Recent zbMATH articles in MSC 51https://zbmath.org/atom/cc/512021-11-25T18:46:10.358925ZWerkzeugBook review of: S. Alexander et al., An invitation to Alexandrov geometry. CAT(0) spaces.https://zbmath.org/1472.000212021-11-25T18:46:10.358925Z"Kunzinger, M."https://zbmath.org/authors/?q=ai:kunzinger.michaelReview of [Zbl 1433.53065].Book review of: U. Daepp et al., Finding ellipses. What Blaschke products, Poncelet's theorem, and the numerical range know about each other.https://zbmath.org/1472.000422021-11-25T18:46:10.358925Z"Zeytuncu, Yunus E."https://zbmath.org/authors/?q=ai:zeytuncu.yunus-ergynReview of [Zbl 1419.51001].Alfonso's \textit{Rectifying the curved}. A fourteenth-century Hebrew geometrical-philosophical treatisehttps://zbmath.org/1472.010122021-11-25T18:46:10.358925Z"Glasner, Ruth"https://zbmath.org/authors/?q=ai:glasner.ruth"Baraness, Avinoam"https://zbmath.org/authors/?q=ai:baraness.avinoamIn the book under review, the authors -- both from the Department of History, Philosophy and Sociology of Sciences at The Hebrew University of Jerusalem (the second author defended his thesis on the present medieval treatise in 2018) -- propose the Hebrew edition (Chapter 3, pp. 201--248), the English translation with commentaries (Chapter 2, pp. 33--199, the main part of this book) on the \textit{Sefer Meyasher 'Aqov} (The rectifying of the curved), a Hebrew medieval geometrical-philosophical text on the foundation of geometry. The authors precise, in their preface, that it is an enigmatic text, a ``mixture of two literary genres: philosophical discussion and formal, Euclidean-type geometrical writing'' (p. vii).
In the first chapter (pp. 1--32), the authors give us a complete introduction with information about the manuscript, historiographical materials and a general presentation of the edited treatise (author and his intellectual environment, mathematical and philosophical content, author's language and, finally, the conventions for the edition and the translation).
Gita Gluskina, from Russia, established the pioneering work on this text, after Solomon Luria became interested in it in the 1930s. Therefore, two papers were published in Russian by \textit{G. Gluskina} [``On the unpublished medieval treatise Meyasher 'Aqov, in the British Museum'' (Russ.), Palestinskiĭ Sb. 25, 152 (1975)] and [``On the authorship of the mathematical treatise Meyasher 'Aqov'' (Russ.), ibid. 26, 66--75 (1978)]. After that, only two studies have been carried out specifically on this text: one by \textit{T. Lévy} on the parallels postulate [Arab. Sci. Philos. 2, No. 1, 39--82 (1992; Zbl 0826.01004)] and the other by \textit{Y. T. Langermann} on the squaring of the lunes [Hist. Math. 23, No. 1, 31--53 (1996; Zbl 0861.01008)]. I still have to notice the presentation by the second author of several geometrical propositions in the sourcebook in history of mathematics edited by \textit{V. J. Katz} (ed.) et al. [Sourcebook in the mathematics of medieval Europe and North Africa. Princeton, NJ: Princeton University Press (2016; Zbl 1356.01001)].
The text of the \textit{Sefer Meyasher 'Aqov} is preserved in a single manuscript held at the British Library. Mordekhai Finzi, the famous 15th-century commentator of the \textit{Algebra} of \textit{Abū Kāmil} [The algebra of Abū Kāmil. Kitāb fī al-jābr wa'l-muqābala in a commentary by Mordecai Finzi (Hebrew). Madison-Milwaukee-London: The University of Wisconsin Press (1966; Zbl 0178.29602)] from the scientific Mantua community, would be the owner of this manuscript. It is a ``product of the Jewish Iberian tradition'' (p.~8). In particular, the mathematical vocabulary was based on that of Abraham Bar-Ḥiyya (\textit{fl}. in the 12th century), a Jewish mathematician and translator from Barcelona.
The author of the \textit{Sefer Meyasher 'Aqov} was identified by Gita Gluskina, and subsequently by \textit{G. Freudenthal} [Science in the medieval Hebrew and Arabic traditions. Aldershot: Ashgate (2005)], as Abner of Burgos (\textit{ca}. 1260--1347), the Jewish apostate otherwise known as Alfonso of Valladolid after his conversion to Christianity (about 1320). The authors are convinced by this first identification (p.~3).
The intellectual environment and the description of knowledge available in the \textit{Sefer Meyasher 'Aqov} are well described. The purpose is really clear with a distinction between Jewish (Hebrew), Arabic and Christian (Latin) backgrounds. We also have a quite impressive list of authors explicitly cited by name, whether Greek, Arab, Jewish, Christian. It is, for me, one of the non-negligible interests of this text. In addition, the authors also mention, for each category, the other authors who could have inspired Alfonso of Valladolid, without being cited. This list is obviously very useful.
The use of motion and superposition in geometry is central in this treaty, without Alfonso neglecting the work of his predecessors. The \textit{Sefer Meyasher 'Aqov} is divided into five major parts (the main objectives of which are detailed by the author himself at the very beginning of the book). Nevertheless, unfortunately, the single known manuscript is incomplete, thus we do not have the entire text of the \textit{Sefer Meyasher 'Aqov}. Parts I and II are complete. The first one could be considered as an introductive part of the whole treatise. Alfonso inquires ``whether the imaging of equally divided motion can qualify as one of the first principles of geometry'' (p.~33). He ends this first part justifying the title of his treatise by its main objectives (pp. 82--83). The second part is ``on accidents associated with motions, which prevent its imagining from being a first principle of geometry, and how several early sages were confused by it'' (p.~84). Part III is a compilation of 33 geometrical propositions, 8 of which are missing (there is a lacuna in the manuscript from the end of Proposition 1 to Proposition 9 included), on ``rectilinear magnitudes and areas that are useful'' in geometry (p.~33). For the end of the treaty, we have only the first 32 lines of Part IV and nothing for Part V.
The modern commentaries are directly included in the English translation, throughout the pages, which allows a comprehensive reading of the original text and associated comments. They are announced, for example, by ``commentary'' in general or ``mathematical commentary'' more specifically, by ``note'' or by ``terms'' when the authors wish to have a lexical discussion. All these comments are useful, even essential, to fully understand the text, which is sometimes difficult to access, in particular due to the Alfonso's philosophico-geometric approach. The Hebrew text end the book before a complete Hebrew-English glossary (pp. 249--261) with reference to the paragraph in which the term is used, an extensive bibliography distinguishing primary sources from secondary studies (pp. 263--274), and a general index (pp. 275--282).
