Recent zbMATH articles in MSC 51https://zbmath.org/atom/cc/512023-01-20T17:58:23.823708ZWerkzeugBrianchon and Poncelet's joint memoir, the nine-point circle, and beyondhttps://zbmath.org/1500.010052023-01-20T17:58:23.823708Z"Del Centina, Andrea"https://zbmath.org/authors/?q=ai:del-centina.andreaThis informative paper provides an account of Brianchon and Poncelet's joint memoir on equilateral hyperbolas subject to four given conditions, published in 1821 and mostly famous for containing a complete proof of the so-called nine-point circle theorem (for the history of this problem in the second half of the 19th century, see: [\textit{M. A. Vaccaro}, Hist. Math. 51, 26--48 (2020; Zbl 1445.01013)]).
The author offers a detailed analysis of the mathematical content of the memoir and an interesting reconstruction of its genesis. In fact, apart from its mathematical relevance, Brianchon and Poncelet's paper is an interesting example of a co-authored work at a time where mathematical collaborations were rare.
With respect to this issue, the author concludes that the genesis of the memoir is twofold: the first part derives from Brianchon's studies on hyperbolas as expounded in a memoir written in 1817, the second part was due to Poncelet, and it was actually a draft of another future memoir composed by the latter.
The nature of the collaboration between Poncelet and Brianchon is also made more precise: Poncelet and Brianchon never met in person, their contacts were limited to letter exchanges, and their ``joint work'' appears to be ``the fruit of particular circumstances rather than of a close collaboration''.
Reviewer: Davide Crippa (Praha)Projective duality and the rise of modern logichttps://zbmath.org/1500.030012023-01-20T17:58:23.823708Z"Eder, Günther"https://zbmath.org/authors/?q=ai:eder.guntherA remarkable fact about projective geometry is the principle of duality, which in its plane form states that for each theorem in plane projective geometry there is another theorem which is like the first except in having the words ``point'' and ``line'' interchanged. The desire to understand how this could be does much to explain how attractive projective geometry was to mathematicians in the nineteenth century. The present paper uses the resources of modern model theory to define three notions of duality, a functional one in terms of isomorphism (Duality 1), an axiomatic one in terms of a semantically closed system of axioms (Duality 2), and a proof-theoretic one in terms of a syntactically closed system of axioms and a formal logical calculus of logical axioms and inference rules (Duality 3). It then uses these precisely defined concepts to clarify various statements and approaches of nineteenth century geometers, notably Poncelet, Gergonne, Jakob Steiner, von Staudt, Pasch, Julius Plücker, Otto Hesse, Dedekind, Arthur Cayley, and finally Hilbert (with whom modern axiomatics began) in exploring the phenomena of duality and the concepts such as transfer (Übertragung) or reinterpretation which they employed in explaining what they took themselves to be doing. It is notable that the functional concept (duality 1) was at least as important as those relative to an axiomatic system (duality 2 and duality 3).
Reviewer: Jim Mackenzie (Sydney)Classifying planar monomials over fields of order a prime cubedhttps://zbmath.org/1500.110862023-01-20T17:58:23.823708Z"Bergman, Emily"https://zbmath.org/authors/?q=ai:bergman.emily"Coulter, Robert S."https://zbmath.org/authors/?q=ai:coulter.robert-s"Villa, Irene"https://zbmath.org/authors/?q=ai:villa.ireneConsider \(\mathbb{F}_q\), a finite field with \(q\) elements. A polynomial \(f \in \mathbb{F}_q[x]\) is said to be a \textit{permutation polynomial} over \(\mathbb{F}_q\) if \(f\) induces a permutation over \(\mathbb{F}_q\). Moreover, we say that a polynomial \(f \in \mathbb{F}_q[x]\) is \textit{planar} if for every \(a \in \mathbb{F}_q^*\) the polynomial \(f(x+a)-f(x)\) is a permutation polynomial over \(\mathbb{F}_q\).
In Theorem 1, the authors are able to prove that when \(q=p^3\) with \(p\) an odd prime, the monomial \(x^n\) is planar over \(\mathbb{F}_q\) if and only if \(n=p^i+p^j \pmod{q-1}\), with \(0\leq i,j<3\). In this way, they establish the Dembowski-Ostrom conjecture in this case. The proof uses Hermite's criteria and non-trivial manipulations.
Reviewer: Ferdinando Zullo (Caserta)On the construction of chaotic dynamical systems on the box fractalhttps://zbmath.org/1500.280042023-01-20T17:58:23.823708Z"Aslan, N."https://zbmath.org/authors/?q=ai:aslan.nihal-kilic|aslan.nurgul|aslan.nisa|aslan.necdet"Saltan, M."https://zbmath.org/authors/?q=ai:saltan.mustafa|saltan.mehmetIn the paper two dynamical systems on the box fractal are constructed. Using a coding of the fractal it is shown that both systems are chaotic in the sense of Devaney and an algorithm to compute periodic points is given.
