Recent zbMATH articles in MSC 52https://zbmath.org/atom/cc/522023-11-13T18:48:18.785376ZWerkzeugCounting tripods on the torushttps://zbmath.org/1521.050072023-11-13T18:48:18.785376Z"Athreya, Jayadev S."https://zbmath.org/authors/?q=ai:athreya.jayadev-s"Aulicino, David"https://zbmath.org/authors/?q=ai:aulicino.david"Richman, Harry"https://zbmath.org/authors/?q=ai:richman.david-harrySummary: Motivated by the problem of counting finite BPS webs, we count certain immersed metric graphs, tripods, on the flat torus. Classical Euclidean geometry turns this into a lattice point counting problem in \(\mathbb{C}^2\), and we give an asymptotic counting result using lattice point counting techniques.Cube packings in Euclidean spaceshttps://zbmath.org/1521.050142023-11-13T18:48:18.785376Z"Yu, Han"https://zbmath.org/authors/?q=ai:yu.hanSummary: In this paper, we study some cube packing problems. In particular, we are interested in compact subsets of \(\mathbb{R}^n\), \(n \geqslant 2\), which contain boundaries of cubes with all side lengths in (0,1). We show here that such sets must have lower box dimension at least \(n - 0.5\), and we will also provide sharp examples. We also show here that such sets must be large in general in a precise sense which is also introduced in this paper.
{{\copyright} 2021 The Authors. \textit{Mathematika} is copyright {\copyright} University College London.}Punctured intervals tile \(\mathbb{Z}^3\)https://zbmath.org/1521.050172023-11-13T18:48:18.785376Z"Cambie, Stijn"https://zbmath.org/authors/?q=ai:cambie.stijnSummary: Extending the methods of \textit{H. Metrebian} [Mathematika 65, No. 2, 181--189 (2019; Zbl 1398.05055)], we prove that punctured intervals tile \(\mathbb{Z}^3\). This solves two questions of Metrebian [loc. cit.] and completely resolves a question of \textit{V. Gruslys} et al. [Proc. Lond. Math. Soc. (3) 112, No. 6, 1019--1039 (2016; Zbl 1347.05027)]. We also pose a question that asks whether there is a relation between the genus \(g\) (number of holes) in a one-dimensional tile \(T\) and a uniform bound \(d\) such that \(T\) tiles \(\mathbb{Z}^d\). An affirmative answer would generalize a conjecture of Gruslys et al. [loc. cit.].
{{\copyright} 2021 The Authors. \textit{Mathematika} is copyright {\copyright} University College London.}Geometric and combinatorial properties of planar graphs with nonnegative curvaturehttps://zbmath.org/1521.050352023-11-13T18:48:18.785376Z"Hua, Bobo"https://zbmath.org/authors/?q=ai:hua.boboSummary: For a planar graph, we endow its ambient space with a piecewise flat metric by replacing each face with a regular Euclidean polygon. We survey some geometric and combinatorial results on planar graphs with various curvature notions defined via this piecewise flat metric.
For the entire collection see [Zbl 1454.00057].Root polytopes and Jaeger-type dissections for directed graphshttps://zbmath.org/1521.050572023-11-13T18:48:18.785376Z"Kálmán, Tamás"https://zbmath.org/authors/?q=ai:kalman.tamas"Tóthmérész, Lilla"https://zbmath.org/authors/?q=ai:tothmeresz.lillaSummary: We associate root polytopes to directed graphs and study them by using ribbon structures. Most attention is paid to what we call the semi-balanced case, that is, when each cycle has the same number of edges pointing in the two directions. Given a ribbon structure, we identify a natural class of spanning trees and show that, in the semi-balanced case, they induce a shellable dissection of the root polytope into maximal simplices. This allows for a computation of the \(h^\ast\)-vector of the polytope and for showing some properties of this new graph invariant, such as a product formula and that in the planar case, the \(h^\ast\)-vector is equivalent to the greedoid polynomial of the dual graph. We obtain a general recursion relation as well. We also work out the case of layer-complete directed graphs, where our method recovers a previously known triangulation. Indeed our dissection is often but not always a triangulation; we address this with a series of examples.
{{\copyright} 2022 The Authors. The publishing rights in this article are licensed to University College London under an exclusive licence. \textit{Mathematika} is published by the London Mathematical Society on behalf of University College London.}Geometric bijections between spanning subgraphs and orientations of a graphhttps://zbmath.org/1521.050692023-11-13T18:48:18.785376Z"Ding, Changxin"https://zbmath.org/authors/?q=ai:ding.changxinSummary: Let \(G\) be a connected finite graph. \textit{S. Backman} et al. [Forum Math. Sigma 7, Paper No. e45, 37 p. (2019; Zbl 1429.52017)] have constructed a family of explicit and easy-to-describe bijections between spanning trees of \(G\) and \((\sigma ,\sigma^\ast)\)-compatible orientations, where the \((\sigma ,\sigma^\ast)\)-compatible orientations are the representatives of equivalence classes of orientations up to cycle-cocycle reversal that are determined by a cycle signature \(\sigma\) and a cocycle signature \(\sigma^\ast\). Their bijections are geometric because the construction comes from zonotopal subdivisions. In this paper, we extend the geometric bijections to subgraph-orientation correspondences. Moreover, we extend the geometric constructions accordingly. Our proofs are combinatorial, which do not make use of the zonotopes. We also provide geometric proofs for partial results, which make use of zonotopal tiling, relate to Backman, Baker, and Yuen's method [loc. cit.], and motivate our combinatorial constructions. Finally, we explain that the main results hold for regular matroids.
{{\copyright} 2023 The Authors. \textit{Journal of the London Mathematical Society} is copyright {\copyright} London Mathematical Society.}On local operations that preserve symmetries and on preserving polyhedrality of mapshttps://zbmath.org/1521.051202023-11-13T18:48:18.785376Z"Brinkmann, Gunnar"https://zbmath.org/authors/?q=ai:brinkmann.gunnar"den Camp, Heidi Van"https://zbmath.org/authors/?q=ai:den-camp.heidi-vanSummary: We prove that local operations that preserve all symmetries, as e.g. dual, truncation, medial, or join, as well as local operations that are only guaranteed to preserve all orientation-preserving symmetries, as e.g. gyro or snub, preserve the polyhedrality of simple maps. This generalizes a result by \textit{B. Mohar} [Math. Slovaca 47, No. 1, 35--63 (1997; Zbl 0958.05034)] proving this for the operation dual. We give the proof based on an abstract characterization of these operations, prove that the operations are well defined, and also demonstrate the close connection between these operations and Delaney-Dress symbols.Embedding bipartite distance graphs under Hamming metric in finite fieldshttps://zbmath.org/1521.051262023-11-13T18:48:18.785376Z"Xu, Zixiang"https://zbmath.org/authors/?q=ai:xu.zixiang"Yu, Wenjun"https://zbmath.org/authors/?q=ai:yu.wenjun"Ge, Gennian"https://zbmath.org/authors/?q=ai:ge.gennianLet \(\mathbb{F}_q\) be a finite field of order \(q\) and let \(\mathbb{F}_q^n\) be equipped with the Hamming distance. The question investigated in this paper is, for a given graph \(H\), how large the size of \(A\subseteq \mathbb{F}_q^n\) provides that \(A\) contains a positive proportion of all possible isometric copies of \(H\). Let \(\mathrm{ex}(n,H)\) denote the maximum number of edges in an \(n\)-vertex \(H\)-free graph and let \(H_q(x) = x\log_q(q-1) - x\log_qx - (1-x)x\log_q(1-x)\). The first main theorem of the paper asserts the following. Let \(H\) be a bipartite graph with \(\mathrm{ex}(m, H) \le cm\) for some \(c > 0\). Let \(\lambda > 0\) be a sufficiently small real number, \(0 < \beta < 1/2\) and \(1/2 < \gamma < 1 - 1/q\). If \(A\subseteq \mathbb{F}_q^n\) satisfies \(|A| > q^{(1 - \frac{\beta^4(1 - H_q(\gamma))^4}{24})}\), then \(A\) contains an isometric copy of \(H\) whose edges are assigned by the same Hamming distance \(d\), for every integer \(d \in ((\beta + \lambda)n, (\gamma - \lambda)n)\). The second main theorem of the paper asserts that for a given bipartite graph \(H\), if \(A\subseteq \mathbb{F}_q^n\) satisfies \(|A| > q^{(1 - c_6)n}\) for some \(c_6 = c_6(q, H, \alpha)\), then \(A\) contains \(\alpha n\) distinct isometric copies \(H\) for some \(\alpha > 0\). The main techniques used to derive these results are the dependent random choice and a new extension of the modular version of Delsarte's inequality.
