On the Multiplicative Monoid of n×n Matrices Over Artinian Chain Rings

Let R be an Artinian chain ring with a principal maximal ideal. We investigate properties of matrices over R and give matrix representations of R-submodules of R ⁿ first, then consider Green's relations, Green's relation equivalent classes, Schützenberger groups of 𝒟-classes, principal factors, and group ?-classes of the multiplicative monoid M _n (R) of n × n matrices over R. Furthermore, we show that M _n (R) is an eventually regular semigroup and derive basic numerical information of M _n (R) when R is finite.     

On (∈, ∈ ∨ q)‐fuzzy filters of R0‐algebras

In this paper, we introduce the notions of (∈, ∈ ∨ q)‐fuzzy filters and (∈, ∈ ∨ q)‐fuzzy Boolean (implicative) filters in R₀‐algebras and investigate some of their related properties. Some characterization theorems of these generalized fuzzy filters are derived. In particular, we prove that a fuzzy set in R₀‐algebras is an (∈, ∈ ∨ q)‐fuzzy Boolean filter if and only if it is an (∈, ∈ ∨ q)‐fuzzy implicative filter. Finally, we consider the concepts of implication‐based fuzzy Boolean (implicative) filters of R₀‐algebras (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Pointwise supercloseness of tensor‐product block finite elements

In this article, we first introduce interpolation operator of projection type in three dimensions, from which we then derive weak estimates for tensor‐product block finite elements of degree m ≥ 1. Finally, using estimates for the discrete Green's function and the discrete derivative Green's function, we prove that both of the gradient and the function value of the finite element solution u_h and the corresponding interpolant Π_mu of projection type and degree m are superclose in the pointwise sense of the L^∞‐norm. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Rational and semi‐rational solutions of a nonlocal (2 + 1)‐dimensional nonlinear Schrödinger equation

We consider the fully parity‐time (PT) symmetric nonlocal (2 + 1)‐dimensional nonlinear Schrödinger (NLS) equation with respect to x and y. By using Hirota's bilinear method, we derive the N‐soliton solutions of the nonlocal NLS equation. By using the resulting N‐soliton solutions and employing long wave limit method, we derive its nonsingular rational solutions and semi‐rational solutions. The rational solutions act as the line rogue waves. The semi‐rational solutions mean different types of combinations in rogue waves, breathers, and periodic line waves. Furthermore, in order to easily understand the dynamic behaviors of the nonlocal NLS equation, we display some graphics to analyze the characteristics of these solutions.     

Definable group extensions in semi‐bounded o‐minimal structures

In this note we show: Let R = 〈R, <, +, 0, …〉 be a semi‐bounded (respectively, linear) o‐minimal expansion of an ordered group, and G a group definable in R of linear dimension m ([2]). Then G is a definable extension of a bounded (respectively, definably compact) definable group B by 〈R^m, +〉 (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Boolean products of R0‐algebras

In this paper, the (weak) Boolean representation of R₀‐algebras are investigated. In particular, we show that directly indecomposable R₀‐algebras are equivalent to local R₀‐algebras and any nontrivial R₀‐algebra is representable as a weak Boolean product of local R₀‐algebras (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Pattern formations of 2D Rayleigh–Bénard convection with no‐slip boundary conditions for the velocity at the critical length scales

We study the Rayleigh–Bénard convection in a 2D rectangular domain with no‐slip boundary conditions for the velocity. The main mathematical challenge is due to the no‐slip boundary conditions, because the separation of variables for the linear eigenvalue problem, which works in the free‐slip case, is no longer possible. It is well known that as the Rayleigh number crosses a critical threshold R_c, the system bifurcates to an attractor, which is an (m ? 1)‐dimensional sphere, where m is the number of eigenvalues, which cross zero as R crosses R_c. The main objective of this article is to derive a full classification of the structure of this bifurcated attractor when m = 2. More precisely, we rigorously prove that when m = 2, the bifurcated attractor is homeomorphic to a one‐dimensional circle consisting of exactly four or eight steady states and their connecting heteroclinic orbits. In addition, we show that the mixed modes can be stable steady states for small Prandtl numbers. Copyright © 2014 John Wiley & Sons, Ltd.     

On the best constant in a Wente‐type inequality for the fractional Laplace operator

In this paper, we consider the solution to Wente's problem with the fractional Laplace operator (?Δ)^α/2, where 0 < α < 2. We derive a Wente‐type inequality for this problem. Next, we compute the optimal constant in such inequality. Copyright © 2015 John Wiley & Sons, Ltd.     