Reviewer: Jörg Neunhäuserer (Goslar)Simplicial complexes from finite projective planes and colored configurationshttps://zbmath.org/1500.510012023-01-20T17:58:23.823708Z"Superdock, Matt"https://zbmath.org/authors/?q=ai:superdock.mattSummary: In the 7-vertex triangulation of the torus, the 14 triangles can be partitioned as \(T_1 \sqcup T_2\), such that each \(T_i\) represents the lines of a copy of the Fano plane \(\mathrm{PG}(2, \mathbb{F}_2)\). We generalize this observation by constructing, for each prime power \(q\), a simplicial complex \(X_q\) with \(q^2 + q + 1\) vertices and \(2( q^2 + q + 1)\) facets consisting of two copies of \(\mathrm{PG}(2, \mathbb{F}_q)\). Our construction works for any \textit{colored} \(k\)-\textit{configuration}, defined as a \(k\)-configuration whose associated bipartite graph \(G\) is connected and has a \(k\)-edge coloring \(\chi : E(G) \to [k]\), such that for all \(v \in V(G)\), \(a, b, c \in [k]\), following edges of colors \(a, b, c, a, b, c\) from \(v\) brings us back to \(v\). We give one-to-one correspondences between (1) Sidon sets of order 2 and size \(k + 1\) in groups with order \(n\), (2) linear codes with radius 1 and index \(n\) in the lattice \(A_k\), and (3) colored \((k + 1)\)-configurations with \(n\) points and \(n\) lines. (The correspondence between (1) and (2) is known.) As a result, we suggest possible topological obstructions to the existence of Sidon sets, and in particular, planar difference sets.Quantum number towers for the Hubbard and Holstein modelshttps://zbmath.org/1500.810452023-01-20T17:58:23.823708Z"Freericks, James K."https://zbmath.org/authors/?q=ai:freericks.james-kSummary: In 1989, \textit{E. H. Lieb} published [``Two theorems on the Hubbard model'', Phys. Rev. Lett. 62, No. 10, 1201--1204 (1989; \url{doi:10.1103/PhysRevLett.62.1201})] proving two theorems about the Hubbard model. This paper used the concept of spin-reflection positivity to prove that the ground state of the attractive Hubbard model was always a nondegenerate spin singlet and to also prove that the ground state for the repulsive model on a bipartite lattice had spin \(\big{|}|\Lambda_A|-|\Lambda_B|\big{|}/2\), corresponding to the difference in number of lattice sites for the two sublattices. In addition, this work relates to quantum number towers -- where the minimal energy state with a given quantum number, such as spin, or pseudospin, is ordered, according to the spin or pseudospin values. It was followed up in 1995 by a second paper that extended some of these results to the Holstein model (and more general electron-phonon models). These works prove results about the quantum numbers of these many-body models in condensed matter physics and have been very influential. In this chapter, I will discuss the context for these proofs, what they mean, and the remaining open questions related to the original work. In addition, I will briefly discuss some of the additional work that this methodology inspired.
For the entire collection see [Zbl 1491.46002].Positron position operators. I: A natural optionhttps://zbmath.org/1500.810462023-01-20T17:58:23.823708Z"Tumulka, Roderich"https://zbmath.org/authors/?q=ai:tumulka.roderichSummary: By ``position operators,'' we mean here a POVM (positive- operator-valued measure) on a suitable configuration space acting on a suitable Hilbert space that serves as defining the position observable of a quantum theory, and by ``positron position operators,'' we mean a joint treatment of positrons and electrons. We consider the standard free second-quantized Dirac field in Minkowski space-time or in a box. On the associated Fock space (i.e., the tensor product of the positron Fock space and the electron Fock space), there acts an obvious POVM \(P_{\mathrm{obv}}\), but we propose a different one that we call the natural POVM, \(P_{\mathrm{nat}}\). In fact, it is a PVM (projection-valued measure); it captures the sense of locality corresponding to the field operators \(\Psi_s(\boldsymbol{x})\) and to the algebra of local observables. The existence of \(P_{\mathrm{nat}}\) depends on a mathematical conjecture which at present we can neither prove nor disprove; here we explore consequences of the conjecture. We put up for consideration the possibility that \(P_{\mathrm{nat}}\), and not \(P_{\mathrm{obv}}\), is the physically correct position observable and defines the Born rule for the joint distribution of electron and positron positions. We describe properties of \(P_{\mathrm{nat}}\), including a strict no-superluminal-signaling property, and how it avoids the Hegerfeldt-Malament no-go theorem. We also point out how to define Bohmian trajectories that fit together with \(P_{\mathrm{nat}}\), and how to generalize \(P_{\mathrm{nat}}\) to curved space-time.Maximal speed of propagation in open quantum systemshttps://zbmath.org/1500.810532023-01-20T17:58:23.823708Z"Breteaux, Sébastien"https://zbmath.org/authors/?q=ai:breteaux.sebastien"Faupin, Jérémy"https://zbmath.org/authors/?q=ai:faupin.jeremy"Lemm, Marius"https://zbmath.org/authors/?q=ai:lemm.marius"Sigal, Israel Michael"https://zbmath.org/authors/?q=ai:sigal.israel-michaelSummary: We prove a maximal velocity bound for the dynamics of Markovian open quantum systems. The dynamics is described by one-parameter semigroups of quantum channels satisfying the von Neumann-Lindblad equation. Our result says that dynamically evolving states are contained inside a suitable light cone up to polynomial errors. We also give a bound on the slope of the light cone, i.e. the maximal propagation speed. The result implies an upper bound on the speed of propagation of local perturbations of stationary states in open quantum systems.
