arXiv Math @arxiv_math@qoto.org

We study {\it permeable} sets. These are sets \(Θ\subset \mathbb{R}^d\) which have the property that each two points \(x,y\in \mathbb{R}^d\) can be connected by a short path \(γ\) which has small (or even empty, apart from the end points of \(γ\)) intersection with \(Θ\). We investigate relations between permeability and Lebesgue measure and establish theorems on the relation of permeability with several notions of dimension. It turns out that for most notions of dimension each subset of \(\mathbb{R}^d\) of dimension less than \(d-1\) is permeable. We use our permeability result on the Nagata dimension to characterize permeability properties of self-similar sets with certain finiteness properties.

In this paper, two classes of quantum MDS codes are constructed. The main tools are multiplicative structures on finite fields. Carefully choosing different cosets can make the corresponding generalized Reed-Solomon codes Hermitian self-orthogonal, which yields quantum MDS codes by utilizing Hermitian type CSS construction. In some cases, these codes have minimal distances larger than $q/2$. Also, the parameters of these codes have never been reported before.

This paper presents a method to establish functional inequalities via fuzzy decomposition on the state space, which generalizes earlier results dealing with exact partitions of the state space. Given a reversible Markov chain on a finite state space, we define its projection chain and restriction chains from classes of a fuzzy partition on the state space. The Poincaré, log-Sobolev and modified log-Sobolev inequalities associated with the original chain can be estimated from those of its projection chain and restriction chains which tend to have simpler structures and hence easier to work with. An application of this generalization is presented in which the fuzzy decomposition method applies but its exact partitioning counterpart does not.

This paper presents a model for a vehicle routing problem in which customer demands are stochastic and vehicles are divided into compartments. The problem is motivated by the needs of certain agricultural cooperatives that produce various types of livestock food. The vehicles and their compartments have different capacities, and each compartment can only contain one type of feed. Additionally, certain farms can only be accessed by specific vehicles, and there may be urgency constraints. To solve the problem, a two-step heuristic algorithm is proposed. First, a constructive heuristic is applied, followed by an improvement phase based on iterated tabu search. The designed algorithm is tested on several instances, including an analysis of real-world datasets where the results are compared with those provided by the model. Furthermore, multiple benchmark instances are created for this problem and an extensive simulation study is conducted. Results are presented for different model parameters, and it is shown that, despite the problems' complexity, the algorithm performs efficiently. Finally, the proposed heuristic is compared to existing solution algorithms for similar problems using benchmark instances from the literature, achieving competitive results.

The number of points on a certain one parameter family of algebraic surface over a finite field $\F_p$ can be expressed as $p^2+A_p(λ),$ where $A_p(λ)$ is a character sum and $λ$ is an element of the finite field $\F_p.$ In this paper, we study the distribution of the term $A_p(λ)$ as the surface varies over a large family of algebraic surfaces of fixed genus and growing $p.$ The power moments of $A_p$'s are weighted sums of Catalan numbers. As a consequence of these results, we obtain limiting distributions of certain families of hypergeometric functions over large finite fields.

We study a fractional birth-death process with state dependent birth and death rates. It is defined using a system of fractional differential equations that generalizes the classical birth-death process introduced by Feller (1939). We obtain the closed form expressions for its transient probabilities using Adomian decomposition method. In this way, we obtain the unknown transient probabilities for the classical birth-death process (see Feller (1968), p. 454). Its various distributional properties are studied. For the case of linear birth and death rates, the obtained results are verified with the existing results. Also, we discuss the cumulative births in the fractional linear birth-death process. Later, we consider a time-changed linear birth-death process where we discuss the asymptotic behaviour of the distribution function of its extinction time at zero.

We analyze the mixing time of a popular shuffling machine known as the shelf shuffler. It is a modified version of a $2m$-handed riffle shuffle ($m=10$ in casinos) in which a deck of $n$ cards is split multinomially into $2m$ piles, the even-numbered piles are reversed, and then cards are dropped from piles proportionally to their sizes. We prove that $\frac{5}{4} \log_{2m} n$ shuffles are necessary and sufficient to mix in total variation, and a cutoff occurs with constant window size. We also determine the cutoff profile in terms of the total variation distance between two shifted normal random variables.

We discuss efficient computation of $f$-centralities and nonbacktracking centralities for time-evolving networks with nonnegative weights. We present a node-level formula for its combinatorially exact computation which proves to be more tractable than previously existing formulae at edge-level for dense networks. Additionally, we investigate the impact of the addition of a final time frame to such a time-evolving network, analyzing its effect on the resulting nonbacktracking Katz centrality. Finally, we demonstrate by means of computational experiments that the node-level formula presented is much more efficient for dense networks than the previously known edge-level formula. As a tool for our goals, in an appendix of the paper, we develop a spectral theory of matrices whose elements are vectors.

