Recent zbMATH articles in MSC 51https://www.zbmath.org/atom/cc/512022-05-16T20:40:13.078697ZWerkzeugTHE BOOK. Book review of: M. Aigner and G. M. Ziegler, Proofs from THE BOOK. 6th ed.https://www.zbmath.org/1483.000252022-05-16T20:40:13.078697Z"Ullman, Daniel H."https://www.zbmath.org/authors/?q=ai:ullman.daniel-hReview of [Zbl 1392.00001].Optical illusions in Rome. A mathematical travel guide. Translated from the Danish by Viktor Blåsjöhttps://www.zbmath.org/1483.000302022-05-16T20:40:13.078697Z"Andersen, Kirsti"https://www.zbmath.org/authors/?q=ai:andersen.kirstiThis book of perspective and optical illusions, originally published in Danish and recently translated into English by Viktor Bläsjö, is a real gem for those who appreciate mathematics. It is also very useful for a better understanding of the techniques of perspective and trompe l'oeil in our trip to Rome. The book consists of 79 pages and 74 figures and is divided into one introduction and six chapters, plus an appendix in which the author makes some notes for the travellers. In the introduction, the author says that she hopes that the book will be an inspiration for travellers to Rome who are interested in mathematics and in the history of mathematics and art history. She deals with two types of optical illusions, the first called trompe l'oeil, in the plane and in space, and the second called anamorphoses.
The first chapter, entitled ``Trompe l'oeil on walls'', deals with works by Baldassare Peruzzi (1481--1536) and analyses the ``Sala delle prospettive'' in the Villa Farnesina. She explains the construction, and the trompe l'oeil wall paintings, through 14 seminal images. She ends this first chapter with the analysis of the decor of the Teatro Olimpico, which opened in 1585 and was designed by Andrea Palladio (1508--1580). The second chapter, entitled ``Three-dimensional trompe l'oeil'', focusses on works by Francesco Borromini (1599--1667) and Giovanni Lorenzo Bernini (1598--1680). From Borromini she analyses the convergence of colonnades in the Palazzo Spada (Rome), and from Bernini she explains the mathematical construction of the oval in St. Peter's Square. Bernini began his project around 1656 and finished it 10 years later. In addition, from Bernini she analyses the construction of the royal staircase (called scala regia) and the geometrical effect of trompe l'oeil in St. Peter's Basilica. The third chapter, entitled ``The anamorphosis in Trinità dei Monti'', deals with other optical effects through Niceron's work and that of his disciple Maignan in the Trinity of the Mounts church and the Minim Monastery, in Rome. The fourth chapter, entitled ``Ceilings as image surfaces'', deals with the Jesuits, and specifically with the works by Andrea Pozzo (1642--1709) in the St. Ignazio Church in Rome. The fifth chapter, entitled ``Some results from perspective theory'', addresses the principles of perspective explained by means of geometrical designs. The author remarks that this chapter can be read first or only by those who wish to learn about the geometrical foundations of perspective. The last chapter contains interesting exercises for students on these optical illusions.
The reading of this book provides a source of ideas for research in other constructions or paintings in other buildings and places. In addition, it inspires readers to travel to Rome in order to see for themselves all the fantastic optical illusions analysed in this seminal book.
Reviewer: Maria Rosa Massa Esteve (Barcelona)Rolling acrobatic apparatushttps://www.zbmath.org/1483.000322022-05-16T20:40:13.078697Z"Segerman, Henry"https://www.zbmath.org/authors/?q=ai:segerman.henry(no abstract)Columella's formulahttps://www.zbmath.org/1483.010052022-05-16T20:40:13.078697Z"Lévy-Leblond, Jean-Marc"https://www.zbmath.org/authors/?q=ai:levy-leblond.jean-marc(no abstract)A Pascal's theorem for rational normal curveshttps://www.zbmath.org/1483.140042022-05-16T20:40:13.078697Z"Caminata, Alessio"https://www.zbmath.org/authors/?q=ai:caminata.alessio"Schaffler, Luca"https://www.zbmath.org/authors/?q=ai:schaffler.lucaPascal's Theorem provides a geometric criterion in terms of span, intersection, and collinearity for six points to lie on a conic section. This article provides conditions of similar flavor for \(d+4\) points in projective space \(\mathbb{P}^d\) to lie on a rational normal curve of degree~\(d\).
