To create a merge request for myNewFeature, visit: +remote: +https://gitlab.com/myProfileName/trunk/-/merge_requests/new?merge_request +�→%5Bsource_branch%5D=myNewFeature +remote: +4. When you visit the link mentioned above, you will have to select “Change branches” and make +sure that correct target branch is selected. Usually that will be yade-dev/trunk:master, because +this is the official YADE repository. +5. Fill in the title and description then click “Create merge request” at the bottom of the page. +6. After we review the merge request we can click on it to run in our Continuous Integration (CI) +pipeline. This pipeline can’t start automatically for security reasons. It will be merged after the +pipeline checks pass. +Alternatively, create a patch from your commit via: +git format-patch origin +#create patch file in current folder) +and send to the developers mailing list (yade-dev@lists.launchpad.net) as attachment. In either way, +after reviewing your changes they will be added to the main trunk. +When the pull request has been reviewed and accepted, your changes are integrated in the main trunk. +Everyone will get them via git fetch. +3.2.4 Guidelines for pushing +These are general guidelines for pushing to yade-dev/trunk. +1. Set autorebase globaly on the computer (only once see above), or at least on current local branch. +Non-rebased pull requests will not be accepted on the upstream. This is to keep history linear, and +avoid the merge commits. +2. Inspect the diff to make sure you will not commit junk code (typically some “cout<<” left here +and there), using in terminal: +git diff file1 +Or using your preferred difftool, such as kdiff3: +git difftool -t kdiff3 file1 +Or, alternatively, any GUI for git: gitg, git-cola… +3. Commit selectively: +git commit file1 file2 file3 -m "message" # is good +git commit -a -m "message" +# is bad. It is the best way to commit␣ +�→things that should not be commited +4. Be sure to work with an up-to-date version launching: +git pull --rebase upstream master +5. Make sure it compiles and that regression tests pass: try yade --test and yade --check. +Thanks a lot for your cooperation to Yade! +588 +Chapter 3. +Yade for programmers + +Chapter 4 +Theoretical background and +extensions +4.1 DEM formulation +The DEM formulation is presented in earlier chapter 2.1 DEM formulation as a common ground for all +DEM calculations. +4.2 CFD-DEM coupled simulations with Yade and OpenFOAM +The FoamCoupling engine provides a framework for Euler-Lagrange fluid-particle simulation with the +open source finite volume solver OpenFOAM. The coupling relies on the Message Passing Interface +library (MPI), as OpenFOAM is a parallel solver, furthermore communication between the solvers are +realised by MPI messages. The FoamCoupling engine must be enabled with the ENABLE_MPI flag +during compilation: +cmake -DCMAKE_INSTALL_PREFIX=/path/to/install /path/to/source -DENABLE_MPI=1 +Yade sends the particle information (particle position, velocity, etc. +) +to all the OpenFOAM pro- +cesses. Each OpenFOAM process searches the particle in the local mesh, if the particle is found, the +hydrodynamic drag force and torque are calculated using the fluid velocity at the particle position (two +interpolation methods are available) and the particle velocity. The hydroynamic force is sent to the Yade +process and it is added to the force container. The negative of the particle hydrodynamic force (interpo- +lated back to the fluid cell center) is set as source term in the Navier-Stokes equations. The OpenFOAM +solver must also be installed to facilitate the MPI connection between Yade and OpenFOAM. Technical +details on the coupling methodology can be found in [Kunhappan2017] and [Kunhappan2018]. +4.2.1 Background +In the standard Euler-Lagrange modelling of particle laden multiphase flows, the particles are treated as +point masses. Two approaches are implemented in the present coupling: +1. Point force coupling +2. Volume fraction based force coupling. +In both of the approaches the flow at the particle scale is not resolved and analytical/empirical hydro- +dynamic force models are used to describe the fluid-particle interactions. For accurate resolution of the +particle volume fraction and hydrodynamic forces on the fluid grid the particle size must be smaller than +the fluid cell size. +589 + +Yade Documentation, Release 3rd ed. +Point force coupling (icoFoamYade) +In the point force coupling, the particles are assumed to be smaller than the smallest fluid length scales, +such that the particle Reynolds Number is Rep < 1.0. +The particle Reynolds number is defined as +the ratio of inertial forces to viscous forces. For a sphere, the associated length-scale is the diameter, +therefore: +Rep = ρf|Ur|dp +µ +(4.1) +where in (4.1) ρf is the fluid density, |Ur| is the norm of the relative velocity between the particle and +the fluid, dp is the particle diameter and µ the fluid dynamic viscosity. In addition to the Reynolds +number, another non-dimensional number that characterizes the particle inertia due to it’s mass called +Stokes number is defined as: +Stk = τp |Uf| +dp +(4.2) +where in equation (4.2) τp is the particle relaxation time defined as: +τp = ρpd2 +p +18µ +For Rep < 1 and Stk < 1, the hydrodynamic force on the particle can be represented as a point force. +This force is calculated using the Stoke’s drag force formulation: +Fh = 3πµdp(Uf − Up) +(4.3) +The force obtained from (4.3) is applied on the particle and in the fluid side (in the cell where the particle +resides), this hydrodynamic force is formulated as a body/volume force: +fh = −Fh +Vcρf +(4.4) +where in equation (4.4) Vc is the volume of the cell and ρf is the fluid density. Hence the Navier-Stokes +equations for the combined system is: +∂U +∂t + ∇ · (UU) = −∇p +ρ + ∇¯¯τ + fh +(4.5) +Along with the continuity equation: +∇ · U = 0 +(4.6) +Volume averaged coupling (pimpleFoamYade) +In the volume averaged coupling, the effect of the particle volume fraction is included. The Navier-Stokes +equations take the following form: +∂(εfUf) +∂t ++ ∇ · (εfUfUf) = −∇p +ρ + εf∇¯¯τ − K (Uf − Up) + Su + εfg +(4.7) +590 +Chapter 4. +Theoretical background and extensions + +Yade Documentation, Release 3rd ed. +Along with the continuity equation: +∂εf +∂t + ∇ · (εfUf) = 0 +(4.8) +where in equations (4.7) and (4.8) εf is the fluid volume fraction. Note that, we do not solve for εf +directly, but obtain it from the local particle volume fraction εs, εf = 1−εs . K is the particle drag force +parameter, Uf and Up are the fluid and particle velocities respectively. Su denotes the explicit source +term consisting the effect of other hydrodynamic forces such as the Archimedes/ambient force, added +mass force etc. Details on the formulation of these forces are presented in the later parts of this section. +The interpolation and averaging of the Eulerean and Lagrangian quantities are based on a Gaussian +envelope G⋆. In this method, the the effect of the particle is ‘seen’ by the neighbouring cells of the cell +in which it resides. Let xc and xp be the fluid cell center and particle position respectively, then the +Gaussian filter G⋆ (xc − xp) defined as: +G⋆ (xc − xp) = +� +2πσ2� 3 +2 exp +� +−||xc − xp||2 +2σ2 +� +(4.9) +with σ being the standard deviation of the filter defined as: +σ = δ/ +� +2 +√ +2 ln 2 +� +(4.10) +where in equation (4.10) δ is the cut-off range (at present it’s set to 3∆x, with ∆x being the fluid cell +size.) and follows the rule: +G⋆ (||xc − xp|| = δ/2) = 1 +2G⋆ (||xc − xp|| = 0) +The particle volume fraction εs,c for a fluid cell c is calculated by: +εs,c = +�Np +i=1 Vp,iG⋆(i,c) +Vc +(4.11) +where in (4.11) Np is the number of particle contributions on the cell c, G⋆(i,c) is the Gaussian weight +obtained from (4.9), Vp,iG⋆(i,c) forms the individual particle volume contribution. Vc is the fluid cell +volume and εf + εs = 1 +The averaging and interpolation of an Eulerean quantity φ from the grid (cells) to the particle position +is performed using the following expression: +�φ = +Nc +� +i=1 +φiG⋆(i,p) +(4.12) +Hydrodynamic Force +In equation (4.7) the term K is the drag force parameter. In the present implementation, K is based on +the Schiller Naumman drag law, which reads as: +K = 3 +4Cd +ρf +dp +��� +��� �Uf − Up +��� +��� ε−hexp +f +(4.13) +4.2. +CFD-DEM coupled simulations with Yade and OpenFOAM +591 + +Yade Documentation, Release 3rd ed. +In equation (4.13) ρf is the fluid density, dp the particle diameter, hexp is defined as the ‘hindrance +coefficient’ with the value set as hexp = 2.65. The drag force force coefficient Cd is valid for particle +Reynolds numbers up to Rep < 1000. The expression for Cd reads as: +Cd = 24 +Rep +� +1 + 0.15Re0.687 +p +� +(4.14) +The expression of hydrodynamic drag force on the particle is: +Fdrag = VpK( �Uf − Up) +In the fluid equations, negative of the drag parameter (−K) is distributed back to the grid based on +equation (4.11). Since the drag force includes a non-linear dependency on the fluid velocity Uf, this +term is set as an implicit source term in the fluid solver. +The Archimedes/ambient force experienced by the particle is calculated as: +Fby = +� +� +−∇p + � +∇¯¯τ +� +Vp +(4.15) +where in (4.15), � +∇p is the averaged pressure gradient at the particle center and � +∇¯¯τ is the averaged +divergence of the viscous stress at the particle position. +Added mass force: +Fam = Cm +� +D� +Uf +Dt − dUp +dt +� +Vp +(4.16) +where in eqaution (4.16), D�Uf +Dt is the material derivative of the fluid velocity. +Therefore the net hydrodynamic force on the particle reads as: +Fhyd = Fdrag + Fby + Fam +And on the fluid side the explicit source term Su,c for a fluid cell c is expressed as : +Su,c = +�Np +i=1 −Fhyd,iεs,cG⋆(i,c) +ρfVc +4.2.2 Setting up a case +In Yade +Setting a case in the Yade side is fairly straight forward. The python script describing the scene in Yade +is based on this method. Make sure the exact wall/periodic boundary conditions are set in Yade as well +as in the OpenFOAM. The particles should not leave the fluid domain. In case a particle has ‘escaped’ +the domain, a warning message would be printed/written to the log file and the simulation will break. +The example in examples/openfoam/scriptYade.py demonstrates the coupling. A symbolic link to Yade +is created and it is imported in the script. The MPI environment is initialized by calling the initMPI() +function before instantiating the coupling engine +592 +Chapter 4. +Theoretical background and extensions + +Yade Documentation, Release 3rd ed. +initMPI() +fluidCoupling = FoamCoupling() +fluidCoupling.getRank() +A list of the particle ids and number of particle is passed to the coupling engine +sphereIDs = [b.id for b in O.bodies if type(b.shape)==Sphere] +numparts = len(sphereIDs); +fluidCoupling.setNumParticles(numparts) +fluidCoupling.setIdList(sphereIDs) +fluidCoupling.isGaussianInterp = False +The type of force/velocity interpolation mode has to be set. For Gaussian envelope interpolation, the +isGaussianInterp flag has to be set, also the solver pimpleFoamYade must be used. The engine is added +to the O.engines after the timestepper +O.engines = [ +ForceResetter(), +..., +GlobalStiffnessTimeStepper, +fluidCoupling ... +newton ] +Substepping/data +exchange +interval +is +set +automatically +based +on +the +ratio +of +timesteps +as +foamDt/yadeDt (see exchangeDeltaT for details). +In OpenFOAM +There are two solvers available in this git repository. The solver icoFoamYade is based on the point force +coupling method and the solver pimpleFoamYade is based on the volume averaged coupling. They are +based on the existing icoFoam and pimpleFoam solvers respectively. Any OpenFOAM supported mesh +can be used, for more details on the mesh options and meshing see here. In the present example, the +mesh is generated using blockMesh utility of OpenFOAM. The case is set up in the usual OpenFOAM +way with the directories 0, system and constant +0/ +U +## velocity boundary conditions +p +## pressure boundary conditions +uSource +## source term bcs (usually set as calculated). +system/ +controlDict +## simulation settings : start time, end time, delta T, solution␣ +�→write control etc. +blockMeshDict +## mesh setup for using blockMesh utility : define coordinates of␣ +�→geometry and surfaces. (used for simple geometries -> cartesian mesh.) +decomposeParDict +## dictionary for setting domain decomposition, (in the present␣ +�→example scotch is used) +fvSchemes +## selection of finite volume schemes for calculations of␣ +�→divergence, gradients and interpolations. +fvSolution +## linear solver selection, setting of relaxation factors and␣ +�→tolerance criterion, +constant/ +polymesh/ +## mesh information, generated by blockMesh or other mesh utils. +transportProperties +## set the fluid and particle properties. (just density of the␣ +�→particle) +Note: Always set the timestep less than the particle relaxation time scale, this is not claculated au- +tomatically yet! Turbulence modelling based on the RANS equations have not been implemented yet. +4.2. +CFD-DEM coupled simulations with Yade and OpenFOAM +593 + +Yade Documentation, Release 3rd ed. +However it is possible to use the present formulations for fully resolved turbulent flow simulations via +DNS. Dynamic/moving mesh problems are not supported yet. +(Let me know if you’re interested in +implementing any new features.) +To prepare a simulation, follow these steps: +blockMesh +## generate the mesh +decomposePar +## decompose the mesh +Any type of mesh that is supported by OpenFOAM can be used. +Dynamic mesh is currently not +supported. +Execution +The simulation is executed via the following command: +mpiexec -n 1 python3 scriptYade.py : -n NUMPROCS icoFoamYade -parallel +The video below shows the steps involved in compiling and executing the coupled CFD-DEM simulation +4.2.3 Post-Processing +Paraview can be used to visulaize both the Yade solution (use VTKRecorder) and OpenFOAM solution. +To visulaize the fluid solution, create an empty file as name.foam , open this file in Paraview and in the +properties section below the pipeline, change “Reconstructed case” to “Decomposed case” , or you can +use the reconstructed case itself but after running the reconstructPar utility, but this is time consuming. +4.3 FEM-DEM hierarchical multiscale modeling with Yade and Es- +cript +Authors: Ning Guo and Jidong Zhao +Institution: Hong Kong University of Science and Technology +Escript download page: https://launchpad.net/escript-finley +mpi4py download page (optional, require MPI): https://bitbucket.org/mpi4py/mpi4py +Tested platforms: Desktop with Ubuntu 10.04, 32 bit; Server with Ubuntu 12.04, 14.04, 64 bit; Cluster +with Centos 6.2, 6.5, 64 bit; +4.3.1 Introduction +The code is built upon two open source packages: Yade for DEM modules and Escript for FEM modules. +It implements the hierarchical multiscale model (FEMxDEM) for simulating the boundary value problem +(BVP) of granular media. FEM is used to discretize the problem domain. Each Gauss point of the FEM +mesh is embedded a representative volume element (RVE) packing simulated by DEM which returns +local material constitutive responses to FEM. Typically, hundreds to thousands of RVEs are involved in +a medium-sized problem which is critically time consuming. Hence parallelization is achieved in the code +through either multiprocessing on a supercomputer or mpi4py on a HPC cluster (require MPICH or Open +MPI). The MPI implementation in the code is quite experimental. The “mpipool.py” is contributed by +Lisandro Dalcin, the author of mpi4py package. Please refer to the examples for the usage of the code. +594 +Chapter 4. +Theoretical background and extensions + +Yade Documentation, Release 3rd ed. +4.3.2 Finite element formulation +Note: +This and the following section are a short excerpt from [Guo2014] to provide some theoretical +background. Yade users of FEM-DEM coupling are welcome to improve the following two sections. +In this coupled FEM/DEM framework on hierarchical multiscale modelling of granular media, the geo- +metric domain Ω of a given BVP is first discretised into a suitable FEM mesh. After the finite element +discretisation, one ends up with the following equation system to be solved, +Ku = f, +(4.17) +where K is the stiffness matrix, u is the unknown displacement vector at the FEM nodes and f is the +nodal force vector lumped from the applied boundary traction. For a typical linear elastic problem, K +can be formulated from the elastic modulus, and equation (4.17) can be solved directly. Whilst in the +case involving nonlinearity such as for granular media where K depends on state parameters and loading +history, Newton–Raphson iterative method needs to be adopted and the stiffness matrix is replaced with +the tangent matrix Kt, which is assembled from the tangent operator: +Kt = +� +Ω +BTDBdV, +(4.18) +where B is the deformation matrix (i.e. gradient of the shape function), and D is the matrix form of +the rank four tangent operator tensor D. During each Newton–Raphson iteration, both Kt and internal +stress σ are updated, and the scheme tries to minimise the residual force R to find a converged solution: +R = +� +Ω +BTσdV − f. +(4.19) +The tangent operator and the stress tensor at each local Gauss integration point are pivotal variables +in the aforementioned calculation and need to be evaluated before each iteration and loading step. A +continuum-based conventional FEM usually assumes a constitutive relation for the material and derives +the tangent matrix and the stress increment based on this constitutive assumption (e.g. using the elasto- +plastic modulus Dep in equation (4.18) to assemble Kt and to integrate stress). The coupled FEM/DEM +multiscale approach obtains the two quantities from the embedded discrete element assembly at each +Gauss point and avoids the needs for phenomenological assumptions. +4.3.3 Multiscale solution procedure +The hierarchical multiscale modelling procedure is schematically summarised in the following steps: +1. Discretise the problem domain by suitable FEM mesh and attach each Gauss point with a DEM +assembly prepared with suitable initial state. +2. Apply one global loading step, that is, imposed by FEM boundary condition on ∂Ω. +a) Determine the current tangent operator for each RVE. +b) Assemble the global tangent matrix using equation (4.18) and obtain a trial solution of dis- +placement u by solving Equation (4.17) with FEM. +c) Interpolate the deformation ∇u at each Gauss point of the FEM mesh and run the DEM +simulation for the corresponding RVE using ∇u as the DEM boundary conditions. +d) Derive the updated total stress for each RVE and use it to evaluate the residual by equation +(4.19) for the FEM domain. +e) Repeat the aforementioned steps from (a) to (d) until convergence is reached and finish the +current loading step. +3. Proceed to the next loading step and repeat Step 2. +4.3. +FEM-DEM hierarchical multiscale modeling with Yade and Escript +595 + +Yade Documentation, Release 3rd ed. +In interpolating the deformation u from the FEM solution for DEM boundary conditions in Step 2(c), +we consider both the infinitesimal strain ε and rotation ω +∇u = 1 +2(∇u + ∇uT) +� +�� +� +ε ++ 1 +2(∇u − ∇uT) +� +�� +� +ω +(4.20) +The corresponding RVE packing will deform according to this prescribed boundary condition. +It is also instructive to add a few remarks on the evolution of stress from the RVE in Step 2(d). In +traditional FEM, the stress is updated based on an incremental manner to tackle the nonlinear material +response. If small strain is assumed, the incremental stress–strain relation may potentially cause inac- +curate numerical results when large deformation occurs in the material, which calls for an alternative +formulation for large deformation. This issue indeed can be naturally circumvented in the current hier- +archical framework. In our framework, the DEM assembly at each Gauss point will memorise its past +state history (e.g. pressure level, void ratio and fabric structure) and will be solved with the current +applied boundary condition (including both stretch and rotation) at each loading and iteration step. +Towards the end of each loading step, instead of using an incremental stress update scheme, the total +true stress (Cauchy stress) is derived directly over the solved DEM assembly through homogenisation +and is then returned to the FEM solver for the global solution. In this way, we do not have to resort to +the use of other objective stress measures to deal with large deformation problems. However, we note +that a proper strain measurement is still required and the FEM mesh should not be severely distorted, +otherwise, remeshing of the FEM domain will be required. +More detailed description of the solution procedure can be found in [Guo2013], [Guo2014], [Guo2014b], +[Guo2014c], [Guo2015]. +4.3.4 Work on the YADE side +The version of YADE should be at least rev3682 in which Bruno added the stringToScene function. +Before installation, I added some functions to the source code (in “yade” subfolder). But only one func- +tion (“Shop::getStressAndTangent” in “./pkg/dem/Shop.cpp”) is necessary for the FEMxDEM coupling, +which returns the stress tensor and the tangent operator of a discrete packing. The former is homoge- +nized using the Love’s formula and the latter is homogenized as the elastic modulus. After installation +and we get the executable file: yade-versionNo. We then generate a .py file linked to the executable +file by “ln yade-versionNo yadeimport.py”. This .py file will serve as a wrapped library of YADE. Later +on, we will import all YADE functions into the python script through “from yadeimport import *” (see +simDEM.py file). +Open a python terminal. Make sure you can run +import sys +sys.path.append('where you put yadeimport.py') +from yadeimport import * +Omega().load('your initial RVE packing, e.g. 0.yade.gz') +If you are successful, you should also be able to run +from simDEM import * +4.3.5 Work on the Escript side +No particular requirement. But make sure the modules are callable in python, which means the main +folder of Escript should be in your PYTHONPATH and LD_LIBRARY_PATH. The modules are +wrapped as a class in msFEM*.py. +Open a python terminal. Make sure you can run: +596 +Chapter 4. +Theoretical background and extensions + +Yade Documentation, Release 3rd ed. +from esys.escript import * +from esys.escript.linearPDEs import LinearPDE +from esys.finley import Rectangle +(Note: Escript is used for the current implementation. It can be replaced by any other FEM package +provided with python bindings, e.g. +FEniCS (http://fenicsproject.org). +But the interface files “ms- +FEM*.py” need to be modified.) +4.3.6 Example tests +After Steps 1 & 2, one should be able to run all the scripts for the multiscale analysis. The initial +RVE packing (default name “0.yade.gz”) should be provided by the user (e.g. using YADE to prepare a +consolidated packing), which will be loaded by simDEM.py when the problem is initialized. The sample +is initially uniform as long as the same RVE packing is assigned to all the Gauss points in the problem +domain. It is also possible for the user to specify different RVEs at different Gauss points to generate an +inherently inhomogeneous sample. +While simDEM.py is always required, only one msFEM*.py is needed for a single test. For example, in +a 2D (3D) dry test, msFEM2D.py (msFEM3D.py) is needed; similarly for a coupled hydro-mechanical +problem (2D only, saturated), msFEMup.py is used which incorporates the u-p formulation. Multipro- +cessing is used by default. To try MPI parallelization, please set useMPI=True when constructing the +problem in the main script. Example tests given in the “example” subfolder are listed below. Note: The +initial RVE packing (named 0.yade.gz by default) needs to be generated, e.g. using prepareRVE.py in +“example” subfolder for a 2D packing (similarly for 3D). +1. 2D drained biaxial compression test on dry dense sand (biaxialSmooth.py) Note: Test +description and result were presented in [Guo2014] and [Guo2014c]. +2. 2D passive failure under translational mode of dry sand retained by a rigid and fric- +tionless wall (retainingSmooth.py) Note: Rolling resistance model (CohFrictMat) is used in the +RVE packing. Test description and result were presented in [Guo2015]. +3. 2D half domain footing settlement problem with mesh generated by Gmsh (footing.py, +footing.msh) Note: Rolling resistance model (CohFrictMat) is used in the RVE packing. Six-node +triangle element is generated by Gmsh with three Gauss points each. Test description and result +were presented in [Guo2015]. +4. 