arXiv Math @arxiv_math@qoto.org

I prove the statement in the title using results from arXiv:2404.07646(2). This shows that Question~1.1 in [1] has negative answer for certain expansions of a valued field.

For each orientation-preserving homotopy equivalence between two closed oriented smooth manifolds, there are mainly two different approaches to the higher $ρ$ invariant associated to this homotopy equivalence. In this article, we show that these two definitions of the higher $ρ$ invariant are equivalent.

Given a graph $A$ on a group $G$ and an equivalence relation $B$ on $G$, the $B$ super$A$ graph, whose vertex set is $G$ and two vertices $g$, $h$ are adjacent if and only if there exist $g^{\prime} \in[g]$ and $h^{\prime} \in[h]$ such that $g^{\prime}$ and $h^{\prime}$ are adjacent in $A$. Recently, Dalal \emph{et al.} (Spectrum of super commuting graphs of some finite groups, \textit{Computational and Applied Mathematics}, 43(6):348, 2024) obtain the Laplacian spectrum of supercommuting graphs of certain non-abelian groups including the dihedral group and the generalized quaternion group. In this paper, we continue the study of Laplacian spectrum of certian $B$ super$A$ graphs. We obtain the Laplacian spectrum of conjugacy superenhanced power graphs of certain non-abelian groups, namely: dihedral group, generalized quaternion group and semidihedral group. Moreover to enhance the work of Dalal \emph{et al}, we obtain the Laplacian spectrum of conjugacy supercommuting graph of semidihedral group. We prove that graphs considered in this paper are $L$-integral.

In this paper, we propose some accelerated methods for solving optimization problems under the condition of relatively smooth and relatively Lipschitz continuous functions with an inexact oracle. We consider the problem of minimizing the convex differentiable and relatively smooth function concerning a reference convex function. The first proposed method is based on a similar triangles method with an inexact oracle, which uses a special triangular scaling property for the used Bregman divergence. The other proposed methods are non-adaptive and adaptive (tuning to the relative smoothness parameter) accelerated Bregman proximal gradient methods with an inexact oracle. These methods are universal in the sense that they are applicable not only to relatively smooth but also to relatively Lipschitz continuous optimization problems. We also introduced an adaptive intermediate Bregman method which interpolates between slower but more robust algorithms non-accelerated and faster, but less robust accelerated algorithms. We conclude the paper with the results of numerical experiments demonstrating the advantages of the proposed algorithms for the Poisson inverse problem.

Frequently, when dealing with many machine learning models, optimization problems appear to be challenging due to a limited understanding of the constructions and characterizations of the objective functions in these problems. Therefore, major complications arise when dealing with first-order algorithms, in which gradient computations are challenging or even impossible in various scenarios. For this reason, we resort to derivative-free methods (zeroth-order methods). This paper is devoted to an approach to minimizing quasi-convex functions using a recently proposed comparison oracle only. This oracle compares function values at two points and tells which is larger, thus by the proposed approach, the comparisons are all we need to solve the optimization problem under consideration. The proposed algorithm to solve the considered problem is based on the technique of comparison-based gradient direction estimation and the comparison-based approximation normalized gradient descent. The normalized gradient descent algorithm is an adaptation of gradient descent, which updates according to the direction of the gradients, rather than the gradients themselves. We proved the convergence rate of the proposed algorithm when the objective function is smooth and strictly quasi-convex in $\mathbb{R}^n$, this algorithm needs $\mathcal{O}\left( \left(n D^2/\varepsilon^2 \right) \log\left(n D / \varepsilon\right)\right)$ comparison queries to find an $\varepsilon$-approximate of the optimal solution, where $D$ is an upper bound of the distance between all generated iteration points and an optimal solution.

We describe Martin boundary of the path space of $r$-differential version of Young--Fibonacci graph. Also we establish ergodicity of the corresponding measures.

We study automorphisms and invariants for the algebra $\mathbb{O}$ of octonions and octonionic slice regular functions $f:\mathbb{O} \to \mathbb{O}$.

We introduce a new class of singular complex manifolds and we develop a degenerate Kodaira-Hodge theory for this class of singular manifolds. As an application we prove the existence of Lagranien cycles which do not meet the canonical divisor in complex surfaces of general type.

