Recent zbMATH articles in MSC 51https://zbmath.org/atom/cc/512024-09-27T17:47:02.548271ZWerkzeugBook review of: D. Acheson, The wonder book of geometry. A mathematical storyhttps://zbmath.org/1541.000042024-09-27T17:47:02.548271Z"Aarts, Ronald"https://zbmath.org/authors/?q=ai:aarts.ronaldReview of [Zbl 1460.00006].Book review of: C. H. Clemens, Two-dimensional geometries a problem-solving approachhttps://zbmath.org/1541.000092024-09-27T17:47:02.548271Z"Heckman, Gert"https://zbmath.org/authors/?q=ai:heckman.gertReview of [Zbl 1422.51001].Book review of: D. Crippa, The impossibility of squaring the circle in the 17th centuryhttps://zbmath.org/1541.000122024-09-27T17:47:02.548271Z"Kleijne, Wim"https://zbmath.org/authors/?q=ai:kleijne.wimReview of [Zbl 1411.01003].Book review of: G. Van Brummelen, Trigonometry. A very short introductionhttps://zbmath.org/1541.000242024-09-27T17:47:02.548271Z"Sterk, Hans"https://zbmath.org/authors/?q=ai:sterk.hansReview of [Zbl 1430.51001].Book review of: C. Alsina and R. B. Nelsen, A cornucopia of quadrilateralshttps://zbmath.org/1541.000262024-09-27T17:47:02.548271Z"van der Vaart, Joop"https://zbmath.org/authors/?q=ai:van-der-vaart.joopReview of [Zbl 1443.51001].Book review of: P. McMullen, Geometric regular polytopeshttps://zbmath.org/1541.000292024-09-27T17:47:02.548271Z"van der Waall, Rob"https://zbmath.org/authors/?q=ai:van-der-waall.robert-willemReview of [Zbl 1454.51002].Algebra in cuneiform. Introduction to an Old Babylonian geometrical technique. Translated from the English by Franz Lemmermeyer and students of High School `St. Gertrudis', Ellwangenhttps://zbmath.org/1541.010012024-09-27T17:47:02.548271Z"Høyrup, Jens"https://zbmath.org/authors/?q=ai:hoyrup.jensPublisher's description: Um 1930 hat man entdeckt, dass einige babylonische Keilschrifttexte Rechnungen enthielten, wie sie beim Lösen quadratischer Gleichungen auftreten. Weil die Bedeutung der Terminologie im wesentlichen von den im Text enthaltenen Zahlen abgeleitet werden musste, hat dies dazu geführt, dass diese Texte als numerische Algebra interpretiert wurden. Diese Interpretation wurde erst angezweifelt, als der Autor des vorliegenden Buchs um 1982 entdeckte, dass sie mit der globalen Struktur der Terminologie inkompatibel ist. Es stellte sich heraus, dass zwei verschiedene und nicht synonyme Operationen beide als Addition aufgefasst wurden; entsprechend wurden zwei subtraktive Operationen vermengt, und vier verschiedene Operationen wurden allesamt als ein und dieselbe Multiplikation aufgefasst. Stattdessen verweist die Struktur auf eine Technik, die auf der Geometrie von Quadraten und Rechtecken mit messbaren Seiten und Flächen aufbaut. Das vorliegende Buch analysiert verschiedene Texte in konformaler Übersetzung, also in einer Übersetzung, in welcher derselbe babylonische Ausdruck immer mit demselben Wort übersetzt wird und, was noch wichtiger ist, in welcher verschiedene Ausdrücke auch auf unterschiedliche Arten übersetzt sind. Philologische Details, die nur solchen Lesern nutzen, welche mit der Assyriologie vertraut sind, werden vermieden; allerdings werden diesen solche Informationen in einem eigenen Anhang zur Verfügung gestelt. Alle vorgestellten Texte stammen aus der zweiten Hälfte der altbabylonischen Periode, also von 1800 bis 1600 v.Chr. Gerade in dieser Periode erreichte die Algebra im Besonderen und die babylonische Mathematik im Allgemeinen ihren Höhepunkt. Selbst wenn einige jüngere Texte Ähnlichkeiten mit denen aus der altbabylonischen Zeit aufweisen, sind sie doch nur Überbleibsel. Außer der Analyse von Texten liefert das Buch eine allgemeine Charakterisierung der auftretenden Mathematik und stellt es in den Kontext der altbabylonischen Schreiberschule und deren besonderen Kultur. Endlich beschreibt das Buch den Ursprung der Disziplin und deren Wirkung auf die spätere Mathematik, nicht zuletzt die euklidische Geometrie und die echte Algebra, die im mittelalterlichen Islam erschaffen wurde und von der europäischen mittelalterlichen und Renaissance-Mathematik übernommen wurde.
See the review of the original English edition in [Zbl 1392.01003].Alternating groups and point-primitive linear spaces with number of points being squarefreehttps://zbmath.org/1541.050222024-09-27T17:47:02.548271Z"Guan, Haiyan"https://zbmath.org/authors/?q=ai:guan.haiyan"Zhou, Shenglin"https://zbmath.org/authors/?q=ai:zhou.shenglinSummary: This paper is a further contribution to the classification of point-primitive finite regular linear spaces. Let \(\mathcal{S}\) be a nontrivial finite regular linear space whose number of points \(v\) is squarefree. We prove that if \(G \leq \Aut (\mathcal{S})\) is point-primitive with an alternating socle, then \(\mathcal{S}\) is the projective space \(\mathrm{PG}(3, 2)\).