3D drained conventional triaxial compression test on dry dense sand using MPI par- +allelism (triaxialRough.py) Note 1: The simulation is very time consuming. It costs ~4.5 days on +one node using multiprocessing (16 processes, 2.0 GHz CPU). When useMPI is switched to True +(as in the example script) and four nodes are used (80 processes, 2.2 GHz CPU), the simulation +costs less than 24 hours. The speedup is about 4.4 in our test. Note 2: When MPI is used, mpi4py +is required to be installed. The MPI implementation can be either MPICH or Open MPI. The file +“mpipool.py” should also be placed in the main folder. Our test is based on openmpi-1.6.5. This +is an on-going work. Test description and result will be presented later. +5. 2D globally undrained biaxial compression test on saturated dense sand with changing +permeability using MPI parallelism (undrained.py) Note: This is an on-going work. Test +description and result will be presented later. +4.3.7 Disclaim +This work extensively utilizes and relies on some third-party packages as mentioned above. Their con- +tributions are acknowledged. Feel free to use and redistribute the code. But there is NO warranty; not +even for MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. +4.3. +FEM-DEM hierarchical multiscale modeling with Yade and Escript +597 + +Yade Documentation, Release 3rd ed. +4.4 Simulating Acoustic Emissions in Yade +Suggested citations: +Caulk, R. (2018), Stochastic Augmentation of the Discrete Element Method for Investigation of Tensile +Rupture in Heterogeneous Rock. Yade Technical Archive. DOI 10.5281/zenodo.1202039. download full +text +Caulk, Robert A. (2020), Modeling acoustic emissions in heterogeneous rocks during tensile fracture +with the Discrete Element Method. Open Geomechanics, Volume 2, article no. 2, 19 p. doi : +10.5802/ogeo.5. full text +4.4.1 Summary +This document briefly describes the simulation of acoustic emissions (AE) in Yade. Yade’s clustered +strain energy based AE model follows the methods introduced by [Hazzard2000] and [Hazzard2013]. A +validation of Yade’s method and a look at the effect of rock heterogeneity on AE during tensile rock +failure is discussed in detail in [Caulk2018] and [Caulk2020]. +4.4.2 Model description +Numerical AE events are simulated by assuming each broken bond (or cluster of broken bonds) repre- +sents an event location. Additionally, the associated system strain energy change represents the event +magnitude. +Once a bond breaks, the strain energies (Ei) are summed for all intact bonds within a +predefined spatial radius (λ): +Ei = 1 +2 +�F2 +n +kn ++ F2 +s +ks +� +Eo = +N +� +i +Ei +where Fn, Fs and kn, ks are the normal and shear force (N) and stiffness (N/m) components of the +interaction prior to failure, respectively. Yade’s implementation uses the maximum change of strain +energy surrounding each broken bond to estimate the moment magnitude of the AE. As soon as the +bond breaks, the total strain energy (Eo = �N +i Ei) is computed for the radius (set by the user as no. +of avg particle diameters, λ. Eo is used as the reference strain energy to compute ∆E = E − Eo during +subsequent time steps. Finally, max(∆E) is used in the empirical equation derived by [Scholz2003]: +Me = 2 +3 log ∆E − 3.2 +Events are clustered if they occur within spatial and temporal windows of other events, similar to the +approach presented by [Hazzard2000] and [Hazzard2013]. The spatial window is simply the user defined +λ and the temporal window Tmax is computed as: +Tmax = int +� +Davgλ +max(vp1, vp2)∆t +� +where Davg is the average diameter of the particles comprising the failed event (m), vp1 and vp2 are the +P-Wave velocities (m/s) of the particle densities, and ∆t is the time step of the simulation (seconds/time +step). As shown in fig-cluster, the final location of a clustered event is simply the average of the clustered +event centroids. Here the updated reference strain energy is computed by adding the strain energy of +the unique interactions surrounding the new broken bond to the original reference strain energy (Eo): +• Original bond breaks, sum strain energy of broken bonds (Norig) within spatial window Eorig,o = +�Norig +i=1 +Ei +598 +Chapter 4. +Theoretical background and extensions + +Yade Documentation, Release 3rd ed. +• New broken bond detected within spatial and temporal window of original bond break +• Update reference strain Eo by adding unique bonds (Nnew) within new broken bond spatial window +Enew,o = Eorig,o + �Nnew +i=1 +Ei +This method maintains a physical reference strain energy for the calculation of ∆E = E − Enew,o and +depends strongly on the spatial window size. Ultimately, the clustering increases the number of larger +events, which yields more comparable b-values to typical Guttenberg Richter curves [Hazzard2013]. +Fig. 1: Example of clustered broken bonds (colored lines) and the final AE events (colored circles) with +their event magnitudes. +For a detailed look at the underlying algorithm, please refer to the source code. +4.4.3 Activating the algorithm within Yade +The simulation of AE is available as part of Yade’s Jointed Cohesive Frictional particle model (JCFpm) +. +As such, your simulation needs to make use of JCFpmMat , JCFpmPhys , and Law2_ScGeom_- +JCFpmPhys +Your material assignment and engines list might look something like this: +4.4. +Simulating Acoustic Emissions in Yade +599 + +0 +-0.04 +-0.02 +0 +0.03 +0.03 +0.02 +0.02 +0.01 +0.01 +αw) +αm) +-0.01 +-0.01 +-0.02 +-0.02 +8.70 +-0.03 +8980° +-0.03 +-0.04 +-0.02 +0 +0Yade Documentation, Release 3rd ed. +JCFmat = O.materials.append(JCFpmMat(young=young, cohesion=cohesion, +density=density, frictionAngle=radians(finalFricDegree), +tensileStrength=sigmaT, poisson=poisson, label='JCFmat', +jointNormalStiffness=2.5e6,jointShearStiffness=1e6,jointCohesion=1e6)) +O.engines=[ +ForceResetter(), +InsertionSortCollider([Bo1_Box_Aabb(),Bo1_Sphere_Aabb +,Bo1_Facet_Aabb()]), +InteractionLoop( +[Ig2_Sphere_Sphere_ScGeom(), Ig2_Facet_Sphere_ScGeom()], +[Ip2_FrictMat_FrictMat_FrictPhys(), +Ip2_JCFpmMat_JCFpmMat_JCFpmPhys( \ +xSectionWeibullScaleParameter=xSectionScale, +xSectionWeibullShapeParameter=xSectionShape, +weibullCutOffMin=weibullCutOffMin, +weibullCutOffMax=weibullCutOffMax)], +[Law2_ScGeom_JCFpmPhys_JointedCohesiveFrictionalPM(\ +recordCracks=True, recordMoments=True, +Key=identifier,label='interactionLaw'), +Law2_ScGeom_FrictPhys_CundallStrack()] +), +GlobalStiffnessTimeStepper(), +VTKRecorder(recorders=['jcfpm','cracks','facets','moments'] \ +,Key=identifier,label='vtk'), +NewtonIntegrator(damping=0.4) +] +Most of this simply enables JCFpm as usual, the AE relevant commands are: +Law2_ScGeom_JCFpmPhys_JointedCohesiveFrictionalPM(... +recordMoments=True ...) +VTKRecorder(... recorders=[... 'moments' ...]) +There are some other commands necessary for proper activation and use of the acoustic emissions algo- +rithm: +clusterMoments +tells Yade to cluster new broken interactions within the user set spatial radius as +described above in the model description. This value is set to True by default. +momentRadiusFactor +is λ from the above model description. The momentRadiusFactor changes the +number of particle radii beyond the initial interaction that Yade computes the strain energy change. +Additionally, Yade uses λ to seek additional broken bonds for clustering. This value is set to 5 by default +( [Hazzard2013] concluded that this value yields accurate strain energy change approximations for the +total strain energy change of the system entire system). +neverErase +allows old interactions to be stored in memory despite no longer affecting the simulation. +This value must be set to True for stable operation of Yade’s AE cluster model. +4.4.4 Visualizing and post processing acoustic emissions +AE are visualized and post processed in a similar manner to JCFpm cracks. +As long as recordMo- +ments=True +and recorder=[‘moments’] , the simulation will produce timestamped .vtu files for easy +Paraview post processing. Within Paraview, the AE can be filtered according to magnitude, number of +constitiuent interactions, and event time. fig-aeexample shows AE collected during a three point bending +test and filtered according to magnitude and time +600 +Chapter 4. +Theoretical background and extensions + +Yade Documentation, Release 3rd ed. +Fig. 2: Example of AE simulated during three point bending test and filtered by magnitude and time. +4.4.5 Consideration of rock heterogeneity +[Caulk2018] and [Caulk2020] hypothesize that heterogeneous rock behavior depends on the distribution +of interacting grain edge lengths. In support of the hypothesis, [Caulk2018] and [Caulk2020] show how +rock heterogeneity can be modeled using cathodoluminescent grain imagery. A Weibull distribution is +constructed based on the so called grain edge interaction length distribution. In Yade’s JCFpm , the +Weibull distribution is used to modify the interaction strengths of contacting particles by correcting the +interaction area Aint: +Aint = π(αw × min(Ra, Rb))2 +where αw is the Weibull correction factor, which is distributed as shown in fig-weibullDist. The corre- +sponding tensile strength distributions for various Weibull shape parameters are shown in fig-strengthDist. +Note: a Weibull shape factor of ∞ is equivalent to the unaugmented JCFpm model. +In Yade, the application of rock heterogeneity is as simple as passing a Weibull shape parameter to +JCFpmPhys : +Ip2_JCFpmMat_JCFpmMat_JCFpmPhys( +xSectionWeibullScaleParameter=xSectionScale, +xSectionWeibullShapeParameter=xSectionShape, +weibullCutOffMin=weibullCutOffMin, +weibullCutOffMax=weibullCutOffMax) +where the xSectionWeibullShapeParameter is the desired Weibull shape parameter. The scale parameter +can be assigned in similar fashion. If you want to control the minimum allowable correction factor, you +can feed it weibullCutoffMin . The maximum correction factor can be controlled in similar fashion. +4.5 Using YADE 1D vertical VANS fluid resolution +The goal of the present note is to detail how the DEM-fluid coupling can be used in practice in YADE. +It is complementary with the three notes [Maurin2018_VANSbasis], [Maurin2018_VANSfluidResol] and +[Maurin2018_VANSvalidations] detailing respectively the theoretical basis of the fluid momentum bal- +ance equation, the numerical resolution, and the validation of the code. +All the coupling and the fluid resolution relies only on the engine HydroForceEngine, which use is detailed +here. Examples scripts using HydroForceEngine for different purposes can be found in YADE source +code in the folder trunk/examples/HydroForceEngine/. In order to get familiar with this engine, it is +recommended to read the present note and test/modify the examples scripts. +4.5. +Using YADE 1D vertical VANS fluid resolution +601 + +0.12 +0.1 +0.08 +0.06 +-0.1 +-0.12 +0.02 +0.02 +-8 +Height (m)0 +OHeight (m) +-10 +-0.02 +-0.02 +-1.1e+01 +Z +-0.04 +0.04 +0.12 +0.1 +0.08 +0.06 +0.040.02 +nQ.02-0.04-0.06-0.08 +3-0.1 +-0.12Yade Documentation, Release 3rd ed. +Fig. 3: Weibull distributions for varying shape parameters used to generate αw. +Fig. 4: Maximum DEM particle bond tensile strength distributions for varying Weibull shape parameters. +602 +Chapter 4. +Theoretical background and extensions + +3.0 +Normalized frequency of occurence +8 +2.5 +2.0 +9 +1.5 +1.0 +2 +0.5 +1 +0.0 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +Interaction area correction factor αw (-)900 +Weibull shape param. +800 +Frequency of occurrence +X +700 +600 +500 +400 +300 +200 +100 +0 +100 +200 +300 +400 +500Yade Documentation, Release 3rd ed. +4.5.1 DEM-fluid coupling and fluid resolution in YADE +In YADE, the fluid coupling with the DEM is done through the engine called HydroForceEngine, which is +coded in the source in the files trunk/pkg/common/HydroForceEngine.cpp and hpp. HydroForceEngine +has three main functions: +• It applies drag and buoyancy to each particle from a 1D vertical fluid velocity profile (Hydro- +ForceEngine::action) +• It can evaluates the average drag force, particle velocity and solid volume fraction profiles (Hydro- +ForceEngine::averageProfile) +• It can solves the fluid velocity equation detailed in the first section, from given average drag force, +particle velocity and solid volume fraction profiles (HydroForceEngine::fluidResolution) +We clearly see the link between the three functions. +The idea is to evaluate the average profiles +from the DEM, put it as input to the fluid resolution, and apply the fluid forces corresponding to +the obtained fluid velocity profile to the particles. +In the following, the three points will be de- +tailed separately with precision and imaging with the example scripts available in yade source code +at trunk/examples/HydroForceEngine/. +4.5.2 Application of drag and buoyancy forces (HydroForceEngine::action) +By default, when adding HydroForceEngine to the list of engine, it applies drag and buoyancy to all the +particles which IDs have been passed in argument to HydroForceEngine through the ids variable. This +is done for example, in the example script trunk/examples/HydroForceEngine/, in the engine lists: +O.engines = [ +ForceResetter(), +... +HydroForceEngine(densFluid = densFluidPY,...,ids = idApplyForce), +... +NewtonIntegrator(gravity=gravityVector, label='newtonIntegr') +] +where idApplyForce corresponds to a list of particle ID to which the hydrodynamic forces should be +applied. The expression of the buoyancy and drag force applied to the particles contained in the id list +is detailed below. +In case where the fluid is at rest (HydroForceEngine.steadyFlow = False), HydroForceEngine applies +buoyancy on a particle p from the fluid density and the acceleration of gravity g as: +fp +b = −ρfVpg. +Meanwhile, if the fluid flow is steady and turbulent, the buoyancy which is related to the fluid pressure +gradient does not have a term in the streamwise direction (see discussion p. 5 of [Maurin2018]). Puting +the option HydroForceEngine.steadyFlow to True turns the expression of the buoyancy into: +fp +b = −ρfVp(g.ex)ex. +Also, HydroForceEngine applies a drag force to each particles contained in the ids list. This drag force +depends on the velocity of the particles and on the fluid velocity, which is defined by a 1D fluid velocity +profile, HydroForceEngine.vxFluid. This fluid velocity profile can be evaluated from the fluid model, but +can also be imposed by the user and stay constant. From this 1D vertical fluid velocity profile, the drag +force applied to particle p reads: +fp +D = 1 +2CdAρf||uf +pex − vp|| +� +uf +pex − vp� +, +where uf +p is the fluid velocity at the center of particle p, vp is the particle velocity, ρf is the fluid density, +A = πd2/4 is the area of the sphere submitted to the flow, and Cd is the drag coefficient accounts for the +4.5. +Using YADE 1D vertical VANS fluid resolution +603 + +Yade Documentation, Release 3rd ed. +effects of particle Reynolds number [Dallavalle1948] and of increased drag due to the presence of other +particles (hindrance, [Richardson1954]: +Cd = +� +0.44 + 24 +Rep +� +(1 − φp)−γ = +� +0.44 + 24 +νf +||ufpex − vp||d +� +(1 − φp)−γ +with φp the solid volume fraction at the center of the particle evaluated from HydroForceEngine.phiPart, +and γ the Richardson-Zaki exponent, which can be set through the parameter HydroForceEngine.expoRZ +(3.1 by default). +HydroForceEngine can also apply a lift force, but this is not done by default (HydroForceEngine.lift = +False), and this is not recommended by the author considering the uncertainty on the actual formulation +(see discussion p. 6 of [Maurin2015] and [Schmeeckle2007]). +As the fluid velocity profile (HydroForceEngine.vxFluid) and solid volume fraction profile (Hy- +droForceEngine.phiPart) can be imposed by the user, +the application of drag and buoyancy +to the particles through HydroForceEngine can be done without using the function average- +Profile and the fluid resolution. +Examples of such use can be found in the source code: +trunk/examples/HydroForceEngine/oneWayCouplingfootnote{In this case, we talk about a one-way cou- +pling as the fluid influence the particles but is not influenced back}. +4.5.3 Solid phase averaging (HydroForceEngine::averageProfile) +In order to solve the fluid equation, we have seen that it is necessary to compute from the DEM +the solid volume fraction, the solid velocity, and the averaged drag profiles. +The function Hydro- +ForceEngine.averageProfile() has been set up in order to do so. It is designed to evaluate the average +profiles over a regular grid, at the position between two mesh nodes. In order to match the fluid velocity +profile numerotation, the averaged vector are of size ndimz + 1 even though the quantities at the top +and bottom boundaries are not evaluated and set to zero by defaultfootnote{It is not necessary to eval- +uate the solid DEM quantities at the boundaries are they are not considered in the fluid resolution, see +subsection boundaries of [Maurin2018_VANSfluidResol]}. textcolor{red}{You should do that} +The solid volume fraction profile is evaluated by considering the volume of particles contained in the +layer considered. The layer is defined by the mesh step along the wall-normal direction, but extend +over the whole length and width of the sample. We perform such an averaging only discretized over +the wall-normal direction in order to match the fluid resolution. Meanwhile, this is also physical as, at +steady state the problem is unidirectional on average, so that the only variation we should observe in +the measured averaged quantities should be along the vertical direction, z. Therefore, the solid volume +fraction is evaluated by considering the volume of particles which is contained inside the layer considered +i + 1/2: +φi+1/2 = +� +p∈[idz;(i+1)dz] +Vp +i+1/2; +where the sum is over the particles p which have at least a part of their volume inside the layer i + 1/2, +i.e. in between an elevation of i∗dz and (i+1)∗dz, and Vp +i+1/2 is the volume of the particles considered +which is contained inside the layer considered. The latter correspond to the integral between two points +of a slice of sphere and can be evaluated analytically in cylindrical coordinate. Following this formulation +and the formalism of [Jackson2000] with a weighting step function, any particle-associated quantity K +can be averaged with the following formulation: +⟨K⟩p�� +i+1/2 = +� +p∈[idz;(i+1)dz] Vp +i+1/2Kp +� +p∈[idz;(i+1)dz] Vp +i+1/2 +, +Where Kp is the quantity associated with particle p, e.g. the particle streamwise velocity. In this case, +we can write: +⟨vx⟩p|i+1/2 = +� +p∈[idz;(i+1)dz] Vp +i+1/2vp +x +� +p∈[idz;(i+1)dz] Vp +i+1/2 +, +604 +Chapter 4. +Theoretical background and extensions + +Yade Documentation, Release 3rd ed. +where vp +x is the velocity of particle p. Regarding the evaluation of the average streamwise drag force +transmitted by the fluid to the particles, it can be written similarly as: +⟨fD,x⟩p|i+1/2 = +� +p∈[idz;(i+1)dz] Vp +i+1/2fp +D,x +� +p∈[idz;(i+1)dz] Vp +i+1/2 +, +where fp +D,x is the drag force on particle p. +As will be detailed in the next part, these averaged profile can be used for the fluid resolution, but they +can also be used for analysis as done for example for bedload transport in [Maurin2015b] [Maurin2018]. +4.5.4 Fluid resolution\HydroForceEngine::fluidResolution +In order to use the fluid resolution inside the fluid-DEM coupling framework, it is necessary to call +the function HydroForceEngine.averageProfile() in order to evaluate the averaged solid volume fraction +profile, streamwise velocity and streamwise drag force. The latter is necessary in order to evaluate the +terms β taken into account in the fluid equation (see [Maurin2018_VANSfluidResol] for details). β is +defined as: +n +� +ff +x +�p��� +i+1/2 = βi+1/2 +� +⟨ux⟩f��� +i+1/2 − ⟨vx⟩p�� +i+1/2 +� +so that it can be evaluated directly from the averaged drag, particle velocity and the fluid velocity at +the last iteration (explicited the term β in the fluid resolution): +βn +i+1/2 = +n +� +ff +x +�p��� +n−1 +i+1/2 +⟨ux⟩f��� +n−1 +i+1/2 − ⟨vx⟩p��n−1 +i+1/2 +where the solid variables have been denoted with a superscript n − 1 as they are known and not re- +evaluated at each time stepfootnote{In a way βn should probably be better written as βn−1}. This +terms is called taufsi and is directly evaluated inside the code. +i-1 +i +i+1 +i+2 +i-2 +i+1/2 +i+3/2 +i-1/2 +Quantities evaluated +in the DEM +Fig. 5: Schematical picture of the numerical fluid resolution and variables definition with a regular mesh. +All the definitions still holds for a mesh with variable spatial step. +All +the +quantities +needed +in +order +to +solve +the +fluid +resolution +- +highlighted +in +[Maurin2018_VANSfluidResol] and recalled in figure fig-scheme - are now explicited. +They can +be directly evaluated in YADE with the function HydroForceEngine.averageProfile(). From there, the +fluid resolution can be performed over a given time tresol with a given time step ∆t by calling directly +4.5. +Using YADE 1D vertical VANS fluid resolution +605 + +Yade Documentation, Release 3rd ed. +the function HydroForceEngine.fluidResolution (tresol,∆t). +This will perform the fluid resolution +described in [Maurin2018_VANSfluidResol], N = tresol/∆t times, with a time step ∆t, considering the +vertical profiles of β, ⟨vx⟩ and φ as constant in time. Therefore, one should not only be carefull about +the time step, but also about the period of coupling, which should not be too large in order to avoid +unphysical behavior in the DEM due to a drastic change of velocity profile not compensated by an +increased transmitted drag force. +In the example script in YADE source code, trunk/examples/HydroForceEngine/twoWayCoupling/sedimentTransportExample_- +1DRANSCoupling.py, the DEM and fluid resolution are coupled with a period of fluidResolPeriod = +10−2s by default, and with a fluid time step of dtFluid = 10−5s. This means that the DEM is let evolved +for 10−2s, and frozen during the fluid resolution which is made over fluidResolPeriod/dtFluid = 103 +step with ∆t = 10−5. Then, the DEM is let evolved again but with a new fluid velocity profile for 10−2s, +and frozen…etc. This period between two fluid resolution should be tested and taken not too long (see +appendix of [Maurin2015b]). +Meanwhile, the fluid resolution can be used in itself, without DEM coupling, in particular to ver- +ify the fluid resolution in known cases. +This is done in the example folder of YADE source code, +trunk/examples/HydroForceEngine/fluidValidation/, where the cases of a poiseuille flow and a log layer +have been considered and validated. +4.6 Potential Particles and Potential Blocks +The origins of scientific development regarding the algorithms described in this section are traced back +to: [Boon2012] (Potential Blocks code), [Boon2013b] (Potential Particles code) and [Boon2015] (Block +Generation code). +4.6.1 Introduction +This section discusses two codes to simulate (i) non-spherical particles using the concept of the Potential +Particles [Houlsby2009], with the solution procedures in [Boon2013] for 3-D and (ii) polyhedral blocks +using planar linear inequalities, based on linear programming concepts [Boon2012]. These codes define +two shape classes in YADE, namely PotentialParticle and PotentialBlock. Besides some similarities in +syntax, they have distinct differences, concerning morphological characteristics of the particles and the +methods used to facilitate contact detection. +The Potential Particles code (abbreviated herein as PP) is detailed in [Boon2013], where non-spherical +particles are assembled as a combination of 2nd degree polynomial functions and a fraction of a sphere, +while their edges are rounded with a user-defined radius of curvature. +The Potential Blocks code (abbreviated herein as PB) is used to simulate polyhedral particles with flat +surfaces, based on the work of [Boon2012], where a smooth, inner potential particle is used to calculate +the contact normal vector. This code is compatible with the Block Generation algorithm defined in +[Boon2015], in which Potential Blocks can be generated by intersections of original, intact blocks with +discontinuity planes. +These two codes are independent, in the sense that either one of them can be compiled/used separately, +without enabling the other, while they do not interact with each other (i.e. we cannot establish contact +between a PP and a PB). Enabling the PB code causes an automatic compilation of the Block Generation +algorithm. +4.6.2 Potential Particles code (PP) +The concept of Potential Particles was introduced and developed by [Houlsby2009]. The problem of +contact detection between a pair of potential particles was cast as a constrained optimization problem, +where