Search Results (83)

We provide sufficient conditions for the existence of $(\omega ,c)$-periodic solutions of a general class of nonlinear functional integral equations. This study extends and generalizes previous contributions in the literature. As an application of the developed theory, we establish the existence of $(\omega ,c)$-periodic solutions for recurrent neural networks with time-varying coefficients and mixed delays, as well as for a class of nonlinear Volterra–Stieltjes integral equations with infinite delay. Full article

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 corresponding operators, denoted by ${\mathbb{Y}}_{(a,s,t)}^{\prime }$ and ${\mathbb{Y}}_{(a,s,t)}^{\prime \prime }$, are defined via a functional equation involving the Cauchy–Stieltjes transform (CST). This framework recovers the classical cases as particular instances, specifically ${\mathbb{Y}}_{(0,s,t)}^{\prime }={\mathbb{Y}}_{(0,s,t)}^{\prime \prime }=U^{\mathbb{T}}$ and ${\mathbb{Y}}_{(a,1,1)}^{\prime }={\mathbb{Y}}_{(a,1,1)}^{\prime \prime }=T_a$. We analyze the analytic and structural properties of the operators ${\mathbb{Y}}_{(a,s,t)}^{\prime }$ and ${\mathbb{Y}}_{(a,s,t)}^{\prime \prime }$ within the concept of Cauchy–Stieltjes kernel (CSK) families, with particular emphasis on their action on variance functions (VFs). In particular, we derive explicit formulas for the VFs associated with measures deformed by ${\mathbb{Y}}_{(a,s,t)}^{\prime }$ and ${\mathbb{Y}}_{(a,s,t)}^{\prime \prime }$. As an application, we establish an invariance property showing that the class of free Meixner family (FMF) is stable under both deformations. Furthermore, by restricting the parameters to ${\mathbb{Y}}_{(a,1,t)}^{\prime }$ and ${\mathbb{Y}}_{(a,1,t)}^{\prime \prime }$, we obtain two new characterizations of the semicircle law. These results highlight the role of symmetry in the analytic deformation and in the stability properties of fundamental distributions in free probability. Full article

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, written as ${\mathcal{W}}_{(a,c)}$, is introduced via a functional relation involving the Cauchy–Stieltjes transform and is constructed so as to recover the initial deformations as special cases, namely ${\mathcal{W}}_{(0,c)}=T_c$ and ${\mathcal{W}}_{(a,0)}=V_a$. Working within the concept of Cauchy–Stieltjes kernel families, we analyze the action of ${\mathcal{W}}_{(a,c)}$ on variance functions and establish an explicit expression for the variance function induced by this deformation. This approach leads to a structural invariance property demonstrating that the free Meixner class is preserved under the action of ${\mathcal{W}}_{(a,c)}$. In addition, the operator provides a new perspective on the semicircle distribution, yielding a characterization that reflects the symmetric nature of the deformation and its compatibility with fundamental distributions in free probability. Full article

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 introduced through a functional relation for the Cauchy–Stieltjes transform. We have ${\mathcal{X}}_{(a,b,0)}={\tilde{U}}^{(a,b)}$ and ${\mathcal{X}}_{(1,1,c)}=T_c$. We examine the structural properties of this transformation within the setting of Cauchy–Stieltjes kernel (CSK) families, with special emphasis on the behavior of the associated variance functions (VFs). An explicit formula for the VF corresponding to measure deformed by ${\mathcal{X}}_{(a,b,c)}$ is established. This result allows us to demonstrate a key invariance property: the free Meixner class of probability measures remains stable under the ${\mathcal{X}}_{(a,b,c)}$-transformation. Furthermore, a novel characterization of the semicircle law is obtained through the action of ${\mathcal{X}}_{(a,1,c)}$, highlighting the role of symmetry in the deformation and preservation of free-probabilistic distributions. Full article

We establish a version of Pontryagin’s maximum principle for optimal control problems with impulses and phase constraints. Using the Dubovitskii–Milyutin theory, we construct a conic variational framework that handles impulsive dynamics and general state constraints. The main difficulty lies in working with piecewise continuous functions, required by the impulsive nature of the system. This setting also demands an extension of the classical result on the existence of non-negative Borel measures, which leads to an adjoint equation formulated as a Stieltjes integral. Theoretical results are illustrated with examples, and key results by I. Girsanov are extended to the impulsive context. Full article