{{\copyright} 2023 Wiley Periodicals LLC.}Annulus graphs in \(\mathbb{R}^d\)https://zbmath.org/1541.050382024-09-27T17:47:02.548271Z"Lichev, Lyuben"https://zbmath.org/authors/?q=ai:lichev.lyuben"Mihaylov, Tsvetomir"https://zbmath.org/authors/?q=ai:mihaylov.tsvetomirSummary: A \(d\)-dimensional annulus graph with radii \(R_1\) and \(R_2\) (here \(R_2 \geq R_1 \geq 0\)) is a graph embeddable in \(\mathbb{R}^d\) so that two vertices \(u\) and \(v\) form an edge if and only if their images in the embedding are at distance 1 in the interval \([R_1, R_2]\). In this paper we show that the family \(\mathcal{A}_d (R_1, R_2)\) of \(d\)-dimensional annulus graphs with radii \(R_1\) and \(R_2\) is uniquely characterised by \(R_2/R_1\) when this ratio is sufficiently large. Moreover, as a step towards a better understanding of the structure of \(\mathcal{A}_d (R_1,R_2)\), we show that \(\sup_{G \in \mathcal{A}_d (R_1, R_2)} \chi (G)/\omega (G)\) is given by \(\exp (O(d))\) for all \(R_1, R_2\) satisfying \(R_2 \geq R_1 > 0\) and also \(\exp (\Omega (d))\) if moreover \(R_2/R_1 \geq 1.2\).\(K_7\)-minors in optimal 1-embedded graphs on the projective planehttps://zbmath.org/1541.051712024-09-27T17:47:02.548271Z"Sone, Katsuya"https://zbmath.org/authors/?q=ai:sone.katsuya"Suzuki, Yusuke"https://zbmath.org/authors/?q=ai:suzuki.yusukeSummary: Every simple optimal 1-embedded graph on a closed surface has a \(K_t\)-minor for \(t \leq 6\) as \textit{W. Mader} [Math. Ann. 178, 154--168 (1968; Zbl 0165.57401)] proved that every simple graph with \(n\) vertices and at least \(4n - 9\) edges has a \(K_6\)-minor. In this paper, we characterize simple optimal 1-embedded graphs on the projective plane that have no \(K_7\)-minor. It follows from the aforementioned result that every simple optimal 1-embedded graph on the projective plane which is 5-connected or whose quadrangular subgraph is 4-representative has a \(K_7\)-minor.Characterizations of a class of planar functions over finite fieldshttps://zbmath.org/1541.111012024-09-27T17:47:02.548271Z"Chen, Ruikai"https://zbmath.org/authors/?q=ai:chen.ruikai"Mesnager, Sihem"https://zbmath.org/authors/?q=ai:mesnager.sihemLet \(\mathbb F_{q^n}\) be an extension of the field \(\mathbb F_q\) of odd characteristic. The function induced by a polynomial \(f\) over \(\mathbb F_{q^n}\) is called a planar function if \(f(x+c)-f(x)\) is a permutation on \(\mathbb F_{q^n}\) for every \(c\in \mathbb F_{q^n}^*\). The paper introduces a new class of planar functions of the form \(\mathrm{Tr}(ax^{q+1})+l(x^2)\), where \(a\in \mathbb F_{q^n}^*\), \(\mathrm{Tr}\) is the trace function from \(\mathbb F_{q^n}\) to \(\mathbb F_q\), and \(l\) is an arbitrary linearized polynomial over \(\mathbb F_{q^n}\) (a polynomial over \(\mathbb F_{q^n}\) that induces a linear endomorphism of \(\mathbb F_{q^n}\) over its prime field). These planar functions are studied according to the degree \(n\). In particular, when \(n=2\) many instances of those planar functions exist and are shown in the paper. When \(n=3\) all those planar functions are characterized explicitly. While, as the degree \(n\) gets higher, those planar functions tend not to exist.
Reviewer: Gloria Rinaldi (Reggio Emilia)Closed-form solution of conic in point-line enumerative problem of conichttps://zbmath.org/1541.140212024-09-27T17:47:02.548271Z"Guo, Yang"https://zbmath.org/authors/?q=ai:guo.yang.1|guo.yangThis article considers the problem of determining the up to \(2^{\min(m,m-5)}\) conics in the real projective plane that contain \(m\) points and are tangent to \(m-5\) lines. As the author is well aware, many solutions to this problem are known. The author's solution impresses with its conciseness also for the cases with \(0 < m < 5\). It also yield necessary and sufficient algebraic and geometric criteria to determine the number of real conics. Derivations in this article are pleasantly short and elegant.
Reviewer: Hans-Peter Schröcker (Innsbruck)Some characterizations of the complex projective space via Ehrhart polynomialshttps://zbmath.org/1541.140712024-09-27T17:47:02.548271Z"Loi, Andrea"https://zbmath.org/authors/?q=ai:loi.andrea"Zuddas, Fabio"https://zbmath.org/authors/?q=ai:zuddas.fabioFor a lattice polytope \(\Delta\subset\mathbb{R}^n\), \textit{E. Ehrhart} [C. R. Acad. Sci., Paris 254, 616--618 (1962; Zbl 0100.27601)] proved that the number lattice points in the scaled polytope \(m\Delta\) for \(m\in\mathbb{N}\) is a polynomial \(P_\Delta(m)\) in \(m\) of degree \(n\), with leading coefficient \(Vol(\Delta)\). \(P_\Delta\) is usually called the ``Ehrhart polynomial'' of \(\Delta\). When \(\Delta\) is a Delzant polytope, then \(\Delta\) correspond to a polarized toric manifold, and Ehrhart's theorem is just the Riemann-Roch-Hirzebruch theorem for this polarized toric manifold. When \(\Delta=\Sigma_n\) is the standard simplex, the corresponding polarized toric manifold is simply \((\mathbb{C}P^n,\mathcal{O}(1))\). We call two Delzant polytopes \(\Delta\) and \(\tilde\Delta\) ``Ehrhart equivalent'' if they have the same Ehrhart polynomial.
In this paper, the authors investigate the following problem: if \(\Delta\) is Ehrhart equivalent to \(\lambda\Sigma_n\) for some \(\lambda\in\mathbb{Z}^+\), when can one conclude that the polarized toric manifold associated to \(\Delta\) is \((\mathbb{C}P^n,\mathcal{O}(\lambda))\)? This is not true in general, and counterexamples are provided in the paper. The main theorem of the paper says that this is indeed true in the following three cases:
\begin{itemize}
\item[(i)] \(\lambda=1\);
\item[(ii)] \(n=2\) and \(\lambda=3\);
\item[(iii)] \(\lambda=n+1\) and the associated polarized pair \((M,L)\) is asymptotically Chow semistable.
\end{itemize}
The proofs of the first two cases are quite simple and direct. For the third case, the author uses Nill-Paffenholz's characterization [\textit{B. Nill} and \textit{A. Paffenholz}, Adv. Geom. 14, No. 4, 579--586 (2014; Zbl 1302.52010)] of the equality case of Ehrhart's volume conjecture.
Reviewer: Yalong Shi (Nanjing)Line-bound vectors, plane-bound bivectors and tetrahedra in the conformal model of three-dimensional spacehttps://zbmath.org/1541.150292024-09-27T17:47:02.548271Z"Havel, Timothy F."https://zbmath.org/authors/?q=ai:havel.timothy-franklinSummary: The representation of some elementary geometric concepts in the conformal geometric algebra of three dimensions are reviewed, and their connections to a recently discovered extension of Heron's formula for the area of a triangle to the volume of a tetrahedron are discussed.