For the entire collection see [Zbl 1491.46002].Transversely trapping surfaces: dynamical versionhttps://zbmath.org/1500.830112023-01-20T17:58:23.823708Z"Yoshino, Hirotaka"https://zbmath.org/authors/?q=ai:yoshino.hirotaka"Izumi, Keisuke"https://zbmath.org/authors/?q=ai:izumi.keisuke"Shiromizu, Tetsuya"https://zbmath.org/authors/?q=ai:shiromizu.tetsuya"Tomikawa, Yoshimune"https://zbmath.org/authors/?q=ai:tomikawa.yoshimuneSummary: We propose new concepts, a dynamically transversely trapping surface (DTTS) and a marginally DTTS, as indicators for a strong gravity region. A DTTS is defined as a two-dimensional closed surface on a spacelike hypersurface such that photons emitted from arbitrary points on it in transverse directions are acceleratedly contracted in time, and a marginally DTTS is reduced to the photon sphere in spherically symmetric cases. (Marginally) DTTSs have a close analogy with (marginally) trapped surfaces in many aspects. After preparing the method of solving for a marginally DTTS in the time-symmetric initial data and the momentarily stationary axisymmetric initial data, some examples of marginally DTTSs are numerically constructed for systems of two black holes in the Brill-Lindquist initial data and in the Majumdar-Papapetrou spacetimes. Furthermore, the area of a DTTS is proved to satisfy the Penrose-like inequality \(A_0\le 4\pi (3GM)^2\), under some assumptions. Differences and connections between a DTTS and the other two concepts proposed by us previously, a loosely trapped surface [\textit{T. Shiromizu} et al., PTEP, Prog. Theor. Exper. Phys. 2017, No. 3, Article ID 033E01, 6 p. (2017; Zbl 1477.83009)] and a static/stationary transversely trapping surface [\textit{H. Yoshino} et al., PTEP, Prog. Theor. Exper. Phys. 2017, No. 6, Article ID 063E01, 23 p. (2017; Zbl 1477.83070)], are also discussed. A (marginally) DTTS provides us with a theoretical tool to significantly advance our understanding of strong gravity fields. Also, since DTTSs are located outside the event horizon, they could possibly be related with future observations of strong gravity regions in dynamical evolutions.Extending two families of maximum rank distance codeshttps://zbmath.org/1500.940832023-01-20T17:58:23.823708Z"Neri, Alessandro"https://zbmath.org/authors/?q=ai:neri.alessandro"Santonastaso, Paolo"https://zbmath.org/authors/?q=ai:santonastaso.paolo"Zullo, Ferdinando"https://zbmath.org/authors/?q=ai:zullo.ferdinandoThis paper deals with the construction of MRD codes. In particular, the authors provide a new family of \(2\)-dimensional \(\mathbb{F}_{q^n}\)-linear MRD codes that properly contains the two families found by \textit{G. Longobardi} and \textit{C. Zanella} [J. Algebr. Comb. 53, No. 3, 639--661 (2021; Zbl 1465.94113)] and by \textit{G. Longobardi} et al. [``A large family of maximum scattered linear sets of \(\mathrm{PG}(1,qn)\) and their associated MRD codes'', Preprint, \url{arXiv:2102.08287}]. The crucial part is proving that the constructed codes are in fact MRD. Since the authors focus on square matrices, they can identify the \(\mathbb{F}_q\)-algebra of \(n\times n\) matrices as the algebra of \(\sigma\)-polynomials, i.e. polynomials of the form
\[
f(x)=\sum_{i=0}^{n-1} f_ix^{\sigma^i}, \quad f_i\in\mathbb{F}_{q^n},
\]
where \(\sigma\) is a generator of the Galois group \(\mathrm{Gal}(\mathbb{F}_{q^n}/\mathbb{F}_q)\), and \(x^{\sigma}\) denotes the action of \(\sigma\) on \(x\). The authors give a general argument which allows to extend any construction of MRD codes based on \(\sigma\)-polynomials to any other generator \(\theta\) of the Galois group \(\mathrm{Gal}(\mathbb{F}_{q^n}/\mathbb{F}_q)\) under certain assumptions. Furthermore, they are able to show that the new codes they introduced are inequivalent to all the other known \(\mathbb{F}_{q^n}\)-linear MRD codes. Moreover, they investigate when two codes in this new wider family are equivalent, which turns to provide a lower bound on the number of equivalence classes of codes in this new family.
Reviewer: Gianira N. Alfarano (Zürich)