Search Results (49)

This paper generalizes stochastic sampled-data synchronization of reaction–diffusion neural networks (RDNNs) from sampling distributions with finite support to infinite discrete distributions, whose sampling intervals are drawn from a countably infinite set ${\left\{q_i\right\}}_{i=1}^{\infty }$ with probabilities ${\left\{{\alpha }_i\right\}}_{i=1}^{\infty }$. Four regularity conditions are identified—finite interval mean, summability of the weighted absolute decay parameters, strict positivity of the probability-averaged decay rate, and boundedness above of the decay parameters, which makes the Lyapunov–Krasovskii functional (LKF) mode-independent in its delay weight and hence continuous across sampling instants. A mode-dependent LKF is constructed, the weak-law step is replaced by the strong law of large numbers combined with a Cesàro-type argument, and the infinite family of design linear matrix inequalities (LMIs) is handled by a truncation-with-tail-verification procedure, for which a rigorous certificate is given: when the tail decay parameters are non-positive, the tail conditions are affine and are certified by corner-point feasibility checks with shared variables; otherwise, a convergent-tail argument applies. Almost-sure exponential synchronization with an explicit rate is established, which implies synchronization in probability, and controller design conditions are derived in terms of LMIs; three examples (two neural networks and one reaction–diffusion neural network) corroborate the design through computed gains, verified tail conditions, stochastic closed-loop simulations, an N-sensitivity analysis, and a comparison with a fixed-period controller. The proposed framework removes the finite-support restriction on the sampling distribution and yields an almost-sure, rate-explicit synchronization guarantee with a numerically verifiable LMI design. Full article

This paper identifies and characterizes the background driving Lévy process (BDLP) associated with the Generalized Tempered Stable (GTS) distribution, a flexible seven-parameter family of infinitely divisible distributions with applications in physics and quantitative finance. We show that the corresponding BDLP is a finite-variation, infinite-activity Type B Lévy process and derive its explicit background driving characteristic exponent function (BDCEF). The resulting BDLP provides a unified representation that encompasses several important special cases, including the bilateral stable, bilateral Gamma, and Variance Gamma distributions. Building on these results, we develop a simulation framework based on a stationary Ornstein–Uhlenbeck (OU)-type process driven by the GTS BDLP. The mean-reversion speed parameter of the OU process is calibrated using maximum likelihood estimation applied to daily return data from the SPY ETF and Ethereum over the period 2010–2024. The proposed simulation methodology produces realistic daily cumulative return trajectories, and comprehensive numerical error analyses demonstrate the accuracy and efficiency of the resulting discretization scheme. These findings provide both a theoretical extension of Lévy-driven OU models and a practical framework for simulating complex financial return dynamics. Full article

We consider an impulsive BVP related to the Hilfer fractional derivatives and the nonlinear p-Laplacian operator. The type parameter $\vartheta \in [0,1]$ is kept unchanged in the formulation, and the Riemann–Liouville and Caputo cases are obtained as limiting cases of the formulation, not as separate cases. The variational functional is then built by adding the point-impulse contribution to the distributed potential and the use of an appropriate space of the Hilfer fractional derivative. Using variants of the fountain theorem, we prove the existence of two infinite sequences of weak solutions, one of which is of unbounded energy and another of which is of small energy and tends to zero from below. The weak residual based stability analysis is further developed, and local generalized Hyers–Ulam and Hyers–Ulam–Rassias stability estimates are obtained. Because of multiplicity of solutions, a uniqueness-based argument for stability, Ulam’s approach, is not possible and stability is instead achieved by providing residual-based arguments.The assumptions are verified through illustrative examples. Lastly, we examine the convergence behavior, residual decay, and effect of the Hilfer type parameter in conjunction with a Hilfer-type parameter neural surrogate with boundary constraints based on a discrete Hilfer scheme. The study, in general, proves a link between the solution multiplicity, residual stability, and the numerical realization in one impulsive fractional p-Laplacian framework. Full article

