Semilinear wave equation with time dependent potential

We consider the following semilinear wave equation: (1) for (t,x) ∈ ?_t × ?. We prove that if the potential V(t,x) is a measurable function that satisfies the following decay assumption: ∣V(t,x)∣?C(1+t)(1+∣x∣) for a.e. (t,x) ∈ ?_t × ? where C, σ₀>0 are real constants, then for any real number λ that satisfies there exists a real number ρ(f,g,λ)>0 such that the equation has a global solution provided that 0<ρ?ρ(f,g,λ). Copyright © 2004 John Wiley & Sons, Ltd.     

On the Lp–Lq maximal regularity for the linear thermoelastic plate equation in a bounded domain

This paper is concerned with the thermoelastic plate equations in a domain Ω: subject to the boundary condition: u|=D_νu|=θ|=0 and initial condition: (u, u_t, θ)|_t=0=(u₀, v₀, θ₀). Here, Ω is a bounded domain in ?ⁿ(n≧2). We assume that the boundary ?Ω of Ω is a C⁴ hypersurface. We obtain an L_p–L_q maximal regularity theorem. Copyright © 2008 John Wiley & Sons, Ltd.     

Reversible and DRAD difference sets in

The existence of Hadamard difference sets has been a central question in design theory. Reversible difference sets have been studied extensively. Dillon gave a method for finding reversible difference sets in groups of the form (C)². DRAD difference sets are a newer concept. Davis and Polhill showed the existence of DRAD difference sets in the same groups as Dillon. This article determines the existence of reversible and DRAD difference sets in groups of the form (C)³. These are the only abelian 2‐groups outside of direct products of C₄ and (C)² known to contain reversible and DRAD difference sets. © 2011 Wiley Periodicals, Inc. J Combin Designs 20:58–67, 2012     

Law of large numbers for increasing subsequences of random permutations

Let the random variable Z_n,k denote the number of increasing subsequences of length k in a random permutation from S_n, the symmetric group of permutations of {1,…,n}. We show that Var(Z) = o((EZ)²) as n → ∞ if and only if . In particular then, the weak law of large numbers holds for Z if ; that is, We also show the following approximation result for the uniform measure U_n on S_n. Define the probability measure μ on S_n by where U denotes the uniform measure on the subset of permutations that contain the increasing subsequence {x₁,x₂,…,x}. Then the weak law of large numbers holds for Z if and only if where ∣∣˙∣∣ denotes the total variation norm. In particular then, (*) holds if . In order to evaluate the asymptotic behavior of the second moment, we need to analyze occupation times of certain conditioned two‐dimensional random walks. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

About smoothness of solutions of the heat equations in closed,smooth space‐time domains

We consider the probabilistic solutions of the heat equation u = u + f in D, where D is a bounded domain in ?² = {(x¹, x²)} of class C^2k. We give sufficient conditions for u to have k^th‐order continuous derivatives with respect to (x¹, x²) in D? for integers k ≥ 2. The equation is supplemented with C^2k boundary data, and we assume that f ? C^2(k?1). We also prove that our conditions are sharp by examples in the border cases. © 2005 Wiley Periodicals, Inc.     

Star factorizations of graph products

A k‐star is the graph K_1,k. We prove a general theorem about k‐star factorizations of Cayley graphs. This is used to give necessary and sufficient conditions for the existence of k‐star factorizations of any power (K_q)^s of a complete graph with prime power order q, products C × C ×··· × C of k cycles of arbitrary lengths, and any power (C_r)^s of a cycle of arbitrary length. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 59–66, 2001     

The Calderón approach to an elliptic boundary problem

We consider a boundary problem for an elliptic system in a bounded region Ω ? ?ⁿ and where the spectral parameter is multiplied by a discontinuous weight function ω (x) = diag(ω₁(x), …, ω_N (x)). The problem is considered under limited smoothness assumptions and under an ellipticity with parameter condition. Recently, this problem was studied under the assumption that the ω_j (x)^–1 are essentially bounded in Ω. In this paper we suppose that ω (x) vanishes identically in a proper subregion Ω of Ω and that the ω_j (x)^–1 are essentially bounded in . Then by using methods which are a variant of those used in constructing the Calderón projectors for the boundary Γ of Ω, we shall derive results here which will enable us in a subsequent work to apply the ideas of Calderón to develop the spectral theory associated with the problem under consideration here (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global attractors for 2‐D wave equations with displacement‐dependent damping

In this paper the long‐time behaviour of the solutions of 2‐D wave equation with a damping coefficient depending on the displacement is studied. It is shown that the semigroup generated by this equation possesses a global attractor in H(Ω) × L₂(Ω) and H²(Ω)∩H(Ω) × H(Ω). Copyright © 2009 John Wiley & Sons, Ltd.     