We give a simple presentation of the six quaternionic equiangular lines in $\mathbb{H}^2$ as an orbit of the primitive quaternionic reflection group of order 720 (which is isomorphic to 2.A_6 the double cover of $A_6)$. Other orbits of this group are also seen to give optimal spherical designs (packings) of 10, 15 and 20 lines in $\mathbb{H}^2$, with angles { 1/3, 2/3 }, { 1/4, 5/8 } and { 0, 1/3, 2/3 }, respectively. We consider the origins of this reflection group as one of Blichfeldt's "finite collineation groups" for lines in $\mathbb{C}^4$, and general methods for finding nice systems of quaternionic lines.

Recently Shekhar Suman [arXiv: 2407.07121v6 [math.GM] 3 Aug 2024] made an attempt to prove the irrationality of $ζ(5)$. But unfortunately the proof is not correct. In this note, we discuss the fallacy in the proof.

In this paper, we shall compare two metrics in terms of orderly dependence, a notion developed in exponential vector space in the article 'Basis and Dimension of Exponential Vector Space' by Jayeeta Saha and Sandip Jana in Transactions of A. Razmadze Mathematical Institute Vol. 175 (2021), issue 1, 101-115. Exponential vector space, in short 'evs', is a partially ordered space associated with a commutative semigroup structure and a compatible scalar multiplication. In the present paper we shall show that the collection $\mathcal{D(\mathbf X)}$ of all metrics on a non-empty set $\mathbf X$, together with the constant function zero $O$, forms a topological exponential vector space. We shall discuss the orderly dependence of two metrics through our findings of a basis of $\mathcal{D(\mathbf X)}\smallsetminus\{O\}$ in different scenario. We shall characterise orderly independence of two elements of a topological evs in terms of the comparing function, another mechanism developed in topological exponential vector space, which can measure the degree of comparability of two elements of an evs. Finally, we shall discuss existence of orderly independent norms on a linear space. For an infinite dimensional linear space we shall construct a large number of orderly independent norms depending on the dimension of the linear space. Orderly independent norms are precisely those which are totally non-equivalent, in the sense that they produce incomparable topologies.

This paper aims to make a mark in the future of sustainable robotics, where efficient algorithms are required to carry out tasks like environmental monitoring and precision agriculture efficiently. We proposed a hybrid algorithm that combines Artificial Bee Colony (ABC) with Levy flight to optimize adaptive sensor placement alongside an important notion of hotspots from domain knowledge experts. By enhancing exploration and exploitation, our approach significantly improves the identification of critical hotspots. This algorithm also finds its usecases for broader search and rescue operations applications, demonstrating its potential in optimization problems across various domains.

We establish interior regularity results for first-order, stationary, local mean-field game (MFG) systems. Specifically, we study solutions of the coupled system consisting of a Hamilton-Jacobi-Bellman equation $H(x, Du, m) = 0$ and a transport equation $-\operatorname{div}(m D_pH(x, Du, m)) = 0$ in a domain $Ω\subset \mathbb{R}^d$. Under suitable structural assumptions on the Hamiltonian $H$, without requiring monotonicity of the system, convexity of the Hamiltonian, separability in variables, or smoothness beyond basic continuity in $(p,m)$, we introduce a notion of weak solutions that allows the application of techniques from elliptic regularity theory. Our main contribution is to prove that the value function $u$ is locally Hölder continuous in $Ω$. The proof leverages the connection between first-order MFG systems and quasilinear equations in divergence form, adapting classical techniques to handle the specific structure of MFG systems.

We construct a series of classic vorticity solutions for incompressible Euler equation on $\mathbb S^2$, which constitute the $C^1$ type regularization for a general traveling point vortex system. The construction is accomplished by applying tangent mapping on $\mathbb S^2$ and Lyapunov--Schmidt reduction argument. Using the fixed-point theorem and a finite dimensional equation on vortex dynamics, we prove that the vortices are located near a nondegenerate critical point of Kirchhoff--Routh function. Moreover, in the tangent space at each vortex center, the scaled stream function is verified as a perturbation of the ground state for generalized plasma problem. Some other qualitative and quantitative estimates for the regularization series are also obtained in this paper.