Business analytics increasingly relies on analytical performance models to support real-time, data-driven decision-making in multi-stage service processes. Discrete-event simulation and process mining offer valuable insights but face scalability and explainability limits when closed-form performance indicators are needed for optimization. This paper develops a systematic procedure for constructing BCMP queueing-network models directly from the structural properties of a business process formalized in BPMN. The framework introduces a six-level state-space hierarchy (V0–V5) that ranges from a full path encoding to a single system-level counter, and three allocation strategies (D1–D3) that govern how the actions of an organizational unit are distributed among its workers. A product-form stationary distribution is derived for the resulting open network with FCFS and infinite-server nodes, yielding closed-form expressions for the mean sojourn time at a node and for the mean process flow time. We validate the framework on the university admissions campaign at RUDN University (2023 cycle, more than 85,000 applications). Numerical experiments show that the baseline mean flow time of about 174 min can be reduced to 111 min—a 33% improvement—by only a 3% increase in total staffing, allocated by greedy coordinate descent. The closed-form model agrees with discrete-event simulation under exponential service times to within 1%, and its analytical tractability enables real-time staffing recommendations that simulation alone cannot deliver. Full article

The classical earth pressure theories, such as Coulomb’s and Rankine’s, are derived based on the assumption that the backfill soil extends infinitely behind a retaining wall. However, in many practical engineering situations, the backfill width is limited, and the failure wedge cannot fully develop as assumed in these classical solutions, which leads to inaccuracies in the estimation of the active earth pressure. To address this issue, this study investigates the active earth pressure under a narrow backfill width through numerical simulations and theoretical analysis. A finite element model was developed in PLAXIS 2D to examine the effects of parameters such as the backfill width-to-height ratio and the internal friction angle of the soil on the active earth pressure. Based on the numerical results, a theoretical model was formulated by discretizing the soil into blocks, formulating equilibrium equations for each block, and considering the interactions between adjacent blocks. The distribution and point of application of the active earth pressure were then computed using the finite difference method. Finally, the theoretical predictions were validated through comparisons with numerical results and previous studies. Full article

We investigate discrete-time deterministic systems whose trajectories are indexed by the positive cone of a countable linearly ordered group G and evolve on a σ-finite measure space $(\Omega ,\mathcal{B},m)$. The paper operates at two levels. At the measure-theoretic level, the ambient space $(\Omega ,m)$ may be non-atomic and infinite, where strong recurrence, characterized by the absence of weakly wandering sets of positive measure, governs structural properties such as syndeticity of return-time sets, invariance under measure equivalence, and inheritance to subgroups. At the combinatorial level, the dynamics is compressed onto the finite effective state space ${\Omega }_{\mathrm{eff}}$, the set of states actually visited by a trajectory, on which a canonical atomic probability measure μ is constructed via a deliberate support-switch from m. At this level, the entropy results additionally require G to be amenable, where Følner sequences are employed to show that μ is the limiting empirical distribution along almost every trajectory. When G acts transitively on ${\Omega }_{\mathrm{eff}}$, the measure μ is necessarily uniform and the stationary Shannon entropy $H\left(X\right)=-{\sum }_x\mu \left(\left\{x\right\}\right)\log \mu \left(\left\{x\right\}\right)$ achieves its maximum $\log |{\Omega }_{\mathrm{eff}}|$ from structural constraints alone. A generalized orbit-decomposition formula covers the non-transitive case. Results at both levels are illustrated through cyclic shifts, rational rotations, Sturmian shifts, and ${\mathbb{Z}}^2$-actions. Full article

The entropic region is formed by the collection of the Shannon entropies of all subvectors of finitely many jointly distributed discrete random variables. For four or more variables, the structure of the entropic region is mostly unknown. We utilize a variant of the Maximum Entropy Method to obtain five-variable non-Shannon entropy inequalities, which delimit the five-variable entropy region. This method adds copies of some of the random variables in generations. A significant reduction in computational complexity, achieved through theoretical considerations and by harnessing the inherent symmetries, allowed us to calculate all five-variable non-Shannon inequalities provided by the first nine generations. Based on the results, we define two infinite collections of such inequalities and prove them to be entropy inequalities. We investigate downward-closed subsets of non-negative lattice points that parameterize these collections, and based on this, we develop an algorithm to enumerate all extremal inequalities. The discovered set of entropy inequalities is conjectured to characterize the applied method completely. Full article