In the previous work [\textit{A. Caminata} et al., Adv. Math. 340, 653--683 (2018; Zbl 1408.14006)] multihomogeneous equations for the Zariski closure of the variety of \((d+4)\)-tuples of points on rational normal curves were derived. In the present article, these are lifted to equations in Grassmann-Cayley algebra which allow for direct geometric interpretations. This lift is an instance of the generally rather difficult ``Cayley-factorization.''
Numerous equivalent formulations in Grassmann-Cayley algebra give rise to equally many geometric characterizations for \(d+4\) points on a rational normal curve of degree \(d\). In case of dimension \(d = 3\), one of them reads as follows: The seven points \(P_1\), \(P_2\), \(\dots\), \(P_7\) lie on a twisted cubic if and only if the four points
\[
P_1P_2 \cap P_4P_5P_7,\quad P_2P_3 \cap P_5P_6P_7,\quad P_3P_4 \cap P_6P_1P_7,\quad P_7
\]
are coplanar.
Reviewer: Hans-Peter Schröcker (Innsbruck)Some properties of rational conic fibrationshttps://www.zbmath.org/1483.140122022-05-16T20:40:13.078697Z"Lanteri, Antonio"https://www.zbmath.org/authors/?q=ai:lanteri.antonio"Mallavibarrena, Raquel"https://www.zbmath.org/authors/?q=ai:mallavibarrena.raquelThe authors study rational conic fibrations (RCFs) of sectional genus \(g=3\). In order to conduct their investigations, the authors study the interplay between rational conic fibrations and their enveloping bundles. Slightly more precisely, if \(X \subset \mathbb{P}^{N}\) is a rational conic fibration of sectional genus \(g\), then the enveloping projective bundle \(P\) is the rational \(\mathbb{P}^{2}\)-bundle giving rise to the ruled variety in \(\mathbb{P}^{N}\) swept out by the planes spanned by conics which are the fibres of \(X\). In Theorem 4 therein the authors provide possibilities for a RCF \((X, \mathcal{L})\) with \(\mathcal{L}\) being the hyperplane bundle of \(X\) of sectional genus \(g=3\) and its enveloping bundle. Moreover, the authors provide a complete description of rational conic fibrations containing a line or a conic, or a smooth twisted cubic transverse to the fibres.
Reviewer: Piotr Pokora (Kraków)A rarity in geometry: a septic curvehttps://www.zbmath.org/1483.140552022-05-16T20:40:13.078697Z"Odehnal, Boris"https://www.zbmath.org/authors/?q=ai:odehnal.borisMotivated by the rarity of curves of degree \(7\) that arise as geometric loci of real interest, the author studies various properties of the locus \(C\) of all points in the plane whose pedal points on the six sides of a complete quadrangle are located on some conic. It so happens that, in the generic case, \(C\) is an algebraic curve of degree \(7\) and genus \(5\), rather than one of degree 12 as could have been expected. The questions of interest are: singularities, focal points, the points on \(C\) with degenerate pedal conics, and the cases in which \(C\) degenerates or is of lower degree, depending on the shape of the initial quadrilateral.
Reviewer: Victor V. Pambuccian (Glendale)A lower bound for \(\chi (\mathcal{O}_S)\)https://www.zbmath.org/1483.140622022-05-16T20:40:13.078697Z"Di Gennaro, Vincenzo"https://www.zbmath.org/authors/?q=ai:di-gennaro.vincenzoSummary: Let \((S,\mathcal{L})\) be a smooth, irreducible, projective, complex surface, polarized by a very ample line bundle \(\mathcal{L}\) of degree \(d > 25\). In this paper we prove that \(\chi (\mathcal{O}_S)\geq -\frac{1}{8}d(d-6)\). The bound is sharp, and \(\chi (\mathcal{O}_S)=-\frac{1}{8}d(d-6)\) if and only if \(d\) is even, the linear system \(|H^0 (S,\mathcal{L})|\) embeds \(S\) in a smooth rational normal scroll \(T\subset\mathbb{P}^5\) of dimension 3, and here, as a divisor, \(S\) is linearly equivalent to \(\frac{d}{2}Q\), where \(Q\) is a quadric on \(T\). Moreover, this is equivalent to the fact that a general hyperplane section \(H\in |H^0 (S,\mathcal{L})|\) of \(S\) is the projection of a curve \(C\) contained in the Veronese surface \(V\subseteq\mathbb{P}^5\), from a point \(x\in V\backslash C\).Recognizing Cartesian products of matrices and