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 priori structural information on Stieltjes series, offers a superior framework with respect to Padé. In the present work, the following fundamental question is addressed: Is the $\delta$-transformation, once it is applied to a typical Stieltjes series, capable of correctly simulating the branch cut structure of the corresponding Stieltjes function? Here, it is proved that the intrinsic log-convexity of the Stieltjes moment sequence (guaranteed via the positivity of Hankel’s determinants) allows the necessary condition for $\delta$ to have all real poles to be satisfied. The same condition, however, is not sufficient to guarantee this. In attempting to bridge such a gap, we propose a mechanism rooted in the iterative action of a specific linear differential operator acting on a class of suitable auxiliary log-concave polynomials. To this end, we show that the denominator of the $\delta$-approximants can always be recast as a high-order derivative of a log-concave polynomial. Then, on invoking the Gauss–Lucas theorem, a consistent geometrical justification of the $\delta$ pole positioning is proposed. Through such an approach, the pole alignment along the negative real axis can be viewed as the result of the progressive restriction of the convex hull under differentiation. Since a fully rigorous proof of this conjecture remains an open challenge, in order to substantiate it, a comprehensive numerical investigation across an extensive catalog of Stieltjes series is proposed. Our results provide systematic evidence of the potential $\delta$-transformation ability to mimic the singularity structure of several target functions, including those involving superfactorial divergences. Full article

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 critical metric for real-time applications enabled by Ultra-Reliable Low Latency Communication (URLLC). While highly useful for system evaluation, the direct analysis of this metric is complicated by the correlation between the random variables constituting the PAoI. Thus, it is often evaluated using only the mean value rather than the full distribution. Furthermore, since CPS communication technologies like Wi-Fi or DECT-2020 involve multiple processing stages, modeling them as tandem queueing systems is essential for accurate PAoI analysis. In this paper, we develop an analytical model for a DECT-2020 network segment represented as a two-phase tandem queueing system, enabling detailed PAoI analysis via Laplace–Stieltjes transforms (LST). We circumvent the dependence between generation and sojourn times by classifying updates into four mutually exclusive groups. This approach allows us to derive the LST of the PAoI and determine the exact Probability Density Function (PDF) for $M\left|M\right|1\to M\left|M\right|1$ system. We also calculate the mean and variance of the PAoIs and validate our results through numerical experiments. Additionally, we evaluate the impact of different service time distributions on PAoI variability. These findings contribute to the theoretical understanding of PAoI in tandem queueing systems and provide practical insights for optimizing DECT-2020-based communication systems. Full article

In this paper, we investigate the existence of positive solutions of the eigenvalue problem for a singular tempered fractional equation with a Riemann–Stieltjes integral boundary condition and signed measures. By establishing the Green function and its properties, an eigenvalue interval for the existence of positive solutions is outlined based on Schauder’s fixed-point theorem and the upper and lower solutions method. An interesting feature of this paper is that f may be singular in both the time and space variables, and the Riemann–Stieltjes integral may involve signed measures. Full article

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 characterizing ${\mathcal{T}}_{(a,t)}$ through the Cauchy–Stieltjes transform and explicitly show how it specializes to known deformations when $a=0$ or $t=1$. Within the setting of Cauchy-Stieltjes kernel families, we prove structural symmetry and invariance properties of the transformation, demonstrating in particular that both the free Meixner family and the free analog of the Letac-Mora class remain invariant under ${\mathcal{T}}_{(a,t)}$. Furthermore, we obtain several new limiting theorems that uncover symmetric relationships among fundamental free distributions, including the semicircular, Marchenko–Pastur, and free binomial laws. Full article

This paper proves a new lemma that characterizes controllability for linear Volterra control systems and shows that the usual controllability assumption for the variational linearized system near an optimal pair is superfluous. Building on this, it establishes a Pontryagin-type maximum principle for Volterra optimal control with general control and state constraints (fixed terminal constraints and time-dependent state bounds), where the cost combines a terminal term with a state-dependent and integral term. Using the Dubovitskii–Milyutin framework, we construct conic approximations for the cost, dynamics, and constraints and derive necessary optimality conditions under mild regularity: (i) a classical adjoint system when only terminal constraints are present and (ii) a Stieltjes-type adjoint with a non-negative Borel measure when pathwise state constraints are active. Furthermore, under convexity of the cost functional and linear Volterra dynamics, the maximum principle becomes a sufficient criterion for global optimality (recovering the classical sufficiency in the differential case). The differential case recovers the classical PMP, and an SIR example illustrates the results. A key theme is symmetry/duality: the adjoint differentiates in the state while the maximum condition differentiates in the control, reflecting operator transposition and the primal–dual geometry of Dubovitskii–Milyutin cones. Full article

