arXiv Math @arxiv_math@qoto.org

A famous conjecture of Chowla on the least primes in arithmetic progressions implies that the abscissa of convergence of the Weil representation zeta function for a procyclic group $G$ only depends on the set $S$ of primes dividing the order of $G$ and that it agrees with the abscissa of the Dedekind zeta function of $\mathbb{Z}[p^{-1}\mid p \not\in S]$. Here we show that these consequences hold unconditionally for random procyclic groups in a suitable model. As a corollary, every real number $1 \leq β\leq 2$ is the Weil abscissa of some procyclic group.

In this paper we prove a reverse Hölder inequality for the variable exponent Muckenhoupt weights $\mathcal{A}_{p(\cdot)}$, introduced by the first author, Fiorenza, and Neugeabauer. All of our estimates are quantitative, showing the dependence of the exponent function on the $\mathcal{A}_{p(\cdot)}$ characteristic. As an application, we use the reverse Hölder inequality to prove that the matrix $\mathcal{A}_{p(\cdot)}$ weights, introduced in our previous paper, have both a right and left-openness property. This result is new even in the scalar case.

We establish the independence of multipliers for polynomial endomorphisms of $\mathbb C^n$ and endomorphisms of $\mathbb P^n.$ This allows us to extend results about the bifurcation measure and the critical height obtained in \cite{arXiv:2305.02246} to the case of polynomial endomorphisms of $\mathbb C^n$ for $n\geq 3$. An important step in the proof is the irreducibility of the spaces of endomorphisms with $N$ marked periodic points, which is of independent interest.

In this expository article, we study and discuss invariants of vector fields and holomorphic foliations that intertwine the theories of complex analytic singular varieties and singular holomorphic foliations on complex manifolds: two different settings with many points in common.

Based on a recent representation of the psi function due to Guillera and Sondow and independently Boyadzhiev, new closed forms for various series involving harmonic numbers and inverse factorials are derived. A high point of the presentation is the rediscovery, by much simpler means, of a famous quadratic Euler sum originally discovered in 1995 by Borwein and Borwein.

Consider that $\mathbb{G}=(\mathbb{X}, \mathbb{Y})$ is a simple, connected graph with $\mathbb{X}$ as the vertex set and $\mathbb{Y}$ as the edge set. The atom-bond connectivity ($ABC$) index is a novel topological index that Estrada introduced in Estrada et al. (1998). It is defined as $$ A B C(\mathbb{G})=\sum_{xy \in Y(\mathbb{G})} \sqrt{\frac{ζ_x+ζ_y-2}{ζ_x ζ_y}} $$ where $ζ_x$ and $ζ_x$ represent the degrees of the vertices $x$ and $y$, respectively. In this work, we explore the behavior of the $A B C$ index for tree graphs. We establish both lower and upper bounds for the $A B C$ index, expressed in terms of the graph's order and its Roman domination number. Additionally, we characterize the tree structures that correspond to these extremal values, offering a deeper understanding of how the Roman domination number ($RDN$) influences the $A B C$ index in tree graphs.

Consider a simple graph $\mathbb{G} = (\mathcal{V}, \mathcal{E}) $, where $ \mathcal{V} $ are the vertices and $ \mathcal{E} $ are the edges. The first Zagreb index, $\mathbb{M}_{1}(\mathbb{G}) = \sum_{v \in \mathcal{V}} ψ_\mathbb{G}(v)^2$. The second Zagreb index, $\mathbb{M}_{2}(\mathbb{G}) = \sum_{uv \in \mathcal{E}} ψ_\mathbb{G}(u) ψ_\mathbb{G}(v)$. The metric dimension of a graph refers to the smallest subset of vertices in a resolving set such that the distances from these vertices to all others in the graph uniquely identify each vertex. In this paper, we characterize bounds for the Zagreb indices of trees, based on the order of the tree and its metric dimension. Furthermore, we identify the trees that achieve these extremal bounds, offering valuable insights into how the metric dimension influences the behavior of the Zagreb indices in tree structures.

