Recent zbMATH articles in MSC 52https://www.zbmath.org/atom/cc/522021-03-30T15:24:00+00:00WerkzeugOn the number of vertices of the convex hull of random points in a square and a triangle.https://www.zbmath.org/1455.520042021-03-30T15:24:00+00:00"Buchta, Christian"https://www.zbmath.org/authors/?q=ai:buchta.christianSummary: Assume that \(n\) points are chosen independently and according to the uniform distribution from a convex polygon \(C\). Consider the convex hull of the randomly chosen points. The probabilities \(p_k^{(n)}(C)\) that the convex hull has exactly \(k\) vertices are stated for all \(k\) in the cases that \(C\) is a square (equivalently a parallelogram) or a triangle.On the invariants of the cohomology of complements of Coxeter arrangements.https://www.zbmath.org/1455.200032021-03-30T15:24:00+00:00"Douglass, J. Matthew"https://www.zbmath.org/authors/?q=ai:douglass.j-matthew"Pfeiffer, Götz"https://www.zbmath.org/authors/?q=ai:pfeiffer.gotz"Röhrle, Gerhard"https://www.zbmath.org/authors/?q=ai:rohrle.gerhard-eLet \(W\) be a finite Coxeter group with a (Coxeter) generating set \(S\) of order \(\ell\), \(V_\mathbb{R}\) be an \(\ell\)-dimensional real vector space that affords the reflection representation of \(W\), \(V=
\mathbb{C} \otimes_{\mathbb{R}} V_{\mathbb{R}}\) be the complexification of \(V_{\mathbb{R}}\) and let \(R\) be the set of reflections of \(W\). If \(W\) is seen as a subgroup \(\mathrm{GL}(V)\), for every \(r\in R\) let \(V^{r}\) be the hyperplane of fixed points of \(r\) in \(V\) and let \(\mathcal{A} =\{ V^{r} \mid r \in R \}\). Then \((V, \mathcal{A})\) is the complexification of a Coxeter arrangement.
The group \(W\) acts naturally on the complement \(M_{W}\) of the hyperplanes in \(\mathcal{A}\), and hence on the cohomology of \(M_{W}\) as algebra automorphisms. For \(p \geq 0\) let \(H^{p}(M_{W})\) denote the \(p^{\mathrm{th}}\) de Rahm cohomology space of \(M_{W}\).
\textit{G. Felder} and \textit{A. P. Veselov} [J. Eur. Math. Soc. (JEMS) 7, No. 1, 101--116 (2005; Zbl 1070.20045)] have conjectured an explicit construction of \(H^{p}(M_{W})^{W}\), the space of \(W\)-invariants in \(H^{p} (M_{W})\), in terms of so-called special involutions and they have verified their conjecture for all Coxeter groups except those with irreducible components of type \(E_{7}\), \(E_{8}\), \(F_{4}\), \(H_{3}\) or \(H_{4}\).
The purpose of the paper under review is to complete the proof of the conjecture of Felder-Veselov (Theorem 2.1). The authors get their result by refining \textit{E. Brieskorn}'s [Lect. Notes Math. 317, 21--44 (1973; Zbl 0277.55003)] study of the cohomology of the complement of the reflection arrangement of the Coxeter group \(W\).
Reviewer: Egle Bettio (Venezia)Diagonal Minkowski classes, zonoid equivalence, and stable laws.https://www.zbmath.org/1455.520032021-03-30T15:24:00+00:00"Molchanov, Ilya"https://www.zbmath.org/authors/?q=ai:molchanov.ilya-s"Nagel, Felix"https://www.zbmath.org/authors/?q=ai:nagel.felixPlus minus analogues for affine Tverberg type results.https://www.zbmath.org/1455.520232021-03-30T15:24:00+00:00"Blagojević, Pavle V. M."https://www.zbmath.org/authors/?q=ai:blagojevic.pavle-v-m"Ziegler, Günter M."https://www.zbmath.org/authors/?q=ai:ziegler.gunter-mFor a face \(\mu\) of a simplex \(\Delta\), denote by \(\mu^{\prime}\) the complementary face of \(\Delta\). The Tverberg plus minus theorem is due to \textit{I. Bárány} and \textit{P. Soberón} [Discrete Comput. Geom. 60, No. 3, 588--598 (2018; Zbl 1401.52014)] and it has the following enounce:
Let \(d \geq 1\), \(r \geq 2\) be integers, \(N = (r - 1)(d + 1)\) and let \(a\) be an affine map of the \(N\)-dimensional simplex \(\Delta_N\) into
the Euclidean space \(\mathbb{R}^d\). Furthermore, let \(\mu\) be a face of \(\Delta_N\) of dimension at most \(r - 2\)
with \(a(\mu) \cap a(\mu^{\prime}) = \emptyset\). Then there exist \(r\) pairwise disjoint proper faces \(\sigma_1, \dots, \sigma_r\) of \(\Delta_N\) and there exists a point \(b \in \operatorname{aff}(a(\sigma_1)) \cap \dots \cap \operatorname{aff}(a(\sigma_r))\) such that for its \(r\) affine representations:
\[
b = \sum_{v \in \operatorname{vert} \sigma_i} \alpha_v a(v), \sum_{v \in \operatorname{vert} \sigma_i} \alpha_v = 1, \; i = 1, \dots, r
\]
the following implications hold:
\[
v \in \operatorname{vert} \mu \Longrightarrow a(v)\leq 0\; \text{ and } v \in \operatorname{vert} \mu^{\prime} \Longrightarrow a(v)\geq 0.
\]
Using an appropriate projective transformation, the authors show that the Tverberg plus minus theorem is a corollary of the classical Tverberg theorem.
The new proof allows to directly derive plus minus analogues of other known affine Tverberg type results.
