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In this paper, we present the ideal convergence of triple sequences for rough variables. Furthermore, sequence convergence plays an extremely important role in the fundamental theory of mathematics. This paper presents two types of ideal convergence of rough triple sequence: Convergence in trust and convergence in mean. Some mathematical properties of those new convergence concepts are also given. In addition, we introduce ideal Cauchy triple sequence in rough spaces. [ABSTRACT FROM AUTHOR]

In this paper, the Galerkin finite element method is applied to solve the generalized delay reaction-diffusion equation, where the spatial and temporal variables are discretized by the Galerkin finite method and the Crank–Nicolson method, respectively. The numerical treatment shows that the proposed numerical scheme is asymptotically stable. Moreover, some a priori estimates of L 2 and L ∞ -norms and with optimal order of convergence C (h 2 + (Δ t) 2 )) are obtained. The numerical experiments illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

This paper investigates the asymptotic behaviour of solutions to certain infinite systems of coupled recurrence relations. In particular, we obtain a characterisation of those initial values which lead to a convergent solution, and for initial values satisfying a slightly stronger condition we obtain an optimal estimate on the rate of convergence. By establishing a connection with a related problem in continuous time, we are able to use this optimal estimate to improve the rate of convergence in the continuous setting obtained by the authors in a previous paper. We illustrate the power of the general approach by using it to study several concrete examples, both in continuous and in discrete time. [ABSTRACT FROM AUTHOR]

Motivated by truncated Euler–Maruyama (EM) method introduced by Mao (2015), a new explicit numerical method named modified truncated Euler–Maruyama method is developed in this paper. Strong convergence rates of the given numerical scheme to the exact solutions to stochastic differential equations are investigated under given conditions in this paper. Compared with truncated EM method, the given numerical simulation strongly converges to the exact solution at fixed time T and over a time interval [ 0 , T ] under weaker sufficient conditions. Meanwhile, the convergence rates are also obtained for both cases. Two examples are provided to support our conclusions. [ABSTRACT FROM AUTHOR]

The problem of consistency of a solution over time remains an important issue in cooperative dynamic games. Payoffs to players prescribed by an inconsistent solution may not be achievable since such a solution is extremely sensitive to its revision in the course of a game developing along an agreed upon cooperative behavior. The paper proposes a strong time-consistent payment scheme which is stable to a revision of cooperative set solutions, e.g., the core. Using a linear transformation of the solution, it becomes possible to obtain non-negative payments to players. In the paper, we also deal with a limit linear transformation of the solution whose convergence is proved. Developing a non-negative strong time-consistent payment scheme in a closed form, we guarantee that the solution supported by the scheme will not be revised over time. [ABSTRACT FROM AUTHOR]

We extend in this paper an optimal family of three-step eighth-order methods developed by Džunić et al. (2011) with higher-order weight functions employed in the second and third sub-steps and investigate their dynamics under the relevant extraneous fixed points among which purely imaginary ones are specially treated for the analysis of the rich dynamics. Their theoretical and computational properties are fully investigated along with a main theorem describing the order of convergence and the asymptotic error constant as well as proper choices of special cases. A wide variety of relevant numerical examples are illustrated to confirm the underlying theoretical development. In addition, this paper investigates the dynamics of selected existing optimal eighth-order iterative maps with the help of illustrative basins of attraction for various polynomials. [ABSTRACT FROM AUTHOR]

This paper investigates identification of Wiener systems with quantized inputs and binary-valued output observations. By parameterizing the static nonlinear function and incorporating both linear and nonlinear parts, we begin by investigating system identifiability under the input and output constraints. Then a three-step algorithm is proposed to estimate the unknown parameters by using the empirical measure, input persistent patterns, and information on noise statistics. Convergence properties of the algorithm, including strong convergence and mean-square convergence rate, are established. Furthermore, by selecting a suitable transformation matrix, the asymptotic efficiency of the algorithm is proved in terms of the Cramér–Rao lower bound. Finally, numerical simulations are presented to illustrate the main results of this paper. [ABSTRACT FROM AUTHOR]