Even if I am not qualified to give an opinion on the edition of the Hebrew text, the book under review is very complete and useful for historians of medieval mathematics. It offers us, in a modern language more accessible than Russian for our community, one of the rare geometric texts of the Hebrew medieval tradition with very appreciable mathematical comments.Graphs cospectral with \(\operatorname{NU}(n + 1,q^2)\), \(n \neq 3\)https://zbmath.org/1472.050972021-11-25T18:46:10.358925Z"Ihringer, Ferdinand"https://zbmath.org/authors/?q=ai:ihringer.ferdinand"Pavese, Francesco"https://zbmath.org/authors/?q=ai:pavese.francesco"Smaldore, Valentino"https://zbmath.org/authors/?q=ai:smaldore.valentinoSummary: Let \(\mathcal{H}(n, q^2)\) be a non-degenerate Hermitian variety of \(\operatorname{PG}(n, q^2)\), \(n \geq 2\). Let \(\operatorname{NU}(n + 1, q^2)\) be the graph whose vertices are the points of \(\operatorname{PG}(n, q^2) \setminus \mathcal{H}(n, q^2)\) and two vertices \(P_1\), \(P_2\) are adjacent if the line joining \(P_1\) and \(P_2\) is tangent to \(\mathcal{H}(n, q^2)\). Then \(\operatorname{NU}(n + 1, q^2)\) is a strongly regular graph. In this paper we show that \(\operatorname{NU}(n + 1, q^2)\), \(n \neq 3\), is not determined by its spectrum.The simplex geometry of graphshttps://zbmath.org/1472.051132021-11-25T18:46:10.358925Z"Devriendt, Karel"https://zbmath.org/authors/?q=ai:devriendt.karel"Van Mieghem, Piet"https://zbmath.org/authors/?q=ai:van-mieghem.pietSummary: Graphs are a central object of study in various scientific fields, such as discrete mathematics, theoretical computer science and network science. These graphs are typically studied using combinatorial, algebraic or probabilistic methods, each of which highlights the properties of graphs in a unique way. Here, we discuss a novel approach to study graphs: the simplex geometry (a simplex is a generalized triangle). This perspective, proposed by \textit{M. Fiedler} [Matrices and graphs in geometry. Cambridge: Cambridge University Press (2011; Zbl 1225.51017)], introduces techniques from (simplex) geometry into the field of graph theory and conversely, via an exact correspondence. We introduce this graph-simplex correspondence, identify a number of basic connections between graph characteristics and simplex properties, and suggest some applications as example.A short and easy proof of Morley's congruence theoremhttps://zbmath.org/1472.110072021-11-25T18:46:10.358925Z"Aebi, Christian"https://zbmath.org/authors/?q=ai:aebi.christianSummary: Morley's theorem (1899): The three points of intersection of adjacent trisectors of the angles of any triangle form an equilateral triangle.Improved dispersion bounds for modified Fibonacci latticeshttps://zbmath.org/1472.112152021-11-25T18:46:10.358925Z"Kritzinger, Ralph"https://zbmath.org/authors/?q=ai:kritzinger.ralph"Wiart, Jaspar"https://zbmath.org/authors/?q=ai:wiart.jasparConsider point sets \(\mathcal P\) in the unit square \([0,1]^2\) consisting of \(N\) not necessarily distinct elements and let \(\mathcal B\) be the set of all axes-parallel boxes in the unit square. The authors study dispersion of \(\mathcal P\) defined as \( \text{disp }(\mathcal P):=\sup_{B\in\mathcal B, B\cap\mathcal P=\emptyset}\lambda(B) \), where \(\lambda(B)\) denotes the area of the box \(B\). Known previous results: \textit{A. Dumitrescu} and \textit{M. Jiang} [Algorithmica 66, No. 2, 225--248 (2013; Zbl 1262.68186)]
proved that \( \text{disp }(\mathcal P)\ge\max(1/(N+1),5/(4(N+5))). \) \textit{S. Breneis} and \textit{A. Hinrichs} [Radon Ser. Comput. Appl. Math. 26, 117--132 (2020; Zbl 1471.11050)] constructed points with small dispersion using the Fibonacci numbers \(F_m\). The dispersion of the Fibonacci lattice \( \mathcal F_m=\{(k/F_m,\{kF_{m-2}/F_m\}):k=0,1,2,\dots,F_m-1\} \) is given by \( \text{disp }\mathcal F_m=(2(F_m-1)/F^2_m) \) for \(m\ge8\), where \(\{x\}\) denotes the fractional part of \(x\). Then the limit \(\lim_{m\to\infty}|\mathcal F_m|\text{disp }(\mathcal F_m)=2\). The central result of this paper is replacing the limit 2 by a smaller constant 1.894427\dots for the modified Fibonacci lattice \(\widetilde{\mathcal F}_m\) defined in the following way: \par \(\pi(k)=kF_{m-2}\pmod {F_m}\), \par \(s(i)=(\sqrt{5}+1)/2 \text{ if } \pi(i)<\pi(i+1) \text{ and } 1 \text{ otherwise }\), \par \(L=\sum_{i=0}^{F_m-1}s(i)\), \par \(x_k=\sum_{i=0}^{k-1}, k=0,1,\dots,F_m-1\), \par \(\widetilde{\mathcal F}_m=\{(x_k/L,x_{\pi(k)}/L),k=0,1,\dots,F_m-1\}\). \par\noindent In this case, the authors prove \par \(\lim_{m\to\infty}|\widetilde{\mathcal F}_m|\text{disp }(\widetilde{\mathcal F}_m) =1+\frac{2}{\sqrt{5}}\).
For proof they use, e.g., \(F^2_{m-2}+F^2_m-3F_mF_{m-2}=(-1)^m\). Using the theory of uniform distribution of sequences it is also conjectured that the Fibonacci lattice \(\mathcal F_m\) have a smallest possible \(L_2\) discrepancy, since for \(m\le 7\) this is known. Note that in uniform distribution theory the \(s\)-dimensional dispersion is defined as follows: If \(\mathbf x_1,\mathbf x_2,\dots,\mathbf x_N\) belong to \([0,1]^s\) then the dispersion \(d_N\) of \(\mathbf x_n\) is defined by \(d_N(\mathbf x_1,\dots,\mathbf x_N)=\sup_{\mathbf x\in[0,1]^s} \min_{1\le n\le N}|\mathbf x-\mathbf x_n|\), where \(|\mathbf x-\mathbf x_n|\) is the Euclidean distance. Basic results of such dispersion are in [\textit{H. Niederreiter}, Random number generation and quasi-Monte Carlo methods. Philadelphia, PA: SIAM (1992; Zbl 0761.65002)].Potential theory on minimal hypersurfaces. I: Singularities as Martin boundarieshttps://zbmath.org/1472.300272021-11-25T18:46:10.358925Z"Lohkamp, Joachim"https://zbmath.org/authors/?q=ai:lohkamp.joachimThe author develops a detailed potential theory on (almost) minimizing hypersurfaces applicable to large classes of linear elliptic second-order operators.
