Lie perfect, Lie central extension and generalization of nilpotency in multiplicative Lie algebras

summary:This paper aims to introduce and explore the concept of Lie perfect multiplicative Lie algebras, with a particular focus on their connections to the central extension theory of multiplicative Lie algebras. The primary objective is to establish and provide proof for a range of results derived from Lie perfect multiplicative Lie algebras. Furthermore, the study extends the notion of Lie nilpotency by introducing and examining the concept of local nilpotency within multiplicative Lie algebras. The paper presents an innovative adaptation of the Hirsch-Plotkin theorem specifically tailored for multiplicative Lie algebras.\looseness -

Polyanalytic Besov spaces and approximation by dilatations

summary:Using partial derivatives $\partial f / \partial z$ and $\partial f / \partial \bar {z}$, we introduce Besov spaces of polyanalytic functions in the open unit disk, as well as in the upper half-plane. We then prove that the dilatations of functions in certain weighted polyanalytic Besov spaces converge to the same functions in norm. When restricted to the open unit disk, we prove that each polyanalytic function of degree $q$ can be approximated in norm by polyanalytic polynomials of degree at most $q$

New equivalent conditions for Hardy-type inequalities

summary:We consider a Hardy-type inequality with Oinarov's kernel in weighted Lebesgue spaces. We give new equivalent conditions for satisfying the inequality, and provide lower and upper estimates for its best constant. The findings are crucial in the study of oscillation and non-oscillation properties of differential equation solutions, as well as spectral properties

Asymptotic modeling of the transient response of nonlinear Kelvin-Voigt viscoelastic thin plates with Norton or Tresca friction by Trotter theory

summary:We study the dynamic response of a thin viscoelastic plate made of a nonlinear Kelvin-Voigt material in bilateral contact with a rigid body along a part of its lateral boundary with Norton or Tresca friction. We opt for a direct use of the Trotter theory of convergence of semi-groups of operators acting on variable spaces. Depending on the various relative behaviors of the physical and geometrical data of the problem, the asymptotic analysis of its unique solution leads to different limit models whose properties are detailed. We highlight the appearance of an additional state variable that allows us to write these limit systems of equations in the same form as the genuine problem

On the stability analysis of Darboux problem on both bounded and unbounded domains

summary:In this paper, we first investigate the existence and uniqueness of solution for the Darboux problem with modified argument on both bounded and unbounded domains. Then, we derive different types of the Ulam stability for the proposed problem on these domains. Finally, we present some illustrative examples to support our results

On feebly nil-clean rings

summary:A ring $R$ is feebly nil-clean if for any $a\in R$ there exist two orthogonal idempotents $e,f\in R$ and a nilpotent $w\in R$ such that $a=e-f+w$. Let $R$ be a 2-primal feebly nil-clean ring. We prove that every matrix ring over $R$ is feebly nil-clean. The result for rings of bounded index is also obtained. These provide many classes of rings over which every matrix is the sum of orthogonal idempotent and nilpotent matrices

Some properties of generalized distance eigenvalues of graphs

summary:Let $G$ be a simple connected graph with vertex set $V(G)=\{v_1,v_2,\dots ,v_n \}$ and edge set $E(G)$, and let $d_{v_{i}}$ be the degree of the vertex $v_i$. Let $D(G)$ be the distance matrix and let $T_r(G)$ be the diagonal matrix of the vertex transmissions of $G$. The generalized distance matrix of $G$ is defined as $D_\alpha (G)=\alpha T_r(G)+(1-\alpha )D(G)$, where $0\leq \alpha \leq 1$. Let $\lambda _1(D_{\alpha }(G))\geq \lambda _2(D_{\alpha }(G)) \geq \ldots \geq \lambda _n(D_{\alpha }(G))$ be the generalized distance eigenvalues of $G$, and let $k$ be an integer with $1\leq k\leq n$. We denote by $S_{k}(D_{\alpha }(G))=\lambda _{1}(D_{\alpha }(G)) +\lambda _{2}(D_{\alpha }(G))+\ldots +\lambda _{k}(D_{\alpha }(G))$ the sum of the $k$ largest generalized distance eigenvalues. The generalized distance spread of a graph $G$ is defined as $D_{\alpha }S(G)=\lambda _{1}(D_{\alpha }(G))-\lambda _{n}(D_{\alpha }(G))$. We obtain some bounds on $S_k((D_{\alpha }(G)))$ and $D_{\alpha }S(G)$ of graph $G$, respectively

Ridge estimation of covariance matrix from data in two classes

summary:This paper deals with the problem of estimating a covariance matrix from the data in two classes: (1) good data with the covariance matrix of interest and (2) contamination coming from a Gaussian distribution with a different covariance matrix. The ridge penalty is introduced to address the problem of high-dimensional challenges in estimating the covariance matrix from the two-class data model. A ridge estimator of the covariance matrix has a uniform expression and keeps positive-definite, whether the data size is larger or smaller than the data dimension. Furthermore, the ridge parameter is tuned through a cross-validation procedure. Lastly, the proposed ridge estimator is verified with better performance than the existing estimator from the data in two classes and the traditional ridge estimator only from the good data