Reviewer: Mircea Balaj (Oradea)Correction to: ``Cohomology of the toric arrangement associated with \(A_n\)''.https://www.zbmath.org/1455.200262021-03-30T15:24:00+00:00"Bergvall, Olof"https://www.zbmath.org/authors/?q=ai:bergvall.olofCorrection to the author's paper [ibid. 21, No. 1, Paper No. 15, 14 p. (2019; Zbl 1454.20081)].A probabilistic proof of the spherical excess formula.https://www.zbmath.org/1455.520012021-03-30T15:24:00+00:00"Klain, Daniel A."https://www.zbmath.org/authors/?q=ai:klain.daniel-aSummary: A probabilistic proof of Girard's angle excess formula for the area of a spherical triangle emerges from the observation that an unbounded 3-dimensional convex cone, with single vertex at the origin, has only three kinds of 2-dimensional orthogonal projections: a 2-dimensional convex cone with one vertex, a 2-dimensional half-plane (an outcome with probability zero), and a 2-dimensional plane.On overlays and minimization diagrams.https://www.zbmath.org/1455.520272021-03-30T15:24:00+00:00"Koltun, Vladlen"https://www.zbmath.org/authors/?q=ai:koltun.vladlen"Sharir, Micha"https://www.zbmath.org/authors/?q=ai:sharir.michaSummary: The overlay of \(2\leq m\leq d\) minimization diagrams of \(n\) surfaces in \(\mathbb R^{d}\) is isomorphic to a substructure of a suitably constructed minimization diagram of \(mn\) surfaces in \(\mathbb R^{d+m-1}\). This elementary observation leads to a new bound on the complexity of the overlay of minimization diagrams of collections of \(d\)-variate semi-algebraic surfaces, a tight bound on the complexity of the overlay of minimization diagrams of collections of hyperplanes, and faster algorithms for constructing such overlays. Further algorithmic implications are discussed.Convex lattice polygons with all lattice points visible.https://www.zbmath.org/1455.520132021-03-30T15:24:00+00:00"Morrison, Ralph"https://www.zbmath.org/authors/?q=ai:morrison.ralph"Tewari, Ayush Kumar"https://www.zbmath.org/authors/?q=ai:tewari.ayush-kumarSummary: Two lattice points are visible to one another if there exist no other lattice points on the line segment connecting them. In this paper we study convex lattice polygons that contain a lattice point such that all other lattice points in the polygon are visible from it. We completely classify such polygons, show that there are finitely many of lattice width greater than 2, and computationally enumerate them. As an application of this classification, we prove new obstructions to graphs arising as skeleta of tropical plane curves.The complete enumeration of 4-polytopes and 3-spheres with nine vertices.https://www.zbmath.org/1455.520082021-03-30T15:24:00+00:00"Firsching, Moritz"https://www.zbmath.org/authors/?q=ai:firsching.moritzSummary: We describe an algorithm to enumerate polytopes. This algorithm is then implemented to give a complete classification of combinatorial spheres of dimension 3 with 9 vertices and decide polytopality of those spheres. In order to decide polytopality, we generate polytopes by adding suitable points to polytopes with less than 9 vertices and therefore realize as many as possible of the combinatorial spheres as polytopes. For the rest, we prove non-realizability with techniques from oriented matroid theory. This yields a complete enumeration of all combinatorial types of 4-dimensional polytopes with 9 vertices. It is shown that all of those combinatorial types are rational: They can be realized with rational coordinates. We find 316 014 combinatorial spheres on 9 vertices. Of those, 274 148 can be realized as the boundary complex of a four-dimensional polytope and the remaining 41 866 are non-polytopal.Distinct volume subsets.https://www.zbmath.org/1455.520152021-03-30T15:24:00+00:00"Conlon, David"https://www.zbmath.org/authors/?q=ai:conlon.david"Fox, Jacob"https://www.zbmath.org/authors/?q=ai:fox.jacob"Gasarch, William"https://www.zbmath.org/authors/?q=ai:gasarch.william-i"Harris, David G."https://www.zbmath.org/authors/?q=ai:harris.david-g"Ulrich, Douglas"https://www.zbmath.org/authors/?q=ai:ulrich.douglas"Zbarsky, Samuel"https://www.zbmath.org/authors/?q=ai:zbarsky.samuelLecture \# 1: Kakeya problem, Kakeya maximal operator, and the restriction phenomenon: connections and relationships.https://www.zbmath.org/1455.520182021-03-30T15:24:00+00:00"Iosevich, Alex"https://www.zbmath.org/authors/?q=ai:iosevich.alexSummary: The purpose of this lecture is to introduce the Kakeya set problem, the Kakeya maximal operator, and the restriction problem, and to discuss the connections between those three.Local orientation-preserving symmetry preserving operations on polyhedra.https://www.zbmath.org/1455.050632021-03-30T15:24:00+00:00"Goetschalckx, Pieter"https://www.zbmath.org/authors/?q=ai:goetschalckx.pieter"Coolsaet, Kris"https://www.zbmath.org/authors/?q=ai:coolsaet.kris"Van Cleemput, Nico"https://www.zbmath.org/authors/?q=ai:van-cleemput.nicolasSummary: Unifying approaches by amongst others Archimedes, Kepler, Goldberg, Caspar and Klug, Coxeter, and Conway, and extending on a previous formalization of the concept of local symmetry preserving (lsp) operations, we introduce a formal definition of local operations on plane graphs that preserve orientation-preserving symmetries, but not necessarily orientation-reversing symmetries. This operations include, e.g., the chiral Goldberg and Conway operations as well as all lsp operations. We prove the soundness of our definition as well as introduce an invariant which can be used to systematically construct all such operations. We also show sufficient conditions for an operation to preserve the connectedness of the plane graph to which it is applied.Localization of plus-one generated arrangements.https://www.zbmath.org/1455.520242021-03-30T15:24:00+00:00"Palezzato, Elisa"https://www.zbmath.org/authors/?q=ai:palezzato.elisa"Torielli, Michele"https://www.zbmath.org/authors/?q=ai:torielli.micheleIn the paper under review the authors study some properties of localizations of plus-one generated arrangements. Let \(V\) be a vector space of dimension \(l\) over a field \(\mathbb{K}\) and consider a hyperplane arrangement \(\mathcal{A} = \{H_{1},\dots, H_{n}\}\) such that \(H_{i} = \operatorname{ker}(a_{i})\) for \(\alpha_{i} \in S = \mathbb{K}[x_{1},\dots,x_{l}]\). Denote by \(Q\) the defining polynomial, i.e, \(Q(x,y,z) = \prod_{i=1}^{n}\alpha_{i}\). Denote by \(L(\mathcal{A}) = \{ \bigcap_{H \in \mathcal{B}} H : \mathcal{B} \subset \mathcal{A}\}\) the intersection lattice of \(\mathcal{A}\) and we define \(L(\mathcal{A})_{p} = \{X \in L(\mathcal{A}) : \operatorname{codim}(X) = p \}\). Moreover, for any flat \(X \in L(\mathcal{A})\) we define the localization of \(\mathcal{A}\) to \(X\) as the subarrangement
\[
\mathcal{A}_{X} = \{H \in \mathcal{A} : X \subseteq H\}.
\]
In the first part of the paper, the authors focus on the associated primes of the Milnor algebra \(M(Q) = S / J_{Q}\), where \(J_{Q}\) is the Jacobian ideal of a plus-one generated arrangement \(\mathcal{A}\) defined by \(Q\).