the equations are solved using the Newton-Raphson method in 2-D. In [Boon2013] it was extended +to 3-D and more robust solutions were proposed. Many numerical optimization solvers generally can- +not cope with discontinuities, ill-conditioned gradients (Jacobians) or curvatures (Hessians), and these +606 +Chapter 4. +Theoretical background and extensions + +Yade Documentation, Release 3rd ed. +obstacles were overcome in [Boon2013], by re-formulating the problem and solving the equations using +conic optimization solvers. Previous versions made use of MOSEK (using its academic licence), while +currently an in-house code written by [Boon2013] is used to solve the conic optimization problem. A +potential particle is defined as in (4.21) [Houlsby2009]: +f = (1 − k) +� N +� +i=1 +⟨aix + biy + ciz − di⟩2 − r2 +� ++ k(x2 + y2 + z2 − R2) +(4.21) +where (ai, bi, ci) is the normal vector of the ith plane, defined with respect to the particle’s local +coordinate system and di is the distance of the plane to the local origin. ⟨ ⟩ are Macaulay brackets, i.e., +〈x〉 = x for x > 0; ⟨x⟩ = 0 for x ≤ 0. The planes are assembled such that their normal vectors point +outwards. They are summed quadratically and expanded by a distance r, which is also related to the +radius of the curvature at the corners. Furthermore, a “shadow” spherical particle is added; R is the +radius of the sphere, with 0 < k ≤ 1, denoting the fraction of sphericity of the particle. The geometry +of some cuboidal potential particles is displayed in Fig. fig-pp, for different values of the parameter k. +The potential function is normalized for computational reasons in the form (4.22) [Houlsby2009]: +f = (1 − k) +� N +� +i=1 +⟨aix + biy + ciz − di⟩2 +r2 +− 1 +� ++ k +� +x2 + y2 + z2 +R2 +− 1 +� +(4.22) +This potential function takes values: +• f = 0: on the particle surface +• f < 0: inside the particle +• f > 0: outside the particle +To ensure numerical stability, it is not advised to use values approaching k=0. In particular, the extreme +value k=0 cannot be used from a theoretical standpoint, since the Potential Particles were formulated +for strictly convex shapes (curved faces). +4.6.3 Potential Blocks code (PB) +The Potential Blocks code was developed during the D.Phil. +thesis of CW Boon [Boon2013b] and +discussed in [Boon2012]. It was developed originally for rock engineering applications, to model polygonal +and polyhedral blocks with flat surfaces. The blocks are defined with linear inequalities only and unlike +the PotentialParticle shape class, no spherical term is considered (so, practically k=0). Although k and +R are input parameters of the PotentialBlock shape class, their existence during computation is null. In +particular, R is used within the source code, denoting a characteristic dimension of the blocks, but does +not reflect the radius of a “shadow particle”, like it does for the Potential Particles. This value of R is +used in the Potential Blocks code to calculate the initial bi-section step size for line search, to obtain a +point on the particle, which in turn is used to calculate the overlap distance during contact. +For a convex particle defined by N planes, the space that it occupies can be defined using the following +inequalities (4.23): +aix + biy + ciz ≤ di, i = 1 : N +(4.23) +where (ai, bi, ci) is the unit normal vector of the ith plane, defined with respect to the particle’s local +coordinate system, and di is the distance of the plane to the local origin. According to [Boon2012], an +inner, smooth potential particle is used to calculate the contact normal, formulated as in (4.24): +4.6. +Potential Particles and Potential Blocks +607 + +Yade Documentation, Release 3rd ed. +Fig. 6: Construction of potential particles (a) constituent planes are squared and expanded by a constant +r. A fraction of sphere is added. Particles with the spherical term are visible in (b) k=0.9, (c) k=0.7, +and (d) k=0.4 (after [Boon2013]). +608 +Chapter 4. +Theoretical background and extensions + +(a) +(b) Before adding +After adding +constituent planes +spherical term +spherical term +(c) +(d) +Before adding +After adding +Before adding +After adding +spherical term +spherical,term +spherical term +spherical termYade Documentation, Release 3rd ed. +f = +N +� +i=1 +⟨aix + biy + ciz − di + r⟩2 +(4.24) +This potential particle is defined inner by a distance r inside the actual particle, with edges rounded by +a radius or curvature r, as well (see Fig. fig-pbInner). +Fig. 7: A potential particle is defined inside the actual particle. The normal vector of the particle at any +point can be calculated from the first derivative of the potential particle. (after [Boon2012]). +In YADE, the Potential Blocks have a slightly different mathematical expression, since their shape is +generated as an assembly of planes as in (4.25): +aix + biy + ciz − di − r = 0, i = 1 : N +(4.25) +while the inner Potential Particle used to calculate the contact normal is defined as in (4.26): +f = +N +� +i=1 +⟨aix + biy + ciz − di⟩2. +(4.26) +Now, the Potential Block surface is at a distance of (di +r) from the local particle center, while the inner +potential particle is at a distance d from the local particle center. +It is worth to emphasize on the fact that the shape of a Potential Block is defined using an assembly of +planes and not a single, implicit potential function, like we have for the Potential Particles code. The +inner potential particle in the Potential Blocks code is only used to calculate the contact normal. +The problem of establishing intersection between a pair of blocks is cast as a standard linear programming +problem of finding a feasible region which satisfies all the linear inequalities defining both blocks. The +contact point is calculated as the analytic centre of the feasible region, a well-known concept of interior- +point methods in convex optimization calculations. The contact normal is obtained from the gradient +of a smooth “potential particle” defined inside the block. The overlap distance is calculated through +bi-section searching along the contact normal, within the overlap region. +The linear programming solver for Potential Blocks was originally CPLEX, but has been updated to +CLP, developed by COIN-OR, since the latter can be downloaded from Ubuntu or Debian’s distributions +without requiring an academic licence. +4.6.4 Engines +The PP and PB codes use their own classes to handle bounding volumes, contact geometry & physics and +recording of outputs in vtk format, while they derive the interparticle friction angle from the frictional +material class FrictMat. The syntax used to invoke these classes is similar, unless if specified otherwise. +4.6. +Potential Particles and Potential Blocks +609 + +Actualparticle +Inner potential particleYade Documentation, Release 3rd ed. +Fig. 8: A potential block. The normal vectors of the faces point outwards (after [Boon2013b]). +Shape +PotentialParticle +PotentialBlock +Material +FrictMat +FrictMat +BoundFunctor +PotentialParticle2AABB +PotentialBlock2AABB +IGeom +ScGeom +ScGeom +IGeomFunctor +Ig2_PP_PP_ScGeom +Ig2_PB_PB_ScGeom +IPhys +KnKsPhys +KnKsPBPhys +IPhysFunctor +Ip2_FrictMat_FrictMat_KnKsPhys +Ip2_FrictMat_FrictMat_KnKsPBPhys +LawFunctor +Law2_SCG_KnKsPhys_KnKsLaw +Law2_SCG_KnKsPBPhys_KnKsPBLaw +VTK Recorder +PotentialParticleVTKRecorder +PotentialBlockVTKRecorder +A simple simulation loop using the Potential Blocks reads as: +O.engines=[ +ForceResetter(), +InsertionSortCollider([PotentialBlock2AABB()], verletDist=0.01), +InteractionLoop( +[Ig2_PB_PB_ScGeom(twoDimension=True, unitWidth2D=1.0, calContactArea=True)], +[Ip2_FrictMat_FrictMat_KnKsPBPhys(kn_i=1e8, ks_i=1e7, Knormal=1e8, Kshear=1e7,␣ +�→viscousDamping=0.2)], +[Law2_SCG_KnKsPBPhys_KnKsPBLaw(label='law', neverErase=False,␣ +�→allowViscousAttraction=False)] +), +NewtonIntegrator(damping=0.2, exactAsphericalRot=True, gravity=[0,0,-9.81]), +PotentialBlockVTKRecorder(fileName='./vtk/file_prefix', iterPeriod=1000,␣ +�→twoDimension=True, sampleX=30, sampleY=30, sampleZ=30, maxDimension=0.2, label='vtkRecorder') +] +Attention should be given to the twoDimension parameter, which defines whether a contact should be +handled as 2-D or 3-D. +4.6.5 Contact Law +In both codes, the normal force is calculated as: +Fn = Knormal · Ac · un · n +(4.27) +where Knormal the normal stiffness coefficient [kN/m3]; Ac the contact area [m2] and un the overlap +distance. The normal stiffness of each contact [kN/m] is thus kn = Knormal · Ac, where Ac is updated +in every timestep. +610 +Chapter 4. +Theoretical background and extensions + +Yade Documentation, Release 3rd ed. +The shear force is calculated incrementally, using a similar logic. The increment of the shear force vector +before slipping of the contact is calculated as: +∆Fs = −Kshear · Ac · ∆us +(4.28) +where Kshear the shear stiffness coefficient [kN/m3] and ∆us the current relative shear displacement. +Contact Area +The contact area is calculated using a heuristic algorithm to detect points on the surface of the overlap +volume, searching along the contact shear direction. In essence, it is calculated as the area of a 2D slice +of the overlap volume along the shear direction, passing from the contact point. If twoDimension=True, +the contactArea parameter is calculated as: +if(twoDimension) { phys->contactArea = phys->jointLength*unitWidth2D;} +The unitWidth2D parameter is defined by the user (usually equal to 1.0), denoting the out-of-plane width +in 2-D simulations. The contactArea and jointLength parameters are calculated if calContactArea =True. +In the opposite case, they are considered equal to 1.0 and the contact law is degenerated to a linear law +with constant stiffness. A minimum value is considered for the contactArea, to represent cases where the +overlap volume is practically a point. +Overlap distance +The overlap distance un is calculated using a bracketed bisection search algorithm along the contact +normal direction, to find two opposite points on the surface of the overlap region, starting from the +contact point. It is stored in the parameter penetrationDepth, as the distance between these two opposite +points. +4.6.6 Shape definition of a PP and a PB +A strong merit of the Potential Particles and the Potential Blocks codes lies in the fact that the geometric +definition of the particle shape and the contact detection problem are resolved using only the equations of +the faces of the particles. In this way, using a single data structure, there is no need to store information +about the vertices or their connectivity to establish contact, a feature that makes them computationally +affordable, while all contacts are handled in the same way (there is no need to distinguish among face- +face, face-edge, face-vertex, edge-edge, edge-vertex or vertex-vertex contacts). Due to this, the geometry +of a particle is defined in the shape class using the values of the normal vectors of the faces and the +distances of the faces from the local origin. +For example, to define a cuboid (6 faces) with rounded edges, an edge length of D, centred to its local +centroid and aligned to its principal axes, using the Potential Particles code, we set: +r=D/10. +k=0.3 +R=D/2. +b=Body() +b.shape=PotentialParticle( r=r, k=k, R=R, +a=[ +1.0, +-1.0, +0.0, +0.0, +0.0, +0.0], +b=[ +0.0, +0.0, +1.0, +-1.0, +0.0, +0.0], +c=[ +0.0, +0.0, +0.0, +0.0, +1.0, +-1.0], +d=[D/2.-r, +D/2.-r, +D/2.-r, +D/2.-r, +D/2.-r, +D/2.-r], ...) +The first element of the vector parameters a, b, c, d refers to the normal vector of the first plane and its +distance from the local origin, the second element to the second plane, and so on. +4.6. +Potential Particles and Potential Blocks +611 + +Yade Documentation, Release 3rd ed. +Using the Potential Particles code, this is not a perfect cube, since the particle geometry is defined by +a potential function as in (4.22). It is reminded that within this potential function, these planes are +summed quadratically, the particle edges are rounded by a radius of curvature r and then the particle +faces are curved by the addition of a “shadow” spherical particle with a radius R, to a percentage defined +by the parameter k. A value r is deducted from each element of the vector parameter d, to compensate +for expanding the potential particle by r. +The parameters ai, bi, ci, di stated above correspond to the planes used in (4.25): +1.0x + 0.0y + 0.0z = D/2 ⇔ +x = D/2 +−1.0x + 0.0y + 0.0z = D/2 ⇔ −x = D/2 +0.0x + 1.0y + 0.0z = D/2 ⇔ +y = D/2 +0.0x − 1.0y + 0.0z = D/2 ⇔ −y = D/2 +0.0x + 0.0y + 1.0z = D/2 ⇔ +z = D/2 +0.0x + 0.0y − 1.0z = D/2 ⇔ −z = D/2 +To model a cube with an edge of D, using the Potential Blocks code, we define: +r=D/10. +R=D/2.*sqrt(3) +b=Body() +b.shape=PotentialBlock( r=r, R=R, +a=[ +1.0, +-1.0, +0.0, +0.0, +0.0, +0.0], +b=[ +0.0, +0.0, +1.0, +-1.0, +0.0, +0.0], +c=[ +0.0, +0.0, +0.0, +0.0, +1.0, +-1.0], +d=[D/2.-r, +D/2.-r, +D/2.-r, +D/2.-r, +D/2.-r, +D/2.-r], ...) +Using the Potential Blocks code, this particle will have sharp edges and flat faces in what regards its +geometry (i.e. the space it occupies), defined by the given planes, while for the calculation of the contact +normal, an inner potential particle with rounded edges is used, formulated as in (4.26), located fully inside +the actual particle. The distances of the planes from the local origin, stored in the vector parameter d, +are reduced by r to achieve an exact edge length of D, using (4.25). The value of r must be sufficiently +small, so that dmin − r > 0, while it should be sufficiently large, to allow for a proper calculation of the +gradient of the inner Potential Particle at the contact point. A recommended value is r ≈ 0.5 ∗ dmin. +To ensure numerical stability, it is advised to normalize the normal vector of each plane, so that ai2 + +bi +2 + ci2 = 1. There is no limit to the number of the particle faces that can be used, a feature that +allows the modelling of a variety of convex particle shapes. +In practice, it is usual for the geometry of a particle to be given in terms of vertices & their connectivity +(e.g. in the form of a surface mesh, like in .stl files). In such cases, the user can calculate the normal +vector of each face, which will give the coefficients ai, bi, ci and using a vertex of each face, then calculate +the coefficients di. A python routine to perform this without any additional effort by the user is currently +being developed. +4.6.7 Body definition of a PP and a PB +To define a body using the PotentialParticle or PotentialBlock shape classes, it has to be assembled using +the _commonBodySetup function, which can be found in the file py/utils.py. For example, to define a +PotentialParticle: +O.materials.append(FrictMat(young=-1,poisson=-1,frictionAngle=radians(0.0),density=2650,label= +�→'frictionless')) +b=Body() +b.shape=PotentialParticle(...) +b.aspherical=True # To be used in conjunction with exactAsphericalRot=True in the␣ +�→NewtonIntegrator +(continues on next page) +612 +Chapter 4. +Theoretical background and extensions + +Yade Documentation, Release 3rd ed. +(continued from previous page) +# V: Volume +# I11, I22, I33: Principal inertias +utils._commonBodySetup(b,V,Vector3(I11,I22,I33), material='frictionless', pos=(0,0,0),␣ +�→fixed=False) +b.state.pos=Vector3(xPos,yPos,zPos) +b.state.ori=Quaternion((random.random(),random.random(),random.random()),random.random()) +b.shape.volume=V; +O.bodies.append(b) +The PotentialParticle must be initially defined, so that the local axes coincide with its principal axes, +for which the inertia tensor is diagonal. More specifically, the plane coefficients (ai, bi, ci) defining the +plane normals must be rotated, so that when the orientation of the particle is zero, the PotentialParticle +is oriented to its principal axes. +It should be noted that the principal inertia values I11, I22, I33 mentioned here are divided with +the density of the considered material, since they are multiplied with the density inside the _- +commonBodySetup function. The mass of the particle is calculated within the same function as well, +so we do not need to set manually b.mass=V*density. +For the Potential Particles, the volume and inertia must be calculated manually and assigned to the +body as demonstrated above. For the Potential Blocks, an automatic calculation has been implemented +for the volume and inertia tensor, the user does not have to define the particle to its principal axes, since +this is handled automatically within the source code, while if no value is given for the parameter R, it is +calculated as half the distance of the farthest vertices. +For example, to define a PotentialBlock: +O.materials.append(FrictMat(young=-1,poisson=-1,frictionAngle=radians(0.0),density=2650,label= +�→'frictionless')) +b=Body() +b.shape=PotentialBlock(R=0.0, ...) #here we set R=0.0 to trigger automatic calculation of R +b.aspherical=True # To be used in conjunction with exactAsphericalRot=True +utils._commonBodySetup(b,b.shape.volume,b.shape.inertia, material='frictionless',␣ +�→pos=Vector3(xPos,yPos,zPos), fixed=False) +b.state.ori=b.shape.orientation # this will rotate the particle to its initial random system.␣ +�→If b.state.ori=Quaternion.Identity, the PB is oriented to its principal axes +O.bodies.append(b) +4.6.8 Boundary Particles +The PP & PB codes support the definition of boundary particles, which interact only with non-boundary +ones. These particles can have a variety of uses, e.g. to model loading plates acting on a granular sample, +while different uses can emerge for different applications. A particle can be set as a boundary one in +both codes, using the boolean parameter isBoundary inside the shape class. +In the PP code, all particles interact with the same normal and shear contact stiffness Knormal and +Kshear, defined in the Ip2_FrictMat_FrictMat_KnKsPhys functor. +The PB code supports the definition of different contact stiffness values for interactions between boundary +and non-boundary or non-boundary and non-boundary particles. When isBoundary=False, the Poten- +tialBlock in question is handled to interact with normal and shear stiffness coefficients Knormal and +Kshear, respectively, with other non-boundary particles. When isBoundary=True, the PotentialBlock in +question is handled to interact with normal and shear stiffness coefficients kn_i and ks_i, respectively, +with non-boundary particles. +4.6. +Potential Particles and Potential Blocks +613 + +Yade Documentation, Release 3rd ed. +4.6.9 Visualization +Visualization of the PotentialParticle and PotentialBlock shape classes is offered using the qt environ- +ment (OpenGL). Additionally, the export.VTKExporter.exportPotentialBlocks function and Potential- +ParticleVTKRecorder and PotentialBlockVTKRecorder engines can be used to export geometrical and +interaction information of the analyses in vtk format (visualized in Paraview). It should be noted that +currently the PotentialBlockVTKRecorder records a rounded approximation of the particle, rather than +the actual particle with sharp corners and edges. +In the qt environment, the PotentialParticle shape class is visualized using the Marching Cubes algorithm, +and the level of display accuracy can be determined by the user. This is controlled by the parameters: +# Potential Particles +Gl1_PotentialParticle.sizeX=20 +Gl1_PotentialParticle.sizeY=20 +Gl1_PotentialParticle.sizeZ=20 +A similar choice exists for output in vtk format, using the PotentialParticleVTKRecorder or Potential- +BlockVTKRecorder, syntaxed as: +# Potential Particles +PotentialParticleVTKRecorder(sampleX=30, sampleY=30, sampleZ=30, maxDimension=20) +# Potential Blocks +PotentialBlockVTKRecorder(sampleX=30, sampleY=30, sampleZ=30, maxDimension=20) +The parameters sizeX,Y,Z (for OpenGL visualization) and sampleX,Y,Z (for output in vtk format) +represent the number of subdivisions of the Aabb of the particle to a grid, which will be used to draw +its geometry, in respect to the global axes X, Y, Z. Larger values will result to a more accurate display +of the particles’ shape, but will slow down the visualization speed in qt and writing speed of the .vtk +files and increase the size of the .vtk files. For output in vtk format, users can also define the parameter +maxDimension, which overrides the selected sampleX,Y,Z values if they are too small, as described below: +if | xmax − xmin | /sampleX > maxDimension ⇒ sampleX =| xmax − xmin | /maxDimension +if | ymax − ymin | /sampleY > maxDimension ⇒ sampleY =| ymax − ymin | /maxDimension +if | zmax − zmin | /sampleZ > maxDimension ⇒ sampleZ =| zmax − zmin | /maxDimension +The PotentialParticleVTKRecorder and PotentialBlockVTKRecorder also support optionally the record- +ing of the particles’ velocities (linear and angular), interaction information (contact point and forces), +colors and ids, using: +# Potential Particles +PotentialParticleVTKRecorder(..., REC_VELOCITY=True, REC_INTERACTION=True, REC_COLORS=True,␣ +�→REC_ID=True) +# Potential Blocks +PotentialBlockVTKRecorder(..., REC_VELOCITY=True, REC_INTERACTION=True, REC_COLORS=True, REC_ +�→ID=True) +Force chains and other visual outputs are available in qt by default, while they can be extracted in vtk +format using the classic VTKRecorder or the export.VTKExporter class. +A boolean parameter twoDimension exists to specify whether the particles will be rendered as 2-D or +3-D in the vtk output: +# Potential Particles +PotentialParticleVTKRecorder(..., twoDimension=False) +# Potential Blocks +PotentialBlockVTKRecorder(..., twoDimension=False) +614 +Chapter 4. +Theoretical background and extensions + +Yade Documentation, Release 3rd ed. +This parameter should not be mixed up with the Ip2_FrictMat_FrictMat_KnKsPBPhys.twoDimension +parameter, which is used to define how the contact forces are calculated, as described in the Engines +section. +4.6.10 Axis-Aligned Bounding Box +The PP & PB codes use their own BoundFunctors, called PotentialParticle2AABB and Potential- +Block2AABB, respectively, to define the Axis-Aligned Bounding Box of each particle. In both bound +functors, a boolean parameter AabbMinMax exists, allowing the user to choose between an approximate +cubic Aabb or a more accurate one. +In particular, if AabbMinMax=False, a cubic Aabb is considered with dimensions 1.0*R. This is imple- +mented for both the PP and PB codes, even though the Potential Blocks do not have a spherical term. +In this case, the radius R is used as a reference length, denoting half the diagonal of the cubic Aabb. +Usage of this approximate cubic Aabb is not advised in general, since it can increase the number of +empty contacts, adding thus to the time needed to facilitate the approximate contact detection, while it +relies on the radius R, the value of which should enclose the whole particle if this option is activated. +If AabbMinMax=True, a more accurate Aabb can be defined. Currently, the initial Aabb of a Poten- +tialParticle has to be defined manually by the user, in the particle local coordinate system and for the +initial orientation of the particle. To do so, the user has to manually specify the two extreme points of +the Aabb: minAabbRotated, maxAabbRotated inside the shape class. The Aabb for a PotentialBlock, on +the other hand, is calculated and updated automatically from the vertices of the particle, if the boolean +parameter AabbMinMax =True. +As discussed in the subsection Visualization, the dimensions of the Aabb are used as a drawing space +in the code implementing rendering of the particles in the qt environment (for the PP code) and for +the creation of the output files in vtk format (for both codes). This is achieved by using two auxiliary +parameters: minAabb and maxAabb. For the Potential Blocks code only, if these parameters are left +unassigned, the drawing space is configured automatically inside the PotentialBlockVTKRecorder using +the Aabb of the particle. For the particles to be properly rendered as closed surfaces in both qt and +vtk outputs using the available codes, we need to define a drawing space slightly larger than the actual +one. Here, this drawing space is represented by the Aabb of the particles, and thus the differentiation +between the minAabb, maxAabb and minAabbRotated, maxAabbRotated stems from the need to satisfy +two conditions: 1. The Aabb used for primary contact detection must be as tight as possible, in order +to have the least number of empty contacts and 2. The Aabb used as a rendering space must be slightly +larger, in order to have proper rendering. If a dimension of the Aabb used for visualization purposes +is defined smaller than the actual one, the faces on that side of the particle are rendered as hollow +and only the edges are visualised, a functionality that can be