64 Results Found

In this paper, we explore the properties of the generalized Lambert transform, the L-transform, the generalized Stieltjes transform, and the Stieltjes–Poisson transform within the framework of Lebesgue spaces. We establish Parseval-type relatio...

In this paper, we study a two-parameter family of Stieltjes transformations related to holomorphic Lambert–Tsallis functions, which are a two-parameter generalization of the Lambert function. Such Stieltjes transformations appear in the study o...

This article is devoted to exploring the properties on the logarithmic growth of entire functions represented by Laplace–Stieltjes transforms of zero order. In order to describe the growth of Laplace–Stieltjes transforms more finely, we i...

The classical Stieltjes transform is modified in such a way as to generalize both Stieltjes and Fourier transforms. This transform allows the introduction of new classes of commutative and non-commutative generalized convolutions. A particular case o...

This paper establishes explicit inversion formulae for the Widder–Lambert transform and the Stieltjes–Poisson transform, extending their applicability to function spaces and compactly supported distributions. Utilizing a generalized Lambe...

In the framework of linear viscoelasticity, the authors have previously calculated a novel inverse Laplace transform involving the Mittag–Leffler function in order to calculate the relaxation modulus in the Andrade model. Here, we generalize th...

This paper aims to develop exchange formulae for the Stieltjes–Poisson transform by using Mellin-type convolutions in the context of weighted Lebesgue spaces. A key result is the introduction of bilinear and continuous Mellin-type convolutions,...

We prove that the Wiener integral, the analytic Wiener integral and the analytic Feynman integral of the first variation of $F\left(x\right)=exp\left\{{\int }_0^T\theta (t,x\left(t\right))dt\right\}$ successfully exist under the certain condition, where $\theta (t,u)={\int }_{R}exp\left\{iuv\right\}d{\sigma }_t($...

This paper presents two analytic mappings defined on probability measures that extend and unify the $(a,b)$- and $(s,t)$-deformations arising in free probability for s, $b>0$ and a, $t\in \mathbb{R}$. These unified operators, denoted ${\mathbf{U}}_{(a,b,s,t)}^{\prime }$ and ${\mathbf{U}}_{(a,b,}^{}$...

This paper establishes a comprehensive analytical framework for a new transformation of probability measures, denoted by ${\mathcal{T}}_{(a,t)}$, which unifies the classical t- and $T_a$-transformations in free probability. We derive the functional equation characterizi...

The article is devoted to reflexive Müntz spaces $M_{\Lambda ,p}$...

In this paper, we propose an analytic deformation acting on probability measures, designed to encompass and extend two fundamental operators in free probability: the $(a,b)$- and the $T_c$-deformations. This unified operator, indicated by ${\mathcal{X}}_{(a,b,c)}$, is int...

The concept of the $(\mathbb{T}=(s,t\left)\right)$-transformation of probability measures, introduced for $s>0$ and $t\in \mathbb{R}$, is examined in this work from the perspective of Cauchy–Stieltjes kernel (CSK) families and their related variance functions (VFs). We calc...

We consider the limit of the empirical spectral distribution of Laplace matrices of generalized random graphs. Applying the Stieltjes transform method, we prove under general conditions that the limit spectral distribution of Laplace matrices converg...

This article proposes a single-deformation scheme for probability measures that simultaneously encompasses two classical deformations from free probability: the $V_a$ deformation ($a\in \mathbb{R}$) and the $T_c$ deformation ($c\in \mathbb{R}$). The associated operator, wri...

Both the stretched exponential function and the Prony series are prominent mathematical frameworks extensively employed to characterize monotonically decaying phenomena across physics and engineering disciplines. Despite being fundamentally distinct...

We study power-mixture type functional equations in terms of Laplace–Stieltjes transforms of probability distributions on the right half-line $[0,\infty ).$ These equations arise when studying distributional equations of the type $Z\stackrel{\mathrm{d}}{=}X+TZ$, where the random va...