We say that \(\mathcal{A} = \{H_{1},\dots, H_{n}\} \subset \mathbb{K}^{l}\) is plus-one generated with exponents \(\operatorname{POexp}(\mathcal{A}) = (d_{1},\dots, d_{l})_{\leq}\) and level \(d\) if and only if \(M(Q)\) has a minimal free resolution of the form
\[0 \rightarrow (S-n-d) \rightarrow S(-n-d+1) \oplus\bigg(\bigoplus_{i=k}^{l}S(-n-d_{i}+1)\bigg)\rightarrow S(-n+1)^{l-k+2} \rightarrow S.\]
Observe that each \(X \in L(\mathcal{A})_{k}\) corresponds a prime ideal \(I(X) = \langle a_{i_{1}},\dots, \alpha_{i_{k}} \rangle\) of codimension \(k\), where \(X = H_{i_{1}} \cap\dots \cap H_{i_{k}}\)
Theorem A. Let \(\mathcal{A}\) be an arrangement in \(\mathbb{K}^{\ell}\) such that \(M(Q)\) has projective dimension \(3\). Then
\[
\operatorname{Ass}_{S}(M(Q)) = \{ I(X) : X \in L(\mathcal{A})_{2}\} \cup \{I(X) : X \in L(\mathcal{A})_{3}, \, \mathcal{A}_{X} \, \text{non-free}\}.
\]
The second result of the paper is devoted to a natural question regarded to localizations of plus-one generated arrangements.
Theorem B. Let \(\mathcal{A}\) be a plus-one generated arrangement in \(\mathbb{K}^{l}\) and \(X \in L(\mathcal{A})\). The \(\mathcal{A}_{X}\) is free or plus-one generated.
Reviewer: Piotr Pokora (Kraków)Fault-free tilings of the \(3 \times n\) rectangle with squares and dominos.https://www.zbmath.org/1455.050152021-03-30T15:24:00+00:00"Alabi, Oluwatobi Jemima"https://www.zbmath.org/authors/?q=ai:alabi.oluwatobi-jemima"Dresden, Greg"https://www.zbmath.org/authors/?q=ai:dresden.gregSummary: A (completely) fault-free tiling of a board is a tiling with no vertical or horizontal faults. We find a fourth-order recurrence relation for the number of ways to tile such a board using squares and dominos, and we do the same for a vertical-fault-free board (which could have horizontal faults).Spherically convex sets and spherically convex functions.https://www.zbmath.org/1455.520072021-03-30T15:24:00+00:00"Guo, Qi"https://www.zbmath.org/authors/?q=ai:guo.qi"Peng, Yanling"https://www.zbmath.org/authors/?q=ai:peng.yanlingSummary: We define first the spherical convexity of sets and functions on general curved surfaces by an analytic approach. Then we study several kinds of properties of spherically convex sets and functions. Several analogies of the results for convex sets and convex functions on Euclidean spaces are established or rediscovered for spherically convex sets and spherically convex functions, such as the Radon-type, Helly-type, Carathéodory-type and Minkowski-type theorems for spherically convex sets, and the Jensen's inequality for spherically convex functions etc. The results obtained here might have applications in some areas, e.g. in the optimization theory on general spherical spaces.On the structure of two-periodic cube tilings of the 4-dimensional Euclidean space.https://www.zbmath.org/1455.520202021-03-30T15:24:00+00:00"Dutour Sikirić, Mathieu"https://www.zbmath.org/authors/?q=ai:dutour-sikiric.mathieu"Łysakowska, Magdalena"https://www.zbmath.org/authors/?q=ai:lysakowska.magdalenaEditorial remark: This is a duplication of [ibid. 113, 69--108 (2019; Zbl 1455.52019)].An excursion through discrete differential geometry. AMS short course on discrete differential geometry, San Diego, CA, USA, January 8--9, 2018.https://www.zbmath.org/1455.530452021-03-30T15:24:00+00:00"Crane, Keenan (ed.)"https://www.zbmath.org/authors/?q=ai:crane.keenanPublisher's description: Discrete Differential Geometry (DDG) is an emerging discipline at the boundary between mathematics and computer science. It aims to translate concepts from classical differential geometry into a language that is purely finite and discrete, and can hence be used by algorithms to reason about geometric data. In contrast to standard numerical approximation, the central philosophy of DDG is to faithfully and exactly preserve key invariants of geometric objects at the discrete level. This process of translation from smooth to discrete helps to both illuminate the fundamental meaning behind geometric ideas and provide useful algorithmic guarantees.
This volume is based on lectures delivered at the 2018 AMS Short Course ``Discrete Differential Geometry,'' held January 8--9, 2018, in San Diego, California.
The papers in this volume illustrate the principles of DDG via several recent topics: discrete nets, discrete differential operators, discrete mappings, discrete conformal geometry, and discrete optimal transport.
The articles of this volume will be reviewed individually.\( \Phi \)-functions of 2D objects with boundaries being second-order curves.https://www.zbmath.org/1455.510072021-03-30T15:24:00+00:00"Gil, M. I."https://www.zbmath.org/authors/?q=ai:gil.michael-iosif"Patsuk, V. M."https://www.zbmath.org/authors/?q=ai:patsuk.v-mLet \(A\) and \(B\) be two sets on the plane. Determining the relative position of \(A\) with respect to \(B\) requires to maximize or minimize some expressions depending on some parameters. This optimization can be presented as a so-called \(\Phi\)-function.
The authors present the \(\Phi\)-functions for the following situations and sets: non-intersection of a pair of ellipses; intersection of an ellipse and an area bounded by a parabola; inclusion of an ellipse in other ellipse; inclusion of an ellipse in an area bounded by a parabola.
Reviewer: Pedro Martín Jiménez (Badajoz)The computation of overlap coincidence in Taylor-Socolar substitution tiling.https://www.zbmath.org/1455.520212021-03-30T15:24:00+00:00"Akiyama, Shigeki"https://www.zbmath.org/authors/?q=ai:akiyama.shigeki"Lee, Jeong-Yup"https://www.zbmath.org/authors/?q=ai:lee.jeong-yupSummary: Recently Taylor and Socolar introduced an aperiodic mono-tile. The associated tiling can be viewed as a substitution tiling. We use the substitution rule for this tiling and apply the algorithm of the authors [Adv. Math. 226, No. 4, 2855--2883 (2011; Zbl 1219.37013)] to check overlap coincidence. It turns out that the tiling has overlap coincidence. So the tiling dynamics has pure point spectrum and we can conclude that this tiling has a quasicrystalline structure.On classification of toric surface codes of dimension seven.https://www.zbmath.org/1455.140552021-03-30T15:24:00+00:00"Hussain, Naveed"https://www.zbmath.org/authors/?q=ai:hussain.naveed"Luo, Xue"https://www.zbmath.org/authors/?q=ai:luo.xue"Yau, Stephen S.-T."https://www.zbmath.org/authors/?q=ai:yau.stephen-shing-toung"Zhang, Mingyi"https://www.zbmath.org/authors/?q=ai:zhang.mingyi"Zuo, Huaiqing"https://www.zbmath.org/authors/?q=ai:zuo.huaiqingFollowing previous work on the classification of low-dimensional toric surface codes by \textit{S. S. T. Yau} and \textit{H. Zuo} [Appl. Algebra Eng. Commun. Comput. 20, No. 2, 175--185 (2009; Zbl 1174.94029)] and \textit{X. Luo} et al. [Finite Fields Appl. 33, 90--102 (2015; Zbl 1394.14017)] the main results of the paper under review extend the classification, up to monomial equivalence, of toric surface codes of dimension less than or equal to \(7\), with the monomial equivalence of some pairs of toric codes in this range still undetermined.