used to e.g. see through boundaries, like +demonstrated in the vtk output of the examples/PotentialParticles/cubePPscaled.py example. +To recap, in the Potential Particles code, the minAabbRotated and maxAabbRotated parameters define +the initial Aabb used to facilitate primary contact detection, while the minAabb and maxAabb parameters +are used for visualization of the particles in qt and vtk outputs. In the Potential Blocks code, the Aabb +used to facilitate primary contact detection is calculated automatically from the particles’ vertices, which +are also used for visualization in qt, while the parameters minAabb and maxAabb are used for visualization +in vtk outputs and can be left unassigned, to trigger an automatic configuration of the drawing space of +the particle in the PotentialBlockVTKRecorder. +Two brief examples demonstrating the syntax of these features can be found below. +For the Potential Particles code: +b=Body() +b.shape=PotentialParticle(AabbMinMax=True, +minAabbRotated=Vector3(xmin,ymin,zmin), +maxAabbRotated=Vector3(xmax,ymax,zmax), +minAabb=Vector3(xmin,ymin,zmin), +maxAabb=Vector3(xmax,ymax,zmax), ...) +For the Potential Blocks code: +4.6. +Potential Particles and Potential Blocks +615 + +Yade Documentation, Release 3rd ed. +b=Body() +b.shape=PotentialBlock(AabbMinMax=True, +minAabb=Vector3(xmin,ymin,zmin), +maxAabb=Vector3(xmax,ymax,zmax), ...) +4.6.11 Block Generation algorithm +The Potential Blocks code is compatible with the Block Generation algorithm introduced in [Boon2015], +which can split particles by their intersection with discontinuity planes, initially developed for the study +of rock-masses. This code is hardcoded in YADE in the form of a Preprocessor. Using a single data +structure for the definition of the particle shape and the definition of the discontinuities, as well, allows the +generation of a large number of particles at a reasonable computational cost. The sequential subdivision +concept is used along with a linear programming framework. Non-persistent joints can be modelled by +introducing more constraints. +An example to demonstrate the usage of this code exists in examples/PotentialBlocks/WedgeYADE.py +The +discontinuity +planes +used +in +this +script +are +included +in +a +csv +format +in +exam- +ples/PotentialBlocks/joints/jointC.csv. +The documentation on how to use this code is currently being written. +4.6.12 Examples +Examples can be found in the folders examples/PotentialParticles and examples/PotentialBlocks/, where +the syntax of the codes is demonstrated. +4.6.13 Disclaimer +These codes were developed for academic purposes. Some variables are no longer in use, as the PhD +thesis of the original developer spanned over many years, with numerous trials and errors. As this piece +of code has many dependencies within the YADE ecosystem, user discretion is advised. +4.6.14 References +To acknowledge our scientific contribution, please cite the following: +Potential Blocks +• Boon CW (2013) Distinct Element Modelling of Jointed Rock Masses: Algorithms and Their +Verification. D.Phil. Thesis, University of Oxford +• Boon CW, Houlsby GT, Utili S (2012) A new algorithm for contact detection between convex +polygonal and polyhedral particles in the discrete element method. Computers and Geotechnics, +44: 73-82 +Potential Particles +• Houlsby GT (2009) Potential particles: a method for modelling non-circular particles in DEM. +Computers and Geotechnics, 36(6):953-959 +• Boon CW, Houlsby GT, Utili S (2013) A new contact detection algorithm for three dimensional +non-spherical particles. Powder Technology, S.I. on DEM, 248: 94-102 +Block Generation +• Boon CW, Houlsby GT, Utili S (2015) A new rock slicing method based on linear programming. +Computers and Geotechnics, 65:12-29 +616 +Chapter 4. +Theoretical background and extensions + +Chapter 5 +Performance enhancements +5.1 Accelerating Yade’s FlowEngine with GPU +(Note: we thank Robert Caulk for preparing and sharing this guide) +5.1.1 Summary +This document contains instructions for adding Suite Sparse’s GPU acceleration to Yade’s Pore Finite +Volume (PFV) scheme as demonstrated in [Caulk2019]. The guide is intended for intermediate to ad- +vanced Yade users. As such, the guide assumes the reader knows how to modify and compile Yade’s +source files. +Readers will find that this guide introduces system requirements, installation of neces- +sary prerequisites, and installation of the modified Yade. Lastly, the document shows the performance +enhancement expected by acceleration of the factorization of various model sizes. +5.1.2 Hardware, Software, and Model Requirements +• Hardware: +– CUDA-capable GPU with >3 GB memory recommended (64 mb required) +• Software: +– NVIDIA CUDA Toolkit +– SuiteSparse (CHOLMOD v2.0.0+) +– Metis (comes with SuiteSparse) +– CuBlas +– OpenBlas +– Lapack +• Model: +– Fluid coupling (Pore Finite Volume aka Yade’s “FlowEngine”) +– >10k particles, but likely >30k to see significant speedups +– Frequent remeshing requirements +617 + +Yade Documentation, Release 3rd ed. +5.1.3 Install CUDA +The following instructions to install CUDA are a boiled down version of these instructions. +lspci | grep -i nvidia #Check your graphics card +# Install kernel headers and development packages +sudo apt-get install linux-headers-$(uname -r) +#Install repository meta-data (see **Note below): +sudo dpkg -i cuda-repo-__.deb +sudo apt-get update +#update the Apt repository cache +sudo apt-get install cuda #install CUDA +# Add the CUDA library to your path +export PATH=/usr/local/cuda/bin${PATH:+:${PATH}} +export LD_LIBRARY_PATH=/usr/local/cuda/lib64\ ${LD_LIBRARY_PATH:+:${LD_LIBRARY_PATH}} +Note: use this tool to determine your __ values. +Restart your computer. +Verify your CUDA installation by navigating to /usr/local/cuda/samples and executing the make com- +mand. Now you can navigate to /usr/local/cuda/samples/1_Utilities/deviceQuery/ and execute +./deviceQuery . Verify the Result = PASS. +5.1.4 Install OpenBlas, and Lapack +Execute the following command: +sudo apt-get install libopenblas-dev liblapack-dev +5.1.5 Install SuiteSparse +Download the SuiteSparse package and extract the files to /usr/local/. Run make config and ver- +ify CUDART_LIB and CUBLAS_LIB point to your cuda installed libraries. +The typical paths will fol- +low CUDART_LIB=/usr/local/cuda-x.y/lib64 and CUBLAS_LIB=/usr/local/cuda-x.y/lib64. If the +paths are blank, you may need to navigate to to CUDA_PATH in /usr/local/SuiteSparse/SuiteSparse_- +config/SuiteSparse_config.mk and modify it manually to point to your cuda installation. Navigate +back to the main SuiteSparse folder and execute make. SuiteSparse is now compiled and installed on +your machine. +Test CHOLMOD’s GPU functionality by navigating to SuiteSparse/CHOLMOD/Demo and executing sh +gpu.sh. Note: you will need to download the nd6k.mtx from here and put it in your home directory. +5.1.6 Compile Yade +Following the instructions outlined here, run cmake with -DCHOLMOD_GPU=ON and -DSUITESPARSEPATH=/ +usr/local/SuiteSparse (or your other custom path). +Check the output to verify the paths to +CHOLMOD (and dependencies such as AMD), SuiteSparse, CuBlas, and Metis are all identified as +the paths we created when we installed these packages. Here is an example of the output you need to +inspect: +-- Found Cholmod in /usr/local/SuiteSparse/lib/libcholmod.so +-- Found OpenBlas in /usr/lib/libopenblas.so +-- Found Metis in /usr/local/SuiteSparse/lib/libmetis.so +-- Found CuBlas in /usr/local/cuda-x.y/libcublas.so +-- Found Lapack in /usr/lib/liblapack.so +If you have multiple versions of any of these packages, it is possible the system finds the wrong one. +In this case, you will need to either uninstall the old libraries (e.g. sudo apt-get remove libcholmod +618 +Chapter 5. +Performance enhancements + +Yade Documentation, Release 3rd ed. +if the other library was installed with apt-get) or edit the paths within cMake/Find_____.cmake. If +you installed a version of Cuda in a different location than /usr/local, you will need to edit cMake/ +FindCublas.cmake to reflect these changes before compilation. +Metis is compiled with SuiteSparse, so the Metis library and Metis include should link to files within +usr/local/SuiteSparse/. When ready, complete installation with make -jX install. Keep in mind +that adding CHOLMOD_GPU alters useSolver=4 so to work with the GPU and not the CPU. If you wish to +useSolver=4 with the CPU without unintended side effects (possible memory leaks), it is recommended +to recompile with CHOLMOD_GPU=OFF. Of course, useSolver=3 should always work on the CPU. +5.1.7 Controlling the GPU +The GPU accelerated solver can be activated within Yade by setting flow.useSolver=4`. There are sev- +eral environment variables that control the allowable memory, allowable GPU matrix size, etc. These are +highlighted within the CHOLMOD User Guide, which can be found in SuiteSparse/CHOLMOD/Doc. At +the minimum, the user needs to set the environment variable by executing export CHOLMOD_USE_GPU=1. +It is also recommended that you designate half of your available GPU memory with export CHOLMOD_- +GPU_MEM_BYTES=3000000000 (for a 6GB graphics card), if you wish to use the multithread=True func- +tionality. If you have a multi-gpu setup, you can tell Yade to use one (or both GPUs with SuiteSparse- +4.6.0-beta) by executing export CUDA_VISIBLE_DEVICES=1, where 1 is the GPU you wish to use. +5.1.8 Performance increase +[Catalano2012] demonstrated the performance of DEM+PFV coupling and highlighted its strengths and +weaknesses. A significant strength of the DEM+PFV coupling is the asymptotic nature of triangulation +costs, volume calculation costs, and force calculation costs ( [Catalano2012], Figure 5.4). In other words, +increasing the number of particles beyond ~200k results in negligible additional computational costs. The +main weakness of the DEM+PFV coupling is the exponential increase of computational cost of factoring +and solving increasingly larger systems of linear equations ( [Catalano2012], Figure 5.7). As shown in +Fig. fig-cpuvsgpu, the employment of Tesla K20 GPU decreases the time cost of factorization by up to +75% for 2.1 million DOFs and 356k particles. +0 +80 160 240 320 400 +0 +20 +40 +60 +80 +100 +120 +140 +160 +Time (s) +Factorize +1050 Ti GPU +10-core CPU +Tesla K20 GPU +0 +80 160 240 320 400 +Analyze +0 +80 160 240 320 400 +Total +0.0 +0.6 +1.2 +1.8 +2.4 +0.0 +0.6 +1.2 +1.8 +2.4 +0.0 +0.6 +1.2 +1.8 +2.4 +Thousands of particles +Millions of degrees of freedom +Fig. 1: Time required to factorize and analyze various sized matrices for 10-core CPU, 1050Ti GPU, and +Tesla K20 GPU [Caulk2019]. +Note: Tesla K20 5GB CPU + 10-core Xeon E5 2.8 GHz CPU +5.1. +Accelerating Yade’s FlowEngine with GPU +619 + +Yade Documentation, Release 3rd ed. +5.2 MPI parallelization +The module mpy implements parallelization by domain decomposition (distributed memory) using the +Message Passing Interface (MPI) implemented by OpenMPI. It aims at exploiting large numbers of com- +pute nodes by running independent instances of Yade on them. The shared memory and the distributed +memory approaches are compatible, i.e. it is possible to run hybrid jobs using both, and it may well be +the optimal solution in some cases. +Most (initially all) calls to OpenMPI library are done in Python using mpi4py. However for the sake +of efficiency some critical communications are triggered via python wrappers of C++ functions, wherein +messages are produced, sent/received, and processed. +This module development was started in 2018. It received contributions during a HPC hackathon. An +extension enables parallel coupling with OpenFoam. +Note: +see also reference documentation of the mpy module. +Note: +Disclaimer: even though the yade.mpy module provides the function mpirun, which may seem +as a simple replacement for O.run(), setting up a simulation with mpy might be deceptively triavial. +As of now, it is anticipated that, in general, a simple replacement of “run” by “mpirun” in an arbitrary +script will not speedup anything and may even fail miserably (it could be improved in the future). To +understand why, and to tackle the problems, basic knowledge of how MPI works will certainly help +(specifically mpi4py). +5.2.1 Concepts +subdomain: a (sub)set of bodies attached to one MPI process after domain decomposition - with +or without spatial coherence. The corresponding class in Yade is Subdomain, a Shape instance with +helper functions for MPI communications. In some sense Subdomain is to subscribed bodies what Clump +(another Shape) is to clump members. +rank: subdomain index from 0 to N-1 (with N the number of mpi processes) to identify subdomains. +The rank of the subdomain a body belongs to can be retrieved as Body.subdomain. Each subdomain +corresponds to an instance of Yade and a specific scene during parallel execution. The rank of the scene +is given by Scene.subdomain. +master: refers to subdomain with rank =0. This subdomain does not behave like others. In general +master will handle boundary conditions and it will control transitions and termination of the whole +simulation. Unlike standard subdomains it may not contain a large number of raw bodies (i.e. not +beyond objects bounding the scene such as walls or boxes). In interactive execution master is the process +responding to the python prompt. +splitting and merging: cutting a master Scene into a set of smaller, distributed, scenes is called +“splitting”. The split is undone by a ‘merge’, by which all bodies and (optionally) all interactions are +sent back to the master thread. Splitting, running, then merging, should leave the scene as if no MPI had +been used at all (i.e. as if the same number of iterations had been executed in single-thread). Therefore +normal O.run() after that should work as usual. +intersections: subsets of bodies in a subdomain intersected by the bounding box of other subdomains +(see fig-subdomains). intersection(i,j) refers to the bodies owned by current (i) subdomain and intersect- +ing subdomain j (retrieved as O._sceneObj.subD.intersections[j]); mirrorIntersection(i,j) refers to bodies +owned by j and intersecting current domain (retrieved as O._sceneObj.subD.mirrorIntersections[j]). The +bodies are listed by Body.id. By definition intersection(i,j)=mirrorIntersection(j,i). +The intersections and mirror intersections are updated automatically as part of parallel collision detec- +tion. They define which body states need to be communicated. The bodies in intersections need to be +620 +Chapter 5. +Performance enhancements + +Yade Documentation, Release 3rd ed. +sent to other subdomains (in pratice only updated position and velocity are sent at every iteration), the +bodies in mirrorIntersections need to be received from other subdomains. +Two +overlapping +subdomains +and +their +intersections. +In +this +sit- +uation +we +have +SubD1.intersections[SubD2.subdomain]=[id4,id5] +and +SubD1.mirrorIntersections[SubD2.subdomain]=[id1], with SubD1 and SubD2 instances of Subdomain. +5.2.2 Walkthrough +For demonstrating the main internal steps in the implemented parallel algorithm let us consider the +example script examples/mpi/testMPI_2D.py. Executing this script (interactive or passive mode) with +three MPI processes generates the scene as shown in fig-scene-mpi. It then executes mpirun, which +triggers the steps described hereafter. +In this scene, we have three MPI processes (three subdomains) and the raw bodies are partitioned among +the subdomains/ranks 1 and 2. The master process with subdomain=0 holds the boundary/wall type +body. Bodies can be manually assigned or automatically assigned via a domain decomposition algorithm. +Details on the dommain decomposition algorithm is presented in the later section of this document. +Scene splitting : +In the function mpy.splitScene, called at the beginning of mpi execution, specific engines are added +silently to the scene in order to handle what will happen next. That very intrusive operation can even +change settings of some pre-existing engines, in particular InsertionSortCollider, to make them behave +with MPI-friendlyness. InsertionSortCollider.verletDist is an important factor controlling the efficiency +of the simulations. The reason for this will become evident in the later steps. +Bounds dispatching : In the next step, the Body.bound is dispatched with the Aabb extended as shown +in figure fig-regularbounds (in dotted lines). Note that the Subdomain Aabb is obtained from taking the +min and max of the owned bodies, see figure fig-subDBounds with solid coloured lines for the subdomain +Aabb. At this time, the min and max of other subdomains are unknown. +Update of Domain bounds : Once the bounds for the regular bodies and the local subdomain +has been dispatched, information on the other subdomain bounds are obtained via the function +5.2. +MPI parallelization +621 + +X +X +id4 +id2 +SubD 1 +SubD2 +id1 +X +X +X +id5 +id3subdomain=1 +subdomain=2 +subdomain=0subdomain=1 +subdomain=2 +subdomain=0Yade Documentation, Release 3rd ed. +mpy.updateDomainBounds. In this collective communication, each subdomain broadcasts its Aabb.min +and Aabb.max to other subdomains. +Figure fig-subdomain-bounds shows a schematic in which each +subdomain has received the Aabb.min and Aabb.max of the other subdomains. +Parallel Collision detection : +• Once the Aabb.min and Aabb.max of the other subdomains are obtained, the collision detection +algorithm is used to determine the bodies that have intersections with the remote subdomains. +The ids of the identified bodies are then used to build the Subdomain.intersections list. +• Next step involves obtaining the ids of the remote bodies intersecting with the current subdomain +(Subdomain.mirrorIntersections). +Each subdomain sends its list of local body intersections to +the respective remote subdomains and also receives the list of intersecting ids from the other +subdomains. If the remote bodies do not exist within the current subdomain’s BodyContainer, +the subdomain then requests these remote bodies from the respective subdomain. A schematic +of this operation is shown in figure fig-mirrorIntersections, in which subdomain=1 receives three +bodies from subdomain=2, and 1 body from subdomain=0. subdomain=2 receives three bodies +from subdomain=1. subdomain=0 only sends its bodies and does not receive from the worker +subdomains. This operation sets the stage for communication of the body states to/from the other +subdomains. +Update states : +Once the subdomains and the associated intersecting bodies, and remote bodies are identified, State of +these bodies are sent and received every timestep, by peer-to-peer communications between the interact- +ing subdomains. In the case of an interaction with the master subdomain (subdomain=0), only the total +force and torque exerted on master’s bodies by a given subdomain are sent. Figure fig-sendRecvStates +622 +Chapter 5. +Performance enhancements + +Set Subdomain bound min & maxIn subdomain= +n subdomain=2 +In subdomain=0n subdomain=1 +In subdomain=2Yade Documentation, Release 3rd ed. +shows a schematic in which the states of the remote bodies between subdomain=1 and subdomain=2 +are communicated. Subdomain=0 receives forces and torques from subdomain=1 and subdomain=2. +5.2.3 MPI initialization and communications +The mpy modules tries to retain one of Yade’s most important features: interactive access to the objects +of scene (or of multiple scenes in this case), as explained below. Interactive execution does not use the +mpiexec command of OpenMPI, instead, a pool of workers is spawned by the mpy module after Yade +startup. In production one may use passive jobs, and in that case mpiexec will preceed the call to Yade. +Note: +Most examples in this page use 4 mpi processes. It is not a problem, in principle, to run the +examples even if the number of available cores is less than 4 (this is called oversubscribing (it may also fail +depending on OS and MPI implementation). There is no performance gain to expect from oversubscribing +but it is useful for experiments (e.g. for testing the examples in this page on a single-core machine). +Interactive mode +The interactive mode aims primarily at inspecting the simulation after some MPI execution for debugging. +Functions shown here (especially sendCommand) may also be usefull in the general case, to achieve +advanced tasks such as controlling transitions between phases of a simulation, collecting and processing +results. +Explicit initialization from python prompt +A pool of Yade instances can be spawned with mpy.initialize() as illustrated hereafter. Mind that the +next sequences of commands are supposed to be typed directly in the python prompt after starting Yade, +5.2. +MPI parallelization +623 + +from SD=2 to SD=1 +In subdomain=2 +In subdomain=1 +from SD=1 to SD=2 +from SD=0 to SD=1 +from SD=0 to SD=2Send states +In subdomain=1 +In subdomain=2 +Recv states +Send states +Recv states +In subdomain=o +Recv partial sums of forces & torques +from SD=2 +from SD=1Yade Documentation, Release 3rd ed. +it will not give exactly the same result if it is pasted into a script executed by Yade (see the next section +on automatic initialization): +@suppress +Yade [1]: from yade.utils import * +@suppress +Yade [1]: O.engines=yade.utils.defaultEngines +Yade [2]: wallId=O.bodies.append(box(center=(0,0,0),extents=(2,0,1),fixed=True)) +Yade [3]: for x in range(-1,2): +...: +O.bodies.append(sphere((x,0.5,0),0.5)) +...: +Yade [5]: from yade import mpy as mp +@suppress +Yade [5]: mp.COLOR_OUTPUT=False +@doctest +Yade [6]: mp.initialize(4) +Master: I will spawn +3 +workers +-> +[6]: (0, 4) +After mp.initialize(np) the parent instance of Yade takes the role of master process (rank=0). +It is +the only one executing the commands typed directly in the prompt. The other instances (rank=1 to +rank=np-1) are idle and they wait for commands sent from master. Sending commands to the other +instances can be done with mpy.sendCommand(), which by default returns the result or the list of results. +We use that command below to verify that the spawned workers point to different (still empty) scenes: +Yade [8]: len(O.bodies) +-> +[8]: 4 +Yade [10]: mp.sendCommand(executors="all",command="len(O.bodies)",wait=True) #check content +-> +[10]: [4, 0, 0, 0] +Yade [9]: mp.sendCommand(executors="all",command="str(O)") # check scene pointers +-> +[9]: +['', +'', +'', +''] +Sending commands makes it possible to manage all types of message passing using calls to the underlying +mpi4py (see mpi4py documentation). Be carefull with sendCommand “blocking” behavior by default. +Next example would hang without “wait=False” since both master and worker would be waiting for a +message from each other. +Yade [3]: mp.sendCommand(executors=1,command="message=comm.recv(source=0); print('received', +�→message)",wait=False) +Yade [4]: mp.comm.send("hello",dest=1) +received hello +Every picklable python object (namely, nearly all Yade objects) can be transmitted this way. Remark +hereafter the use of mpy.mprint +(identifies the worker by number and by font colors). Note also that +the commands passed via sendCommand are executed in the context of the mpy module, for this reason +comm, mprint, rank and all objects of the module are accessed without the mp. prefix. +Yade [3]: mp.sendCommand(executors=1,command="O.bodies.append(comm.recv(source=0))", +�→wait=False) # leaves the worker idle waiting for an argument to append() +(continues on next page) +624 +Chapter 5. +Performance enhancements + +Yade Documentation, Release 3rd ed. +(continued from previous page) +Yade [4]: b=Body(shape=Sphere(radius=0.7)) +# now create body in the context of master +Yade [5]: mp.comm.send(b,dest=1) # send it to worker 1 +Yade [6]: mp.sendCommand(executors="all",command="mprint('received',[b.shape.radius if␣ +�→hasattr(b.shape,'radius') else None for b in O.bodies])") +Master: received [None, 0.5, 0.5, 0.5] +Worker1: received [0.7] +Worker3: received [] +Worker2: received [] +-> +[5]: [None, None, None, None] # printing yields no return value, hence that empty list of␣ +�→returns, "wait=False" argument to sendCommand would suppress it +Explicit initialization from python script +Though usefull for advanced operations, the function sendCommand() is limited. Basic features of the +python language are missing, e.g. +function definitions and loops are a problem - in fact every code +fragment which can’t fit on a single line is. +In practice the mpy module provides a mechanism to +initialize from a script, where functions and variables will be declared. +Whenever Yade is started with a script as an argument, the script name will be remembered, and if +mpy.initialize() is