When the quantities of interest remain underdetermined a posteriori, we would like to draw inferences for at least one particular solution. Can we do so in a Bayesian way? What is a probability distribution over an underdetermined quantity? How do we get a posterior for one particular solution from a posterior for infinitely many underdetermined solutions? Guided by discrete invariant underdetermined ill-posed inverse problems, we find that a probability distribution over an underdetermined quantity is non-absolutely continuous, partially improper with respect to the initial reference measure but proper with respect to its restriction to its support. Thus, it is necessary and sufficient to choose the prior restricted reference measure to assign partially improper priors using e.g., the principle of maximum entropy and the posterior restricted reference measure to obtain the proper posterior for one particular solution. We can then work with underdetermined models like Hoeffding–Sobol expansions seamlessly, especially to effectively counter the curse of dimensionality within discrete nonparametric inverse problems. We show Singular Bayesian Inference (SBI) at work in an advanced Bayesian optimization application: dynamic pricing. Such a nice generalization of Bayesian–maxentropic inference could motivate many theoretical and practical developments. Full article

We consider the scenario in which the source sends status updates, or packets, to the receiver through an infinite capacity transmitter, where the transmitted packets are subject to potential illegal eavesdropping. Time is discretized into identical time slots. In recent years, the age of information (AoI) metric, which was defined as the time has elapsed since the generation instant of the latest received packet, has been widely applied to characterize the freshness of obtained packets. Due to the presence of eavesdroppers, some packets may be eavesdropped during their transmissions, causing information leakages. To assess an infinite-capacity system’s performance of securely transmitting status updates, in this paper, we define an AoI-related metric called the freshness advantage of the legitimate receiver, F, to be average instantaneous gap between eavesdropper’s and legitimate receiver’s AoI. For arbitrarily distributed packet interarrival times, and assuming that in each time slot with probabilities ${\gamma }_d$, ${\gamma }_E$, the transmitted packet is received by the legitimate receiver and the eavesdropper, we derive the explicit formula of F. The concise expression shows that F is fully determined by the average interarrival time and the ratio of ${\gamma }_d$ to ${\gamma }_E$. For special cases where the interarrival time follows geometric distributions, we first determine the explicit distribution of instantaneous AoI gap. Then, given ${\gamma }_d$ and ${\gamma }_E$, we derive the optimal packet generation rate $p^{\ast }$ that minimizes the combined performance Q, which is constructed as the average AoI minus the freshness advantage F. When imposing timeliness and security constraints at the same time, the feasible regions of p and ${\gamma }_d$ such that both two required performances can be satisfied are depicted and discussed. Finally, we investigate the impacts of different parameters on F and show the tradeoffs between timeliness performance and security performance through numerical simulations. Full article

Quantization for a Borel probability measure refers to the idea of estimating a given probability by a discrete probability with support containing a finite number of elements. If in the quantization some of the elements in the finite support are preselected, then the quantization is called a conditional quantization. In this paper, we have determined the conditional quantization, first for two different finite discrete distributions with a same conditional set, and for a finite discrete distribution with two different conditional sets. Next, we have determined the conditional and unconditional quantization for an infinite discrete distribution with support $\{{\displaystyle \frac{1}{2^n}}:n\in \mathbb{N}\}$. We have also investigated the conditional quantization for an infinite discrete distribution with support $\{{\displaystyle \frac{1}{n}}:n\in \mathbb{N}\}$. At the end of the paper, we have given a conjecture and discussed about some open problems based on the conjecture. Full article