Let \(H\) be an (almost) minimizing hypersurface containing the singularity set \(\Sigma \subset H\). By \(\mathcal S\)-uniformity, we can regard \(H \setminus \Sigma\) as a generalized convex set and \(\Sigma\) as its boundary. Then the author proves a generalized boundary Harnack inequality, and use it to deduce other interesing results concerning Martin theory, the Dirichlet problem and Hardy inequalities.On commuting billiards in higher-dimensional spaces of constant curvaturehttps://zbmath.org/1472.370312021-11-25T18:46:10.358925Z"Glutsyuk, Alexey"https://zbmath.org/authors/?q=ai:glutsyuk.alexey-aSummary: We consider two nested billiards in \(\mathbb{R}^d\), \(d\geq3\), with \(C^2\)-smooth strictly convex boundaries. We prove that if the corresponding actions by reflections on the space of oriented lines commute, then the billiards are confocal ellipsoids. This together with the previous analogous result of the author in two dimensions solves completely the commuting billiard conjecture due to \textit{S. Tabachnikov} [Geom. Dedicata 53, No. 1, 57--68 (1994; Zbl 0813.52003)]. The main result is deduced from the classical theorem due to \textit{M. Berger} [Geometry. I, II. Transl. from the French by M. Cole and S. Levy. Berlin: Springer (2009; Zbl 1153.51001)]
which says that in higher dimensions only quadrics may have caustics. We also prove versions of Berger's theorem and the main result for billiards in spaces of constant curvature (space forms).Nonasymptotic densities for shape reconstructionhttps://zbmath.org/1472.490622021-11-25T18:46:10.358925Z"Ibrahim, Sharif"https://zbmath.org/authors/?q=ai:ibrahim.sharif"Sonnanburg, Kevin"https://zbmath.org/authors/?q=ai:sonnanburg.kevin"Asaki, Thomas J."https://zbmath.org/authors/?q=ai:asaki.thomas-j"Vixie, Kevin R."https://zbmath.org/authors/?q=ai:vixie.kevin-rSummary: In this work, we study the problem of reconstructing shapes from simple nonasymptotic densities measured only along shape boundaries. The particular density we study is also known as the integral area invariant and corresponds to the area of a disk centered on the boundary that is also inside the shape. It is easy to show uniqueness when these densities are known for all radii in a neighborhood of \(r = 0\), but much less straightforward when we assume that we only know the area invariant and its derivatives for only one \(r > 0\). We present variations of uniqueness results for reconstruction (modulo translation and rotation) of polygons and (a dense set of) smooth curves under certain regularity conditions.The Reye configuration in a \(\mathrm{MOL}(6)\)https://zbmath.org/1472.510012021-11-25T18:46:10.358925Z"Betten, Dieter"https://zbmath.org/authors/?q=ai:betten.dieterIn the paper under review the author shows that even if a \(\mathrm{MOL}(6)\) (pair of mutually othogonal latin squares) does not exist, some interesting geometries may be described as its substructures, if one considers a \(\mathrm{MOL}(6)\) as a suitable \(36\)-subset of the four dimensional cube of edge length \(6\) and uses decompositions of the \(4\)-cube.
The author defines the structural invariant \(\mu_4\) as {\em the maximal number of distinguished elements in subcubes of edge lenght }\(3\) and proves that for evey \(\mathrm{MOL}(6)\) one has \(\mu_4 = 6\). Then, the structure of the subcube with \(\mu_4 =6\) is studied and, in the last part of the paper, by weakening the decompositions obtained he gets the Reye configuration and the \(12_3\)-symmetric configuration with highest group order \(72\).
Also, he gets another proof of the non existence of a \(\mathrm{MOL}(6)\).An infinite family of \(m\)-ovoids of \(Q(4,q)\)https://zbmath.org/1472.510022021-11-25T18:46:10.358925Z"Feng, Tao"https://zbmath.org/authors/?q=ai:feng.tao"Tao, Ran"https://zbmath.org/authors/?q=ai:tao.ranLet \(Q(4,q)\) denote the point line geometry associated to a non-singular quadratic form on the \(5\)-dimensional vector space over the finite field of order \(q\), \(q=p^h\), \(p\) prime, \(h \geq 1\). It is well known that the Witt index of such a form equals \(2\). The singular subspaces of dimension \(1\) and \(2\), are considered as points and lines, respectively, in the projective space \(\mathrm{PG}(4,q)\), and it is well known that this point line geometry is an example of a generalized quadrangle, or order \(q\), embedded in \(\mathrm{PG}(4,q)\). This GQ is one of the so-called finite classical generalized quadrangles.
Let \(\mathcal{S}\) be a GQ. An {\em ovoid} of \(\mathcal{S}\) is a set \(\mathcal{O}\) of points such that every line of \(\mathcal{S}\) contains exactly one point of \(\mathcal{O}\). Let \(m \geq 1\). An {\em \(m\)-ovoid} of \(\mathcal{S}\) is a set \(\mathcal{O}\) of points such that every line contains exactly \(m\) points of \(\mathcal{O}\). The concept of an \(m\)-ovoid might look as a straightforward generalization of an ovoid. However, as ovoids of GQs are relatively rare, \(m\)-ovoids are quite exceptional. Some non-existence results are available, but construction results are, so far, quite rare. The nice paper under review provides a construction of \(\frac{q-1}{2}\)-ovoids of \(Q(4,q)\), \(q \equiv 1 \pmod{4}\), \(q > 5\), admitting \(C_{\frac{q^2-1}{2}} \rtimes C_2\) as automorphism group.Equivalence classes of Niho bent functionshttps://zbmath.org/1472.510032021-11-25T18:46:10.358925Z"Abdukhalikov, Kanat"https://zbmath.org/authors/?q=ai:abdukhalikov.kanat-sAuthor's abstract: Equivalence classes of Niho bent functions are in one-to-one correspondence with equivalence classes of ovals in a projective plane. Since a hyperoval can produce several ovals, each hyperoval is associated with several inequivalent Niho bent functions. For all known types of hyperovals we described the equivalence classes of the corresponding Niho bent functions. For some types of hyperovals the number of equivalence classes of the associated Niho bent functions are at most 4. In general, the number of equivalence classes of associated Niho bent functions increases exponentially as the dimension of the underlying vector space grows. In small dimensions the equivalence classes were considered in detail.On line colorings of finite projective spaceshttps://zbmath.org/1472.510042021-11-25T18:46:10.358925Z"Araujo-Pardo, Gabriela"https://zbmath.org/authors/?q=ai:araujo-pardo.gabriela"Kiss, György"https://zbmath.org/authors/?q=ai:kiss.gyorgy"Rubio-Montiel, Christian"https://zbmath.org/authors/?q=ai:rubio-montiel.christian"Vázquez-Ávila, Adrián"https://zbmath.org/authors/?q=ai:vazquez-avila.adrianA \textit{colouring} of a finite linear space \(S\) is an assignment of the lines of S to a set of colours, say \([k]=\{1,2,\ldots,k\}\). A colouring of \(S\) is called \textit{proper} if any two intersecting lines have a different color, while a colouring is called \textit{complete} if each pair of colours appears on at least one point of \(S\).
The \textit{chromatic index} \(\chi'(S)\) is the smallest \(k\) such that there exists a proper colouring with \(k\) colours, the \textit{achromatic index} \(\alpha'(S)\) is the largest \(k\) such that there exists a proper and complete colouring with \(k\) colours, and the \textit{pseudoachromatic index} \(\phi'(S)\) is the largest \(k\) such that there exist a complete (not necessarily proper) colouring with \(k\) colours.
It is clear that \(\chi'(S)\leq \alpha'(S)\leq \psi'(S)\) and it is also not too hard to deduce that for \(S=\mathrm{PG}(2,q)\), \(\chi'(S)= \alpha'(S)= \psi'(S)=q^2+q+1\). At present, these values are not known for \(\mathrm{PG}(n,q)\), \(n\geq 3\).
In this paper, the authors contribute to this study by showing the following:, where \(\theta_n\) denotes the number of points in \(\mathrm{PG}(n,q)\).
They show that for \(n=3.2^i-1\), \(i>0\), \(\alpha'(S)>c_n\frac{1}{q}\theta_n^{\frac{4}{3}+\frac{1}{3n}}\) where \(\frac{1}{2^{7/5}}<c_n<\frac{1}{2^{4/3}}\) is a constant depending only on \(n\). Furthermore, \(\psi'(S)<\frac{1}{q}\theta_n^{3/2}\) for all \(n\geq 2\).