In this study, we introduce a novel transformation of probability measures that unifies two significant transformations in free probability theory: the t-transformation and the $V_a$-transformation. Our unified transformation, denoted ${\mathcal{U}}_{(a,t)}$, is defined...

We compare the classical viscoelastic models due to Becker and Lomnitz with respect to a recent viscoelastic model based on the Lambert W function. We take advantage of this comparison to derive new analytical expressions for the relaxation spectrum...

In this paper, we introduce two analytic deformations of probability measures that unify and extend two classical deformations from free probability theory, namely the $\mathbb{T}=(s,t)$-deformation $U^{\mathbb{T}}$ and the $T_a$-deformation, where $a,t\in \mathbb{R}$ and $s>0$. The co...

The resummation of Stieltjes series remains a key challenge in mathematical physics, especially when Padé approximants fail, as in the case of superfactorially divergent series. Weniger’s $\delta$-transformation, which incorporates a prio...

A notion of generalized (two-parameterized) t-transformation of free convolution, also called $(\mathbf{t}=(a,b\left)\right)$-deformed free convolution, is introduced for $a\in \mathbb{R}$ and $b>0$. In this article, some results of $\mathbf{t}$-deformed free convolution are given within th...

The objective of this work is to investigate the $\mathbb{T}=(s,t)$-deformed free convolution ${⊞}_{\mathbb{T}}$ for $s>0$ and $t\in \mathbb{R}$ and to clarify its structural and asymptotic properties within the framework of Cauchy–Stieltjes kernel (CSK) families. The meth...

The Peak Age of Information (PAoI) quantifies the freshness of updates used in cyber-physical systems (CPSs), realized within the Internet of Things (IoT) paradigm, encompassing devices, networks, and control algorithms. Consequently, PAoI is a criti...

In non-Markovian tandem queueing networks the output process of one site, which is the input process to the next site, is not renewal. Consequently, the correlation analysis of that output processes is essential when studying such networks. A correla...

The purpose of this note is to propose an application of the Lambert W function in linear viscoelasticity based on the Bernstein and Stieltjes properties of this function. In particular, we recognize the role of its main branch, $W_0\left(t\right)$, in a peculiar...

Hyperbolic complete monotonicity property ($\mathrm{HCM}$) is a way to check if a distribution is a generalized gamma ($\mathrm{GGC}$), hence is infinitely divisible. In this work, we illustrate to which extent the Mittag-Leffler functions $E_{\alpha },\alpha \in (0,2]$...

The study of the stability of measure families under measure transformations, as well as the accompanying limit theorems, is motivated by both fundamental and applied probability theory and dynamical systems. Stability analysis tries to uncover invar...

In this article, we study $V_a$-transformation of a measure and of a convolution (denoted by $\boxed{a}$) defined for $a\in \mathbb{R}$. We provide significant insights into the stability of the free Meixner family of probability measures under $V_a$-transformation. We show...

Cyberattacks against the elements of technological data transmission networks represent a rather significant threat of disrupting the management of regional electric power complexes. Therefore, evaluating the functioning quality of data transmission...

We proved the local Marchenko–Pastur law for sparse sample covariance matrices that corresponded to rectangular observation matrices of order $n\times m$ with $n/m\to y$ (where $y>0$) and sparse probability $n{p}_n>{log}^{\beta }n$ (where $\beta >0$)....

In this article, we provide some new limiting laws related to the free multiplicative law of large numbers and involving free and Boolean additive convolutions. Some examples of these limiting laws are presented within the framework of non-commutativ...

In the setting of Cauchy–Stieltjes kernel (CSK) families, this study provides some features of free Poisson, free Gamma, and free Binomial laws, as well as some innovative limit theorems linked to Fermi convolution. These findings highlight the...

There are various types of sharing economy services, such as ride-sharing and shared-taxi rides. Motivated by these services, we consider a single-server queue in which customers probabilistically select the type of service, that is, the single servi...