polytopeshttps://www.zbmath.org/1483.150232022-05-16T20:40:13.078697Z"Aprile, Manuel"https://www.zbmath.org/authors/?q=ai:aprile.manuel"Conforti, Michele"https://www.zbmath.org/authors/?q=ai:conforti.michele"Faenza, Yuri"https://www.zbmath.org/authors/?q=ai:faenza.yuri"Fiorini, Samuel"https://www.zbmath.org/authors/?q=ai:fiorini.samuel"Huynh, Tony"https://www.zbmath.org/authors/?q=ai:huynh.tony"Macchia, Marco"https://www.zbmath.org/authors/?q=ai:macchia.marcoSummary: The 1-\textit{product} of matrices \(S_1 \in \mathbb{R}^{m_1 \times n_1}\) and \(S_2 \in \mathbb{R}^{m_2 \times n_2}\) is the matrix in \(\mathbb{R}^{(m_1+m_2) \times (n_1n_2)}\) whose columns are the concatenation of each column of \(S_1\) with each column of \(S_2\). Our main result is a polynomial time algorithm for the following problem: given a matrix \(S\), is \(S\) a 1-product, up to permutation of rows and columns? Our main motivation is a close link between the 1-product of matrices and the Cartesian product of polytopes, which relies on the concept of slack matrix. Determining whether a given matrix is a slack matrix is an intriguing problem whose complexity is unknown, and our algorithm reduces the problem to irreducible instances. Our algorithm is based on minimizing a symmetric submodular function that expresses mutual information in information theory. We also give a polynomial time algorithm to recognize a more complicated matrix product, called the 2-\textit{product}. Finally, as a corollary of our 1-product and 2-product recognition algorithms, we obtain a polynomial time algorithm to recognize slack matrices of 2-level matroid base polytopes.
For the entire collection see [Zbl 1465.05002].Intersection pairings for higher laminationshttps://www.zbmath.org/1483.300862022-05-16T20:40:13.078697Z"Le, Ian"https://www.zbmath.org/authors/?q=ai:le.ianSummary: One can realize higher laminations as positive configurations of points in the affine building [the author, Geom. Topol. 20, No. 3, 1673--1735 (2016; Zbl 1348.30023)]. The duality pairings of \textit{V. Fock} and \textit{A. Goncharov} [Publ. Math., Inst. Hautes Étud. Sci. 103, 1--211 (2006; Zbl 1099.14025)] give pairings between higher laminations for two Langlands dual groups \(G\) and \(G^{\vee}\). These pairings are a generalization of the intersection pairing between measured laminations on a topological surface.
We give a geometric interpretation of these intersection pairings in a wide variety of cases. In particular, we show that they can be computed as the minimal weighted length of a network in the building. Thus we relate the intersection pairings to the metric structure of the affine building. This proves several of the conjectures from [the author and \textit{E. O'Dorney}, Doc. Math. 22, 1519--1538 (2017; Zbl 1383.51009)]. We also suggest the next steps toward giving geometric interpretations of intersection pairings in general.
The key tools are linearized versions of well-known classical results from combinatorics, like Hall's marriage lemma, König's theorem, and the Kuhn-Munkres algorithm, which are interesting in themselves.Frames over finite fields: basic theory and equiangular lines in unitary geometryhttps://www.zbmath.org/1483.420192022-05-16T20:40:13.078697Z"Greaves, Gary R. W."https://www.zbmath.org/authors/?q=ai:greaves.gary"Iverson, Joseph W."https://www.zbmath.org/authors/?q=ai:iverson.joseph-w"Jasper, John"https://www.zbmath.org/authors/?q=ai:jasper.john"Mixon, Dustin G."https://www.zbmath.org/authors/?q=ai:mixon.dustin-gSummary: We introduce the study of frames and equiangular lines in classical geometries over finite fields. After developing the basic theory, we give several examples and demonstrate finite field analogs of equiangular tight frames (ETFs) produced by modular difference sets, and by translation and modulation operators. Using the latter, we prove that Gerzon's bound is attained in each unitary geometry of dimension \(d=2^{2l+1}\) over the field \(\mathbb{F}_{3^2}\). We also investigate interactions between complex ETFs