Interior numerical approximation of boundary value problems with a distributional data

We study the approximation properties of a harmonic function u ∈ H^1?k(Ω), k > 0, on a relatively compact subset A of Ω, using the generalized finite element method (GFEM). If Ω = ??, for a smooth, bounded domain ??, we obtain that the GFEM‐approximation u_S ∈ S of u satisfies ‖u ? u_S‖ ≤ Ch^γ‖u‖, where h is the typical size of the “elements” defining the GFEM‐space S and γ ≥ 0 is such that the local approximation spaces contain all polynomials of degree k + γ. The main technical ingredient is an extension of the classical super‐approximation results of Nitsche and Schatz (Applicable Analysis 2 (1972), 161–168; Math Comput 28 (1974), 937–958). In addition to the usual “energy” Sobolev spaces H¹(??), we need also the duals of the Sobolev spaces H^m(??), m ∈ ?₊. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Uniqueness of weak solutions in critical space of the 3‐D time‐dependent Ginzburg‐Landau equations for superconductivity

We prove the uniqueness of weak solutions of the 3‐D time‐dependent Ginzburg‐Landau equations for super‐conductivity with initial data (ψ₀, A₀)∈ L² under the hypothesis that (ψ, A) ∈ L^s(0, T; L^r,∞) × (0, T; with Coulomb gauge for any (r, s) and satisfying + = 1, + = 1, ≥ , ≥ and 3 < r ≤ 6, 3 < ≤ ∞. Here L^r,∞ ≡ is the Lorentz space. As an application, we prove a uniqueness result with periodic boundary condition when ψ₀ ∈ , A₀ ∈ L³ (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Robust characterizations of k‐wise independence over product spaces and related testing results

A discrete distribution D over Σ₁ ×··· ×Σ_n is called (non‐uniform) k ‐wise independent if for any subset of k indices {i₁,…,i_k} and for any z₁∈Σ,…,z_k∈Σ, Pr_X～D[X···X = z₁···z_k] = Pr_X～D[X = z₁]···Pr_X～D[X = z_k]. We study the problem of testing (non‐uniform) k ‐wise independent distributions over product spaces. For the uniform case we show an upper bound on the distance between a distribution D from k ‐wise independent distributions in terms of the sum of Fourier coefficients of D at vectors of weight at most k. Such a bound was previously known only when the underlying domain is {0,1}ⁿ. For the non‐uniform case, we give a new characterization of distributions being k ‐wise independent and further show that such a characterization is robust based on our results for the uniform case. These results greatly generalize those of Alon et al. (STOC'07, pp. 496–505) on uniform k ‐wise independence over the Boolean cubes to non‐uniform k ‐wise independence over product spaces. Our results yield natural testing algorithms for k ‐wise independence with time and sample complexity sublinear in terms of the support size of the distribution when k is a constant. The main technical tools employed include discrete Fourier transform and the theory of linear systems of congruences.© 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

Non-linear parabolic problem associated with the penetration of a magnetic field into a substance

We study the following initial and boundary value problem: In section 1, with u₀ in L²(Ω), f continuous such that f(u) + ? non-decreasing for ? positive, we prove the existence of a unique solution on (0,T), for each T > 0. In section 2 it is proved that the unique soluition u belongs to L²(0, T; H ∩ H²) ∩ L^∞(0, T; H) if we assume u₀ in H and f in C¹(?,?). Numerical results are given for these two cases.     

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.     

Global and blow‐up solutions for non‐linear degenerate parabolic systems

In this paper the degenerate parabolic system u_t=u(u_xx+av). vt=v(v_xx+bu) with Dirichlet boundary condition is studied. For , the global existence and the asymptotic behaviour (α₁=α₂) of solution are analysed. For , the blow‐up time, blow‐up rate and blow‐up set of blow‐up solution are estimated and the asymptotic behaviour of solution near the blow‐up time is discussed by using the ‘energy’ method. Copyright © 2003 John Wiley & Sons, Ltd.     