Convergence rate result for a modified steepest descent method and a modified minimal error method for the solution of nonlinear ill-posed operator equation have been proved with noisy data. To our knowledge, convergence rate result for the steepest descent method and minimal error method with noisy data are not known. We provide a convergence rate results for these methods with noisy data. The result in this paper are obtained under less computational cost when compared to the steepest descent method and minimal error method. We present an academic example which satisfies the assumptions of this paper. [ABSTRACT FROM AUTHOR]

Abstract The purpose of this paper is to establish Bogoliubov averaging principle of stochastic reaction–diffusion equation with a stochastic process and a small parameter. The solutions to stochastic reaction–diffusion equation can be approximated by solutions to averaged stochastic reaction–diffusion equation in the sense of convergence in probability and in distribution. Namely, we establish a weak law of large numbers for the solution of stochastic reaction–diffusion equation. [ABSTRACT FROM AUTHOR]

In this paper, we consider the quasi-neutral limit of the compressible Navier-Stokes-Fourier-Poisson system in a periodic domain with the well-prepared initial data. We prove that the weak solution of the compressible Navier-Stokes-Fourier-Possion system converges to the strong solution of the compressible Navier-Stokes-Fourier system as long as the latter exists. [ABSTRACT FROM AUTHOR]

In 1936 Zygmunt Zalcwasser asked, with respect to the trigonometric system, how "rare" can a sequence of strictly monotone increasing integers (nj) be such that the almost everywhere relation 1N∑j=1NSnjf→f is fulfilled for each integrable function f. In this paper, we give an answer to this question. It follows from the main result that this a.e. relation holds for every integrable function f and lacunary sequence (nj) of natural numbers. [ABSTRACT FROM AUTHOR]

In this paper, we study locally and globally small Riemann sums with respect to α for n-dimensional fuzzy number valued functions. And we prove that a fuzzy-number-valued functions in n-dimensional is Henstock-Stieltjes (HS) integrable on [a; b] if and only if it has (LSRS) with respect to α on [a, b]. Also we shall prove that a fuzzy number valued functions in n-dimensional is Henstock-Stieltjes (HS) integrable on [a, b] if and only if it has (GSRS) with respect to α on [a; b]. [ABSTRACT FROM AUTHOR]

Abstract. In this paper, we study Δ-convergence and strong convergence of the sequence generated by the extragradient method for pseudo-monotone equilibrium problems in Hadamard spaces. We first show Δ-convergence of the generated sequence to a solution of the equilibrium problem, then the strong convergence of Halpern regularization method is proved. Finally we give some examples where the main results can be applied. [ABSTRACT FROM AUTHOR]

In this paper, we use a good technique to construct a high-order numerical scheme for the impulsive fractional ordinary differential equations (IFODEs). This technique is based on the so-called block-by-block method, which is a common method for the integral equations. In our approach, the classical block-by-block method is improved so as to avoid the coupling of the unknown solutions at each block step with an exception in the first two steps between two adjacent pulse points. The convergence and stability analysis of the scheme are given. It proves that the numerical solution converges to the exact solution with order 3+q for , where q is the order of the fractional derivative. A series of numerical examples are provided to support the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, statistical epi-limit is defined by using epigraphs in order to increase sensitivity by eliminating outliers for mathematical problems. Various characterizations of statistical epi-convergence and their relations are given and it is compared with statistical convergence. Also, its connections with level sets and monotone increasing or decreasing cases are studied. Moreover, statistical equi-lower semicontinuity and its relation with statistical epilimit is examined. [ABSTRACT FROM AUTHOR]

In this paper, we propose the modified Newton-PSS method for solving systems of nonlinear equations with positive definite Jacobian matrices. Local convergence theorem is established under the Hölder continuous condition. Numerical results demonstrate the effectiveness of our method. [ABSTRACT FROM AUTHOR]