The proof of the first statement hinges on the construction of a particular geometric \(2^i-1\)-spread while the proof of the second statement is an easy counting argument.
Finally, the authors study the case \(\mathrm{PG}(3,2)\) in detail and provide a quite intricate computer-free proof of the fact that \(\psi'(\mathrm{PG}(3,2))=18\).Opposition diagrams for automorphisms of small spherical buildingshttps://zbmath.org/1472.510052021-11-25T18:46:10.358925Z"Parkinson, James"https://zbmath.org/authors/?q=ai:parkinson.james"Van Maldeghem, Hendrik"https://zbmath.org/authors/?q=ai:van-maldeghem.hendrik-jSummary: An automorphism \(\theta\) of a spherical building \(\Delta\) is called \textit{capped} if it satisfies the following property: if there exist both type \(J_1\) and \(J_2\) simplices of \(\Delta\) mapped onto opposite simplices by \(\theta\) then there exists a type \(J_1\cup J_2\) simplex of \(\Delta\) mapped onto an opposite simplex by~\(\theta\). In previous work we showed that if \(\Delta\) is a thick irreducible spherical building of rank at least \(3\) with no Fano plane residues then every automorphism of \(\Delta\) is capped. In the present work we consider the spherical buildings with Fano plane residues (the \textit{small buildings}). We show that uncapped automorphisms exist in these buildings and develop an enhanced notion of ``opposition diagrams'' to capture the structure of these automorphisms. Moreover we provide applications to the theory of ``domesticity'' in spherical buildings, including the complete classification of domestic automorphisms of small buildings of types \(\mathsf{F}_4\) and \(\mathsf{E}_6\).Some geometric characterizations of Pythagorean and Euclidean fieldshttps://zbmath.org/1472.510062021-11-25T18:46:10.358925Z"Gröger, Detlef"https://zbmath.org/authors/?q=ai:groger.detlefThe results presented in this note concern Euclidean planes. These structures, defined with incidence and congruence relations, can be considered as coordinate planes derived from a separable quadratic extension \(E\) over a (commutative) field \(K\) of characteristic \(\neq 2\), endowed with an involutory automorphism.
Setting \(K^*:=K \setminus \{0\}\), \ \(K^{(2)}:=\{\xi^2 : \xi \in K^*\}\), \ \(K_0^{(2)}:=K^{(2)} \cup \{0\}\), the field \(K\) is called \textit{Pythagorean} if \(K^{(2)}+ K^{(2)} \subseteq K^{(2)}\), is called \textit{semi Euclidean} if \(K^*= K^{(2)}\dot\cup -K^{(2)}\) (disjoint union) and is called \textit{Euclidean} if \(K\) is Pythagorean and semi Euclidean.
The author characterizes the algebraic properties: \(-1\) is a square in \(K\), \(K\) is Pythagorean and \(K\) is Euclidean in terms of incidence conditions of lines and circles in \(E\). Moreover, he characterizes these properties by comparing four betweenness relations.A novel construction of Urysohn universal ultrametric space via the Gromov-Hausdorff ultrametrichttps://zbmath.org/1472.510072021-11-25T18:46:10.358925Z"Wan, Zhengchao"https://zbmath.org/authors/?q=ai:wan.zhengchaoThis paper shows that the collection of all compact ultrametric spaces \(\mathcal U\) equipped with the Gromov-Hausdorff metric is universal. This means that any Polish ultrametric space is isometrically embeddable into \(\mathcal U\). This space is in addition ultra-homogeneous which means that for any finite ultrametric space \(B\), a subset \(A\subset B\) and an isometric embedding \(\varphi:A\to\mathcal U\) there exists an isometric extension \(\psi:B\to \mathcal U\) of \(\varphi\).
If \(R\subset \mathbb R_{\ge 0}\) is a countable set containing \(0\) an ultrametric space \(X\) is called \(R\)-ultrametric if \(d\) takes only values in \(R\). The author shows the collection of all compact \(R\)-ultrametric spaces equipped with the Gromov-Hausdorff metric is a Polish ultrametric space universal for Polish \(R\)-ultrametric spaces and homogeneous for all finite \(R\)-ultrametric spaces.Closed subsets of a \(\mathrm{CAT}(0)\) 2-complex are intrinsically \(\mathrm{CAT}(0)\)https://zbmath.org/1472.510082021-11-25T18:46:10.358925Z"Ricks, Russell"https://zbmath.org/authors/?q=ai:ricks.russellSummary: Let \(\kappa\leq 0\), and let \(X\) be a complete, locally finite \(\mathrm{CAT}(\kappa)\) polyhedral \(2\)-complex \(X\), each face with constant curvature \(\kappa\). Let \(E\) be a closed, rectifiably connected subset of \(X\) with trivial first singular homology. We show that \(E\), under the induced path metric, is a complete \(\mathrm{CAT}(\kappa)\) space.On mappings that preserve Fermat-Torricelli pointshttps://zbmath.org/1472.510092021-11-25T18:46:10.358925Z"Demirel, Oğuzhan"https://zbmath.org/authors/?q=ai:demirel.oguzhanIn this paper, the author presented a new characterization of affine transformations by use of Fermat-Torricelli points of triangles. The author denoted with $\Delta$ the set of all triple points $\{A,B,C\}$ in \(\mathbb{R}^n\) such that the largest angle of the triangle ABC is less than \(2\pi/3\). The author proved that if a mapping \(f:\mathbb{R}^n\to\mathbb{R}^n\) preserves the Fermat-Torricelli points of the triangles in $\Delta$, then $f$ is an affine transformation.The rectangular peg problemhttps://zbmath.org/1472.510102021-11-25T18:46:10.358925Z"Greene, Joshua Evan"https://zbmath.org/authors/?q=ai:greene.joshua-evan"Lobb, Andrew"https://zbmath.org/authors/?q=ai:lobb.andrewUsing a theorem of \textit{V. Shevchishin} [Izv. Math. 73, No. 4, 797--859 (2009); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 73, No. 4, 153--224 (2009; Zbl 1196.57021)] and \textit{S. Yu. Nemirovski} [Izv. Math. 66, No. 1, 151--164 (2002); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 66, No. 1, 153--166 (2002; Zbl 1041.53049)] -- that the Klein bottle does not admit a smooth Lagrangian embedding in \({\mathbb C}^2\) --, a proposition proved in this paper regarding a Lagrangian smoothing, involving a \(4\)-manifold with a symplectic form, and the equivariant Darboux-Weinstein theorem, the authors prove the long-standing conjecture that
for every smooth Jordan curve \(\gamma\) and rectangle \(R\) in the Euclidean plane, there exists a rectangle similar to \(R\) whose vertices lie on \(\gamma\).