called (by the script itself or interactively in the prompt), all Yade instances will be +initialized with that same script. It makes distributing function definitions and simulation parameters +trivial (and even distributing scene constructions as seen below). +This behaviour is what happens usually with MPI: all processes execute the same program. It is also +what happens with “mpiexec -np N yade …”. +If the first commands above are pasted into a script used to start Yade, all workers insert the same +bodies as master (with interactive execution only master was inserting). Here is the script: +# script 'test1.py' +wallId=O.bodies.append(box(center=(0,0,0),extents=(2,0,1),fixed=True)) +for x in range(-1,2): +O.bodies.append(sphere((x,0.5,0),0.5)) +from yade import mpy as mp +mp.initialize(4) +print( mp.sendCommand(executors="all",command="str(O)",wait=True) ) +print( mp.sendCommand(executors="all",command="len(O.bodies)",wait=True) ) +and the output reads: +yade test1.py +... +Running script test1.py +Master: will spawn +3 +workers +None +None +None +None +None +None +['', '', '', ''] +[4, 4, 4, 4] +That’s because all instances execute the script in the initialize() phase. “None” is printed 2x3 times +because the script contains print( mp.sendCommand(…)) twice, the workers try to execute that too, but +5.2. +MPI parallelization +625 + +Yade Documentation, Release 3rd ed. +for them sendCommand returns by default, hence the None. +Though logical, this result is not what we want if we try to split a simulation into pieces. The solution +(typical of all mpi programs) is to use the rank of the process in conditionals. Different parts of the +script can then be executed, differently, by each worker, depending on its rank. In order to produce the +same result as before, for instance, the script can be modified as follows: +# script 'test2.py' +from yade import mpy as mp +mp.initialize(4) +if mp.rank==0: # only master +wallId=O.bodies.append(box(center=(0,0,0),extents=(2,0,1),fixed=True)) +for x in range(-1,2): +O.bodies.append(sphere((x,0.5,0),0.5)) +print( mp.sendCommand(executors="all",command="str(O)",wait=True) ) +print( mp.sendCommand(executors="all",command="len(O.bodies)",wait=True) ) +print( mp.sendCommand(executors="all",command="str(O)",wait=True) ) +Resulting in: +Running script test2.py +Master: will spawn +3 +workers +['', '', '', ''] +[4, 0, 0, 0] +We could also use rank to assign bodies from different regions of space to different workers, as found in +example examples/mpi/helloMPI.py, with rank-dependent positions: +# rank is accessed without "mp." prefix as it is interpreted in mpy module's scope +mp.sendCommand(executors=[1,2],command= "ids=O.bodies.append([sphere((xx,1.5+rank,0),0.5) for␣ +�→xx in range(-1,2)])") +Keep in mind that the position of the call mp.initialize(N) relative to the other commands has no +consequence for the execution by the workers (for them initialize() just returns), hence program logic +should not rely on it. The workers execute the script from begin to end with the same MPI context, +already set when the first line is executed. It can lead to counter intuitive behavior, here is a script: +# testInit.py +# script.py +O.bodies.append([Body() for i in range(100)]) +from yade import mpy as mp +mp.mprint("before initialize: rank ", mp.rank,"/", mp.numThreads,"; ",len(O.bodies)," bodies") +mp.initialize(2) +mp.mprint("after initialize: rank ", mp.rank,"/", mp.numThreads,"; ",len(O.bodies)," bodies") +and the output: +Running script testInit.py +Master: before initialize: rank +0 / 1 ; +100 +bodies +Master: will spawn +1 +workers +Master: after initialize: rank +0 / 2 ; +100 +bodies +Worker1: before initialize: rank +1 / 2 ; +100 +bodies +Worker1: after initialize: rank +1 / 2 ; +100 +bodies +626 +Chapter 5. +Performance enhancements + +Yade Documentation, Release 3rd ed. +mpirun (automatic initialization) +Effectively running a distributed DEM simulation on the basis of the previously described commands +would be tedious. The mpy module thus provides the function mpy.mpirun +to automate most of the +steps, as described in introduction. Mainly, splitting the scene into subdomains based on rank assigned to +bodies and handling collisions between the subdomains as time integration proceeds (includes changing +the engine list agressively to make this all happen). +If needed, the first execution of mpirun will call the function initialize(), which can therefore be omitted +on the user’s side. The subdomains will be merged into a centralized scene on the master process at the +end of the iterations depending on the argument withMerge. +Here is a concrete example where a floor is assigned to master and multiple groups of spheres are assigned +to subdomains: +import os +from yade import mpy as mp +NSTEPS=5000 #turn it >0 to see time iterations, else only initilization +numThreads = 4 # number of threads to be spawned, (in interactive mode). +#materials +young = 5e6 +compFricDegree = 0.0 +O.materials.append(FrictMat(young=young, poisson=0.5, frictionAngle = radians(compFricDegree),␣ +�→density= 2600, label='sphereMat')) +O.materials.append(FrictMat(young=young*100, poisson = 0.5, frictionAngle = compFricDegree,␣ +�→density =2600, label='wallMat')) +#add spheres +mn,mx=Vector3(0,0,0),Vector3(90,180,90) +pred = pack.inAlignedBox(mn,mx) +O.bodies.append(pack.regularHexa(pred,radius=2.80,gap=0, material='sphereMat')) +#walls (floor) +wallIds=aabbWalls([Vector3(-360,-1,-360),Vector3(360,360,360)],thickness=10.0, material= +�→'wallMat') +O.bodies.append(wallIds) +#engines +O.engines=[ +ForceResetter(), +InsertionSortCollider([ +Bo1_Sphere_Aabb(), +Bo1_Box_Aabb()], label = 'collider'), # always add labels. +InteractionLoop( +[Ig2_Sphere_Sphere_ScGeom(),Ig2_Box_Sphere_ScGeom()], +[Ip2_FrictMat_FrictMat_FrictPhys()], +[Law2_ScGeom_FrictPhys_CundallStrack()], +label="interactionLoop" +), +GlobalStiffnessTimeStepper(timestepSafetyCoefficient=0.3, +timeStepUpdateInterval=100,␣ +�→parallelMode=True, label = 'timeStepper'), +NewtonIntegrator(damping=0.1,gravity = (0, -0.1, 0), label='newton'), +VTKRecorder(fileName='spheres/3d-vtk-', recorders=['spheres', 'intr', 'boxes'],␣ +�→parallelMode=True,iterPeriod=500), #use .pvtu to open spheres, .pvtp for ints, and .vtu for␣ +�→boxes. +] +(continues on next page) +5.2. +MPI parallelization +627 + +Yade Documentation, Release 3rd ed. +(continued from previous page) +#set a custom verletDist for efficiency. +collider.verletDist = 1.5 +######### +RUN +########## +# customize mpy +mp.ERASE_REMOTE_MASTER = True +#keep remote bodies in master? +mp.DOMAIN_DECOMPOSITION= True +#automatic splitting/domain decomposition +#mp.mpirun(NSTEPS) +#passive mode run +mp.MERGE_W_INTERACTIONS = False +mp.mpirun(NSTEPS,numThreads,withMerge=True) # interactive run, numThreads is the number of␣ +�→workers to be initialized, see below for withMerge explanation. +mp.mergeScene() +#merge scene after run. +if mp.rank == 0: O.save('mergedScene.yade') +#demonstrate getting stuff from workers, here we get kinetic energy from worker subdomains,␣ +�→notice that the master (mp.rank = 0), uses the sendCommand to tell workers to compute␣ +�→kineticEnergy. +if mp.rank==0: +print("kinetic energy from workers: "+str(mp.sendCommand([1,2],"kineticEnergy()", +�→True))) +The script is then executed: +yade script.py +For running further timesteps, the mp.mpirun command has to be executed in yade prompt: +Yade [0]: mp.mpirun(100,4,withMerge=False) #run for 100 steps and no scene merge. +Yade [1]: mp.sendCommand([1,2],"kineticEnergy()",True) # get kineticEnergy from workers 1 and␣ +�→2. +Yade [2]: mp.mpirun(1,4,withMerge=True) # run for 1 step and merge scene into master. Repeat␣ +�→multiple time to watch evolution in QGL view +Non-interactive execution +Instead of spawning mpi processes after starting Yade, it is possible to run Yade with the classical +“mpiexec” from OpenMPI. Importantly, it may be the only method allowed through HPC job submission +systems. When using mpiexec there is no interactive shell, or a broken one (which is ok in general in +production). The job needs to run (or “mpirun”) and terminate by itself. +The functions initialize and mpirun described above handle both interactive and passive executions +transparently, and the user scripts should behave the same in both cases. “Should”, since what happens +behind the scenes is not exactly the same at startup, and it may surface in some occasions (let us know). +Provided that a script calls mpy.mpirun with a number of timesteps, the simulation (see e.g. exam- +ples/mpi/vtkRecorderExample.py) is executed with the following command: +mpiexec -np NUMSUBD+1 yade vtkRecorderExample.py +where NUMSUBD corresponds to the required number of subdomains. +Note: +Remember that the master process counts one while it does not handle an ordinary subdomain, +therefore the number of processes is always NUMSUBD +1. +628 +Chapter 5. +Performance enhancements + +Yade Documentation, Release 3rd ed. +5.2.4 Splitting +Splitting an initial scene into subdomains and updating the subdomains after particle motion are two +critical issues in terms of efficiency. The decomposition can be prescribed on users’s side (first section +below), but mpy module also provides algorithms for both. +Note: +The mpy module has no requirement in terms of how the subdomains are defined, and using +the helper functions described here is not a requirement. Even assigning the bodies randomly from a +large cloud to a number of subdomains (such that the subdomains overlap each other and the scene +entirely) would work. It would only be suboptimal as the number of interactions between subdomains +would increase compared to a proper partition of space. +Split by yourself +In order to impose a decomposition it is enough to assign Body.subdomain a value corresponding to the +process rank it should belong to. This can be done either in one centralized scene that is later split, +or by inserting the correct subsets of bodies independently in each subdomain (see section on scene +construction) +In the example script examples/mpi/testMPI_2D.py the spheres are generated as follows (centralized +construction in this example, easily turned into distributed one). For each available worker a bloc of +spheres is generated with a different position in space. The spheres in each block are assigned a subdomain +rank (and a color for visualisation) so that they will be picked up by the right worker after mpirun().: +for sd in range(0,numThreads-1): +col = next(colorScale) +ids=[] +for i in range(N):#(numThreads-1) x N x M spheres, one thread is for master and will␣ +�→keep only the wall, others handle spheres +for j in range(M): +id = O.bodies.append(sphere((sd*N+i+j/30.,j,0),0.500,color=col)) #a␣ +�→small shift in x-positions of the rows to break symmetry +ids.append(id) +for id in ids: O.bodies[id].subdomain = sd+1 +Don’t know how to split? Leave it to mpirun +Initial split +mpirun will decide by itself how to distribute the bodies across several subdomains if DO- +MAIN_DECOMPOSITION =True. In such case the difference between the sequential script +and its mpi version is limited to importing mpy and calling mpirun after turning the DO- +MAIN_DECOMPOSITION flag. +The automatic splitting of bodies to subdomains is based on the Orthogonal Recursive Bi- +section Algortithm of Berger [Berger1987], and [Fleissner2007]. The partitioning is based on +bisecting the space at several levels, with the longest axis in each level chosen as the bisection +axis. The number of levels is determined as int(log2(Nw)) with Nw being the number of +worker subdomains. A schematic of this decomposition is shown in fig-bisectionAlgo, with 4 +worker subdomains. At the initial stage (level = 0), we assume that subdomain=1 contains +the information of the body positions (and bodies), the longest axis is first determined, this +forms the bisectioning axis/plane. The list containing the body positions is sorted along the +bisection axis, and the median of this sorted list is determined. The bodies with positions (bi- +section coordinate) less than the median is coloured with the current subdomain, (SD=1) and +the other half is coloured with SD = 2, the subdomain colouring at each level is determined +using the following rule: +5.2. +MPI parallelization +629 + +Yade Documentation, Release 3rd ed. +if (subdomain < +1< +1<0. +The algorithm is not centralized, which preserves scalability. Additionally, it only engages peer-to-peer +communications between MPI workers that share an intersection. The re-assignment depends on a filter +for making local decisions. At the moment, there is one filter available called mpy.medianFilter. Custom +filters can be used instead. +630 +Chapter 5. +Performance enhancements + +SD1 +split +at level = 0 +SD1 +SD2 +split at level = 1 +SD1 +SD3 +SD2 +SD4b) +(e + subdomain +subdomain +5 +6Yade Documentation, Release 3rd ed. +The median filter body re-allocation criterion criterion involves finding the position of a median plane +between two subdomains such that after discriminating bodies on the “+” and “-” side of that plane +the total number in each subdomain is preserved. It results in the type of split shown in the video +hereafter. Even though the median planes seem to rotate rather quickly at some point in this video, +there are actually five collision detections between each re-allocation, i.e. thousands of time iterations to +effectively rotate the split between two different colors. These progressive rotations are beneficial since +the initial split would have resulted in flat discs otherwise. +Note: +This is not a load balancing in the sense of achieving an equal amount of work per core. In +fact that sort of balancing is achieved by definition already as soon as each worker is assigned the same +amount of bodies (and because a subdomain is really ultimately a list of bodies, not a specific region of +space). Instead the objective is to decrease the communication times overall. +Centralized versus distributed scene construction +For the centralized scene construction method, the master process creates all of the bodies of a scene +and assigns subdomains to them. As part of mpy initialization some engines will be modified or inserted, +then the scene is broadcasted to the workers. Each worker receives the entire scene, identifies its assigned +bodies via Body.subdomain (if worker’s rank==b.subdomain the bodies are retained) and erase the others. +Such a scene construction was used in the previous example and it is by far the simplest. It makes no +real difference with building a scene for non-MPI execution besides calling mp.mpirun instead or just +O.run. +For large number of bodies and processes, though, the centralized scene construction and distribution +can consume a significant amount of time. It can also be memory bound since the memory usage is +quadratic: suppose N bodies per thread on a 32-core node, centralized construction implies that 32 +copies of the entire scene exist simultaneously in memory at some point in time (during the split), i.e. +322N bodies on one single node. For massively parallel applications distributed construction should be +prefered. +In distributed mode each worker instantiates its own bodies and insert them in the local BodyCon- +tainer. Attention need to be paid to properly assign bodies ids since no index should be owned by two +different workers initially. Insertion of bodies in BodyContainer with imposed ids is done with BodyCon- +tainer.insertAtId. The distributed mode is activated by setting the DISTRIBUTED_INSERT flag ON, the +user is in charge of setting up the subdomains and partitioning the bodies. An example of distributed +insertion can be found in examples/mpi/parallelBodyInsert3D.py. +The relevant fragment, where the filtering is done by skipping all steps of a loop except the one with +proper rank (keep in mind that all workers will run the same loop but they all have a different rank +each), reads: +#add spheres +subdNo=0 +import itertools +_id = 0 #will be used to count total number of bodies regardless of subdomain attribute, so␣ +�→that same ids are not reused for different bodies +for x,y,z in itertools.product(range(int(Nx)),range(int(Ny)),range(int(Nz))): +subdNo+=1 +if mp.rank!=subdNo: continue +ids=[] +for i in range(L):#(numThreads-1) x N x M x L spheres, one thread is for master and␣ +�→will keep only the wall, others handle spheres +for j in range(M): +for k in range(N): +dxOndy = 1/5.; dzOndy=1/15. +# shifts in x/y-positions to make␣ +�→columns inclines +px= x*L+i+j*dxOndy; pz= z*N+k+j*dzOndy; py = (y*M+j)*(1 - +�→dxOndy**2 -dzOndy**2)**0.5 #so they are always nearly touching initialy +(continues on next page) +5.2. +MPI parallelization +631 + +Yade Documentation, Release 3rd ed. +(continued from previous page) +id = O.bodies.insertAtId(sphere((px,py,pz),0.500),_ +�→id+(N*M*L*(subdNo-1))) +_id+=1 +ids.append(id) +for id in ids: O.bodies[id].subdomain = subdNo +if mp.rank==0: #the wall belongs to master +WALL_ID=O.bodies.insertAtId(box(center=(Nx*L/2,-0.5,Nz*N/2),extents=(2*Nx*L,0,2*Nz*N), +�→fixed=True),(N*M*L*(numThreads-1))) +The bissection algorithm can be used for defining the initial split, in the distributed case too, since +it takes a points dataset as input. Provided that all workers work with the same dataset (e.g. the +same sequence of a random number generator) they will all reach the same partitioning, and they can +instantiate their bodies on this basis. +5.2.5 Merging +The possibility of a “merge”, shown in the previous example, can be performed using an optional argu- +ment of mpirun or as a standalone function mpy.mergeScene . +If withMerge=True in mpirun then the bodies in master scene are updated to reflect the evolution of +their distributed clones. This is done once after finishing the required number of iterations in mpirun. +This merge operation can include updating interactions. mpy.mergeScene +does the same within the +current iteration. Merging is an expensive task which requires the communication of large messages and, +therefore, it should be done purposely and at a reasonable frequency. It can even be the main bottleneck +for massively parallel scenes. Nevertheless, it can be useful for debugging with the 3D view, or for various +post-processing tasks. The MERGE_W_INTERACTIONS provides a full merge, i.e. the interactions +in the worker subdomains and between the subdomains are included, otherwise, only the position and +states of the bodies are used. Merging with interactions should result in a usual Yade scene, ready for +further time-stepping in non-mpi mode or (more useful) for some post-processing. The merge operation +is not required for a proper time integration in general. +5.2.6 Hints and problems to expect +MPI support in engines +For MPI cases, the parallelMode flag for GlobalStiffnessTimeStepper and VTKRecorder have to be turned +on. They are the only two engines upgraded with MPI support at the moment. +For other things. Read next section and be careful. If you feel like implementing MPI support for other +engines, that would be great, consider using the two available examples as guides. Let us know! +Reduction (partial sums) +Quantities such as kinetic energy cannot be obtained for the entire scene just by summing the return value +of kineticEnergy() from each subdomain. This is because each subdmomain may contain also images of +bodies from intersecting subdomains and they may add their velocity, mass, or whatever is summed, to +what is returned by each worker. Although some most-used functions of Yade may progressively get mpi +support to filter out bodies from remote domains, it is not standard yet and therefore partial sums may +need to be implemented on a case-by-case basis, with proper filtering in the user script. +This is just an example of why many things may go wrong if run is directly replaced by mpirun in a +complex script. +632 +Chapter 5. +Performance enhancements + +Yade Documentation, Release 3rd ed. +Miscellaneous +• sendCommand() has a hardcoded latency of 0.001s to not keep all cores 100% busy waiting for a +command (with possibly little left to OS). If sendCommand() is used at high frequency in complex +algorithms it might be beneficial to decrease that sleep time. +5.2.7 Control variables +• VERBOSE_OUTPUT +: +Details +on +each +operation/step +(such +as +mpy.splitScene, +mpy.parallelCollide etc) is printed on the console, useful for debugging purposes +• ACCUMULATE_FORCES : Control force summation on bodies owned by the master. +• ERASE_REMOTE_MASTER : Erase remote bodies in the master subdomain or keep them as +unbounded ? Useful for fast merge. +• OPTIMIZE_COM, USE_CPP_MPI : Use optimized communication functions and MPI functions +from Subdomain class +• YADE_TIMING : Report timing statistics, prints time spent in communications, collision detection +and other operations. +• DISTRIBUTED_INSERT : Bodies are created and inserted by each subdomain, used for dis- +tributed scene construction. +• DOMAIN_DECOMPOSITION : If true, the bisection decomposition algorithm is used to assign +bodies to the workers/subdomains. +• MINIMAL_INTERSECTIONS : Reduces the size of position/velocity communications (at the end +of the colliding phase, we can exclude those bodies with no interactions besides body<->subdomain +from intersections). +• REALLOCATE_FREQUENCY : if > 0, bodies are migrated between subdomains for efficient load +balancing. If =1 realloc. happens each time collider is triggered, else every N collision detection +• REALLOCATE_MINIMAL : Intersections are minimized before reallocations, hence minimizing +the number of reallocated bodies +• USE_CPP_REALLOC : Use optimized C++ functions to perform body reallocations +• FLUID_COUPLING : Flag for coupling with OpenFOAM. +5.2.8 Benchmark +5.2. +MPI parallelization +633 + +1E+08 +MPY throughput on Dahu cluster (UMS Gricad-Grenoble) +24.6x10°particles +400cores +~3iter/sec +1E+07 +1E+06 +OpenMP -j28 -- 8k particle / core +MPI -- 1k particle/core +MPI -- 8k particle / core ++MPI -- 64k particle / core +Cu~320k*N^0.9 +1E+05 +1E+00 +1E+01 +1E+02 +1E+03 +Numberofcores +Script: parallelBodylnsert3D.py (dense column collapse) with git.6f028a573e. +nodes Dell C6420 bi-xeon SKL Gold 6130 (16 cores, 2.1Ghz) ; connectivity 10GbE et OPA (100Gb/s)Yade Documentation, Release 3rd ed. +Comments: +• From 1k particles/core to 8k particles/core there is a clear improvement. Obviously 1k is too small +and most of the time is spent in comunications. +• From 8k/core to 64k/core the throughput per core is more or less the same, and the performance +is not too far from linear. The data includes elimination of random noise, and overall it is not clear +to me which non-linearity comes from the code and which one comes from the hardware. +• Conclusion, if you don’t have at least 8k spheres/core (maybe less for more compex shapes) mpi is +not your friend. This in line with the estimate of 10k by Dion Weatherley (DEM8+beer) +• It looks like OpenMP sucks, but be aware that the benchmark script is heavily tuned for MPI. It +includes huges verletDist and more time wasted on virtual interactions to minimize global updates. +• I believe tuning for OpenMP could make -j26 (or maybe 2xMPIx -j13) on par or faster than 26 +MPI threads for less than a million particle. Given the additional difficulty, MPI’s niche is for more +than a million particles or more than one compute node. +• the nominal per-core throughput is not impressive. On an efficient script my laptop can approach +1e6Cu while we get 0.3e6Cu per core on Dahu. MPI is not to blame here, my laptop would also +outperform Dahu on a single core. +5.3 Using YADE with cloud computing on Amazon EC2 +(Note: we thank Robert Caulk for preparing and sharing this guide) +5.3.1 Summary +This guide is intended to help YADE users migrate their simulations to Amazon Web Service (AWS) +EC2. Two of the most notable benefits of using scalable cloud computing for YADE include decreased +upfront cost and increased productivity. The entire process, from launching an instance, to installing +YADE, to running a YADE simulation on the cloud can be executed in under 5 minutes. Once the +EC2 instance is running, you can submit YADE scripts the same way you would submit jobs on a local +workstation. +5.3.2 Launching an EC2 instance +Start by signing into the console on Amazon EC2. This will require an existing or new Amazon account. +Once you’ve signed in, you should find the EC2 console by clicking on ‘services’ in the upper left hand +corner of the AWS homepage. Start by clicking on the launch an instance blue button (Fig. fig- +console). Select the Amazon Machine Image (AMI): Ubuntu Server 16.04 LTS (Fig. fig-ubuntu). +You will now select the instance type. It is worth