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In this paper, the split feasibility problem is studied in real Banach spaces. Through the W-mapping, a new iterative algorithm with the inertial technique for solving the split feasibility problem is proposed, which the step size is self-adaptive and no prior estimation of operator norm is required. We prove that the proposed algorithm converges weakly to a solution of the split feasibility problem under some mild conditions. Finally, the effectiveness of the proposed algorithm is indicated by numerical experiments. Our results are innovative and can enrich recently announced related results in the literature.

We study the properties of two subsampling procedures for networks, vertex subsampling and |$p$| -subsampling, under the sparse graphon model. The consistency of network subsampling is demonstrated under the minimal assumptions of weak convergence of the corresponding network statistics and an expected subsample size growing to infinity more slowly than the number of vertices in the network. Furthermore, under appropriate sparsity conditions, we derive limiting distributions for the nonzero eigenvalues of an adjacency matrix under the sparse graphon model. Our weak convergence result implies the consistency of our subsampling procedures for eigenvalues under appropriate conditions. [ABSTRACT FROM AUTHOR]

We consider stochastic inviscid dyadic models with energy-preserving noise. It is shown that the models admit weak solutions which are unique in law. Under a certain scaling limit of the noise, the stochastic models converge weakly to a deterministic viscous dyadic model, for which we provide explicit convergence rates in terms of the parameters of noise. A central limit theorem underlying such scaling limit is also established. In case that the stochastic dyadic model is viscous, we show the phenomenon of dissipation enhancement for suitably chosen noise. [ABSTRACT FROM AUTHOR]

In this present work, we study new algorithms to solve the split equality fixed-point problems for total quasi-asymptotically nonexpansive mappings in Hilbert spaces. In addition, we have established the convergence criteria for the proposed algorithms, at the end, we provided numerical results that justified our theoretical results. The results presented in this paper, provided a unified framework for studying this kind of problem involving different classes of mappings. [ABSTRACT FROM AUTHOR]

In this paper, we consider discounted penalty functions, also called Gerber-Shiu functions, in a Markovian shot-noise environment. At first, we exploit the underlying structure of piecewise-deterministic Markov processes (PDMPs) to show that these penalty functions solve certain partial integro-differential equations (PIDEs). Since these equations cannot be solved exactly, we develop a numerical scheme that allows us to determine an approximation of such functions. These numerical solutions can be identified with penalty functions of continuous-time Markov chains with finite state space. Finally, we show the convergence of the corresponding generators over suitable sets of functions to prove that these Markov chains converge weakly against the original PDMP. That gives us that the numerical approximations converge to the discounted penalty functions of the original Markovian shot-noise environment. [ABSTRACT FROM AUTHOR]

Suppose that Q is a positive defined n × n matrix, and Q [ x ̲ ] = x ̲ T Q x ̲ with x ̲ ∈ Z n . The Epstein zeta-function ζ (s ; Q) , s = σ + i t , is defined, for σ > n 2 , by the series ζ (s ; Q) = ∑ x ̲ ∈ Z n ∖ { 0 ̲ } (Q [ x ̲ ]) − s , and it has a meromorphic continuation to the whole complex plane. Let n ⩾ 4 be even, while φ (t) is an increasing differentiable function with a continuous monotonic bounded derivative φ ′ (t) such that φ (2 t) (φ ′ (t)) − 1 ≪ t , and the sequence { a φ (k) } is uniformly distributed modulo 1. In the paper, it is obtained that 1 N # N ⩽ k ⩽ 2 N : ζ (σ + i φ (k) ; Q) ∈ A , A ∈ B (C) , for σ > n − 1 2 , converges weakly to an explicitly given probability measure on (C , B (C)) as N → ∞ . [ABSTRACT FROM AUTHOR]

In this paper, we consider the simultaneous approximation of tuples of analytic functions by tuples of shifts of Lerch zeta-functions with arbitrary parameters. We prove that there exists a closed set of tuples of functions analytic in the right-hand side of the critical strip, which is approximated by the above tuples of shifts. Further, a generalization for some compositions of tuples of Lerch zeta-functions is given. [ABSTRACT FROM AUTHOR]