The paper ends with an in-depth history of the problem, going back to Toeplitz in 1911.Inscribed rectangles in a smooth Jordan curve attain at least one third of all aspect ratioshttps://zbmath.org/1472.510112021-11-25T18:46:10.358925Z"Hugelmeyer, Cole"https://zbmath.org/authors/?q=ai:hugelmeyer.coleSummary: We prove that for every smooth Jordan curve \(\gamma\), if \(X\) is the set of all \(r \in [0,1]\) so that there is an inscribed rectangle in \(\gamma\) of aspect ratio \(\tan(r\cdot\pi/4)\), then the Lebesgue measure of \(X\) is at least \(1/3\). To do this, we study sets of disjoint homologically nontrivial projective planes smoothly embedded in \(\mathbb{R}\times\mathbb{R}P^3\). We prove that any such set of projective planes can be equipped with a natural total ordering. We then combine this total ordering with Kemperman's theorem in \(S^1\) to prove that \(1/3\) is a sharp lower bound on the probability that a Möbius strip filling the \((2,1)\)-torus knot in the solid torus times an interval will intersect its rotation by a uniformly random angle.Seidel's conjectures in hyperbolic 3-spacehttps://zbmath.org/1472.510122021-11-25T18:46:10.358925Z"Cussy, Omar Chavez"https://zbmath.org/authors/?q=ai:cussy.omar-chavez"Grossi, Carlos H."https://zbmath.org/authors/?q=ai:grossi.carlos-hThis paper answers several conjectures raised by \textit{J. J. Seidel} [Stud. Sci. Math. Hung. 21, 243--249 (1986; Zbl 0561.52010)] for hyperbolic \(3\)-space.
Given a \(4\)-dimensional \({\mathbb R}\)-linear space \(V\) equipped with a bilinear symmetric form of signature \(---+\), the hyperbolic \(3\)-space is the open ball of positive points \({\mathbb H}^3 = \{\mathbf{p} \in {\mathbb P}V : \langle p, p\rangle > 0\}\), with \(p\in V\) denoting the representative of a point \(\mathbf{p}\in {\mathbb P}V\).
If \(S := (\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{v}_4)\), with \(\mathbf{v}_i\in \partial {\mathbb H}^3\) is an ideal tetrahedron, choosing representatives \(v_i \in V\) we obtain a Gram matrix \(G\) of the vertices of \(S\), where \(G:= [\langle v_i, v_j\rangle\). Among all the Gram matrices of the vertices of \(S\), one, \(G_{ds}\), is doubly stochastic. Seidel's first conjecture states that the volume of \(S\) is determined by a preferably natural algebraic functions of the entries of \(G_{ds}\). The authors prove more than was expected, by obtaining, using Milnor's volume formula for an ideal tetrahedron, an explicit formula for the volume of an ideal tetrahedron as a function of the permanent and the determinant of \(G_{ds}\).
While Seidel's fourth conjecture says that the volume is a decreasing function of the permanent, the authors show that this not the case, that the volume is in fact decreasing in the determinant and increasing in (the square root of) the permanent, whenever the determinant is non-zero.
They also prove Seidel's third conjecture for hyperbolic \(3\)-space.Problem of shadow in the Lobachevskii spacehttps://zbmath.org/1472.510132021-11-25T18:46:10.358925Z"Kostin, A. V."https://zbmath.org/authors/?q=ai:kostin.andrey-viktorovichIn [\textit{G. Khudaiberganov}, ``On a homogeneous polynomially convex hull of the union of balls'', VINITI 21, 1772--1785 (1982)], it was proved that it is sufficient to have two disks in order to guarantee that any straight line passing through the center of a circle in the Euclidean plane crosses at least one disk centered in this circle (or, in other words, in order that the center of the circle belong to the convex 1-hull of the disks [\textit{Y. B. Zelinskii} et al., ``Problem of shadow and related problems,'' Proc. Int. Geom. Cent. 9, No. 3--4, 50--58 (2016; \url{doi:10.15673/tmgc.v9i3-4.319 })]). The author present the main steps of this generalization with indication of all distinctions caused by the specific features of the hyperbolic plane. This problem can be regarded as a problem of finding conditions guaranteeing that points belong to a generalized convex hull of the family of balls.A note on empty balanced tetrahedra in two-colored point sets in \(\mathbb{R}^3\)https://zbmath.org/1472.510142021-11-25T18:46:10.358925Z"Díaz-Bañez, José-Miguel"https://zbmath.org/authors/?q=ai:diaz-banez.jose-miguel"Fabila-Monroy, Ruy"https://zbmath.org/authors/?q=ai:fabila-monroy.ruy"Urrutia, Jorge"https://zbmath.org/authors/?q=ai:urrutia.jorge-lSummary: Let \(S\) be a set of \(n\) red and \(n\) blue points in general position in \(\mathbb{R}^3\). Let \(\tau\) be a tetrahedron with vertices in \(S\). We say that \(\tau\) is \textit{empty} if it does not contain any point of \(S\) in its interior. We say that \(\tau\) is \textit{balanced} if two of its vertices are blue, and two of its vertices are red. In this paper we show that \(S\) spans \(\Omega( n^{5 / 2})\) empty balanced tetrahedra.Realizations of the \(120\)-cellhttps://zbmath.org/1472.510152021-11-25T18:46:10.358925Z"McMullen, Peter"https://zbmath.org/authors/?q=ai:mcmullen.peterRealizations provide geometric pictures of abstract regular polytopes, and thereby help to investigate their structures. In several articles and monographs [the author, Geometric regular polytopes. Cambridge: Cambridge University Press (2020; Zbl 1454.51002); the author and \textit{E. Schulte}, Abstract regular polytopes. Cambridge: Cambridge University Press (2002; Zbl 1039.52011)] the realization spaces of all the classical regular polytops have been described except that of the 120-cell \(\{5,3,3\}\). For several reasons, the realization domain of the 120-cell is more complicated. In this article, the author describes the realization of the 120-cell by imploying many of the techniques of the theory and the notion of cosine vectors introduced here for the first time.
For the entire collection see [Zbl 1467.52001].On geometric construction of some power meanshttps://zbmath.org/1472.510162021-11-25T18:46:10.358925Z"Hoibakk, Ralph"https://zbmath.org/authors/?q=ai:hoibakk.ralph"Lukkassen, Dag"https://zbmath.org/authors/?q=ai:lukkassen.dag"Persson, Lars-Erik"https://zbmath.org/authors/?q=ai:persson.lars-erik"Meidell, Annette"https://zbmath.org/authors/?q=ai:meidell.annetteSummary: In the homogenization theory, there are many examples where the effective conductivities of composite structures are power means of the local conductivities. The main aim of this paper is to initiate research concerning geometric construction of some power means of three or more variables. We contribute by giving methods for the geometric construction of the harmonic mean \( P_{-1} \) and the arithmetic mean \(P_{1}\) of three variables \(a,b\) and \(c\).The Feuerbach theorem and cyclography in universal geometryhttps://zbmath.org/1472.510172021-11-25T18:46:10.358925Z"Beare, William"https://zbmath.org/authors/?q=ai:beare.william"Wildberger, N. J."https://zbmath.org/authors/?q=ai:wildberger.norman-johnThe nine-point circle of a triangle passes through the three midpoints of the sides, the three feet of the altitudes, and the three midpoints between the vertices and the orthocentre. The Feuerbach theorem tells us that the nine-point circle of a triangle is tangent to the four incircles (excircles) of that triangle. In the paper under review, the authors obtain the so-called ``oriented Feuerbach theorem'' generalizing the Feuerbach theorem to finite fields by using the purely algebraic approach of rational trigonometry and universal geometry.Existence of continuous right inverses to linear mappings in finite-dimensional geometryhttps://zbmath.org/1472.510182021-11-25T18:46:10.358925Z"Kiselman, Christer Oscar"https://zbmath.org/authors/?q=ai:kiselman.christer-oscar"Melin, Erik"https://zbmath.org/authors/?q=ai:melin.erikIf \(f: \mathbb{R}^n \to \mathbb{R}^m\) is a linear mapping, \(A \subseteq \mathbb{R}^n\) is a non-empty subset and \(B = f(A)\), the authors study the problem of the existence of a continuous right inverse of the mapping \(f |_A: A \to B\). Section 2 contains examples of \((f,A,B)\) for which the restriction \(f |_A : A \to B\) does not admit a continuous right inverse, with \(A\) compact and convex (in the third example in that section, which is a bit more complex than the first two examples, \(A\) has even smooth boundary).