Finite fields and the 1‐chromatic number of orientable surfaces

The 1‐chromatic number χ₁(S_p) of the orientable surface S_p of genus p is the maximum chromatic number of all graphs which can be drawn on the surface so that each edge is crossed by no more than one other edge. We show that if there exists a finite field of order 4m+1, m≥3, then 8m+2≤χ₁(S)≤8m+3, where 8m+3 is Ringel's upper bound on χ₁(S). © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 179–184, 2010     

On a Relation between Intrinsic and Extrinsic Dirichlet Forms for Interacting Particle Systems

In this paper we extend the result obtained in [AKR98] (see also [AKR96a]) on the representation of the intrinsic pre–Dirichlet form ℰ^Γ of the Poisson measure π_σ in terms of the extrinsic one ℰ^P. More precisely, replacing π_σ by a Gibbs measure μ on the configuration space Γ_X we derive a relation between the intrinsic prend–Dirichlet form ℰ^Γ_μ of the measure μ and the extrinsic one ℰ^P. As a consequence we prove the closability of ℰ^Γ_μ on L²(Γ_X, μ) under very general assumptions on the interaction potential of the Gibbs measures μ.     

Tangent Conics at Quartic Surfaces and Conics in Quartic Double Solids

For a quartic double solid we study the parameter space of conics (i.e. of smooth rational curves C ⊂ Z such that C · φ*O(1) = 2). This parameter space is naturally fibred (with disconnected fibres) over Pˇ³. We study the monodromy of the fibres and determine this way the irreducible components of the parameter space.     

Global existence of solutions for 2‐D semilinear wave equations with dissipation localized near infinity in an exterior domain

We shall derive some global existence results to semilinear wave equations with a damping coefficient localized near infinity for very special initial data in H×L². This problem is dealt with in the two‐dimensional exterior domain with a star‐shaped complement. In our result, a power p on the non‐linear term |u|^p is strictly larger than the two‐dimensional Fujita‐exponent. Copyright © 2005 John Wiley & Sons, Ltd.     

A continuation principle for a class of periodically perturbed autonomoussystems

The paper deals with a T ‐periodically perturbed autonomous system in ℝⁿ of the form ((PS)) with ε > 0 small. The main goal of the paper is to provide conditions ensuring the existence of T ‐periodic solutions to (PS) belonging to a given open set W ⊂ C ([0, T ],ℝⁿ ). This problem is considered in the case when the boundary ∂W of W contains at most a finite number of nondegenerate T ‐periodic solutions of the autonomous system = ϕ (x). The starting point of our approach is the following property due to Malkin: if for any T ‐periodic limit cycle x 0 of = ϕ (x) belonging to ∂W the so‐called bifurcation function f (θ), θ ∈ [0, T ], associated to x₀, see (1.11), satisfies the condition f(0) ≠ 0 then the integral operator does not have fixed points on ∂W for all ε > 0 sufficiently small. By means of the Malkin's bifurcation function we then establish a formula to evaluate the Leray–Schauder topological degree of I – Q_ε on W. This formula permits to state existence results that generalize or improve several results of the existing literature. In particular, we extend a continuation principle due to Capietto, Mawhin and Zanolin where it is assumed that ∂W does not contain any T ‐periodic solutions of the unperturbed system. Moreover, we obtain generalizations or improvements of some existence results due to Malkin and Loud. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the asymptotic behaviour of the discrete spectrum in buckling problems for thin plates

We consider the buckling problem for a family of thin plates with thickness parameter ε. This involves finding the least positive multiple λ of the load that makes the plate buckle, a value that can be expressed in terms of an eigenvalue problem involving a non‐compact operator. We show that under certain assumptions on the load, we have λ = ??(ε²). This guarantees that provided the plate is thin enough, this minimum value can be numerically approximated without the spectral pollution that is possible due to the presence of the non‐compact operator. We provide numerical computations illustrating some of our theoretical results. Copyright © 2005 John Wiley & Sons, Ltd.