We present a model of classical binary time series derived from a matrix product state (MPS) Ansatz widely used in one-dimensional quantum systems. We discuss how this quantum Ansatz allows us to generate classical time series in a sequential manner. Our time series are built in two steps: First, a lower-level series (the driving noise or the increments) is created directly from the MPS representation, which is then integrated to create our ultimate higher-level series. The lower- and higher-level series have clear interpretations in the quantum context, and we elaborate on this correspondence with specific examples such as the spin-1/2 Ising model in a transverse field (ITF model), where spin configurations correspond to the increments of discrete-time, discrete-level stochastic processes with finite or infinite autocorrelation lengths, Gaussian or non-Gaussian limit distributions, nontrivial Hurst exponents, multifractality, asymptotic self-similarity, etc. Our time series model is a parametric model, and we investigate how flexible the model is in some synthetic and real-life calibration problems. Full article

For a given $r\in (0,+\infty )$, the quantization dimension of order r, if it exists, denoted by $D_r(\mu )$, represents the rate at which the nth quantization error of order r approaches zero as the number of elements n in an optimal set of n-means for $\mu$ tends to infinity. If $D_r(\mu )$ does not exist, we define ${\underline{D}}_r(\mu )$ and ${\overline{D}}_r(\mu )$ as the lower and the upper quantization dimensions of $\mu$ of order r, respectively. In this paper, we investigate the quantization dimension of the condensation measure $\mu$ associated with a condensation system $({\left\{S_j\right\}}_{j=1}^N, {(p_j)}_{j=0}^N,\nu ).$ We provide two examples: one where $\nu$ is an infinite discrete distribution on $\mathbb{R}$, and one where $\nu$ is a uniform distribution on $\mathbb{R}$. For both the discrete and uniform distributions $\nu$, we determine the optimal sets of n-means, calculate the quantization dimensions of condensation measures $\mu$, and show that the $D_r(\mu )$-dimensional quantization coefficients do not exist. Moreover, we demonstrate that the lower and upper quantization coefficients are finite and positive. Full article

The final goal of this paper is to organize the tools needed to study digital Quantum Communications, where classical information is entrusted to quantum states represented by density operators. A density operator is usually defined starting from a set of kets in the Hilbert space and a probability distribution. A fundamental problem in Quantum Communications is the factorization of such operators of the form $\rho =\gamma {\gamma }^{*}$, where $\gamma$ is a matrix called a density factor (DF). The environments considered are finite dimensional Hilbert space (discrete variables) and infinite dimensional Hilbert space (continuous variables). Using discrete variables, the multiplicity and the variety of DFs are investigated using the tools of matrix analysis, arriving in particular to establish the DF with minimal size. With continuous variables, the target is the closed-form factorization, which is achieved with recent results for the important class of Gaussian states. Finally, an application is carried out in Quantum Communications with noisy Gaussian states. Full article

Orthogonal polynomials on radial rays in the complex plane were introduced and studied intensively in several papers almost three decades ago. This paper presents an account of such kinds of orthogonality in the complex plane, as well as a number of new results and examples. In addition to several types of standard orthogonality, the concept of orthogonality on arbitrary radial rays is introduced, some or all of which may be infinite. A general method for numerical constructing, the so-called discretized Stieltjes–Gautschi procedure, is described and several interesting examples are presented. The main properties, zero distribution and some applications are also given. Special attention is paid to completely symmetric cases. Recurrence relations for such kinds of orthogonal polynomials and their zero distribution, as well as a connection with the standard polynomials orthogonal on the real line, are derived, including the corresponding linear differential equation of the second order. Finally, some applications in physics and electrostatics are mentioned. Full article

We consider an irreducible discrete-time Markov process with states represented as (k, i) where k is an M-dimensional vector with non-negative integer entries, and i indicates the state (phase) of the external environment. The number n of phases may be either finite or infinite. One-step transitions of the process from a state (k, i) are limited to states (n, j) such that n ≥ k−1, where 1 represents the vector of all 1s. We assume that for a vector k ≥ 1, the one-step transition probability from a state (k, i) to a state (n, j) may depend on i, j, and n − k, but not on the specific values of k and n. This process can be classified as a Markov chain of M/G/1 type, where the minimum entry of the vector n defines the level of a state (n, j). It is shown that the first passage distribution matrix of such a process, also known as the matrix G, can be expressed through a family of nonnegative square matrices of order n, which is a solution to a system of nonlinear matrix equations. Full article