This study is devoted to the detailed examination of the concept of $(a,b)$-deformation, defined for parameters $a\in \mathbb{R}$ and $b>0$. The analysis is conducted within the framework of Cauchy–Stieltjes kernel (CSK) families of probability measures, with particular attention given to the role of their variance functions (VFs). Using the VF as the main analytical tool, it is shown that the $(a,b)$-deformation of any measure belonging to the free Meixner family (FMF) remains within the same family. Moreover, the VF framework provides a powerful and flexible means for establishing new limit theorems associated with $(a,b)$-deformed free convolution. In particular, several novel limiting behaviors are derived, which naturally encompass both free and Boolean additive convolutions as special cases. Full article

This paper analyzes a finite-capacity $GI/M/2/N$ queue with two heterogeneous servers operating under a multiple working-vacation policy, Bernoulli feedback, and customer impatience. Using the supplementary-variable technique in tandem with a tailored recursive scheme, we derive the stationary distributions of the system size as observed at pre-arrival instants and at arbitrary epochs. From these, we obtain explicit expressions for key performance metrics, including blocking probability, average reneging rate, mean queue length, mean sojourn time, throughput, and server utilizations. We then embed these metrics in an economic cost function and determine service-rate settings that minimize the total expected cost via the Bat Algorithm. Numerical experiments implemented in R validate the analysis and quantify the managerial impact of the vacation, feedback, and impatience parameters through sensitivity studies. The framework accommodates general renewal arrivals ($GI$), thereby extending classical ($M/M/2/N$) results to more realistic input processes while preserving computational tractability. Beyond methodological interest, the results yield actionable design guidance: (i) they separate Palm and time-stationary viewpoints cleanly under non-Poisson input, (ii) they retain heterogeneity throughout all formulas, and (iii) they provide a cost–optimization pipeline that can be deployed with routine numerical effort. Methodologically, we (i) characterize the generator of the augmented piecewise–deterministic Markov process and prove the existence/uniqueness of the stationary law on the finite state space, (ii) derive an explicit Palm–time conversion formula valid for non-Poisson input, (iii) show that the boundary-value recursion for the Laplace–Stieltjes transforms runs in linear time $\mathcal{O}\left(N\right)$ and is numerically stable, and (iv) provide influence-function (IPA) sensitivities of performance metrics with respect to $({\mu }_1,{\mu }_2,\nu ,\alpha ,\varphi ,\beta )$. Full article

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 analytically via a modified functional equation involving the Cauchy transform, and reduces to the t-transformation when $a=0$, and to the $V_a$-transformation when $t=1$. We investigate some properties of this new transformation from the lens of Cauchy–Stieltjes kernel (CSK) families and the corresponding variance functions (VFs). We derive a general expression for the VF resulting from the ${\mathcal{U}}_{(a,t)}$-transformation. This new expression is applied to prove a central result: the free Meixner family (FMF) of measures is invariant under this transformation. Furthermore, novel limiting theorems involving ${\mathcal{U}}_{(a,t)}$-transformation are proved providing new insights into the relations between some important measures in free probability such as the semicircle, Marchenko–Pastur, and free binomial measures. Full article

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 fundamental links between noncommutative probability and analytic function theory, demonstrating the usefulness of CSK families for advancing the computational and theoretical aspects of free harmonic analysis. Full article

This paper examines solutions to mathematical models based on functional-differential equations, which have applications in immunology. This new approach allows us to study discontinuous solutions that more accurately depict real-world phenomena. It also enables us to exploit the information contained in the initial function. We discuss immunology models by generalizing existing impulsive delay differential equation models to the proposed form. The new phase space introduced here enables a unified approach to continuous and impulsive solutions that were previously studied, as well as the development of new properties that depend on the initial function. To illustrate our work, we present extensions of current immunological models and demonstrate some applications in fields beyond immunology. This paper focuses on establishing the theoretical basis for modifying models based on delayed differential equations, which are not limited to immunology. It also provides some examples. Full article