and those in finite unitary geometries, and we show that every complex ETF implies the existence of ETFs with the same size over infinitely many finite fields.Paley-Wiener type theorem for generalized spherical transform on the hyperbolic planehttps://www.zbmath.org/1483.440042022-05-16T20:40:13.078697Z"Vasylianska, V. S."https://www.zbmath.org/authors/?q=ai:vasylianska.v-s"Volchkov, Vit. V."https://www.zbmath.org/authors/?q=ai:volchkov.vitalii-vladimirovichSummary: Analogues of the spherical transform on the hyperbolic plane are studied. The inversion formula and Paley-Wiener type theorem for the specified transform are obtained.Homeomorphisms with the transmutation property with respect to weighted convolutionhttps://www.zbmath.org/1483.440052022-05-16T20:40:13.078697Z"Volchkov, V. V."https://www.zbmath.org/authors/?q=ai:volchkov.valerii-vladimirovich"Volchkov, Vit. V."https://www.zbmath.org/authors/?q=ai:volchkov.vitalii-vladimirovichSummary: A generalized Laplacian \(\mathfrak{L}\) on hyperbolic plane invariant over weighted shifts is studied. For integral transform induced by eigenfunctions of \(\mathbb{H}^2\) we obtain the analogues of the inversion formula and the Paley-Wiener theorem. Transmutation mappings for a weighted convolution on a hyperbolic plane induced by weighted shifts are constructed.The group action on the finite projective planes of orders 53, 61, 64https://www.zbmath.org/1483.510012022-05-16T20:40:13.078697Z"Makhrib Al-Seraji, Najm Abdulzahra"https://www.zbmath.org/authors/?q=ai:makhrib-al-seraji.najm-abdulzahra"Alnussairy, Esam A."https://www.zbmath.org/authors/?q=ai:alnussairy.esam-a"Jafar, Zainab Sadiq"https://www.zbmath.org/authors/?q=ai:jafar.zainab-sadiqLet PG\((2,q)\) be the projective plane over the finite field of \(q\) elements. It is well-known that there exists a cyclic group \(G\) of automorphisms of PG\((2,q)\) acting regularly on points as well as on lines; sometimes called Singer group. The authors use GAP to compute orbits of subgroups of \(G\) in the cases \(q\in\{53, 61, 64\}\).
Reviewer: Hubert Kiechle (Hamburg)Characterising elliptic and hyperbolic hyperplanes of the parabolic quadric \(\mathcal{Q}(2n, q)\)https://www.zbmath.org/1483.510022022-05-16T20:40:13.078697Z"Schillewaert, Jeroen"https://www.zbmath.org/authors/?q=ai:schillewaert.jeroen"Van de Voorde, Geertrui"https://www.zbmath.org/authors/?q=ai:van-de-voorde.geertruiThe authors provide combinatorial characterizations of the sets of elliptic or hyperbolic hyperplanes on a parabolic quadric \(Q(2n,q)\) in the finite projective space \(\mathrm{PG}(2n,q)\), where \(n\ge 2\) and \(q>2\) is a power of \(2\), or \(n=2=q\). The combinatorial conditions restrict the number of hyperplanes through a point and through a subspace of codimension \(2\). The main theorem generalizes results of \textit{S. G. Barwick} et al. [Discrete Math. 343, No. 6, Article ID 111857, 8 p. (2020; Zbl 1447.51006); Des. Codes Cryptography 88, No. 1, 33--39 (2020; Zbl 1430.51010)].
Reviewer: Theo Grundhöfer (Würzburg)The type of a point and a characterization of the set of external points of a conic in \(PG (2, q)\), \(q\) oddhttps://www.zbmath.org/1483.510032022-05-16T20:40:13.078697Z"Tondini, Daniela"https://www.zbmath.org/authors/?q=ai:tondini.danielaUsing the definition of the type of a point, the author characterizes the set of external points of a conic in \(\mathrm{PG}(2,q)\) with \(q\) odd within a class of three-character sets. Specifically, if \(K\) is a \(\frac{q(q+1)}{2}\) set of \(\mathrm{PG}(2,q)\) with \(q\) odd having at most \(3\) different intersection numbers \(m\), \(n\) and \(q\) with \(m<n<q+1\), then \(K\) is of type \((n,m,q)\) with \(m \leq \frac{q-1}{2} < \frac{q+1}{2} \leq n <q.\) If \(t_q\leq q\) and \(K\) has exactly one type of inner points, then \(K\) is the point set of \(\frac{q+1}{2}\) concurrent lines except for the point of intersection. Finally, \(t_q=q+1\) if and only if \(K\) is the set of external points of a non-degenerate conic giving \(K\) exactly one type of inner points.