Reviewer: Sandi Klavžar (Ljubljana)Excluded minors are almost fragile. II: Essential elementshttps://zbmath.org/1521.051932023-11-13T18:48:18.785376Z"Brettell, Nick"https://zbmath.org/authors/?q=ai:brettell.nick"Oxley, James"https://zbmath.org/authors/?q=ai:oxley.james-g"Semple, Charles"https://zbmath.org/authors/?q=ai:semple.charles"Whittle, Geoff"https://zbmath.org/authors/?q=ai:whittle.geoffrey-pSummary: Let \(M\) be an excluded minor for the class of \(\mathbb{P} \)-representable matroids for some partial field \(\mathbb{P} \), let \(N\) be a 3-connected strong \(\mathbb{P} \)-stabilizer that is non-binary, and suppose \(M\) has a pair of elements \(\{a, b \}\) such that \(M \setminus a, b\) is 3-connected with an \(N\)-minor. Suppose also that \(| E(M) | \geq | E(N) | + 11\) and \(M \setminus a\), \(b\) is not \(N\)-fragile. In the prequel to this paper, we proved that \(M \setminus a, b\) is at most five elements away from an \(N\)-fragile minor. An element \(e\) in a matroid \(M^\prime\) is \(N\)-essential if neither \(M^\prime / e\) nor \(M^\prime \setminus e\) has an \(N\)-minor. In this paper, we prove that, under mild assumptions, \(M \setminus a, b\) is one element away from a minor having at least \(r(M) - 2\) elements that are \(N\)-essential.
For Part I see [\textit{N. Brettell} et al., ibid. 140, 263--322 (2020; Zbl 1430.05121)].On Borsuk-Ulam theorems and convex setshttps://zbmath.org/1521.052182023-11-13T18:48:18.785376Z"Crabb, M. C."https://zbmath.org/authors/?q=ai:crabb.michael-cSummary: The Intermediate Value Theorem is used to give an elementary proof of a Borsuk-Ulam theorem of \textit{H. Adams} et al. [Mathematika 66, No. 1, 79--102 (2020; Zbl 1443.05188)] that if \(f: S^1\rightarrow{\mathbb{R}}^{2k+1}\) is a continuous function on the unit circle \(S^1\) in \({\mathbb{C}}\) such that \(f(-z)=-f(z)\) for all \(z\in S^1\), then there is a finite subset \(X\) of \(S^1\) of diameter at most \(\pi -\pi /(2k+1)\) (in the standard metric in which the circle has circumference of length \(2 \pi)\) such the convex hull of \(f(X)\) contains \(0\in{\mathbb{R}}^{2k+1}\).
{{\copyright} 2023 The Authors. The publishing rights in this article are licensed to University College London under an exclusive licence. \textit{Mathematika} is published by the London Mathematical Society on behalf of University College London.}Perfectly contractile graphs and quadratic toric ringshttps://zbmath.org/1521.130402023-11-13T18:48:18.785376Z"Ohsugi, Hidefumi"https://zbmath.org/authors/?q=ai:ohsugi.hidefumi"Shibata, Kazuki"https://zbmath.org/authors/?q=ai:shibata.kazuki"Tsuchiya, Akiyoshi"https://zbmath.org/authors/?q=ai:tsuchiya.akiyoshiSummary: Perfect graphs form one of the distinguished classes of finite simple graphs. \textit{M. Chudnovsky} et al. [Ann. Math. (2) 164, No. 1, 51--229 (2006; Zbl 1112.05042)] proved that a graph is perfect if and only if it has no odd holes and no odd antiholes as induced subgraphs, which was conjectured by Berge. We consider the class \({\mathcal{A}}\) of graphs that have no odd holes, no antiholes, and no odd stretchers as induced subgraphs. In particular, every graph belonging to \({\mathcal{A}}\) is perfect. Everett and Reed conjectured that a graph belongs to \({\mathcal{A}}\) if and only if it is perfectly contractile. In the present paper, we discuss graphs belonging to \({\mathcal{A}}\) from a viewpoint of commutative algebra. In fact, we conjecture that a perfect graph \(G\) belongs to \({\mathcal{A}}\) if and only if the toric ideal of the stable set polytope of \(G\) is generated by quadratic binomials. Especially, we show that this conjecture is true for Meyniel graphs, perfectly orderable graphs, and clique separable graphs, which are perfectly contractile graphs.
{{\copyright} 2022 The Authors. \textit{Bulletin of the London Mathematical Society} is copyright {\copyright} London Mathematical Society.}On polynomial images of a closed ballhttps://zbmath.org/1521.140992023-11-13T18:48:18.785376Z"Fernando, José F."https://zbmath.org/authors/?q=ai:fernando.jose-f"Ueno, Carlos"https://zbmath.org/authors/?q=ai:ueno.carlosThe authors characterize semialgebraic subsets of \(\mathbb R^n\) that are the image of a real morphism (i.e. real polynomial map) of a closed unit ball \(\mathbb R^m\). In particular, they proved:
If \(S\subset \mathbb R^n\) is a finite union of \(n\)-dimensional compact and convex polyhedra then the following are equivalent
\begin{itemize}
\item \(S\) is connected by real analytic paths
\item There is a real morphism \(f:\mathbb R^{n+1} \to \mathbb R^n\) such that \(f(B_{n+1})= S\)
\item There is a real morphism \(f:\mathbb R^n \to \mathbb R^n\) such that \(f(B_n)= S\)
\end{itemize}
where \(B_k\subset \mathbb R^k\) is a closed unit ball in the Euclidean topology.
This is a special case of their main result, where \(S\) above can be replaced with a finite union of \textit{\(m\)-bricks} and the morphism is \(f:\mathbb R^{m+1}\to \mathbb R^n\). An \textit{\(m\)-brick} is a set \(T\subset \mathbb R^n\) such that there is a homotopy \[H_\lambda :B_m \to T \quad \lambda \in [0,1]\] that deforms \(H_0(B_m)=T\) to a point \(H_1(B_m)\) and has intermediate images contained in the Euclidean interior of \(T\).
The authors also pose an open problem asking the minimum degrees of such polynomial maps.
Reviewer: Jose Capco (Linz)Characterization of tropical plane curves up to genus sixhttps://zbmath.org/1521.141052023-11-13T18:48:18.785376Z"Tewari, Ayush Kumar"https://zbmath.org/authors/?q=ai:tewari.ayush-kumarSummary: We provide a new forbidden criterion for the realizability of smooth tropical plane curves. This in turn provides us a complete classification of smooth tropical plane curves up to genus six.Approximating tensor norms via sphere covering: bridging the gap between primal and dualhttps://zbmath.org/1521.150202023-11-13T18:48:18.785376Z"He, Simai"https://zbmath.org/authors/?q=ai:he.simai"Hu, Haodong"https://zbmath.org/authors/?q=ai:hu.haodong"Jiang, Bo"https://zbmath.org/authors/?q=ai:jiang.bo"Li, Zhening"https://zbmath.org/authors/?q=ai:li.zheningSummary: The matrix spectral norm and nuclear norm appear in enormous applications. The generalization of these norms to higher-order tensors is becoming increasingly important, but unfortunately they are NP-hard to compute or even approximate. Although the two norms are dual to each other, the best-known approximation bound achieved by polynomial-time algorithms for the tensor nuclear norm is worse than that for the tensor spectral norm. In this paper, we bridge this gap by proposing deterministic algorithms with the best bound for both tensor norms. Our methods not only improve the approximation bound for the nuclear norm but also are data independent and easily implementable compared to existing approximation methods for the tensor spectral norm. The main idea is to construct a selection of unit vectors that can approximately represent the unit sphere, in other words, a collection of spherical caps to cover the sphere. For this purpose, we explicitly construct several collections of spherical caps for sphere covering with adjustable parameters for different levels of approximations and cardinalities. These readily available constructions are of independent interest, as they provide a powerful tool for various decision-making problems on spheres and related problems. We believe the ideas of constructions and the applications to approximate tensor norms can be useful to tackle optimization problems over other sets, such as the binary hypercube.Self-dual polyhedral cones and their slack matriceshttps://zbmath.org/1521.150292023-11-13T18:48:18.785376Z"Gouveia, João"https://zbmath.org/authors/?q=ai:gouveia.joao.1|gouveia.joao"Lourenço, Bruno F."https://zbmath.org/authors/?q=ai:lourenco.bruno-fSummary: We analyze self-dual polyhedral cones and prove several properties about their slack matrices. In particular, we show that self-duality is equivalent to the existence of a positive semidefinite (PSD) slack. Beyond that, we show that if the underlying cone is irreducible, then the corresponding PSD slacks are not only doubly nonnegative matrices (DNN) but are extreme rays of the cone of DNN matrices, which correspond to a family of extreme rays not previously described. More surprisingly, we show that, unless the cone is simplicial, PSD slacks not only fail to be completely positive matrices but they also lie outside the cone of completely positive semidefinite matrices. Finally, we show how one can use semidefinite programming to probe the existence of self-dual cones