The authors then study sufficient conditions under which \(f |_A\) has a continuous right inverse. Without loss of generality, it is enough to consider the case where \(m = n-1\) and \(f\) is the projection mapping
\[
(x_1, \ldots, x_n) \mapsto (x_1, \ldots, x_{n-1}),
\]
essentially because a linear projection from \(\mathbb{R}^n\) onto a subspace of codimension \(k\) can be obtained as the composition of \(k\) such projections onto a codimension \(1\) subspace at every step.
Given a non-empty subset \(A \subseteq \mathbb{R}^m \times \mathbb{R}\), the authors define two functions \(u_A\) and \(w_A: \mathbb{R}^m \to [-\infty, \infty]\) by
\begin{align*}
u_A(x) &= \operatorname{inf}_{t \in \mathbb{R}} \{ t \in \mathbb{R}: (x,t) \in A \} \\
w_A(x) &= \operatorname{sup}_{t \in \mathbb{R}} \{ t \in \mathbb{R}: (x,t) \in A \}
\end{align*}
Note that \(u_A\) and \(w_A\) are not necessarily continuous. If one assumes that \(f^{-1}(x)\) is connected, for any \(x \in B\), then one can see that the problem of the existence of a continuous right inverse for \(f |_A\) is closely related to the problem of the existence of a continuous function \(w\) whose values are squeezed between \(u_A\) and \(w_A\).
Sufficient conditions for the existence of a continuous right inverse are obtained in Sections 4, 5 and 6. For instance, Corollary 2 in Section 6 states the following.
Corollary 2 Let \(A\) be a compact convex subset of \(\mathbb{R}^m \times \mathbb{R}\), let \(f\) be the restriction to \(A\) of the projection \(\mathbb{R}^m \times \mathbb{R}\) onto its first factor \(\mathbb{R}^m\), thus mapping \((x,t) \mapsto x\) and let \(B = f(A)\). Assume there exist two functions \(u, w: B \to [-\infty, \infty]\) such that
\[
A = \{ (x,t) \in \mathbb{R}^m \times \mathbb{R}: u(x) \leq t \leq w(x) \}.
\]
Then \(f\) admits a continuous right inverse iff \(u^\sharp \leq w^\flat\) on \(\partial B\), where \(f^\sharp\) (\(f^\flat\)) denotes the upper (lower) regularization of a function \(f: B \to [-\infty, \infty]\).
In Section 7, the authors consider the special case of polyhedra, and in Section 9, they formulate and prove a result which shows that projections which do not have a continuous right inverse are exceptional (rather than generic).Leaves decompositions in Euclidean spaceshttps://zbmath.org/1472.520052021-11-25T18:46:10.358925Z"Ciosmak, Krzysztof J."https://zbmath.org/authors/?q=ai:ciosmak.krzysztof-jSummary: We partly extend the localisation technique from convex geometry to the multiple constraints setting.
For a given 1-Lipschitz map \(u:\mathbb{R}^n\to\mathbb{R}^m\), \(m\leq n\), we define and prove the existence of a partition of \(\mathbb{R}^n\), up to a set of Lebesgue measure zero, into maximal closed convex sets such that restriction of \(u\) is an isometry on these sets.
We consider a disintegration, with respect to this partition, of a log-concave measure. We prove that for almost every set of the partition of dimension \(m\), the associated conditional measure is log-concave. This result is proven also in the context of the curvature-dimension condition for weighted Riemannian manifolds. This partially confirms a conjecture of Klartag.The equivariant Ehrhart theory of the permutahedronhttps://zbmath.org/1472.520162021-11-25T18:46:10.358925Z"Ardila, Federico"https://zbmath.org/authors/?q=ai:ardila.federico"Supina, Mariel"https://zbmath.org/authors/?q=ai:supina.mariel"Vindas-Meléndez, Andrés R."https://zbmath.org/authors/?q=ai:vindas-melendez.andres-rIn [\textit{A. Stapledon}, Adv. Math. 226, No. 4, 3622--3654 (2011; zbl 1218.52014)] Stapledon introduced \textit{equivariant Ehrhart theory}, a variant of Ehrhart theory that takes group actions into account. For a lattice polytope \(P\) whose vertices lie in the lattice \(M\) and a group \(G\) acting on \(M\), one can define the \textit{equivariant \(H^*\)-series} \(H^*[z]\) which can be written as \(\sum_{i\geq 0} H_i^*z^i\) for appropriate virtual characters \(H_i^*\). Stapledon asks whether or not this series is effective, i.e, whether all the \(H_i^*\) are characters of representations of \(G\), and proposes the \textit{effectiveness conjecture} which states that the effectiveness of the equivariant \(H^*\)-series is equivalent to two other properties, namely
\begin{itemize}
\item[(i)] the toric variety of \(P\) admits a \(G\)-invariant non-degenerate hypersurface,
\item[(ii)] the equivariant \(H^*\)-series is a polynomial.
\end{itemize}
It is already known that (i) is a sufficient and (ii) is a necessary condition.
The present paper proves the effectiveness conjecture and three minor conjectures in the case of permutahedra under the action of the symmetric group.Continuous flattening of all polyhedral manifolds using countably infinite creaseshttps://zbmath.org/1472.520202021-11-25T18:46:10.358925Z"Abel, Zachary"https://zbmath.org/authors/?q=ai:abel.zachary-r"Demaine, Erik D."https://zbmath.org/authors/?q=ai:demaine.erik-d"Demaine, Martin L."https://zbmath.org/authors/?q=ai:demaine.martin-l"Ku, Jason S."https://zbmath.org/authors/?q=ai:ku.jason-s"Lynch, Jayson"https://zbmath.org/authors/?q=ai:lynch.jayson"Itoh, Jin-ichi"https://zbmath.org/authors/?q=ai:itoh.jin-ichi"Nara, Chie"https://zbmath.org/authors/?q=ai:nara.chieSummary: We prove that any finite polyhedral manifold in 3D can be continuously flattened into 2D while preserving intrinsic distances and avoiding crossings, answering a 19-year-old open problem, if we extend standard folding models to allow for countably infinite creases. The most general cases previously known to be continuously flattenable were convex polyhedra and semi-orthogonal polyhedra. For non-orientable manifolds, even the existence of an instantaneous flattening (flat folded state) is a new result. Our solution extends a method for flattening semi-orthogonal polyhedra: slice the polyhedron along parallel planes and flatten the polyhedral strips between consecutive planes. We adapt this approach to arbitrary nonconvex polyhedra by generalizing strip flattening to nonorthogonal corners and slicing along a countably infinite number of parallel planes, with slices densely approaching every vertex of the manifold. We also show that the area of the polyhedron that needs to support moving creases (which are necessary for closed polyhedra by the Bellows Theorem) can be made arbitrarily small.Smoothness filtration of the magnitude complexhttps://zbmath.org/1472.550062021-11-25T18:46:10.358925Z"Gomi, Kiyonori"https://zbmath.org/authors/?q=ai:gomi.kiyonoriLet \((X, d)\) be a metric space. A sequence of points \(\langle x_0, x_1, \dots, x_n\rangle\) (\(x_i\in X\)) is said to be a proper \(n\)-chain of length \(\ell\) if \(x_{i-1}\neq x_i\) (\(i=1, \dots, n\)) and \(d(x_0, x_1)+d(x_1, x_2)+\dots +d(x_{n-1}, x_n)=\ell\). Let \(C_n^\ell(X)\) be the free abelian group generated by proper \(n\)-chains of length \(\ell\). One can naturally define the boundary map \(\partial : C_n^\ell(X)\longrightarrow C_{n-1}^\ell(X)\) whose homology group \(H_n^\ell(X):=H_n(C_*^\ell(X))\) is called the \textit{magnitude homology} of \((X, d)\).
