Recent zbMATH articles in MSC 51https://zbmath.org/atom/cc/512022-11-17T18:59:28.764376ZWerkzeugPythagoras, and the napkin ringhttps://zbmath.org/1496.000102022-11-17T18:59:28.764376Z"Wright, C. D."https://zbmath.org/authors/?q=ai:wright.colin-dFor the entire collection see [Zbl 1496.00073].A graphical analysis of ``The geographer''https://zbmath.org/1496.000452022-11-17T18:59:28.764376Z"Sato, Noriko"https://zbmath.org/authors/?q=ai:sato.norikoSummary: This research conducts a graphical analysis of The Geographer, a seventeenth-century painting by Dutch artist Johannes Vermeer. Produced in 1669, it creates a contrast between light and shadow by depicting a close-up view in dark tones with a long-distance view on a bright floor. This contrast creates a sense of depth in the painting's virtual space. A rectangle like the seat of a backless chair at the bottom right of the screen creates a difference in height with respect to the floor in the back, and functions as a motif to give a sense of depth to the picture. This sense of depth would be lost without the bright flooring. Vermeer painted the flooring when The Milk Maid was created, which would indicate that the flooring configuration was done in the early stages of his career. However, The Geographer belongs to the latter period. Furthermore, checkered tiles were brilliantly placed on the floor of the interior paintings produced before and after 1669. Nevertheless, the flooring in this work does not constitute tiles. Therefore, this study analyzes whether the backless chair depicted in the foreground of this work was a part of the tiled floor that Vermeer intended to depict. As a result, the seat of this backless chair was equivalent to two square tiles. And if we assume that the construction method for those tiles that cover the floor of this painting was the first step, it suggests that a trigonometric ratio may have been used there. From this analysis, a hypothesis is formulated about Vermeer constructing floors of checkered tiles.Triangulation algorithms for generating as-is floor planshttps://zbmath.org/1496.000492022-11-17T18:59:28.764376Z"da Silva Brandão, Filipe Jorge"https://zbmath.org/authors/?q=ai:da-silva-brandao.filipe-jorge"Paio, Alexandra"https://zbmath.org/authors/?q=ai:paio.alexandra"Lopes, Adriano"https://zbmath.org/authors/?q=ai:lopes.adrianoSummary: Precisely capturing context is a fundamental first step in dealing with built environments. Previous research has demonstrated that existing methods for generating as-is floor plans of non-orthogonal rooms by non-expert users do not produce geometrically accurate results. The present paper proposes the adaptation of empirical triangulation methods, traditionally used by architects and other building professionals in surveying building interiors, to the development of semi-automated workflow of room survey. A set of triangulation algorithms that automate the plan drawing stage are presented.Axiomatic theory of betweennesshttps://zbmath.org/1496.030332022-11-17T18:59:28.764376Z"Azimipour, Sanaz"https://zbmath.org/authors/?q=ai:azimipour.sanaz"Naumov, Pavel"https://zbmath.org/authors/?q=ai:naumov.pavel-gThis paper looks at set-betweenness from an axiomatic perspective. As one of several motivating examples presented by the authors, we have the notion of a subset $C$ of a metric space $\langle X,d\rangle$ being \textit{strictly between} two subsets $A$ and $B$-in symbols, $A|C|B$-if for each $a\in A$ and $b\in B$, there is a $c\in C\setminus\{a,b\}$ with $d(a,c)+d(c,b)=d(a,b)$. (This is a direct generalization of the notion of metric point-betweenness introduced by K.~Menger in the 1920s.) The principal focus of the paper is set-betweenness for simple graphs (with a possible loop at a vertex). Here $A|C|B$ holds if for each $a\in A$ and $b\in B$, and each edge path joining $a$ and $b$, there is an internal vertex of the path that belongs to $C$. The authors deliniate a quantifier-free first order language involving the ternary predicate $\cdot |\cdot |\cdot$ and constant symbols, a finite set of axioms in that language, and a straightforward semantics involving finite graphs. They than prove a soundness/completeness result for this system.