Let ${\mathcal{K}}_{+}\left({\mu }_i\right)=\{Q_{s_i}^{{\mu }_i},s_i\in (m_0^{{\mu }_i},m_{+}^{{\mu }_i})\}$, $i=1,2$, be two CSK families generated by the nondegenerate probability measures ${\mu }_1$ and ${\mu }_2$ with support bounded from above. Define the set of measures $\mathcal{L}={\mathcal{K}}_{+}\left({\mu }_1\right)•{\mathcal{K}}_{+}\left({\mu }_2\right)=\{Q_{s_1}^{{\mu }_1}•Q_{}^{}$...

Several authors are interested in the study of limiting distributions for large symmetrical random matrices. Our approach of studying limiting distributions is different and is related to the new concept of variance functions of Cauchy–Stieltje...

In free probability, the theory of Cauchy–Stieltjes Kernel (CSK) families has recently been introduced. This theory is about a set of probability measures defined using the Cauchy kernel similarly to natural exponential families in classical pr...

The investigation of free additive convolution is a key concept in free probability theory, offering a framework for studying the sum of freely independent random variables. This paper uses free additive convolution and measure dilations to investiga...

Let $\mathfrak{F}\left({\nu }_j\right)=\{Q_{m_j}^{{\nu }_j},m_j\in (m_{-}^{{\nu }_j},m_{+}^{{\nu }_j})\}$, $j=1,2$, be two Cauchy–Stieltjes Kernel (CSK) families induced by non-degenerate compactly supported probability measures ${\nu }_1$ and ${\nu }_2$. Introduce the set of measures $\mathcal{F}=\mathfrak{F}\left({\nu }_1\right)⊞\mathfrak{F}({\&}_{}$...

The concept of r-free convolution (which is represented by $\begin{array}{|c|}\hline r\\ \hline\end{array}$) was introduced for $0\le r\le 1$. It is equal to the Boolean additive convolution ⊎ if $r=0$ and reduced to the free additive convolution ⊞ when $r=1$. This paper presents certain fe...

We investigate the onset of hyperbolicity in Jensen polynomials $J_{d,n}$ associated with the Riemann $\mathsf{\Xi }$-function and identify a robust parity-driven bifurcation with a natural geometric interpretation. Numerical analysis for degrees $5\le d\le 16$ reveal...

Padé approximants are computational tools customarily employed for resumming divergent Stieltjes series. However, they become ineffective or even fail when applied to Stieltjes series whose moments do not satisfy the Carleman condition. Differ...

The practical usefulness of Levin-type nonlinear sequence transformations as numerical tools for the summation of divergent series or for the convergence acceleration of slowly converging series is nowadays beyond dispute. The Weniger transformation,...

In this study, the linear renewal equation is obtained by using the integral equation, the renewal function and the Fourier–Stieltjes transform. It is proven that the linear renewal equation can be obtained by taking the derivative of the integ...

The free Meixner family ($\mathbb{FMF}$) is the family of measures that produces quadratic Cauchy–Stieltjes Kernel (CSK) families (i.e., meaning that the associated variance function (VF) is a polynomial with degree $\le 2$ in the mean). Furthermore, a cubi...

The paper provides reliability analysis of a cold double redundant renewable system assuming that both life-time and repair time distributions are arbitrary. The proposed approach is based on the theory of decomposable semi-regenerative processes. We...

A general system of fractional differential equations with coupled fractional Stieltjes integrals and a Riemann–Liouville fractional integral in boundary conditions is studied in the context of pattern formation. We need to transform the fracti...

In this paper, we study the k-Kummer hypergeometric function $M_k(a,c;w)$, a generalization of the classical Kummer confluent hypergeometric function, arising as a solution of the k-confluent hypergeometric differential equation. We establish a Kummer-t...

In this paper a two component redundant renewable system operating under the Marshall–Olkin failure model is considered. The purpose of the study is to find analytical expressions for the time dependent and the steady state characteristics of t...

In this paper, a cold standby repairable system comprising two heterogeneous components, each characterized by multiple types of mutually independent failure modes, is investigated. The operational lifetimes of the components follow exponential distr...