For the entire collection see [Zbl 1539.68035].Inner product of two oriented points in conformal geometric algebrahttps://zbmath.org/1541.150302024-09-27T17:47:02.548271Z"Hitzer, Eckhard"https://zbmath.org/authors/?q=ai:hitzer.eckhardSummary: We study the inner product of oriented points in conformal geometric algebra and its geometric meaning. The notion of oriented point is introduced and the inner product of two general oriented points is computed, and analyzed (including symmetry) in terms of point to point distance, and angles between the distance vector and the local orientation planes of the two points. Seven examples illustrate the results obtained. Finally, the results are extended from dimension three to arbitrary dimensions \(n\).
For the entire collection see [Zbl 1539.68035].Geometric algebra and distance matriceshttps://zbmath.org/1541.150332024-09-27T17:47:02.548271Z"Riter, Vinicius"https://zbmath.org/authors/?q=ai:riter.vinicius"Alves, Rafael"https://zbmath.org/authors/?q=ai:alves.rafael"Lavor, Carlile"https://zbmath.org/authors/?q=ai:lavor.carlile-camposSummary: We present a new approach to the problem of recognizing an Euclidean distance matrix, based on Conformal Geometric Algebra. Such matrices are symmetric and hollow with non negative entries that are equal to the squared distances among the set of points. In addition to find these points, the method presented here also provides the minimal dimension of the related space. A comparison with a linear algebra approach is also provided.
For the entire collection see [Zbl 1539.68035].Line-cyclide intersection and colinear point quadruples in the double conformal modelhttps://zbmath.org/1541.150352024-09-27T17:47:02.548271Z"Yao, Huijing"https://zbmath.org/authors/?q=ai:yao.huijing"Mann, Stephen"https://zbmath.org/authors/?q=ai:mann.stephen"Li, Qinchuan"https://zbmath.org/authors/?q=ai:li.qinchuanSummary: In this paper, we look at using the double conformal model for ray tracing. In particular, we explore the intersection of a line with a cyclide in the double conformal model, and how to extract the four points from the resulting colinear point quadruple. Further, we show how to directly construct a colinear point quadruple from four points, and we show how to find the line containing the points of a colinear point quadruple. We also briefly touch on barycentric coordinates in DCGA.
For the entire collection see [Zbl 1539.68035].Shadows of the circle. From conic sections to planetary motionhttps://zbmath.org/1541.510012024-09-27T17:47:02.548271Z"Hansen, Vagn Lundsgaard"https://zbmath.org/authors/?q=ai:hansen.vagn-lundsgaardPublisher's description: The ancient Greeks were the first to seriously ask for scientific explanations of the panorama of the heavens based on mathematical ideas. Ever since, mathematics has played a major role for human perception and description of the outside physical world, and in a larger perspective for comprehending the universe. This second edition pays tribute to this line of thought and takes the reader on a journey in the mathematical universe from conic sections to mathematical modelling of planetary systems.
In the second edition, the four chapters in the first edition on conic sections (two chapters), isoperimetric problems for plane figures, and non-Euclidean geometry, are treated in four revised chapters with many new exercises added. In three new chapters, the reader is taken through mathematics in curves, mathematics in a Nautilus shell, and mathematics in the panorama of the heavens. In all chapters of the book, the circle plays a prominent role.
This book is addressed to undergraduate and graduate students as well as researchers interested in the geometry of conic sections, including the historical background and mathematical methods used. It features selected important results, and proofs that not only proves but also 'explains' the results.
See the review of the first edition in [Zbl 0913.51008].Plane Euclidean geometry. Algebraization, axiomatization and interfaces to school mathematicshttps://zbmath.org/1541.510022024-09-27T17:47:02.548271Z"Hoffmann, Max"https://zbmath.org/authors/?q=ai:hoffmann.max"Hilgert, Joachim"https://zbmath.org/authors/?q=ai:hilgert.joachim"Weich, Tobias"https://zbmath.org/authors/?q=ai:weich.tobiasPublisher's description: n diesem Lehrbuch stellen die Autoren einen axiomatischen Zugang zur ebenen Geometrie dar, der im Vergleich zu den Hilbertaxiomen und anderen oft gewählten Zugängen strukturelle und didaktische Vorteile bietet. Dieser auf metrischen Räumen basierende Zugang wird ausführlich motiviert und didaktisch aufbereitet. Ein besonderes Augenmerk liegt auf der besseren Verzahnung der Mathematikausbildung der Lehramtsstudierenden mit dem Schulstoff. In Ergänzung des axiomatischen Zugangs erklären die Autoren auch, wie man sich der ebenen Geometrie mit Mitteln der linearen Algebra nähern kann und stellen so den Bezug zur analytischen Geometrie der Oberstufe her.
Als weitere Schnittstellen zwischen Schulmathematik und axiomatischer Geometrie werden die Begriffe Kongruenz und Symmetrie vertieft und so wichtigen Zusammenhänge zwischen den Begriffen Isometrie, Kongruenz und Symmetrie transparent gemacht und in schultypische Kontexte eingebettet.In square circle: geometric knowledge of the Indus civilizationhttps://zbmath.org/1541.510032024-09-27T17:47:02.548271Z"Sinha, Sitabhra"https://zbmath.org/authors/?q=ai:sinha.sitabhra"Yadav, Nisha"https://zbmath.org/authors/?q=ai:yadav.nisha"Vahia, Mayank"https://zbmath.org/authors/?q=ai:vahia.m-nFor the entire collection see [Zbl 1242.00057].Rational quadrilaterals from Brahmagupta to Kummerhttps://zbmath.org/1541.510042024-09-27T17:47:02.548271Z"Sridharan, R."https://zbmath.org/authors/?q=ai:sridharan.ramanujan|sridharan.raja|sridharan.ramaiyengarFor the entire collection see [Zbl 1242.00057].Some remarks on a theorem of Greenhttps://zbmath.org/1541.510052024-09-27T17:47:02.548271Z"Jalled, Abdessami"https://zbmath.org/authors/?q=ai:jalled.abdessami-ben-hmida"Haggui, Fathi"https://zbmath.org/authors/?q=ai:haggui.fathi(no abstract)3-uniform hypergraphs from vector spaceshttps://zbmath.org/1541.510062024-09-27T17:47:02.548271Z"Meulewaeter, Jeroen"https://zbmath.org/authors/?q=ai:meulewaeter.jeroen"Van Maldeghem, Hendrik"https://zbmath.org/authors/?q=ai:van-maldeghem.hendrik-jConsider a projective space of finite dimension \(n\) over a skew field \(\mathbb{L}\). The fundamental theorem of projective geometry states that any collineation (a bijection of the point set onto itself that preserves collinearity) is induced by a semi-linear automorphism of the underlying vector space.