Reviewer: Paul Bankston (Milwaukee)The Eckardt point configuration of cubic surfaces revisitedhttps://zbmath.org/1496.050152022-11-17T18:59:28.764376Z"Betten, Anton"https://zbmath.org/authors/?q=ai:betten.anton"Karaoglu, Fatma"https://zbmath.org/authors/?q=ai:karaoglu.fatmaSummary: The classification problem for cubic surfaces with 27 lines is concerned with describing a complete set of the projective equivalence classes of such surfaces. Despite a long history of work, the problem is still open. One approach is to use a coarser equivalence relation based on geometric invariants. The Eckardt point configuration is one such invariant. It can be used as a coarse-grain case distinction in the classification problem. We provide an explicit parametrization of the equations of cubic surfaces with a given Eckardt point configuration over any field. Our hope is that this will be a step towards the bigger goal of classifying all cubic surfaces with 27 lines.Constructing saturating sets in projective spaces using subgeometrieshttps://zbmath.org/1496.050162022-11-17T18:59:28.764376Z"Denaux, Lins"https://zbmath.org/authors/?q=ai:denaux.linsSummary: A \(\varrho\)-saturating set of \(\mathrm{PG}(N,q)\) is a point set \({\mathcal{S}}\) such that any point of \(\mathrm{PG}(N,q)\) lies in a subspace of dimension at most \(\varrho\) spanned by points of \({\mathcal{S}}\). It is generally known that a \(\varrho\)-saturating set of \(\mathrm{PG}(N,q)\) has size at least \(c\cdot \varrho \,q^\frac{N-\varrho}{\varrho +1}\), with \(c>\frac{1}{3}\) a constant. Our main result is the discovery of a \(\varrho\)-saturating set of size roughly \(\frac{(\varrho +1)(\varrho +2)}{2}q^\frac{N-\varrho}{\varrho +1}\) if \(q=(q')^{\varrho +1}\), with \(q'\) an arbitrary prime power. The existence of such a set improves most known upper bounds on the smallest possible size of \(\varrho\)-saturating sets if \(\varrho <\frac{2N-1}{3}\). As saturating sets have a one-to-one correspondence to linear covering codes, this result improves existing upper bounds on the length and covering density of such codes. To prove that this construction is a \(\varrho\)-saturating set, we observe that the affine parts of \(q'\)-subgeometries of \(\mathrm{PG}(N,q)\) having a hyperplane in common, behave as certain lines of \(\mathrm{AG}\big (\varrho +1,(q')^N\big)\). More precisely, these affine lines are the lines of the linear representation of a \(q'\)-subgeometry \(\mathrm{PG}(\varrho ,q')\) embedded in \(\mathrm{PG}\big (\varrho +1,(q')^N\big)\).Combinatorial invariants for nets of conics in \(\mathrm{PG}(2,q)\)https://zbmath.org/1496.050172022-11-17T18:59:28.764376Z"Lavrauw, Michel"https://zbmath.org/authors/?q=ai:lavrauw.michel"Popiel, Tomasz"https://zbmath.org/authors/?q=ai:popiel.tomasz"Sheekey, John"https://zbmath.org/authors/?q=ai:sheekey.johnSummary: The problem of classifying linear systems of conics in projective planes dates back at least to \textit{C. Jordan}, who classified pencils (one-dimensional systems) of conics over \({\mathbb{C}}\) and \(\mathbb{R}\) in [Journ. de Math. (6) 2, 403--438 (1906; JFM 37.0136.01); ibid. (6) 3, 5--51 (1907; JFM 38.0151.01)]. The analogous problem for finite fields \(\mathbb{F}_q\) with \(q\) odd was solved by \textit{L. E. Dickson} [Quart. J. 39, 316--333 (1907; JFM 39.0148.01)]. \textit{A. H. Wilson} [Am. J. Math. 36, 187--210 (1914; JFM 45.0228.01)] attempted to classify nets (two-dimensional systems) of conics over finite fields of odd characteristic, but his classification was incomplete and contained some inaccuracies. In a recent article, we completed Wilson's classification (for \(q\) odd) of nets of rank one, namely those containing a repeated line. The aim of the present paper is to introduce and calculate certain combinatorial invariants of these nets, which we expect will be of use in various applications. Our approach is geometric in the sense that we view a net of rank one as a plane in \(\mathrm{PG}(5,q), q\) odd, that meets the quadric Veronesean in at least one point; two such nets are then equivalent if and only if the corresponding planes belong to the same orbit under the induced action of \(\mathrm{PGL}(3,q)\) viewed as a subgroup of \(\mathrm{PGL}(6,q)\). Since \(q\) is odd, the orbits of lines in \(\mathrm{PG}(5,q)\) under this action correspond to the aforementioned pencils of conics in \(\mathrm{PG}(2,q)\). The main contribution of this paper is to determine the line-orbit distribution of a plane \(\pi\) corresponding to a net of rank one, namely, the number of lines in \(\pi\) belonging to each line orbit. It turns out that this list of invariants completely determines the orbit of \(\pi\), and we will use this fact in forthcoming work to develop an efficient algorithm for calculating the orbit of a given net of rank one. As a more immediate application, we also determine the stabilisers of nets of rank one in \(\mathrm{PGL}(3,q)\), and hence the orbit sizes.On sets of subspaces with two intersection dimensions and a geometrical junta boundhttps://zbmath.org/1496.050182022-11-17T18:59:28.764376Z"Longobardi, Giovanni"https://zbmath.org/authors/?q=ai:longobardi.giovanni"Storme, Leo"https://zbmath.org/authors/?q=ai:storme.leo"Trombetti, Rocco"https://zbmath.org/authors/?q=ai:trombetti.roccoSummary: In this article, constant dimension subspace codes whose codewords have subspace distance in a prescribed set of integers, are considered. The easiest example of such an object is a junta [\textit{I. Dinur} and \textit{E. Friedgut}, Comb. Probab. Comput. 