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In this paper, we establish approximate representations of solutions to stochastic Volterra integral equations (SVIEs, for short), by virtue of solutions to a family of stochastic differential equations. As an application, we present two algorithms for solving SVIEs: stochastic theta method and splitting method, and prove the global 1/2 order convergence rates in L 2 norm. [ABSTRACT FROM AUTHOR]

We show that if u is a Leray solution to the Navier–Stokes equations in the full three-dimensional space with an initial condition from W 0 , σ 1 , 2 , T > 0 and u ∈ L t ( 0 , T ; L s ) , where 2 / t + 3 / s = 59 / 30 for s ∈ ( 2 , 30 / 13 ] and 2 / t + 3 / s = 7 / 4 + 1 / ( 2 s ) for s ∈ ( 30 / 13 , 3 ) then u is regular on ( 0 , T ) . We prove our result as a special case of a more general method which might possibly bring a further improvement. [ABSTRACT FROM AUTHOR]

We prove that if the numerical range of a Hilbert space contraction T is in a certain closed convex set of the unit disk which touches the unit circle only at 1, then ‖ T n ( I − T ) ‖ = O ( 1 / n β ) with β ∈ [ 1 2 , 1 ) . For normal contractions the condition is also necessary. Another sufficient condition for β = 1 2 , necessary for T normal, is that the numerical range of T be in a disk { z : | z − δ | ≤ 1 − δ } for some δ ∈ ( 0 , 1 ) . As a consequence of results of Seifert, we obtain that a power-bounded T on a Hilbert space satisfies ‖ T n ( I − T ) ‖ = O ( 1 / n β ) with β ∈ ( 0 , 1 ] if and only if sup 1 < | λ | < 2 ⁡ | λ − 1 | 1 / β ‖ R ( λ , T ) ‖ < ∞ . When T is a contraction on L 2 satisfying the numerical range condition, it is shown that T n f / n 1 − β converges to 0 a.e. with a maximal inequality, for every f ∈ L 2 . An example shows that in general a positive contraction T on L 2 may have an f ≥ 0 with lim sup T n f / log ⁡ n n = ∞ a.e. [ABSTRACT FROM AUTHOR]

We prove the last remaining case of the original 13 Ramanujan-type supercongruence conjectures due to Van Hamme from 1997. The proof utilizes classical congruences and a WZ pair due to Guillera. Additionally, we mention some future directions concerning this type of supercongruence. [ABSTRACT FROM AUTHOR]

Abstract: This paper studies the limit of solutions to the Aw–Rascle model as the pressure tends to the Chaplygin gas pressure. For concreteness, the pressure is taken as a modified Chaplygin gas pressure. Firstly, the Riemann problem for the Aw–Rascle model with the modified Chaplygin gas pressure is solved constructively. Secondly, it is shown that as the pressure tends to the Chaplygin gas pressure, some Riemann solutions containing a shock and a contact discontinuity tend to a delta-shock solution, whose propagation speed and strength are different from those of delta-shock solution to the Aw–Rascle model with a Chaplygin gas pressure. Besides, it is also proven that the rest Riemann solutions converge to a two-contact-discontinuity solution, which is exactly the solution to the Aw–Rascle model with a Chaplygin gas pressure. Thirdly, some numerical results are presented to exhibit the process of formation of delta-shocks. [Copyright &y& Elsevier]

Abstract: In this paper we obtain the weighted estimates of the gradients of weak solutions for the general elliptic -Laplacian equations under some proper assumptions on a, p, f. We prove that [Copyright &y& Elsevier]

Abstract: We study global dynamics of a mass conserved reaction–diffusion system. First, we show the global-in-time existence of the solution with compact orbit. Then the dynamical stability of local minima associated with a variational function is proven. [Copyright &y& Elsevier]