This article asks for generalizations in the following sense: Fix integers \(i\), \(j\), \(k\) and \(\ell\) (subject to some obvious inequalities) and denote by \(V_i\), \(V_j\), and \(V_k\) the sets of \(i\)-spaces, \(j\)-spaces, and \(k\)-spaces, respectively. Is any bijection of the disjoint union of \(\mathcal{V} = V_i \sqcup V_j \sqcup V_k\) that maps a triple of subspaces that (a) span a subspace of dimension \(\ell\) or (b) whose span is contained in a subspace of dimension \(\ell\) to a like triple of subspaces induced by a semi-linear vector space automorphism? A complete answer for this question in terms of \(i\), \(j\), \(k\), \(\ell\), and \(n\) is given.
The main tool of proof is a translation of the problem into the language of graphs and hypergraphs. Denote by \(\mathcal{T}_{i,j,k;\ell}\) the set of triples \((I,J,K)\) with \(I \in V_i\), \(J \in V_j\), and \(K \in V_k\) that span an \(\ell\)-space and by \(\mathcal{T}_{i,j,k;\le\ell}\) the set of triples whose span is contained in an \(\ell\)-space. Then \((\mathcal{V}, \mathcal{T}_{i,j,k;\ell})\) and \((\mathcal{V}, \mathcal{T}_{i,j,k;\le\ell})\) are \(3\)-uniform hypergraphs and the research question boils down determine under which conditions the hypergraphs' automorphism groups equal the automorphism group of the projective space. For this, see the results of \textit{A. De Schepper} and \textit{H. Van Maldeghem} [Linear Algebra Appl. 449, 435--464 (2014; Zbl 1302.15002)] and earlier results by Chow and by Lim are used.
Reviewer: Hans-Peter Schröcker (Innsbruck)The pentagon theorem in Miquelian Möbius planeshttps://zbmath.org/1541.510072024-09-27T17:47:02.548271Z"Halbeisen, Lorenz"https://zbmath.org/authors/?q=ai:halbeisen.lorenz-j"Hungerbühler, Norbert"https://zbmath.org/authors/?q=ai:hungerbuhler.norbert"Loureiro, Vanessa"https://zbmath.org/authors/?q=ai:loureiro.vanessaA simple algebraic proof which is based on the cross ratio, is given for the pentagon theorem in arbitrary miquelian Möbius planes $\mathfrak{M}(K,q)$ obtained from a separable quadratic field extension (including all finite miquelian Möbius planes).
Reviewer: Dirk Keppens (Gent)Translational and great Darboux cyclideshttps://zbmath.org/1541.510082024-09-27T17:47:02.548271Z"Lubbes, Niels"https://zbmath.org/authors/?q=ai:lubbes.nielsThis article characterizes real irreducible algebraic surfaces in \({\mathbb R}^3\) that contain at least two circles through each point.
Two surfaces in \({\mathbb R}^3\) are said to be Möbius equivalent if one surface is mapped to the other by a composition of inversions.
With \(\mu: {\mathbb S}^3\rightarrow {\mathbb R}^3\) the stereographic projection from the point \((0, 0, 0, 1)\) on the 3-dimensional unit-sphere \({\mathbb S}^3\subset {\mathbb R}^4\) -- \(\mu(y):= (y_1, y_2, y_3)/(1-y_4)\) -- the Möbius degree of a real irreducible algebraic surface \(Z \subset {\mathbb R}^3\) is defined as \(\deg \mu^{-1}(Z)\) and \(Z\) is called \(\lambda\)-circled if the Zariski closure of \(\mu^{-1}(Z)\) contains at least \(\lambda\in Z_{\geq 0} \cup \{\infty\}\) circles through a general point. If \(\lambda\in Z_{\geq 0}\), then by \(\lambda\)-circled one understands that \(Z\) is not \((\lambda + 1)\)-circled. If \(\lambda\geq 2\), then \(Z\) is called \textit{celestial}.
If \(A\) and \(B\) are curves in \({\mathbb R}^3\) or \({\mathbb S}^3\), one identifies the unit-sphere \({\mathbb S}^3\subset {\mathbb R}^4\) with the unit quaternions and, denoting the Hamiltonian product by \(*\) one can define, \(A + B\) and \(A*B\) in the usual manner. A real irreducible algebraic surface is said to be \textit{Bohemian} or \textit{Cliffordian} if there exist generalized circles \(A\) and \(B\) such that \(Z\) is the Zariski closure of \(A + B\) and \(\mu(A*B)\), respectively. A surface that is either Bohemian or Cliffordian is called \textit{translational}. If \(A\) and \(B\) are great circles such that \(A*B\subset {\mathbb S}^3\) is a real irreducible algebraic surface, then \(A*B\) is called a \textit{Clifford torus}. A \textit{Darboux cyclide} in \({\mathbb R}^3\) is a real irreducible algebraic surface of Möbius degree four. A \textit{\(Q\) cyclide} is a Darboux cyclide that is Möbius equivalent to a quadric \(Q\). With the following abbreviations for quadrics, \(E\) = elliptic/ellipsoid, \(P\) = parabolic/paraboloid, \(O\) = cone, \(C\) = circular, \(H\) = hyperbolic/hyperboloid, \(Y\) = cylinder, with a \textit{\(CH1\) cyclide} denoting one that is Möbius equivalent to a Circular Hyperboloid of 1 sheet, with a \textit{ring cyclide}, \textit{Perseus cyclide} or \textit{Blum cyclide} designating a Darboux cyclide without real singularities that is 4-circled, 5-circled and 6-circled, respectively, and with a real irreducible algebraic surface \(Z \subset {\mathbb R}^3\) called \textit{great} if its inverse stereographic projection \(\mu^{-1}(Z)\) is covered by great circular arcs, the main theorem states that:
For a \(\lambda\)-circled surface \(Z \subset {\mathbb R}^3\) of Möbius degree \(d\), with \(\lambda\geq 2\) and \((d, \lambda)\neq (8, 2)\), we have:
\(\bullet\) \(Z\) is Bohemian if and only if \(Z\) is either a plane, \(CY\) or \(EY\).
\(\bullet\) If \(Z\) is Cliffordian, then \(Z\) is either a Perseus cyclide, ring cyclide or \(CH1\) cyclide. Conversely, if \(Z\) is a ring cyclide, then \(Z\) is Möbius equivalent to a Cliffordian surface.