Infinite‐dimensional exponential attractors for fourth‐order nonlinear parabolic equations in unbounded domains

We study in this article the long‐time behavior of solutions of fourth‐order parabolic equations in bfR³. In particular, we prove that under appropriate assumptions on the nonlinear interaction function and on the external forces, these equations possess infinite‐dimensional exponential attractors whose Kolmogorov's ε‐entropy satisfies an estimate of the same type as that obtained previously for the ε‐entropy of the global attractor. Copyright © 2011 John Wiley & Sons, Ltd.     

Existence and modulation of traveling waves in particles chains

We consider an infinite particle chain whose dynamics are governed by the following system of differential equations: where q_n(t) is the displacement of the n^th particle at time t along the chain axis and denotes differentiation with respect to time. We assume that all particles have unit mass and that the interaction potential V between adjacent particles is a convex C∞ function. For this system, we prove the existence of C∞, time‐periodic, traveling‐wave solutions of the form q_n(t) = q(wt kn + where q is a periodic function q(z) = q(z+1) (the period is normalized to equal 1), ω and k are, respectively, the frequency and the wave number, is the mean particle spacing, and can be chosen to be an arbitrary parameter. We present two proofs, one based on a variational principle and the other on topological methods, in particular degree theory. For small‐amplitude waves, based on perturbation techniques, we describe the form of the traveling waves, and we derive the weakly nonlinear dispersion relation. For the fully nonlinear case, when the amplitude of the waves is high, we use numerical methods to compute the traveling‐wave solution and the non‐linear dispersion relation. We finally apply Whitham's method of averaged Lagrangian to derive the modulation equations for the wave parameters α, β, k, and ω. © 1999 John Wiley & Sons, Inc.     

Limit Processes for Age-Dependent Branching Particle Systems

We consider systems of spatially distributed branching particles in R ^d . The particle lifelengths are of general form, hence the time propagation of the system is typically not Markov. A natural time-space-mass scaling is applied to a sequence of particle systems and we derive limit results for the corresponding sequence of measure-valued processes. The limit is identified as the projection on R ^d of a superprocess in R ₊×R ^d . The additive functional characterizing the superprocess is the scaling limit of certain point processes, which count generations along a line of descent for the branching particles.     

On the existence of steady flows of a Navier–Stokes liquid around a moving rigid body

We prove the existence of a strong solution to the three‐dimensional steady Navier–Stokes equations in the exterior of an obstacle undergoing a rigid motion. Unlike the classical exterior problem for the Navier–Stokes equations, that only takes into account the translational motion of the obstacle, is this case, the obstacle can also rotate. Assuming the total flux of the velocity field through the boundary to be sufficiently small, we first construct approximating solutions in bounded regions Ω_R = Ω∩ {x ∈ ?³:∣x∣< R} invading the liquid domain Ω. A set of estimates independent of R are shown to hold for the approximating solutions which allows to obtain a strong solution by taking the limit R→∞. Copyright © 2004 John Wiley & Sons, Ltd.     

On Saint‐Venant's principle in a poroelastic arch‐like region

In this paper we consider the state of plane strain in an elastic material with voids occupying a curvilinear strip as an arch‐like region described by R: a<r<b, 0<θ<ω, where r and θ are polar coordinates and a, b, and ω (<2π) are prescribed positive constants. Such a curvilinear strip is maintained in equilibrium under self‐equilibrated traction and equilibrated force applied on one of the edges, whereas the other three edges are traction free and subjected to zero volumetric fraction or zero equilibrated force. In fact, we study the case when one right or curved edge is loaded. Our aim is to derive some explicit spatial estimates describing how some appropriate measures of a specific Airy stress function and volume fraction evolve with respect to the distance to the loaded edge. The results of the present paper prove how the spatial decay rate varies with the constitutive profile. For the problem corresponding to a loaded right edge, we are able to establish an exponential decay estimate with respect to the angle θ. Whereas for the problem corresponding to a loaded curved edge, we establish an algebraical spatial decay with respect to the polar distance r, provided the angle ω is lower than the critical value $\pi\sqrt{2}$. The intended applications of these results concern various branches of medicine as for example the bone implants. Copyright © 2010 John Wiley & Sons, Ltd.     

The WKB method for the quantum mechanical two-Coulomb-center problem

Using a modified perturbation theory, we obtain asymptotic expressions for the two-center quasiradial and quasiangular wave functions for large internuclear distances R. We show that in each order of 1/R, corrections to the wave functions are expressed in terms of a finite number of Coulomb functions with a modified charge. We derive simple analytic expressions for the first, second, and third corrections. We develop a consistent scheme for obtaining WKB expansions for solutions of the quasiangular equation in the quantum mechanical two-Coulomb-center problem. In the framework of this scheme, we construct semiclassical two-center wave functions for large distances between fixed positively charged particles (nuclei) for the entire space of motion of a negatively charged particle (electron). The method ensures simple uniform estimates for eigenfunctions at arbitrary large internuclear distances R, including R ≥ 1. In contrast to perturbation theory, the semiclassical approximation is not related to the smallness of the interaction and hence has a wider applicability domain, which permits investigating qualitative laws for the behavior and properties of quantum mechanical systems.     