Reviewer: Steven T. Dougherty (Scranton)Gromov-Hausdorff distance between interval and circlehttps://www.zbmath.org/1483.510042022-05-16T20:40:13.078697Z"Ji, Yibo"https://www.zbmath.org/authors/?q=ai:ji.yibo"Tuzhilin, Alexey A."https://www.zbmath.org/authors/?q=ai:tuzhilin.alexey-aThe authors introduce the new notions of round metric spaces and nonlinearity degree of a metric space.
A metric space \((X, d)\) is called round if, for every \(b \in (0, \operatorname{diam} X)\) and each \(x \in X\), there exists \(y \in X\) such that \(d(x, y) \geqslant b\). The nonlinearity degree \(c(X)\) of \((X, d)\) is defined as
\[
c(X) := \inf_{f \in \operatorname{Lip}_1(X)} \sup \{d(x, y) - |f(x) - f(y)| : x, y \in X\},
\]
where \(\operatorname{Lip}_1(X)\) is the set of all function \(f \colon X \to \mathbb{R}\) which satisfy the inequality \(|f(x) - f(y)| \leqslant d(x, y)\) for all \(x\), \(y \in X\).
Using these concepts, the authors develop an original technique enabled to obtain the exact values of the Gromov-Hausdorff distance \(d_{GH}(I_{\lambda}, S^1)\) between the interval \(I_{\lambda} = [0, \lambda] \subseteq \mathbb{R}\) and the one-dimensional sphere \(S^1 = \{z \in \mathbb{C} : |z| = 1\}\) each of which is equipped with the standard Euclidean metric,
\[
d_{GH}(I_{\lambda}, S^1) = \begin{cases} \frac{\pi}{2} - \frac{\lambda}{4} & \text{if } 0 \leqslant \lambda < \frac{2}{3} \pi,\\
\frac{\pi}{3} & \text{if } \frac{2}{3} \pi \leqslant \lambda \leqslant \frac{5}{3} \pi,\\
\frac{\lambda - \pi}{2} & \text{otherwise}. \end{cases}
\]
The last formula is one of the few exact values of the Gromov-Hausdorff distance between given metric spaces.
Reviewer: Aleksey A. Dovgoshey (Slovyansk)A theorem on equiareal triangles with a fixed basehttps://www.zbmath.org/1483.510052022-05-16T20:40:13.078697Z"Pambuccian, Victor"https://www.zbmath.org/authors/?q=ai:pambuccian.victorInspired by a theorem from spherical geometry, \textit{A. Papadopoulos} and \textit{W. Su} [Adv. Stud. Pure Math. 73, 225--253 (2017; Zbl 1446.51005)] proved the following result in the hyperbolic plane: if we fix two points \(A, B\), and consider all those points \(C\) so that the triangle \(ABC\) has the same (fixed) area and lies on a given side of the geodesic through \(A,B\), then the midpoints of the segments \(BA, BC\) all lie on a line.
The purpose of this article is to prove (an equivalent reformulation of) this result in a purely axiomatic way, in the setting of non-elliptic metric planes in which every segment has a midpoint.
For the entire collection see [Zbl 1412.51001].
Reviewer: Sebastian Hensel (München)Mollweide's formula and circumcevians of the incenterhttps://www.zbmath.org/1483.510062022-05-16T20:40:13.078697Z"Lukarevski, Martin"https://www.zbmath.org/authors/?q=ai:lukarevski.martinIn this paper, the author, using Mollwaide's formula, gives some inequalities for the sum of the circumcevians of the incenter in terms of the circumradius and inradius of the triangle.
Reviewer: Cătălin Barbu (Bačau)Some generalizations of the shadow problem in the Lobachevsky spacehttps://www.zbmath.org/1483.520082022-05-16T20:40:13.078697Z"Kostin, A. V."https://www.zbmath.org/authors/?q=ai:kostin.andrey-viktorovichA set \(U\) in the \(n\)-dimensional Euclidean space \(E^n\) is called \(m\)-convex with respect to a point \(P \notin E^n \setminus U\) for some \(1 \leq m < n\) if there is no \(m\)-dimensional subspace containing \(P\) and disjoint from \(U\), and \(U\) is called \(m\)-semiconvex with respect to \(P\) if there is no \(m\)-dimensional half subspace containing \(P\) and disjoint from \(U\). The shadow problem asks for the minimum number \(N\) of open/closed balls mutually disjoint and disjoint from a point \(P\) such that their union is \(1\)-convex with respect to \(P\) and their centers lie on the same sphere centered at \(P\).