A quasi-infinitely divisible distribution on R is a probability distribution whose characteristic function allows a L'evy-Khintchine type representation with a "signed L'evy measure", rather than a L'evy measure. Quasiinfinitely divisible distributions appear naturally in the factorization of infinitely divisible distributions. Namely, a distribution μ is quasi-infinitely divisible if and only if there are two infinitely divisible distributions μ1 and μ2 such that μ1 * μ = μ2. The present paper studies certain properties of quasiinfinitely divisible distributions in terms of their characteristic triplet, such as properties of supports, finiteness of moments, continuity properties, and weak convergence, with various examples constructed. In particular, it is shown that the set of quasi-infinitely divisible distributions is dense in the set of all probability distributions with respect to weak convergence. Further, it is proved that a distribution concentrated on the integers is quasi-infinitely divisible if and only if its characteristic function does not have zeroes, with the use of the Wiener-L'evy theorem on absolutely convergent Fourier series. A number of fine properties of such distributions are proved based on this fact. A similar characterization is not true for nonlattice probability distributions on the line. [ABSTRACT FROM AUTHOR]

We develop a unfied and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate di ffusion equations ∂u-L[φ(u)] = f(x; t) in ℝN×(0; T); where L is a general symmetric Lévy-type diffusion operator. Included are both local and nonlocal problems with, e.g., L = Δ or L = -(-Δ) ff2, ff ϵ (0; 2), and porous medium, fast diffusion, and Stefan-type nonlinearities φ . By robust methods we mean that they converge even for nonsmooth solutions and under very weak assumptions on the data. We show that they are Lp-stable for p ϵ[1,∞], compact, and convergent in C([0; T];Lploc(ℝN)) for p 2 [1,∞). The first part of this project is given in [F. del Teso, J. Endal, and E. R. Jakobsen, preprint, arXiv:1801.07148v1 [math.NA], 2018] and contains the unified and easy to use theoretical framework. This paper is devoted to schemes and testing. We study many different problems and many different concrete discretizations, proving that the results of Part I apply and testing the schemes numerically. Our examples include fractional diffusions of different orders and Stefan problems, porous medium, and fast diffusion nonlinearities. Most of the convergence results and many schemes are completely new for nonlocal versions of the equation, including results on high order methods, the powers of the discrete Laplacian method, and discretizations of fast diffusions. Some of the results and schemes are new even for linear and local problems. [ABSTRACT FROM AUTHOR]

In this paper, we consider the varying stepsize gradient projection algorithm (GPA) for solving the split equality problem (SEP) in Hilbert spaces, and study its linear convergence. In particular, we introduce a notion of bounded linear regularity property for the SEP, and use it to establish the linear convergence property for the varying stepsize GPA. We provide some mild sufficient conditions to ensure the bounded linear regularity property, and then conclude the linear convergence rate of the varying stepsize GPA. To the best of our knowledge, this is the first work to study the linear convergence for the SEP. [ABSTRACT FROM AUTHOR]

In this paper, we define m-uniform I-statistical convergence, (θ,m)-uniform I-lacunary statistical convergence and (Iθ,m)-uniform strongly p-lacunary summability of functions on an arbitrary time scale. Also, by using m-uniform and (λ,m)-uniform density of the subset of the time scale, we will focus on constructing concepts of (Iλ,m)-uniform statistically convergence and (Iλ,m)-uniformly strongly p-summability of functions on time scale. Some inclusion relations about these new concepts are also presented. [ABSTRACT FROM AUTHOR]

In this paper, we present two penalty-splitting inspired iteration schemes (PFFSA) and (PFFSA) for hierarchical equilibrium problems in Hilbert space. Based on the Opial-Passty lemma, we propose weak ergodic convergence and weak convergence of the iterative sequences generated by the Forward-Forward algorithm (PFFSA) and the Forward-Backward algorithm (PFFSA), which are proved under quite mild conditions: the bifunction of the two level equilibrium problems are supposed pseudomonotone. For strong convergence, we first add a strong monotonicity condition on the objective bifunction. We present after, a strong convergence result of algorithm (PFFSA) by adding a topological assumption, i.e. the objective bifunction is of class . Some examples are given to illustrate our results. The first example deals with pseudomonotone variational inequalities and convex minimization problem in the upper level problem. In the second one, we propose a convex minimization in the lower-level problem, where strong convergence of (PFFSA) to a minimum point is insured under infcompactness condition for objective function. These convergence results are new and generalize some recent results in this field. [ABSTRACT FROM AUTHOR]

