Oscillation of solutions of impulsive vector hyperbolic differential equations with delays

In this article, we investigate a class of impulsive vector hyperbolic differential equation with delays. Several sufficient conditions are established for H-oscillation of solutions of such systems subject to the Neumann boundary condition by employing certain second-order impulsive differential inequality, where H is a unit vector in ? ^M . The main results are illustrated by two examples.     

Boundary value problem for second-order differential operators with integral conditions

In this article, we study a second-order differential operator with mixed nonlocal boundary conditions combined weighting integral boundary condition with another two-point boundary condition. Under certain conditions on the weighting functions and on the coefficients in the boundary conditions, called regular and nonregular cases, we prove that the resolvent decreases with respect to the spectral parameter in L ^p ?(0,?1), but there is no maximal decreasing at infinity for p?>?1. Furthermore, the studied operator generates in L ^p ?(0,?1) an analytic semigroup for p?=?1 in regular case, and an analytic semigroup with singularities for p?>?1 in both cases, and for p?=?1 in the nonregular case only. The obtained results are then used to show the correct solvability of a mixed problem for a parabolic partial differential equation with nonregular boundary conditions.     

<Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-convergence of complex double fourier series

It is proved that the complex double Fourier series of an integrable functionf(x, y) with coefficients c_jk satisfying certain conditions, will converge in L¹-norm. The conditions used here are the combinations of Tauberian condition of Hardy-Karamata kind and its limiting case. This paper extends the result of Bray [1] to complex double Fourier series. An erratum to this article is available at .     

Second‐order sufficient optimality conditions for the optimal control of instationary Navier‐Stokes equations

In this article sufficient optimality conditions are established for optimal control of evolutionary Navier‐Stokes equations. The second‐order condition requires coercivity of the Lagrange function on a suitable subspace together with first‐order necessary conditions. It ensures local optimality of a reference function in a L^s‐neighborhood, whereby the underlying analysis allows to use weaker norms than L^∞. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

An Impact of Stochastic Dynamic Boundary Conditions on the Evolution of the Cahn-Hilliard System

Abstract

Nonlinear systems are often subject to random influences. Sometimes the noise enters the system through physical boundaries and this leads to stochastic dynamic boundary conditions. A dynamic, as opposed to static, boundary condition involves the time derivative as well as spatial derivatives for the system state variables on the boundary. Although stochastic static (Neumann or Dirichet type) boundary conditions have been applied for stochastic partial differential equations, not much is known about the dynamical impact of stochastic dynamic boundary conditions. The purpose of this article is to study possible impacts of stochastic dynamic boundary conditions on the long term dynamics of the Cahn-Hilliard equation arising in the materials science. We show that the dimension estimation of the random attractor increases as the coefficient for the dynamic term in the stochastic dynamic boundary condition decreases. However, the dimension of the random attractor is not affected by the corresponding stochastic static boundary condition.     

A New Condition and Approach for Zero-Sum Stochastic Games with Average Payoffs

Abstract

This article deals with discrete-time two-person zero-sum stochastic games with Borel state and action spaces. The optimality criterion to be studied is the long-run expected average payoff criterion, and the (immediate) payoff function may have neither upper nor lower bounds. We first replace the optimality equation widely used in the previous literature with two so-called optimality inequalities, and give a new set of conditions for the existence of solutions to the optimality inequalities. Then, from the optimality inequalities we ensure the existence of a pair of average optimal stationary strategies. Our new condition is slightly weaker than those in the previous literature, and as a byproduct some interesting results such as the convergence of a value iteration scheme to the value of the discounted payoff game is obtained. Finally, we first apply the main results in this article to generalized inventory systems, and then further provide an example of controlled population processes for which all of our conditions are satisfied, while some of conditions in some of previous literature fail to hold.     

Generalized Semi-Infinite Programming: Optimality Conditions Involving Reverse Convex Problems

This article deals with a generalized semi-infinite programming problem (S). Under appropriate assumptions, for such a problem we give necessary and sufficient optimality conditions via reverse convex problems. In particular, a necessary and sufficient optimality condition reduces the problem (S) to a min-max problem constrained with compact convex linked constraints.     