\(\bullet\) \(Z\) is Möbius equivalent to a great celestial surface if and only if \(Z\) is either a plane, sphere, Blum cyclide, Perseus cyclide, ring cyclide, \(EO\) cyclide or \(CO\) cyclide.
Reviewer: Victor V. Pambuccian (Glendale)On \(m\)-ovoids of finite classical polar spaces with an irreducible transitive automorphism grouphttps://zbmath.org/1541.510092024-09-27T17:47:02.548271Z"Feng, Tao"https://zbmath.org/authors/?q=ai:feng.tao"Li, Weicong"https://zbmath.org/authors/?q=ai:li.weicong"Tao, Ran"https://zbmath.org/authors/?q=ai:tao.ran.2|tao.ran.1|tao.ranAn \(m\)-ovoid of a polar space \(\mathcal{P}\) is a set \(\Omega\) of points of \(\mathcal{P}\) meeting each maximal projective subspace of the same (i.e. an element of the dual polar space) in exactly \(m\) points.
A polar space is \textit{classical} if it can be embedded in a projective space; see [\textit{J. Tits}, Buildings of spherical type and finite BN-pairs. Berlin-Heidelberg-New York: Springer-Verlag (1974; Zbl 0295.20047)]. As the embeddings of finite classical polar spaces are all homogeneous, the automorphism group of the polar space acts on the projective space hosting the embedding as a collineation group (and turn the corresponding vector space into a module). In the paper under review, the authors classify the transitive \(m\)-ovoids of embedded finite classical polar spaces with a group acting irreducibly on the ambient vector space, obtaining also several new infinite families of transitive \(m\)-ovoids.
Reviewer: Luca Giuzzi (Brescia)Bi-directional models of ``radically synthetic'' differential geometryhttps://zbmath.org/1541.510102024-09-27T17:47:02.548271Z"Menni, Matías"https://zbmath.org/authors/?q=ai:menni.matiasSummary: The radically synthetic foundation for smooth geometry formulated in [\textit{F. W. Lawvere}, West. Ont. Ser. Philos. Sci. 75, 249--254 (2011; Zbl 1251.03098)] postulates a space \(T\) with the property that it has a unique point and, out of the monoid \(T^T\) of endomorphisms, it extracts a submonoid \(R\) which, in many cases, is the (commutative) multiplication of a rig structure. The rig \(R\) is said to be \textit{bi-directional} if its subobject of invertible elements has two connected components. In this case, \(R\) may be equipped with a pre-order compatible with the rig structure. We adjust the construction of ``well-adapted'' models of Synthetic Differential Geometry in order to build the first pre-cohesive toposes with a bi-directional \(R\). We also show that, in one of these pre-cohesive variants, the pre-order on \(R\), \textit{derived} radically synthetically from bi-directionality, coincides with that \textit{defined} in the original model.Double balanced quadrilateralshttps://zbmath.org/1541.510112024-09-27T17:47:02.548271Z"Berele, Allan"https://zbmath.org/authors/?q=ai:berele.allan"Catoiu, Stefan"https://zbmath.org/authors/?q=ai:catoiu.stefanSummary: The \(k\)-centroids \(G_k\) of a polygon, for \(k = 0, 1, 2\), are the centroids of the polygon when the mass is equally distributed respectively between the vertices, along the perimeter, or across the area. A fundamental theorem by \textit{A. Al-Sharif} et al. in [Result. Math. 55, No. 3--4, 231--247 (2009; Zbl 1191.51004)] asserts that the quadrilaterals with either \(G_0 = G_1\) or \(G_0 = G_2\) are precisely all parallelograms. Our main result describes the non-parallelograms with \(G_1 = G_2\) by providing formulas for their diagonals in terms of the sides, as well as formulas for the ratios determined on the diagonals by their intersection point. In this way, we complete a fifteen-year-old problem by these three authors on characterizing all double balanced quadrilaterals. As an application, we show how our main theorem can be used to deduce their characterizations of double balanced circumscribed and cyclic quadrilaterals.Hyperbolic width functions and characterizations of bodies of constant width in the hyperbolic spacehttps://zbmath.org/1541.510122024-09-27T17:47:02.548271Z"Böröczky, Károly J."https://zbmath.org/authors/?q=ai:boroczky.karoly-jun"Csépai, András"https://zbmath.org/authors/?q=ai:csepai.andras"Sagmeister, Ádám"https://zbmath.org/authors/?q=ai:sagmeister.adamIn this paper, the authors analyze the basic properties of several width functions in \(n\)-dimensional hyperbolic space, such as continuity. Additionally, a new hyperbolic width is defined as an extension of Leichtweiss' width function. Then, the authors prove a characterization theorem for bodies of constant width concerning the aforementioned notions of hyperbolic width.
Reviewer: Sonia Pérez Díaz (Madrid)Euler's inequality for circumradius and inradiushttps://zbmath.org/1541.510132024-09-27T17:47:02.548271Z"Oliveira Costa Filho, Wagner"https://zbmath.org/authors/?q=ai:oliveira-costa-filho.wagnerThis note presents a brief proof of Euler's inequality for the circumradius \(R\) and the inradius \(r\) of a triangle, \(R\geq 2r\), based on the use of trigonometric functions. The proof is elementary in nature and can be accessible to high-school students.
Reviewer: Victor Oxman (Haifa)Counterintuitive patterns on angles and distances between lattice points in high dimensional hypercubeshttps://zbmath.org/1541.510142024-09-27T17:47:02.548271Z"Anderson, Jack"https://zbmath.org/authors/?q=ai:anderson.jack-m"Cobeli, Cristian"https://zbmath.org/authors/?q=ai:cobeli.cristian"Zaharescu, Alexandru"https://zbmath.org/authors/?q=ai:zaharescu.alexandruSummary: Let \(\mathcal{S}\) be a finite set of integer points in \(\mathbb{R}^d\), which we assume has many symmetries, and let \(P\in\mathbb{R}^d\) be a fixed point. We calculate the distances from \(P\) to the points in \(\mathcal{S}\) and compare the results. In some of the most common cases, we find that they lead to unexpected conclusions if the dimension is sufficiently large. For example, if \(\mathcal{S}\) is the set of vertices of a hypercube in \(\mathbb{R}^d\) and \(P\) is any point inside, then almost all triangles \textit{PAB} with \(A,B\in\mathcal{S}\) are almost equilateral. Or, if \(P\) is close to the center of the cube, then almost all triangles \textit{PAB} with \(A\in\mathcal{S}\) and \(B\) anywhere in the hypercube are almost right triangles.On the odd area of the unit dischttps://zbmath.org/1541.510152024-09-27T17:47:02.548271Z"Pinchasi, Rom"https://zbmath.org/authors/?q=ai:pinchasi.romGiven a family \({\mathcal F}\) of measurable sets in the plane, one denotes by \(OA({\mathcal F })\) the area of the set of all points in the plane that belong to an odd number of members in \({\mathcal F}\). The infimum of \(OA({\mathcal F })\) over all families \({\mathcal F}\) consisting of an odd number of translates of a given measurable set \(B\) is called \textit{the odd area of \(B\)} and is denoted by \(OA(B)\).