looking at the specifications for each of the instances +so you can properly select the power you need for you YADE simulation. This document will not go into +detail in the selection of size, but you can find plenty of YADE specific performance reports that will help +you decide. However, the instance type is an important selection. The Compute Optimized instances +are necessary for most YADE simulations because they provide access to high performing processors and +guaranteed computing power. The C3.2xlarge (Fig. fig-type) is equivalent to an 8 core 2.8ghz Xeon E5 +with 25 mb of cache, which is likely the best option for medium-large scale YADE simulations. +Before launching, you will be asked to select an existing key pair or create a new key pair. +Create a new one, download it, and place it in a folder that you know the path to. Modify the permissions +on the file by navigating to the same directory in the terminal and typing: +chmod 400 KeyPair.pem +Now the instance is launched, you will need to connect to it via SSH. On unix systems this is as easy as +typing: +634 +Chapter 5. +Performance enhancements + +Yade Documentation, Release 3rd ed. +Fig. 2: Amazon Web Services (AWS) Console +Fig. 3: Select Ubuntu server 16.04 LTS AMI +5.3. +Using YADE with cloud computing on Amazon EC2 +635 + +Services +Resource Groups +EC2 Dashboard +Resources +Events +You are using the following Amazon Ec2 resources in the Us West (Oregon) region: +Tags +Reports +0 Running Instances +Limits +0 Dedicated Hosts +2 Volumes +INSTANCES +3Key Pairs +Instances +0 Placement Groups +Spot Requests +Reserved Instances +Scheduled Instances +Just need a simple virtual private server? Get everything you need to jumpstart yol +Lightsail for free. +Dedicated Hosts + IMAGES +Create Instance +AMIs +Bundle Tasks +To start using Amazon Ec2 you will want to launch a virtual server, known as an Amazo +ELASTICBLOCKSTORE +Launch Instance +Volumes +Snapshots +Note: Your instances will launch in the Us West (Oregon) region +NETWORK&SECURITY +Service Health +Security Groups +Elastic IPs +Service Status: +Placement Groups +US West (Oregon): +Key Pairs +This service is operating normally +Network Interfaces +Availability Zone Status: + LOAD BALANCING +us-west-2a: +Load Balancers +Availability zoneis operating normally +Target Groups +us-west-2b: +Availability zone is operating normally +AUTO SCALING +Launch Configurations +us-west-2c: +Auto Scaling Groups +Availability zone is operating normally +Service HealthDashboard +SYSTEMSMANAGER +SERVICESUbuntu Server 16.04 LTS (HVM), SSD Volume Type - ami-b7a114d7 +Ubuntu Server 16.04 LTS (HVM),EBS General Purpose (SSD) Volume Type. Support available from Canonical (http://www.ubuntu.com/cloud/services) +Free tier eligible +Rootdevicetype:ebs +Virtualization type: hvmYade Documentation, Release 3rd ed. +Fig. 4: Compute optimized (C3) instance tier +ssh -i path/to/KeyPair.pem ubuntu@ec2-XX-XXX-XX-XX.us-west-2.compute.amazon.com +into the terminal. There are other options such as using PuTTY, or even a java based terminal on the +AWS website. You can find the necessary information by navigating to Instances in the left menu of +the AWS console. Right click on the instance as shown in Fig. fig-connect and click connect. +Fig. 5: Connecting to the instance +You will be presented with the public DNS, which should look something like Fig. fig-dns. +5.3.3 Installing YADE and managing files +After you’ve connected to the instance through SSH, you will need to install YADE. The following +commands should be issued to install yadedaily, python, and some other useful tools: +636 +Chapter 5. +Performance enhancements + +C3 +Features: +Mem +SSD Storage +Model +vCPU +High Frequency Intel Xeon E5-2680 v2 (lvy Bridge) Processors +(GiB) +(GB) +.Supportfor EnhancedNetworking +c3.large +2 +3.75 +2 x 16 +Supportforclustering +c3.xlarge +4 +7.5 +2 x 40 +.SsD-backedinstancestorage +c3.2xlarge +8 +15 +2 x 80 +c3.4xlarge +16 +30 +2 x 160 +c3.8xlarge +32 +60 +2 x 320 +Use Cases +High performance front-end fleets, web-servers, batch processing, distributed analytics, high performance science and engineering +applications, ad serving, MMo gaming, and video-encoding.Launch Instance +Connect +Actions V +Q Filter by tags and attributes or search by keyword +Name +InstanceID[ +Instance Type +Availability Zone +Instance State +Status Checks +Alarm +i-02a7c5661... +t2.micro +us-west-2a +running +Connect +one +i-0b8cfd978f... +c4.2xlarge +us-west-2c +stopped +Get Windows Password +one +Launch More Like This +Instance State +Instance Settings +Image +Networking +CloudWatch MonitoringYade Documentation, Release 3rd ed. +Fig. 6: Public DNS +#install yadedaily +sudo bash -c 'echo "deb http://www.yade-dem.org/packages/ xenial/" >> /etc/apt/sources.list' +wget -O - http://www.yade-dem.org/packages/yadedev_pub.gpg | sudo apt-key add - +sudo apt-get update +sudo apt-get install -y yadedaily +# install python +sudo apt-get -y install python +sudo apt-get -y install python-pip python-dev build-essential +# install htop +sudo apt-get -y install htop +Note that ..packages/ xenial/ should match the Ubuntu distribution. 16.04 LTS is Xenial, but if you +chose to start Ubuntu 14.04, you will need to change ‘xenial’ to ‘trusty’. +Finally, you will need to upload the necessary YADE files. If you have a folder with the contents of your +simulation titled yadeSimulation you can upload the folder and its contents by issuing the following +command: +scp -r -i path/to/KeyYADEbox.pem path/to/yadeSimulation ubuntu@ec2-XX-XXX-XX-XX.us-west-2. +�→compute.amazonaws.com:~/yadeSimulation +You should now be able to run your simulation by changing to the proper directory and typing: +yadedaily nameOfSimulation.py +In order to retrieve the output files (folder titled ‘out’ below) for post processing purposes, you will use +the same command that you used to upload the folder, but the remote and local file destinations should +be reversed: +scp -r -i path/to/KeyYADEbox.pem ubuntu@ec2-XX-XXX-XX-XX.us-west-2.compute.amazonaws.com:~/ +�→yadeSimulation/out/ path/to/yadeSimulation/out +5.3.4 Plotting output in the terminal +One of the main issues encountered with cloud computing is the lack of graphical feedback. There is +an easy solution for graphically checking the status of your simulations which makes use of gnuplot’s +wonderful ‘terminal dumb’ feature. Any data can be easily plotted by navigating to the subfolder where +the simulation is saving its output and typing: +gnuplot +set terminal dumb +plot ``data.txt" using 1:2 with lines +Where ‘1:2’ refers to the columns in data.txt that you wish to plot against one another. Your output +should look something like this: +5.3. +Using YADE with cloud computing on Amazon EC2 +637 + +4. Connect to your instance using its Public DNS: +ec2-35-163-62-84.us-west-2.compute.amazonaws.comYade Documentation, Release 3rd ed. +Fig. 7: Gnuplot output +5.3.5 Comments +• Amazon AWS allows you to stop your instance and restart it again later with the same files and +package installations. If you wish to create several instances that all contain the same installation +and file directory you can create a snapshot of your default image which you will be able to use to +create various volumes that you can attach to new instances. These actions are all performed very +easily and graphically through the EC2 console +• You can use Spot Instances, which are a special type of instance that allow you to bid on unused +servers. The price is heavily discounted and worth looking into for any YADE user that wishes to +run hundreds of hours of simulations. +• For most simulations, your computational efficiency will decrease if you use above 8 cores per +simulation. It is preferable to use yadedaily-batch to distribute your cores accordingly so that you +always dedicate 8 cores to each simulation and ensure 100% of the processor is running. +• Create a tmux session to avoid ending YADE simulations upon disconnecting from the server. +tmux +# starts a new session +tmux attach -t 0 +# attach session 0 +tmux kill -t 0 +# kill session +## cntrl - b - d to move back to home +## cntrl - b - [ to navigate within the session +5.4 High precision calculations +Yade supports high and arbitrary precision Real type for performing calculations. All tests and checks +pass but still the current support is considered experimental. The backend library is boost multiprecision +along with corresponding boost math toolkit. +The supported types are following: +638 +Chapter 5. +Performance enhancements + +2.5e+08 +'stressstrain40mpa.txt" +****************门 +2e+08 +*************** +****** +**** +1.5e+08 +++ +**** +**** +*** +1e+08 +-+ +**** +*** +** +5e+07 +** +5e+07 +0.002 +0.004 +0.006 +0.008 +0.01 +8.812 +0.014Yade Documentation, Release 3rd ed. +type +bits +decimal places1 +notes +float +32 +6 +hardware accelerated (not useful, it is only for testing +purposes) +double +64 +15 +hardware accelerated +long double +80 +18 +hardware accelerated +boost +float128 +128 +33 +depending on processor type it may be hardware accel- +erated, wrapped by boost +boost mpfr +Nbit Nbit/(log(2)/ +log(10)) +uses external mpfr library, wrapped by boost +boost cpp_- +bin_float +Nbit Nbit/(log(2)/ +log(10)) +uses boost only, but is slower +The last two types are arbitrary precision, and their number of bits Nbit or decimal places is specified +as argument during compilation. +Note: +See file Real.hpp for details. All Real types pass the real type concept test from boost concepts. +The support for Eigen and CGAL is done with numerical traits. +5.4.1 Installation +The precompiled Yade package uses double type by default. In order to use high precision type Yade has +to be compiled and installed from source code by following the regular installation instructions. With +extra following caveats: +1. Following +packages +are +required +to +be +installed: +python3-mpmath +libmpfr-dev +libmpfrc++-dev libmpc-dev (the mpfr and mpc related packages are necessary only to use +boost::multiprecision::mpfr type). These packages are already listed in the default require- +ments. +2. A g++ compiler version 9.2.1 or higher is required. It shall be noted that upgrading only the +compiler on an existing linux installation (an older one, in which packages for different versions of +gcc were not introduced) is difficult and it is not recommended. A simpler solution is to upgrade +entire linux installation. +3. During cmake invocation specify: +1. either number of bits as REAL_PRECISION_BITS=……, +2. or number of requested decimal places as REAL_DECIMAL_PLACES=……, but not both +3. to use MPFR specify ENABLE_MPFR=ON (is OFF by default). +The arbitrary precision (mpfr or cpp_bin_float) types are used only when more than 128 bits +or more than 39 decimal places are requested. In such case if ENABLE_MPFR=OFF then the slower +cpp_bin_float type is used. The difference in decimal places between 39 and 33 stems from the +fact that 15 bits are used for exponent. Note: a fast quad-double (debian package libqd-dev) +implementation with 62 decimal places is in the works with boost multiprecision team. +5.4.2 Supported modules +During compilation several Yade modules can be enabled or disabled by passing an ENABLE_* command +line argument to cmake. The following table lists which modules are currently working with high precision +1 The amount of decimal places in this table is the amount of places which are completely determined by the binary +represenation. +Few additional decimal digits is necessary to fully reconstruct binary representation. +A simple python +example to demonstrate this fact: for a in range(16): print(1./pow(2.,a)), shows that every binary digit produces +“extra” …25 at the end of decimal representation, but these decimal digits are not completely determined by the binary +representation, because for example …37 is impossible to obtain there. More binary bits are necessary to represent …37, but +the …25 was produced by the last available bit. +5.4. +High precision calculations +639 + +Yade Documentation, Release 3rd ed. +(those marked with “maybe” were not tested): +ENABLE_* module name +HP support +cmake default setting +notes +ENABLE_GUI +yes +ON +native support2 +ENABLE_CGAL +yes +ON +native support2 +ENABLE_VTK +yes +ON +supported3 +ENABLE_OPENMP +partial +ON +partial support4 +ENABLE_MPI +maybe +OFF +not tested5 +ENABLE_GTS +yes +ON +supported6 +ENABLE_GL2PS +yes +ON +supported6 +ENABLE_LINSOLV +no +OFF +not supported7 +ENABLE_PARTIALSAT +no +OFF +not supported7 +ENABLE_PFVFLOW +no +OFF +not supported7 +ENABLE_TWOPHASEFLOW +no +OFF +not supported7 +ENABLE_THERMAL +no +OFF +not supported7 +ENABLE_LBMFLOW +yes +ON +supported6 +ENABLE_SPH +maybe +OFF +not tested8 +ENABLE_LIQMIGRATION +maybe +OFF +not tested8 +ENABLE_MASK_ARBITRARY +maybe +OFF +not tested8 +ENABLE_PROFILING +maybe +OFF +not tested8 +ENABLE_POTENTIAL_BLOCKS +no +OFF +not supported9 +ENABLE_POTENTIAL_PARTICLES +yes +ON +supported10 +ENABLE_DEFORM +maybe +OFF +not tested8 +ENABLE_OAR +maybe +OFF +not tested8 +ENABLE_FEMLIKE +yes +ON +supported6 +ENABLE_ASAN +yes +OFF +supported6 +ENABLE_MPFR +yes +OFF +native support2 +ENABLE_LS_DEM +no +ON +not supported11 +The unsupported modules are automatically disabled during a high precision cmake stage. +5.4.3 Double, quadruple and higher precisions +Sometimes a critical section of the calculations in C++ would work better if it was performed in the +higher precision to guarantee that it will produce the correct result in the default precision. A simple +example is solving a system of linear equations (basically inverting a matrix) where some coefficients are +very close to zero. Another example of alleviating such problem is the Kahan summation algorithm. +If requirements are satisfied, Yade supports higher precision multipliers in such a way that RealHP<1> is +the Real type described above, and every higher number is a multiplier of the Real precision. RealHP<2> +is double precision of RealHP<1>, RealHP<4> is quadruple precision and so on. The general formula for +2 This feature is supported natively, which means that specific numerical traits were written for Eigen and for CGAL, +as well as GUI and python support was added. +3 VTK is supported via the compatibility layer which converts all numbers down to double type. See below. +4 The OpenMPArrayAccumulator is experimentally supported for long double and float128. For types mpfr and cpp_- +bin_float the single-threaded version of accumulator is used. File lib/base/openmp-accu.hpp needs further testing. If in +doubt, compile yade with ENABLE_OPENMP=OFF. In all other places OpenMP multithreading should work correctly. +5 MPI support has not been tested and sending data over network hasn’t been tested yet. +6 The module was tested, the yade --test and yade --check pass, as well as most of examples are working. But it +hasn’t been tested extensively for all possible use cases. +7 Not supported, the code uses external cholmod library which supports only double type. To make it work a native +Eigen solver for linear equations should be used. +8 This feature is OFF by default, the support of this feature has not been tested. +9 Potential blocks use external library coinor for linear programming, this library uses double type only. To make it +work a linear programming routine has to be implemented using Eigen or coinor library should start using C++ templates +or a converter/wrapper similar to LAPACK library should be used. +10 The module is enabled by default, the yade --test and yade --check pass, as well as most of examples are working. +However the calculations are performed at lower double precision. A wrapper/converter layer for LAPACK library +has +been implemented. To make it work with full precision these routines should be reimplemented using Eigen. +11 Possible future enchancement. See comments there . +640 +Chapter 5. +Performance enhancements + +Yade Documentation, Release 3rd ed. +amount of decimal places is implemented in RealHP.hpp file and the number of decimal places used is +simply a multiple N of decimal places in Real precision, it is used when native types are not available. +The family of available native precision types is listed in the RealHPLadder type list. +All types listed in MathEigenTypes.hpp follow the same naming pattern: Vector3rHP<1> is the regular +Vector3r and Vector3rHP for any supported N uses the precision multiplier N. One could then use an +Eigen algorithm for solving a system of linear equations with a higher N using MatrixXrHP to obtain +the result with higher precision. Then continuing calculations in default Real precision, after the critical +section is done. The same naming convention is used for CGAL types, e.g. CGAL_AABB_treeHP which +are declared in file AliasCGAL.hpp. +Before we fully move to C++20 standard, one small restriction is in place: the precision multipliers +actually supported are determined by these two defines in the RealHPConfig.hpp file: +1. #define YADE_EIGENCGAL_HP (1)(2)(3)(4)(8)(10)(20) - the multipliers listed here will work in +C++ for RealHP in CGAL and Eigen. They are cheap in compilation time, but have to be listed +here nonetheless. After we move code to C++20 this define will be removed and all multipliers +will be supported via single template constraint. This inconvenience arises from the fact that both +CGAL and Eigen libraries offer template specializations only for a specific type, not a generalized +family of types. Thus this define is used to declare the required template specializations. +Hint: +The highest precision available by default N= (20) corresponds to 300 decimal places when +compiling Yade with the default settings, without changing REAL_DECIMAL_PLACES=…… cmake compilation +option. +2. #define YADE_MINIEIGEN_HP (1)(2) - the precision multipliers listed here are exported to +python, they are expensive: each one makes compilation longer by 1 minute. Adding more can be +useful only for debugging purposes. The double RealHP<2> type is by default listed here to allow +exploring the higher precision types from python. Also please note that mpmath supports only one +precision at a time. Having different mpmath variables with different precision is poorly supported, +albeit mpmath authors promise to improve that in the future. Fortunately this is not a big problem +for Yade users because the general goal here is to allow more precise calculations in the critical +sections of C++ code, not in python. This problem is partially mitigated by changing mpmath +precision each time when a C++ ￿ python conversion occurs. So one should keep in mind that the +variable mpmath.mp.dps always reflects the precision used by latest conversion performed, even if +that conversion took place in GUI (not in the running script). Existing mpmath variables are not +truncated to lower precision, their extra digits are simply ignored until mpmath.mp.dps is increased +again, however the truncation might occur during assignment. +On some occasions it is useful to have an intuitive up-conversion between C++ types of different pre- +cisions, say for example to add RealHP<1> to RealHP<2> type. The file UpconversionOfBasicOperator- +sHP.hpp +serves this purpose. This header is not included by default, because more often than not, +adding such two different types will be a mistake (efficiency–wise) and compiler will catch them and +complain. After including this header this operation will become possible and the resultant type of such +operation will be always the higher precision of the two types used. This file should be included only in +.cpp files. If it was included in any .hpp file then it could pose problems with C++ type safety and will +have unexpected consequences. An example usage of this header is in the following test routine. +Warning: +Trying to use N unregistered in YADE_MINIEIGEN_HP for a Vector3rHP type inside the +YADE_CLASS_BASE_DOC_ATTRS_* macro to export it to python will not work. Only these N listed in +YADE_MINIEIGEN_HP will work. However it is safe (and intended) to use these from YADE_EIGENCGAL_- +HP in the C++ calculations in critical sections of code, without exporting them to python. +5.4.4 Compatibility +5.4. +High precision calculations +641 + +Yade Documentation, Release 3rd ed. +Python +To declare python variables with Real and RealHP precision use functions math.Real(…), +math.Real1(…), math.Real2(…). Supported are precisions listed in YADE_MINIEIGEN_HP, but please note +the mpmath-conversion-restrictions. +Python has native support for high precision types using mpmath package. Old Yade scripts that use +supported modules can be immediately converted to high precision by switching to yade.minieigenHP. +In order to do so, the following line: +from minieigen import * +has to be replaced with: +from yade.minieigenHP import * +Respectively import minieigen has to be replaced with import yade.minieigenHP as minieigen, the +old name as minieigen being used only for the sake of backward compatibility. Then high precision +(binary compatible) version of minieigen is used when non double type is used as Real. +The RealHP higher precision vectors and matrices can be accessed in python by using the .HPn module +scope. For example: +import yade.minieigenHP as mne +mne.HP2.Vector3(1,2,3) # produces Vector3 using RealHP<2> precision +mne.Vector3(1,2,3) +# without using HPn module scope it defaults to RealHP<1> +The respective math functions such as: +import yade.math as mth +mth.HP2.sqrt(2) # produces square root of 2 using RealHP<2> precision +mth.sqrt(2) +# without using HPn module scope it defaults to RealHP<1> +are supported as well and work by using the respective C++ function calls, which is usually faster than +the mpmath functions. +Warning: +There may be still some parts of python code that were not migrated to high precision +and may not work well with mpmath module. See debugging section for details. +C++ +Before introducing high precision it was assumed that Real is actually a POD double type. It was +possible to use memset(…), memcpy(…) and similar functions on double. This was not a good approach +and even some compiler #pragma commands were used to silence the compilation warnings. To make +Real work with other types, this assumption had to be removed. A single memcpy(…) still remains in file +openmp-accu.hpp and will have to be removed. In future development such raw memory access functions +are to be avoided. +All remaining double were replaced with Real and any attempts to use double type in the code will fail +in the gitlab-CI pipeline. +Mathematical functions of all high precision types are wrapped using file MathFunctions.hpp, these are +the inline redirections to respective functions of the type that Yade is currently being compiled with. The +code will not pass the pipeline checks if std:: is used. All functions that take Real argument should +now call these functions in yade::math:: namespace. Functions which take only Real arguments may +omit math:: specifier and use ADL instead. Examples: +1. Call to std::min(a,b) is replaced with math::min(a,b), because a or b may be int (non Real) +therefore math:: is necessary. +642 +Chapter 5. +Performance enhancements + +Yade Documentation, Release 3rd ed. +2. Call to std::sqrt(a) can be replaced with either sqrt(a) or math::sqrt(a) thanks to ADL, +because a is always Real. +If a new mathematical function is needed it has to be added in the following places: +1. lib/high-precision/MathFunctions.hpp +or +lib/high-precision/MathComplexFunctions.hpp +or +lib/high-precision/MathSpecialFunctions.hpp, depending on function type. +2. py/high-precision/_math.cpp, see math module for details. +3. py/tests/testMath.py +4. py/tests/testMathHelper.py +The tests for a new function are to be added in py/tests/testMath.py in one of these functions: +oneArgMathCheck(…):, twoArgMathCheck(…):, threeArgMathCheck(…):. A table of approximate ex- +pected error tolerances in self.defaultTolerances is to be supplemented as well. To determine toler- +ances with better confidence it is recommended to temporarily increase number of tests in the test loop. +To determine tolerances for currently implemented functions a range(1000000) in the loop was used. +Note: +When passing arguments in C++ in function calls it is preferred to use const Real& rather than +to make a copy of the argument as Real. The reason is following: in non high-precision regular case both +the double type and the reference have 8 bytes. However float128 is 16 bytes large, while its reference +is still only 8 bytes. So for regular precision, there is no difference. For all higher precision types it +is beneficial to use const Real& as the function argument. Also for const Vector3r& arguments the +speed gain is larger, even