Analysis of a two‐stage least‐squares finite element method for the planar elasticity problem

A new first‐order formulation for the two‐dimensional elasticity equations is proposed by introducing additional variables which, called stresses here, are the derivatives of displacements. The resulted stress–displacement system can be further decomposed into two dependent subsystems, the stress system and the displacement system recovered from the stresses. For constructing finite element approximations to these subsystems with appropriate boundary conditions, a two‐stage least‐squares procedure is introduced. The analysis shows that, under suitable regularity assumptions, the rates of convergence of the least‐squares approximations for all the unknowns are optimal both in the H¹‐norm and in L²‐norm. Also, numerical experiments with various Poisson's ratios are examined to demonstrate the theoretical estimates. Copyright © 1999 John Wiley & Sons, Ltd.     

Stochastic functional differential equations with infinite delay driven by G‐Brownian motion

In this paper, we consider a class of stochastic functional differential equations with infinite delay at phase space BC ( ? ∞ ,0]; R^d) driven by G‐Brownian motion (SFDEGs) in the framework of sublinear expectation spaces . We prove the existence and uniqueness of the solutions to SFDEGs with the coefficients satisfying the linear growth condition and the classical Lipschitz condition. In addition, we establish the exponential estimate of the solution. Copyright © 2013 John Wiley & Sons, Ltd.     

Estimation of a class of quasi‐resonances generated by multiple small particles with high surface impedances

We deal with the stationary acoustic waves propagating in a cluster of small particles enjoying high contrasts. Such contrasts allow the appearance of (complex valued) resonances that are close to the real line as the size of the particles becomes small. For single (but not necessarily small) particles, we derive the characteristic equation that generates a class of these resonances (the ones for which the corresponding eigenfunctions are uniformly constant). For multiple and small particles, we provide sufficient conditions on the contrasts that generates quasi‐resonances for which the corresponding eigenfunctions are uniformly constant. Precisely, we show that, if we distribute the particles on a uniform line, then the existence of such quasi‐resonances is related to the eigenvalues of the Harary matrix. To show these results, we take, as the small contrasted particles, small obstacles with high surface impedances λ of the form λ: = βa^?1 ? αβa^{?1 + h} where a is the maximum radi of the particles, with a < <1, and β is a universal and positive constant depending only on the shape of the particles (but not on their size). In this case, if the relative constant α is an eigenvalue of the Harary matrix, then the used frequency is a quasi resonance of the cluster of the small particles where the error of approximation is of the order for h ∈ (0,1) as a < <1.     

Superconvergence analysis for time‐dependent Maxwell's equations in metamaterials

In this article, we consider the time‐dependent Maxwell's equations modeling wave propagation in metamaterials. One‐order higher global superclose results in the L² norm are proved for several semidiscrete and fully discrete schemes developed for solving this model using nonuniform cubic and rectangular edge elements. Furthermore, L^∞ superconvergence at element centers is proved for the lowest order rectangular edge element. To our best knowledge, such pointwise superconvergence result and its proof are original, and we are unaware of any other publications on this issue. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential 2011     

Multiple non semi‐trivial solutions of systems of Kirchhoff‐type equations with discontinuous nonlinearities in RN

In the present paper, we study the problem of multiple non semi‐trivial solutions for the following systems of Kirchhoff‐type equations with discontinuous nonlinearities (1.1) where F∈C¹(R^N×R⁺×R⁺,R),V∈C(R^N,R), and By establishing a new index theory, we obtain some multiple critical point theorems on product spaces, and as applications, three multiplicity results of non semi‐trivial solutions for (1.1) are obtained. Copyright © 2015 John Wiley & Sons, Ltd.     

Obtaining equations of motion for charged particles in the (<Emphasis Type="Italic">v/c</Emphasis>)<Superscript>3</Superscript>-approximation by the Einstein-Infeld-Hoffmann method

We consider some principal methodological problems that appear when the Einstein-Infeld-Hoffmann method is used to find approximate solutions of the general relativity equations and to obtain information about the motion of particles whose interaction force is much greater than the gravitational attraction force. Among these problems are normalizing approximate expressions by expanding exact solutions written in the same coordinate conditions used in the Einstein-Infeld-Hoffmann method, assigning the smallness orders depending on relations between the smallness parameters in play, and verifying cancellations of divergent terms arising in surface integrals. Solving these questions in accordance with the internal logic of the Einstein-Infeld-Hoffmann method results in new tools and techniques for applying the method. We demonstrate these tools and techniques in the example of the problem of the motion of two electrically charged pointlike particles in the (v/c)³-approximation.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 142, No. 1, pp. 160–176, January, 2005.