18, No. 1--2, 107--122 (2009; Zbl 1243.05235)]; i.e. a subspace code in which all codewords go through a common subspace. We focus on the case when only two intersection values for the codewords, are assigned. In such a case we determine an upper bound for the dimension of the vector space spanned by the elements of a non-junta code. In addition, if the two intersection values are consecutive, we prove that such a bound is tight, and classify the examples attaining the largest possible dimension as one of four infinite families.The connected graphs obtained from finite projective planeshttps://zbmath.org/1496.050352022-11-17T18:59:28.764376Z"Akpinar, Atilla"https://zbmath.org/authors/?q=ai:akpinar.atillaSummary: In this paper, we give a method of obtaining graphs from finite projective planes, by using an approach based method of taking each line of such a plane as a path graph. All the graphs obtained with the help of this method are connected and some properties of these graphs are determined.Van Lint-MacWilliams' conjecture and maximum cliques in Cayley graphs over finite fieldshttps://zbmath.org/1496.050682022-11-17T18:59:28.764376Z"Asgarli, Shamil"https://zbmath.org/authors/?q=ai:asgarli.shamil"Yip, Chi Hoi"https://zbmath.org/authors/?q=ai:yip.chi-hoiSummary: A well-known conjecture due to \textit{J. H. van Lint} and \textit{F. J. MacWilliams} [IEEE Trans. Inf. Theory 24, 730--737 (1978; Zbl 0395.94025)] states that if \(A\) is a subset of \(\mathbb{F}_{q^2}\) such that \(0, 1 \in A\), \(| A | = q\), and \(a - b\) is a square for each \(a\), \(b \in A\), then \(A\) must be the subfield \(\mathbb{F}_q\). This conjecture is often phrased in terms of the maximum cliques in Paley graphs. It was first proved by \textit{A. Blokhuis} [Indag. Math. 46, 369--372 (1984; Zbl 0561.12009)] and later extended by \textit{P. Sziklai} [Discrete Math. 208--209, 547--555 (1999; Zbl 0945.51004)] to generalized Paley graphs. In this paper, we give a new proof of the conjecture and its variants, and show how this Erdős-Ko-Rado property of Paley graphs extends to a larger family of Cayley graphs, which we call Peisert-type graphs. These results resolve the conjectures by \textit{N. Mullin} [Self-complementary arc-transitive graphs and their imposters. Waterloo, ON: University of Waterloo (Master Thesis) (2009)] and \textit{C. H. Yip} [J. Algebr. Comb. 56, No. 2, 323--333 (2022; Zbl 07572732)].An infinite family of incidence geometries whose incidence graphs are locally \(X^\ast\)https://zbmath.org/1496.051512022-11-17T18:59:28.764376Z"Colin, Natalia Garcia"https://zbmath.org/authors/?q=ai:garcia-colin.natalia"Leemans, Dimitri"https://zbmath.org/authors/?q=ai:leemans.dimitriSummary: We construct a new infinite family of incidence geometries of arbitrarily large rank. These geometries are thick and residually connected and their type-preserving automorphism groups are symmetric groups. We also compute their Buekenhout diagram. The incidence graphs of these geometries are locally \(X\) graphs, but more interestingly, the automorphism groups act transitively, not only on the vertices, but more strongly on the maximal cliques of these graphs.An algebraic approach to Erdős-Ko-Rado sets of flags in spherical buildingshttps://zbmath.org/1496.051822022-11-17T18:59:28.764376Z"De Beule, Jan"https://zbmath.org/authors/?q=ai:de-beule.jan"Mattheus, Sam"https://zbmath.org/authors/?q=ai:mattheus.sam"Metsch, Klaus"https://zbmath.org/authors/?q=ai:metsch.klausSummary: In this paper, oppositeness in spherical buildings is used to define an EKR-problem for flags in projective and polar spaces. A novel application of the theory of buildings and Iwahori-Hecke algebras is developed to prove sharp upper bounds for EKR-sets of flags. In this framework, we can reprove and generalize previous upper bounds for EKR-problems in projective and polar spaces. The bounds are obtained by the application of the Delsarte-Hoffman coclique bound to the opposition graph. The computation of its eigenvalues is due to earlier work by \textit{A. E. Brouwer} [Contemp. Math. 531, 1--10 (2010; Zbl 1232.05122)] and an explicit algorithm is worked out. For the classical geometries, the execution of this algorithm boils down to elementary combinatorics. Connections to building theory, Iwahori-Hecke algebras, classical groups and diagram geometries are briefly discussed. Several open problems are posed throughout and at the end.Delandtsheer-Doyen parameters for block-transitive point-imprimitive 2-designshttps://zbmath.org/1496.051902022-11-17T18:59:28.764376Z"Amarra, Carmen"https://zbmath.org/authors/?q=ai:amarra.carmen"Devillers, Alice"https://zbmath.org/authors/?q=ai:devillers.alice"Praeger, Cheryl E."https://zbmath.org/authors/?q=ai:praeger.cheryl-eSummary: \textit{A. Delandtsheer} and \textit{J. Doyen} [Geom. Dedicata 29, No. 3, 307--310 (1989; Zbl 0673.05010)] bounded, in terms of the block size, the number of points of a point-imprimitive, block-transitive 2-design. To do this they introduced two integer parameters \(m\), \(n\), now called Delandtsheer-Doyen parameters, linking the block size with the parameters of an associated imprimitivity system on points. We show that the Delandtsheer-Doyen parameters provide upper bounds on the permutation ranks of the groups induced on the imprimitivity system and on a class of the system. We explore extreme cases where these bounds are attained, give a new construction for a family of designs achieving these bounds, and pose several open questions concerning the Delandtsheer-Doyen parameters.Tensor trigonometry. Theory and applicationshttps://zbmath.org/1496.510012022-11-17T18:59:28.764376Z"Ninul, Anatoliĭ Sergeevich"https://zbmath.org/authors/?q=ai:ninul.anatolii-sergeevichFor the English edition see [Zbl 1482.51002].Burnside-type problems in discrete geometryhttps://zbmath.org/1496.510022022-11-17T18:59:28.764376Z"Kuz'min, Leonid V."https://zbmath.org/authors/?q=ai:kuzmin.l-vSummary: The paper is concerned with systems of incidence involving a space of points \(X\) and lines consisting of \(q\) points each. A free space \(X\) is defined. For a space \(X\) an analogue of the Burnside problem (solved in the negative) and an analogue of the weakened Burnside problem are formulated. In the case \(q = 3\) the positive answer to the analogue of the weakened Burnside problem is equivalent to the existence of a universal finite geometry.On the sunflower bound for \(k\)-spaces, pairwise intersecting in a pointhttps://zbmath.org/1496.510032022-11-17T18:59:28.764376Z"Blokhuis, A."https://zbmath.org/authors/?q=ai:blokhuis.aart"De Boeck, M."https://zbmath.org/authors/?q=ai:de-boeck.maarten"D'haeseleer, J."https://zbmath.org/authors/?q=ai:dhaeseleer.jozefienSummary: A \(t\)-intersecting constant dimension subspace code \(C\) is a set of \(k\)-dimensional subspaces in a projective space \(\mathrm{PG}(n,q)\), where distinct subspaces intersect in exactly a \(t\)-dimensional subspace. A classical example of such a code is the sunflower, where all subspaces pass through the same \(t\)-space. The sunflower bound states that such a code is a sunflower if \(|C| > \left(\frac{q^{k + 1} - q^{t + 1}}{q - 1} \right)^2 + \left(\frac{q^{k + 1} - q^{t + 1}}{q - 1} \right) + 1\). In this article we will look at the case \(t=0\) and we will improve this bound for \(q\ge 9\): a set \(\mathcal{S}\) of \(k\)-spaces in \(\mathrm{PG}(n,q)\), \(q\ge 9\), pairwise intersecting in a point is a sunflower if \(|\mathcal{S}|> \left(\frac{2}{\sqrt[6]{q}}+\frac{4}{\sqrt[3]{q}}- \frac{5}{\sqrt{q}}\right) \left(\frac{q^{k + 1} - 1}{q - 1}\right)^2\).On Severi varieties as intersections of a minimum number of quadricshttps://zbmath.org/1496.510042022-11-17T18:59:28.764376Z"van Maldeghem, Hendrik"https://zbmath.org/authors/?q=ai:van-maldeghem.hendrik-j"Victoor, Magali"https://zbmath.org/authors/?q=ai:victoor.magaliSummary: Let \(\mathscr{V}\) be a variety related to the second row of the Freudenthal-Tits Magic square in \(N\)-dimensional projective space over an arbitrary field. We show that there exist \(M\leq N\) quadrics intersecting precisely in \(\mathscr{V}\) if and only if there exists a subspace of projective dimension \(N-M\) in the secant variety disjoint from the Severi variety. We present some examples of such subspaces of relatively large dimension. In particular, over the real numbers we show that the Cartan variety (related to the exceptional group \(\mathsf{E_6}(\mathbb{R}))\) is the set-theoretic intersection of 15 quadrics.A solution to two old problems by Menger concerning angle spaceshttps://zbmath.org/1496.510052022-11-17T18:59:28.764376Z"Prieto-Martínez, Luis Felipe"https://zbmath.org/authors/?q=ai:prieto-martinez.luis-felipeSummary: Around 1930, Menger expressed his interest in the concept of abstract angle function. He introduced a general definition of this notion for metric and semi-metric spaces. He also proposed two problems concerning conformal embeddability of spaces endowed with an angle function into Euclidean spaces. These problems received attention in later years but only for some particular cases of metric spaces. In this article, we first update the definition of angle function to apply to the larger class of spaces with a notion of betweenness, which seem to us a more natural framework. In this new general setting, we solve the two problems proposed by Menger.Poncelet triangles: a theory for locus ellipticityhttps://zbmath.org/1496.510062022-11-17T18:59:28.764376Z"Helman, Mark"https://zbmath.org/authors/?q=ai:helman.mark"Laurain, Dominique"https://zbmath.org/authors/?q=ai:laurain.dominique"Garcia, Ronaldo"https://zbmath.org/authors/?q=ai:garcia.ronaldo-a"Reznik, Dan"https://zbmath.org/authors/?q=ai:reznik.dan-sSummary: We present a theory which predicts if the locus of a triangle center over certain Poncelet triangle families is a conic or not. We consider families interscribed in (i) the confocal pair and (ii) an outer ellipse and an inner concentric circular caustic. Previously, determining if a locus was a conic was done on a case-by-case basis. In the confocal case, we also derive conditions under which a locus degenerates to a segment or a circle. We show the locus' turning number is \(\pm 3\), while predicting its monotonicity with respect to the motion of a vertex of the triangle family.Any equilateral tetrahedron is equifacialhttps://zbmath.org/1496.510072022-11-17T18:59:28.764376Z"Lévy-Leblond, Jean-Marc"https://zbmath.org/authors/?q=ai:levy-leblond.jean-marc"Marmorat, Jean-Paul"https://zbmath.org/authors/?q=ai:marmorat.jean-paulA tetrahedron is called \textit{equifacial} if all of its faces are congruent triangles and it is called \textit{equilateral} if all of its faces have the same area. The authors give a proof of the well-known fact that any equilateral tetrahedron is equifacial.
Reviewer: Martin Lukarevski (Skopje)Some new properties of the complete quadrilateral analogous to classical propertieshttps://zbmath.org/1496.510082022-11-17T18:59:28.764376Z"Pasquay, Jean-Nicolas"https://zbmath.org/authors/?q=ai:pasquay.jean-nicolasIn this nice paper, many remarkable results for a complete quadrilateral analogous to classical properties are given. They include the circle of Miquel, Steiner's line and the theorem of Kantor-Harvey. Additionally, the author gives a description of two new circles associated to the complete quadrilateral. All the proofs are based on use of complex numbers.
Reviewer: Martin Lukarevski (Skopje)Group of isometries of Hilbert ball equipped with the Carathéodory metrichttps://zbmath.org/1496.510092022-11-17T18:59:28.764376Z"Mishra, Mukund Madhav"https://zbmath.org/authors/?q=ai:mishra.mukund-madhav"Aggarwal, Rachna"https://zbmath.org/authors/?q=ai:aggarwal.rachnaThis paper is concerned with the study of properties of certain isometries of infinite dimensional hyperbolic spaces.