The famous Selberg class is defined axiomatically and consists of Dirichlet series satisfying four axioms (Ramanujan hypothesis, analytic continuation, functional equation, multiplicativity). The Selberg–Steuding class S is a complemented Selberg class by an arithmetic hypothesis related to the distribution of prime numbers. In this paper, a joint universality theorem for the functions L from the class S on the approximation of a collection of analytic functions by shifts L (s + i a 1 τ) , ... , L (s + i a r τ) , where a 1 , ... , a r are real algebraic numbers linearly independent over the field of rational numbers, is obtained. It is proved that the set of the above approximating shifts is infinite, its lower density and, with some exception, density are positive. For the proof, a probabilistic method based on weak convergence of probability measures in the space of analytic functions is applied together with the Backer theorem on linear forms of logarithms and the Mergelyan theorem on approximation of analytic functions by polynomials. [ABSTRACT FROM AUTHOR]

Let θ (t) denote the increment of the argument of the product π − s / 2 Γ (s / 2) along the segment connecting the points s = 1 / 2 and s = 1 / 2 + i t , and t n denote the solution of the equation θ (t) = (n − 1) π , n = 0 , 1 , ... . The numbers t n are called the Gram points. In this paper, we consider the approximation of a collection of analytic functions by shifts in the Riemann zeta-function (ζ (s + i t k α 1) , ... , ζ (s + i t k α r)) , k = 0 , 1 , ... , where α 1 , ... , α r are different positive numbers not exceeding 1. We prove that the set of such shifts approximating a given collection of analytic functions has a positive lower density. For the proof, a discrete limit theorem on weak convergence of probability measures in the space of analytic functions is applied. [ABSTRACT FROM AUTHOR]

In this paper, we establish a new iterative process for approximation of fixed points for contraction mappings in closed, convex metric space. We conclude that our iterative method is more accurate and has very fast results from previous remarkable iteration methods like Picard-S, Thakur new, Vatan Two-step and K-iterative process for contraction. Stability of our iteration method and data dependent results for contraction mappings are exact, correspondingly on testing our iterative method is advanced. Finally, we prove enquiring results for some weak and strong convergence theorems of a sequence which is generated from a new iterative method, Suzuki generalized non-expansive mappings with condition (C) in uniform convexity of metric space. Our results are addition, enlargement over and above generalization for some well-known conclusions with literature for theory of fixed point. [ABSTRACT FROM AUTHOR]

We give a multi-valued version of M iterative process [K. Ullah and M. Arshad, Numerical reckoning fixed points Suzuki's generalized nonexpansive mappings via new iteration process, Filomat 32 (2018) 187–196] in the setting of Banach spaces. After this, we prove some weak and strong convergence results for the class of multi-valued quasi-nonexpansive mappings (which contains the class of multi-valued nonexpansive mappings) with proper numerical examples using multi-valued version of M-iteration process. These results generalize several well-known comparable results in the literature. [ABSTRACT FROM AUTHOR]

Multiple-try Metropolis (MTM) is a popular Markov chain Monte Carlo method with the appealing feature of being amenable to parallel computing. At each iteration, it samples several candidates for the next state of the Markov chain and randomly selects one of them based on a weight function. The canonical weight function is proportional to the target density. We show both theoretically and empirically that this weight function induces pathological behaviours in high dimensions, especially during the convergence phase. We propose to instead use weight functions akin to the locally-balanced proposal distributions of Zanella (2020), thus yielding MTM algorithms that do not exhibit those pathological behaviours. To theoretically analyse these algorithms, we study the high-dimensional performance of ideal schemes that can be thought of as MTM algorithms which sample an infinite number of candidates at each iteration, as well as the discrepancy between such schemes and the MTM algorithms which sample a finite number of candidates. Our analysis unveils a strong distinction between the convergence and stationary phases: in the former, local balancing is crucial and effective to achieve fast convergence, while in the latter, the canonical and novel weight functions yield similar performance. Numerical experiments include an application in precision medicine involving a computationally-expensive forward model, which makes the use of parallel computing within MTM iterations beneficial. [ABSTRACT FROM AUTHOR]

We study a perturbed inertial Krasnoselskii-Mann-type algorithm and prove that the algorithm is an approximate fixed point sequence for Lipschitz pseudocontractive maps in arbitrary real Banach spaces. Strong convergence results are then established for our inertial iteration scheme for approximation of fixed points of Lipschitz pseudocontractive maps and solutions of certain important accretive-type operator equations in certain real Banach spaces. Implementation of our algorithm is illustrated using numerical examples in both finite and infinite dimensional Banach spaces. Our results improve rate of convergence and extend several related recent results. [ABSTRACT FROM AUTHOR]