The notion of magnitude homology was first introduced by \textit{R. Hepworth} and \textit{S. Willerton} [Homology Homotopy Appl. 19, No. 2, 31--60 (2017; Zbl 1377.05088)] for a finite metric space defined by a graph. Later, it was generalized to a metric space (furthermore, enriched category) by \textit{T. Leinster} and \textit{M. Shulman} [``Magnitude homology of enriched categories and metric spaces'', Preprint, \url{arXiv:1711.00802}]. The computation of magnitude homology is, in general, difficult. In particular, if there exists a \textit{\(4\)-cut}, that is a chain \(\langle x_0, x_1, x_2, x_3\rangle\) satisfying
\[
\begin{split} d(x_0, x_3) &< d(x_0, x_1) + d(x_1, x_2) + d(x_2, x_3)\\
&= d(x_0, x_2) + d(x_2, x_3) = d(x_0, x_1) + d(x_1, x_3), \end{split}
\]
then the computation becomes complicated. Indeed, a previous work by \textit{R. Kaneta} and \textit{M. Yoshinaga} [Bull. Lond. Math. Soc. 53, No. 3, 893--905 (2021; Zbl 1472.55007)] showed that if the metric space \((X, d)\) does not have \(4\)-cuts, then the computation of the magnitude homology is reduced to that of the order complexes for posets.
The present paper extends and refines the previous works by using spectral sequences. In a proper chain \(\langle x_0, x_1, \dots, x_n\rangle\), a point \(x_i\) is said to be a \textit{smooth point} if \(d(x_{i-1}, x_{i})+d(x_{i}, x_{i+1})=d(x_{i-1}, x_{i+1})\) (otherwise, it is called a \textit{singular point}). Denote the number of smooth points in \(x\) by \(\sigma(x)\) and the submodule of \(C_n^\ell(X)\) generated by proper chains with \(\sigma(x)\leq p\) by \(F_pC_n^\ell(X)\). Then, \(F_pC_*^\ell(X)\) defines a filtration on the magnitude chain complex \(C_*^\ell(X)\). The associated spectral sequence is the main object of the present paper. The author completely describes at which page the spectral sequence degenerates. Precise results are as follows.
\begin{itemize}
\item[(a)] The spectral sequence always degenerates at \(E^4\).
\item[(b)] The spectral sequence degenerates at \(E^2\) if and only if the metric space \((X, d)\) does not contain \(4\)-cuts.
\item[(c)] The spectral sequence degenerates at \(E^3\) if and only if there does not exist a chain \(x=\langle x_0, x_1, x_2, x_3, x_4\rangle\) such that both \(\langle x_0, x_1, x_2, x_3\rangle\) and \(\langle x_1, x_2, x_3, x_4\rangle\) are \(4\)-cuts.
\end{itemize}
The author also applies the spectral sequence to a number of concrete examples.Generating random hyperbolic graphs in subquadratic timehttps://zbmath.org/1472.681232021-11-25T18:46:10.358925Z"von Looz, Moritz"https://zbmath.org/authors/?q=ai:von-looz.moritz"Meyerhenke, Henning"https://zbmath.org/authors/?q=ai:meyerhenke.henning"Prutkin, Roman"https://zbmath.org/authors/?q=ai:prutkin.romanSummary: Complex networks have become increasingly popular for modeling various real-world phenomena. Realistic generative network models are important in this context as they simplify complex network research regarding data sharing, reproducibility, and scalability studies. \textit{Random hyperbolic graphs} are a very promising family of geometric graphs with unit-disk neighborhood in the hyperbolic plane. Previous work provided empirical and theoretical evidence that this generative graph model creates networks with many realistic features.
In this work, we provide the first generation algorithm for random hyperbolic graphs with subquadratic running time. We prove a time complexity of \(O((n^{3/2}+m) \log n)\) with high probability for the generation process. This running time is confirmed by experimental data with our implementation. The acceleration stems primarily from the reduction of pairwise distance computations through a polar quadtree, which we adapt to hyperbolic space for this purpose and
which
can be of independent interest. In practice we improve the running time of a previous implementation (which allows more general neighborhoods than the unit disk) by at least two orders of magnitude this way. Networks with billions of edges can now be generated in a few minutes.
For the entire collection see [Zbl 1326.68015].Dependence of the density of states outer measure on the potential for deterministic Schrödinger operators on graphs with applications to ergodic and random modelshttps://zbmath.org/1472.811042021-11-25T18:46:10.358925Z"Hislop, Peter D."https://zbmath.org/authors/?q=ai:hislop.peter-d"Marx, Christoph A."https://zbmath.org/authors/?q=ai:marx.christoph-aSummary: We prove quantitative bounds on the dependence of the density of states on the potential function for discrete, deterministic Schrödinger operators on infinite graphs. While previous results were limited to random Schrödinger operators with independent, identically distributed potentials, this paper develops a deterministic framework, which is applicable to Schrödinger operators independent of the specific nature of the potential. Following ideas by Bourgain and Klein, we consider the density of states outer measure (DOSoM), which is well defined for all (deterministic) Schrödinger operators. We explicitly quantify the dependence of the DOSoM on the potential by proving a modulus of continuity in the \(\ell^\infty \)-norm. The specific modulus of continuity so obtained reflects the geometry of the underlying graph at infinity. For the special case of Schrödinger operators on \(\mathbb{Z}^d\), this implies the Lipschitz continuity of the DOSoM with respect to the potential. For Schrödinger operators on the Bethe lattice, we obtain log-Hölder dependence of the DOSoM on the potential. As an important consequence of our deterministic framework, we obtain a modulus of continuity for the density of states measure (DOSm) of ergodic Schrödinger operators in the underlying potential sampling function. Finally, we recover previous results for random Schrödinger operators on the dependence of the DOSm on the single-site probability measure by formulating this problem in the ergodic framework using the quantile function associated with the random potential.Spinorial \(R\) operator and algebraic Bethe ansatzhttps://zbmath.org/1472.811202021-11-25T18:46:10.358925Z"Karakhanyan, D."https://zbmath.org/authors/?q=ai:karakhanyan.david"Kirschner, R."https://zbmath.org/authors/?q=ai:kirschner.rolandSummary: We propose a new approach to the spinor-spinor R-matrix with orthogonal and symplectic symmetry. Based on this approach and the fusion method we relate the spinor-vector and vector-vector monodromy matrices for quantum spin chains. We consider the explicit spinor \(R\) matrices of low rank orthogonal algebras and the corresponding \(RTT\) algebras. Coincidences with fundamental \(R\) matrices allow to relate the Algebraic Bethe Ansatz for spinor and vector monodromy matrices.Background magnetic field and quantum correlations in the Schwinger effecthttps://zbmath.org/1472.811362021-11-25T18:46:10.358925Z"Bhattacharya, Sourav"https://zbmath.org/authors/?q=ai:bhattacharya.sourav.1"Chakrabortty, Shankhadeep"https://zbmath.org/authors/?q=ai:chakrabortty.shankhadeep"Hoshino, Hironori"https://zbmath.org/authors/?q=ai:hoshino.hironori"Kaushal, Shagun"https://zbmath.org/authors/?q=ai:kaushal.shagunSummary: In this work we consider two complex scalar fields distinguished by their masses coupled to constant background electric and magnetic fields in the \((3 + 1)\)-dimensional Minkowski spacetime and subsequently investigate a few measures quantifying the quantum correlations between the created particle-antiparticle Schwinger pairs. Since the background magnetic field itself cannot cause the decay of the Minkowski vacuum, our chief motivation here is to investigate the interplay between the effects due to the electric and magnetic fields. We start by computing the entanglement entropy for the vacuum state of a single scalar field. Second, we consider some maximally entangled states for the two-scalar field system and compute the logarithmic negativity and the mutual information. Qualitative differences of these results pertaining to the charge content of the states are emphasised. Based upon these results, we suggest some possible effects of a background magnetic field on the degradation of entanglement between states in an accelerated frame, for charged quantum fields.Spectral representation of lattice gluon and ghost propagators at zero temperaturehttps://zbmath.org/1472.813092021-11-25T18:46:10.358925Z"Dudal, David"https://zbmath.org/authors/?q=ai:dudal.david"Oliveira, Orlando"https://zbmath.org/authors/?q=ai:oliveira.orlando-anibal"Roelfs, Martin"https://zbmath.org/authors/?q=ai:roelfs.martin"Silva, Paulo"https://zbmath.org/authors/?q=ai:silva.paulo-roberto|silva.paulo-m-p|silva.paulo-h-d|silva.paulo-j-s|silva.paulo-f|silva.paulo-a-sSummary: We consider the analytic continuation of Euclidean propagator data obtained from 4D simulations to Minkowski space. In order to perform this continuation, the common approach is to first extract the Källén-Lehmann spectral density of the field. Once this is known, it can be extended to Minkowski space to yield the Minkowski propagator. However, obtaining the Källén-Lehmann spectral density from propagator data is a well known ill-posed numerical problem. To regularise this problem we implement an appropriate version of Tikhonov regularisation supplemented with the Morozov discrepancy principle. We will then apply this to various toy model data to demonstrate the conditions of validity for this method, and finally to zero temperature gluon and ghost lattice QCD data. We carefully explain how to deal with the IR singularity of the massless ghost propagator. We also uncover the numerically different performance when using two -- mathematically equivalent -- versions of the Källén-Lehmann spectral integral.Dynamically generated inflationary two-field potential via non-Riemannian volume formshttps://zbmath.org/1472.830722021-11-25T18:46:10.358925Z"Benisty, D."https://zbmath.org/authors/?q=ai:benisty.david"Guendelman, E. I."https://zbmath.org/authors/?q=ai:guendelman.eduardo-i"Nissimov, E."https://zbmath.org/authors/?q=ai:nissimov.emil"Pacheva, S."https://zbmath.org/authors/?q=ai:pacheva.svetlanaSummary: We consider a simple model of modified gravity interacting with a single scalar field \(\phi\) with weakly coupled exponential potential within the framework of non-Riemannian spacetime volume-form formalism. The specific form of the action is fixed by the requirement of invariance under global Weyl-scale symmetry. Upon passing to the physical Einstein frame we show how the non-Riemannian volume elements create a second canonical scalar field \(u\) and dynamically generate a non-trivial two-scalar-field potential \(U_{\mathrm{eff}}(u, \varphi)\) with two remarkable features: (i) it possesses a large flat region for large \(u\) describing a slow-roll inflation; (ii) it has a stable low-lying minimum w.r.t. \((u, \varphi)\) representing the dark energy density in the ``late universe''. We study the corresponding two-field slow-roll inflation and show that the pertinent slow-roll inflationary curve \(\varphi = \varphi(u)\) in the two-field space \((u, \varphi)\) has a very small curvature, \textit{i.e.}, \(\phi\) changes very little during the inflationary evolution of \(u\) on the flat region of \(U_{\mathrm{eff}}(u, \varphi)\). Explicit expressions are found for the slow-roll parameters which differ from those in the single-field inflationary counterpart. Numerical solutions for the scalar spectral index and the tensor-to-scalar ratio are derived agreeing with the observational data.NMR protein structure calculation and sphere intersectionshttps://zbmath.org/1472.921672021-11-25T18:46:10.358925Z"Lavor, Carlile"https://zbmath.org/authors/?q=ai:lavor.carlile-campos"Alves, Rafael"https://zbmath.org/authors/?q=ai:alves.rafael"Souza, Michael"https://zbmath.org/authors/?q=ai:souza.michael"José, Luis Aragón"https://zbmath.org/authors/?q=ai:jose.luis-aragonSummary: Nuclear magnetic resonance (NMR) experiments can be used to calculate 3D protein structures and geometric properties of protein molecules allow us to solve the problem iteratively using a combinatorial method, called branch-and-prune (BP). The main step of BP algorithm is to intersect three spheres centered at the positions for atoms \(i - 3, i - 2, i - 1\), with radii given by the atomic distances \(d_{i-3,i}, d_{i-2,i}, d_{i-1,i}\), respectively, to obtain the position for atom \(i\). Because of uncertainty in NMR data, some of the distances \(d_{i-3,i}\) should be represented as interval distances \([\underline{d}_{i - 3,i},\bar d_{i - 3,i}]\), where \(\underline{d}_{i - 3,i} \leq d_{i - 3,i} \leq\bar d_{i - 3,i}\). In the literature, an extension of the BP algorithm was proposed to deal with interval distances, where the idea is to sample values from \([\underline{d}_{i - 3,i},\bar d_{i - 3,i}]\). We present a new method, based on conformal geometric algebra, to reduce the size of \([\underline{d}_{i - 3,i},\bar d_{i - 3,i}]\), before the sampling process. We also compare it with another approach proposed in the literature.On correlation of hyperbolic volumes of fullerenes with their propertieshttps://zbmath.org/1472.922922021-11-25T18:46:10.358925Z"Egorov, A. A."https://zbmath.org/authors/?q=ai:egorov.aleksandr-anatolevich|egorov.andrey-aleksandrovich"Vesnin, A. Yu."https://zbmath.org/authors/?q=ai:vesnin.andrei-yu|vesnin.andrei-yurevichSummary: We observe that fullerene graphs are one-skeletons of polyhedra, which can be realized with all dihedral angles equal to \(\pi /2\) in a hyperbolic 3-dimensional space. One of the most important invariants of such a polyhedron is its volume. We are referring this volume as a hyperbolic volume of a fullerene. It is known that some topological indices of graphs of chemical compounds serve as strong descriptors and correlate with chemical properties. We demonstrate that hyperbolic volume of fullerenes correlates with few important topological indices and so, hyperbolic volume can serve as a chemical descriptor too. The correlation between hyperbolic volume of fullerene and its Wiener index suggested few conjectures on volumes of hyperbolic polyhedra. These conjectures are confirmed for the initial list of fullerenes.