A bounded domain in a (possibly infinite dimensional) complex Banach space can be given a pseudo-metric known as the \textit{Carathéodory metric}. The Carathéodory metric on the unit disk of the complex plane coincides with the Poincaré metric, and it is hence a model for the hyperbolic plane. If \(B\) denotes the unit ball of an infinite dimensional Hilbert space \(H\), it is then legitimate to think of \(B\) with Carathéodory metric as an infinite dimensional hyperbolic space.
\textit{T. Franzoni} and \textit{E. Vesentini} studied the group \(G\) of bi-holomorphic isometries \(\Aut(B)\) in [Holomorphic maps and invariant distances. Amsterdam - New York - Oxford: North-Holland Publishing Company (1980; Zbl 0447.46040)]. They showed that any isometry in \(G\) can be described in terms of linear isomorphisms of \(H\otimes \mathbb{C}\) that preserves a sesquilinear form.
It is also known [\textit{T. L. Hayden} and \textit{T. J. Suffridge}, Pac. J. Math. 38, 419--422 (1971; Zbl 0229.47043)] that the elements of \(G\) can also be categorised as elliptic, hyperbolic or parabolic in terms of fixed points of their extension to the closed ball \(\overline{B}\).
This is the starting point of the paper under consideration. The main focus of the authors is to understand more explicitly properties of the linear operator \(S\in\mathcal{L}(H\otimes \mathbb{C})\) associated with an isometry in \(G\) and try to give them a geometric meaning.
More in detail:
\begin{itemize}
\item they study when \(S\) is a normal or unitary operator. They show that if \(S\) is normal then the isometry is hyperbolic (Theorem 3).
\item they study when \(S\) is self-adjoint (Proposition 5) or involutory (Proposition 6).
\end{itemize}
They also verify that if \(H=\ell_2(\mathbb{N})\) is the infinite dimensional separable Hilbert space, then both \(G\) and and the set of self-adjoint elements of \(G\) have the cardinality of the continuum.
Reviewer: Federico Vigolo (Münster)Geodesic folding of regular tetrahedronhttps://zbmath.org/1496.510102022-11-17T18:59:28.764376Z"Nishimoto, Seri"https://zbmath.org/authors/?q=ai:nishimoto.seri"Horiyama, Takashi"https://zbmath.org/authors/?q=ai:horiyama.takashi"Tachi, Tomohiro"https://zbmath.org/authors/?q=ai:tachi.tomohiroSummary: We show geometric properties of a family of polyhedra obtained by folding a regular tetrahedron along triangular grids. Each polyhedron is identified by a pair of nonnegative integers. The polyhedron can be cut along a geodesic strip of triangles to be decomposed and unfolded into one or multiple bands. We show that the number of bands is the greatest common divisor of the two integers. By a proper choice of pairs of numbers, a common triangular band that folds into different multiple polyhedra can be created. We construct the configuration of the polyhedron algebraically and numerically through angular and truss models respectively. We discuss the volumes of the obtained folded states and provide relevant open problems regarding the existence of popped-up state. We also show some geometric connections to other art forms.On the Gaussian isoperimetric inequalityhttps://zbmath.org/1496.510112022-11-17T18:59:28.764376Z"Thomas, Erik"https://zbmath.org/authors/?q=ai:thomas.erik-g-fThe author first gives an elementary proof of an isoperimetric inequality for a Gaussian measure \(\gamma_1\) on \(\mathbb R\). Then as an application he proves one inequality of Pisier.
Reviewer: Martin Lukarevski (Skopje)Packings with geodesic and translation balls and their visualizations in \(\widetilde{\mathbf{SL}_2\mathbf{R}}\) spacehttps://zbmath.org/1496.520202022-11-17T18:59:28.764376Z"Molnár, Emil"https://zbmath.org/authors/?q=ai:molnar.emil"Szirmai, Jenő"https://zbmath.org/authors/?q=ai:szirmai.jenoSummary: Remembering on our friendly cooperation between the Geometry Departments of Technical Universities of Budapest and Vienna (also under different names) a nice topic comes into consideration: the ``Gum fibre model'' (see Fig. 1).
One point of view is the so-called kinematic geometry by Vienna colleagues, e.g., as in [\textit{H. Stachel}, Math. Appl., Springer 581, 209--225 (2006; Zbl 1100.52005)], but also in very general context. The other point is the so-called \(\mathbf{H}^2 \times \mathbf{R}\) geometry and \(\widetilde{\mathbf{SL}_2\mathbf{R}}\) geometry where -- roughly -- two hyperbolic planes as circle discs are connected with gum fibres, first: in a simple way, second: in a twisted way. This second homogeneous (Thurston) geometry will be our topic (initiated by Budapest colleagues, and discussed also in international cooperations).