In this work, we introduce and study the modified inertial Mann’s algorithm and inertial hybrid algorithm for approximating some fixed points of a k-strict pseudo-contractive mapping in Hilbert spaces. Weak convergence to a solution of fixed-point problems for a k-strict pseudo-contractive mapping is obtained by using the modified inertial Mann’s algorithm. In order to obtain strong convergence, we introduce an inertial hybrid algorithm by using the inertial extrapolation method mixed with the convex combination of three iterated vectors and forcing for strong convergence by the hybrid projection method for a k-strict pseudo-contractive mapping in Hilbert spaces. The strong convergence theorem of the proposed method is proved under mild assumptions on the scalars. For illustrating the performance of the proposed algorithms, we provide some new nonlinear k-strict pseudo-contractive mappings which are not nonexpansive to create some numerical experiments to show the advantage of the two new inertial algorithms for a k-strict pseudo-contractive mapping. [ABSTRACT FROM AUTHOR]

In the paper, we construct an absolutely convergent Dirichlet series which in the mean is close to the periodic Hurwitz zeta-function, and has the universality property on the approximation of a wide class of analytic functions. [ABSTRACT FROM AUTHOR]

In this paper, weak convergence of balanced stochastic one-step methods and especially balanced stochastic Runge–Kutta (SRK) methods for Itô multidimensional stochastic differential equations is analyzed. Generalizing a corresponding result obtained by H. Schurz for the standard Euler method, it is shown that under certain conditions, balanced one-step methods preserve the weak convergence properties of their underlying methods. As an application, this allows to prove the weak convergence order of the balanced SRK methods presented in earlier work by A. Rathinasamy, P. Nair and D. Ahmadian. [ABSTRACT FROM AUTHOR]

The usual stochastic order and the likelihood ratio order between probability distributions on the real line are reviewed in full generality. In addition, for the distribution of a random pair (X, Y), it is shown that the conditional distributions of Y, given X = x, are increasing in x with respect to the likelihood ratio order if and only if the joint distribution of (X, Y) is totally positive of order two (TP2) in a certain sense. It is also shown that these three types of constraints are stable under weak convergence, and that weak convergence of TP2 distributions implies convergence of the conditional distributions just mentioned. [ABSTRACT FROM AUTHOR]

For the stochastic differential equation with piecewise continuous arguments, multiplicative noises and dissipative drift coefficients, we show that the solution at integer time is a Markov chain and admits a unique invariant measure. In order to numerically preserve the invariant measure, we apply the backward Euler method to the equation, and prove that the numerical solution at integer time is also a Markov chain and possesses a unique numerical invariant measure. By establishing several a priori estimations, we present the time-independent weak error analysis for the method via Malliavin calculus, which implies that the numerical invariant measure converges to the original one with weak order 1. Numerical experiments verify the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we derive a new projection algorithm by incorporating inertial effects for solving a split feasibility problem in real Hilbert spaces. We then establish a weak convergence theorem under some suitable conditions. As an application, we apply our result to image restoration. [ABSTRACT FROM AUTHOR]

In this paper, a new accelerated fixed point algorithm for finding a common fixed point of two countable families of nonexpansive mappings in a Hilbert space is introduced. The weak convergence of the proposed algorithm is analyzed under some suitable conditions. We apply our proposed algorithm to solve the convex optimization problems in the form of the sum of two lower semi-continuous and convex functions. Some numerical experiments are conducted to demonstrate the efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new iteration scheme, named as the S**-iteration scheme, for approximation of fixed points of Suzuki's generalized non-expansive mappings. We also put forward some weak and strong convergence theorems for Suzuki's generalized nonexpansive mappings in the setting of uniformly convex Banach spaces. We prove the stability of our instigated scheme and give numerical examples to show that this scheme is faster than Agarwal, Abbas, Thakur and Ullah iteration schemes. Our results comprehend, improve and consolidate many results in the existing literature. [ABSTRACT FROM AUTHOR]

Let a={am:m∈N} be a periodic multiplicative sequence of complex numbers and L(s;a), s=σ+it a Dirichlet series with coefficients am. In the paper, we obtain a theorem on the approximation of non-vanishing analytic functions defined in the strip 1/2<σ<1 via discrete shifts L(s+ihtk;a), h>0, k∈N, where {tk:k∈N} is the sequence of Gram points. We prove that the set of such shifts approximating a given analytic function is infinite. This result extends and covers that of [Korolev, M.; Laurinčikas, A. A new application of the Gram points. Aequat. Math. 2019, 93, 859–873]. For the proof, a limit theorem on weakly convergent probability measures in the space of analytic functions is applied.

