Simple eigenvalues for the Steklov problem in a domain with a small hole. A functional analytic approach

Let be a bounded open domain of . Let denote the outward unit normal of . We assume that the Steklov problem Δu = 0 in and on has a simple eigenvalue of . Then we consider an annular domain obtained by removing from a small‐cavity size of ε > 0, and we show that under proper assumptions there exists a real valued and real analytic function defined in an open neighborhood of (0,0) in and such that is a simple eigenvalue for the Steklov problem Δu = 0 in and on for all ε > 0 small enough, and such that . Here denotes the outward unit normal of , and δ_2,2 ≡ 1 and δ_2,n ≡ 0 if n ≥ 3. Then related statements have been proved for corresponding eigenfunctions. Copyright © 2013 John Wiley & Sons, Ltd.     

Riemann boundary value problems on half space in Clifford analysis

In this paper, we discuss some properties of the Cauchy type integral operator defined over the half space of . As applications, we study a type of Riemann boundary value problem for solutions to polynomially generalized Cauchy–Riemann equations including with and as special cases over the half space of . Making use of Fischer‐type decomposition and the Clifford calculus for solutions to these equations, we will obtain explicit expressions of solutions to the kind of boundary value problems over the half space of . Copyright © 2012 John Wiley & Sons, Ltd.     

On the monotonicity of some singular integral operators

The paper deals with the monotonicity of singular integral operators of the form where is the Cauchy singular integral operator on the interval (0,1) of the real axis and q is a power or logarithmic function. Under suitable assumptions, such singular integral operators are proved to be monotone and maximal monotone in spaces with power weights. Moreover, two related integral equations with weakly singular kernels of logarithmic type are studied. Copyright © 2012 John Wiley & Sons, Ltd.     

Self‐organized criticality of cellular automata model; absorbtion in finite‐time of supercritical region into the critical one

In this work, it is studied the evolution and time behavior of solutions to nonlinear diffusion equation in where , d ≥ 1, and H is the Heaviside function. For d = 1,2,3, this equation describes the dynamics of self‐organizing sandpile process with critical state ρ_c. The main conclusion is that the supercritical region is absorbed in a finite‐time in the critical region . Copyright © 2013 John Wiley & Sons, Ltd.     

On the well‐posedness for the heat flow of harmonic maps and the hydrodynamic flow of nematic liquid crystals in critical spaces

In recent paper, we prove the well‐posedness for the heat flow of harmonic maps with initial data and for the hydrodynamic flow of nematic liquid crystals with initial data . Copyright © 2011 John Wiley & Sons, Ltd.     

Harmonic and monogenic potentials in low dimensional Euclidean half‐space

In the framework of Clifford analysis, a chain of harmonic and monogenic potentials in the upper half of Euclidean space was constructed recently, including a higher dimensional analogue of the logarithmic function in the complex plane. Their distributional limits at the boundary were also determined. In this paper, the potentials and their distributional boundary values are calculated in dimensions 3 and 4, dimensions for which the expressions in general dimension break down. Copyright © 2013 John Wiley & Sons, Ltd.     

On 3D orthogonal prolate spheroidal monogenics

S. G. Georgiev, Complete orthogonal systems of monogenic polynomials over 3D prolate spheroids have recently experienced an upsurge of interest because of their many remarkable properties. These generalized polynomials and their applications to the theory of quasi‐conformal mappings and approximation theory have played a major role in this development. In particular, the underlying functions of three real variables take on values in the reduced quaternions (identified with ) and are generally assumed to be null‐solutions of the well‐known Riesz system in . The present paper introduces and explores a new complete orthogonal system of monogenic functions as solutions to this system for the space exterior of a 3D prolate spheroid. This will be made in the linear spaces of square integrable functions over . The representations of these functions are explicitly given. Some important properties of the system are briefly discussed, from which several recurrence formulae for fast computer implementations can be derived. Copyright © 2015 John Wiley & Sons, Ltd.     

Regular orthogonal basis on Heisenberg group and application to function spaces

In this paper, we construct some compactly supported orthogonal regular wavelet basis on Heisenberg group . Because of the regularity of wavelets, we could use such wavelets to characterize function spaces on , such as bounded mean oscillation space (BMO) space, Hardy space, Besov spaces and Besov–Morrey spaces. Copyright © 2014 John Wiley & Sons, Ltd.     

Perturbations preserving conditioned invariant subspaces

Given the set of matrix pairs keeping a subspace invariant, we obtain a miniversal deformation of a pair belonging to an open dense subset of . It generalizes the known results when S is a supplementary subspace of the unobservable one. Copyright © 2011 John Wiley & Sons, Ltd.     

Interface approximation of Darcy flow in a narrow channel

A mixed formulation is introduced for the singular problem of Darcy flow in a porous medium in a region containing a narrow fracture or channel with width and high permeability . The solution converges as ε → 0 to that of Darcy flow coupled to tangential flow on the lower‐dimensional interface or boundary. Copyright © 2012 John Wiley & Sons, Ltd.     

The Gelfand–Tsetlin bases for Hodge–de Rham systems in Euclidean spaces

The main aim of this paper is to construct explicitly orthogonal bases for the spaces of k‐homogeneous polynomial solutions of the Hodge–de Rham system in the Euclidean space , which take values in the space of s‐vectors. Actually, we describe even the so‐called Gelfand–Tsetlin bases for such spaces in terms of Gegenbauer polynomials. As an application, we obtain an algorithm on how to compute an orthogonal basis of the space of homogeneous solutions for an arbitrary generalized Moisil–Théodoresco system in . Copyright © 2012 John Wiley & Sons, Ltd.     