We use for the computation and visualization of \(\widetilde{\mathbf{SL}_2\mathbf{R}}\) its projective model, as in our previous papers. We found seemingly extremal geodesic ball packing for \(\widetilde{\mathbf{SL}_2\mathbf{R}}\) group \(\mathbf{pq}_k \mathbf{o}_\ell\) (\(p = 9\), \(q = 3\), \(k = 1\), \(o = 2\), \(\ell = 1\)) with density
\(\approx 0.787758\) (Table 2). Much better translation ball packing is for group \(\mathbf{pq}_k \mathbf{o}_\ell\) (\(p = 11\), \(q = 3\), \(k = 1\), \(o = 2\), \(\ell = 1\)) with density \(\approx 0.845306\) (Table 3).Sharing pizza in \(n\) dimensionshttps://zbmath.org/1496.520242022-11-17T18:59:28.764376Z"Ehrenborg, Richard"https://zbmath.org/authors/?q=ai:ehrenborg.richard"Morel, Sophie"https://zbmath.org/authors/?q=ai:morel.sophie"Readdy, Margaret"https://zbmath.org/authors/?q=ai:readdy.margaret-aSummary: We introduce and prove the \(n\)-dimensional Pizza Theorem: Let \(\mathcal{H}\) be a hyperplane arrangement in \(\mathbb{R}^n \). If \(K\) is a measurable set of finite volume, the pizza quantity of \(K\) is the alternating sum of the volumes of the regions obtained by intersecting \(K\) with the arrangement \(\mathcal{H} \). We prove that if \(\mathcal{H}\) is a Coxeter arrangement different from \(A_1^n\) such that the group of isometries \(W\) generated by the reflections in the hyperplanes of \(\mathcal{H}\) contains the map \(-\text{id} \), and if \(K\) is a translate of a convex body that is stable under \(W\) and contains the origin, then the pizza quantity of \(K\) is equal to zero. Our main tool is an induction formula for the pizza quantity involving a subarrangement of the restricted arrangement on hyperplanes of \(\mathcal{H}\) that we call the even restricted arrangement. More generally, we prove that for a class of arrangements that we call even (this includes the Coxeter arrangements above) and for a sufficiently symmetric set \(K\), the pizza quantity of \(K+a\) is polynomial in \(a\) for \(a\) small enough, for example if \(K\) is convex and \(0\in K+a\). We get stronger results in the case of balls, more generally, convex bodies bounded by quadratic hypersurfaces. For example, we prove that the pizza quantity of the ball centered at \(a\) having radius \(R\geq \|a\|\) vanishes for a Coxeter arrangement \(\mathcal{H}\) with \(|\mathcal{H}|-n\) an even positive integer. We also prove the Pizza Theorem for the surface volume: When \(\mathcal{H}\) is a Coxeter arrangement and \(|\mathcal{H}| - n\) is a nonnegative even integer, for an \(n\)-dimensional ball the alternating sum of the \((n-1)\)-dimensional surface volumes of the regions is equal to zero.A discrete version of Liouville's theorem on conformal mapshttps://zbmath.org/1496.530152022-11-17T18:59:28.764376Z"Pinkall, Ulrich"https://zbmath.org/authors/?q=ai:pinkall.ulrich"Springborn, Boris"https://zbmath.org/authors/?q=ai:springborn.boris-andreIn the paper under review, a discrete version of Liouville's theorem for simplicial complexes is given and proved using Cauchy's rigidity theorem for convex polyhedra and its higher-dimensional generalization.
Reviewer: Andreea Olteanu (Bucureşti)Projective points over matrices and their separability propertieshttps://zbmath.org/1496.540072022-11-17T18:59:28.764376Z"Agnew, Alfonso F."https://zbmath.org/authors/?q=ai:agnew.alfonso-f"Rathbun, Matt"https://zbmath.org/authors/?q=ai:rathbun.matt"Terry, William"https://zbmath.org/authors/?q=ai:terry.williamThe authors focus their work on topological quotients of real and complex matrices, by various subgroups, and they study their separation properties, as this finds an immediate application to twistor spaces in mathematical physics. As a main conclusion, the authors find that as the group one quotients by gets smaller, the separability properties of the quotient improve. The authors then pose a number of topological questions. For example, the quotient by the general linear group is compact, but the others are not; is there a ground of a further topological study, apart from the separation properties? What about the homotopy or homology properties?
Reviewer: Kyriakos Papadopoulos (Madīnat al-Kuwait)A universal coregular countable second-countable spacehttps://zbmath.org/1496.540192022-11-17T18:59:28.764376Z"Banakh, Taras"https://zbmath.org/authors/?q=ai:banakh.taras-o"Stelmakh, Yaryna"https://zbmath.org/authors/?q=ai:stelmakh.yarynaIn this interesting paper, the authors present a topological characterization of the infinite rational projective space \({\mathbb Q}P^\infty\). It is topologically, the unique countable, second countable space that possesses a superskeleton. Among its properties are the Hausdorff property, it is coregular and has very strong homogeneity properties. Moreover, it is a universal object for the class of all countable, second countable coregular spaces. The proof of the characterization theorem is quite involved and long. As the paper demonstrates, there are many spaces homeomorphic to \({\mathbb Q}P^\infty\) that surface in several seemingly unrelated situations. It is unknown whether the famous Golomb (or Kirch) space contains a subspace homeomorphic to \({\mathbb Q}P^\infty\). This paper is an absolute must for anybody interested in countable connected Hausdorff spaces.