Fixed point theory is a branch of mathematics that studies solutions that remain unchanged under a given transformation or operator, and it has numerous applications in fields such as mathematics, economics, computer science, engineering, and physics. In the present article, we offer a quicker iteration technique, the D** iteration technique, for approximating fixed points in generalized α-nonexpansive mappings and nearly contracted mappings. In uniformly convex Banach spaces, we develop weak and strong convergence results for the D** iteration approach to the fixed points of generalized α-nonexpansive mappings. In order to demonstrate the effectiveness of our recommended iteration strategy, we provide comprehensive analytical, numerical, and graphical explanations. Here, we also demonstrate the stability consequences of the new iteration technique. We approximately solve a fractional Volterra–Fredholm integro-differential problem as an application of our major findings. Our findings amend and expand upon some previously published results.

The aim of this paper is to conduct a systematic and theoretical analysis of estimation and inference for a class of functional mixed effects models (FMEM). Such FMEMs consist of fixed effects that characterize the association between longitudinal functional responses and covariates of interest and random effects that capture the spatial-temporal correlations of longitudinal functional responses. We propose local linear estimates of refined fixed effect functions and establish their weak convergence along with a simultaneous confidence band for each fixed-effect function. We propose a global test for the linear hypotheses of varying coefficient functions and derive the associated asymptotic distribution under the null hypothesis and the asymptotic power under the alternative hypothesis are derived. We also establish the convergence rates of the estimated spatial-temporal covariance operators and their associated eigenvalues and eigenfunctions. We conduct extensive simulations and apply our method to a white-matter fiber data set from a national database for autism research to examine the finite-sample performance of the proposed estimation and inference procedures.

We investigate unimodular random networks. Our motivations include their characteri- zation via reversibility of an associated random walk and their similarities to unimodular quasi-transitive graphs. We extend various theorems concerning random walks, percolation, spanning forests, and amenability from the known context of unimodular quasi-transitive graphs to the more general context of unimodular random networks. We give properties of a trace associated to unimodular random networks with applications to stochastic comparison of continuous-time random walk. © 2007 Applied Probability Trust.

In this paper, we proposed a new iterative process to approximate fixed point of generalized $\alpha$-nonexpansivemappings and show that the coefficient used in the proposed iterative process play a fundamental role in the rate of convergence. We compare the speed of convergence of new iterative process with other well-known iterative process by using numerical examples. Finally, by using new iterative process, we obtained some weak and strong convergence theorems for generalized $\alpha$-nonexpansive mappings in a Banach space.

Abstract In this paper, we define a new concept of left and right coordinate affine of a directed graph and then employ it to introduce a new accelerated common fixed point algorithm for a countable family of G-nonexpansive mappings in a real Hilbert space with a graph. We prove, under certain conditions, weak convergence theorems for the proposed algorithm. As applications, we also apply our results to solve convex minimization and image restoration problems. Moreover, we show that our algorithm provides better convergence behavior than other methods in the literature.

In this paper, we propose a new iteration process which is faster than the leading S [J. Nonlinear Convex Anal. 8, no. 1 (2007), 61-79], Thakur et al. [App. Math. Comp. 275 (2016), 147-155] and M [Filomat 32, no. 1 (2018), 187-196] iterations for numerical reckoning fixed points. Using new iteration process, some fixed point convergence results for generalized α-nonexpansive mappings in the setting of uniformly convex Banach spaces are proved. At the end of paper, we offer a numerical example to compare the rate of convergence of the proposed iteration process with the leading iteration processes.

Davie’s exact coupling technique for stochastic differential equations may be used to enhance the convergence of the multilevel Monte Carlo (MC) methodology. Giles developed the multilevel MC technique, which is based on executing the MC method several times with various time increments. It cuts computing costs significantly by executing most simulations at a low cost. The essential concept behind the multilevel MC approach with the exact coupling is discussed in this article. Numerical implementation reveals significant computational savings, which supports the analysis.