Reviewer: Felipe Zaldívar (Ciudad de México)Formal adventures in convex and conical spaces.https://www.zbmath.org/1455.682542021-03-30T15:24:00+00:00"Affeldt, Reynald"https://www.zbmath.org/authors/?q=ai:affeldt.reynald"Garrigue, Jacques"https://www.zbmath.org/authors/?q=ai:garrigue.jacques"Saikawa, Takafumi"https://www.zbmath.org/authors/?q=ai:saikawa.takafumiSummary: Convex sets appear in various mathematical theories, and are used to define notions such as convex functions and hulls. As an abstraction from the usual definition of convex sets in vector spaces, we formalize in Coq an intrinsic axiomatization of convex sets, namely convex spaces, based on an operation taking barycenters of points. A convex space corresponds to a specific type that does not refer to a surrounding vector space. This simplifies the definitions of functions on it. We show applications including the convexity of information-theoretic functions defined over types of distributions. We also show how convex spaces are embedded in conical spaces, which are abstract real cones, and use the embedding as an effective device to ease calculations.
For the entire collection see [Zbl 1452.68005].From joint convexity of quantum relative entropy to a concavity theorem of Lieb.https://www.zbmath.org/1455.520062021-03-30T15:24:00+00:00"Tropp, Joel A."https://www.zbmath.org/authors/?q=ai:tropp.joel-aSummary: This paper provides a succinct proof of a 1973 theorem of Lieb that establishes the concavity of a certain trace function. The development relies on a deep result from quantum information theory, the joint convexity of quantum relative entropy, as well as a recent argument due to Carlen and Lieb.On the structure of two-periodic cube tilings of the 4-dimensional Euclidean space.https://www.zbmath.org/1455.520192021-03-30T15:24:00+00:00"Dutour Sikirić, Mathieu"https://www.zbmath.org/authors/?q=ai:dutour-sikiric.mathieu"Łysakowska, Magdalena"https://www.zbmath.org/authors/?q=ai:lysakowska.magdalenaSummary: Two-periodic unit cube tilings of the 4-dimensional Euclidean space are classified. This paper contains the algorithm one used to do the classification and obtained results.Lattice simplices with a fixed positive number of interior lattice points: a nearly optimal volume bound.https://www.zbmath.org/1455.520102021-03-30T15:24:00+00:00"Averkov, Gennadiy"https://www.zbmath.org/authors/?q=ai:averkov.gennadiy"Krümpelmann, Jan"https://www.zbmath.org/authors/?q=ai:krumpelmann.jan-alexander"Nill, Benjamin"https://www.zbmath.org/authors/?q=ai:nill.benjaminGiven a \textit{lattice polytope} \(P \subseteq \mathbb{R}^d\) with a positive number \(k = \#\left(\mathrm{int}(P) \cap \mathbb{Z}^d\right) \geq 1\) of interior lattice points in \(P\), the problem of deriving a sharp upper bound on its volume can be traced back to a work of \textit{D. Hensley} [Pac. J. Math. 105, 183--191 (1983; Zbl 0471.52006)].
In recent years, this problem has received a lot of attention, not only because of its natural appeal but also from its connections to integer programming and toric geometry.
For the precise formulation of the main conjecture that is tackled in the paper at hand, define \(p(d,k)\) and \(s(d,k)\) as the maximal volume of a lattice polytope and lattice simplex in \(\mathbb{R}^d\), respectively, with a given number \(k\geq1\) of interior lattice points.
The conjecture is that \(p(d,k) = s(d,k) = \mathrm{vol}(S_{d,k})\), where \(S_{d,k}\) denotes the so-called Zaks-Perles-Wills-simplex.
The volume of these simplices \(S_{d,k}\) is a linear function in \(k\), but a quadratic function in the number \(s_d\), which denotes an element in the double-exponentially growing \textit{Sylvester sequence} \((s_i)_{i=1,2,\dots}\).
The main result of the paper is to show that \(\mathrm{vol}(S_{d,k}) \leq s(d,k) \leq (d+1)\mathrm{vol}(S_{d,k})\).
This major advance solves said problem for the class of lattice simplices up to a linear factor in the dimension \(d\), and thus further supports the main conjecture.
The methods are based on the combination of the many insights regarding \(p(d,k)\) and \(s(d,k)\) that have been collected over the years, and in particular on a generalization of the elegant \textit{sum-product-inequalities} on the barycentric coordinates of an interior lattice point of a lattice simplex.
Reviewer: Matthias Schymura (Lausanne)An FPTAS for the volume of some \(\mathcal{V} \)-polytopes -- it is hard to compute the volume of the intersection of two cross-polytopes.https://www.zbmath.org/1455.682732021-03-30T15:24:00+00:00"Ando, Ei"https://www.zbmath.org/authors/?q=ai:ando.ei"Kijima, Shuji"https://www.zbmath.org/authors/?q=ai:kijima.shujiA \(\mathcal{V} \)-polytope is a convex hull of a finite point set in \(\mathbb{R}^n\). A particular type of \(\mathcal{V} \)-polytope, called ``knapsack dual polytope'' is defined as \(P_{a}=\text{conv}\{\pm e_1, \pm e_2, ..., \pm e_n, a\}\), \(a\in \mathbb{Z}^n_{\geq 0}\). The authors constructively prove the following theorem: For any \(\varepsilon\), \((0 <\varepsilon <1)\), there is a deterministic algorithm that computes \(\hat{V}\) satisfying \(|1- \frac{\text{Vol}(P_a)}{\hat{V}}| \leq \varepsilon\) in \(O(n^{10}\varepsilon^{-6})\) time. This is the first result on designing a fully polynomial time approximation scheme for computing the volume of a \(\mathcal{V} \)-polytope, which is known to be \#P-hard.
Reviewer: Gabriela Cristescu (Arad)Computing the integer points of a polyhedron. I: Algorithm.https://www.zbmath.org/1455.520112021-03-30T15:24:00+00:00"Jing, Rui-Juan"https://www.zbmath.org/authors/?q=ai:jing.rui-juan"Moreno Maza, Marc"https://www.zbmath.org/authors/?q=ai:moreno-maza.marcSummary: Let \(K\) be a polyhedron in \(\mathbb R^d\), given by a system of \(m\) linear inequalities, with rational number coefficients bounded over in absolute value by \(L\). In this series of two papers, we propose an algorithm for computing an irredundant representation of the integer points of \(K\), in terms of ``simpler'' polyhedra, each of them having at least one integer point. Using the terminology of W. Pugh: for any such polyhedron \(P\), no integer point of its grey shadow extends to an integer point of \(P\). We show that, under mild assumptions, our algorithm runs in exponential time w.r.t. \(d\) and in polynomial w.r.t \(m\) and \(L\). We report on a software experimentation. In this series of two papers, the first one presents our algorithm and the second one [Zbl 1455.52012] discusses our complexity estimates.
