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This paper presents a two inertial technique with a projection Mann forward-backward splitting algorithm for solving variational inclusion problems that exhibit weak convergence under suitable conditions in Hilbert spaces. Furthermore, we provide a numerical example in infinitely dimensional spaces to support the main result. Finally, we provide an application for data classification using an extreme learning machine. According to data provided by theWorld Health Organization (WHO), stroke is recognized as the predominant contributor to mortality and disability on a global scale. To appraise the efficacy of our algorithm, we procured a dependable dataset for stroke prediction from the Kaggle website. The best algorithm that performed this task is ours compared to other machine learning methods. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce a new inertial iterative algorithm involving the linesearch rule for the split variational inclusion in real Hilbert spaces. We use two inertial terms to accelerate the convergence speed of the proposed algorithm. Under suitable conditions, we prove its weak convergence theorem. Finally, in applications, we apply our algorithm to image deblurring and data classification problem in predicting diabetes mellitus. [ABSTRACT FROM AUTHOR]

In this work, we propose inclusion problems based on a novel class of forward-backwardforward algorithms. Our approach incorporates multi-inertial extrapolations and utilizes a self-adaptive technique to eliminate the need for explicitly selecting Lipschitz assumptions to enhance the speed convergence of the algorithm. We establish a weak convergence theorem under suitable assumptions. Furthermore, we conduct numerical tests on image deblurring as a practical application. The experimental results demonstrate that our algorithm surpasses some existing methods in the literature, which shows its superior performance and effectiveness. [ABSTRACT FROM AUTHOR]

In this work, we propose a novel class of forward‐backward‐forward algorithms for solving monotone inclusion problems. Our approach incorporates a self‐adaptive technique to eliminate the need for explicitly selecting Lipschitz assumptions and utilizes two‐step inertial extrapolations to enhance the convergence rate of the algorithm. We establish a weak convergence theorem under mild assumptions. Furthermore, we conduct numerical tests on image deblurring and data classification as practical applications. The experimental results demonstrate that our algorithm surpasses some existing methods in the literature which shows its superior performance and effectiveness. [ABSTRACT FROM AUTHOR]

This work is motivated by a gap in the functional data analysis literature, particularly in the context of neuroimaging, regarding the ability of functional models to robustly accommodate intra‐observation dependence. In response, we propose an M‐estimator based on generalized empirical likelihood for the varying‐coefficient model with a functional response. We develop statistical inference procedures, simultaneous confidence regions, and a global general linear hypothesis test for the model's functional coefficient. Our theoretical results establish the weak convergence of the log‐likelihood ratio process, a nonparametric version of Wilks' theorem for the log‐likelihood ratio, and asymptotic properties of the proposed estimator. Through a simulation study, we show that the proposed confidence sets have close‐to‐nominal coverage probabilities. In a real‐world application to a neuroimaging dataset, we show that mini‐mental state examination score and apolipoprotein E genotype have significant associations with fractional anisotropy, while associations with gender and age are only present at high quantile levels. [ABSTRACT FROM AUTHOR]

We consider optimal control problems for a system governed by a stochastic differential equation driven by a d-dimensional Brownian motion where both the drift and the diffusion coefficient are controlled. It is well known that without additional convexity conditions the strict control problem does not admit an optimal control. To overcome this difficulty, we consider the relaxed model, in which admissible controls are measure-valued processes and the relaxed state process is governed by a stochastic differential equation driven by a continuous orthogonal martingale measure. This relaxed model admits an optimal control that can be approximated by a sequence of strict controls by the so-called chattering lemma. We establish optimality necessary conditions, in terms of two adjoint processes, extending Peng's maximum principle to relaxed control problems. We show that relaxing the drift and diffusion martingale parts directly as in deterministic control does not lead to a true relaxed model as the obtained controlled dynamics is not continuous in the control variable. [ABSTRACT FROM AUTHOR]

Our study in this paper is focused on the split equality fixed-point problem with firmly quasi-non-expansive operators in infinite-dimensional Hilbert spaces. A self-adaptive simultaneous scheme is introduced, and its weak convergence is established under mild and standard assumptions. The new proposed scheme generalizes and extends some related works in the literature, and its simple structure makes it easy for implementation and numerical testing. Primary experiments presented in this paper, in finite- and infinite-dimensional spaces, emphasize their practical advantages over existing results. [ABSTRACT FROM AUTHOR]

In this work, we consider the stochastic Cahn-Hilliard/Allen-Cahn equation with fractional noise, which is fractional in time and white in space. We obtain the existence, uniqueness, and Hölder regularity of the solution. Also, using a weak convergence approach, we prove that the law of solution associated with the above equation with a small perturbation satisfies the large deviation principle in the Hölder norm. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the difference between the deterministic and stochastic models of nonlocal diffusion. We use a nonlocal reaction-diffusion equation and a multi-dimensional jump Markov process to analyze these mathematical models. First, we demonstrate that the difference converges to 0 in probability with a supremum norm for a sizeable network. Next, we consider the rescaled difference and show that it converges to a stochastic process in distribution on the Skorokhod space. [ABSTRACT FROM AUTHOR]