This paper aims to investigate the finite element weak convergence rate for semilinear parabolic stochastic partial differential equations(SPDEs) driven by additive noise. In contrast to many results in the current scientific literature, we investigate the more general case where the nonlinearity is allowed to be of Nemytskii-type and the linear operator is not necessarily self-adjoint, which is more challenging and models more realistic phenomena such as convection–reaction–diffusion processes. Using Malliavin calculus, Kolmogorov equations and by splitting the linear operator into a self-adjoint and non self-adjoint parts, we prove the convergence of the finite element approximation and obtain a weak convergence rate that is twice the strong convergence rate.

In the paper, we construct an absolutely convergent Dirichlet series which in the mean is close to the periodic Hurwitz zeta-function, and has the universality property on the approximation of a wide class of analytic functions.

In this work, we establish two new self-adaptive parallel algorithms to solve the generalized split common fixed point problem which is to find a point which belongs to the intersection of finite family of fixed point sets of demimetric mappings such that its image under a finite number of linear transformations belongs to the intersection of another finite family of fixed point sets of demimetric mappings in the image space. Under suitable assumptions, the weak and strong convergence theorems are analysed. The obtained results generalize and improve the recent results announced by many other authors in the framework of split inverse problem. As a direct consequence of our two main algorithms, we obtain several new algorithms. Preliminary numerical experiments are provided to illustrate the efficiency and implementation of our new methods and also to compare with others. [ABSTRACT FROM AUTHOR]

The Lerch zeta function is defined by a Dirichlet series depending on two fixed parameters. In the paper, we consider the approximation of analytic functions by discrete shifts of the Lerch zeta function, and we prove that, for arbitrary parameters and a step of arithmetic progression, there is a closed non-empty subset of the space of analytic functions defined in the critical strip such that its functions can be approximated by discrete shifts of the Lerch zeta function. The set of those shifts is infinite, and it has a positive density. For the proof, the weak convergence of probability measures in the space of analytic functions is applied. [ABSTRACT FROM AUTHOR]

The lift zonoid is a convenient representation of an integrable measure by a convex set in a higher-dimensional space. It is known that, under appropriate conditions, a uniformly integrable sequence of measures converges weakly if and only if the corresponding sequence of lift zonoids converges in the Hausdorff metric. We provide a new proof of this essential result. Our proof technique allows us to eliminate the unnecessary conditions previously considered in the literature. As a by-product, we obtain a characterization of uniform integrability, and a simple sufficient condition for tightness, of a sequence of integrable measures in terms of their lift zonoids. [ABSTRACT FROM AUTHOR]

This paper derives an approximation for a generalized Langevin equation driven by a force with random oscillation in time and periodic oscillation in space. By a diffusion approximation and the weak convergence of periodic oscillation function, the solution of the generalized Langevin equation is shown to converge in distribution to the solution of a stochastic partial differential equations (SPDEs) driven by time white noise. [ABSTRACT FROM AUTHOR]

We study the forward-backward-forward splitting method with alternated inertial step in this paper. We consider under-relaxation and over-relaxation approaches to the proposed method and obtain weak convergence results in both cases in real Hilbert spaces. Linear convergence alongside priori and apriori error estimates are obtained under some standard assumptions. Finally, we give some numerical implementations and comparisons to support the theoretical analysis. [ABSTRACT FROM AUTHOR]

The forward–backward–forward (FBF) splitting method is a popular iterative procedure for finding zeros of the sum of maximal monotone and Lipschitz continuous monotone operators. In this paper, we introduce a modification of the forward–backward splitting method with an adaptive step size rule for inclusion problems in real Hilbert spaces. Under standard assumptions, such as Lipschitz continuity and monotonicity (also maximal monotonicity), we establish weak convergence of the proposed algorithm. Moreover, if the single-valued operator is cocoercivity, then the proposed algorithm strongly converges to the unique solution of the problem with an R-linear rate. Finally, we give several numerical experiments to illustrate the convergence of the proposed algorithm and also to compare them with others. [ABSTRACT FROM AUTHOR]