Walsh Multipliers for Dyadic Hardy Spaces

In this article we are interested in conditions on the coefficients of a Walsh multiplier operator that imply the operator is bounded on certain dyadic Hardy spaces H ^p , 0 < p < ∞. In particular, we consider two classical coefficient conditions, originally introduced for the trigonometric case, the Marcinkiewicz and the Hörmander–Mihlin conditions. They are known to be sufficient in the spaces L ^p , 1 < p < ∞. Here we study the corresponding problem on dyadic Hardy spaces, and find the values of p for which these conditions are sufficient. Then, we consider the cases of H ¹ and L ¹ which are of special interest. Finally, based on a recent integrability condition for Walsh series, a new condition is provided that implies that the multiplier operator is bounded from L ¹ to L ¹, and from H ¹ to H ¹. We note that existing multiplier theorems for Hardy spaces give growth conditions on the dyadic blocks of the Walsh series of the kernel, but these growth are not computable directly in terms of the coefficients.     

Bounds on the Segal-Bargmann transform ofL p functions

This article gives necessary conditions and slightly stronger sufficient conditions for a holomorphic function to be the Segal-Bargmann transform of a function inL ^p (ℝ ^d , ρ) where ρ is a Gaussian measure. The proof relies on a family of inversion formulas for the Segal-Bargmann transform, which can be “tuned” to give the best estimates for a given value of p. This article also gives a single necessary-and-sufficient condition for a holomorphic function to be the transform of a function f such that any derivative of f multiplied by any polynomial is in L^p ( ^d , ρ). Finally, I give some weaker but dimension-independent conditions.     

L 1-Convergence for Weighted Sums of Some Dependent Random Variables

In this article, the authors discuss the L ₁-convergence for weighted sums of some dependent random variables under the condition of h-integrability with respect to an array of weights. The dependence structure of the random variables includes pairwise lower case negative dependence and conditions on the mixing coefficient, the maximal correlation coefficient, or the ρ*-mixing coefficient. They prove that all the weighted sums have similar limiting behaviour.     

The existence of uniform 5-GDDs

In this article, we construct group divisible designs (GDDs) with block size five, group-type g^u and index unity. The necessary condition for the existence of such a GDD is u ≷ 5, (u - 1)g ≡ 0 (mod 4) and u(u - 1)g² ≡ 0 (mod 20). It is shown that these necessary conditions are also sufficient, except possibly in a few cases. Additionally, a new construction to obtain GDDs using holey TDs is presented. © 1997 John Wiley & Sons, Inc. J Combin Designs 5: 275–299, 1997     

On the existence of unitarily invariant norm under some conditions

Let r ₁, …, r _m be positive real numbers and A ₁, …, A _m be n × n matrices with complex entries. In this article, we present a necessary and sufficient condition for the existence of a unitarily invariant norm ‖·‖, such that ‖A _i ‖ = r _i , for i = 1, …, m. Then we identify the greatest unitarily invariant norm which satisfies this condition. Using this, we get an approximation of unitarily invariant norms. Although the minimum unitarily invariant norm which satisfies this condition does not exist in general, we find conditions over A _i s and r _i s which are sufficient for the existence of such a norm. Finally, we get a characterization of unitarily invariant norms.     

On some class of nonseparable continuous wavelet transforms

It is well-known that only a single condition (called the admissibility condition) is sufficient for L ₂-convergence of multiple continuous wavelet transforms (MCWT). However known results suggest that to guarantee the pointwise convergence of MCWT for L _p -functions wavelets should vanish quite rapidly at infinity. In this article, we consider the class of nonseparable multiple wavelets with square-symmetric Fourier transforms. For these wavelets ψ we prove that the admissibility condition and the condition ψ∈L ₁(R ⁿ ) are sufficient for the convergence of corresponding MCWT almost everywhere.     

A Dynamical Approach to Constrained Nonsmooth Convex Minimization Problem Coupling with Penalty Function Method in Hilbert Space

This article is concerned with a class of nonsmooth constrained convex optimization in a real Hilbert space. Coupling with the penalty method, we propose an automatic system (AS) and a nonautomatic system (NS) modeled by differential inclusions. Under a suitable assumption on the feasible region and a proper condition on the objective and constrained functions, some valuable convergence properties of (AS) are obtained. In order to obtain strong convergence result in general cases, based on evolution differential inclusion, we propose a nonautomatic system (NS). When the control item ?(t) of (NS) satisfies some basal conditions, global and unique existence of the solution, finite time convergence to the feasible region and slow solution choice are obtained. Moreover, under different conditions of ?(t), we give some strong convergence results of (NS). Furthermore, we end the article by numerical experiments to illustrate the efficiency and good performance of the proposed systems in this article.     