For the entire collection see [Zbl 1371.68008].Computing the integer points of a polyhedron. II: Complexity estimates.https://www.zbmath.org/1455.520122021-03-30T15:24:00+00:00"Jing, Rui-Juan"https://www.zbmath.org/authors/?q=ai:jing.rui-juan"Moreno Maza, Marc"https://www.zbmath.org/authors/?q=ai:moreno-maza.marcSummary: Let \(K\) be a polyhedron in \(\mathbb R^d\), given by a system of \(m\) linear inequalities, with rational number coefficients bounded over in absolute value by \(L\). In this series of two papers, we propose an algorithm for computing an irredundant representation of the integer points of \(K\), in terms of ``simpler'' polyhedra, each of them having at least one integer point. Using the terminology of W. Pugh: for any such polyhedron \(P\), no integer point of its grey shadow extends to an integer point of \(P\). We show that, under mild assumptions, our algorithm runs in exponential time w.r.t. \(d\) and in polynomial w.r.t \(m\) and \(L\). We report on a software experimentation. In this series of two papers, the first one [the authors, ibid. 10490, 225--241 (2017; Zbl 1455.52011)] presents our algorithm and the second one discusses our complexity estimates.
For the entire collection see [Zbl 1371.68008].On some properties of certain discrete point-sets in Euclidean spaces.https://www.zbmath.org/1455.520162021-03-30T15:24:00+00:00"Kirtadze, A."https://www.zbmath.org/authors/?q=ai:kirtadze.aleks-p"Kasrashvili, T."https://www.zbmath.org/authors/?q=ai:kasrashvili.tamar(no abstract)Modular construction of free hyperplane arrangements.https://www.zbmath.org/1455.520252021-03-30T15:24:00+00:00"Tsujie, Shuhei"https://www.zbmath.org/authors/?q=ai:tsujie.shuheiSummary: In this article, we study freeness of hyperplane arrangements. One of the most investigated arrangement is a graphic arrangement. Stanley proved that a graphic arrangement is free if and only if the corresponding graph is chordal and Dirac showed that a graph is chordal if and only if the graph is obtained by ``gluing'' complete graphs. We will generalize Dirac's construction to simple matroids with modular joins introduced by Ziegler and show that every arrangement whose associated matroid is constructed in the manner mentioned above is divisionally free. Moreover, we apply the result to arrangements associated with gain graphs and arrangements over finite fields.Fat-wedge filtration and decomposition of polyhedral products.https://www.zbmath.org/1455.550072021-03-30T15:24:00+00:00"Iriye, Kouyemon"https://www.zbmath.org/authors/?q=ai:iriye.kouyemon"Kishimoto, Daisuke"https://www.zbmath.org/authors/?q=ai:kishimoto.daisukeLet \(K\) be a finite simplicial complex with vertex set \([m]=\{1,\dots,m\}\).
For a sequence \((X_i,A_i)\) of pairs of spaces indexed by \(i\in[m]\), the polyhedral product \(\mathcal{Z}_K(X,A)\) is defined as \(\bigcup_{\sigma\in K}(X,A)^\sigma\), where \((X,A)^\sigma=Y_1\times\cdots\times Y_m\subset X_1\times\cdots\times X_m\); here \(Y_i=X_i\) if \(i\in\sigma\) and \(Y_i=A_i\) if \(i\not\in\sigma\).
This technical construction is interesting since, e.g., moment angle complexes, arrangements of real coordinate subspaces and their complements arise as special cases; moreover, cohomology rings and fundamental groups of polyhedral products occur in commutative algebra and geometric group theory.
The authors obtain general structural information about the homotopy types of polyhedral products, strengthening results of Bahri, Bendersky, Cohen, and Gitler [\textit{A. Bahri} et al., Adv. Math. 225, No. 3, 1634--1668 (2010; Zbl 1197.13021)].
The main focus is on the case where each \(X_i\) is a cone over the base space \(A_i\).
In a second train of thought the authors investigate what happens if the Alexander dual of \(K\) satisfies certain combinatorial properties (e.g., shellability) which imply (sequential) Cohen-Macaulayness.
Reviewer: Michael Joswig (Berlin)On levels in arrangements of surfaces in three dimensions.https://www.zbmath.org/1455.520262021-03-30T15:24:00+00:00"Chan, Timothy M."https://www.zbmath.org/authors/?q=ai:chan.timothy-m-ySummary: A favorite open problem in combinatorial geometry is to determine the worst-case complexity of a \(level\) in an arrangement. Up to now, nontrivial upper bounds in three dimensions are known only for the linear cases of planes and triangles. We propose the first technique that can deal with more general surfaces in three dimensions. For example, in an arrangement of \(n\) ``pseudo-planes'' or ``pseudo-spherical patches'' (where the main criterion is that each triple of surfaces has at most two common intersections), we prove that there are at most \(O(n ^{2.997})\) vertices at any given level.Counting integral points in polytopes via numerical analysis of contour integration.https://www.zbmath.org/1455.520092021-03-30T15:24:00+00:00"Hirai, Hiroshi"https://www.zbmath.org/authors/?q=ai:hirai.hiroshi"Oshiro, Ryunosuke"https://www.zbmath.org/authors/?q=ai:oshiro.ryunosuke"Tanaka, Ken'ichiro"https://www.zbmath.org/authors/?q=ai:tanaka.kenichiroSummary: In this paper, we address the problem of counting integer points in a rational polytope described by \(P(\mathbf{y}) = \{\mathbf{x} \in \mathbb{R}^m: A\mathbf{x}= \mathbf{y}, \, \mathbf{x} \geq 0\}\), where \(A\) is an \(n \times m\) integer matrix and \(\mathbf{y}\) is an \(n\)-dimensional integer vector. We study the \(Z\) transformation approach initiated in works by Brion and Vergne, Beck, and Lasserre and Zeron from the numerical analysis point of view and obtain a new algorithm on this problem. If \(A\) is nonnegative, then the number of integer points in \(P(\mathbf{y})\) can be computed in \(O(\operatorname{poly} (n,m, \| \mathbf{y} \|_\infty)(\| \mathbf{y} \|_\infty + 1)^n)\) time and \(O(\operatorname{poly} (n,m, \| \mathbf{y} \|_\infty))\) space. This improves, in terms of space complexity, a naive DP algorithm with \(O((\| \mathbf{y} \|_\infty + 1)^n)\)-size dynamic programming table. Our result is based on the standard error analysis of the numerical contour integration for the inverse \(Z\) transform and establishes a new type of an inclusion-exclusion formula for integer points in \(P(\mathbf{y})\).
