arXiv Math @arxiv_math@qoto.org

In this paper, we introduce the notions of crossed module of anti-pre-Lie algebras. We explore the non-abelian cohomology of anti-pre-Lie algebras to classify their non-abelian extensions. Additionally, we investigate the inducibility of a pair of automorphisms for anti-pre-Lie algebras and construct a Wells exact sequence. Finally, we show how these results fit with abelian extensions of anti-pre-Lie algebras.

This paper introduces a computational approach to identify performance constraints in the time-domain based on optimizing the excitation waveform. The method builds on an optimization algorithm that has been employed for decades to establish fundamental limits in the frequency domain and this paper showcases its first comprehensive application to time-domain pulses. The method is applied to arbitrarily polarized multiport antennas and arrays. The demonstration performed is based on finding an antenna's maximum peak radiation intensity in a given direction and time with limited total input energy available. To highlight the generality of the approach, an analysis on finding optimal illumination for antiferromagnetic memory switching is conducted.

In the field of mathematics, a purely combinatorial equivalent to a simplicial complex, or more generally, a down-set, is an abstract structure known as a family of sets. This family is closed under the operation of taking subsets, meaning that every subset of a set within the family is also included in the family. The purpose of this paper is two-fold. Firstly, it aims to present a comprehensive survey of recent results in the field. This survey intends to provide an overview of the advancements made in codes constructed from simplicial complexes. Secondly, the paper seeks to propose open problems that are anticipated to stimulate further research in this area. By highlighting these open problems, the paper aims to encourage and inspire future investigations and developments in the field of codes derived from simplicial complexes.

For a graph $G$ define the parameters $\ell(G)$ and $L(G)$ as the minimum and maximum value of $ν(G\backslash F)$, where $F$ is a maximum matching of $G$ and $ν(G)$ is the number of edges in it. In this paper, we show that there is a small constant $c>0$, such that the following decision problem is NP-complete: given a graph $G$ and $k\leq \frac{|V|}{2}$, check whether there is a maximum matching $F$ in $G$, such that $|ν(G\backslash F)-k|\leq c\cdot |V|$. Note that when $c=1$, this problem is polynomial time solvable as we observe in the paper. Since in any graph $G$, we have $L(G)\leq 2\ell(G)$, any polynomial time algorithm constructing a maximum matching of a graph is a 2-approximation algorithm for $\ell(G)$ and $\frac{1}{2}$-approximation algorithm for $L(G)$. We complement these observations by presenting two inapproximability results for $\ell(G)$ and $L(G)$.

We provide an exposition of identities containing q-multiple sums given by Dilcher, Prodinger, Fu and Lascoux, Zeng, Guo and Zhang. Meanwhile, for each of these identities, a more powerful statement will be derived through our exposition. Our exposition shall also provide a generalization of the duality relation for finite multiple harmonic $q$-series given by Bradley.

In this paper, we study the mixing time of the simple exclusion process with $k$ particles in the line segment $[1, N]$ with conductances $c^{(N)}(x, x+1)_{1\le x<N}$ where $c^{(N)}(x, x+1)>0$ is the rate of swapping the contents of the two sites $x$ and $ x+1$. Writing $r^{(N)}(x, x+1) := 1/c^{(N)}(x, x+1)$, under the assumption \begin{equation*} \limsup_{N\to \infty}\, \frac{1}{N}\sup_{1< m \le N}\, \left| \sum_{x=2}^m r^{(N)}(x-1, x)- (m-1) \right|\;=\;0\,, \end{equation*} and some further assumptions on $r^{(N)}(x, x+1)_{x \in \mathbb{N} }$ and $k$, we prove that around time $(1+o(1)) (2 π^2)^{-1} N^2 \log k$, the total variation distance to equilibrium of the simple exclusion process drops abruptly from $1$ to $0$.

In this article, we develop tools for computing $G$-crossed extensions of braided tensor categories, including all coherence data (associator, braiding, tensor structures). Their equivariantisations describe representation categories of fixed-point vertex operator algebras, also called orbifolds. The seminal work of Etingof, Nikshych and Ostrik asserts the existence and uniqueness of the $G$-crossed extensions we construct. In particular, for the group $G=\mathbb{Z}_2$ acting as inversion on a discriminant form or metric group, we explicitly construct a braided $\mathbb{Z}_2$-crossed tensor category with a nondegenerate braiding that generalises the Tambara-Yamagami category, now with more than one simple object in the twisted sector. Its equivariantisation yields the modular tensor category of the orbifold of a lattice vertex operator algebra under a lift of $-\mathrm{id}$. In order to derive the above result, we show that $G$-crossed extensions and condensations by commutative algebras commute appropriately, leading to an effective method for constructing new $G$-crossed extensions. Based on this, we also outline a strategy for addressing the general problem of lattice orbifolds.