with given combinatorics. Our results are given for polyhedral cones but we also discuss some consequences for negatively self-polar polytopes.Eulerian representations for real reflection groupshttps://zbmath.org/1521.200792023-11-13T18:48:18.785376Z"Brauner, Sarah"https://zbmath.org/authors/?q=ai:brauner.sarahSummary: The Eulerian idempotents, first introduced for the symmetric group and later extended to all reflection groups, generate a family of representations called the Eulerian representations that decompose the regular representation. In Type \(A\), the Eulerian representations have many elegant but mysterious connections to rings naturally associated with the braid arrangement. Here, we unify these results and show that they hold for any reflection group of coincidental type -- that is, \(S_n\), \(B_n\), \(H_3\) or the dihedral group \(I_2(m)\) -- by giving six characterizations of the Eulerian representations, including as components of the associated graded Varchenko-Gelfand ring \(\mathcal{V}\). As a consequence, we show that Solomon's descent algebra contains a commutative subalgebra generated by sums of elements with a fixed number of descents if and only if \(W\) is coincidental. More generally, for any finite real reflection group, we give a case-free construction of a family of Eulerian representations described by a flat decomposition of the ring \(\mathcal{V}\).A description of the minimal elements of Shi regions in classical Weyl groupshttps://zbmath.org/1521.200802023-11-13T18:48:18.785376Z"Charles, Balthazar"https://zbmath.org/authors/?q=ai:charles.balthazarSummary: In this extended abstract, we show how a bijection between parking functions and regions of the Shi arrangement from \textit{C. A. Athanasiadis} and \textit{S. Linusson} [Discrete Math. 204, No. 1--3, 27--39 (1999; Zbl 0959.52019)] (in type \(A_n\)) and \textit{D. Armstrong} et al. [Adv. Math. 269, 647--706 (2015; Zbl 1347.20039)] (in type \(B_n\), \(C_n\), \(D_n\)) allows for the computation of the minimal elements of the Shi regions. This gives a combinatorial interpretation of these minimal elements: they can be seen as counting non-crossing arcs in non-nesting arc diagrams.A class of self-affine tiles in \(\mathbb{R}^d\) that are \(d\)-dimensional tame ballshttps://zbmath.org/1521.280082023-11-13T18:48:18.785376Z"Deng, Guotai"https://zbmath.org/authors/?q=ai:deng.guotai"Liu, Chuntai"https://zbmath.org/authors/?q=ai:liu.chuntai"Ngai, Sze-Man"https://zbmath.org/authors/?q=ai:ngai.sze-manSummary: We study a family of self-affine tiles in \(\mathbb{R}^d (d\geq 2)\) with noncollinear digit sets, which naturally generalizes a two-dimensional class studied originally by Deng and Lau and its extension to \(\mathbb{R}^3\) by the authors. By using Brouwer's invariance of domain theorem, along with a tool which we call horizontal distance, we obtain necessary and sufficient conditions for the tiles to be \(d\)-dimensional tame balls. This answers positively the conjecture in an earlier paper by the authors stating that a member in a certain class of self-affine tiles is homeomorphic to a \(d\)-dimensional ball if and only if its interior is connected.Homogenization of random quasiconformal mappings and random Delauney triangulationshttps://zbmath.org/1521.300292023-11-13T18:48:18.785376Z"Ivrii, Oleg"https://zbmath.org/authors/?q=ai:ivrii.oleg-v"Marković, Vladimir"https://zbmath.org/authors/?q=ai:markovic.vladimir|markovic.vladimir-mLet \(\lambda\) be a probability measure on the standard unit disk of the complex plane \({\mathbb{C}}\). The authors randomly assign a complex number in the unit disk for each cell in a square grid in the complex plane according to the measure \(\lambda\). The collection of these numbers defines a Beltrami coefficient \(\mu(z)\) on \({\mathbb{C}}\) which is constant on the cells of the grid. The Beltrami equation \(\bar\partial w(z)=\mu(z)\partial w(z)\) has a unique injective solution \(w^\mu\) that fixes \(0\), \(1\) and \(\infty\). The authors call \(w^\mu\) a random quasiconformal mapping, but point out that \(w^\mu\) may not be quasiconformal. The first main result of the paper says that if the mesh size of the grid is small, then with high probability, \(w^{\mu}\) is close to an affine transformation \(A_\lambda\) determined by the measure \(\lambda\).
A circle packing is a collection of circles in \({\mathbb{C}}\) with disjoint interiors. By the Koebe-Andreev-Thurston circle packing theorem, any finite triangulation of a topological disk admits a maximal circle packing whose boundary circles are horocycles. A discrete set \(V\) of points in \({\mathbb{C}}\) determines a Voronoi tessellation. This means that \({\mathbb{C}}\) can be written as a union of sets \(F_x\), \(x\in V\), where \(F_x\) consists of all points \(z\in {\mathbb{C}}\) for which \(\min_{y\in V}\vert y-z\vert=\vert x-z\vert\). If the points in \(V\) are in general position, then the Delauney triangulation is the dual graph to the Voronoi tessellation. The union of all the triangles in the Delauney triangulation is the convex hull of \(V\), and hence a topological disk.
Let \(\Omega\subset {\mathbb{C}}\) be a simply connected domain bounded by a \(\mathrm{C}^1\)-curve. The authors randomly choose \(N\geq 1\) points in \(\Omega\) with respect to a Lebesgue measure. For technical reasons, they also choose \(\asymp \sqrt{N}\) equally spaced points on the boundary \(\partial\Omega\).They define the random Delauney triangulation as the union of the Delauney triangles contained in \(\Omega\). Let \(\varphi_{\mathcal{P}}\) denote the circle packing map of the suitably normalized maximal circle packing of the random Delauney triangulation. Kenneth Stephenson has suggested that when \(N\) is large, then with high probability, \(\varphi_{\mathcal{P}}\) approximates a conformal map \(\varphi\colon \Omega \to {\mathbb{D}}\). The second main result of the paper is a proof of this conjecture.
Reviewer: Marja Kankaanrinta (Helsinki)Polarization and covering on sets of low smoothnesshttps://zbmath.org/1521.310202023-11-13T18:48:18.785376Z"Anderson, A."https://zbmath.org/authors/?q=ai:anderson.aaron|anderson.andy-b|anderson.alastair|anderson.anders|anderson.austin|anderson.adam-w|anderson.alexander-g|anderson.adam-l|anderson.anthony-m|anderson.a-j-b|anderson.ann|anderson.alan-ross.1|anderson.abby|anderson.alexander-r-a|anderson.annelise-g|anderson.arthur-e-jun|anderson.aparna-b|anderson.amos-g|anderson.asalie|anderson.alejandro|anderson.alex|anderson.arlen|anderson.aemer-d|anderson.axel|anderson.anne-h|anderson.andrew-a"Reznikov, A."https://zbmath.org/authors/?q=ai:reznikov.andre|reznikov.a-n|reznikov.aleksandr|reznikov.a-b"Vlasiuk, O."https://zbmath.org/authors/?q=ai:vlasiuk.oleksandr-v"White, E."https://zbmath.org/authors/?q=ai:white.emma|white.emett-r|white.edward-b|white.edward-t|white.eugene|white.edna-m|white.emily|white.edward-dalton-iii|white.ella|white.ethan-patrickSummary: In this paper we study the asymptotic properties of point configurations that achieve optimal covering of sets lacking smoothness. Our results include the proofs of existence of asymptotics of best covering and maximal polarization for \((\mathcal{H}_d,d)\)-rectifiable sets and maximal polarization on self-similar fractals.Temperate distributions with locally finite support and spectrum on Euclidean spaceshttps://zbmath.org/1521.420122023-11-13T18:48:18.785376Z"Favorov, Sergii Yu."https://zbmath.org/authors/?q=ai:favorov.sergii-yuLet \(\mu\) be a measure of \(\mathbb{R}^d\) such that \(\mu\) and its Fourier transform \(\widehat{\mu}\) have discrete supports. In the theory of Fourier quasicrystals, there are some results giving sufficient conditions for \(\operatorname{supp}\mu\) to be a finite union of full-rank lattices (or contained in a finite union of full-rank lattices). In the paper, these results are generalised to a wide class of temperate distributions.
Reviewer: Anton Shutov (Vladimir)Complete sets in normed linear spaceshttps://zbmath.org/1521.460082023-11-13T18:48:18.785376Z"He, Chan"https://zbmath.org/authors/?q=ai:he.chan"Martini, Horst"https://zbmath.org/authors/?q=ai:martini.horst"Wu, Senlin"https://zbmath.org/authors/?q=ai:wu.senlinFrom the abstract: ``A bounded set of a (finite or infinite-dimensional) normed linear space is said to be \textit{complete} (or \textit{diametrically complete}) if it cannot be enlarged without increasing its diameter. Any bounded subset \(A\) of a normed linear space is contained in a complete set having the same diameter, which is called a \textit{completion} of \(A\).''
This paper presents a wide, excellent, selfcontained survey on the state of the art. The headings of the sections are: complete sets and sets of constant width (elementary properties and characterizations), completions of bounded sets (wide and tight spherical hulls, the diametric completion map, sets having unique completions, Eggleston's construction, the Maehara set of a bounded set, generalized Bückner completion, completions related to hyperplanes, a stochastic construction, completions to sets of constant width with preassigned shape), relations to sets of constant width and to balls, further related set families, interior and boundary, asymmetry of complete sets in Minkowski spaces, structure of the space of diametrically complete sets.
The matter is enriched by many suitable examples. References to the current literature are exhaustive, so the paper turns out to be a precious source for anybody who plans to work in the field.
Reviewer: Clemente Zanco (Milano)Multi-marginal maximal monotonicity and convex analysishttps://zbmath.org/1521.470842023-11-13T18:48:18.785376Z"Bartz, Sedi"https://zbmath.org/authors/?q=ai:bartz.sedi"Bauschke, Heinz H."https://zbmath.org/authors/?q=ai:bauschke.heinz-h"Phan, Hung M."https://zbmath.org/authors/?q=ai:phan.hung-m"Wang, Xianfu"https://zbmath.org/authors/?q=ai:wang.xianfuThe authors study extensions of classical monotone operator theory and convex analysis to the multi-marginal setting. Multi-marginal \(c\)-monotonicity is characterized in terms of classical monotonicity and firmly nonexpansive mappings. Minty type, continuity and conjugacy criteria for multi-marginal maximal monotonicity are presented. Partition of the identity into a sum of firmly nonexpansive mappings, as well as Moreau's decomposition of the quadratic function into envelopes and proximal mappings, is extended in this framework. Examples and applications are presented. Many open questions are posed.
Reviewer: K. C. Sivakumar (Chennai)Kirszbraun's theorem via an explicit formulahttps://zbmath.org/1521.470872023-11-13T18:48:18.785376Z"Azagra, Daniel"https://zbmath.org/authors/?q=ai:azagra.daniel"Le Gruyer, Erwan"https://zbmath.org/authors/?q=ai:le-gruyer.erwan-y"Mudarra, Carlos"https://zbmath.org/authors/?q=ai:mudarra.carlosThe problem of extending maps from a smaller to a larger set, keeping their properties, is of utmost importance and has an impact on many application-oriented questions. Thus, \textit{M. D. Kirszbraun} [Fundam. Math. 22, 77--108 (1934; Zbl 0009.03904)] proved that a Lipschitz map \(f: E\to\mathbb{R}^n\) admits an extension from an arbitrary subset \(E\subseteq\mathbb{R}^m\) to a map \(F: \mathbb{R}^m\to\mathbb{R}^n\) with the same minimal Lipschitz constant. Subsequently, this result was generalized to maps between two Hilbert spaces by \textit{F. A. Valentine} [Am. J. Math. 67, 83--93 (1945; Zbl 0061.37507)]. However, these results have the flaw that they are not constructive and not transparent.
In this interesting paper, the authors give an explicit formula for the extension in the Kirszbraun-Valentine theorem and discuss related questions.
Reviewer: Jürgen Appell (Würzburg)Duality between Lagrangians and Rockafellianshttps://zbmath.org/1521.490262023-11-13T18:48:18.785376Z"De Lara, Michel"https://zbmath.org/authors/?q=ai:de-lara.michelSummary: In his monograph [CBMS-NSF Reg. Conf. Ser. Appl. Math. 16, 74 p. (1974; Zbl 0296.90036)], \textit{R. T. Rockafellar} puts forward a ``perturbation + duality'' method to obtain a dual problem for an original minimization problem. First, one embeds the minimization problem into a family of perturbed problems (thus giving a so-called perturbation function); the perturbation of the original function to be minimized has recently been called a Rockafellian. Second, when the perturbation variable belongs to a primal vector space paired, by a bilinear form, with a dual vector space, one builds a Lagrangian from a Rockafellian; one also obtains a so-called dual function (and a dual problem). The method has been extended from Fenchel duality to generalized convexity: when the perturbation belongs to a primal set paired, by a coupling function, with a dual set, one also builds a Rockafellian from a Lagrangian. Following these paths, we highlight a duality between Lagrangians and Rockafellians. Where the material mentioned above mostly focuses on moving from Rockafellian to Lagrangian, we treat them equally and display formulas that go both ways. We propose a definition of Lagrangian-Rockafellian couples. We characterize these latter as dual functions, with respect to a coupling, and also in terms of generalized convex functions. The duality between perturbation and dual functions is not as clear cut.Optimal design of sensors via geometric criteriahttps://zbmath.org/1521.490292023-11-13T18:48:18.785376Z"Ftouhi, Ilias"https://zbmath.org/authors/?q=ai:ftouhi.ilias"Zuazua, Enrique"https://zbmath.org/authors/?q=ai:zuazua.enriqueGiven a set \(\Omega\subset\mathbb{R}^N\), and an amount of volume \(c\in (0, |\Omega|)\), the authors study the min-max problem
\[
\inf\left\{\, \sup_{x\in\Omega}\{\, \mathrm{dist}(x; \omega)\,\}\,:\, |\omega|=c,\, \omega\subset\Omega\,\right\}
\]
Without any additional constraint on \(\omega\), the authors show that the problem is ill-posed as the infimum is zero and it is asymptotically attained by a sequence of disconnected sets with an increasing number of connected components. Assuming the convexity of the competitors \(\omega\), they show that there exist minima and that the function
\[
f\colon (0, |\Omega|)\ni c\mapsto \inf\left\{\, \sup_{x\in\Omega} \mathrm{dist}\,(x; \omega)\,:\, |\omega|=c,\, \omega\subset\Omega\,\right\}
\]
is continuous and strictly decreasing (ref.\ first part of Theorem 1). One might naively expect the solution(s) \(\omega\) to inherit the symmetries of the ambient set \(\Omega\). The authors show this not to be the case, and that symmetry-breaking phenomena might happen already for the choice \(\Omega=[0,1]\times[0,1]\) (ref.\ Theorem 2).
It is worth mentioning, that they also show the problem to be equivalent to few other shape optimization problems (ref.\ second part of Theorem 1), which are more suitable to be treated numerically, and some qualitative and numerical results in the planar case appear in Sections 4 and 5.
Reviewer: Giorgio Saracco (Firenze)All polytopes are coset geometries: characterizing automorphism groups of \(k\)-orbit abstract polytopeshttps://zbmath.org/1521.510092023-11-13T18:48:18.785376Z"Hubard, Isabel"https://zbmath.org/authors/?q=ai:hubard.isabel"Mochán, Elías"https://zbmath.org/authors/?q=ai:mochan.eliasSummary: Abstract polytopes generalize the classical notion of convex polytopes to more general combinatorial structures. The most studied ones are regular and chiral polytopes, as it is well-known, they can be constructed as coset geometries from their automorphism groups. This is also known to be true for 2- and 3-orbit 3-polytopes. In this paper we show that every abstract \(n\)-polytope can be constructed as a coset geometry. This construction is done by giving a characterization, in terms of generators, relations and intersection conditions, of the automorphism group of a \(k\)-orbit polytope with given symmetry type graph. Furthermore, we use these results to show that for all \(k \neq 2\), there exist \(k\)-orbit \(n\)-polytopes with Boolean automorphism groups, for all \(n \geq 3\).Edge connectivity of simplicial polytopeshttps://zbmath.org/1521.510102023-11-13T18:48:18.785376Z"Pilaud, Vincent"https://zbmath.org/authors/?q=ai:pilaud.vincent"Pineda-Villavicencio, Guillermo"https://zbmath.org/authors/?q=ai:pineda-villavicencio.guillermo"Ugon, Julien"https://zbmath.org/authors/?q=ai:ugon.julienSummary: We show that the graph of a simplicial polytope of dimension \(d \geq 3\) has no nontrivial minimum edge cut with fewer than \(d(d+1)/2\) edges, hence the graph is \(\min \{ \delta, d(d+1)/2\}\)-edge-connected where \(\delta\) denotes the minimum degree. When \(d = 3\), this implies that every minimum edge cut in a plane triangulation is trivial. When \(d \geq 4\), we construct a simplicial \(d\)-polytope whose graph has a nontrivial minimum edge cut of cardinality \(d(d+1)/2\), proving that the aforementioned result is best possible.On the existence of solutions to the Orlicz-Minkowski problem for torsional rigidityhttps://zbmath.org/1521.520012023-11-13T18:48:18.785376Z"Hu, Zejun"https://zbmath.org/authors/?q=ai:hu.zejun"Li, Hai"https://zbmath.org/authors/?q=ai:li.hai|li.hai.1|li.hai.2The Orlicz-Minkowski problem for the torsional rigidity reads as follows. Let \(h_K\) denote the support function of the convex body (convex and compact subset) in \(\mathbb{R}^n\), and let \(\mu_T\) denote the torsion measure of \(K\). Given a suitable continuous function \(\varphi: (0,\infty)\longrightarrow (0,\infty)\) and a finite Borel measure \(\mu\) on the \((n-1)\)-dimensional unit sphere \(S^{n-1}\), does there exist a convex body \(K\) in the Euclidean \(n\)-space \(\mathbb{R}^n\) such that \(\mu=c\phi(h_K)\mu_T(K,\cdot)\) for some \(c>0\)?
\textit{N. Li} and \textit{B. Zhu} [J. Differ. Equations 269, No. 10, 8549--8572 (2020; Zbl 1443.52006)] proved the existence of solutions to the Orlicz-Minkowski problem for the torsional rigidity, regarding a continuous function \(\varphi\), such that \(\lim_{x\to 0^+}\varphi(x)=\infty\). More precisely, the mentioned result of Li and Zhu is the following one, numbered according to the actual paper.
Theorem 1.1 (Li and Zhu [loc. cit.]). Suppose \(\varphi: (0,\infty)\longrightarrow (0,\infty)\) is a continuous function with \(\lim_{x\to 0^+}\varphi(x)=\infty\), such that \(\phi(t)=\int_{0}^t \frac{1}{\varphi(x)}dx\) exists for all positive \(t\) and is unbounded as \(t\to \infty\). Let \(\mu\) be a finite Borel measure on \(S^{n-1}\), that is not concentrated on any closed hemisphere. Then there exist a convex body \(K\) containing the origin and \(c>0\) such that \(\mu=c\varphi(h_K)\mu_T(K,\cdot)\).
\textit{J. Hu} et al. [J. Geom. Anal. 32, No. 2, Paper No. 63, 28 p. (2022; Zbl 1484.35250)] studied this problem via the Gauss curvature flow, under the additional assumption that the measure is even.
In this paper, the authors consider a complementary outcome of the latter Theorem 1.1., obtaining as a main result the following existence result for the Orlicz-Minkowski problem for the torsional rigidity, in the case of a strictly increasing, continuously differentiable function \(\varphi\) satisfying \(\lim_{x\to 0^+}\varphi(x)=0\).
Theorem 1.2. Let \(\varphi: (0,\infty)\longrightarrow (0,\infty)\) be a strictly increasing, continuously differentiable function with \(\lim_{x\to 0^+}\varphi(x)=\infty\). Assume that \(\phi(t)=\int_{0}^t \frac{1}{\varphi(x)}dx\) exists for all positive \(t\) and \(\lim_{t\to\infty} \phi(t)=\infty\). Then, for any finite Borel measure \(\mu\) on \(S^{n-1}\), that is not concentrated on any closed hemisphere, there exist a constant \(c>0\), and a convex body \(K\) containing the origin in its interior, such that
\[
\mu=c\varphi(h_K)\mu_T(K,\cdot).
\]
The particular case of \(\varphi(x)=x^{1-p}\), for \(0<p<1\), in the previous result retrieves the following existence result of \textit{J. Hu} and \textit{J. Liu} [Adv. Appl. Math. 128, Article ID 102188, 22 p. (2021; Zbl 1479.52008)]:
Corollary 1.1 (Hu and Liu [loc. cit.]). Let \(\mu\) be a finite Borel measure on \(S^{n-1}\) that is not concentrated on any closed hemisphere. Tehn, for \(0<p<1\), there exists a convex body \(K\) containing the origin in its interior such that \(\mu= h_K^{1-p}\mu_T(K,\cdot)\).
Reviewer: Eugenia Saorín Gómez (Bremen)On typical triangulations of a convex \(n\)-gonhttps://zbmath.org/1521.520022023-11-13T18:48:18.785376Z"Mansour, Toufik"https://zbmath.org/authors/?q=ai:mansour.toufik"Rastegar, Reza"https://zbmath.org/authors/?q=ai:rastegar.rezaSummary: Let \(f_n\) be a function assigning weight to each possible triangle whose vertices are chosen from vertices of a convex polygon \(P_n\) of \(n\) sides. Suppose \(\mathcal{T}_n\) is a random triangulation, sampled uniformly out of all possible triangulations of \(P_n\). We study the sum of weights of triangles in \(\mathcal{T}_n\) and give a general formula for average and variance of this random variable. In addition, we look at several interesting special cases of \(f_n\) in which we obtain explicit forms of generating functions for the sum of the weights. For example, among other things, we give new proofs for already known results such as the degree of a fixed vertex and the number of ears in \(\mathcal{T}_n\), as well as, provide new results on the number of ``blue'' angles and refined information on the distribution of angles at a fixed vertex. We note that our approach is systematic and can be applied to many other new examples while generalizing the existing results.Quiver combinatorics and triangulations of cyclic polytopeshttps://zbmath.org/1521.520032023-11-13T18:48:18.785376Z"Williams, Nicholas J."https://zbmath.org/authors/?q=ai:williams.nicholas-jThe primary objective of the paper under consideration is to associate quivers to triangulations of even-dimensional cyclic polytopes. The author investigates the combinatorics of these quivers. He proves two results that attest to the information encapsulated within the quiver with respect to the given triangulation. Firstly, he proves that the cut quivers of \textit{O. Iyama} and \textit{S. Oppermann} [Trans. Am. Math. Soc. 363, No. 12, 6575--6614 (2011; Zbl 1264.16015)] are in precise correspondance with \(2d\)-dimensional triangulations without interior \((d+1)\)-simplices. As a consequence, these particular triangulations constitute a connected subgraph of the flip graph. Subsequently, the author's second result emphasizes how the quiver of a triangulation can be useful in identifying mutable internal \(d\)-simplices. This outcome paves the way for the development of a theory of higher-dimensional quiver mutation and provides a new approach to comprehending the flips of triangulations of even-dimensional cyclic polytopes.
Reviewer: Alireza Nasr-Isfahani (Isfahan)On a convex polyhedron in a regular point systemhttps://zbmath.org/1521.520042023-11-13T18:48:18.785376Z"Shtogrin, Mikhail I."https://zbmath.org/authors/?q=ai:shtogrin.mikhail-ivanovichSummary: Faceting with a `filling': An ideal crystal structure consists of finitely many equal and parallel translational point lattices. In \(\mathbb{R}^3\) it extends unboundedly in all directions. We distinguish in it a finite part situated in a closed convex polyhedron every face of which contains nodes of a translational point lattice involved in the structure not belonging to the same straight line. Such a polyhedron is called a possible faceting of the ideal crystal structure.
There are 32 well-known crystal classes, or 32 crystallographic point groups. Among them is the symmetry group of the possible faceting calculated taking account of the nodes of the ideal crystal structure belonging to it. A cyclic subgroup \(C_n\) of the symmetry group of any possible faceting has order \(n\le 4\) or \(n=6\).
Faceting without `filling': In this paper we construct two crystal structures in which there are crystal polyhedra whose symmetry groups, calculated without taking account of the nodes of the crystal structure belonging to it, have rotation axes of orders \(n=8\) and \(n=12\). In both cases, the crystal polyhedron is a right prism of finite height. Without taking account of the internal structure, a possible faceting of a crystal structure in three-dimensional Euclidean space cannot have an axes of rotation of order \(n\) satisfying \(6<n<\infty \).
The proposed constructions are accompanied by a detailed analysis of ideal crystal structures, as well as Delone sets \(S\) of type \((r, R)\) in \(\mathbb{R}^2\) and \(\mathbb{R}^3\). In particular, we produce an expanded proof of one of the theorems stated in 2010 at an international conference dedicated to the 120th anniversary of B. N. Delone.Boundary \(H^\ast\)-polynomials of rational polytopeshttps://zbmath.org/1521.520052023-11-13T18:48:18.785376Z"Bajo, Esme"https://zbmath.org/authors/?q=ai:bajo.esme"Beck, Matthias"https://zbmath.org/authors/?q=ai:beck.matthiasSummary: If \(P\) is a \textit{lattice polytope} (i.e., \(P\) is the convex hull of finitely many integer points in \(\mathbb{R}^d\)) of dimension \(d\), \textit{E. Ehrhart}'s famous theorem [C. R. Acad. Sci., Paris 254, 616--618 (1962; Zbl 0100.27601)] asserts that the integer-point counting function \(|nP\cap \mathbb{Z}^d|\) is a degree-\(d\) polynomial in the integer variable \(n\). Equivalently, the generating function \(1+\sum_{n\geq 1}|nP\cap\mathbb{Z}^d| \, z^n\) is a rational function of the form \(\frac{h^\ast (z)}{(1-z)^{d+1}}\); we call \(h^\ast (z)\) the \(h^\ast\)-\textit{polynomial} of \(P\). There are several known necessary conditions for \(h^\ast\)-polynomials, including results by \textit{T. Hibi} [Discrete Math. 83, No. 1, 119--121 (1990; Zbl 0708.52005)], \textit{R. P. Stanley} [J. Pure Appl. Algebra 73, No. 3, 307--314 (1991; Zbl 0735.13010)] and \textit{A. Stapledon} [Trans. Am. Math. Soc. 361, No. 10, 5615--5626 (2009; Zbl 1181.52024)], who used an interplay of arithmetic (integer-point structure) and topological (local \(h\)-vectors of triangulations) data of a given polytope. We introduce an alternative \textit{ansatz} to understand Ehrhart theory through the \(h^\ast\)-polynomial of the \textit{boundary} of a polytope, recovering all of the above results and their extensions for rational polytopes in a unifying manner. We include applications for (rational) Gorenstein polytopes and rational Ehrhart dilations.The distribution of roots of Ehrhart polynomials for the dual of root polytopes of type \(C\)https://zbmath.org/1521.520062023-11-13T18:48:18.785376Z"Higashitani, Akihiro"https://zbmath.org/authors/?q=ai:higashitani.akihiro"Yamada, Yumi"https://zbmath.org/authors/?q=ai:yamada.yumiConsider an integer polytope of type \(C\) of dimension \(d\) and let \(C^\ast_d\) denotes the dual of the root polytope of such polytope. The authors provide an explicit formula for the values of the Ehrhart polynomial of \(C^\ast_d\) and prove that its roots have the same real part \(-1/2\). They also show that the Ehrhart polynomial of \(C^\ast_d\) satisfies the interlacing property (with respect to dimension \(d\)).
Reviewer: Oleg Karpenkov (Liverpool)The constant of point-line incidence constructionshttps://zbmath.org/1521.520072023-11-13T18:48:18.785376Z"Balko, Martin"https://zbmath.org/authors/?q=ai:balko.martin"Sheffer, Adam"https://zbmath.org/authors/?q=ai:sheffer.adam"Tang, Ruiwen"https://zbmath.org/authors/?q=ai:tang.ruiwenThe Szemerédi-Trotter theorem asserts that for a set of \(L\) lines and a set of \(P\) points in the plane, the number of incidences between the points and the lines is at most \(O(|P|^{2/3}|L|^{2/3}+|P|+|L|)\). The interesting range is \(|L|=o(|P|^2)\), \(|P|=o(|L|^2)\), as outside of this range the linear terms are dominant. Much work was done on the upper bound. The current best upper bound \(2.44|P|^{2/3}|L|^{2/3}+|P|+|L|\) is due to \textit{E. Ackerman} [Comput. Geom. 85, Article ID 101574 31 p. (2019; Zbl 1439.05163)]. For a long time Erdős' grid construction provided for a lower bound, then Elekes gave a different grid construction. There is also a recent construction of Guth and Silier. The paper under review computes the (asymptotically) best constant for some ranges, which include both the Erdős and Elekes constructions: if \(1/3\leq \alpha\leq 1/2\), \(|L|=o(|P|^2)\), and \(|L|/|P|^{2-3\alpha}\rightarrow \infty\), then the number of incidences can reach \((c+o(1))|P|^{2/3}|L|^{2/3}\), where \(c=3\cdot (3\pi^2/4)^{1/3}\approx 1.27.\) Working out this constant requires substantial work on Euler's totient function.
Reviewer: László A. Székely (Columbia)On the two-parameter Erdős-Falconer distance problem in finite fieldshttps://zbmath.org/1521.520082023-11-13T18:48:18.785376Z"Clément, Francois"https://zbmath.org/authors/?q=ai:clement.francois"Mojarrad, Hossein Nassajian"https://zbmath.org/authors/?q=ai:mojarrad.hossein-nassajian"Pham, Duc Hiep"https://zbmath.org/authors/?q=ai:pham.duc-hiep"Shen, Chun-Yen"https://zbmath.org/authors/?q=ai:shen.chun-yenSummary: Given \(E \subseteq \mathbb{F}_q^d \times \mathbb{F}_q^d\), with the finite field \(\mathbb{F}_q\) of order \(q\) and the integer \(d\geq 2\) , we define the two-parameter distance set \(\Delta_{d,d} (E) = \{(\|x-y\|, \|z-t\|) : (x, z), (y, t) \in E \}\). \textit{P. Birklbauer} and \textit{A. Iosevich} [Bull. Hell. Math. Soc. 61, 21--30 (2017; Zbl 1425.42010)] proved that if \(|E| \gg q^{{(3d+1)}/{2}}\), then \(|\Delta_{d, d}(E)| = q^2\). For \(d=2\), they showed that if \(|E| \gg q^{{10}/{3}}\), then \(|\Delta_{2, 2}(E)| \gg q^2\). In this paper, we give extensions and improvements of these results. Given the diagonal polynomial \(P(x)=\sum_{i=1}^da_ix_i^s\in \mathbb{F}_q[x_1,\dots , x_d]\), the distance induced by \(P\) over \(\mathbb{F}_q^d\) is \(\|x-y\|_s:=P(x-y)\), with the corresponding distance set \(\Delta^s_{d, d}(E)=\{(\|x-y\|_s, \|z-t\|_s) : (x, z), (y, t) \in E \}\). We show that if \(|E| \gg q^{{(3d+1)}/{2}}\), then \(|\Delta_{d, d}^s(E)| \gg q^2\). For \(d=2\) and the Euclidean distance, we improve the former result over prime fields by showing that \(|\Delta_{2,2}(E)| \gg p^2\) for \(|E| \gg p^{{13}/{4}}\).Almost-rigidity of frameworkshttps://zbmath.org/1521.520092023-11-13T18:48:18.785376Z"Holmes-Cerfon, Miranda"https://zbmath.org/authors/?q=ai:holmes-cerfon.miranda-c"Theran, Louis"https://zbmath.org/authors/?q=ai:theran.louis"Gortler, Steven J."https://zbmath.org/authors/?q=ai:gortler.steven-jSummary: We extend the mathematical theory of rigidity of frameworks (graphs embedded in \(d\)-dimensional space) to consider nonlocal rigidity and flexibility properties. We provide conditions on a framework under which (I) as the framework flexes continuously it must remain inside a small ball, a property we call ``almost-rigidity''; (II) any other framework with the same edge lengths must lie outside a much larger ball; (III) if the framework deforms by some given amount, its edge lengths change by a minimum amount; (IV) there is a nearby framework that is prestress stable, and thus rigid. The conditions can be tested efficiently using semidefinite programming. The test is a slight extension of the test for prestress stability of a framework, and gives analytic expressions for the radii of the balls and the edge length changes. Examples illustrate how the theory may be applied in practice, and we provide an algorithm to test for rigidity or almost-rigidity. We briefly discuss how the theory may be applied to tensegrities.Realizability of some Böröczky arrangements over the rational numbershttps://zbmath.org/1521.520102023-11-13T18:48:18.785376Z"Janasz, Marek"https://zbmath.org/authors/?q=ai:janasz.marek"Lampa-Baczyńska, Magdalena"https://zbmath.org/authors/?q=ai:lampa-baczynska.magdalena"Wójcik, Daniel"https://zbmath.org/authors/?q=ai:wojcik.daniel-kThe authors study the parameter spaces for Böröczky arrangements \(B_n\) of \(n\) lines, with \(n<12\). Here, \(B_n\) is the configuration of \(n\) lines arranged according to the following construction: considering an \(2n\)-gon inscribed in a circle, and fixing one of the \(2n\) vertices, which will be denoted by \(Q_0\), then \(Q_\alpha\) is the point constructed by rotating \(Q_0\) around the center of a circle by some angle \(\alpha\). Next the following \(n\) lines are considered
\[
B_n=\left\{Q_\alpha Q_{\pi-2\alpha}, \text{ were }\alpha=\frac{2k\pi}{n} \text{ for } k=0,1,\dots,n-1\right\}.
\]
If \(\alpha\equiv (\pi-2\alpha) \mod 2\pi\), then \(Q_\alpha Q_{\pi-2\alpha}\) is the tangent to the circle at the point \(Q_\alpha\).
The aim of this paper is to \textit{complete the picture} for a number of lines between 3 and 11 in Böröczky arrangements, and to establish the realizability of these configurations over the rational numbers. The latter means, that there exists a configuration of that number of lines with the same incidences between the lines and the intersection points, such that all points have rational coordinates.
The interest in these configurations seems to have been recently renewed, due to its connection to the containment problem in Commutative Algebra (see [\textit{A. Czapliński} et al., Adv. Geom. 16, No. 1, 77--82 (2016; Zbl 1333.13005)] and [\textit{Ł. Farnik} et al., J. Algebr. Comb. 50, No. 1, 39--47 (2019; Zbl 1419.52021)]).
The authors mention the works [\textit{M. Lampa-Baczyńska} and \textit{J. Szpond}, Geom. Dedicata 188, 103--121 (2017; Zbl 1366.14048)], and [\textit{Ł. Farnik} et al., Int. J. Algebra Comput. 28, No. 7, 1231--1246 (2018; Zbl 1403.52012)], where the parameter spaces of certain Böröczky arrangements are considered, as their inspiration for the actual work. Indeed, in [Lampa-Baczyńska and Szpond, loc. cit.] it was shown that the \(B_{12}\) arrangement is realizable over the rational numbers.
The authors use an algorithm based on ideas of [\textit{B. Sturmfels}, J. Symb. Comput. 11, No. 5--6, 595--618 (1991; Zbl 0766.14043)], combined with methods established in [Lampa-Baczyńska and Szpond, loc. cit.], from which they can conclude that all arrangements \(B_n\), \(n\leq 10\), are realizable over the rationals. They obtain, further, that \(B_{11}\) is not realizable over the rationals. As mentioned above, a proof that \(B_{12}\) is realizable over the rational numbers can be found in [Lampa-Baczyńska and Szpond, loc. cit.]. Thus, \(B_{11}\) is the only Böröczky configuration, up to \(n = 12\), which is nonrealizable over the rational numbers. Other results on realizability of \(B_n\) are known in the literature for \(n\in\{ 13, 14,15,16,18,24\}\).
For the entire collection see [Zbl 1509.14002].
Reviewer: Eugenia Saorín Gómez (Bremen)On the existence of mass minimizing rectifiable \(G\) chains in finite dimensional normed spaceshttps://zbmath.org/1521.530532023-11-13T18:48:18.785376Z"De Pauw, Thierry"https://zbmath.org/authors/?q=ai:de-pauw.thierry"Vasilyev, Ioann"https://zbmath.org/authors/?q=ai:vasilyev.ioannSummary: We introduce the notion of density contractor of dimension \(m\) in a finite dimensional normed space \(X\). If \(m+1=\dim X\), this includes the area contracting projectors on hyperplanes whose existence was established by \textit{H. Buseman} [Ann. Math. (2) 48, 234--267 (1947; Zbl 0029.35301)]. If \(m=2\), density contractors are an ersatz for such projectors and their existence, established here, follows from works by \textit{D. Burago} and \textit{S. Ivanov} [Geom. Funct. Anal. 14, No. 3, 469--490 (2004; Zbl 1067.53042); Geom. Funct. Anal. 22, No. 3, 627--638 (2012; Zbl 1266.52008)]. Once density contractors are available, the corresponding Plateau problem admits a solution among rectifiable \(G\) chains, regardless of the group of coefficients \(G\). This is obtained as a consequence of the lower semicontinuity of the \(m\) dimensional Hausdorff mass, of which we offer two proofs. One of these is based on a new type of integral geometric measure.Coisotropic Hofer-Zehnder capacities of convex domains and related resultshttps://zbmath.org/1521.530622023-11-13T18:48:18.785376Z"Jin, Rongrong"https://zbmath.org/authors/?q=ai:jin.rongrong"Lu, Guangcun"https://zbmath.org/authors/?q=ai:lu.guangcunSummary: We prove representation formulas for the coisotropic Hofer-Zehnder capacities of bounded convex domains with special coisotropic submanifolds and the leaf relation (introduced by \textit{S. Lisi} and \textit{A. Rieser} [J. Symplectic Geom. 18, No. 3, 819--865 (2020; Zbl 1478.53126)] recently), study their estimates and relations with the Hofer-Zehnder capacity, give some interesting corollaries, and also obtain corresponding versions of a Brunn-Minkowski type inequality by \textit{S. Artstein-Avidan} and \textit{Y. Ostrover} [Int. Math. Res. Not. 2008, Article ID rnn044, 31 p. (2008; Zbl 1149.52006)] and a theorem by \textit{E. Neduv} [Math. Z. 236, No. 1, 99--112 (2001; Zbl 0967.37031)].Asymptotics of quantum \(6j\) symbolshttps://zbmath.org/1521.570132023-11-13T18:48:18.785376Z"Chen, Qingtao"https://zbmath.org/authors/?q=ai:chen.qingtao"Murakami, Jun"https://zbmath.org/authors/?q=ai:murakami.jun.1|murakami.junThis paper gives the asymptotics of the quantum \(6j\) symbols corresponding to a hyperbolic tetrahedron with at least one ideal or ultra-ideal vertex. The authors show that the quantum \(6j\) symbol corresponding to a hyperbolic tetrahedron grows exponentially and the leading term is given by the volume of the tetrahedron. This point is quite different from the asymptotics given in previous publications [\textit{J. Roberts}, Geom. Topol. 3, 21--66 (1999; Zbl 0918.22014) and \textit{Y. U. Taylor} and \textit{C. T. Woodward}, Sel. Math., New Ser. 11, No. 3--4, 539--571 (2005; Zbl 1162.17306)].
Reviewer: Meili Zhang (Dalian)Lower bound for Buchstaber invariants of real universal complexeshttps://zbmath.org/1521.570252023-11-13T18:48:18.785376Z"Shen, Qifan"https://zbmath.org/authors/?q=ai:shen.qifanSummary: In this article, we prove that Buchstaber invariant of 4-dimensional real universal complex is no less than 24 as a follow-up to the work of \textit{A. Ayzenberg} [Osaka J. Math. 53, No. 2, 377--395 (2016; Zbl 1339.05439) and ``The problem of Buchstaber number and its combinatorial aspects'', Preprint, \url{arXiv:1003.0637}] and \textit{Y. Sun} [Chin. Ann. Math., Ser. B 38, No. 6, 1335--1344 (2017; Zbl 1387.57052)]. Moreover, a lower bound for Buchstaber invariants of \(n\)-dimensional real universal complexes is given as an improvement of result of \textit{N. Yu. Erokhovets} [Proc. Steklov Inst. Math. 286, 128--187 (2014; Zbl 1317.52019); translation from Tr. Mat. Inst. Steklova 286, 144--206 (2014)].On intersection probabilities of four lines inside a planar convex domainhttps://zbmath.org/1521.600072023-11-13T18:48:18.785376Z"Martirosyan, Davit"https://zbmath.org/authors/?q=ai:martirosyan.davit-m"Ohanyan, Victor"https://zbmath.org/authors/?q=ai:ohanyan.v-kIn this paper, the authors consider the problem of computing the probabilities \(p_{nk}\) that \(n\) random lines produce exactly \(k\) intersection points inside a planar convex domain \(D\), and their approach is based on the use of Ambartzumian's combinatorial algorithm.
To illustrate this approach, the authors first re-derive existing results for \(n=2,3\). They then introduce new geometric invariants, which are subsequently used for the formulae for \(p_{4k}\), obtained by a careful study of events that lead to different number of intersections. The paper concludes with some explicit formulae in the case where \(D\) is a disc.