We apply our result to hypergraph \(\mathbf{b}\)-matching and obtain an \(O(\operatorname{poly}(n,m,\|\mathbf{b}\|_\infty)(\|\mathbf{b}\|_\infty + 1)^{(1-1/k)n}\) time algorithm for counting \(\mathbf{b}\)-matchings in a \(k\)-partite hypergraph with \(n\) vertices and \(m\) hyperedges. This result is viewed as a \(\mathbf{b}\)-matching generalization of the classical result by Ryser for \(k= 2\) and its multipartite extension by Björklund and Husfeldt.The Pagoda sequence: a ramble through linear complexity, number walls, D0L sequences, finite state automata, and aperiodic tilings.https://www.zbmath.org/1455.681542021-03-30T15:24:00+00:00"Lunnon, Fred"https://www.zbmath.org/authors/?q=ai:lunnon.fred.1Summary: We review the concept of the number wall as an alternative to the traditional linear complexity profile (LCP), and sketch the relationship to other topics such as linear feedback shift-register (LFSR) and context-free Lindenmayer (D0L) sequences. A remarkable ternary analogue of the Thue-Morse sequence is introduced having deficiency 2 modulo 3, and this property verified via the re-interpretation of the number wall as an aperiodic plane tiling.
For the entire collection see [Zbl 1392.68029].Toric log del Pezzo surfaces with one singularity.https://www.zbmath.org/1455.140982021-03-30T15:24:00+00:00"Dais, Dimitrios I."https://www.zbmath.org/authors/?q=ai:dais.dimitrios-iSummary: This paper focuses on the classification (up to isomorphism) of all toric log Del Pezzo surfaces with exactly one singularity, and on the description of how they are embedded as intersections of finitely many quadrics into suitable projective spaces.A periodic isoperimetric problem related to the unique games conjecture.https://www.zbmath.org/1455.600292021-03-30T15:24:00+00:00"Heilman, Steven"https://www.zbmath.org/authors/?q=ai:heilman.steven-mSummary: We prove the endpoint case of a conjecture of \textit{S. Khot} and \textit{D. Moshkovitz} [in: Proceedings of the 48th annual ACM SIGACT symposium on theory of computing, STOC '16, Cambridge, MA, USA, June 19--21, 2016. New York, NY: Association for Computing Machinery (ACM). 63--76 (2016; Zbl 1373.68240)] related to the unique games conjecture, less a small error. Let \(n\geq 2\). Suppose a subset \(\Omega\) of \(n\)-dimensional Euclidean space \(\mathbb{R}^n\) satisfies \(-\Omega=\Omega^c\) and \(\Omega+v=\Omega^c\) (up to measure zero sets) for every standard basis vector \(v\in\mathbb{R}^n\). For any \(x=(x_1,\dots,x_n)\in\mathbb{R}^n\) and for any \(q\geq 1\), let \(||x||^q_q=|x_1|^q+\cdots+|x_n|^q\) and let \(\gamma_n(x)=(2\pi)^{-n/2}e^{-||x||^2_2/2}\). For any \(x\in\partial\Omega\), let \(N(x)\) denote the exterior normal vector at \(x\) such that \(\Vert N(x)\Vert_2=1\). Let \(B=\{x\in\mathbb{R}^n:\sin(\pi(x_1+\cdots+x_n))\geq 0\}\). Our main result shows that \(B\) has the smallest Gaussian surface area among all such subsets \(\Omega\), less a small error: \(\int_{\partial\Omega}\gamma_n(x)\mathrm{d}x\geq (1-6\times 10^{-9})\int_{\partial B}\gamma_n(x)\mathrm{d}x+\int_{\partial\Omega}\left(1-\frac{||N(x)||_1}{\sqrt n}\right)\gamma_n(x)\mathrm{d}x\). In particular, \(\int_{\partial\Omega}\gamma_n(x)\mathrm{d}x\geq(1-6\times 10^{-9})\int_{\partial B}\gamma_n(x)\mathrm{d}x\). Standard arguments extend these results to a corresponding weak inequality for noise stability. Removing the factor \(6\times 10^{-9}\) would prove the endpoint case of the Khot-Moshkovitz conjecture. Lastly, we prove a Euclidean analogue of the Khot and Moshkovitz conjecture. The full conjecture of Khot and Moshkovitz provides strong evidence for the truth of the unique games conjecture, a central conjecture in theoretical computer science that is closely related to the P versus NP problem. So, our results also provide evidence for the truth of the unique games conjecture. Nevertheless, this paper does not prove any case of the unique games conjecture.On the nonlinear Brascamp-Lieb inequality.https://www.zbmath.org/1455.420242021-03-30T15:24:00+00:00"Bennett, Jonathan"https://www.zbmath.org/authors/?q=ai:bennett.jonathan-m"Bez, Neal"https://www.zbmath.org/authors/?q=ai:bez.neal"Buschenhenke, Stefan"https://www.zbmath.org/authors/?q=ai:buschenhenke.stefan"Cowling, Michael G."https://www.zbmath.org/authors/?q=ai:cowling.michael-g"Flock, Taryn C."https://www.zbmath.org/authors/?q=ai:flock.taryn-cSummary: We prove a nonlinear variant of the general Brascamp-Lieb inequality. Our proof consists of running an efficient, or ``tight'', induction-on-scales argument, which uses the existence of Gaussian near-extremizers to the underlying linear Brascamp-Lieb inequality (Lieb's theorem) in a fundamental way. A key ingredient is an effective version of Lieb's theorem, which we establish via a careful analysis of near-minimizers of weighted sums of exponential functions. Instances of this inequality are quite prevalent in mathematics, and we illustrate this with some applications in harmonic analysis.Reverse Loomis-Whitney inequalities via isotropicity.https://www.zbmath.org/1455.520022021-03-30T15:24:00+00:00"Alonso-Gutiérrez, David"https://www.zbmath.org/authors/?q=ai:alonso-gutierrez.david"Brazitikos, Silouanos"https://www.zbmath.org/authors/?q=ai:brazitikos.silouanosSummary: Given a centered convex body \(K\subseteq\mathbb{R}^n\), we study the optimal value of the constant \(\tilde{\Lambda}(K)\) such that there exists an orthonormal basis \(\{w_i\}_{i=1}^n\) for which the following reverse dual Loomis-Whitney inequality holds:
\[
\vert K\vert^{n-1}\leqslant\tilde{\Lambda}(K)\prod\limits_{i=1}^n\vert K\cap w_i^\perp \vert.
