arXiv Math @arxiv_math@qoto.org

In this article, firstly, some simple and smoothness properties of the weighted numerical radius and the weighted Crawford number functions are investigated. Then, some generalization formulas for lower and upper bounds of the weighted numerical radius function are obtained. Later on, some evaluations for lower and upper bounds of the weighted numerical index are given. The obtained results are generalized some well-known famous results about the special weighted numerical radius and the special weighted Crawford number functions in the recently literature. Also, the different and useful results are provided to the literature.

Defining a family of recurrences, we generalize Comtet's formula for the generating function of the enumeration of indecomposable permutations. Consequently, we generalize Panaitopol's asymptotic expansion for the prime counting function, obtaining asymptotic expansions salient to the first Hardy-Littlewood conjecture.

Two time scale stochastic approximation is analyzed when the iterates on either or both time scales do not necessarily converge.

We present a broader framework for the Cauchy identity derived from the determinant expansion of collocation matrices. This approach yields an infinite family of identities, where the original Cauchy identity stands as a particular case. To illustrate the versatility and depth of this approach, we provide a range of compelling examples, showcasing the connections and applications of these novel identities.

Let $(M,g)$ be a Riemannian manifold, $L(M)$ be its frame bundle, $O(M)$ its orthonormal frame bundle. For a distribution $D$ on $M$ we define a subbundle $L(D)\subset L(M)$ or $O(D)\subset O(M)$ in a natural way. This allows us to consider a lift $Lφ$ of a map $φ:M\to N$ not necessarily being a local diffeomorphism. More precisely, if $φ:M\to N$ is a submersion, then $Lφ:L(\mathcal{H}^φ)\to L(N)$ or $Lφ:O(\mathcal{H}^φ)\to L(N)$, where $\mathcal{H}^φ$ is a horizontal distribution of $φ$. Equipping $L(M)$ and $L(N)$ with the Mok metrics, we study conformality and harmonicity of lifts $Lφ$.

This work aims to establish the asymptotic behavior of Carleman weight functions when discrete difference and average operators are applied. We provide a characterization of the error term in arbitrary order and dimension, extending previously known results. This generalization is of independent interest due to its applications in deriving discrete deterministic and stochastic Carleman estimates where the asymptotic behavior of the Carleman weight functions is crucial. The aforementioned asymptotic behavior holds for Carleman weight functions used for parabolic, hyperbolic, and elliptic operators, which are applied to obtain control and inverse problems results for those operators.

In this note we evaluate c-Entropy of perturbed L-systems introduced in [5]. Explicit formulas relating the c-Entropy of the L-systems and the perturbation parameter are established. We also show that c-Entropy attains its maximum value (finite or infinite) whenever the perturbation parameter vanishes so that the impedance function of such a L-system belongs to one of the generalized (or regular) Donoghue classes.

This work concerns results on conditions guaranteeing that certain banded $M$-matrices have banded inverses. As a first goal, a graph theoretic characterization for an off-diagonal entry of the inverse of an $M$-matrix to be positive, is presented. This result, in turn, is used in providing alternative graph theoretic proofs of the following: (1) a characterization for a tridiagonal $M$-matrix to have a tridiagonal inverse. (2) a necessary condition for an $M$-matrix to have a pentadiagonal inverse. The results are illustrated by several numerical examples.

This paper explores the operational principles and monomiality principle that significantly shape the development of various special polynomial families. We argue that applying the monomiality principle yields novel results while remaining consistent with established findings. The primary focus of this study is the introduction of degenerate multidimensional Hermite-based Appell polynomials (DMHAP), denoted as ${}_{\mathbb{H}}\mathbb{A}^{[r]}_n(l_1, l_2, l_3, \dots, l_r; \vartheta)$. These DMHAP form essential families of orthogonal polynomials, demonstrating strong connections with classical Hermite and Appell polynomials. Additionally, we derive symmetric identities and examine the fundamental properties of these polynomials. Finally, we establish an operational framework to investigate and develop these polynomials further.

We offer an insight into our mathematical endeavors, which aim to advance the foundational understanding of energy systems in a broad context, encompassing facets such as charge transport, energy storage, markets, and collective behavior. Our working techniques include a combination of well-posed mathematical models (both deterministic and stochastic), mathematical analysis arguments (mostly concerned with model dimension reduction and averaging, periodic homogenization), and simulation tools (numerical approximation techniques, computational statistics, high-performance computing).