Recently, the Randomized Kaczmarz algorithm (RK) draws much attention because of its low computational complexity and less requirement on computer memory. Many existing results on analysis focus on the behavior of RK in Euclidean spaces, and typically derive exponential converge rates with the base tending to one, as the condition number of the system increases. The dependence on the condition number largely restricts the application of these estimates. There are also results using relaxation (i.e., small step-sizes tending to zero as the sample size increases) to achieve polynomial convergence rates of RK in Hilbert spaces. In this paper, we prove the weak convergence (with polynomial rates) of RK with constant step-size in Hilbert spaces. As a consequence, the strong convergence of RK in Euclidean spaces is obtained with condition number-free polynomial rates. In the setting with noisy data, we study the relaxation technique and obtain a strong convergence of RK in Hilbert spaces, with the rates arbitrarily close to the minimax optimal ones. We apply the analysis to reproducing kernel-based online gradient descent learning algorithms and improve the state-of-the-art convergence estimate in the literature. [ABSTRACT FROM AUTHOR]

Let t τ be a solution to the equation θ (t) = (τ − 1) π , τ > 0 , where θ (t) is the increment of the argument of the function π − s / 2 Γ (s / 2) along the segment connecting points s = 1 / 2 and s = 1 / 2 + i t . t τ is called the Gram function. In the paper, we consider the approximation of collections of analytic functions by shifts of the Riemann zeta-function (ζ (s + i t τ α 1) , ... , ζ (s + i t τ α r)) , where α 1 , ... , α r are different positive numbers, in the interval [ T , T + H ] with H = o (T) , T → ∞ , and obtain the positivity of the density of the set of such shifts. Moreover, a similar result is obtained for shifts of a certain absolutely convergent Dirichlet series connected to ζ (s) . Finally, an example of the approximation of analytic functions by a composition of the above shifts is given. [ABSTRACT FROM AUTHOR]

In the paper, a theorem on approximation of a wide class of analytic functions by generalized shifts uT (s + if(t )) of an absolutely convergent Dirichlet series uT (s) which in the mean is close to the Riemann zetafunction is obtained. Here f(t) is a monotonically increasing differentiable function having a monotonic continuous derivative such that f(2t) max tt2t 1 f'(t) « t as t 8, and uT 8 and uT « T2 as T 8. [ABSTRACT FROM AUTHOR]

A nested occupancy scheme in a random environment is a generalization of the classical Karlin infinite balls-in-boxes occupancy scheme in a random environment (with random probabilities). Unlike the Karlin scheme in which the collection of boxes is unique, there is a nested hierarchy of boxes, and the hitting probabilities of boxes are defined in terms of iterated fragmentation of a unit mass. In the present paper, we assume that the random fragmentation law is given by stick-breaking in which case the infinite occupancy scheme defined by the first level boxes is known as the Bernoulli sieve. Assuming that n balls have been thrown, denote by K n (j) the number of occupied boxes in the jth level and call the level j intermediate if j = j n → ∞ and j n = o (log ⁡ n) as n → ∞. We prove a multidimensional central limit theorem for the vector (K n (⌊ j n u 1 ⌋) , ... , K n (⌊ j n u ℓ ⌋) , properly normalized and centred, as n → ∞ , where j n → ∞ and j n = o ((log ⁡ n) 1 / 2). The present paper continues the line of investigation initiated in the article [D. Buraczewski, B. Dovgay, and A. Iksanov, On intermediate levels of nested occupancy scheme in random environment generated by stick-breaking I, Electron. J. Probab. 25(123) (2020), pp. 1–24] in which the occupancy of intermediate levels j n → ∞ , j n = o ((log ⁡ n) 1 / 3) was analysed. [ABSTRACT FROM AUTHOR]

We explore an exchangeably weighted bootstrap of the general function-indexed empirical U-processes in the Markov setting, which is a natural higher-order generalization of the weighted bootstrap empirical processes. As a result of our findings, a considerable variety of bootstrap resampling strategies arise. This paper aims to provide theoretical justifications for the exchangeably weighted bootstrap consistency in the Markov setup. General structural conditions on the classes of functions (possibly unbounded) and the underlying distributions are required to establish our results. This paper provides the first general theoretical study of the bootstrap of the empirical U-processes in the Markov setting. Potential applications include the symmetry test, Kendall's tau and the test of independence. [ABSTRACT FROM AUTHOR]