Reviewer: Jan van Mill (Amsterdam)Geometric interpretation of the multi-solution phenomenon in the P3P problemhttps://zbmath.org/1496.683462022-11-17T18:59:28.764376Z"Wang, Bo"https://zbmath.org/authors/?q=ai:wang.bo.2|wang.bo|wang.bo.1"Hu, Hao"https://zbmath.org/authors/?q=ai:hu.hao"Zhang, Caixia"https://zbmath.org/authors/?q=ai:zhang.caixiaSummary: It is well known that the P3P problem could have 1, 2, 3 and at most 4 positive solutions under different configurations among its three control points and the position of the optical center. Since in any real applications, the knowledge on the exact number of possible solutions is a prerequisite for selecting the right one among all the possible solutions, and the study on the phenomenon of multiple solutions in the P3P problem has been an active topic since its very inception. In this work, we provide some new geometric interpretations on the multi-solution phenomenon in the P3P problem, and our main results include: (1) the necessary and sufficient condition for the P3P problem to have a pair of side-sharing solutions is the two optical centers of the solutions both lie on one of the three vertical planes to the base plane of control points; (2) the necessary and sufficient condition for the P3P problem to have a pair of point-sharing solutions is the two optical centers of the solutions both lie on one of the three so-called skewed danger cylinders;(3) if the P3P problem has other solutions in addition to a pair of side-sharing (point-sharing) solutions, these remaining solutions must be a point-sharing (side-sharing ) pair. In a sense, the side-sharing pair and the point-sharing pair are companion pairs; (4) there indeed exist such P3P problems that have four completely distinct solutions, i.e., the solutions sharing neither a side nor a point, closing a long guessing issue in the literature. In sum, our results provide some new insights into the nature of the multi-solution phenomenon in the P3P problem, and in addition to their academic value, they could also be used as some theoretical guidance for practitioners in real applications to avoid occurrence of multiple solutions by properly arranging the control points.Quantum teleportation of unknown seven-qubit entangled state using four-qubit entangled statehttps://zbmath.org/1496.810432022-11-17T18:59:28.764376Z"Zheng, Yundan"https://zbmath.org/authors/?q=ai:zheng.yundan"Li, Dongfen"https://zbmath.org/authors/?q=ai:li.dongfen"Liu, Xiaofang"https://zbmath.org/authors/?q=ai:liu.xiaofang"Liu, Mingzhe"https://zbmath.org/authors/?q=ai:liu.mingzhe"Zhou, Jie"https://zbmath.org/authors/?q=ai:zhou.jie|zhou.jie.1|zhou.jie.3|zhou.jie.4|zhou.jie.2"Yang, Xiaolong"https://zbmath.org/authors/?q=ai:yang.xiaolong"Tan, Yuqiao"https://zbmath.org/authors/?q=ai:tan.yuqiao"Wang, Ruijin"https://zbmath.org/authors/?q=ai:wang.ruijinSummary: Quantum communication is a kind of communication mode which uses quantum physical characteristics to ensure the security of information transmission channel. It is widely concerned because it is different from the traditional cryptographic communication. Among them, quantum teleportation is the main research field and the key technology to achieve secure communication. In this scheme, four-qubit entangled state is used as quantum channel to transmit unknown seven-qubit entangled state. Firstly, Alice will deform the unknown seven-qubit entangled state. After transmission, Alice performs a series of operations on the particles she owns, and performs single-qubit measurement operation, and tells Bob the measurement results through the classical channel. After the measurement, Bob can reconstruct the unknown seven-qubit entangled state by sending the measurement results and the corresponding unitary operation, and with the help of auxiliary particles. We also carried out a security analysis to prove that the scheme is safe, and use the IBM platform for experimental verification, compared with the previous scheme, the scheme is more simple and efficient.
Editorial remark: Due to an oversight during the processing of the new submission of the manuscripts this article was published twice (see also, [Zbl 07542136]).The properties of the polaron in III-V compound semiconductor quantum dots induced by the influence of Rashba spin-orbit interactionhttps://zbmath.org/1496.810642022-11-17T18:59:28.764376Z"Zhang, Wei"https://zbmath.org/authors/?q=ai:zhang.wei.61"Han, Shuang"https://zbmath.org/authors/?q=ai:han.shuang"Ma, Xin-Jun"https://zbmath.org/authors/?q=ai:ma.xinjun"Xianglian"https://zbmath.org/authors/?q=ai:xianglian."Sun, Yong"https://zbmath.org/authors/?q=ai:sun.yong"Xiao, Jing-Lin"https://zbmath.org/authors/?q=ai:xiao.jinglinSummary: We study the ground state energy (GSE) of weak coupling polaron confined in quantum dots (QD) of III-V compound semiconductors using the linear combinatorial operator (LCO) and the Lee-Low-Pines unitary transformation (LLPUT) method. Our calculated results show that the GSE of the polaron splits into two branches due to the Rashba spin-orbit (SO) coupling effect, and spin splitting spacing is influenced by Rashba SO coupling strength and the coupling strength and the effective mass of III-V compound semiconductor material. That reveals the SO coupling properties of weak coupling polaron in the QD of III-V compound semiconductors, which provides a theoretical platform for the fabrication of nanometer devices.The noncommutative space of light-like worldlineshttps://zbmath.org/1496.810692022-11-17T18:59:28.764376Z"Ballesteros, Angel"https://zbmath.org/authors/?q=ai:ballesteros.angel"Gutierrez-Sagredo, Ivan"https://zbmath.org/authors/?q=ai:gutierrez-sagredo.ivan"Herranz, Francisco J."https://zbmath.org/authors/?q=ai:herranz.francisco-joseSummary: The noncommutative space of light-like worldlines that is covariant under the light-like (or null-plane) \(\kappa\)-deformation of the (3+1) Poincaré group is fully constructed as the quantization of the corresponding Poisson homogeneous space of null geodesics. This new noncommutative space of geodesics is five-dimensional, and turns out to be defined