In this paper, we solve in the convergence set, the fractional logistic equation making use of Euler’s numbers. To our knowledge, the answer is still an open question. The key point is that the coefficients can be connected with Euler’s numbers, and then they can be explicitly given. The constrained of our approach is that the formula is not valid outside the convergence set. The idea of the proof consists to explore some analogies with logistic function and Euler’s numbers, and then to generalize them in the fractional case. [ABSTRACT FROM AUTHOR]

This paper is devoted to the development and analysis of a modified family of Falkner-type methods for solving differential systems of second-order initial-value problems. The approaches of collocation and interpolation are adopted to derive the new methods. These modified methods are implemented in block form to obtain the numerical solutions to the considered problems. The study of the properties of the proposed block Falkner-type methods reveals that they are consistent and zero-stable, and thus, convergent. From the stability analysis, it could be seen that the proposed Falkner methods have non-empty stability regions for k = 2 , 3 , 4 . Some numerical test are presented to illustrate the efficiency of the proposed family. [ABSTRACT FROM AUTHOR]

In this paper, we prove the bilaterally almost uniformly convergence of bounded L1(M) -noncommutative quasi-martingales. We also prove Gundy's decomposition for noncommutative quasi-martingales. As an application, we prove that every relatively weakly compact quasi-martingale difference sequence in L1(M; ι) whose sequence of norms is bounded away from zero is 2-co-lacunary. [ABSTRACT FROM AUTHOR]

In this paper, a notion of modified ⊤-convergence spaces (initially defined by Fang and Yue in FSS, 2017) is given. Then the relationships between the modified ⊤-convergence spaces and types of lattice-valued convergence spaces are established. [ABSTRACT FROM AUTHOR]

This paper aims at putting forward several types of convergence concepts of complex uncertain random sequences. The relations among convergence concepts are derived by some limit theorems. In addition, for illustrating of convergence theorems, lots of examples are derived. Finally, a lot of counterexamples about relationships between convergence concepts are stated. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the damped stochastic nonlinear Schrödinger (NLS) equation with multiplicative noise and its splitting-based approximation. When the damped effect is large enough, we prove that the solutions of both the damped stochastic NLS equation and the splitting scheme are exponentially stable and possess some exponential integrability. These properties show that the strong order of the scheme is 1 and independent of time. Additionally, we analyze the regularity of the Kolmogorov equation with respect to the stochastic NLS equation. As a consequence, the weak order of the scheme is shown to be 1 and independent of time. [ABSTRACT FROM AUTHOR]

In the literature, error bound conditions have been widely used to study the linear convergence rates of various first-order algorithms. Most of the literature focuses on how to ensure these error bound conditions, usually posing numerous assumptions or special structures on the model under discussion. In this paper, we focus on the alternating direction method of multipliers (ADMM) and show that the known error bound conditions for studying the ADMM's linear convergence rate can indeed be further weakened if the error bound is studied over the specific iterative sequence it generates. An error bound condition based on ADMM's iterations is thus proposed, and linear convergence under this condition is proved. Furthermore, taking advantage of a specific feature of ADMM's iterative scheme by which part of the perturbation is automatically zero, we propose the so-called partial error bound condition, which is weaker than known error bound conditions in the literature, and we derive the linear convergence rate of ADMM. We further show that this partial error bound condition is useful for interpreting the difference if the two primal variables are updated in different orders when implementing the ADMM. This has been empirically observed in the literature, yet no theory is known. [ABSTRACT FROM AUTHOR]

