arXiv Math @arxiv_math@qoto.org

In acoustical engineering, analytical methodologies are often restricted to two or three dimensions; however, a general-dimensional approach can enhance learning and implementation efficiency while providing a unified understanding of foundational principles. In this manuscript, we present a straightforward derivation of the polar coordinate solutions of the homogeneous Helmholtz equation in general dimensions. We define the radial function as a special function, with coefficients selected to maintain orthonormality within a reproducing kernel Hilbert space, which simplifies its kernel representation. Additionally, we derive an orthonormal basis for this space, thereby demonstrating that the addition theorem arises naturally from a property of the reproducing kernel.

Let $n$ and $k$ be two positive integers with $k\leq n$ and $C$ an $n \times n$ matrix with nonnegative entries. In this paper, the rank-$k$ numerical range in the max algebra setting is introduced and studied. The related notions of the max joint $k$-numerical range and the max joint $C$-numerical range of an entry-wise nonnegative matrix and an $m$-tuple of nonnegative matrices are also introduced. Some interesting algebraic properties of these concepts are investigated.

There is a class of physical filtration processes where the input is adequately modeled by a continuous periodic function f (x) of bounded variation over its period, and the output depends only on certain harmonics of the Fourier expansion of f (x) in the orthogonal basis of trigonometric functions. One example is the discrete spectrum sound generation by a revolving body in a steady fluid flow. This type of sound can be controlled through the amplitudes of certain harmonics of the circular distribution of the inflow velocity. Attainable goals of such a controlled distortion of f (x) are formulated as fuzzy targets and required inequalities relating the integral characteristics of f (x) with the coefficients of the corresponding Fourier expansion are proven.

The JEFT is the acronym for the Joint-Eigenspace Fourier Transform defined on a noncompact symmetric space. It is a consequence of a general construction of a Fourier transform modelled on the Harish-Chandra Fourier transform (on a semi-simple Lie group with finite centre) which (on the corresponding symmetric space of the noncompact type) serves as the Poisson-completion of the famous Helgason Fourier transform

In this paper, our main focus is expressing real numbers on the non-integer bases. We denote those bases as $β$'s, which is also a real number and $β\in (1,2)$. This project has 3 main parts. The study of expansions of real numbers in such bases and algorithms for generating them will contribute to the first part of the paper. In this part, firstly, we will define those expansions as the sums of fractions with $1$'s or $0$'s in the nominator and powers of $β$ in the denominator. Then we will focus on the sequences of $1$'s and $0$'s generated by the nominators of in the sums we mentioned above. Such sequences will be called \textit{coefficient sequences} throughout the paper. In the second half, we will study the results in the first chapter of \cite{erdos1990characterization}, namely the greedy and lazy $β$-expansions . The last part of the paper will be on the characterisation of lazy expansion of 1, which was the first open question at the end of \textit{Erdos and Komornik}. I still don't know if that problem has been solved already. However, the solution that was presented here is the original work of mine.

A brief introduction to universal algebra and the theory of topological algebras, their varieties, and free topological algebras is presented. Free topological Mal'tsev algebras are studied. Their properties, relationship with topological groups and heaps, and the problem of direct limit decomposition are considered.

We present some results related to Hahn-Banach extension theorem for linear operators on asymmetric normed spaces. L. Nachbin, Trans. Amer. Math. Soc. 68 (1950), proved that a Banach space has the extension property for linear operators (a property also called injectivity) if and only if it has the Binary Intersection Property (BIP), meaning that every family of mutually intersecting closed balls has nonempty intersection. Its analog for quasi-metric spaces, called mixed BIP, was considered by Kemajou et al. Topology Appl. 159 (2012). The equivalence of mixed BIP to the injectivity of an asymmetric normed space was proved by Conradie et al., Topology Appl. 231 (2017), derived from some properties of the injective hull of a quasi-metric space. The aim of the present paper is to give a direct proof of this result by adapting Nachbin's ideas to the asymmetric case. Keywords: quasi-metric space, asymmetric normed space, injectivity, binary intersection property, hyperconvexity, Isbell-completeness, Isbell-convexity

Let $k$ be an algebraically closed field of characteristic $p\neq 0$. Let $G$ be a connected reductive group over $k$, $P \subseteq G$ be a parabolic subgroup and $λ: P \longrightarrow G$ be a strictly anti-dominant character. Let $C$ be a projective smooth curve over $k$ with function field $K=k(C)$ and $F$ be a principal $G$-bundle on $C$. Then $F/P \longrightarrow C$ is a flag bundle and $\mathcal{L}_λ=F \times_P k_λ$ on $F/P$ is a relatively ample line bundle. We compute the height filtration and successive minima of the height function $h_{\mathcal{L}_λ}: X(\overline{K}) \longrightarrow \mathbb{R}$ over the flag variety $X=(F/P)_K$.

This paper provides an exploration of parking functions, a classical combinatorial object. We present two viewpoints on their structure and properties: through poset of noncrossing partitions and polytopes.

In this work derivations of definite integrals listed in Prudnikov volume I, Gradshteyn and Ryzhik and a few other tables are produced. Special cases of these integrals in terms of fundamental constants are also evaluated. The method used in these derivations is contour integration.