as a quadratic algebra that can be mapped to a non-central extension of the direct sum of two Heisenberg-Weyl algebras whose noncommutative parameter is just the Planck scale parameter \(\kappa^{-1}\). Moreover, it is shown that the usual time-like \(\kappa\)-deformation of the Poincaré group does not allow the construction of the Poisson homogeneous space of light-like worldlines. Therefore, the most natural choice in order to model the propagation of massless particles on a quantum Minkowski spacetime seems to be provided by the light-like \(\kappa\)-deformation.Darboux vector in four-dimensional space-timehttps://zbmath.org/1496.830112022-11-17T18:59:28.764376Z"Hu, Na"https://zbmath.org/authors/?q=ai:hu.na"Zhang, Tingting"https://zbmath.org/authors/?q=ai:zhang.tingting"Jiang, Yang"https://zbmath.org/authors/?q=ai:jiang.yang(no abstract)Scotogenic \(A_5 \rightarrow A_4\) Dirac neutrinos with freeze-in dark matterhttps://zbmath.org/1496.830192022-11-17T18:59:28.764376Z"Ma, Ernest"https://zbmath.org/authors/?q=ai:ma.ernestSummary: Radiative Dirac neutrino masses and their mixing are linked to dark matter through the non-Abelian discrete symmetry \(A_5\) of the 4-dimensional pentatope, softly broken to \(A_4\) of the 3-dimensional tetrahedron. This unifying understanding of neutrino family structure from dark matter is made possible through the interplay of gauge symmetry, renormalizable Lagrangian field theory, and softly broken discrete symmetries. Dark neutral fermions are produced through Higgs decay.An effective model for the quantum Schwarzschild black holehttps://zbmath.org/1496.830202022-11-17T18:59:28.764376Z"Alonso-Bardaji, Asier"https://zbmath.org/authors/?q=ai:alonso-bardaji.asier"Brizuela, David"https://zbmath.org/authors/?q=ai:brizuela.david"Vera, Raül"https://zbmath.org/authors/?q=ai:vera.raulSummary: We present an effective theory to describe the quantization of spherically symmetric vacuum motivated by loop quantum gravity. We include anomaly-free holonomy corrections through a canonical transformation and a linear combination of constraints of general relativity, such that the modified constraint algebra closes. The system is then provided with a fully covariant and unambiguous geometric description, independent of the gauge choice on the phase space. The resulting spacetime corresponds to a singularity-free (black-hole/white-hole) interior and two asymptotically flat exterior regions of equal mass. The interior region contains a minimal smooth spacelike surface that replaces the Schwarzschild singularity. We find the global causal structure and the maximal analytical extension. Both Minkowski and Schwarzschild spacetimes are directly recovered as particular limits of the model.Uniformly accelerated Brownian oscillator in (2+1)D: temperature-dependent dissipation and frequency shifthttps://zbmath.org/1496.830352022-11-17T18:59:28.764376Z"Moustos, Dimitris"https://zbmath.org/authors/?q=ai:moustos.dimitrisSummary: We consider an Unruh-DeWitt detector modeled as a harmonic oscillator that is coupled to a massless quantum scalar field in the (2+1)-dimensional Minkowski spacetime. We treat the detector as an open quantum system and employ a quantum Langevin equation to describe its time evolution, with the field, which is characterized by a frequency-independent spectral density, acting as a stochastic force. We investigate a point-like detector moving with constant acceleration through the Minkowski vacuum and an inertial one immersed in a thermal reservoir at the Unruh temperature, exploring the implications of the well-known non-equivalence between the two cases on their dynamics. We find that both the accelerated detector's dissipation rate and the shift of its frequency caused by the coupling to the field bath depend on the acceleration temperature. Interestingly enough this is not only in contrast to the case of inertial motion in a heat bath but also to any analogous quantum Brownian motion model in open systems, where dissipation and frequency shifts are not known to exhibit temperature dependencies. Nonetheless, we show that the fluctuating-dissipation theorem still holds for the detector-field system and in the weak-coupling limit an accelerated detector is driven at late times to a thermal equilibrium state at the Unruh temperature.Tensionless strings and Killing(-Yano) tensorshttps://zbmath.org/1496.830422022-11-17T18:59:28.764376Z"Lindström, Ulf"https://zbmath.org/authors/?q=ai:lindstrom.ulf"Sarıoğlu, Özgür"https://zbmath.org/authors/?q=ai:sarioglu.ozgurSummary: We construct invariants for bosonic and spinning tensionless (null) strings in backgrounds that carry Killing tensors or Killing-Yano tensors of mixed type. This is facilitated by the close relation of these strings to point particles. We apply the construction to the Minkowski and to the Kerr-Newman backgrounds.The intersection of two petals: a computer-assisted extension of another old geometric problemhttps://zbmath.org/1496.970012022-11-17T18:59:28.764376Z"Hoseana, Jonathan"https://zbmath.org/authors/?q=ai:hoseana.jonathan(no abstract)Angle-side properties of polygons inscribable in an ellipsehttps://zbmath.org/1496.970022022-11-17T18:59:28.764376Z"Jahangiri, Jay"https://zbmath.org/authors/?q=ai:jahangiri.jay-m"Segal, Ruti"https://zbmath.org/authors/?q=ai:segal.ruti"Stupel, Moshe"https://zbmath.org/authors/?q=ai:stupel.moshe(no abstract)Some inequalities in a triangle in which the length of one side and the inradius are givenhttps://zbmath.org/1496.970042022-11-17T18:59:28.764376Z"Oxman, Victor"https://zbmath.org/authors/?q=ai:oxman.victor(no abstract)Investigation on inscribed circles: one, two, three, four, infinitely manyhttps://zbmath.org/1496.970052022-11-17T18:59:28.764376Z"Oxman, Victor"https://zbmath.org/authors/?q=ai:oxman.victor"Stupel, Moshe"https://zbmath.org/authors/?q=ai:stupel.moshe(no abstract)An old problem with a new twisthttps://zbmath.org/1496.970062022-11-17T18:59:28.764376Z"Wares, Arsalan"https://zbmath.org/authors/?q=ai:wares.arsalan"Reid, Denise Taunton"https://zbmath.org/authors/?q=ai:reid.denise-taunton(no abstract)