The aim of the author is to present solutions for variants of this problem in the hyperbolic space \(H^n\). After defining \(m\)-convexity and \(m\)-semiconvexity analogously to the Euclidean definition, he examines variants of the shadow problem in the \(2\)- and \(3\)-dimensional hyperbolic space. More specifically, he proves that the minimum number of open (closed) disjoint disks in the hyperbolic plane whose centers are at the same distance from some point \(P\), are disjoint from \(P\) and their union is \(1\)-convex with respect to \(P\), is \(2\), and the same holds if we replace disks by horodisks. Furthermore, for both open (closed) disks or horodisks this number is \(3\) if we replace \(1\)-convexity by \(1\)-semiconvexity.
For points in the \(3\)-dimensional hyperbolic space the author remarks that analogously to the Euclidean problem it can be shown that the minimum number of pairwise disjoint open (closed) balls in \(H^3\) whose centers are on a sphere centered at a point \(P\), and are disjoint from \(P\) and their union is \(1\)-convex with respect to \(P\), is \(4\). Furthermore, he proves that if we drop the condition that the centers of the balls lie on a sphere centered at \(P\), or assume that they lie on an ellipsoid centered at \(P\), this number is \(3\). He also proves results about the case when some or all of the balls are replaced by horoballs.
Reviewer: Zsolt Lángi (Budapest)Quasicircle boundaries and exotic almost-isometrieshttps://www.zbmath.org/1483.530652022-05-16T20:40:13.078697Z"Lafont, Jean-François"https://www.zbmath.org/authors/?q=ai:lafont.jean-francois"Schmidt, Benjamin"https://www.zbmath.org/authors/?q=ai:schmidt.benjamin"van Limbeek, Wouter"https://www.zbmath.org/authors/?q=ai:van-limbeek.wouterSummary: We show that the limit set of an isometric and convex cocompact action of a surface group on a proper geodesic hyperbolic metric space, when equipped with a visual metric, is a Falconer-Marsh (weak) quasicircle. As a consequence, the Hausdorff dimension of such a limit set determines its bi-Lipschitz class. We give applications, including the existence of almost-isometries between periodic negatively curved metrics on \(\mathbb{H}^2\) that cannot be realized equivariantly.Quaternionic contact \(4n+3\)-manifolds and their \(4n\)-quotientshttps://www.zbmath.org/1483.530702022-05-16T20:40:13.078697Z"Kamishima, Yoshinobu"https://www.zbmath.org/authors/?q=ai:kamishima.yoshinobuThe author studies quaternionic contact structure (``qc-structure'' for short) on \((4n+3)\)-manifolds to construct quaternionic Hermitian \(4n\)-manifolds as their quotient.
The qc-structure was first introduced by \textit{O. Biquard} [Métriques d'Einstein asymptotiquement symétriques. Paris: Société Mathématique de France (2000; Zbl 0967.53030)], together with a canonical connection (now known as ``Biquard connection''). A qc-structure on a \((4n+3)\)-manifold \(X\) is a distribution \(D\subset TX\) of codimension 3, such that \(D\) admits a \(n\)-dimensional quaternionic structure \(Q\).
In a previous work of the author with \textit{D. Alekseevsky} [Ann. Mat. Pura Appl. (4) 187, No. 3, 487--529 (2008; Zbl 1223.53054)], a special case of qc-structures, called ``quaternionic CR-structure'' was studied, which corresponds to qc-Einstein manifolds with nonzero qc-scalar curvatures. In this paper, the author studies the complementing vanishing qc-scalar curvature case, which has an equivalent description as ``strict qc-structure''.
In the first part of the paper, the author construct a family of simply connected strict qc solvable Lie groups \(\mathcal{M}_{k,l}\), starting with the standard quaternionic Heisenberg nilpotent Lie group \(\mathcal{M}\). These are then characterized as the only contractible unimodular strict qc-groups. Then, both compact and non-compact uniformizable strict qc-manifolds are classified modeling on \(\mathcal{M}\).
In the second part, the author uses the quotients of a conformal deformation of the standard qc-structure on \(\mathcal{M}\) to construct a family of quaternionic Hermitian metrics on a domain of the standard quaternion space, one of which is a Bochner-flat Kähler metric.