The aim of this survey is to outline the state of the art in research on a class of linearized polynomials with coefficients over finite fields, known as scattered polynomials. These have been studied in several contexts, such as in [A. Blokhuis, M. Lavrauw. Scattered spaces with respect to a spread in $\mathrm{PG}(n, q)$. Geometriae Dedicata 81(1) (2000), 231-243] and [G. Lunardon, O. Polverino. Blocking sets and derivable partial spreads. J. Algebraic Combin. 14 (2001), 49-56]. Recently, their connection to maximum rank-metric codes was brought to light in [J. Sheekey. MRD codes: Constructions and connections. In K.-U. Schmidt and A. Winterhof, editors, Combinatorics and Finite Fields, De Gruyter (2019), 255-286]. This link has significantly advanced their study and investigation, sparking considerable interest in recent years. Here, we will explore their relationship with certain subsets of the finite projective line $\mathrm{PG}(1, q^n)$ known as maximum scattered linear sets, as well as with codes made up of square matrices of order $n$ equipped with the rank metric. We will review the known examples of scattered polynomials up to date and discuss some of their key properties. We will also address the classification of maximum scattered linear sets of the finite projective line $\mathrm{PG}(1, q^n)$ for small values of $n$ and discuss characterization results for the examples known so far. Finally, we will retrace how each scattered polynomial gives rise to a translation plane, as discussed in [V. Casarino, G. Longobardi, C. Zanella. Scattered linear sets in a finite projective line and translation planes, Linear Algebra Appl. 650 (2022), 286-298] and in [G. Longobardi, C. Zanella, A standard form for scattered linearized polynomials and properties of the related translation planes, J. Algebr. Comb. 59(4) (2024), 917-937].

The problem of finding the sum of a polynomial's values is considered. In particular, for any $n\geq 3$, the explicit formula for the sum of the $n$th powers of natural numbers $S_n=\sum_{x=1}^{m}x^{n}$ is proved: $$\sum_{x=1}^{m}x^{n}=(-1)^{n}m(m+1)(-\frac{1}{2}+\sum_{i=2}^{n}a_i(m+2)(m+3)...(m+i)),$$ here $a_i=\frac{1}{i+1}\sum_{k=1}^{i}\frac{(-1)^{k}k^{n}}{k!(i-k)!}$, $(i=2,3,...,n-1)$, $a_n=\frac{(-1)^n}{n+1}$. Note that this formula does not contain Bernoulli numbers.

Our aim is to develop a general approach for the dynamics of material bodies of dimension d represented by a mater manifold of dimension (d + 1) embedded into the space-time. It can be declined for d = 0 (pointwise object), d = 1 (arch if it is a solid, flow in a pipe or jet if it is a fluid), d = 2 (plate or shell if it is a solid, sheet of fluid), d = 3 (bulky bodies). We call torsor a skew-symmetric bilinear map on the vector space of affine real functions on the affine tangent space to the space-time. We use the affine connections as originally developed by Élie Cartan, that is the connections associated to the affine group. We introduce a general principle of covariant divergence free torsor from which we deduce 10 balance equations. We show the relevance of this general principle by applying it for d from 1 to 4 in the context of the Galilean relativity.

In 1960, Grünbaum proved that for any convex body $C\subset\mathbb{R}^d$ and every halfspace $H$ containing the centroid of $C$, one has that the volume of $H\cap C$ is at least a $\frac{1}{e}$-fraction of the volume of $C$. Recently, in 2017, Basu and Oertel conjectured that a similar result holds for mixed-integer convex sets. Concretely, they proposed that for any convex body $C\subset \mathbb{R}^{n+d}$, there should exist a point $\mathbf{x} \in S=C\cap(\mathbb{Z}^{n}\times\mathbb{R}^d)$ such that for every halfspace $H$ containing $\mathbf{x}$, one has that \[ \mathcal{H}_d(H\cap S) \geq \frac{1}{2^n}\frac{1}{e}\mathcal{H}_d(S), \] where $\mathcal{H}^d$ denotes the $d$-dimensional Hausdorff measure. While the conjecture remains open, Basu and Oertel proved that the above inequality holds true for sufficiently large sets, in terms of a measure known as the \emph{lattice width} of a set. In this work, by following a geometric approach, we improve this result by substantially reducing the threshold at which a set can be considered large. We reduce this threshold from an exponential to a polynomial dependency on the dimension, therefore significantly enlarging the family of mixed-integer convex sets over which the Basu-Oertel conjecture holds true.

The following is an exposition of a course of algebra that Prof. Aleksandr Aleksandrovich Zykov (1922-2013) distributed among the participants of his seminar in graph theory not far away from Odessa, Ukraine, on September, 1991. It is a privilege for me to be able to reproduce, with some good additions, the English version of this remarkable course that he designed carefully over several years with the help of other colleagues and with the only purpose of making the science of algebra more accessible to the less gifted student. Prof. Zykov was an admirable person and a very clever mathematician and scientist. His scientific interests were wide: from the mathematical sciences at any level and in any of their branches to the theory of relativity as I recall. He kindly wrote a review recommending the results of my doctoral dissertation, which was necessary for the approval to obtain a Ph.D. degree during the meeting of my defense. In that meeting, I was very nervous because I had in front of me fifteen professors all of them doctors of technical or mathematical sciences asking all sort of questions and whom I had to convince of the fact that my results were true. I am sure this algebra course played the role that Prof. Zykov, with his deepness of thought, had primarily intended, that is, one course designed with the right approach to help out many of his students.