\textit{Igor Pak} suggested finding the odd area of the unit disc in the plane, but ``despite a considerable effort by the author and quite a few colleagues and students'' nothing was known about it prior to this paper. In it, the author presents an analytic approach and derives some geometric constraints on families \({\mathcal F}\) of unit discs that minimize \(OA({\mathcal F })\) for a given size of a family \({\mathcal F}\) of translates of the unit disc in the plane. The hope is that these ``will help to conclude nontrivial bounds on the odd area of the unit disc.'' There are five main results, one of which is:
Let \({\mathcal F}\) be a family of \(n\) unit discs in the plane, where \(n\geq 3\) is odd. Assume that the discs in \({\mathcal F}\) have a nonempty intersection that we denote by \(Q\). Assume further that none of the discs in \({\mathcal F}\) contains \(Q\) in its interior. That is, the boundary of every disc in \({\mathcal F}\) contains a boundary point of \(Q\). Then \(OA({\mathcal F})\geq \pi\).
Reviewer: Victor V. Pambuccian (Glendale)Perspectives through a two-slit camerahttps://zbmath.org/1541.510162024-09-27T17:47:02.548271Z"Crannell, Annalisa"https://zbmath.org/authors/?q=ai:crannell.annalisa"Abraham, Ojima"https://zbmath.org/authors/?q=ai:abraham.ojima"Dai, Jihang"https://zbmath.org/authors/?q=ai:dai.jihang"Gong, Yike"https://zbmath.org/authors/?q=ai:gong.yike"McClain, Rebecca"https://zbmath.org/authors/?q=ai:mcclain.rebecca"Ramaswamy, Nithya"https://zbmath.org/authors/?q=ai:ramaswamy.nithya"Reisner, Charles"https://zbmath.org/authors/?q=ai:reisner.charles"Shinn, Evan"https://zbmath.org/authors/?q=ai:shinn.evan"Wang, Shen"https://zbmath.org/authors/?q=ai:wang.shen|wang.shen.1A mathematical two-slit camera consists of three planes in \(\mathbb{R}^3\): a \textit{canvas} \(w\), a planar barrier \(w_h\) containing a linear slit \(h\) and a second planar barrier \(w_v\) containing a linear slit \(v\). Assume that \(w\), \(w_h\), and \(w_v\) are parallel, and \(h\) is Euclidean orthogonal to \(v\). Let \(P\in \mathbb{R}^3\notin (w\cup w_h\cup w_v)\). The image \(P'\) of \(P\) on \(w\) by the two-slit camera is defined as the intersection of \(w\) with the unique line that meets both \(h\) and \(v\) and contains \(P\). Firstly, the authors calculate the image of lines and segments parallel to or perpendicular to \(w\). If \(l\) is a line non parallel to the canvas, then its image is an hyperbola passing through two points and with asymptotes that are precisely described in the last result of the paper. There are fifteen figures illustrating the manuscript.
Reviewer: Pedro Martín Jiménez (Badajoz)Dimension-free estimates on distances between subsets of volume \(\varepsilon\) inside a unit-volume bodyhttps://zbmath.org/1541.520132024-09-27T17:47:02.548271Z"Ismailov, Abdulamin"https://zbmath.org/authors/?q=ai:ismailov.abdulamin"Kanel-Belov, Alexei"https://zbmath.org/authors/?q=ai:kanel-belov.alexei"Ivlev, Fyodor"https://zbmath.org/authors/?q=ai:ivlev.fyodorGiven a family of bounded convex bodies in \(\mathbb{R}^n\) with unit volume, consider the supremum of possible Euclidean distances between two subsets of fixed volume \(\varepsilon\in(0,1/2)\). This quantity is denoted by \(d_n(\varepsilon)\). The authors derive asymptotics for \(d_n(\varepsilon)\) when \(n\to\infty\) for various families of convex bodies. For example, for the family of unit-volume Euclidean balls, the authors show that
\[
\lim_{n\to\infty}d_n(\varepsilon)=-\frac{2}{\sqrt{e}}\Phi^{-1}(\varepsilon)\,,
\]
where \(\Phi(a)=\int_{-\infty}^ae^{-\pi x^2}\text{d}x\). Similarly, for the family of unit cubes,
\[
-2\sqrt{\frac{\pi}{6}}\Phi^{-1}(\varepsilon)\leq\liminf_{n\to\infty}d_n(\varepsilon)\leq\limsup_{n\to\infty}d_(\varepsilon)\leq-2\Phi^{-1}(\varepsilon)\,,
\]
and for the family of unit-volume simplices,
\[
-\frac{\sqrt{2}}{e}\ln(2\varepsilon)\leq\liminf_{n\to\infty}d_n(\varepsilon)\leq\limsup_{n\to\infty}d_(\varepsilon)\leq-c\ln(\varepsilon),
\]
for some constant \(c>0\) independent of \(n\) and \(\varepsilon\). Some analogous results are also presented for unit-volume \(\ell_p\) balls for \(p\geq1\), and for the Manhattan distance in place of the Euclidean distance. Proofs make use of isoperimetric inequalities, and links are drawn with concentration of measure.