In this article, we propose a mathematical model that describes hydrodynamics and deformation mechanics within a solid tumor which is embedded in or adjacent to a healthy (normal) tissue. The tumor and normal tissue regions are assumed to be deformable and the theory of mixtures is adapted to mass and momentum balance equations for fluid flow and tissue deformation mechanics in each region. The momentum balance equations are coupled via forces that interact between the phases (fluid and solid). Continuity of normal velocities, displacements, and normal stresses along with the Beaver–Joseph–Saffman condition are imposed at the interface between the tumor and tissue regions. The physiological transport parameters (such as hydraulic resistivity or permeability) are assumed to be heterogeneous and deformation dependent which makes the model nonlinear. We establish the existence of a weak solution using Galerkin and weak convergence methods. We show further that the solution is unique and depends continuously on the given data. [ABSTRACT FROM AUTHOR]

Consider a general branching process, a.k.a. Crump–Mode–Jagers process, generated by a perturbed random walk η1, ξ1 + η2, ξ1 + ξ2 + η3, ..., where (ξ1, η1), (ξ2, η2), ... are independent identically distributed random vectors with arbitrarily dependent positive components. Denote by Nj(t) the number of the jth generation individuals with birth times ⩽ t. Assume that j = j(t) → ∞ and j(t) = o(ta) as t → ∞ for some explicitly given a > 0 (to be specified in the paper). The corresponding jth generation belongs to the set of intermediate generations. We provide sufficient conditions under which the finite-dimensional distributions of the process (N⌊j(t)u⌋(t))u > 0, properly normalized and centered, converge weakly to those of an integral functional of a stable Lévy process with finite mean. [ABSTRACT FROM AUTHOR]

This paper deals with the weak convergence of nonparametric estimators of the multidimensional and multidimensional-multivariate renewal functions on Skorohod topology spaces. It is an extension of Harel et al. (J Math Anal Appl 189:240–255, 1995) from the one-dimensional case to the multivariate and multidimensional case. The estimators are based on a sequence of non-negative independent and identically distributed (iid) random vectors. They are expressed as infinite sums of k-folds convolutions of the empirical distribution function. Their weak convergence study heavily rests on that of the empirical distribution function. [ABSTRACT FROM AUTHOR]

New iterative algorithms for solving variational inequalities in uniformly convex Banach spaces are analyzed. The first algorithm is a modification of the forward-reflected-backward algorithm, which uses the Alber generalized projection instead of the metric one. The second algorithm is an adaptive version of the first one, where the monotone step size update rule is used, which does not require knowledge of the Lipschitz constants and linear search procedure. For variational inequalities with monotone, Lipschitz continuous operators acting in a 2-uniformly convex and uniformly smooth Banach space, theorems on the weak convergence of the methods are proved. Also, for the first algorithm, an efficiency estimate in terms of the gap function is proved. [ABSTRACT FROM AUTHOR]

Consider the Krasnosel’skiĭ-Mann iteration in real Hilbert spaces. In this note, we suggest weaker assumptions on the involved two factors of an accelerated version, and we use them to prove its weak convergence and analyze its rate of convergence. Furthermore, we suggest a practical way of choosing adaptive acceleration factors and confirm its numerical superiority. [ABSTRACT FROM AUTHOR]

Consider a general branching process, a.k.a. Crump–Mode–Jagers process, generated by a perturbed random walk η1, ξ1 + η2, ξ1 + ξ2 + η3, ..., where (ξ1, η1), (ξ2, η2), ... are independent identically distributed random vectors with arbitrarily dependent positive components. Denote by Nj(t) the number of the jth generation individuals with birth times ⩽ t. Assume that j = j(t) → ∞ and j(t) = o(ta) as t → ∞ for some explicitly given a > 0 (to be specified in the paper). The corresponding jth generation belongs to the set of intermediate generations. We provide sufficient conditions under which the finite-dimensional distributions of the process (N⌊j(t)u⌋(t))u > 0, properly normalized and centered, converge weakly to those of an integral functional of a stable Lévy process with finite mean. [ABSTRACT FROM AUTHOR]

In this work, we are interested in solving a convex minimization problem in real Hilbert spaces. We propose a new modified proximal algorithm using the inertial extrapolation and the linesearch technique. Its weak convergence theorems are established under mild conditions. Numerical experiments are presented to illustrate the performance of the proposed algorithm in image deblurring. [ABSTRACT FROM AUTHOR]