Homoclinic solutions for second‐order discrete Hamiltonian systems with asymptotically quadratic potentials

In this paper, we study the existence of infinitely many homoclinic solutions for the second‐order self‐adjoint discrete Hamiltonian system , where , and are unnecessarily positive definites for all . By using the variant fountain theorem, we obtain an existence criterion to guarantee that the aforementioned system has infinitely many homoclinic solutions under the assumption that W(n,x) is asymptotically quadratic as | x | → + ∞ . Copyright © 2013 John Wiley & Sons, Ltd.     

ψ‐Hyperholomorphic functions and a Kolosov–Muskhelishvili formula

Holomorphic function theory is an effective tool for solving linear elasticity problems in the complex plane. The displacement and stress field are represented in terms of holomorphic functions, well known as Kolosov–Muskhelishvili formulae. In , similar formulae were already developed in recent papers, using quaternionic monogenic functions as a generalization of holomorphic functions. However, the existing representations use functions from to , embedded in . It is not completely appropriate for applications in . In particular, one has to remove many redundancies while constructing basis solutions. To overcome that problem, we propose an alternative Kolosov–Muskhelishvili formula for the displacement field by means of a (paravector‐valued) monogenic, an anti‐monogenic and a ψ‐hyperholomorphic function. Copyright © 2015 John Wiley & Sons, Ltd.     

Miniversal deformations of observable marked matrices

Given the set of vertical pairs of matrices keeping the subspace invariant, we compute miniversal deformations of a given pair when it is observable, and the subspace is marked. Moreover, we obtain the dimension of the orbit, characterize the structurally stable vertical pairs, and study the effect of each deformation parameter. Copyright © 2013 John Wiley & Sons, Ltd.     

The Bergman–Sce transform for slice monogenic functions

In a recent paper, we showed that the classical Bergman theory admits two possible formulations for the class of slice regular functions with quaternionic values. In the so called formulation of the first kind, we provide a Bergman kernel which is defined on and is a reproducing kernel. In the so called formulation of the second kind, we use the Representation Formula for slice regular functions to define a second Bergman kernel; this time the kernel is still defined on U, but the integral representation of f is based on an integral computed only on and the integral does not depend on , (here denotes the sphere unit of purely imaginary quaternions, and represents the complex plane with imaginary unit I). In this paper, we extend the second formulation of the Bergman theory to the case of slice monogenic functions and we focus our attention on the so‐called Bergman–Sce transform. This integral transform is defined by using the Bergman kernel and the Sce mapping theorem and associates to every slice monogenic function f, an axially monogenic function . Copyright © 2011 John Wiley & Sons, Ltd.     

Solutions for perturbed biharmonic equations with critical nonlinearity

In this paper, we study the perturbed biharmonic equations where Δ² is the biharmonic operator, is the Sobolev critical exponent, p ∈ (2,2 ^{* *} ), P(x), and Q(x) are bounded positive functions. Under some given conditions on V, we prove that the problem has at least one nontrivial solution provided that and that for any , it has at least n ^* pairs solutions if , where and are sufficiently small positive numbers. Moreover, these solutions u_ε → 0 in as ε → 0. Copyright © 2013 The authors. Mathematical Methods in the Applied Sciences published by John Wiley & Sons, Ltd.     

(Discrete) Almansi type decompositions: an umbral calculus framework based on symmetries

We introduce the umbral calculus formalism for hypercomplex variables starting from the fact that the algebra of multivariate polynomials shall be described in terms of the generators of the Weyl–Heisenberg algebra. The extension of to the algebra of Clifford‐valued polynomials gives rise to an algebra of Clifford‐valued operators whose canonical generators are isomorphic to the orthosymplectic Lie algebra . This extension provides an effective framework in continuity and discreteness that allow us to establish an alternative formulation of Almansi decomposition in Clifford analysis obtained by Ryan (Zeitschrift für Analysis und ihre Anwendungen 1990) and Malonek & Ren (Mathematical Methods in the Applied Sciences 2002;2007) that corresponds to a meaningful generalization of Fischer decomposition for the subspaces ker(D′)^k. We will discuss afterwards how the symmetries of (even part of ) are ubiquitous on the recent approach of RENDER (Duke Mathematical Journal 2008) showing that they can be interpreted in terms of the method of separation of variables for the Hamiltonian operator in quantum mechanics. Copyright © 2011 John Wiley & Sons, Ltd.     

Analytical solution of variant Boussinesq equations

In this paper, analytic solutions of the variant Boussinesq equations are obtained by the homotopy analysis and the homotopy Pad methods. The obtained approximation using homotopy method contains an auxiliary parameter, which is a simple way to control and adjust the convergence region and rate of solution series. The approximation solutions by [m,m] homotopy Pad technique is often independent of auxiliary parameter , and this technique accelerates the convergence of the related series. Copyright © 2013 John Wiley & Sons, Ltd.     

Multiplicity of nontrivial solutions to a biharmonic equation via Lusternik–Schnirelman theory

In this paper, we are concerned with the multiplicity of nontrivial solutions for the following class of biharmonic problem where is a bounded domain with smooth boundary. Using the Lusternik–Schnirelman theory, we relate the number of solutions with the topology of Ω. Copyright © 2012 John Wiley & Sons, Ltd.     

Modeling and stability analysis for a cholera model with vaccination

In this paper, we consider a cholera model with vaccination. The disease‐free equilibrium of the system is globally asymptotically stable when the basic reproduction number . If , the disease persists and the unique endemic equilibrium is globally asymptotically stable in the interior of the feasible region under some conditions, which is obtained by compound matrices and geometric approaches. We perform sensitivity analysis of on the parameters in order to determine their relative importance to disease transmission and prevalence. Numerical simulations are presented to illustrate the results. Copyright © 2011 John Wiley & Sons, Ltd.