We consider a generalized derivative nonlinear Schr\''odinger equation. We prove existence of wave operator under an explicit smallness of the given asymptotic states. Our method bases on studying the associated system used in \cite{Tinpaper4}. Moreover, we show that if the initial data is small enough in $H^2(\mathbb{R})$ then the associated solution scatters up to a Gauge transformation.

The random population control decision problem asks for the existence of a controller capable of gathering almost-surely a whole population of identical finite-state agents simultaneously in a final state. The controller must be able to satisfy this requirement however large the population, provided that it is finite. The problem was previously known to be decidable and EXPTIME-hard. This paper tackles the exact complexity: the problem is EXPTIME-complete.

We implement the quantum inverse scattering method for the 4-vertex model. In comparison to previous works of the author which examined the 6-vertex, and 20-vertex, models, the 4-vertex model exhibits different characteristics, ranging from L-operators expressed in terms of projectors and Pauli matrices to algebraic and combinatorial properties, including Poisson structure and boxed plane partitions. With far fewer computations with an L-operator provided for the 4-vertex model by Bogoliubov in 2007, in comparison to those for L-operators of the 6, and 20, vertex models, from lower order expansions of the transfer matrix we derive a system of relations from the structure of operators that can be leveraged for studying characteristics of the higher-spin XXX chain in the weak finite volume limit. In comparison to quantum inverse scattering methods for the 6, and 20, vertex models which can be used to further study integrability, and exact solvability, an adaptation of such an approach for the 4-vertex model can be used to approximate, asymptotically in the weak finite volume limit, sixteen brackets which generate the Poisson structure. From explicit relations for operators of the 4-vertex transfer matrix, we conclude by discussing corresponding aspects of the Yang-Baxter algebra, which is closely related to the operators obtained from products of L-operators for approximating the transfer, and quantum monodromy, matrices. The structure of computations from L-operators of the 4-vertex model directly transfers to L-operators of the higher-spin XXX chain, revealing a similar structure of another Yang-Baxter algebra of interest.

This paper, in the first step, develops the system of bipolar fuzzy relational equations (FRE) to the most general case where the bipolar FREs are defined by an arbitrary continuous t-norm. Also, since fuzzy relational equations are special cases of the bipolar FREs, the proposed system can be also interpreted as a generalization of traditional FREs where the fuzzy compositions are generally defined by any continuous t-norm. The consistency of the continuous bipolar FREs is initially investigated and some necessary and sufficient conditions are derived for determining the feasibility of the proposed system. Subsequently, the feasible solutions set of the problem is completely characterized. It is shown that unlike FREs and those bipolar FREs defined by continuous Archimedean t-norms, the feasible solutions set of the generalized bipolar FREs is formed as the union of a finite number of compact sets that are not necessarily connected. Moreover, five techniques have also been introduced with the aim of simplifying the current problem, and then an algorithm is accordingly presented to find the feasible region of the problem. In the second step, a new class of optimization models is studied where the constraints are defined by the continuous bipolar FREs and the objective function covers a wide range of (non)linear functions such as maximum function, geometric mean function, log-sum-exp function, maximum eigenvalue of a symmetric matrix, support function of a set, etc. It is proved that the problem has a finite number of local optimal solutions and a global optimal solution can always be obtained by choosing a point having the minimum objective value compared to all the local optimal solutions. Finally, to illustrate the discussed study, a step-by-step example is described in several sections, whose constraints are a system of the bipolar FRES defined by Dubois-Prade t-norm.

We consider the uniqueness of the following positive solutions of anisotropic elliptic equation: \begin{equation}\nonumber \left\{ \begin{aligned} -Δ^F _p u&=u^q \quad \text{in} \quad Ω, u&=0 \quad \text{on} \quad \partial Ω, \end{aligned} \right. \end{equation} where $p>\frac{3}{2}$ is a constant. We utilize the linearized method to derive the uniqueness results, which extends the conclusion obtained by L. Brasco and E. Lindgren.

We classify real or complex finite-dimensional $C^*$-algebras and their underlying fields from the properties of Birkhoff-James orthogonality. Application to strong Birkhoff-James orthogonality preservers is also given.