In this work, we present a numerical solver for optimal control problems constrained by linear and semi-linear second-order elliptic PDEs. The approach is based on recasting the problem and includes an extension of Uzawa's algorithm to build approximating sequences for these constrained optimal control problems. We prove strong convergence of the iterative scheme in their respective norms, and this convergence is generalised to a class of restricted function spaces. We showcase the algorithm by demonstrating its use numerically with neural network methods that we coin Deep Uzawa Algorithms and show they perform favourably compared with some existing Deep Neural Network approaches.

In this work, we study a two-type critical branching particle system in $\mathbb{R}^{N}$, where particles follow symmetric stable motions, with type-dependent lifetimes and offspring distributions. Our main result is the convergence as $t\to\infty$ of the particle system to a non-trivial limiting population, focusing on two cases: (1) all particle lifetimes have finite mean, or (2) one type has a lifetime distribution with a heavy tail, while the others have finite mean. This complements previous results on extinction \cite{Kevei}. Using the Extended Final Value Theorem, we prove the existence of a limiting distribution for the particle system. The non extinction of the limiting population is demonstrated using a technique inspired in \cite{Fino}. These results describe the long-term behavior of the particle system, highlighting the interaction between mobility, longevity, and offspring variability. Additionally, the study of a particle system with a finite number of types would follow analogously with the techniques presented here. Our approach introduces new techniques for the asymptotic study of critical multitype branching processes.

In this article, we study twisted derivations of cyclic group rings. Let $R$ be a commutative ring with unity, $G$ be a finite cyclic group, and ($σ, τ$) be a pair of $R$-algebra endomorphisms of the group algebra $RG$, which are $R$-linear extensions of the group endomorphisms of $G$. In this article, we give two characterizations concerning $(σ, τ)$-derivations of the group ring $RG$. First, we develop a necessary and sufficient condition for a $(σ, τ)$-derivation of $RG$ to be inner. Second, we provide a necessary and sufficient condition for an $R$-linear map $D: RG \rightarrow RG$ with $D(1) = 0$ to be a $(σ, τ)$-derivation. We also illustrate our theorems with the help of examples. As a consequence of these two characterizations, we answer the well-known twisted derivation problem for $RG$: Under what conditions are all $(σ, τ)$-derivations of $RG$ inner? Or is the space of outer $(σ, τ)$-derivations trivial? More precisely, we give a sufficient condition under which all $(σ, τ)$-derivations of $RG$ are inner and a sufficient condition under which $RG$ has non-trivial outer $(σ, τ)$-derivations. Our result helps in generating several examples of non-trivial outer derivations.

This letter explores the implementation of a safe control law for systems of dynamically coupled cooperating agents. Under a CBF-based collaborative safety framework, we examine how the maximum safety capability for a given agent, which is computed using a collaborative safety condition, influences safety requests made to neighbors. We provide conditions under which neighbors may be resilient to non-compliance of neighbors to safety requests, and compute an upper bound for the total amount of non-compliance an agent is resilient to, given its 1-hop neighborhood state and knowledge of the network dynamics. We then illustrate our results via simulation on a networked susceptible-infected-susceptible (SIS) epidemic model.

Let $X$ be a double EPW sextic, and $ι$ its anti-symplectic involution. We relate the $ι$-anti-invariant part of the Chow group of zero-cycles of $X$ with Voisin's rational orbit filtration. For a general double EPW sextic $X$, we also relate the anti-invariant part of the Chow motive of $X$ with the motive of a Gushel-Mukai fourfold. As an application, we obtain a similar result for certain Fano varieties of lines in cubics with infinite-order birational automorphisms.

Wireless communications using light waves are called visible light communication. A technique called Physical-layer Network coding allows several users to communicate simultaneously over the same channel in a distributed space-time Coding scheme. Herein, we report a new PNC-VLC one-to-many scheme using physical-layer network coding in VLC system for enhancing throughput by boosting data transmission at relay nodes. We develop an integrated PNC-VLC framework including physical-layer signal processing algorithms and medium access control approaches. Our proposed technique is then modelled mathematically and compared with some of the VLC and PNC-VLC schemes under various channel conditions and scenarios. Our investigation revealed that our proposed technique performed better than traditional PNC schemes in achieving better Bit Error Rate performance in different channel

In this paper, the $C^*$-algebra of the seven-dimensional un-decomposable nilpotent Lie group is characterized explicitly for the first time(see \cite{chin}). Furthermore, the topology of its spectrum is described as a preparation for the analysis of its $C^*$-algebra. Then, the operator-valued Fourier transform is employed to translate the given C*-algebra into the algebra of bounded operator fields through its spectrum. We find the conditions satisfied by the image of the goal to characterize them by these conditions, namely the "norm controlled dual limit" (NCDL)-conditions (see \cite{Lud-Reg}). The methods used for the nilpotent Lie groups are different for each. We consider the co-adjoint orbits and for our case, we use the Kirillov theory.