without high precision. +String conversions +On the python side it is recommended to use math.Real(…) math.Real1(…), or math.toHP1(…) to declare +python variables and math.radiansHP1(…) to convert angles to radians using full Pi precision. +On the C++ side it is recommended to use yade::math::toString(…) and yade::math::fromStringReal(…) +conversion functions instead of boost::lexical_cast(…). The toString and its high +precision version toStringHP functions (in file RealIO.hpp) guarantee full precision during conversion. +It is important to note that std::to_string does not guarantee this and boost::lexical_cast does +not guarantee this either. +For higher precision types it is possible to control in runtime the precision of C++ ￿ python during the +RealHP string conversion by changing the math.RealHPConfig.extraStringDigits10 static parameter. +Each decimal digit needs log10(2) ≈ 3.3219 bits. The std::numeric_limits::digits10 provides +information about how many decimal digits are completely determined by binary representation, meaning +that these digits are absolutely correct. However to convert back to binary more decimal digits are +necessary because log2(10) ≈ 0.3010299 decimal digits are used by each bit, and the last digit from +std::numeric_limits::digits10 is not sufficient. In general 3 or more in extraStringDigits10 +is enough to have an always working number round tripping. However if one wants to only extract +results from python, without feeding them back in to continue calculations then a smaller value of +extraStringDigits10 is recommended, like 0 or 1, to avoid a fake sense of having more precision, when +it’s not there: these extra decimal digits are not correct in decimal sense. They are only there to have +working number round tripping. See also a short discussion about this with boost developers. Also see +file RealHPConfig.cpp for more details. +Note: +The parameter extraStringDigits10 does not affect double conversions, +because +boost::python uses an internal converter for this particular type. It might be changed in the future if +the need arises. E.g. using a class similar to ThinRealWrapper. +It is important to note that creating higher types such as RealHP<2> from string representation of +RealHP<1> is ambiguous. Consider following example: +5.4. +High precision calculations +643 + +Yade Documentation, Release 3rd ed. +import yade.math as mth +mth.HP1.getDecomposedReal(1.23)['bits'] +Out[2]: '10011101011100001010001111010111000010100011110101110' +mth.HP2.getDecomposedReal('1.23')['bits'] +# passing the same arg in decimal format to HP2␣ +�→produces nonzero bits after the first 53 bits of HP1 +Out[3]: +�→'10011101011100001010001111010111000010100011110101110000101000111101011100001010001111010111000010100011110101110 +�→' +mth.HP2.getDecomposedReal(mth.HP1.toHP2(1.23))['bits'] # it is possible to use yade.math.HPn. +�→toHPm(…) conversion, which preserves binary representation +Out[4]: +�→'10011101011100001010001111010111000010100011110101110000000000000000000000000000000000000000000000000000000000000 +�→' +Which of these two RealHP<2> binary representations is more desirable depends on what is needed: +1. The best binary approximation of a 1.23 decimal. +2. Reproducing the 53 binary bits of that number into a higher precision to continue the calculations +on the same number which was previously in lower precision. +To achieve 1. simply pass the argument '1.23' as string. To achieve 2. use math.HPn.toHPm(…) or +math.Realn(…) conversion, which maintains binary fidelity using a single static_cast>(…). +Similar problem is discussed in mpmath and boost documentation. +The difference between toHPn and Realn is following: the functions HPn.toHPm create a m × n matrix +converting from RealHP to RealHP. When n < m then extra bits are set to zero (case 2 above, +depending on what is required one might say that “precision loss occurs”). The functions math.Real(…), +math.Real1(…), math.Real2(…) are aliases to the diagonal of this matrix (case 1 above, depending on +what is required one might say that “no conversion loss occurs” when using them). +Hint: +All RealHP function arguments that are of type higher than double can also accept decimal +strings. This allows to preserve precision above python default floating point precision. +Warning: +On the contrary all the function arguments that are of type double can not accept +decimal strings. To mitigate that one can use toHPn(…) converters with string arguments. +Hint: +To make debugging of this problem easier the function math.toHP1(…) will raise RuntimeError +if the argument is a python float (not a decimal string). +Warning: +I cannot stress this problem enough, please try running yade --check (or yade ./ +checkGravityRungeKuttaCashKarp54.py) in precision different than double after changing this line +into g = -9.81. +In this (particular and simple) case the getCurrentPos() function fails on the +python side because low-precision g is multiplied by high-precision t. +Complex types +Complex numbers are supported as well. +All standard C++ functions are available in lib/high- +precision/MathComplexFunctions.hpp and also are exported to python in py/high-precision/_math.cpp. +There is a cmake compilation option ENABLE_COMPLEX_MP which enables using better complex types from +644 +Chapter 5. +Performance enhancements + +Yade Documentation, Release 3rd ed. +boost::multiprecision library for representing ComplexHP family of types: complex128, mpc_- +complex, cpp_complex and complex_adaptor. It is ON by default whenever possible: for boost version +>= 1.71. For older boost the ComplexHP types are represented by std::complex> in- +stead, which has larger numerical errors in some mathematical functions. +When +using +the +ENABLE_COMPLEX_MP=ON +(default) +the +previously +mentioned +lib/high- +precision/UpconversionOfBasicOperatorsHP.hpp is not functional for complex types, it is a reported +problem with the boost library. +When using MPFR type, the libmpc-dev package has to be installed (mentioned above). +Eigen and CGAL +Eigen and CGAL libraries have native high precision support. +• All declarations required by Eigen are provided in files EigenNumTraits.hpp and MathEigen- +Types.hpp +• All declarations required by CGAL are provided in files CgalNumTraits.hpp and AliasCGAL.hpp +VTK +Since VTK is only used to record results for later viewing in other software, such as paraview, the record- +ing of all decimal places does not seem to be necessary (for now). Hence all recording commands in C++ +convert Real type down to double using static_cast command. This has been implemented +via classes vtkPointsReal, vtkTransformReal and vtkDoubleArrayFromReal in file VTKCompatibil- +ity.hpp. Maybe VTK in the future will support non double types. If that will be needed, the interface +can be updated there. +LAPACK +Lapack is an external library which only supports double type. Since it is not templatized it is not +possible to use it with Real type. +Current solution is to down-convert arguments to double upon +calling linear equation solver (and other functions), then convert them back to Real. This temporary +solution omits all benefits of high precision, so in the future Lapack is to be replaced with Eigen or other +templatized libraries which support arbitrary floating point types. +5.4.5 Debugging +High precision is still in the experimental stages of implementation. Some errors may occur during use. +Not all of these errors are caught by the checks and tests. Following examples may be instructive: +1. Trying to use const references to Vector3r members - a type of problem with results in a segmen- +tation fault during runtime. +2. A part of python code does not cooperate with mpmath - the checks and tests do not cover all +lines of the python code (yet), so more errors like this one are expected. The solution is to put +the non compliant python functions into py/high-precision/math.py. Then replace original calls to +this function with function in yade.math, e.g. numpy.linspace(…) is replaced with yade.math. +linspace(…). +The most flexibility in debugging is with the long double type, because special files ThinRealWrap- +per.hpp, ThinComplexWrapper.hpp were written for that. They are implemented with boost::operators, +using partially ordered field. Note that they do not provide operator++. +A couple of #defines were introduced in these two files to help debugging more difficult problems: +1. YADE_IGNORE_IEEE_INFINITY_NAN - it can be used to detect all occurrences when NaN or Inf are +used. Also it is recommended to use this define when compiling Yade with -Ofast flag, without +-fno-associative-math -fno-finite-math-only -fsigned-zeros +5.4. +High precision calculations +645 + +Yade Documentation, Release 3rd ed. +2. YADE_WRAPPER_THROW_ON_NAN_INF_REAL, YADE_WRAPPER_THROW_ON_NAN_INF_COMPLEX - can be +useful for debugging when calculations go all wrong for unknown reason. +Also refer to address sanitizer section, as it is most useful for debugging in many cases. +Hint: If crash is inside a macro, for example YADE_CLASS_BASE_DOC_ATTRS_CTOR_PY, it is useful to know +where inside this macro the problem happens. For this purpose it is possible to use g++ preprocessor to +remove the macro and then compile the postprocessed code without the macro. Invoke the preprocessor +with some variation of this command: +g++ -E -P core/Body.hpp -I ./ -I /usr/include/eigen3 -I /usr/include/python3.7m > /tmp/Body.hpp +Maybe use clang-format so that this file is more readable: +./scripts/clang-formatter.sh /tmp/Body.hpp +Be careful because such files tend to be large and clang-format is slow. So sometimes it is more useful +to only use the last part of the file, where the macro was postprocessed. Then replace the macro in the +original file in question, and then continue debugging. But this time it will be revealed where inside a +macro the problem occurs. +Note: +When asking questions about High Precision it is recommended to start the question title with +[RealHP]. +646 +Chapter 5. +Performance enhancements + +Chapter 6 +Literature +6.1 Yade Technical Archive +6.1.1 About +The Yade Technical Archive (YTA) seeks to improve the reproducibility of Yade related publications +by clarifying the theory that underlies Yade’s opensource code, explaining algorithmic implementations, +and providing practical tutorials. In doing so, YTA removes the opacity that commonly exists between +readers and computational journal articles, strengthens and improves visibility of existing Yade journal +papers, enables academic collaborations, and broadens open access academia. +6.1.2 Contribute +YTA seeks a variety of Yade related materials including, but not limited to: +• theoretical descriptions of code packages +• user guides and tutorials for code packages +• presentations +• course materials +• supplementary materials for journal articles +6.1.3 Contact +If you wish to contribute, please contact rob.caulk@gmail.com. Questions about individual publications +are referred to the email address attached to the document description. If you have general questions +regarding code, we refer you to our Q&A forum. +6.1.4 Archive +Chareyre, Bruno; Caulk, Robert; Chèvremont, William; Guntz, Thomas; Kneib, François; Kunhappen, +Deepak; Pourroy, Jean (2019), Calcul distribué MPI pour la dynamique de systèmes particulaires. Yade +Technical Archive. download full text , watch video summary , read the poster summary +Pirnia, Pouyan; Duhaime Francois; Ethier Yannic; Dubé, Jean-Sébastien (2019), COMSOL-Yade In- +terface (ICY) instruction guide. +Yade Technical Archive. +download full text, send an email seyed- +pouyan.pirnia.1@ens.etsmtl.ca , download helper files +647 + +Yade Documentation, Release 3rd ed. +Maurin, Raphael (2018), YADE 1D vertical VANS fluid resolution: Numerical resolution details. Yade +Technical Archive. download full text, send an email raphael.maurin@imft.fr, follow the tutorial: Using +YADE 1D vertical VANS fluid resolution +Maurin, Raphael (2018), YADE 1D vertical VANS fluid resolution: Theoretical basis. Yade Technical +Archive. download full text, send an email raphael.maurin@imft.fr, follow the tutorial: Using YADE 1D +vertical VANS fluid resolution +Maurin, Raphael (2018), YADE 1D vertical VANS fluid resolution: validations. Yade Technical Archive. +download full text, send an email raphael.maurin@imft.fr, follow the tutorial: Using YADE 1D vertical +VANS fluid resolution +Caulk, Robert (2018), Stochastic Augmentation of the Discrete Element Method for Investigation of +Tensile Rupture in Heterogeneous Rock. Yade Technical Archive. DOI 10.5281/zenodo.1202039. down- +load full text , send an email rob.caulk@gmail.com , follow the tutorial: Simulating Acoustic Emissions +in Yade +6.2 Publications on Yade +Publications on Yade itself or done with Yade are listed on this page. +The first section gives the reference that we kindly ask you to use for citing Yade in publications, as +explained in the “Acknowledging Yade” section. +With the increasing rate of publications using Yade it became difficult to list them all, therefore coverage +of recent years is only partial. You can help us: if you publish or you know publications related to Yade +do not hesitate to add it to this list. If you don’t have direct access to the source code, please send the +reference (as a bibtex item) to Yade developpers. If a pdf PDF is freely available, add url for direct +fulltext downlad. Yade’s web server will gladly host such PDF if legally permitted. +Note: +This file is generated from doc/yade-articles.bib, doc/yade-conferences.bib, doc/yade-theses.bib, +doc/yade-tech-archive.bib, and doc/citing_yade.bib. +6.2.1 Citing Yade +Corresponding bibtex entries here. See also “Acknowledging Yade”. +6.2.2 Journal articles +6.2.3 Conference materials and book chapters +6.2.4 Master and PhD theses +6.2.5 Yade Technical Archive +6.3 References +All external articles referenced in Yade documentation. +Note: +This file is generated from doc/references.bib. +648 +Chapter 6. +Literature + +Chapter 7 +Indices and tables +• genindex +• modindex +• search +649 + +Yade Documentation, Release 3rd ed. +650 +Chapter 7. +Indices and tables + +Bibliography +[yade:doc3] V. Smilauer et al. (2021), Yade Documentation 3rd ed.. The Yade Project. DOI +10.5281/zenodo.5705394 (http://yade-dem.org/doc/) +[Aboul2017] Aboul Hosn, R., Sibille, L., Benahmed, N., Chareyre, B. (2017), Discrete numerical +modeling of loose soil with spherical particles and interparticle rolling friction. +Granular Matter (19). DOI 10.1007/s10035-016-0687-0 +[Albaba2015] Albaba, A, Lambert, S, Nicot, F, Chareyre, B (2015), Relation between microstruc- +ture and loading applied by a granular flow to a rigid wall using DEM modeling. +Granular Matter (17), pages 603–616. DOI 10.1007/s10035-015-0579-8 +[Angelidakis2021] Angelidakis, Vasileios, Nadimi, Sadegh, Utili, Stefano (2021), SHape Analyser for +Particle Engineering (SHAPE): Seamless characterisation and simplification of +particle morphology from imaging data. Computer Physics Communications, pages +107983. +[Bance2014] Bance, S., Fischbacher, J., Schrefl, T., Zins, I., Rieger, G., Cassignol, C. (2014), Micro- +magnetics of shape anisotropy based permanent magnets. Journal of Magnetism +and Magnetic Materials (363), pages 121–124. +[Barbosa2020] Barbosa, Luis Alfredo Pires (2020), Modelling the aggregate structure of a bulk +soil to quantify fragmentation properties and energy demand of soil tillage tools +in the formation of seedbeds. Biosystems Engineering (197), pages 203–215. +[Barbosa2020b] Barbosa, Luis Alfredo Pires, Keller, Thomas, de Oliveira Ferraz, Antonio Carlos (2020), +Scale effect of aggregate rupture: Using the relationship between friability and +fractal dimension to parameterise discrete element models. Powder Technology +(375), pages 327–336. +[Benniou2020] Benniou, H., Accary, A., Malecot, Y., Briffaut, M., Daudeville, L. (2020), Discrete +element modeling of concrete under high stress level: influence of saturation +ratio. Computational Particle Mechanics. DOI 10.1007/s40571-020-00318-5 +[Bonilla2015] Bonilla-Sierra, V., Scholtès, L., Donzé, F.V., Elmouttie, M.K. (2015), Rock slope stabil- +ity analysis using photogrammetric data and DFN–DEM modelling. Acta Geotech- +nica, pages 1–15. DOI 10.1007/s11440-015-0374-z +[Boon2012] Boon, C.W., Houlsby, G.T., Utili, S. (2012), A new algorithm for contact detection be- +tween convex polygonal and polyhedral particles in the discrete element method. +Computers and Geotechnics (44), pages 73–82. DOI 10.1016/j.compgeo.2012.03.012 +[Boon2013] Boon, C.W., Houlsby, G.T., Utili, S. (2013), A new contact detection algorithm for +three-dimensional non-spherical particles. Powder Technology (248), pages 94–102. +DOI 10.1016/j.powtec.2012.12.040 +[Boon2014] Boon, C.W., Houlsby, G.T., Utili, S. (2014), New insights into the 1963 Vajont slide +using 2D and 3D distinct-element method analyses. Géotechnique (64), pages 800– +816. DOI 10.1680/geot.14.P.041 +[Boon2015] Boon, +C.W., +Houlsby, +G.T., +Utili, +S. +(2015), +A +new +rock +slicing +method +based on linear programming. Computers and Geotechnics (65), pages 12–29. DOI +10.1016/j.compgeo.2014.11.007 +651 + +Yade Documentation, Release 3rd ed. +[Boon2015b] Boon, C.W., Houlsby, G.T., Utili, S. (2015), Designing Tunnel Support in Jointed +Rock Masses Via the DEM. Rock Mechanics and Rock Engineering (48), pages 603–632. +DOI 10.1007/s00603-014-0579-8 +[Bourrier2013] Bourrier, F., Kneib, F., Chareyre, B., Fourcaud, T. (2013), Discrete +model- +ing of granular soils reinforcement by plant roots. Ecological Engineering. DOI +10.1016/j.ecoleng.2013.05.002 +[Bourrier2015] Bourrier, F., Lambert, S., Baroth, J. (2015), A reliability-based approach for the +design of rockfall protection fences. Rock Mechanics and Rock Engineering (48), pages +247–259. +[Yuan2017] C. Yuan, B. Chareyre (2017), A pore-scale method for hydromechanical coupling in +deformable granular media. Computer Methods in Applied Mechanics and Engineering. +DOI 10.1016/j.cma.2017.02.024 +[Yuan2016] C. Yuan, B. Chareyre, F. Darve (2016), Pore-scale simulations of drainage in granular +materials: finite size effects and the representative elementary volume. Adv. in +Water Ressources (95), pages 109–124. +[Catalano2014a] Catalano, E., Chareyre, B., Barthélémy, E. (2014), Pore-scale modeling of fluid- +particles interaction and emerging poromechanical effects. International Journal for +Numerical and Analytical Methods in Geomechanics (38), pages 51–71. DOI 10.1002/nag.2198 +(http://arxiv.org/pdf/1304.4895.pdf) +[Caulk2020] Caulk, Robert A. (2020), Modeling acoustic emissions in heterogeneous rocks dur- +ing tensile fracture with the Discrete Element Method. Open Geomechanics (2). +DOI 10.5802/ogeo.5 +[Chalak2017] Chalak, C., Chareyre, B., Nikooee, E., Darve, F. (2017), Partially saturated media: +from DEM simulation to thermodynamic interpretation. European Journal of Envi- +ronmental and Civil Engineering (21), pages 798–820. DOI 10.1080/19648189.2016.1164087 +[Chapelle2021] Chapelle, David, Maynadier, Anne, Bebon, Ludovic, Thi’ebaud, Fr’ed’eric (2021), Hy- +drogen Storage: Different Technologies, Challenges and Stakes. Focus on TiFe +Hydrides. In Advances in Renewable Hydrogen and Other Sustainable Energy Carriers +Springer , +[Chareyre2012a] Chareyre, B., Cortis, A., Catalano, E., Barthélemy, E. (2012), Pore-Scale Modeling +of Viscous Flow and Induced Forces in Dense Sphere Packings. Transport in Porous +Media (92), pages 473–493. DOI 10.1007/s11242-011-9915-6 +[Chassagne2020] Chassagne, Rémi, Frey, Philippe, Maurin, Raphaël, Chauchat, Julien (2020), Mobility +of bidisperse mixtures during bedload transport. Physical Review Fluids (5), pages +114307. +[Chen2007] Chen, F., Drumm, E. C., Guiochon, G. (2007), Prediction/Verification of Particle +Motion in One Dimension with the Discrete-Element Method. International Journal +of Geomechanics, ASCE (7), pages 344–352. 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lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Robert Caulk Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Grenoble Alpes, 3SR lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Bruno Chareyre Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Grenoble Alpes, 3SR lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' William Chèvremont Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Grenoble Alpes, LRP Sergei Dorofeenko IPCP RAS, Chernogolovka Jérôme Duriez INRAE, Aix Marseille Univ, RECOVER, Aix-en-Provence, France Nolan Dyck Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' of Western Ontario Jan Eliáš Brno University of Technology Burak Er Bursa Technical University Alexander Eulitz TU Berlin / Institute for Machine Tools and Factory Management Anton Gladky TU Bergakademie Freiberg Ning Guo Hong Kong Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' of Science and Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Christian Jakob TU Bergakademie Freiberg François Kneib Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Grenoble Alpes, 3SR lab.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' GeM Jan Stránský CVUT Prague Thomas Sweijen Utrecht University Klaus Thoeni The University of Newcastle (Australia) Chao Yuan Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Grenoble Alpes, 3SR lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Citing this document When referring to Yade-DEM software in scientific publication please cite it ”by DOI” as follows: Šmilauer V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' (2021) Yade Documentation 3rd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' The Yade Project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' DOI:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='5705394.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' http://yade-dem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='org See also http://yade-dem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='org/doc/citing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' i ii Contents 1 Guided tour 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='1 Introduction .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' 34 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='6 Examples with tutorial .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' 93 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='3 Postprocessing .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' 109 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='4 Python specialties and tricks .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' 534 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='3 Debugging .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' 535 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='4 Regression tests .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' 541 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='5 Conventions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' 543 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='6 Support framework .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' 603 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='2 Application of drag and buoyancy forces (HydroForceEngine::action) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' 603 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='3 Solid phase averaging (HydroForceEngine::averageProfile) .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' 606 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='2 Potential Particles code (PP) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' 610 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='6 Shape definition of a PP and a PB .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' 618 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='4 Install OpenBlas, and Lapack .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' 648 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='4 Master and PhD theses .