Abstract: We study the numerical radius of Lipschitz operators on Banach spaces via the Lipschitz numerical index, which is an analogue of the numerical index in Banach space theory. We give a characterization of the numerical radius and obtain a necessary and sufficient condition for Banach spaces to have Lipschitz numerical index 1. As an application, we show that real lush spaces and C-rich subspaces have Lipschitz numerical index 1. Moreover, using the Gâteaux differentiability of Lipschitz operators, we characterize the Lipschitz numerical index of separable Banach spaces with the RNP. Finally, we prove that the Lipschitz numerical index has the stability properties for the -, -, and -sums of spaces and vector-valued function spaces. From this, we show that the spaces, -spaces and -spaces have Lipschitz numerical index 1. [Copyright &y& Elsevier]

Abstract: We prove the formula where is the Mertens constant. [Copyright &y& Elsevier]

Abstract: In Huang et al. [17] it was proved that for any Nil d Bohr0-set , there are a minimal system and a non-empty open subset of with , and for any minimal system and any open non-empty , the set is an almost Nil d Bohr0-set. The polynomial form of this problem is considered in this paper. It is shown that the latter is still true in the polynomial case, while the former is not in general. We also consider the special case when the system is a nilsystem. [Copyright &y& Elsevier]

Abstract: In this paper, we prove some symmetry and monotonicity results for classical positive solutions for elliptic Schrödinger systems on half spaces with the Neumann type condition. Our methods employ the Alexandrov–Serrin method of moving planes combined with Hopf’s lemma at a corner. [Copyright &y& Elsevier]

Abstract: The aim of this note is to prove the factorization theorem of Kolmogorov in von Neumann algebras. [Copyright &y& Elsevier]

Abstract: In this paper we prove the existence of multiple solutions for the following nonhomogeneous Klein–Gordon equation coupled with Born–Infeld theory, when . [Copyright &y& Elsevier]

Abstract: We prove the existence of strong solutions of a nonautonomous stochastic inclusion , with Carathéodory-“upper separated” set-valued functions and . [Copyright &y& Elsevier]

Abstract: In this paper, we give new characterization of the classical Morrey space. Complementary global Morrey-type spaces are introduced. It is proved that for particular values of parameters these spaces give new pre-dual space of the classical Morrey space. We also show that our new pre-dual space of the Morrey space coincides with known pre-dual spaces. [Copyright &y& Elsevier]

Abstract: In this paper we consider the long time behavior of solutions of a version of gKdV equation on the real line. The existence of the global attractor in is proved when the force f belongs to . Moreover, we investigate the change of the attractor according to the change of f and the coefficient γ of damping term. Finally, several numerical experiments confirm all the theoretical conclusions. [Copyright &y& Elsevier]

Abstract: In this paper, the authors prove some inequalities for the generalized elliptic integrals with respect to Hölder means. [Copyright &y& Elsevier]

Abstract: We prove that solutions to elliptic equations in two variables in divergence form, possibly non-self-adjoint and with lower order terms, satisfy the strong unique continuation property. [Copyright &y& Elsevier]

Abstract: We consider the simplicity of the -algebra associated to a labelled space , where is a labelled graph and is the smallest accommodating set containing all generalized vertices. We prove that if is simple, then is strongly cofinal, and if, in addition, for every vertex v, then is disagreeable. It is observed that is simple whenever is strongly cofinal and disagreeable, which is recently known for the -algebra associated to a labelled space of the smallest accommodating set . [Copyright &y& Elsevier]

Abstract: In this paper, we consider a discrete version of the following p-Laplacian evolution equation , with , on a network. Using the discrete p-Laplacian operator in graphs which means a nonlinear diffusion phenomenon on networks, we first introduce the p-Laplacian evolution equation on networks. In fact, spatial derivative in p-Laplacian evolution equation, comparing the continuous case, is replaced with the discrete p-Laplacian operator. Thus, the resulting system is semi-discrete, discrete in space and continuous in time. The main concern is to investigate the large time behaviors of nontrivial solutions of this equation whose initial data are nonnegative and the boundary data vanish. In order to do so, we use the analytic approaches such as vector calculus on networks, maximum principle and comparison principle instead of numerical ones. After deriving the basic properties of this equation, we finally prove that the solution becomes extinct for and remains strictly positive for . [Copyright &y& Elsevier]