For $A \in M_m(\mathbb{C})$, Yamamoto's generalization of the spectral radius formula (1967) asserts that $\lim_{n \to \infty} s_j(A^n)^{\frac{1}{n}}$ is equal to the $j^{\textrm{th}}$-largest eigenvalue-modulus of $A$, where $s_j (A^n)$ denotes the $j^{\textrm{th}}$-largest singular value of $A^n$. Let $\mathscr{H}$ be a complex Hilbert space, and for an operator $T \in \mathscr{B}(\mathscr{H})$, let $|T| := (T^*T)^{\frac{1}{2}}$. For $A$ in $\mathscr{B}(\mathscr{H})$, we refer to the sequence, $\{ |A^n|^{\frac{1}{n}} \}_{n \in \mathbb{N} }$, as the $\textit{normalized power sequence}$ of $A$. As our main result, we prove a spatial form of Yamamoto's theorem which asserts that the normalized power sequence of a spectral operator in $\mathscr{B}(\mathscr{H})$ converges in norm, and provide an explicit description of the limit in terms of its idempotent-valued spectral resolution. Our approach substantially generalizes the corresponding result by the first-named author in the case of matrices in $M_m(\mathbb{C})$, and supplements the Haagerup-Schultz theorem on SOT-convergence of the normalized power sequence of an operator in a $II_1$ factor.

We show how to freely add the first layer of quantifier alternation depth to a Boolean doctrine over a small base category. This amounts to a doctrinal version of Herbrand's theorem for formulas with quantifier alternation depth at most one modulo a universal theory. To achieve this, we characterize, within the doctrinal setting, the classes $A$ of quantifier-free formulas for which there is a model $M$ such that $A$ is precisely the class of formulas whose universal closure is valid in $M$.

One of the most important sequences in enumerative combinatorics is OEIS sequence A85, the number of involutions of length n. In the Art of Computer Programming, vol. 3, Don Knuth derived the O(1/n) asymptotic formula for these numbers. In this modest tribute to our two heroes, Neil Sloane who just turned 85, and Don Knuth who was 85 a year ago, we go all the way to an $O(1/n^{85})$ asymptotic formula.

Randomized arcade processes are a class of continuous stochastic processes that interpolate in a strong sense, i.e., omega by omega, between any given ordered set of random variables, at fixed pre-specified times. Utilizing these processes as generators of partial information, a class of continuous-time martingale -- the filtered arcade martingales (FAMs) -- is constructed. FAMs interpolate through a sequence of target random variables, which form a discrete-time martingale. The research presented in this paper treats the problem of selecting the worst martingale coupling for given, convexly ordered, probability measures contingent on the paths of FAMs that are constructed using the martingale coupling. This optimization problem, that we term the information-based martingale optimal transport problem (IB-MOT), can be viewed from different perspectives. It can be understood as a model-free construction of FAMs, in the case where the coupling is not determined, a priori. It can also be considered from the vantage point of optimal transport (OT), where the problem is concerned with introducing a noise factor in martingale optimal transport, similarly to how the entropic regularization of optimal transport introduces noise in OT. The IB-MOT problem is static in its nature, since it is concerned with finding a coupling, but a corresponding dynamical solution can be found by considering the FAM constructed with the identified optimal coupling. Existence and uniqueness of its solution are shown, and an algorithm for empirical measures is proposed.

The $σ_k(A_g)$ curvature and the boundary $\mathcal{B}^k_g$ curvature arise naturally from the Chern--Gauss--Bonnet formula for manifolds with boundary. In this paper, we prove a Liouville theorem for the equation $σ_k(A_g)=1$ in $\overline{\mathbb{R}^n_+}$ with the boundary condition $\mathcal{B}^k_g=c$ on $\partial\mathbb{R}^n_+$, where $g=e^{2v}|dx|^2$ and $c$ is some nonnegative constant. This extends an earlier result of Wei, which assumes the existence of $\lim_{|x|\to\infty}(v(x)+2\log|x|)$. In addition, we establish a local gradient estimate for solutions of such equations, assuming an upper bound on the solution $v$.