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' 648 7 Indices and tables 649 Bibliography 651 Python Module Index 679 vi Chapter 1 Guided tour 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='1 Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='1 Getting started Before you start moving around in Yade, you should have some prior knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Basics of command line in your Linux system are necessary for running yade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Look on the web for tutorials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Python language;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' we recommend the official Python tutorial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Reading further documents on the topic, such as Dive into Python will certainly not hurt either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' You are advised to try all commands described yourself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Don’t be afraid to experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Hint: Sometimes reading this documentation in a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='pdf format can be more comfortable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' For example in okular pdf viewer clicking links is faster than a page refresh in the web browser and to go back press the shortcut Alt Shift ←.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' To try it have a look at the inheritance graph of PartialEngine then go back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Starting yade Yade is being run primarily from terminal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' the name of command is yade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content="1 (In case you did not install from package, you might need to give specific path to the command2): $ yade Welcome to Yade TCP python prompt on localhost:9001, auth cookie `sdksuy' TCP info provider on localhost:21000 (continues on next page) 1 The executable name can carry a suffix, such as version number (yade-0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='20), depending on compilation options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Packaged versions on Debian systems always provide the plain yade alias, by default pointing to latest stable version (or latest snapshot, if no stable version is installed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' You can use update-alternatives to change this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' 2 In general, Unix shell (command line) has environment variable PATH defined, which determines directories searched for executable files if you give name of the file without path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Typically, $PATH contains /usr/bin/, /usr/local/bin, /bin and others;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' you can inspect your PATH by typing echo $PATH in the shell (directories are separated by :).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' If Yade executable is not in directory contained in PATH, you have to specify it by hand, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' by typing the path in front of the filename, such as in /home/user/bin/yade and similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' You can also navigate to the directory itself (cd ~/bin/yade, where ~ is replaced by your home directory automatically) and type .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='/yade then (the .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' is the current directory, so .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='/ specifies that the file is to be found in the current directory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' To save typing, you can add the directory where Yade is installed to your PATH, typically by editing ~/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='profile (in normal cases automatically executed when shell starts up) file adding line like export PATH=/home/user/bin:$PATH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' You can also define an alias by saying alias yade="/home/users/bin/yade" in that file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Details depend on what shell you use (bash, zsh, tcsh, …) and you will find more information in introductory material on Linux/Unix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' 1 Yade Documentation, Release 3rd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' (continued from previous page) [[ ^L clears screen, ^U kills line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' F12 controller, F11 3d view, F10 both, F9 generator, F8␣ �→plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' ]] Yade [1]: These initial lines give you some information about some information for Remote control, which you are unlikely to need now;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' basic help for the command-line that just appeared (Yade [1]:).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Type quit(), exit() or simply press ^D (^ is a commonly used written shortcut for pressing the Ctrl key, so here ^D means Ctrl D) to quit Yade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' The command-line is ipython, python shell with enhanced interactive capabilities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' it features persistent history (remembers commands from your last sessions), searching and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' See ipython’s documentation for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Typically, you will not type Yade commands by hand, but use scripts, python programs describing and running your simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Let us take the most simple script that will just print “Hello world!”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=': print("Hello world!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='") Saving such script as hello.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='py, it can be given as argument to Yade: $ yade hello.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content="py Welcome to Yade TCP python prompt on localhost:9001, auth cookie `askcsu' TCP info provider on localhost:21000 Running script hello." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='py ## the script is being run Hello world!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' ## output from the script [[ ^L clears screen, ^U kills line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' F12 controller, F11 3d view, F10 both, F9 generator, F8␣ �→plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' ]] Yade [1]: Yade will run the script and then drop to the command-line again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='3 If you want Yade to quit immediately after running the script, use the -x switch: $ yade -x script.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='py There is more command-line options than just -x, run yade -h to see all of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Options: v, --version show program’s version number and exit h, --help show this help message and exit j THREADS, --threads=THREADS Number of OpenMP threads to run;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' defaults to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Equivalent to setting OMP_- NUM_THREADS environment variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' --cores=CORES Set number of OpenMP threads (as –threads) and in addition set affinity of threads to the cores given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' --update Update deprecated class names in given script(s) using text search & replace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Changed files will be backed up with ~ suffix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Exit when done without running any simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' --nice=NICE Increase nice level (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' decrease priority) by given number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' 3 Plain Python interpreter exits once it finishes running the script.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' The reason why Yade does the contrary is that most of the time script only sets up simulation and lets it run;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' since computation typically runs in background thread, the script is technically finished, but the computation is running.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' 2 Chapter 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Guided tour Yade Documentation, Release 3rd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' x Exit when the script finishes f Set logging verbosity, default is -f3 (yade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='WARN) for all classes n Run without graphical interface (equivalent to unset- ting the DISPLAY environment variable) --test Run regression test suite and exit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' the exists status is 0 if all tests pass, 1 if a test fails and 2 for an unspecified exception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' --check Run a series of user-defined check tests as described in scripts/checks-and-tests/checks/README and Re- gression tests --performance Starts a test to measure the productivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' --stdperformance Starts a standardized test to measure the productiv- ity, which will keep retrying to run the benchmark until standard deviation of the performance is below 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' A common type of simulation is done: the spheres fall down in a box and are given enough time to settle in there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Note: better to use this with argument -j THREADS (explained above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' --quickperformance Starts a quick test to measure the productivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Same as above, but only two short runs are performed, without the attempts to find the computer perfor- mance with small error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' --no-gdb Do not show backtrace when yade crashes (only effec- tive with –debug)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Quick inline help All of functions callable from ipython shell have a quickly accessible help by appending ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' to the function name, or calling help(…) command on them: Yade [1]: O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='run?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Docstring: run( (Omega)arg1 [, (int)nSteps=-1 [, (bool)wait=False]]) -> None : Run the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' *nSteps* how many steps to run, then stop (if positive);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' *wait* will␣ �→cause not returning to python until simulation will have stopped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Type: method Yade [2]: help(O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='pause) Help on method pause: pause(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=') method of yade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='wrapper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='Omega instance pause( (Omega)arg1) -> None : Stop simulation execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' (May be called from within the loop, and it will stop after␣ �→the current step).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' A quick way to discover available functions is by using the tab-completion mechanism, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' type O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' then press tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Creating simulation To create simulation, one can either use a specialized class of type FileGenerator to create full scene, possibly receiving some parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Generators are written in C++ and their role is limited to well- 4 On some linux systems stack trace will produce Operation not permitted error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' See debugging section for solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Introduction 3 Yade Documentation, Release 3rd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' defined scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' For instance, to create triaxial test scene: Yade [3]: TriaxialTest(numberOfGrains=200).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='load() Yade [4]: len(O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='bodies) Out[4]: 206 Generators are regular yade objects that support attribute access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' It is also possible to construct the scene by a python script;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' this gives much more flexibility and speed of development and is the recommended way to create simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Yade provides modules for streamlined body construction, import of geometries from files and reuse of common code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Since this topic is more involved, it is explained in the User’s manual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Running simulation As explained below, the loop consists in running defined sequence of engines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Step number can be queried by O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='iter and advancing by one step is done by O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='step().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Every step advances virtual time by current timestep, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='dt that can be directly assigned or, which is usually better, automatically determined by a GlobalStiffnessTimeStepper, if present: Yade [5]: O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='iter Out[5]: 0 Yade [6]: O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='time Out[6]: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='0 Yade [7]: O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='dt=1e-4 Yade [8]: O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='dynDt=False #else it would be adjusted automaticaly during first iteration Yade [9]: O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='step() Yade [10]: O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='iter Out[10]: 1 Yade [11]: O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='time Out[11]: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='0001 Normal simulations, however, are run continuously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Starting/stopping the loop is done by O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='run() and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='pause();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' note that O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='run() returns control to Python and the simulation runs in background;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' if you want to wait for it to finish, use O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='wait().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Fixed number of steps can be run with O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='run(1000), O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='run(1000,True) will run and wait.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' To stop at absolute step number, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='stopAtIter can be set and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='run() called normally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Yade [12]: O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='run() Yade [13]: O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='pause() Yade [14]: O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='iter Out[14]: 1715 Yade [15]: O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='run(100000,True) Yade [16]: O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='iter Out[16]: 101715 Yade [17]: O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='stopAtIter=500000 Yade [18]: O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='run() (continues on next page) 4 Chapter 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Guided tour Yade Documentation, Release 3rd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' (continued from previous page) Yade [19]: O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='wait() Yade [20]: O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='iter Out[20]: 500000 Saving and loading Simulation can be saved at any point to a binary file (optionaly compressed if the filename has extensions such as “.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='gz” or “.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='bz2”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Saving to a XML file is also possible though resulting in larger files and slower save/load, it is used when the filename contains “xml”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' With some limitations, it is generally possible to load the scene later and resume the simulation as if it were not interrupted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Note that since the saved scene is a dump of Yade’s internal objects, it might not (probably will not) open with different Yade version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' This problem can be sometimes solved by migrating the saved file using “.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='xml” format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Yade [21]: O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content="save('/tmp/a." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='yade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content="bz2') Yade [22]: O." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='reload() Yade [23]: O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content="load('/tmp/another." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='yade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content="bz2') The principal use of saving the simulation to XML is to use it as temporary in-memory storage for checkpoints in simulation, e." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' for reloading the initial state and running again with different parameters (think tension/compression test, where each begins from the same virgin state).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' The functions O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' saveTmp() and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='loadTmp() can be optionally given a slot name, under which they will be found in memory: Yade [24]: O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='saveTmp() Yade [25]: O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='loadTmp() Yade [26]: O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content="saveTmp('init') ## named memory slot Yade [27]: O." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content="loadTmp('init') Simulation can be reset to empty state by O." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='reset().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' It can be sometimes useful to run different simulation, while the original one is temporarily suspended, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' when dynamically creating packing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='switchWorld() toggles between the primary and secondary simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Graphical interface Yade can be optionally compiled with QT based graphical interface (qt4 and qt5 are supported).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' It can be started by pressing F12 in the command-line, and also is started automatically when running a script.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Introduction 5 Yade Documentation, Release 3rd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' The control window on the left (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' imgQtGui) is called Controller (can be invoked by yade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='qt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Controller() from python or by pressing F12 key in terminal): 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' The Simulation tab is mostly self-explanatory, and permits basic simulation control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' The Display tab has various rendering-related options, which apply to all opened views (they can be zero or more, new one is opened by the New 3D button).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' The Python tab has only a simple text entry area;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' it can be useful to enter python commands while the command-line is blocked by running script, for instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Inside the Inspect window (on the right in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' imgQtGui) all simulation data can be examined and modified in realtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Clicking left mouse button on any of the blue hyperlinks will open documentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Clicking middle mouse button will copy the fully qualified python name into clipboard, which can be pasted into terminal by clicking middle mouse button in the terminal (or pressing Ctrl-V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' 3d views can be controlled using mouse and keyboard shortcuts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' help is displayed if you press the h key while in the 3d view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Note that having the 3d view open can slow down running simulation significantly, it is meant only for quickly checking whether the simulation runs smoothly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Advanced post-processing is described in dedicated section Data mining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='2 Architecture overview In the following, a high-level overview of Yade architecture will be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' As many of the features are directly represented in simulation scripts, which are written in Python, being familiar with this language will help you follow the examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' For the rest, this knowledge is not strictly necessary and you can ignore code examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' 6 Chapter 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Guided tour Yade oX Simulation Display Generate Python Load Save Inspect Primary view real 00:02:20 virt 000s671m552μ639n iter : #3243, 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='0/s At O fixed O timestepper Simulation Inspection 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='000207077594613 Engines Bodies Interactions Cell :memory: 56 V 0 56+0 Body0x466fd80 bound Aabb 0x4684670 color 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='0 clumpld 1 flags 1 groupMask 1 id 56 material FrictMat "defaultMat" density 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='0 New 3D Reference Center Ni X frictionAngle 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='5 id 0 label defaultMat #3243 poisson 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='3 cl0ck 02:20 671m552u639n young 10000000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='0 shape Sphere 0x46845e0 Left mouse button open documentation color 656201236 882652506303750713 Middle mouse buttor copy to clipboard full highlight python name radius 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='0316463982726Yade Documentation, Release 3rd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Data and functions To assure flexibility of software design, yade makes clear distinction of 2 families of classes: data com- ponents and functional components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' The former only store data without providing functionality, while the latter define functions operating on the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' In programming, this is known as visitor pattern (as functional components “visit” the data, without being bound to them explicitly).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' Entire simulation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' both data and functions, are stored in a single Scene object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content=' It is accessible through the Omega class in python (a singleton), which is by default stored in the O global variable: Yade [28]: O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AyT4oBgHgl3EQfuPnr/content/2301.00611v1.pdf'} +page_content='bodies # some data components Out[28]: