Recent zbMATH articles in MSC 51https://www.zbmath.org/atom/cc/512021-04-16T16:22:00+00:00WerkzeugTowards an automatic geometer.https://www.zbmath.org/1456.682272021-04-16T16:22:00+00:00"Botana, Francisco"https://www.zbmath.org/authors/?q=ai:botana.francisco"Kovács, Zoltán"https://www.zbmath.org/authors/?q=ai:kovacs.zoltan"Recio, Tomás"https://www.zbmath.org/authors/?q=ai:recio.tomas"Vélez, M. Pilar"https://www.zbmath.org/authors/?q=ai:velez.m-pilarThe article, which is in Spanish language, describes the history of the authors' work group in the last three decades and the systems they developed for automated reasoning in the field of geometry; it furthermore gives an overview of the general development of the field and how the authors' systems relate to other systems. The systems conjecture, prove, refute and discover theorems and construct geometric objects. For example the authors' system GDI rediscovered a theorem of MacLane. The article gives an overview by decades and explains the systems and the geometric operations which they performed and results they proved or discovered in detail, including screen shots of the corresponding computer programs. The article is for an audience of Spanish-speaking people in Latin America and Spain. It appeared in a column on computational mathematics of a general-audience mathematical journal.
Reviewer: Frank Stephan (Singapore)Imaginary hypercubes.https://www.zbmath.org/1456.510162021-04-16T16:22:00+00:00"Tsuiki, Hideki"https://www.zbmath.org/authors/?q=ai:tsuiki.hideki"Tsukamoto, Yasuyuki"https://www.zbmath.org/authors/?q=ai:tsukamoto.yasuyukiSummary: Imaginary cubes are three-dimensional objects that have square projections in three orthogonal ways, just like a cube has. In this paper, we introduce higher-dimensional extensions of imaginary cubes and study their properties.
For the entire collection see [Zbl 1318.68008].Projections and distances in \(\mathbb{R}^7\), double vector cross product and associated hyperplanes.https://www.zbmath.org/1456.510182021-04-16T16:22:00+00:00"Catarino, Paula"https://www.zbmath.org/authors/?q=ai:catarino.paula-m-m-c"Vitória, José"https://www.zbmath.org/authors/?q=ai:vitoria.joseSummary: The distance between two skew lines in \(\mathbb{R}^7\) is expressed both in terms of a inner product and in terms of a double vector cross product. The best approximation pair of two skew lines in \(\mathbb{R}^7\) is expressed with inner product and double vector cross product, as well. The obtained formulas using a double vector cross product are not valid for \(\mathbb{R}^n\), with \(n\neq 3,7\). Another feature of this paper is to show that the distance from a point to a line and the distance between skew lines are essentially the distance from a point to a hyperplane.Results on partial geometries with an abelian Singer group of rigid type.https://www.zbmath.org/1456.510042021-04-16T16:22:00+00:00"De Winter, Stefaan"https://www.zbmath.org/authors/?q=ai:de-winter.stefaan"Kamischke, Ellen"https://www.zbmath.org/authors/?q=ai:kamischke.ellen"Neubert, Eric"https://www.zbmath.org/authors/?q=ai:neubert.eric"Wang, Zeying"https://www.zbmath.org/authors/?q=ai:wang.zeyingA partial geometry \(pg(s,t,\alpha)\) is a finite partial linear space such that every line contains \(s+1\) points, every point lies on \(t+1\) lines, and given a point \(P\) not incident with a line \(L\), there are exactly alpha lines through \(P\) meeting \(L\).
The partial geometry is said to admit a Singer group if its automorphism group contains a subgroup acting sharply transitively on the points. If the Singer group is an abelian group G, then it was shown in [\textit{S. De Winter}, J. Algebr. Comb. 24, No. 3, 285--297 (2006; Zbl 1106.51003)] that the stabiliser of a line in G is either trivial or has size \(s+1\). The authors call the partial geometry, together with the group G, of ``rigid type'' if all lines have a trivial stabiliser.
The only known example of a partial geometry of rigid type is due to Van Lint and Schrijver and has parameters \((5,5,2)\). In this paper, the authors study the possible parameters of partial geometries of rigid type (and their associated Singer group). By use of different necessary conditions on the parameters and some properties of the associated group, and aided by a computer, the authors show that if \(\alpha\leq 1000\), there are only five parameter sets that could possibly correspond to a partial geometry of rigid type.
The hypothetical partial geometries would have one of the four sets of parameters
\((11,23,3), (39,39,15), (272,272,104)\) or \((2295,4591,615)\).
The authors resist the temptation to conjecture that the Van Lint-Schrijver partial geometry is the only partial geometry of rigid type. Instead they formulate the weaker conjecture that for a partial geometry of rigid type, the number of points is either a power of \(2\) or a power of \(3\). Note that this conjecture does not rule out the existence of a partial geometry of rigid type with parameters \((11,23,3)\) or \((39,39,15)\).
Reviewer: Geertrui Van de Voorde (Canterbury)Thickness design for ambiguous cylinder illusion.https://www.zbmath.org/1456.001042021-04-16T16:22:00+00:00"Sugihara, Kokichi"https://www.zbmath.org/authors/?q=ai:sugihara.kokichiSummary: This paper proposes methods for giving as uniform a thickness as possible to a class of illusion solids called ambiguous cylinders. Ambiguous cylinders are solids that have two quite different appearances when seen from two specific viewpoints, and thus create the impression of impossible objects. In order to realize them as physical objects, we have to give them thickness. However, it is impossible to give a completely uniform thickness despite this being desirable. Instead we have to content ourselves with second-best methods. For this purpose, this paper proposes three alternative strategies for creating objects as uniform as possible. Each strategy has its own merits and demerits, and hence users can choose their method according to their priorities for the visual effects which they want to emphasize.Rotors in triangles and tetrahedra.https://www.zbmath.org/1456.510132021-04-16T16:22:00+00:00"Bracho, Javier"https://www.zbmath.org/authors/?q=ai:bracho.javier"Montejano, Luis"https://www.zbmath.org/authors/?q=ai:montejano.luisSummary: A barycentric formula that involves the curvatures at the contact points of a rotor within a triangle is proved, and the 3-dimensional case of rotors in tetrahedra is considered.Point-free foundation of geometry looking at laboratory activities.https://www.zbmath.org/1456.510092021-04-16T16:22:00+00:00"Gerla, Giangiacomo"https://www.zbmath.org/authors/?q=ai:gerla.giangiacomo"Miranda, Annamaria"https://www.zbmath.org/authors/?q=ai:miranda.annamariaThe authors propose a point-free axiomatization of Euclidean geometry, in which the notions of convexity and half-planes, and with them the Boolean algebra of regular closed subsets, regions, and \(n\)-dimensional prototypes are crucial, with the aim of providing an intuitively convincing axiomatization that could be used in an educational setting.
Reviewer: Victor V. Pambuccian (Glendale)Investigation graph isomorphism problem via entanglement entropy in strongly regular graphs.https://www.zbmath.org/1456.940332021-04-16T16:22:00+00:00"Jafarizadeh, M. A."https://www.zbmath.org/authors/?q=ai:jafarizadeh.mohamad-ali|jafarizadeh.mohammad-ali"Eghbalifam, F."https://www.zbmath.org/authors/?q=ai:eghbalifam.f"Nami, S."https://www.zbmath.org/authors/?q=ai:nami.susanFractional supersymmetric quantum mechanics and lacunary Hermite polynomials.https://www.zbmath.org/1456.812062021-04-16T16:22:00+00:00"Bouzeffour, F."https://www.zbmath.org/authors/?q=ai:bouzeffour.fethi"Garayev, M."https://www.zbmath.org/authors/?q=ai:garayev.mubariz-tapdigogluSummary: We consider a realization of fractional supersymmetric of quantum mechanics of order \(r\), where the Hamiltonian and supercharges involve reflection operators. It is shown that the Hamiltonian has \(r\)-fold degenerate spectrum and the eigenvalues of hermitian supercharges are zeros of the associated Hermite polynomials of Askey and Wimp. Also it is shown that the associated eigenfunctions involve lacunary Hermite polynomials.Constructions in the locus of isogonal conjugates for a quadrilateral.https://www.zbmath.org/1456.510192021-04-16T16:22:00+00:00"Hu, Daniel"https://www.zbmath.org/authors/?q=ai:hu.danielFor any quadrilateral \(ABCD\) in the Euclidean plane, there is an associated cubic plane curve \(\mathcal{C}\) which consists of points \(X\) such that the pairs of lines \(( XA,XC ), \; ( XB,XD )\) have the same pair of angle bisectors. This curve is called the isogonal cubic of \(ABCD\)
and the author studies it in general case when \(A\), \(B\), \(C\), \(D\) are distinct points. In the first half of the paper, geometric properties of \(\mathcal{C}\) are discussed. In the second half of the paper, all isogonal cubics are characterized either by embedding of \(\mathcal{C}\) in the complex projective plane, or by a specific cubic polynomial defined on the Euclidean plane.
Reviewer: Georgi Hristov Georgiev (Shumen)Novel structures for optimal space partitions.https://www.zbmath.org/1456.510152021-04-16T16:22:00+00:00"Opsomer, E."https://www.zbmath.org/authors/?q=ai:opsomer.e"Vandewalle, N."https://www.zbmath.org/authors/?q=ai:vandewalle.nicolasSquare turning maps and their compactifications.https://www.zbmath.org/1456.370422021-04-16T16:22:00+00:00"Schwartz, Richard Evan"https://www.zbmath.org/authors/?q=ai:schwartz.richard-evanSummary: In this paper we introduce some infinite rectangle exchange transformations which are based on the simultaneous turning of the squares within a sequence of square grids. We will show that such noncompact systems have higher dimensional dynamical compactifications. In good cases, these compactifications are polytope exchange transformations based on pairs of Euclidean lattices. In each dimension \(8m+4\) there is a \(4m+2\) dimensional family of them. Here \(m=0,1,2,\dotsc \) We studied the case \(m=0\) in depth in [\textit{M. Keane}, Isr. J. Math. 26, 188--196 (1977; Zbl 0351.28012)].The endomorphisms algebra of translations group and associative unitary ring of trace-preserving endomorphisms in affine plane.https://www.zbmath.org/1456.510012021-04-16T16:22:00+00:00"Zaka, Orgest"https://www.zbmath.org/authors/?q=ai:zaka.orgest"Mohammed, Mohanad A."https://www.zbmath.org/authors/?q=ai:mohammed.mohanad-aSummary: A description of Endomorphisms of the translation group is introduced in an affine plane, will define the addition and composition of the set of endomorphisms and specify the neutral elements associated with these two actions and present the Endomorphism algebra thereof will distinguish the Trace-preserving endomorphism algebra in affine plane, and prove that the set of Trace-preserving endomorphism associated with the `addition' action forms a commutative group. We also try to prove that the set of trace-preserving endomorphism, together with the two actions, in it, `addition' and `composition' forms an associative and unitary ring.50 years of finite geometry, the ``geometries over finite rings'' part.https://www.zbmath.org/1456.510022021-04-16T16:22:00+00:00"Keppens, Dirk"https://www.zbmath.org/authors/?q=ai:keppens.dirkSummary: Whereas for a substantial part, ``finite geometry'' during the past 50 years has focussed on geometries over finite fields, geometries over finite rings that are not division rings have got less attention. Nevertheless, several important classes of finite rings give rise to interesting geometries.
In this paper we bring together some results, scattered over the literature, concerning finite rings and plane projective geometry over such rings. The paper does not contain new material, but by collecting information in one place, we hope to stimulate further research in this area for at least another 50 years of finite geometry.Joints formed by lines and a \(k\)-plane, and a discrete estimate of Kakeya type.https://www.zbmath.org/1456.520262021-04-16T16:22:00+00:00"Carbery, Anthony"https://www.zbmath.org/authors/?q=ai:carbery.anthony"Iliopoulou, Marina"https://www.zbmath.org/authors/?q=ai:iliopoulou.marinaSummary: Let \(\mathcal{L}\) be a family of lines and let \(\mathcal{P}\) be a family of \(k\)-planes in \(\mathbb{F}^n\) where \(\mathbb{F}\) is a field. In our first result we show that the number of joints formed by a \(k\)-plane in \(\mathcal{P}\) together with \((n-k)\) lines in \(\mathcal{L}\) is \(O_n(|\mathcal{L}| |\mathcal{P}|^{1/(n-k)})\). This is the first sharp result for joints involving higher-dimensional affine subspaces, and it holds in the setting of arbitrary fields \(F\). In contrast, for our second result, we work in the three-dimensional Euclidean space \(\mathbb{R}^3\), and we establish the Kakeya-type estimate
\[
\sum\limits_{x\in J}\left(\sum\limits_{\ell\in\mathcal{L}}\chi_\ell(x)\right)^{3/2}\lesssim|\mathcal{L}|^{3/2}
\]
where \(J\) is the set of joints formed by \(\mathcal{L}\); such an estimate fails in the setting of arbitrary fields. This result strengthens the known estimates for joints, including those counting multiplicities. Additionally, our techniques yield significant structural information on quasi-extremisers for this inequality.Relevant implication and ordered geometry.https://www.zbmath.org/1456.030422021-04-16T16:22:00+00:00"Urquhart, Alasdair"https://www.zbmath.org/authors/?q=ai:urquhart.alasdairSummary: This paper shows that model structures for \(\mathbf{R}^+\), the system of positive
relevant implication, can be constructed from ordered geometries. This extends earlier results building such model structures from projective spaces. A final section shows how such models can be extended to models for the full system \(\mathbf{R}\).On volume functions of special flow polytopes associated to the root system of type \(A\).https://www.zbmath.org/1456.520172021-04-16T16:22:00+00:00"Negishi, Takayuki"https://www.zbmath.org/authors/?q=ai:negishi.takayuki"Sugiyama, Yuki"https://www.zbmath.org/authors/?q=ai:sugiyama.yuki"Takakura, Tatsuru"https://www.zbmath.org/authors/?q=ai:takakura.tatsuruBased on a result in [\textit{W. Baldoni} and \textit{M. Vergne}, Transform. Groups 13, No. 3--4, 447--469 (2008; Zbl 1200.52008)], the authors characterize the volume function of a flow polytope for the nice chamber in terms of a system of differential equations, also providing an inductive formula for the volume with respect to the rank of the root system of type \(A\).
Reviewer: Victor V. Pambuccian (Glendale)On the balanced upper chromatic number of finite projective planes.https://www.zbmath.org/1456.050542021-04-16T16:22:00+00:00"Blázsik, Zoltán L."https://www.zbmath.org/authors/?q=ai:blazsik.zoltan-l"Blokhuis, Aart"https://www.zbmath.org/authors/?q=ai:blokhuis.aart"Miklavič, Štefko"https://www.zbmath.org/authors/?q=ai:miklavic.stefko"Nagy, Zoltán Lóránt"https://www.zbmath.org/authors/?q=ai:nagy.zoltan-lorant"Szőnyi, Tamás"https://www.zbmath.org/authors/?q=ai:szonyi.tamasSummary: In this paper, we study vertex colorings of hypergraphs in which all color class sizes differ by at most one (balanced colorings) and each hyperedge contains at least two vertices of the same color (rainbow-free colorings). For any hypergraph \(H\), the maximum number \(k\) for which there is a balanced rainbow-free \(k\)-coloring of \(H\) is called the balanced upper chromatic number of the hypergraph. We confirm the conjecture of \textit{G. Araujo-Pardo} et al. [ibid. 338, No. 12, 2562--2571 (2015; Zbl 1317.05046)] by determining the balanced upper chromatic number of the desarguesian projective plane \(\operatorname{PG} ( 2 , q )\) for all \(q\). In addition, we determine asymptotically the balanced upper chromatic number of several families of non-desarguesian projective planes and also provide a general lower bound for arbitrary projective planes using probabilistic methods which determines the parameter up to a multiplicative constant.On linear sets of minimum size.https://www.zbmath.org/1456.510062021-04-16T16:22:00+00:00"Jena, Dibyayoti"https://www.zbmath.org/authors/?q=ai:jena.dibyayoti"Van de Voorde, Geertrui"https://www.zbmath.org/authors/?q=ai:van-de-voorde.geertruiSummary: An \(\mathbb{F}_q\)-linear set of rank \(k\), \(k \leq h\), on a projective line \(\mathrm{PG} ( 1 , q^h )\), containing at least one point of weight one, has size at least \(q^{k - 1} + 1\) (see [\textit{J. De Beule} and the second author, J. Comb. Theory, Ser. A 164, 109--124 (2019; Zbl 1411.51004)]). The classical example of such a set is given by a club. In this paper, we construct a broad family of linear sets meeting this lower bound, where we are able to prescribe the weight of the heaviest point to any value between \(k \slash 2\) and \(k - 1\). Our construction extends the known examples of linear sets of size \(q^{k - 1} + 1\) in \(\mathrm{PG} ( 1 , q^h )\) constructed for \(k = h = 4\) [\textit{G. Bonoli} and \textit{O. Polverino}, Innov. Incidence Geom. 2, 35--56 (2005; Zbl 1103.51006)] and \(k = h\) in [\textit{G. Lunardon} and \textit{O. Polverino}, J. Comb. Theory, Ser. A 90, No. 1, 148--158 (2000; Zbl 0964.51009)]. We determine the weight distribution of the constructed linear sets and describe them as the projection of a subgeometry. For small \(k\), we investigate whether all linear sets of size \(q^{k - 1} + 1\) arise from our construction. Finally, we modify our construction to define rank \(k\) linear sets of size \(q^{k - 1} + q^{k - 2} + \dots + q^{k - l} + 1\) in \(\mathrm{PG} ( l , q^h )\). This leads to new infinite families of small minimal blocking sets which are not of Rédei type.Non-elliptic webs and convex sets in the affine building.https://www.zbmath.org/1456.051732021-04-16T16:22:00+00:00"Akhmejanov, Tair"https://www.zbmath.org/authors/?q=ai:akhmejanov.tairSummary: We describe the \(\mathfrak{sl}_3\) non-elliptic webs in terms of convex sets in the affine building. Kuperberg defined the non-elliptic web basis in his work on rank-\(2\) spider categories. \textit{B. Fontaine} et al. [Compos. Math. 149, No. 11, 1871--1912 (2013; Zbl 1304.22016)] showed that the \(\mathfrak{s}l_3\) non-elliptic webs are dual to CAT(0) triangulated diskoids in the affine building. We show that each such triangulated diskoid is the intersection of the min-convex and max-convex hulls of a generic polygon in the building. Choosing a generic polygon from each of the components of the Satake fiber produces (the duals of) the non-elliptic web basis. The convex hulls in the affine building were first introduced by \textit{G. Faltings} [Prog. Math. 195, 157--184 (2001; Zbl 1028.14002)] and are related to tropical convexity, as discussed in work by \textit{M. Joswig} et al. [Albanian J. Math. 1, No. 4, 187--211 (2007; Zbl 1133.52003)] and by \textit{L. Zhang} [``Computing convex hulls in the affine building of \(\mathfrak{sl}_d\)'', Preprint, \url{arXiv:1811.08884}].Expected dispersion of uniformly distributed points.https://www.zbmath.org/1456.600422021-04-16T16:22:00+00:00"Hinrichs, Aicke"https://www.zbmath.org/authors/?q=ai:hinrichs.aicke"Krieg, David"https://www.zbmath.org/authors/?q=ai:krieg.david"Kunsch, Robert J."https://www.zbmath.org/authors/?q=ai:kunsch.robert-j"Rudolf, Daniel"https://www.zbmath.org/authors/?q=ai:rudolf.danielSummary: The dispersion of a point set in \([0,1]^d\) is the volume of the largest axis parallel box inside the unit cube that does not intersect the point set. We study the expected dispersion with respect to a random set of \(n\) points determined by an i.i.d. sequence of uniformly distributed random variables. Depending on the number of points \(n\) and the dimension \(d\) we provide an upper and a lower bound of the expected dispersion. In particular, we show that the minimal number of points required to achieve an expected dispersion less than \(\varepsilon\in(0,1)\) depends linearly on the dimension \(d\).Generating polygons with triangles.https://www.zbmath.org/1456.510142021-04-16T16:22:00+00:00"Kuwata, T."https://www.zbmath.org/authors/?q=ai:kuwata.takayasu"Maehara, H."https://www.zbmath.org/authors/?q=ai:maehara.hiroshiSummary: A set of triangles \(\mathcal F\) is said to generate a polygon \(P\) if a homothetic transform \(\lambda P\) of \(P\) can be dissected into triangles each congruent to a triangle in \(\mathcal F\). The simplicial element number of a polygon \(P\) is defined to be the minimum cardinality of a family \(\mathcal F\) of triangles that can generate \(P\). The simplicial element number of a set of polygons \(P_1,P_2,\dots ,P_k\) is defined to be the minimum cardinality of a family \(\mathcal F\) of triangles that can generate all \(P_1,\dots ,P_k\). In this paper, we consider simplicial element numbers for several set of regular polygons and generating relations among triangles.
For the entire collection see [Zbl 1318.68008].On wrapping spheres and cubes with rectangular paper.https://www.zbmath.org/1456.510112021-04-16T16:22:00+00:00"Cole, Alex"https://www.zbmath.org/authors/?q=ai:cole.alex"Demaine, Erik D."https://www.zbmath.org/authors/?q=ai:demaine.erik-d"Fox-Epstein, Eli"https://www.zbmath.org/authors/?q=ai:fox-epstein.eliSummary: What is the largest cube or sphere that a given rectangular piece of paper can wrap? This natural problem, which has plagued gift-wrappers everywhere, remains very much unsolved. Here we introduce new upper and lower bounds and consolidate previous results. Though these bounds rarely match, our results significantly reduce the gap.
For the entire collection see [Zbl 1318.68008].Curved Pythagorean triangles.https://www.zbmath.org/1456.510102021-04-16T16:22:00+00:00"Cabral, Henrique"https://www.zbmath.org/authors/?q=ai:cabral.henrique"Carvalho, Maria"https://www.zbmath.org/authors/?q=ai:carvalho.maria-leonor-da-silva|de-carvalho.maria-pires|carvalho.maria-conceicao|carvalho.maria-luciliaSummary: A Pythagorean triangle is a right angled triangle whose sides have integer lengths. On the Euclidean plane they are completely known, after Euclid's classification. In this note, we discuss the existence of such triangles on surfaces of constant Gaussian curvature.The maximum volume of hyperbolic polyhedra.https://www.zbmath.org/1456.520152021-04-16T16:22:00+00:00"Belletti, Giulio"https://www.zbmath.org/authors/?q=ai:belletti.giulioWith \(\mathbb{H}^3\) the unit ball of \(\mathbb{R}^3\), a projective polyhedron \(P\subseteq\mathbb{R}^3 \subseteq \mathbb{RP}^3\) is a generalized hyperbolic polyhedron if each edge of \(P\) intersects \(\mathbb{H}^3\). A projective polyhedron \(\overline{\Gamma}\) is a rectification of a \(3\)-connected planar graph \(\Gamma\) if the \(1\)-skeleton of \(\overline{\Gamma}\) is equal to \(\Gamma\) and all the edges of
\(\overline{\Gamma}\) are tangent to \(\partial\mathbb{H}^3\) (which is \(\mathbb{S}^2\)). Although \(\overline{\Gamma}\) is not a generalized hyperbolic polyhedron, it is still possible
to provide a definition for the volume of \(\overline{\Gamma}\) as for any proper polyhedron.
The paper's main result states:
For any \(3\)-connected planar graph \(\Gamma\), \(\sup_{P}\mathrm{Vol}(P)=\mathrm{Vol}(\overline{\Gamma})\),
where \(P\) varies among all proper generalized hyperbolic polyhedra with \(1\)-skeleton \(\Gamma\) and
\(\overline{\Gamma}\) is the rectification of \(\Gamma\).
``The theorem is proved by applying a sort of volume-increasing flow to any hyperbolic polyhedron.''
Reviewer: Victor V. Pambuccian (Glendale)A Poncelet criterion for special pairs of conics in \(\mathrm{PG}(2,p^m)\).https://www.zbmath.org/1456.510052021-04-16T16:22:00+00:00"Hungerbühler, Norbert"https://www.zbmath.org/authors/?q=ai:hungerbuhler.norbert"Kusejko, Katharina"https://www.zbmath.org/authors/?q=ai:kusejko.katharinaSummary: We study Poncelet's Theorem in finite projective planes over the field \(\mathrm{GF}(q), q = p^m\) for \(p\) an odd prime and \(m > 0\), for a particular pencil of conics. We investigate whether we can find polygons with \(n\) sides which are inscribed in one conic and circumscribed around the other, so-called Poncelet Polygons. By using suitable elements of the dihedral group for these pairs, we prove that the length \(n\) of such Poncelet Polygons is independent of the starting point. In this sense Poncelet's Theorem is valid. By using Euler's divisor sum formula for the totient function, we can make a statement about the number of different conic pairs, which carry Poncelet Polygons of length \(n\). Moreover, we will introduce polynomials whose zeros in \(\mathrm{GF}(q)\) yield information about the relation of a given pair of conics. In particular, we can decide for a given integer \(n\), whether and how we can find Poncelet \(n\)-gons for pairs of conics in the plane \(\mathrm{PG}(2,q)\).Inscribed radius bounds for lower Ricci bounded metric measure spaces with mean convex boundary.https://www.zbmath.org/1456.510072021-04-16T16:22:00+00:00"Burtscher, Annegret"https://www.zbmath.org/authors/?q=ai:burtscher.annegret-y"Ketterer, Christian"https://www.zbmath.org/authors/?q=ai:ketterer.christian"McCann, Robert J."https://www.zbmath.org/authors/?q=ai:mccann.robert-j"Woolgar, Eric"https://www.zbmath.org/authors/?q=ai:woolgar.eric\textit{A. Kasue} [J. Math. Soc. Japan 35, 117--131 (1983; Zbl 0494.53039)] established a sharp estimate for the inscribed radius,
or inradius denoted \(\mathrm{InRad}\), of a smooth \(n\)-dimensional Riemannian manifold \(M\) with nonnegative Ricci curvature and smooth boundary \(\partial M\)\ whose mean curvature is bounded from below by \(n-1\). Exactly speaking, he concluded that
\[
\mathrm{InRad}_{M}\leq 1.
\]
The result was rediscovered by [\textit{M. M. C. Li}, J. Geom. Anal. 24, No. 3, 1490--1496 (2014; Zbl 1303.53053)], being extended to weighted Riemannian manifolds with Bakry-Émery curvature bounds in [\textit{H. Li} and \textit{Y. Wei}, J. Geom. Anal. 25, No. 1, 421--435 (2015; Zbl 1320.53075); Int. Math. Res. Not. 2015, No. 11, 3651--3668 (2015; Zbl 1317.53065); \textit{Y. Sakurai}, Tohoku Math. J. (2) 71, No. 1, 69--109 (2019; Zbl 1422.53029)]. These results are to be seen either as a manifold-with-boundary analogue of Bonnet and Myers' diameter bound or as a Riemannian analogue of the Hawking singularity theorem [\textit{S. W. Hawking}, Proc. R. Soc. Lond., Ser. A 294, 511--521 (1966; Zbl 0139.45803)], whose generalization to a nonsmooth setting is of paramount interest [\textit{M. Graf}, Commun. Math. Phys. 378, No. 2, 1417--1450 (2020; Zbl 1445.53052); \textit{M. Kunzinger} et al., Classical Quantum Gravity 32, No. 7, Article ID 075012, 19 p. (2015; Zbl 1328.83123); \textit{Y. Lu} et al., ``Geometry of weighted Lorentz-Finsler manifolds. I: Singularity theorems'', Preprint, \url{arXiv:1908.03832}].
This paper generalizes Kasue's [loc. cit.] and Li's [loc. cit.] estimate to subsets \(\Omega\)\ of a possibly nonsmooth space \(X\) abiding by a curvature dimension condition \(\mathrm{CD}(K,N)\) with \(K\in\mathbb{R}\) and \(N>1\), provided the topological boundary \(\partial\Omega\) has a lower bound on its inner mean curvature in the sense of [\textit{C. Ketterer}, Proc. Am. Math. Soc. 148, No. 9, 4041--4056 (2020; Zbl 1444.53028)]. The authors' result not only covers Kasue's [loc. cit.] theorem but also holds for a large class of domains in Alexandrov spaces or in Finsler manifolds. Kasue [loc. cit.] as well as Li [loc. cit.] was able to establish a rigidity result analogous to \textit{S.-Y. Cheng}'s theorem [Math. Z. 143, 289--297 (1975; Zbl 0329.53035)] in the Bonnet-Myers context [\textit{S. B. Myers}, Duke Math. J. 8, 401--404 (1941; JFM 67.0673.01); \textit{S. B. Myers}, Duke Math. J. 8, 401--404 (1941; Zbl 0025.22704)], namely that, among smooth manifolds, their inscribed radious bound is obtained exactly by the Euclidean unit ball. In the nonsmooth case, there are also truncated cones attaining maximal inradius. The authors establish, under an additional hypothesis known as RCD, that these are the only nonsmooth oprimizers provided \(\Omega\)\ is compact and its interior is connected.
Independently and almost simultaneously, \textit{F. Cavalletti} and \textit{A. Mondino} [Commun. Contemp. Math. 19, No. 6, Article ID 1750007, 27 p. (2017; Zbl 1376.53064); Invent. Math. 208, No. 3, 803--849 (2017; Zbl 1375.53053); Anal. PDE 13, 2091--2147 (2020); ``Optimal transport in Lorentzian synthetic spaces, synthetic timelike Ricci curvature lower bounds and applications'', Preprint, \url{arXiv:2004.08934}] have proposed a synthetic new framework for Lorentzian
geometry in which an analogue of the Hawking result is established.
Reviewer: Hirokazu Nishimura (Tsukuba)Calculation of the Voronoi boundary for Lens-shaped particles and spherocylinders.https://www.zbmath.org/1456.820222021-04-16T16:22:00+00:00"Portal, Louis"https://www.zbmath.org/authors/?q=ai:portal.louis"Danisch, Maximilien"https://www.zbmath.org/authors/?q=ai:danisch.maximilien"Baule, Adrian"https://www.zbmath.org/authors/?q=ai:baule.adrian"Mari, Romain"https://www.zbmath.org/authors/?q=ai:mari.romain"Makse, Hernán A."https://www.zbmath.org/authors/?q=ai:makse.hernan-aThe homogeneous pseudo-embeddings and hyperovals of the generalized quadrangle \(H(3,4)\).https://www.zbmath.org/1456.510032021-04-16T16:22:00+00:00"De Bruyn, Bart"https://www.zbmath.org/authors/?q=ai:de-bruyn.bart"Gao, Mou"https://www.zbmath.org/authors/?q=ai:gao.mouLet \(\mathcal{S}=(\mathcal{P},\mathcal{L})\) be a generalized quadrangle and denote by \(\varepsilon:\mathcal{S} \longrightarrow \mathrm{PG}(V)\) the universal pseudo-embedding \(\mathcal{S}.\) The vector dimension of the subspace \(\langle \varepsilon(\mathcal{P})\rangle\) of \(V\) is called the \textit{pseudo-embedding rank} of \(\mathcal{S}.\)
If \(|\mathcal{P}|\) is finite, denote by \(C=C(\mathcal{S})\) the binary code of length \(|\mathcal{P}|\) generated by the characteristic vectors of the lines of \(\mathcal{S}.\) It has been proved in [the first author, Adv. Geom. 13, No. 1, 71--95 (2013; Zbl 1267.51002)] that the pseudo-embedding rank of \(\mathcal{S}\) is \(|\mathcal{P}|- \dim(C).\)
In this paper, the authors prove that the pseudo-embedding rank of the Hermitian quadrangle \(H(3,4)\) is equal to \(24.\) As a consequence, the binary code \(C(H(3,4))\) has dimension \(45-24=21,\) because the generalized quadrangle \(H(3,4)\) has \(45\) points.
They also show that there are, up to isomorphism, four homogeneous pseudo-embeddings of \(H(3,4),\) with respective vector dimensions \(14,\) \(15,\) \(23\) and \(24.\)
Reviewer: Guglielmo Lunardon (Napoli)Hilbert's third problem and Dehn invariant -- an elementarization with spherical triangles.https://www.zbmath.org/1456.510172021-04-16T16:22:00+00:00"Leppmeier, Max"https://www.zbmath.org/authors/?q=ai:leppmeier.maxTo avoid the tensor product of Abelian groups involved in the Dehn invariant, which is considered non-elementary, the author introduces another invariant of a polyhedron \(P\) in \(\mathbb{R}^3\), defined by \(L(P)=\sum_{e\in P} \epsilon_e\), where, for every vertex \(e\) of \(P\), \(\epsilon_e=\sum_{k\in K_e} \alpha(k)-(|K_e|-2)\cdot \pi\), where \(\alpha(k)\) denotes the dihedral angle of edge \(k\), and \(K_e\) the set of all edges emanating from \(e\). Computing the \(L\)-invariant of a cube and of a regular tetrahedron and showing that: (*) ``if two polyhedrons \(P\) have \(Q\) of equal volume have each equal dihedral angles, and those of the former are rational multiples of \(\pi\), whereas those of the latter are irrational multiples of \(\pi\), then \(P\) and \(Q\) are not equidecomposable'' solves Hilbert's third problem. Using (*), one can also provide two tetrahedra with the same basis and congruent altitudes which are not equidecomposable. The Bricard condition can also be stated using (*) in a Dehn-Hadwiger type theorem as:
Let \(P\) and \(Q\) be two polyhedra with dihedral angles of the edges \(\alpha_1, \ldots \alpha_p\) and \(\beta_1, \ldots \beta_q\), respectively. If \(P\) and \(Q\) are equidecomposable, then there exists an integer \(n_{\pi}\) and there are natural numbers \(m_1, \ldots, m_p\) and \(n_1, \ldots, n_q\), so that
\[\sum_{i=1}^pm_i\alpha_i-\sum_{j=1}^qn_j\beta_j = n_{\pi}\cdot \pi\]
Reviewer: Victor V. Pambuccian (Glendale)A geometric interpretation of curvature inequalities on hypersurfaces via Ravi substitutions in the Euclidean plane.https://www.zbmath.org/1456.510122021-04-16T16:22:00+00:00"Suceavă, Bogdan D."https://www.zbmath.org/authors/?q=ai:suceava.bogdan-dragos(no abstract)Combinatorial structures in algebra and geometry. NSA 26, Constanţa, Romania, August 26 -- September 1, 2018.https://www.zbmath.org/1456.050012021-04-16T16:22:00+00:00"Stamate, Dumitru I. (ed.)"https://www.zbmath.org/authors/?q=ai:stamate.dumitru-ioan"Szemberg, Tomasz (ed.)"https://www.zbmath.org/authors/?q=ai:szemberg.tomaszPublisher's description: This proceedings volume presents selected, peer-reviewed contributions from the 26th National School on Algebra, which was held in Constanţa, Romania, on August 26-September 1, 2018. The works cover three fields of mathematics: algebra, geometry and discrete mathematics, discussing the latest developments in the theory of monomial ideals, algebras of graphs and local positivity of line bundles. Whereas interactions between algebra and geometry go back at least to Hilbert, the ties to combinatorics are much more recent and are subject of immense interest at the forefront of contemporary mathematics research. Transplanting methods between different branches of mathematics has proved very fruitful in the past -- for example, the application of fixed point theorems in topology to solving nonlinear differential equations in analysis. Similarly, combinatorial structures, e.g., Newton-Okounkov bodies, have led to significant advances in our understanding of the asymptotic properties of line bundles in geometry and multiplier ideals in algebra.This book is intended for advanced graduate students, young scientists and established researchers with an interest in the overlaps between different fields of mathematics. A volume for the 24th edition of this conference was previously published with Springer under the title ``Multigraded Algebra and Applications''
[\textit{V. Ene} (ed.) and \textit{E. Miller} (ed.), Multigraded algebra and applications. NSA 24, Moieciu de Sus, Romania, August 17--24, 2016. Cham: Springer (2018; Zbl 1400.13003)].
The articles of this volume will be reviewed individually.
Indexed articles:
\textit{Andrei-Ciobanu, Claudia}, Nearly normally torsionfree ideals, 1-13 [Zbl 1453.13007]
\textit{Cimpoeaş, Mircea; Stamate, Dumitru I.}, Gröbner-nice pairs of ideals, 15-29 [Zbl 1453.13077]
\textit{Dumnicki, Marcin; Farnik, Łucja; Harbourne, Brian; Szemberg, Tomasz; Tutaj-Gasińska, Halszka}, Veneroni maps, 31-42 [Zbl 1453.14043]
\textit{Ene, Viviana; Herzog, Jürgen}, On the symbolic powers of binomial edge ideals, 43-50 [Zbl 1456.05183]
\textit{Erey, Nursel}, Multigraded Betti numbers of some path ideals, 51-65 [Zbl 1456.05075]
\textit{Hibi, Takayuki; Tsuchiya, Akiyoshi}, Depth of an initial ideal, 67-71 [Zbl 1453.13081]
\textit{Juhnke-Kubitzke, Martina; Le, Dinh Van; Römer, Tim}, Asymptotic behavior of symmetric ideals: a brief survey, 73-94 [Zbl 1453.13024]
\textit{Pegel, Christoph; Sanyal, Raman}, On piecewise-linear homeomorphisms between distributive and anti-blocking polyhedra, 95-114 [Zbl 07304466]
\textit{Popescu, Dorin}, The Bass-Quillen conjecture and Swan's question, 115-121 [Zbl 1453.13026]
\textit{Rinaldo, Giancarlo; Terai, Naoki; Yoshida, Ken-Ichi}, Licci level Stanley-Reisner ideals with height three and with type two, 123-142 [Zbl 1453.13062]
\textit{Seyed Fakhari, S. A.}, Homological and combinatorial properties of powers of cover ideals of graphs, 143-159 [Zbl 1453.13036]
\textit{Szpond, Justyna}, Fermat-type arrangements, 161-182 [Zbl 1453.14022]On circumcenters of finite sets in Hilbert spaces.https://www.zbmath.org/1456.510082021-04-16T16:22:00+00:00"Bauschke, Heinz H."https://www.zbmath.org/authors/?q=ai:bauschke.heinz-h"Ouyang, Hui"https://www.zbmath.org/authors/?q=ai:ouyang.hui"Wang, Xianfu"https://www.zbmath.org/authors/?q=ai:wang.xianfuSummary: A well-known object in classical Euclidean geometry is the circumcenter of a triangle, i.e., the point that is equidistant from all vertices. The purpose of this paper is to provide a systematic study of the circumcenter of sets containing finitely many points in a Hilbert space. This is motivated by recent works of \textit{R. Behling} et al. [Numer. Algorithms 78, No. 3, 759--776 (2018; Zbl 1395.49023); Oper. Res. Lett. 46, No. 2, 159--162 (2018; Zbl 07064464)] on accelerated versions of the Douglas-Rachford method. We present basic results and properties of the circumcenter. Several examples are provided to illustrate the tightness of various assumptions.