It is known that, given a left continuous t-norm $T$ on $[0,1]$ and a right continuous $\CL$-operation $L$ on $[0,\infty]$, then their tensor product $L\otimes T$ is a triangle function on $\Delp$. Hence, $(\Delp,L\otimes T)$ becomes a partially ordered monoid. In this paper, it is shown that: (1) $L\otimes T$ coincides with the triangle function $τ_{T,L}$ if and only if $L$ satisfies the condition $(LCS)$; (2) let $\CDp$ be the set of all non-defective distance distribution functions, then $(\CDp,L\otimes T)$ is a submonoid of $(\Delp,L\otimes T)$ if and only if $L$ has no zero divisor; (3) let $\CDp_c$ be the set of all continuous distance distribution functions, then for each continuous t-norm $T$, $(\CDp_c, L\otimes T)$ is a subsemigroup of $(\Delp,L\otimes T)$ if and only if $L$ satisfies the condition $(LS)$. Moreover, $(\CDp_c,L\otimes T)$ is an ideal of $(\CDp,L\otimes T)$ if and only if $L$ satisfies the cancellation law.

Examples of algebraic surfaces of general type with maximal Picard number are not abundant in the literature. Moreover, most known examples either possess low invariants, lie near the Noether line $K^2=2χ-6$ or are somewhat scattered. A notable exception is Persson's sequence of double covers of the projective plane with maximal Picard number, whose invariants converge to the Severi line $K^2=4χ$. This note is devoted to the construction of infinitely many new sequences of surfaces of general type with maximal Picard number whose invariants converge to the Severi line.

Terahertz (THz) communication offers the necessary bandwidth to meet the high data rate demands of next-generation wireless systems. However, it faces significant challenges, including severe path loss, dynamic blockages, and beam misalignment, which jeopardize communication reliability. Given that many 6G use cases require both high data rates and strong reliability, robust transmission schemes that achieve high throughput under these challenging conditions are essential for the effective use of high-frequency bands. In this context, we propose a novel mixed-criticality superposition coding scheme for reconfigurable intelligent surface (RIS)-assisted THz systems. This scheme leverages both the strong but intermittent direct line-of-sight link and the more reliable, yet weaker, RIS path to ensure robust delivery of high-criticality data while maintaining high overall throughput. We model a mixed-criticality queuing system and optimize transmit power to meet reliability and queue stability constraints. Simulation results show that our approach significantly reduces queuing delays for critical data while sustaining high overall throughput, outperforming conventional time-sharing methods. Additionally, we examine the impact of blockage, beam misalignment, and beamwidth adaptation on system performance. These results demonstrate that our scheme effectively balances reliability and throughput under challenging conditions, while also underscoring the need for robust beamforming techniques to mitigate the impact of misalignment in RIS-assisted channels.

We study monoidal 2-categories and bicategories in terms of categorical extensions and the cohomological data they determine in appropriate cohomology theories with coefficients in Picard groupoids. In particular, we analyze the hierarchy of possible commutativity conditions in terms of progressive stabilization of these data. We also show that monoidal structures on bicategories give rise to biextensions of a pair of (abelian) groups by a Picard groupoid, and that the progressive vanishing of obstructions determined by the tower of commutative structures corresponds to appropriate symmetry conditions on these biextensions. In the fully symmetric case, which leads us fully into the stable range, we show how our computations can be expressed in terms of the cubical Q-construction underlying MacLane (co)homology.

This paper proves a 2017 conjecture of De Loera, La Haye, Oliveros, and Roldán-Pensado that the "prime grid" $\big\{(p,q) \in \mathbb{Z}^2 : \text{$p$ and $q$ are prime}\big\} \subseteq \mathbb{R}^2$ contains empty polygons with arbitrarily many vertices. This implies that no Helly-type theorem is true for the prime grid.

We revisit the local form subordination condition on the perturbation of a self-adjoint operators with compact resolvent, which is used to show the Riesz basis property of the eigensystem of the perturbed operator. Our new assumptions and new proof allow for establishing the Riesz basis property also in the case of slow and non-monotone decay in this subordination condition.

These lecture notes (in Spanish) are based on a mini-course given by A. Osorno and M. Rivera at the First Colombian Geometry and Topology Meeting that took place at the Universidad Nacional de Colombia in July 2024 in Bogota. They are intended to be a guide for a first encounter with homotopy theory and simplicial methods - emphasizing intuition and important statements - accessible to students with basic knowledge of point set topology. Estas notas surgieron como parte de un mini-curso dictado por A. Osorno y M. Rivera en el primer Encuentro Colombiano de Geometría y Topología (ECOGyT) que se llevó a cabo en la Universidad Nacional de Colombia sede Bogotá en julio del 2024. La idea es que sirvan como una guía para un primer encuentro con la teoría de homotopía y técnicas simpliciales - enfatizando en la intuición y enunciados importantes - accesible a estudiantes con conocimiento básico de topología general y así invitar a indagar más profundamente sobre el tema y sus aplicaciones en distintos campos.