\]
We prove that \(\tilde{\Lambda}(K)\leqslant (CL_K)^n\) for some absolute \(C>1\) and that this estimate in terms of \(L_K\), the isotropic constant of \(K\), is asymptotically sharp in the sense that there exist another absolute constant \(c>1\) and a convex body \(K\) such that \((cL_K)^n\leqslant \tilde{\Lambda }(K)\leqslant (CL_K)^n\). We also prove more general reverse dual Loomis-Whitney inequalities as well as reverse restricted versions of Loomis-Whitney and dual Loomis-Whitney inequalities.Improved bounds for pencils of lines.https://www.zbmath.org/1455.520172021-03-30T15:24:00+00:00"Roche-Newton, Oliver"https://www.zbmath.org/authors/?q=ai:roche-newton.oliver"Warren, Audie"https://www.zbmath.org/authors/?q=ai:warren.audieSummary: We consider a question raised by Rudnev: given four pencils of \(n\) concurrent lines in \(\mathbb{R}^2\), with the four centres of the pencils non-collinear, what is the maximum possible size of the set of points where four lines meet? Our main result states that the number of such points is \(O(n^{11/6})\), improving a result of Chang and Solymosi.
We also consider constructions for this problem. Alon, Ruzsa, and Solymosi constructed an arrangement of four non-collinear \(n\)-pencils which determine \(\Omega (n^{3/2})\) four-rich points. We give a construction to show that this is not tight, improving this lower bound by a logarithmic factor. We also give a construction of a set of \(m n\)-pencils, whose centres are in general position, that determine \(\Omega_m(n^{3/2}) m\)-rich points.BMO-type seminorms from Escher-type tessellations.https://www.zbmath.org/1455.460342021-03-30T15:24:00+00:00"Di Fratta, Giovanni"https://www.zbmath.org/authors/?q=ai:di-fratta.giovanni"Fiorenza, Alberto"https://www.zbmath.org/authors/?q=ai:fiorenza.albertoSummary: The paper is about a representation formula introduced by Fusco, Moscariello, and Sbordone in
[\textit{N.~Fusco} et al., ESAIM, Control Optim. Calc. Var. 24, No.~2, 835--847 (2018; Zbl 1410.46021)].
The formula permits to characterize the gradient norm of a Sobolev function, defined on the whole space \(\mathbb{R}^n\), as the limit of non-local energies (BMO-type seminorms) defined on tessellations of generated by cubic cells with arbitrary orientation. We improve the main result in [loc. cit.] in three different regards: we give a new concise proof of the representation formula, we analyze the case of a generic open subset \(\Omega \subseteq \mathbb{R}^n\), and consider general tessellations of \(\Omega\) by means of cells more general than cubes, again arbitrarily-oriented, inspired by the creative mind of the graphic artist M. C. Escher.Erratum for: ``On the number of lattice points in convex symmetric bodies and their duals''.https://www.zbmath.org/1455.520142021-03-30T15:24:00+00:00"Gillet, Henri"https://www.zbmath.org/authors/?q=ai:gillet.henri-a"Soulé, Christophe"https://www.zbmath.org/authors/?q=ai:soule.christophe(no abstract)Efficient explicit constructions of multipartite secret sharing schemes.https://www.zbmath.org/1455.942072021-03-30T15:24:00+00:00"Chen, Qi"https://www.zbmath.org/authors/?q=ai:chen.qi"Tang, Chunming"https://www.zbmath.org/authors/?q=ai:tang.chunming"Lin, Zhiqiang"https://www.zbmath.org/authors/?q=ai:lin.zhiqiangSummary: Multipartite secret sharing schemes are those having a multipartite access structure, in which the set of participants is divided into several parts and all participants in the same part play an equivalent role. Secret sharing schemes for multipartite access structures have received considerable attention due to the fact that multipartite secret sharing can be seen as a natural and useful generalization of threshold secret sharing.
This work deals with efficient and explicit constructions of ideal multipartite secret sharing schemes, while most of the known constructions are either inefficient or randomized. Most ideal multipartite secret sharing schemes in the literature can be classified as either hierarchical or compartmented. The main results are the constructions for ideal hierarchical access structures, a family that contains every ideal hierarchical access structure as a particular case such as the disjunctive hierarchical threshold access structure and the conjunctive hierarchical threshold access structure, and the constructions for compartmented access structures with upper bounds and compartmented access structures with lower bounds, two families of compartmented access structures.
On the basis of the relationship between multipartite secret sharing schemes, polymatroids, and matroids, the problem of how to construct a scheme realizing a multipartite access structure can be transformed to the problem of how to find a representation of a matroid from a presentation of its associated polymatroid. In this paper, we give efficient algorithms to find representations of the matroids associated to the three families of multipartite access structures. More precisely, based on know results about integer polymatroids, for each of the three families of access structures, we give an efficient method to find a representation of the integer polymatroid over some finite field, and then over some finite extension of that field, we give an efficient method to find a presentation of the matroid associated to the integer polymatroid. Finally, we construct ideal linear schemes realizing the three families of multipartite access structures by efficient methods.
For the entire collection see [Zbl 1428.94009].Quotient functions of dual quermassintegrals.https://www.zbmath.org/1455.520052021-03-30T15:24:00+00:00"Zhao, Chang-Jian"https://www.zbmath.org/authors/?q=ai:zhao.changjianSummary: Motivated by the notion of volume difference functions, we introduce quotient functions of dual quermassintegrals and establish Brunn-Minkowski type inequalities for them, which have several recent results as special cases.The diagonal of the associahedra.https://www.zbmath.org/1455.180142021-03-30T15:24:00+00:00"Masuda, Naruki"https://www.zbmath.org/authors/?q=ai:masuda.naruki"Thomas, Hugh"https://www.zbmath.org/authors/?q=ai:thomas.hugh-ross"Tonks, Andy"https://www.zbmath.org/authors/?q=ai:tonks.andy"Vallette, Bruno"https://www.zbmath.org/authors/?q=ai:vallette.brunoThis paper has a threefold purpose.
\begin{itemize}
\item[1.] to introduce a general machinery to solve the problem of the approximation of the diagonal of \textit{face-coherent families of polytopes} (\S 2),
\item[2.] to give a complete proof for the case of the \textit{associahedra} (Theorem 1), and
\item[3.] to popularize the resulting \textit{magical formula} (Theorem 2).
\end{itemize}
The problem of the approximation of the diagonal of the associahedra lies at the crossroads of three clusters of domains.
\begin{itemize}
\item[1.] There are mathematicians inclined to apply it in their work of computing the homology of fibered spaces in algebraic topology [\textit{E. H. Brown jun.}, Ann. Math. (2) 69, 223--246 (1959; Zbl 0199.58201); \textit{A. Prouté}, Repr. Theory Appl. Categ. 2011, No. 21, 99 p. (2011; Zbl 1245.55007)], to construct tensor products of string field theories [\textit{M. R. Gaberdiel} and \textit{B. Zwiebach}, Nucl. Phys., B 505, No. 3, 569--624 (1997; Zbl 0911.53044); Phys. Lett., B 410, No. 2--4, 151--159 (1997; Zbl 0911.53046)], or to consider the product of Fukaya \(\mathcal{A}_{\infty}\)-categories in symplectic geometry [\textit{P. Seidel}, Fukaya categories and Picard-Lefschetz theory. Zürich: European Mathematical Society (EMS) (2008; Zbl 1159.53001); \textit{L. Amorim}, Int. J. Math. 28, No. 4, Article ID 1750026, 38 p. (2017; Zbl 1368.53057)].