Reviewer: Mo Dick Wong (Durham)Optimal area polygonization by triangulation and visibility searchhttps://zbmath.org/1521.682382023-11-13T18:48:18.785376Z"Lepagnot, Julien"https://zbmath.org/authors/?q=ai:lepagnot.julien"Moalic, Laurent"https://zbmath.org/authors/?q=ai:moalic.laurent"Schmitt, Dominique"https://zbmath.org/authors/?q=ai:schmitt.dominiqueComputing circuit polynomials in the algebraic rigidity matroidhttps://zbmath.org/1521.682672023-11-13T18:48:18.785376Z"Malić, Goran"https://zbmath.org/authors/?q=ai:malic.goran"Streinu, Ileana"https://zbmath.org/authors/?q=ai:streinu.ileanaSummary: We present an algorithm for computing \textit{circuit polynomials} in the algebraic rigidity matroid \(\boldsymbol{\mathcal{A}}(\mathrm{CM}_n)\) associated to the Cayley-Menger ideal \(\mathrm{CM}_n\) for \(n\) points in 2D. It relies on combinatorial resultants, a new operation on graphs that captures properties of the Sylvester resultant of two polynomials in this ideal. We show that every rigidity circuit has a construction tree from \(K_4\) graphs based on this operation. Our algorithm performs an algebraic elimination guided by such a construction tree and uses classical resultants, factorization, and ideal membership. To highlight its effectiveness, we implemented the algorithm in Mathematica: it took less than 15 seconds on an example where a Gröbner basis calculation took 5 days and 6 hours. Additional speed-ups are obtained using non-\(K_4\) generators of the Cayley-Menger ideal and simple variations on our main algorithm.Recognition of affine-equivalent polyhedra by their natural developmentshttps://zbmath.org/1521.830092023-11-13T18:48:18.785376Z"Alexandrov, V. A."https://zbmath.org/authors/?q=ai:alexandrov.victor-a|aleksandrov.vasil-aSummary: The classical Cauchy rigidity theorem for convex polytopes reads that if two convex polytopes have isometric developments then they are congruent. In other words, we can decide whether two convex polyhedra are isometric or not by only using their developments. We study a similar problem of whether it is possible to understand that two convex polyhedra in Euclidean 3-space are affine-equivalent by only using their developments.Loop quantum gravity of a spherically symmetric scalar field coupled to gravity with a clockhttps://zbmath.org/1521.830432023-11-13T18:48:18.785376Z"Gambini, Rodolfo"https://zbmath.org/authors/?q=ai:gambini.rodolfo"Pullin, Jorge"https://zbmath.org/authors/?q=ai:pullin.jorge-aSummary: The inclusion of matter fields in spherically symmetric loop quantum gravity has proved problematic at the level of implementing the constraint algebra including the Hamiltonian constraint. Here we consider the system with the introduction of a clock. Using the abelianizaton technique we introduced in previous papers in the case of gravity coupled to matter, the system can be gauge fixed and rewritten in terms of a restricted set of dynamical variables that satisfy simple Poisson bracket relations. This creates a true Hamiltonian and therefore one bypasses the issue of the constraint algebra. We show how loop quantum gravity techniques may be applied for the vacuum case and show that the Hamiltonian system reproduces previous results for the physical space of states and the observables of a Schwarzchild black hole.Odd-parity perturbations in the most general scalar-vector-tensor theoryhttps://zbmath.org/1521.830852023-11-13T18:48:18.785376Z"Baez, Yolbeiker Rodríguez"https://zbmath.org/authors/?q=ai:baez.yolbeiker-rodriguez"Gonzalez-Espinoza, Manuel"https://zbmath.org/authors/?q=ai:gonzalez-espinoza.manuelSummary: In the context of the most general scalar-vector-tensor theory, we study the stability of static spherically symmetric black holes under linear odd-parity perturbations. We calculate the action to second order in the linear perturbations to derive a master equation for these perturbations. For this general class of models, we obtain the conditions of no-ghost and Laplacian instability. Then, we study in detail the generalized Regge-Wheeler potential of particular cases to find their stability conditions.Vacuum-dual static perfect fluid obeying \(p = -(n-3)\rho/(n+1)\) in \(n(\geqslant 4)\) dimensionshttps://zbmath.org/1521.831372023-11-13T18:48:18.785376Z"Maeda, Hideki"https://zbmath.org/authors/?q=ai:maeda.hidekiSummary: We obtain the general \(n(\geqslant 4)\)-dimensional static solution with an \(n-2\)-dimensional Einstein base manifold for a perfect fluid obeying a linear equation of state \(p = -(n-3)\rho/(n+1)\). It is a generalization of Semiz's four-dimensional general solution with spherical symmetry and consists of two different classes. Through the Buchdahl transformation, the class-I and class-II solutions are dual to the topological Schwarzschild-Tangherlini-(A)dS solution and one of the \(\Lambda\)-vacuum direct-product solutions, respectively. While the metric of the spherically symmetric class-I solution is \(C^\infty\) at the Killing horizon for \(n = 4\) and 5, it is \(C^1\) for \(n \geqslant 6\) and then the Killing horizon turns to be a parallelly propagated curvature singularity. For \(n = 4\) and 5, the spherically symmetric class-I solution can be attached to the Schwarzschild-Tangherlini vacuum black hole with the same value of the mass parameter at the Killing horizon in a regular manner, namely without a lightlike massive thin-shell. This construction allows new configurations of an asymptotically (locally) flat black hole to emerge. If a static perfect fluid hovers outside a vacuum black hole, its energy density is negative. In contrast, if the dynamical region inside the event horizon of a vacuum black hole is replaced by the class-I solution, the corresponding matter field is an anisotropic fluid and may satisfy the null and strong energy conditions. While the latter configuration always involves a spacelike singularity inside the horizon for \(n = 4\), it becomes a non-singular black hole of the big-bounce type for \(n = 5\) if the ADM mass is larger than a critical value.On pseudoconvexity conditions and static spacetimeshttps://zbmath.org/1521.832182023-11-13T18:48:18.785376Z"Vatandoost, Mehdi"https://zbmath.org/authors/?q=ai:vatandoost.mehdi"Pourkhandani, Rahimeh"https://zbmath.org/authors/?q=ai:pourkhandani.rahimeh(no abstract)Study of decoupled gravastars in energy-momentum squared gravityhttps://zbmath.org/1521.850042023-11-13T18:48:18.785376Z"Sharif, M."https://zbmath.org/authors/?q=ai:sharif.muhammad"Naz, Saba"https://zbmath.org/authors/?q=ai:naz.sabaSummary: In this paper, we generate an exact anisotropic gravastar model using gravitational decoupling technique through minimal geometric deformation in the framework of \(f(\Re,T^2)\) gravity. This novel model explains an ultra-compact stellar configuration whose internal region is smoothly matched to the exterior region. The developed stellar model satisfies some of the essential characteristics of a physically acceptable model such as a positive monotonically decreasing profile of energy density from the center to the boundary and monotonically decreasing behavior of the pressure. The anisotropic factor and Schwarzschild spacetime follows physically acceptable behavior. We find that all the energy bounds are satisfied except strong energy condition inside the ultra-compact stellar structure for the coupling constant of this theory, which is compatible with the regularity condition.On the geometry of elementary flux modeshttps://zbmath.org/1521.920552023-11-13T18:48:18.785376Z"Wieder, Frederik"https://zbmath.org/authors/?q=ai:wieder.frederik"Henk, Martin"https://zbmath.org/authors/?q=ai:henk.martin"Bockmayr, Alexander"https://zbmath.org/authors/?q=ai:bockmayr.alexanderSummary: Elementary flux modes (EFMs) play a prominent role in the constraint-based analysis of metabolic networks. They correspond to minimal functional units of the metabolic network at steady-state and as such have been studied for almost 30 years. The set of all EFMs in a metabolic network tends to be very large and may have exponential size in the number of reactions. Hence, there is a need to elucidate the structure of this set. Here we focus on geometric properties of EFMs. We analyze the distribution of EFMs in the face lattice of the steady-state flux cone of the metabolic network and show that EFMs in the relative interior of the cone occur only in very special cases. We introduce the concept of degree of an EFM as a measure how elementary it is and study the decomposition of flux vectors and EFMs depending on their degree. Geometric analysis can help to better understand the structure of the set of EFMs, which is important from both the mathematical and the biological viewpoint.Disjoint placement probability of line segments via geometryhttps://zbmath.org/1521.970122023-11-13T18:48:18.785376Z"Ennis, Christopher"https://zbmath.org/authors/?q=ai:ennis.christopher.1"Shier, John"https://zbmath.org/authors/?q=ai:shier.johnSummary: We have shown that when \textit{any} finite number \(n\), of line segments with total combined length less than one, have their centers placed randomly inside the unit interval \([0, 1]\), the probability of obtaining a mutually disjoint placement of the segments \textit{within} \([0, 1]\), is given by the expression \((1-L)^n\) where \(L = |L_1| + \cdots + |L_n|\), and \(|L_k|\) denotes the length of the \(k\)-\textit{th} segment, \(L_k\). The result is established by a careful analysis of the \textit{geometry} of the event, ``all segments disjoint and contained within \([0, 1]\),'' considered as a subset of the uniform probability space of \(n\) centers, each of which is in \([0, 1]\); that is to say, the unit \(n\)-cube of \(\mathbb{R}^n\). This event has an interesting geometric structure consisting of \(n!\) disjoint, congruent, (up to a mirror image) \textit{polytopes} within the unit \(n\)-cube. It is shown these event polytopes fit together perfectly to form, except for a set of measure zero, a partition of an \(n\)-dimensional cube with common edge length \(1-L\), and hence an \(n\)-volume given by the formula. In the case of \(n = 3\) segments, the polytopes form one of the known tetrahedral partitions of the cube as discussed, for example in [\textit{M. Senechal}, Math. Mag. 54, 227--243 (1981; Zbl 0469.51017)]. In fact for all \(n > 0\), the polytopes comprise a partition of the \(n\)-dimensional \textit{hypercube}, and are therefore \(n\)-dimensional \textit{space filling}.