We find implicit general solutions for modified eikonal equations $u_a u_a=F(u_t)$, where lower indices at dependent variables designate derivatives, $a=1,2,..,n$, and summation is implied over the repeated indices. We will consider the general solutions and symmetries of these equations.

In this paper, we prove a sharp and strong non-uniqueness for a class of weak solutions to the incompressible Navier-Stokes equations in $\R^3$. To be more precise, we exhibit the non-uniqueness result in a strong sense, that is, any weak solution is non-unique in L^p([0,T];L^\infty(\R^3)) with 1\le p<2. Moreover, this non-uniqueness result is sharp with regard to the classical Ladyzhenskaya-Prodi-Serrin criteria at endpoint (2, \infty), which extends the sharp nonuniqueness for the Navier-Stokes equations on torus $\TTT^3$ in the recent groundbreaking work (Cheskidov and Luo, Invent. Math., 229 (2022), pp. 987-1054) to the setting of the whole space. The key ingredient is developing a new iterative scheme that balances the compact support of the Reynolds stress error with the non-compact support of the solution via introducing incompressible perturbation fluid.

In this paper, we consider a fractional p-Laplacian system of equations in the entire space RN with doubly critical singular nonlinearities involving a local critical Sobolev term together with a nonlocal Choquard critical term; the problem also includes a homogeneous singular Hardy term; moreover, all the nonlinearities involve singular critical weights. To prove the main result we use a refinement of Sobolev inequality that is related to Morrey space because our problem involves doubly critical exponentss and a version of the Caffarelli-Kohn-Nirenberg inequality. With the help of these results, we provide sufficient conditions under which a weak nontrivial solution to the problem exists via variational methods.

In the critical beta-splitting model of a random $n$-leaf rooted tree, clades are recursively split into sub-clades, and a clade of $m$ leaves is split into sub-clades containing $i$ and $m-i$ leaves with probabilities $\propto 1/(i(m-i))$. Study of structure theory and explicit quantitative aspects of the model is an active research topic. It turns out that many results have several different proofs, and detailed studies of analytic proofs are given elsdewhere (via analysis of recursions and via Mellin transforms). This article describes two core probabilistic methods for studying $n \to \infty$ asymptotics of the basic finite-$n$-leaf models. (i) There is a canonical embedding into a continuous-time model, that is a random tree CTCS(n) on $n$ leaves with real-valued edge lengths, and this model turns out to be more convenient to study. The family (CTCS(n), $n \ge 2)$ is consistent under a ``delete random leaf and prune" operation. That leads to an explicit inductive construction (the {\em growth algorithm}) of (CTCS(n), $n \ge 2)$ as $n$ increases, and then to a limit structure CTCS$(\infty)$ which can be formalized via exchangeable partitions, in some ways analogous to the Brownian continuum random tree. (ii) There is an explicit description of the limit fringe distribution relative to a random leaf, whose graphical representation is essentially the format of the cladogram representation of biological phylogenies.

We study the asymptotic behavior of the maximum indegree in the spatial preferential attachment model with a choice-based edge step. We prove different types of behavior of maximal indegree based on the model's parameters.

We characterize sums of normal ultrafilters after the Magidor iteration of Prikry forcings over a discrete set of measurable cardinals. We apply this to show that the weak Ultrapower Axiom is not equivalent to the Ultrapower Axiom. We also construct a non-rigid ultrapower and two uniform ultrafilters on different cardinals that have the same ultrapower.

A construction that assigns a Boolean 1D TQFT with defects to a finite state automaton was recently developed by Gustafson, Im, Kaldawy, Khovanov, and Lihn. We show that the construction is functorial with respect to the category of finite state automata with transducers as morphisms. Certain classes of subregular languages correspond to additional cohomological structures on the associated TQFTs. We also show that the construction generalizes to context-free grammars through a categorical version of the Chomsky-Schützenberger representation theorem, due to Melliès and Zeilberger. The corresponding TQFTs are then described as morphisms of colored operads on an operad of cobordisms with defects.

We propose a $C^0$ interior penalty method for the fourth-order stream function formulation of the surface Stokes problem. The scheme utilizes continuous, piecewise polynomial spaces defined on an approximate surface. We show that the resulting discretization is positive definite and derive error estimates in various norms in terms of the polynomial degree of the finite element space as well as the polynomial degree to define the geometry approximation. A notable feature of the scheme is that it does not explicitly depend on the Gauss curvature of the surface. This is achieved via a novel integration-by-parts formula for the surface biharmonic operator.

In this paper we obtain the Wedderburn-Artin decomposition of a semisimple group algebra associated to a direct product of finite groups. We also provide formulae for the number of all possible group codes, and their dimensions, that can be constructed in a group algebra. As particular cases, we present the complete algebraic description of the group algebra of any direct product of groups whose direct factors are cyclic, dihedral, or generalised quaternion groups. Finally, in the specific case of semisimple dihedral group algebras, we give a method to build quantum error-correcting codes, based on the CSS construction.

The aim of this brief note is to demonstrate that the boundary pair of a dissipative operator is determined by the unitary boundary pair of its symmetric part.