In this paper, we establish nonlinear Landau damping below survival threshold for collisionless charged particles following the meanfield Vlasov theory near general radial equilibria. In absence of collisions, the long-range Coulomb pair interaction between particles self-consistently gives rise to oscillations, known in the physical literature as plasma oscillations or Langmuir's oscillatory waves, that disperse in space like a Klein-Gordon's dispersive wave. As a matter of fact, there is a non-trivial survival threshold of wave numbers that characterizes the large time dynamics of a plasma: {\em phase mixing} above the threshold driven by the free transport dynamics and {\em plasma oscillations} below the threshold driven by the collective meanfield interaction. The former mechanism provides exponential damping, while the latter is much slower and dictated by Klein-Gordon's dispersion which gives decay of the electric field precisely at rate of order $t^{-3/2}$. Up to date, all the works in the mathematical literature on nonlinear Landau damping fall into the phase mixing regime, in which plasma oscillations were absent. The present work resolves the problem in the plasma oscillation regime. Our nonlinear analysis includes (1) establishing the existence and dispersion of Langmuir's waves, (2) decoupling oscillations from phase mixing in different time regimes, (3) detailing the oscillatory structure of particle trajectories in the phase space, (4) treating plasma echoes via a detailed analysis of particle-particle, particle-wave, and wave-wave interaction, and (5) designing a nonlinear iterative scheme in the physical space that captures both phase mixing and dispersion in low norms and allows growth in time in high norms. As a result, we establish nonlinear plasma oscillations and Landau damping below survival threshold for data with finite Sobolev regularity.

In this paper, we will use the entropy approach to derive a necessary and sufficient condition for the existence of an element that belongs to at least half of the sets in a finite family of sets.

A set $S\subseteq V$ in an isolate-free graph $G$ is a total restrained dominating set, abbreviated TRD-set, if every vertex in $V$ is adjacent to a vertex in $S$, and every vertex in $V\setminus S$ is adjacent to a vertex in $V\setminus S$. A total restrained coalition is made up of two disjoint sets of vertices $X$ and $Y$ of $G$, neither of which is a TRD-set but their union $X\cup Y$ is a TRD-set. A total restrained coalition partition of a graph $G$ is a partition $Φ=\{V_1, V_2,\dots,V_k\}$ such that for all $i \in [k]$, the set $V_i$ forms a total restrained coalition with another set $V_j$ for some $j$, where $j\in [k]\setminus{i}$. The total restrained coalition number $C_{tr}(G)$ in $G$ equals the maximum order of a total restrained coalition partition in $G$. In this work, we initiate the study of total restrained coalition in graphs and its properties.

Let $Ab_0$ be the class of finite abelian groups and consider the function $f:Ab_0\longrightarrow(0,\infty)$ given by $f(G)=\frac{|{\rm Aut}(G)|}{|G|}$\,, where ${\rm Aut}(G)$ is the automorphism group of a finite abelian group $G$. In this short note, we prove that the image of $f$ is a dense set in $[0,\infty)$.

Let $p\in (1,\infty)$ and let $G$ be a discrete group. G. An, J.-J. Lee and Z.-J. Ruan introduced $p$-nuclearity for $L^p$-operator algebras. They proved that the reduced group $L^p$-operator algebra $F^p_λ(G)$ is $p$-nuclear if $G$ is amenable. In this paper, we show that the converse is true. This answers an open problem concerning the $p$-nuclearity for reduced group $L^p$-operator algebras of N. C. Phillips.

We prove a sufficient condition for a \emph{pattern} $π$ on a \emph{triod} $T$ to have \emph{rotation number} $ρ_π$ coincide with an end-point of its \emph{forced rotation interval} $I_π$. Then, we demonstrate the existence of peculiar \emph{patterns} on \emph{triods} that are neither \emph{triod twists} nor possess a \emph{block structure} over a \emph{triod twist pattern}, but their \emph{rotation numbers} are an end point of their respective \emph{forced rotation intervals}, mimicking the behavior of \emph{triod twist patterns}. These \emph{patterns}, absent in circle maps (see \cite{almBB}), highlight a key difference between the rotation theories for \emph{triods} (introduced in \cite{BMR}) and that of circle maps. We name these \emph{patterns}: ``\emph{strangely ordered}" and show that they are semi-conjugate to circle rotations via a piece-wise monotone map. We conclude by providing an algorithm to construct unimodal \emph{strangely ordered patterns} with arbitrary \emph{rotation pairs}.

We consider the asymptotics of orthogonal polynomials for measures that are differentiable, but not necessarily analytic, multiplicative perturbations of Jacobi-like measures supported on disjoint intervals. We analyze the Fokas--Its--Kitaev Riemann--Hilbert problem using the Deift--Zhou method of nonlinear steepest descent and its $\overline \partial$ extension due to Miller and McLaughlin. Our results extend that of Yattselev in the case of Chebyshev-like measures with error bounds that are slightly improved while less regular perturbations are admissible. For the general Jacobi-like case, we present, what appears to be the first result for asymptotics when the perturbation of the measure is only assumed to be twice differentiable, with bounded third derivative.