Stability study,error estimation,and condition number for semi-implicit schemes using incremental unknowns

The concept of Incremental Unknowns leads to the introduction of large classes of new numerical schemes for which the large scale component Y for the unknown, and its small scale component(s) Z are treated differently, leading to an improved condition number for elliptic problem and to an improved stability condition for evolution problems. In this article, we present a method for studying Y-explicit and Z-implicit schemes for the heat operator. For that purpose, we introduce a basis stemming from the eigenvectors of the Laplace operator, in which the amplification matrix is easy to analyze. This basis allows us to find the stability condition and the error scheme. We show rigorously that these schemes give a better stability condition, which becomes δt ≤ h²/v instead of δt ≤ h²/2v for the classical explicit scheme in the case of two levels of unknowns. We show that it is possible to recover the initial errors of the schemes, if the constant v is small enough. Besides, the number of conditions is better than that of the implicit scheme if δt ≥ ch². © 1996 John Wiley & Sons, Inc.     

A Proof of Sznitman's Conjecture about Ballistic RWRE

We consider random walk in a uniformly elliptic i.i.d. random environment in ℤ^d for d ≥ 2 . It is believed that whenever the random walk is transient in a given direction it is necessarily ballistic. In order to quantify the gap which would be needed to prove this equivalence, several ballisticity conditions have been introduced. In particular, Sznitman defined the so-called conditions (T) and (T′) . The first one is the requirement that certain unlikely exit probabilities from a set of slabs decay exponentially fast with their width L . The second one is the requirement that for all γ ∈ (0, 1) condition (T)_γ is satisfied, which in turn is defined as the requirement that the decay is like for some C > 0 . In this article we prove a conjecture of Sznitman from 2002, stating that (T) and (T′) are equivalent. Hence, this closes the circle proving the equivalence of conditions (T) , (T′) , and (T)_γ for some γ ∈ (0, 1) as conjectured by Sznitman, and also of each of these ballisticity conditions with the polynomial condition (P)_M for M ≥ 15d + 5 introduced by Berger, Drewitz, and Ramı́rez in 2014. © 2019 Wiley Periodicals, Inc.     

Strongly Flat and Condition (P) Covers of Acts Over Monoids

In this article we characterize monoids over which every right S-act has a strongly flat (condition (P)) cover. Similar to the perfect monoids, such monoids are characterized by condition (A) and having strongly flat (condition (P)) cover for each cyclic right S-act. We also give a new characterization for perfect monoids as monoids over which every strongly flat right S-act has a projective cover.     

Numerical Solutions of Stochastic Differential Delay Equations with Jumps

Abstract

In this article, we investigate the strong convergence of the Euler–Maruyama method and stochastic theta method for stochastic differential delay equations with jumps. Under a global Lipschitz condition, we not only prove the strong convergence, but also obtain the rate of convergence. We show strong convergence under a local Lipschitz condition and a linear growth condition. Moreover, it is the first time that we obtain the rate of the strong convergence under a local Lipschitz condition and a linear growth condition, i.e., if the local Lipschitz constants for balls of radius R are supposed to grow not faster than log R.     

On the existence of positive solutions for semilinear elliptic equations with singular lower order coefficients and Dirichlet boundary conditions

We give sufficient conditions for the existence of positive solutions to some semilinear elliptic equations in bounded domains with Dirichlet boundary conditions. We impose mild conditions on the domains and lower order (nonlinear) coefficients of the equations in that the bounded domains are only required to satisfy an exterior cone condition and we allow the coefficients to have singularities controlled by Kato class functions. Our approach uses an implicit probabilistic representation, Schauder's fixed point theorem, and new a priori estimates for solutions of the corresponding linear elliptic equations. In the course of deriving these a priori estimates we show that the Green functions for operators of the form on D are comparable when one modifies the drift term b on a compact subset of D. This generalizes a previous result of Ancona [2], obtained under an condition on b, to a Kato condition on . Received: 21 April 1998 / in final form 26 March 1999     

The least squares g-inverses for sum of matrices

In this article, we exhibit under suitable conditions a neat relationship between the least squares g-inverse for a sum of two matrices and the least squares g-inverses of the individual terms. We give a necessary and sufficient condition for the set equations (A?+?B){1,?3}?=?A{1,?3}?+?B{1,?3} and (A?+?B){1,?4}?=?A{1,?4}?+?B{1,?4}.