Reviewer: Yalong Shi (Nanjing)Configuration spaces of squares in a rectanglehttps://www.zbmath.org/1483.570312022-05-16T20:40:13.078697Z"Plachta, Leonid"https://www.zbmath.org/authors/?q=ai:plachta.leonid-pThe author studies the configuration space \(F_k(Q;r)\) of \(k\) squares of size \(r\) in a rectangle \(Q\), by using the tautological function \(\theta\) defined on the affine polytope complex \(Q^k\). The critical points of the function \(\theta\) are described in geometric and combinatorial terms. It is also proved that under certain conditions, the space \(F_k(Q;r)\) is connected. The paper is organized into three sections dealing with the following aspects: Introduction, affine Morse-Bott functions on affine polytope complexes, configuration spaces of squares in a rectangle (first surgery of \(F_k(Q;r)\), connectivity of \(F_n(Q;r)\)).
Reviewer: Dorin Andrica (Riyadh)Analytical modelling of perforated geometrical domains by the R-functionshttps://www.zbmath.org/1483.650312022-05-16T20:40:13.078697Z"Semerich, Yuriy"https://www.zbmath.org/authors/?q=ai:semerich.yuriySummary: This paper deals with the construction of boundary equations for geometric domains with perforation. Different types of perforated geometric domains are considered. The R-functions method for analytical modelling of perforated geometrical domains is used. For all constructed equations, function plots are obtained.Hyperbolic node embedding for temporal networkshttps://www.zbmath.org/1483.682632022-05-16T20:40:13.078697Z"Wang, Lili"https://www.zbmath.org/authors/?q=ai:wang.lili.4|wang.lili.3|wang.lili.6|wang.lili|wang.lili.7|wang.lili.2|wang.lili.8|wang.lili.5|wang.lili.1"Huang, Chenghan"https://www.zbmath.org/authors/?q=ai:huang.chenghan"Ma, Weicheng"https://www.zbmath.org/authors/?q=ai:ma.weicheng"Liu, Ruibo"https://www.zbmath.org/authors/?q=ai:liu.ruibo"Vosoughi, Soroush"https://www.zbmath.org/authors/?q=ai:vosoughi.soroushSummary: Generating general-purpose vector representations of networks allows us to analyze them without the need for extensive feature-engineering. Recent works have shown that the hyperbolic space can naturally represent the structure of networks, and that embedding networks into hyperbolic space is extremely efficient, especially in low dimensions. However, the existing hyperbolic embedding methods apply to static networks and cannot capture the dynamic evolution of the nodes and edges of a temporal network. In this paper, we present an unsupervised framework that uses temporal random walks to obtain training samples with both temporal and structural information to learn hyperbolic embeddings from continuous-time dynamic networks. We also show how the framework extends to attributed and heterogeneous information networks. Through experiments on five publicly available real-world temporal datasets, we show the efficacy of our model in embedding temporal networks in low-dimensional hyperbolic space compared to several other unsupervised baselines. We show that our model obtains state-of-the-art performance in low dimensions, outperforming all baselines, and has competitive performance in higher dimensions, outperforming the baselines in three of the five datasets. Our results show that embedding temporal networks in hyperbolic space is extremely effective when necessitating low dimensions.Pappus's hexagon theorem in real projective planehttps://www.zbmath.org/1483.684872022-05-16T20:40:13.078697Z"Coghetto, Roland"https://www.zbmath.org/authors/?q=ai:coghetto.rolandSummary: In this article we prove, using Mizar [\textit{G. Bancerek} et al., Lect. Notes Comput. Sci. 9150, 261--279 (2015; Zbl 1417.68201); J. Autom. Reasoning 61, No. 1--4, 9--32 (2018; Zbl 1433.68530)], the Pappus's hexagon theorem in the real projective plane: ``Given one set of collinear points \(A, B, C\), and another set of collinear points \(a, b, c\), then the intersection points \(X, Y, Z\) of line pairs \(Ab\) and \(aB, Ac\) and \(aC, Bc\) and \(bC\) are collinear''.
More precisely, we prove that the structure \texttt{ProjectiveSpace TOP-REAL3} [\textit{W. Leonczuk} and \textit{K. Prazmowski}, ``A construction of analytical projective space'', Formaliz. Math. 1, No. 4, 761--766 (1990)] (where \texttt{TOP-REAL3} is a metric space defined in [\textit{A. Darmochwał}, ``The Euclidean space'', ibid. 2, No. 4, 599--603 (1991)]) satisfies the Pappus's axiom defined in [``Projective spaces. I'', ibid. 1, No. 4, 767--776 (1990)] by \textit{W. Leończuk} and \textit{K. Prażmowski}. \textit{E. Kusak} and \textit{W. Leończuk} formalized the Hessenberg theorem early in the MML [``Hessenberg theorem'', ibid. 2, No. 2, 217--219 (1991)]. With this result, the real projective plane is Desarguesian.