Abstract: In this paper, we study the existence of solutions for a class of resonant difference systems by using the minimax methods and the Morse theory. We prove that there exists at least one nontrivial solution for the systems. [Copyright &y& Elsevier]

Abstract: It was recently proven in Case et al. (2010) that, under mild restrictions, grad-div stabilized Taylor–Hood solutions of Navier–Stokes problems converge to the Scott–Vogelius solution of that same problem. However, even though the analytical rate was only shown to be (where γ is the stabilization parameter), the computational results suggest the rate may be improvable to . We prove herein the analytical rate is indeed , and extend the result to other incompressible flow problems including Leray-α and MHD. Numerical results are given that verify the theory. [Copyright &y& Elsevier]

Abstract: Let A be a unilateral (resp., bilateral) weighted shift with weights , (resp., ). Eckstein and Rácz showed before that A has its numerical range contained in the closed unit disc if and only if there is a sequence (resp., ) in such that for all n. In terms of such ʼs, we obtain a necessary and sufficient condition for to be open. If the ʼs are periodic, we show that the ʼs can also be chosen to be periodic. As a result, we give an alternative proof for the openness of for an A with periodic weights, which was first proven by Stout. More generally, a conjecture of his on the openness of for A with split periodic weights is also confirmed. [Copyright &y& Elsevier]

Abstract: Performance of an improved-Levin quadrature method for oscillatory integrals is studied. In the study, the behavior of the target system of linear equations is analyzed and an error reduction factor is proposed to measure the behaviorʼs impact on the integral result. Numerical investigations show that the error reduction factor is extremely small for ill-conditioned case, and the ill-conditioning has little impact on the final integral result. Therefore, the concerned quadrature method is numerically very stable and it has addressed the Levin methodʼs problem of being susceptible to the ill-conditioning. [Copyright &y& Elsevier]

Abstract: We prove the existence of infinitely many radial solutions for a p-Laplacian Dirichlet problem which is p-superlinear at the origin. The main tool that we use is the shooting method. We extend for more general nonlinearities the results of J. Iaia in [J. Iaia, Radial solutions to a p-Laplacian Dirichlet problem, Appl. Anal. 58 (1995) 335–350]. Previous developments require a behavior of the nonlinearity at zero and infinity, while our main result only needs a condition of the nonlinearity at zero. [ABSTRACT FROM AUTHOR]

Abstract: A new regularized projection method was developed for numerically solving ill-posed equations of the first kind. This method consists of combining the dynamical systems method with an adaptive projection discretization scheme. Optimality of the proposed method was proved on wide classes of ill-posed problems. [Copyright &y& Elsevier]

Abstract: We prove that certain operational quantities q which characterize upper-semi Fredholm operators are supermultiplicative, in the sense of that . Based on the distortion of Banach spaces we show that another is not supermultiplicative. Moreover we introduce two supermultiplicative operational quantities which characterize also the upper-semi Fredholm operators and we prove that they are not equivalent to some operational quantities known. [Copyright &y& Elsevier]

Abstract: Delay functional differential equations are essentially different from ordinary differential equations because their phase space is infinite dimensional. We first establish a sufficient condition for delay functional differential equations with bounded lag to be dissipative. Then we construct a one-leg θ-method to solve such dissipative equations and prove that it is dissipative if . One numerical example is given to confirm our theoretical result. [Copyright &y& Elsevier]

Abstract: In this paper, we prove a theorem on the growth of a solution of a linear differential equation. From this we obtain some uniqueness theorems concerning that a nonconstant entire function and its derivatives sharing a small entire function. The results in this paper improve many known results. Some examples are provided to show that the results in this paper are the best possible. [Copyright &y& Elsevier]