Let P be the set of generalized prime numbers, and ζ P (s) , s = σ + i t , denote the Beurling zeta-function associated with P. In the paper, we consider the approximation of analytic functions by using shifts ζ P (s + i τ) , τ ∈ R . We assume the classical axioms for the number of generalized integers and the mean of the generalized von Mangoldt function, the linear independence of the set { log p : p ∈ P } , and the existence of a bounded mean square for ζ P (s) . Under the above hypotheses, we obtain the universality of the function ζ P (s) . This means that the set of shifts ζ P (s + i τ) approximating a given analytic function defined on a certain strip σ ^ < σ < 1 has a positive lower density. This result opens a new chapter in the theory of Beurling zeta functions. Moreover, it supports the Linnik–Ibragimov conjecture on the universality of Dirichlet series. For the proof, a probabilistic approach is applied. [ABSTRACT FROM AUTHOR]

We consider the Ginzburg–Landau equation in the perforated domain, with rapidly oscillating coefficients. We derive the homogenized Ginzburg–Landau equation with a 'strange term' (potential) and prove that the trajectory attractors of the given equation tend in a weak sense to the trajectory attractors of the homogenized one. Assuming additional conditions to be satisfied for the coefficients, we provide also a convergence of the global attractor. [ABSTRACT FROM AUTHOR]

We establish a large deviation principle (LDP) for a class of stochastic porous media equations driven by Lévy-type noise on a σ-finite measure space (E,B(E),μ), with the Laplacian replaced by a negative definite self-adjoint operator. One of the main contributions of this paper is that we do not assume the compactness of embeddings in the corresponding Gelfand triple, and to compensate for this generalization, a new procedure is provided. This is the first paper to deal with LDPs for stochastic evolution equations with Lévy noise without compactness conditions. The coefficient Ψ is assumed to satisfy nondecreasing Lipschitz nonlinearity, so an important physical problem covered by this case is the Stefan problem. Numerous examples of negative definite self-adjoint operators are applicable to our results, for example, for open E⊂Rd, L= Laplacian or fractional Laplacians, i.e., L=-(-Δ)α, α∈(0,1], generalized Schrödinger operators, i.e., L=Δ+2∇ρ/ρ⋅∇, Laplacians on fractals is also included. [ABSTRACT FROM AUTHOR]

Many authors have exploited the fact that the distribution of the multivariate probability integral transformation (PIT) of a continuous random vector X ∈ R d with cumulative distribution function F X is free of the marginal distributions. While most of these methods are based on the cdf of W = F X (X) , this paper introduces the weighted characteristic function (WCf) of W. A sample version of the WCf of W based on pseudo-observations is proposed and its weak limit in a space of complex functions is formally established. This result can be used to define test statistics for multivariate independence and goodness-of-fit in copula models, whose asymptotic behaviour comes from the weak convergence of the empirical WCf process. Simulations show the good sampling properties of these new tests, and an illustration is given on the multivariate Cook and Johnson dataset. [ABSTRACT FROM AUTHOR]

The Korpelevich method is an algorithm which is used to find solutions to equilibrium problems. These problems are mathematical models which are used in economics, game theory, and engineering. Pseudomonotone equilibrium problems are a specific class of equilibrium problems that involve a weakened form of monotonicity. This work introduces a novel approach to applying the Korpelevich method to solving pseudomonotone equilibrium problems. We present a weak convergence theorem and linear convergence of the proposed method under some suitable conditions. Finally, a numerical example of a Nash-Cournot oligopolistic electricity market equilibrium model is given to complement the theoretical discussion and strengthen the evidence for the capabilities of our new approach. [ABSTRACT FROM AUTHOR]

Can convergence of random variables, random vectors or, more generally, random elements of a metric space be studied using fuzzy integrals, in particular the seminormed integral? Among the positive results we present, it is shown that for large families of t-seminorms convergence in distribution and convergence in probability can be characterized via seminormed integrals. This suggests that, while fuzzy integrals were devised for working with non-additive fuzzy measures, some of their theoretical properties within the realm of probability measures can be comparable to those of the Lebesgue integral. [ABSTRACT FROM AUTHOR]

We propose and study two new methods based on the golden ratio technique for approximating solutions to variational inequality problems in Hilbert space. The first method combines the golden ratio technique with the subgradient extragradient method. In the second method, we incorporate the alternating golden ratio technique into the subgradient extragradient method. Both methods use self-adaptive step sizes which are allowed to increase during the execution of the algorithms, thus limiting the dependence of our methods on the starting point of the scaling parameter. We prove that under appropriate conditions, the resulting methods converge either weakly or R-linearly to a solution of the variational inequality problem associated with a pseudomonotone operator. In order to show the numerical advantage of our methods, we first present the results of several pertinent numerical experiments and then compare the performance of our proposed methods with that of some existing methods which can be found in the literature. [ABSTRACT FROM AUTHOR]

This paper focuses on stability of stochastic hybrid systems with random delay driven by a singularly perturbed Markov chain. By weak convergence and the martingale method, the limit system is obtained. Using the limit system as a bridge, this paper establishes the moment exponential stability of the original system by Razumikhin-type techniques. Finally, an example is given to illustrate this result. [ABSTRACT FROM AUTHOR]