For proving the Pappus's theorem, two different proofs are given. First, we use the techniques developed in the section ``Projective proofs of Pappus's theorem'' in the first chapter of [\textit{J. Richter-Gebert}, Perspectives on projective geometry. A guided tour through real and complex geometry. Berlin: Springer (2011; Zbl 1214.51001)]. Secondly, Pascal's theorem [\textit{R. Coghetto}, Formaliz. Math. 25, No. 2, 107--119 (2017; Zbl 1377.51003)] is used.
In both cases, to prove some lemmas, we use \texttt{Prover9}, the successor of the \texttt{Otter} prover and \texttt{ott2miz} by Josef Urban [\textit{P. Rudnicki} and \textit{J. Urban}, ``Escape to ATP for Mizar'', in: Proceedings of the first international workshop on proof eXchange for theorem proving, PxTP 2011. 46--59 (2011); \textit{A. Grabowski}, Lect. Notes Comput. Sci. 3839, 138--153 (2006; Zbl 1172.03309); J. Autom. Reasoning 55, No. 3, 211--221 (2015; Zbl 1356.68189)].
In \texttt{Coq}, the Pappus's theorem is proved as the application of Grassmann-Cayley algebra [\textit{L. Fuchs} and \textit{L. Théry}, Lect. Notes Comput. Sci. 6877, 51--67 (2011; Zbl 1350.68233)] and more recently in Tarski's geometry [\textit{G. Braun} and \textit{J. Narboux}, J. Autom. Reasoning 58, No. 2, 209--230 (2017; Zbl 1405.03034)].Existence of relativistic dynamics for two directly interacting Dirac particles in \(1+3\) dimensionshttps://www.zbmath.org/1483.810782022-05-16T20:40:13.078697Z"Lienert, Matthias"https://www.zbmath.org/authors/?q=ai:lienert.matthias"Nöth, Markus"https://www.zbmath.org/authors/?q=ai:noth.markusA numbers-based approach to a free particle's proper spacetimehttps://www.zbmath.org/1483.810922022-05-16T20:40:13.078697Z"Ferber, R."https://www.zbmath.org/authors/?q=ai:ferber.reginaldSummary: This paper contains a proposal for a free, nonzero-rest-mass particle's proper spacetime, determined exclusively by the particle's rest mass \(m_0\) and numbers. The approach defines proper time as de Broglie time, which is isomorphic to a sequence of natural numbers \(1, 2, \ldots\) that count de Broglie time units \((h/c^2)(m_0^{-1}\) (see \textit{R. Ferber} in [Found. Phys. Lett. 9, No. 6, 575--586 (1996,; \url{doi:10.1007/BF02190032})]. The approach is based on defining the spatial coordinate as proper following the constructive definition of positive and negative integers as all possible differences of ordered pairs of natural numbers multiplied by the Compton unit \((h/c)(m_0^{-1})\). The spatial and temporal coordinates that form the particle's proper spacetime are constructed as Euclidean projections of the de Broglie time. The corresponding expression in the form of an energy-momentum relation reveals the existence, aside from the rest energy term \(m_0c^2\), of an additional energy term of the same order of magnitude, which is related to large intervals of the \(m_0\)-particle's proper space. The relation of the numbers-based approach to the foundations of the special theory of relativity and of quantum mechanics is discussed.On a conformal Schwarzschild-de Sitter spacetimehttps://www.zbmath.org/1483.830402022-05-16T20:40:13.078697Z"Culetu, Hristu"https://www.zbmath.org/authors/?q=ai:culetu.hristuSummary: On the basis of the C-metric, we investigate the conformal Schwarzschild - deSitter spacetime and compute the source stress tensor and study its properties, including the energy conditions. Then we analyze its extremal version \((b^2=27m^2\), where \(b\) is the deS radius and \(m\) is the source mass), when the metric is nonstatic. The weak-field
version is investigated in several frames, and the metric becomes flat with the special choice \(b=1/a\), \(a\) being the constant acceleration of the Schwarzschild-like mass or black hole. This form is Rindler's geometry in disguise and is also conformal to a de Sitter metric where the acceleration plays the role of the Hubble constant. In its time dependent version, one finds that the proper acceleration of a static observer is constant everywhere, in contrast with the standard Rindler case. The timelike geodesics along the z-direction are calculated and proves to be hyperbolae.