Kaniadakis deformed κ-mathematics is an area of mathematics that has found relevance in the analysis of complex systems. Specifically, the mathematical framework in the context of a first-order decay κ-differential equation is investigated, facilitating an in-depth examination of the κ-mathematical structure. This framework serves as a foundational platform, representing the simplest non-trivial setting for such inquiries which are demonstrated for the first time in the literature. Finally, additional avenues of study are discussed

In a previous paper, one of us interpreted mod 2 Dyer-Lashof operations as explicit A-module extensions between Brown-Gitler modules, and showed these A-modules can be topologically realized by finite spectra occurring as fibers of maps between 2-local dual Brown-Gitler spectra. Starting from these constructions, in this paper, we show that infinite families of these finite spectra are of chromatic type 2, with mod 2 cohomology that is free over A(1). Applications include classifying the dual Brown-Gitler spectra after localization with respect to K-theory.

In this paper, we study the biased random transposition shuffle, a natural generalization of the classical random transposition shuffle studied by Diaconis and Shahshahani. We diagonalize the transition matrix of the shuffle and use these eigenvalues to prove that the shuffle exhibits total variation cutoff at time $t_N = \frac{1}{2b} N \log N$ with window $N$. We also prove that the limiting distribution of the number of fixed cards near the cutoff time is Poisson.

We introduce derangetropy, a novel functional measure designed to characterize the dynamics of information within probability distributions. Unlike scalar measures such as Shannon entropy, derangetropy offers a functional representation that captures the dispersion of information across the entire support of a distribution. By incorporating self-referential and periodic properties, it provides deeper insights into information dynamics governed by differential equations and equilibrium states. Through combinatorial justifications and empirical analysis, we demonstrate the utility of derangetropy in depicting distribution behavior and evolution, providing a new tool for analyzing complex and hierarchical systems in information theory.

Key generation, a pillar in physical-layer security (PLS), is the process of the exchanging signals from two legitimate users (Alice and Bob) to extract a common key from the random, common channels. The drawback of extracting keys from wireless channels is the ample dependence on the dynamicity and fluctuations of the radio channel, rendering the key vulnerable to estimation by Eve (an illegitimate user) in low-entropy environments because of insufficient randomness. Added to that, the lack of channel fluctuations lower the secret key rate (SKR) defined as the number of bits of key generated per channel use. In this work, we aim to address this challenge by using a reconfigurable intelligent surface (RIS) to produce random phases at certain, carefully curated intervals such that it disrupts the channel in low-entropy environments. We propose an RIS assisted key generation protocol, study its performance, and compare with benchmarks to observe the benefit of using an RIS while considering various important metrics such as key mismatch rate and secret key throughput. Furthermore, we characterize a scaling law as a function of the rate of change of RIS phase switching for the average secret information rate under this protocol. Then, we use both the key throughput and information rate to optimize the overall secrecy rate. Simulations are made to validate our theoretical findings and effectiveness of the proposed scheme showing an improvement in performance when an RIS is deployed.

The next-generation wireless network, 6G and beyond, envisions to integrate communication and sensing to overcome interference, improve spectrum efficiency, and reduce hardware and power consumption. Massive Multiple-Input Multiple Output (mMIMO)-based Joint Communication and Sensing (JCAS) systems realize this integration for 6G applications such as autonomous driving, as it requires accurate environmental sensing and time-critical communication with neighboring vehicles. Reinforcement Learning (RL) is used for mMIMO antenna beamforming in the existing literature. However, the huge search space for actions associated with antenna beamforming causes the learning process for the RL agent to be inefficient due to high beam training overhead. The learning process does not consider the causal relationship between action space and the reward, and gives all actions equal importance. In this work, we explore a causally-aware RL agent which can intervene and discover causal relationships for mMIMO-based JCAS environments, during the training phase. We use a state dependent action dimension selection strategy to realize causal discovery for RL-based JCAS. Evaluation of the causally-aware RL framework in different JCAS scenarios shows the benefit of our proposed framework over baseline methods in terms of the beamforming gain.