Reviewer: Fraser Daly (Edinburgh)Polytopality of 2-orbit maniplexeshttps://zbmath.org/1541.520182024-09-27T17:47:02.548271Z"Mochán, Elías"https://zbmath.org/authors/?q=ai:mochan.eliasSummary: \textit{Abstract polytopes} are a combinatorial generalization of convex and skeletal polytopes. Counting how many flag orbits a polytope has under its automorphism group is a way of measuring how symmetric it is. Polytopes with one flag orbit are called \textit{regular} and are very well known. Polytopes with two flag orbits (called \textit{2-orbit polytopes}) are, however, way more elusive. There are \(2^n - 1\) possible classes of 2-orbit polytopes in rank (dimension) \(n\), but for most of those classes, determining whether or not they are empty is still an open problem. In 2019, in their article [J. Comb. Theory, Ser. A 166, 226--253 (2019; Zbl 1416.05274)], \textit{D. Pellicer}, \textit{P. Potočnik} and \textit{M. Toledo} constructed 2-orbit \textit{maniplexes} (objects that generalize abstract polytopes and maps) in all these classes, but the question of whether or not they are also polytopes remained open. In this paper we use the results of a previous paper by the author and Hubard to show that some of these 2-orbit maniplexes are, in fact, polytopes. In particular we prove that there are 2-orbit polytopes in all the classes where exactly two kinds of reflections are forbidden. We use this to show that there are at least \(n^2 - n + 1\) classes of 2-orbit polytopes of rank \(n\) that are not empty. We also show that the maniplexes constructed with this method in the remaining classes satisfy all but (possibly) one of the properties necessary to be polytopes, therefore we get closer to proving that there are 2-orbit polytopes in all the classes.The hyperbolic revolution: from topology to geometry, and backhttps://zbmath.org/1541.530052024-09-27T17:47:02.548271Z"Bonahon, Francis"https://zbmath.org/authors/?q=ai:bonahon.francisFor the entire collection see [Zbl 1326.00083].Pure metric geometryhttps://zbmath.org/1541.540012024-09-27T17:47:02.548271Z"Petrunin, Anton"https://zbmath.org/authors/?q=ai:petrunin.antonThis is a succinct introduction to the tools one needs to understand metric geometry. It starts with the most basic definitions, such as that of a metric space, of completeness (with a proof of the Baire category theorem), of a compact space, of a geodesic, of metric trees, or length spaces (with a proof of Menger's lemma and of the Hopf-Rinow theorem), to move on to more advanced topics, such as the extension property, universality, the Urysohn space, injective spaces, the Hausdorff metric, the Gromov-Hausdorff metric, and ultrafilters. There are exercises of various levels of difficulty with ``semisolutions'' at the back of the book. For complicated solutions, the source of the solution in the literature is also mentioned.
Reviewer: Victor V. Pambuccian (Glendale)Strongly rigid metrics in spaces of metricshttps://zbmath.org/1541.540172024-09-27T17:47:02.548271Z"Ishiki, Yoshito"https://zbmath.org/authors/?q=ai:ishiki.yoshitoA topological space \(X\) is said to be strongly \(0\)-dimensional if for every pair \(A\), \(B\) of disjoint closed subsets of \(X\), there exists a clopen subset \(V\) of \(X\) such that \(A \subset V\) and \(V \cap B = \varnothing\).
Let \((X, \tau)\) be a metrizable topological space and let \(\mathrm{Met}(X)\) be the set of all metrics on \(X\) generating the topology \(\tau\). The author considers the supremum metric \(\mathcal{D}_X\) on \(\mathrm{Met}(X)\),
\[
\mathcal{D}_X(d, e) = \sup_{x, y \in X} |d(x, y) - e(x, y)|,
\]
and defines a set \(\mathrm{LI}(X) \subseteq \mathrm{Met}(X)\) as \(d \in \mathrm{LI}(X)\) if and only if \(d(x, y)\) and \(d(u, v)\) are linearly independent over \(\mathbb Q\) whenever \(d(x, y) \neq 0 \neq d(u, v)\) and \(\{x, y\} \neq \{u, v\}\).
The following theorem is the first main result.
\textbf{Theorem.} Let \(X\) be a strongly \(0\)-dimensional metrizable space with \(\mathrm{Card}(X) \leqslant \mathfrak{c}\). Let \(\varepsilon \in (0, \infty)\) and \(d \in \mathrm{Met}(X)\). Then there exists \(e \in \mathrm{LI}(X)\) such that \(\mathcal{D}_X(d, e) \leqslant \varepsilon\). Moreover, if \(X\) is completely metrizable, we can choose \(e\) as a complete metric.
Also the author obtains the following interesting corollaries.
\textbf{Corollary.} Let \(X\) be a strongly \(0\)-dimensional metrizable space with \(\mathrm{Card}(X) \leqslant \mathfrak{c}\) and let \(SR(X)\) be the set of all strongly rigid metrics \(d \in \mathrm{Met}(X)\). Then the set \(SR(X)\) is dense in \(\mathrm{Met}(X)\). Moreover, if \(X\) is \(\sigma\)-compact, then \(SR(X)\) is dense \(G_\sigma\) in \((\mathrm{Met}(X), \mathcal{D}_X)\).
\textbf{Corollary.} Let \(X\) be a strongly \(0\)-dimensional second-countable locally compact Hausdorff space. Then there exists \(d \in \mathrm{Met}(X)\) such that for every \(\xi \in X\), the map \(F_\xi\colon X \to [0, \infty)\) defined by \(F_\xi(x) = d(x, \xi)\) is a topological embedding.
The author also finds some conditions under which the set of all rigid metrics \(d \in \mathrm{Met}(X)\) is comeager in \((\mathrm{Met}(X), \mathcal{D}_X)\).
Reviewer: Oleksiy Dovgoshey (Turku)Geometric averages of partitioned datasetshttps://zbmath.org/1541.624092024-09-27T17:47:02.548271Z"Needham, Tom"https://zbmath.org/authors/?q=ai:needham.tom"Weighill, Thomas"https://zbmath.org/authors/?q=ai:weighill.thomasSummary: We introduce a method for jointly registering ensembles of partitioned datasets in a way which is both geometrically coherent and partition-aware. Once such a registration has been defined, one can group partition blocks across datasets in order to extract summary statistics, generalizing the commonly used order statistics for scalar-valued data. By modeling a partitioned dataset as an unordered \(k\)-tuple of points in a Wasserstein space, we are able to draw from techniques in optimal transport. More generally, our method is developed using the formalism of local Fréchet means in symmetric products of metric spaces. We establish basic theory in this general setting, including Alexandrov curvature bounds and a verifiable characterization of local means. Our method is demonstrated on ensembles of political redistricting plans to extract and visualize properties of the space of plans for a particular state, using North Carolina as our main example.Interactive theorem proving and finite projective planeshttps://zbmath.org/1541.684252024-09-27T17:47:02.548271Z"Ueberberg, Johannes"https://zbmath.org/authors/?q=ai:ueberberg.johannesFor the entire collection see [Zbl 0853.68027].Geometry machines: from AI to SMChttps://zbmath.org/1541.684262024-09-27T17:47:02.548271Z"Wang, Dongming"https://zbmath.org/authors/?q=ai:wang.dongmingSummary: The existing techniques and software tools for automated geometry theorem proving (GTP) are examined and reviewed. The underlying ideas of various approaches are explained with a set of selected examples. Comments and analyses are provided to illustrate the encouraging success of GTP which interrelates AI and SMC. We also present some technological applications of GTP and discuss its challenges ahead.