The Ensemble Kalman methodology in an inverse problems setting can be viewed as an iterative scheme, which is a weakly tamed discretization scheme for a certain stochastic differential equation (SDE). Assuming a suitable approximation result, dynamical properties of the SDE can be rigorously pulled back via the discrete scheme to the original Ensemble Kalman inversion. The results of this paper make a step towards closing the gap of the missing approximation result by proving a strong convergence result in a simplified model of a scalar SDE. We focus here on a toy model with similar properties to the one arising in the context of an Ensemble Kalman filter. The proposed model can be interpreted as a single particle filter for a linear map and thus forms the basis for further analysis. The difficulty in the analysis arises from the formally derived limiting SDE with nonglobally Lipschitz continuous nonlinearities both in the drift and in the diffusion. Here the standard Euler-Maruyama scheme might fail to provide a strongly convergent numerical scheme and taming is necessary. In contrast to the strong taming usually used, the method presented here provides a weaker form of taming. We present a strong convergence analysis by first proving convergence on a domain of high probability by using a cutoff or localization, which then leads, combined with bounds on moments for both the SDE and the numerical scheme, by a bootstrapping argument to strong convergence. [ABSTRACT FROM AUTHOR]

Our underlying motivation is the iterative method for the implementation of the limit representation of the Moore-Penrose inverse lim αI0 (αI + A*A)-1 A* from [Žukovski, Lipcer, On recurent computation of normal solutions of linear algebraic equations, Ž. Vicisl. Mat. i Mat. Fiz. 12 (1972), 843-857] and [Žukovski, Lipcer, On computation pseudoinverse matrices, Ž. Vicisl. Mat. i Mat. Fiz. 15 (1975), 489-492]. The iterative process for the implementation of the general limit formula lim αI0 (αI + R*S)-1R* was defined in [P.S. Stanimirović, Limit representations of gen- eralized inverses and related methods, Appl. Math. Comput. 103 (1999), 51-68]. In this paper we develop an improvement of this iterative process. The iterative method defined in such a way is able to produce the result in a predefined number of iterative steps. Convergence properties of defined iterations are further investigated. [ABSTRACT FROM AUTHOR]

In this paper we discuss the asymptotic stability of stationary solutions for the non-isentropic Euler-Maxwell system in R. It is known in the authors' previous works [17, 18, 19] that the Euler-Maxwell system verifies the decay property of the regularity-loss type. In this paper we first prove the existence and uniqueness of a small stationary solution. Then we show that the non-stationary problemhas a global solution in a neighborhood of the stationary solution under smallness condition on the initial perturbation. Moreover, we show the asymptotic convergence of the solution toward the stationary solution as time tends to infinity. The crucial point of the proof is to derive a priori estimates by using the energy method. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the strong convergence properties for the Kaplan–Meier estimator and hazard estimator based on censored negatively superadditive dependent data. Under some mild conditions, the strong convergence rate of the Kaplan–Meier estimator and hazard estimator is established. In addition, the strong representation of the Kaplan–Meier estimator and hazard estimator is also obtained with the remainder of orderOur results established in the paper generalize the corresponding ones for independent random variables and negatively associated random variables. [ABSTRACT FROM AUTHOR]

Weak convergence of semi-Markov processes in the diffusive approximation scheme is studied in the paper. This problem is not new and it is studied in many papers, using convergence of random processes. Unlike other studies, we used in this paper concept of the compensating operator. It enables getting sufficient conditions of weak convergence under the conditions on the local characteristics of output semi-Markov process. [ABSTRACT FROM AUTHOR]

This paper studies the problem of split convex feasibility and a strong convergent alternating algorithm is established. According to this algorithm, some strong convergent theorems are obtained and an affirmative answer to the question raised by Moudafi is given. At the same time, this paper also generalizes the problem of split convex feasibility. [ABSTRACT FROM AUTHOR]

In this paper, a three-term conjugate gradient algorithm is developed for solving large-scale unconstrained optimization problems. The search direction at each iteration of the algorithm is determined by rectifying the steepest descent direction with the difference between the current iterative points and that between the gradients. It is proved that such a direction satisfies the approximate secant condition as well as the conjugacy condition. The strategies of acceleration and restart are incorporated into designing the algorithm to improve its numerical performance. Global convergence of the proposed algorithm is established under two mild assumptions. By implementing the algorithm to solve 75 benchmark test problems available in the literature, the obtained results indicate that the algorithm developed in this paper outperforms the existent similar state-of-the-art algorithms. [ABSTRACT FROM AUTHOR]