Using the language of homotopy type theory (HoTT), we 1) prove a synthetic version of the classification theorem for covering spaces, and 2) explore the existence of canonical change-of-basepoint isomorphisms between homotopy groups. There is some freedom in choosing how to translate concepts from classical algebraic topology into HoTT. The final translations we ended up with are easier to work with than the ones we started with. We discuss some earlier attempts to shed light on this translation process. The proofs are mechanized using the Coq proof assistant and closely follow classical treatments like those by Hatcher.

In this paper, we consider the maximum $k$-edge-colorable subgraph problem. In this problem we are given a graph $G$ and a positive integer $k$, the goal to take $k$ matchings of $G$ such that their union contains maximum number of edges. This problem is NP-hard in cubic graphs, and polynomial time solvable in bipartite graphs as we observe in our paper. We present two NP-hardness results for two versions of this problem where we have weights on edges or color constraints on vertices. In fact, we show that these versions are NP-hard already in bipartite graphs of maximum degree three. In order to achieve these results, we establish a connection between our problems and the problem of construction of special maximum matchings considered in the Master thesis of the author and defended back in 2003.

Kennard, Khalili Samani, and Searle showed that for a $\mathbb{Z}_2$-torus acting on a closed, positively curved Riemannian $n$-manifold, $M^{n}$, with a non-empty fixed point set for $n$ large enough and $r$ approximately half the dimension of $M$, then $M^n$ is homotopy equivalent to $S^n$, $\mathbb{R}\mathrm{P}^{n}$, $\mathbb{C}\mathrm{P}^{\frac{n}{2}}$, or a lens space. In this paper, we lower $r$ to approximately $2n/5$ and show that we still obtain the same result.

We survey a selection of Yasha Eliashberg's contributions to the philosophy of the h-principle, with a focus on the simplification of singularities and its applications.

We consider the Brown measure of $a+\mathfrak{c}$, where $a$ lies in a commutative tracial von Neumann algebra $\mathcal{B}$ and $\mathfrak{c}$ is a $\mathcal{B}$-valued circular element. Under certain regularity conditions on $a$ and the covariance of $\mathfrak{c}$ this Brown measure has a density with respect to the Lebesgue measure on the complex plane which is real analytic apart from jump discontinuities at the boundary of its support. With the exception of finitely many singularities this one-dimensional spectral edge is real analytic. We provide a full description of all possible edge singularities as well as all points in the interior, where the density vanishes. The edge singularities are classified in terms of their local edge shape while internal zeros of the density are classified in terms of the shape of the density locally around these points. We also show that each of these countably infinitely many singularity types occurs for an appropriate choice of $a$ when $\mathfrak{c}$ is a standard circular element. The Brown measure of $a+\mathfrak{c}$ arises as the empirical spectral distribution of a diagonally deformed non-Hermitian random matrix with independent entries when its dimension tends to infinity.

Given a graph $H$, we say that a graph $G$ is properly rainbow $H$-saturated if: (1) There is a proper edge colouring of $G$ containing no rainbow copy of $H$; (2) For every $e \notin E(G)$, every proper edge colouring of $G+e$ contains a rainbow copy of $H$. The proper rainbow saturation number $\text{prsat}(n,H)$ is the minimum number of edges in a properly rainbow $H$-saturated graph. In this paper we use connections to the classical saturation and semi-saturation numbers to provide new upper bounds on $\text{prsat}(n,H)$ for general cliques, cycles, and complete bipartite graphs. We also provide some general lower bounds on $\text{prsat}(n,H)$ and explore several other interesting directions.

Let $R$ be a commutative ring. One may ask when a general $R$-module $P$ that satisfies $P \oplus R \cong R^n$ has a free summand of a given rank. M. Raynaud translated this question into one about sections of certain maps between Stiefel varieties: if $V_r(\mathbb{A}^n)$ denotes the Stiefel variety $\textrm{GL}(n) / \textrm{GL}(n-r)$ over a field $k$, then the projection $V_r(\mathbb{A}^n) \to V_1(\mathbb{A}^n)$ has a section if and only if the following holds: any module $P$ over any $k$-algebra $R$ with the property that $P \oplus R \cong R^n$ has a free summand of rank $r-1$. Using techniques from $\mathbb{A}^1$-homotopy theory, we characterize those $n$ for which the map $V_r(\mathbb{A}^n) \to V_1(\mathbb{A}^n)$ has a section in the cases $r=3,4$ under some assumptions on the base field. We conclude that if $P \oplus R \cong R^{24m}$ and $R$ contains a field of characteristic $0$, then $P$ contains a free summand of rank $2$. If $R$ contains a quadratically closed field of characteristic $0$, or the field of real numbers, then $P$ contains a free summand of rank $3$. The analogous results hold for schemes and vector bundles over them.