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Let a = { a m : m ∈ N } be a periodic multiplicative sequence of complex numbers and L (s ; a) , s = σ + i t a Dirichlet series with coefficients a m . 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 + i h t k ; a) , h > 0 , k ∈ N , where { t k : 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. [ABSTRACT FROM AUTHOR]

In this paper, we obtain weak convergence results for a family of Gibbs measures depending on the parameter θ > 0 in the following form d P θ (x) = Z θ exp − H θ (x) / θ d Q (x) , where we show that the limit distribution is concentrated in the set of the global minima of the limit Gibbs potential. We also give an explicit calculus for the limit distribution. Here, we use the above as an alternative to Lyapunov's function or to direct methods for stationary probability convergence and apply it to the repairman problem. Finally, we illustrate this method with a numerical example. [ABSTRACT FROM AUTHOR]

In the paper, it is obtained that there are infinite discrete shifts Ξ (s + i k h) , h > 0 , k ∈ N 0 of the Mellin transform Ξ (s) of the square of the Riemann zeta-function, approximating a certain class of analytic functions. For the proof, a probabilistic approach based on weak convergence of probability measures in the space of analytic functions is applied. [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]

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]

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 paper, we alter Wang's new iterative method as well as apply it to find the common solution of fixed point problem (FPP) and split variational inclusion problem ( S p VIP) in Hilbert space. We discuss the weak convergence for ( S p VIP) and strong convergence for the common solution of ( S p VIP) and (FPP) using appropriate assumptions. Some consequences of the proposed methods are studied. We compare our iterative schemes with other existing related schemes. [ABSTRACT FROM AUTHOR]

Let Q be a positive defined n × n matrix and Q [ x ̲ ] = x ̲ T Q x ̲ . 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 is meromorphically continued on the whole complex plane. Suppose that n ⩾ 4 is even and φ (t) is a differentiable function with a monotonic derivative. In the paper, it is proved that 1 T meas t ∈ [ 0 , T ] : ζ (σ + i φ (t) ; Q) ∈ A , A ∈ B (C) , converges weakly to an explicitly given probability measure on (C , B (C)) as T → ∞ . [ABSTRACT FROM AUTHOR]

In the paper, the simultaneous approximation of a tuple of analytic functions in the strip { s = σ + i t ∈ C : 1 / 2 < σ < 1 } by shifts (ζ (s + i φ 1 (τ)) , ... , ζ (s + i φ r (τ))) of the Riemann zeta-function ζ (s) with a certain class of continuously differentiable increasing functions φ 1 , ... , φ r is considered. This class of functions φ 1 , ... , φ r is characterized by the growth of their derivatives. It is proved that the set of mentioned shifts in the interval [ T , T + H ] with H = o (T) has a positive lower density. The precise expression for H is described by the functions (φ j (τ)) 1 / 3 (log φ j (τ)) 26 / 15 and derivatives φ j ′ (τ) . The density problem is also discussed. An example of the approximation by a composition F (ζ (s + i φ 1 (τ)) , ... , ζ (s + i φ r (τ))) with a certain continuous operator F in the space of analytic functions is given. [ABSTRACT FROM AUTHOR]

We designed a modified projection method with a new condition of the inertial step and the step size for the split feasibility problem in Hilbert spaces. We show that our iterate weakly converged to a solution. Lastly, we give numerical examples and comparisons that could be applied to signal recovery to show the efficiency of our method. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new iterative method using an inertial technique for approximating a common fixed point of an infinite family of nonexpansive mappings in a Hilbert space. The proposed method's weak convergence theorem was established under some suitable conditions. Furthermore, we applied our main results to solve convex minimization problems and image restoration problems. [ABSTRACT FROM AUTHOR]

In this work, we study iterative methods for the approximation of common attractive points of two widely more generalized hybrid mappings in Hilbert spaces and obtain weak and strong convergence theorems without assuming the closedness for the domain. A numerical example supporting our main result is also presented. As a consequence, our main results can be applied to solving a common fixed point problem. [ABSTRACT FROM AUTHOR]

In the paper, a Dirichlet series ζ u N (s) whose shifts ζ u N (s + i k h) , k = 0 , 1 , ⋯ , h > 0 , approximate analytic non-vanishing functions defined on the right-hand side of the critical strip is considered. This series is closely connected to the Riemann zeta-function. The sequence u N → ∞ and u N ≪ N 2 as N → ∞ . [ABSTRACT FROM AUTHOR]

In this research, we study the convex minimization problem in the form of the sum of two proper, lower-semicontinuous, and convex functions. We introduce a new projected forward-backward algorithm using linesearch and inertial techniques. We then establish a weak convergence theorem under mild conditions. It is known that image processing such as inpainting problems can be modeled as the constrained minimization problem of the sum of convex functions. In this connection, we aim to apply the suggested method for solving image inpainting. We also give some comparisons to other methods in the literature. It is shown that the proposed algorithm outperforms others in terms of iterations. Finally, we give an analysis on parameters that are assumed in our hypothesis. [ABSTRACT FROM AUTHOR]

In this paper, we presented a modification of the extragradient method to solve pseudomonotone equilibrium problems involving the Lipschitz-type condition in a real Hilbert space. The method uses an inertial effect and a formula for stepsize evaluation, that is updated for each iteration based on some previous iterations. The key advantage of the algorithm is that it is achieved without previous knowledge of the Lipschitz-type constants and also without any line search procedure. A weak convergence theorem for the proposed method is well established by assuming mild cost bifunction conditions. Many numerical experiments are presented to explain the computational performance of the method and to equate them with others. [ABSTRACT FROM AUTHOR]

In this paper, we propose two modified two-step proximal methods that are formed through the proximal-like mapping and inertial effect for solving two classes of equilibrium problems. A weak convergence theorem for the first method and the strong convergence result of the second method are well established based on the mild condition on a bifunction. Such methods have the advantage of not involving any line search procedure or any knowledge of the Lipschitz-type constants of the bifunction. One practical reason is that the stepsize involving in these methods is updated based on some previous iterations or uses a stepsize sequence that is non-summable. We consider the well-known Nash–Cournot equilibrium models to support our well-established convergence results and see the advantage of the proposed methods over other well-known methods. [ABSTRACT FROM AUTHOR]

A stick of length 1 is broken at random into n smaller sticks. How much inequality does this procedure produce? What happens if, instead of breaking a stick, we break a square? What happens asymptotically? Which is the most egalitarian distribution of the smaller sticks (or rectangles)? Usually, when studying inequality, one uses a Lorenz curve. The more egalitarian a distribution, the closer the Lorenz curve is to the first diagonal of [ 0 ,   1 ] 2 . This is why in the first section we study the space of Lorenz curves. What is the limit of a convergent sequence of Lorenz curves? We try to answer these questions, firstly, in the deterministic case and based on the results obtained there in the stochastic one. [ABSTRACT FROM AUTHOR]

In this work, we study the inclusion problem of the sum of two monotone operators and the fixed-point problem of nonexpansive mappings in Hilbert spaces. We prove the weak and strong convergence theorems under some weakened conditions. Some numerical experiments are also given to support our main theorem. [ABSTRACT FROM AUTHOR]