Text The aim of this paper is to present a method to calculate the rate of growth of divergent series. In the case of the harmonic series, we study an alternate of other classical methods presented by L. Comtet, S.M. Zeyman, W. DeTemple and S.-H. Wang. Moreover, the method presented in this paper is suitable for obtaining similar results on other divergent series, as we will show in the final part of our investigation. Video For a video summary of this paper, please visit http://youtu.be/SFR7zY4pvG4 . [ABSTRACT FROM AUTHOR]

This is the first of two papers, in which we prove some properties of the Webster scalar curvature flow. More precisely, we establish the long-time existence, L p convergence and the blow-up analysis for the solution of the flow. As a by-product, we prove the convergence of the CR Yamabe flow on the CR sphere. The results in this paper will be used to prove a result of prescribing Webster scalar curvature on the CR sphere, which is the main result of the second paper. [ABSTRACT FROM AUTHOR]

In this paper, we consider the preconditioned generalized accelerated overrelaxation (GAOR) methods for solving weighted linear least squares problems. New type of preconditioners are proposed and the convergence rates of the new preconditioned GAOR methods are studied. Comparison results show the effectiveness of the proposed preconditioners in this paper. A numerical example is given to confirm our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we study the pointwise convergence of the function e i t Δ α 2 f (x) whenever f belongs to some Sobolev space. When α = 2, e i t Δ f (x) is the solution to the standard Schrödinger equation and this problem has been widely studied. Here we focus on the case 0 < α ≠ 1 . The same convergence problem for the nonelliptic Schrödinger equation is also discussed. The proof here follows the idea of Carleson's first paper on this subject. But we do reprove some known theorems by this straightforward method and get some new results as well. [ABSTRACT FROM AUTHOR]

The notion of traveling wave, which refers to some particular spatio-temporal connections between two stationary states (typically, entire solutions keeping the same profile's shape through time), is essential in the mathematical analysis of propagation phenomena. They provide insight on the underlying dynamics, and an accurate description of large time behavior of solutions of the Cauchy problem, as we will see in this paper. For instance, in an homogeneous framework, it is well-known that, given a fast decaying initial datum (for instance, compactly supported), the solution of a KPP type reaction-diffusion equation converges in both speed and shape to the traveling wave with minimal speed. The issue at stake in this paper is the generalization of this result to some one-dimensional heterogeneous environments, namely spatially periodic or converging to a spatially periodic medium. This result fairly improves our understanding of the large-time behavior of solutions, as well as of the role of heterogeneity, which has become a crucial challenge in this field over the past few years. [ABSTRACT FROM AUTHOR]

In this paper, we study a spectral method for the triangular prism. We construct an approximation space in the “pole” condition in which the integral singularity is removed in a simple and effective way. We build a quasi-interpolation operator in the approximation space, and analyze its L 2 -error. Based on the quasi-interpolation, a triangular prism spectral method for the elliptic modal problem is studied. Furthermore, we extend this triangular prism spectral method to a triangular prism spectral element method. For the elliptic modal problem, we present the spectral element scheme and analyze the convergence. At last, we do some experiments to test the effectiveness of the method. [ABSTRACT FROM AUTHOR]

In this paper we consider the notion of strongly I-statistically pre-Cauchy double sequences in probabilistic metric spaces in line of Das et. al. (On I-statistically pre-Cauchy sequences, Taiwanese J. Math 18 (1) (2014), 115{126) and introduce the new concept of strongly I*-statistically pre-Cauchy double sequences in probabilistic metric spaces. We mainly study interrelationship among strong I-statistical convergence, strong I-statistical pre-Cauchy condition and strong I*-statistical pre-Cauchy condition for double sequences in probabilistic metric spaces and examine some basic properties of these notions. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the convergence and semi-convergence of a class of SSOR-like methods with four real functions for augmented systems. The class takes most existed SSOR-like methods as its special cases. For nonsingular systems, we obtain the minimum of convergence factors of all the SSOR-like methods in the class, and study when it can be reached by the convergence factors of methods in the class. By considering the equivalence of methods, we show that most of the existed SSOR-like methods have the same minimum of convergence factors. [ABSTRACT FROM AUTHOR]