\item[2.] The analogous result is known, within the kingdom of operad theory and homotopical algebra, in the differential graded context [\textit{S. Saneblidze} and \textit{R. Umble}, Homology Homotopy Appl. 6, No. 1, 363--411 (2004; Zbl 1069.55015); \textit{L. J. Billera} and \textit{B. Sturmfels}, Ann. Math. (2) 135, No. 3, 527--549 (1992; Zbl 0762.52003)].
\item[3.] This result is appreciated conceptualy as a new development in the theory of \textit{fiber polytopes} [loc. cit.] by combinatorists and
discrete geometers.
\end{itemize}
The possible ways of iteraring a binary product are to be encoded by planar binary trees, the associativity relation being interpreted as an order relation, which encouraged Dov Tamari to introduce the so-called \textit{Tamari lattice} [\textit{D. Tamari}, Nieuw Arch. Wiskd., III. Ser. 10, 131--146 (1962; Zbl 0109.24502)]. These lattices are to be realized by associahedra in the sense that their \(1\)-skeleton is the Hasse diagram of the Tamari lattice [\textit{C. Ceballos} et al., Combinatorica 35, No. 5, 513--551 (2015; Zbl 1389.52013)]. For loop spaces, composition fails to be strictly associative due to the different parametrizations, but this failure is governed by an infinite sequence of higher homotopies, which was made precise by \textit{J. D. Stasheff} [Trans. Am. Math. Soc. 108, 275--292, 293--312 (1963; Zbl 0114.39402)], introducing a family of curvilinear polytopes called the \textit{Stascheff polytopes}, whose combinatorics coincides with the associahedra. Stascheff's work opened the door to the study of homotopical algebra by means of operad-like objects, summoning [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. Berlin-Heidelberg-New York: Springer-Verlag (1973; Zbl 0285.55012)] and [\textit{J. P. May}, The geometry of iterated loop spaces. Berlin-Heidelberg-New York: Springer-Verlag (1972; Zbl 0244.55009)] in particular, the latter of which introduced the \textit{little disks} operads playing a key role in many domains nowadays. In dimension \(1\), this gives the \textit{little intervals} operad, a finite-dimensional topological operad pervious to Stascheff's theory, whose operad structure is given by scaling a configuration of intervals in order to insert it into another interval. So far there has been no operad structure on any family of convex polytopal realizations of the associahedra in the literature, though there should be a rainbow bridge
between operad theory as well as homotopy theories on the one hand and combinatorics as well as discrete geometry on the other.
Since the set-theoretic diagonal of a polytope fails to be cellular in general, the authors need to find a \textit{cellular approximation to the
diagonal}, that is to say, a cellular map from the polytope to its cartesian square homotopic to the diagonal. For a coherent family of polytopes, it is highly challenging to find a family of diagonals compatible with the combinatorics of faces. In the case of the first face-coherent family of polytopes, the geometric simplices, such a diagronal map is given by the classical \textit{Alexander-Whitney map} [\textit{S. Eilenberg} and \textit{J. A. Zilber}, Am. J. Math. 75, 200--204 (1953; Zbl 0050.17301); \textit{N. E. Steenrod}, Ann. Math. (2) 48, 290--320 (1947; Zbl 0030.41602)]. For the next family given by cubes, a coassociative approximation to the diagonal is straightforward [\textit{J.-P. Serre}, Ann. Math. (2) 54, 425--505 (1951; Zbl 0045.26003)]. The associahedra form the face-coherent family of polytopes coming next in terms of further truncations of the simplices or of combinatorial complexity. While a face of a simplex or a cube is a simplex or a cube of lower dimension, a face of an associahedra is a product of associahedra of lower dimensions, which makes the problem of the
approximation of the diagonal in this turn highly intricate. The two-fold principal result of the paper (Theorem 1) is an explicit operad structure on the \textit{Loday realizations} of the associahedra together with a compatible approximation to the diagonal.
There is a dichotomy between pointwise and cellular formulas. To investigate
their relationship and to make precise the various face-coherent properties,
the authors introduce the \textit{category of polytopes with subdivision}. The
definition of the diagonal maps comes from the theory of fiber polytopes so
that an induced polytopal subdivision of the associahedra is obtained, for
which the authors establishes a magical formula in the verbalism of Jean-Louis
Loday. It is made up of the pairs of cells of matching dimensions and
comparable order under the Tamari order (Theorem 2).
A synopsis of the paper consisting of four sections goes as follows. \S 1
recalls the main relevant notions, introducing the category of polytopes in
which the authors work. \S 2 gives a canonical definition of the diagonal map
for positively oriented polytopes, addressing their cellular properties. \S 3
endows the family of Loday realizations of the associahedra with a
nonsymmetric operad structure compatible with the diagonal maps. \S 4
establishes the magical celluar formula for the diagonal map of the associahedra.
Reviewer: Hirokazu Nishimura (Tsukuba)Barrier functions in subdifferential theory.https://www.zbmath.org/1455.490102021-03-30T15:24:00+00:00"Ivanov, Milen"https://www.zbmath.org/authors/?q=ai:ivanov.milen"Zlateva, Nadia"https://www.zbmath.org/authors/?q=ai:zlateva.nadia-pThe authors present a new proof for the equivalence between the convexity and the monotonicity of the subdifferential of a given function, a result known as Correa-Jofré-Thibault's Theorem. Instead of using Zagrodny's Mean Value Theorem, the authors use in their proof a new approach based on a different technique involving \textit{barrier functions}.
Reviewer: Chadi Nour (Byblos)Symmetry perspectives on some auxetic body-bar frameworks.https://www.zbmath.org/1455.520222021-03-30T15:24:00+00:00"Fowler, Patrick W."https://www.zbmath.org/authors/?q=ai:fowler.patrick-w"Guest, Simon D."https://www.zbmath.org/authors/?q=ai:guest.simon-d"Tarnai, Tibor"https://www.zbmath.org/authors/?q=ai:tarnai.tiborSummary: Scalar mobility counting rules and their symmetry extensions are reviewed for finite frameworks and also for infinite periodic frameworks of the bar-and-joint, body-joint and body-bar types. A recently published symmetry criterion for the existence of equiauxetic character of an infinite framework is applied to two long known but apparently little studied hinged-hexagon frameworks, and is shown to detect auxetic behaviour in both. In contrast, for double-link frameworks based on triangular and square tessellations, other affine deformations can mix with the isotropic expansion mode.