For the entire collection see [Zbl 0853.68027].Completely discretized, finite quantum mechanicshttps://zbmath.org/1541.810022024-09-27T17:47:02.548271Z"Carroll, Sean M."https://zbmath.org/authors/?q=ai:carroll.sean-mSummary: I propose a version of quantum mechanics featuring a discrete and finite number of states that is plausibly a model of the real world. The model is based on standard unitary quantum theory of a closed system with a finite-dimensional Hilbert space. Given certain simple conditions on the spectrum of the Hamiltonian, Schrödinger evolution is periodic, and it is straightforward to replace continuous time with a discrete version, with the result that the system only visits a discrete and finite set of state vectors. The biggest challenges to the viability of such a model come from cosmological considerations. The theory may have implications for questions of mathematical realism and finitism.Decoherence as a high-dimensional geometrical phenomenonhttps://zbmath.org/1541.810232024-09-27T17:47:02.548271Z"Soulas, Antoine"https://zbmath.org/authors/?q=ai:soulas.antoineSummary: We develop a mathematical formalism that allows to study decoherence with a great level generality, so as to make it appear as a geometrical phenomenon between reservoirs of dimensions. It enables us to give quantitative estimates of the level of decoherence induced by a purely random environment on a system according to their respectives sizes, and to exhibit some links with entanglement entropy.Homological quantum rotor codes: logical qubits from torsionhttps://zbmath.org/1541.810482024-09-27T17:47:02.548271Z"Vuillot, Christophe"https://zbmath.org/authors/?q=ai:vuillot.christophe"Ciani, Alessandro"https://zbmath.org/authors/?q=ai:ciani.alessandro"Terhal, Barbara M."https://zbmath.org/authors/?q=ai:terhal.barbara-mSummary: We formally define homological quantum rotor codes which use multiple quantum rotors to encode logical information. These codes generalize homological or CSS quantum codes for qubits or qudits, as well as linear oscillator codes which encode logical oscillators. Unlike for qubits or oscillators, homological quantum rotor codes allow one to encode both logical rotors and logical qudits in the same block of code, depending on the homology of the underlying chain complex. In particular, a code based on the chain complex obtained from tessellating the real projective plane or a Möbius strip encodes a qubit. We discuss the distance scalling for such codes which can be more subtle than in the qubit case due to the concept of logical operator spreading by continuous stabilizer phase-shifts. We give constructions of homological quantum rotor codes based on 2D and 3D manifolds as well as products of chain complexes. Superconducting devices being composed of islands with integer Cooper pair charges could form a natural hardware platform for realizing these codes: we show that the \(0 - \pi\) qubit as well as Kitaev's current-mirror qubit -- also known as the Möbius strip qubit -- are indeed small examples of such codes and discuss possible extensions.On the origin of black hole paradoxeshttps://zbmath.org/1541.810522024-09-27T17:47:02.548271Z"Hajian, Kamal"https://zbmath.org/authors/?q=ai:hajian.kamalSummary: Black hole firewall paradox is an inconsistency between four postulates in black hole physics: (1) the unitary evolution in quantum systems, (2) application of the semi-classical field theory in low curvature backgrounds, (3) statistical mechanical origin of the black hole entropy, and (4) the equivalence principle in the version of no drama for free-falling observers in the vicinity of the horizon. Based on the existence of the Hawking radiation for the static observers standing outside a Schwarzschild black hole, we show a direct contradiction between the postulates (2) and (4). If there is not a way out of this new problem, it implies the necessity of relaxing one of these two assumptions for resolving the black hole firewall paradox.Time evolution and the Schrödinger equation on time dependent quantum graphshttps://zbmath.org/1541.810562024-09-27T17:47:02.548271Z"Smilansky, Uzy"https://zbmath.org/authors/?q=ai:smilansky.uzy"Sofer, Gilad"https://zbmath.org/authors/?q=ai:sofer.giladSummary: The purpose of the present paper is to discuss the time dependent Schrödinger equation on a metric graph with time-dependent edge lengths, and the proper way to pose the problem so that the corresponding time evolution is unitary. We show that the well posedness of the Schrödinger equation can be guaranteed by replacing the standard Kirchhoff Laplacian with a magnetic Schrödinger operator with a harmonic potential. We then generalize the result to time dependent families of vertex conditions. We also apply the theory to show the existence of a geometric phase associated with a slowly changing quantum graph.
{{\copyright} 2024 The Author(s). Published by IOP Publishing Ltd}Trace distance ergodicity for quantum Markov semigroupshttps://zbmath.org/1541.810902024-09-27T17:47:02.548271Z"Bertini, Lorenzo"https://zbmath.org/authors/?q=ai:bertini.lorenzo-bertini"De Sole, Alberto"https://zbmath.org/authors/?q=ai:de-sole.alberto"Posta, Gustavo"https://zbmath.org/authors/?q=ai:posta.gustavoSummary: We discuss the quantitative ergodicity of quantum Markov semigroups in terms of the trace distance from the stationary state, providing a general criterion based on the spectral decomposition of the Lindblad generator. We then apply this criterion to the bosonic and fermionic Ornstein-Uhlenbeck semigroups and to a family of quantum Markov semigroups parametrized by semisimple Lie algebras and their irreducible representations, in which the Lindblad generator is given by the adjoint action of the Casimir element.Quantum heat machines enabled by twisted geometryhttps://zbmath.org/1541.820012024-09-27T17:47:02.548271Z"Filgueiras, Cleverson"https://zbmath.org/authors/?q=ai:filgueiras.cleverson"Rojas, Moises"https://zbmath.org/authors/?q=ai:rojas.moises"Silva, Edilberto O."https://zbmath.org/authors/?q=ai:silva.edilberto-o"Romero, Carlos"https://zbmath.org/authors/?q=ai:romero.carlos-santiuste|romero.carlos|romero.carlos-barronSummary: In this paper, we analyze the operation of an Otto cycle heat machine driven by a non-interacting two-dimensional electron gas on a twisted geometry. We show that due to both the energy quantization on this structure and the adiabatic transformation of the number of complete twists per unit length of a helicoid, the machine performance in terms of output work, efficiency, and operation mode can be altered. We consider the deformations as in a spring, which is either compressed or stretched from its resting position. The realization of classically inconceivable Otto machines, with an incompressible sample, can be realized as well. The energy-level spacing of the system is the quantity that is being either compressed or stretched. These features are due to the existence of an effective geometry-induced quantum potential which is of pure quantum-mechanical origin.