Attractors of the reaction-diffusion systems with rapidly oscillating coefficients and their homogenization

The following reaction-diffusion system in spatially non-homogeneous almost-periodic media is considered in a bounded domain : (1)∂_tu=Au−f(u)+g, u|_∂Ω=0. Here u=(u¹,…,u^k) is an unknown vector-valued function, f is a given nonlinear interaction function and the second order elliptic operator A has the following structure: where a_ij^l(y) are given almost-periodic functions. We prove that, under natural assumptions on the nonlinear term f(u), the longtime behavior of solutions of (1) can be described in terms of the global attractor of the associated dynamical system and that the attractors  , 0<<₀1, converge to the attractor of the homogenized problem (1) as →0. Moreover, in the particular case of periodic media, we give explicit estimates for the distance between the non-homogenized and the homogenized attractors in terms of the parameter .     

Uniqueness and stability of regional blow-up in a porous-medium equation

We study the blow-up phenomenon for the porous-medium equation in R^N, N1, u_t=Δu^m+u^m, m>1, for nonnegative, compactly supported initial data. A solution u(x,t) to this problem blows-up at a finite time . Our main result asserts that there is a finite number of points x₁,…,x_kR^N, with |x_i−x_j|2R^* for i≠j, such that Here w_*(|x|) is the unique nontrivial, nonnegative compactly supported, radially symmetric solution of the equation in R^N and R^* is the radius of its support. Moreover u(x,t) remains uniformly bounded up to its blow-up time on compact subsets of . The question becomes reduced to that of proving that the ω-limit set in the problem consists of a single point when its initial condition is nonnegative and compactly supported.     

Strong characterizing sequences for subgroups of compact groups

In [A. Biró, V.T. Sós, Strong characterizing sequences in simultaneous Diophantine approximation, J. Number Theory 99 (2003) 405–414] we proved that if Γ is a subgroup of the torus R/Z generated by finitely many independent irrationals, then there is an infinite subset AZ which characterizes Γ in the sense that for γR/Z we have ∑_aAaγ<∞ if and only if γΓ. Here we consider a general compact metrizable Abelian group G instead of R/Z, and we characterize its finitely generated free subgroups Γ by subsets AG^*, where G^* is the Pontriagin dual of G. For this case we prove stronger forms of the analogue of the theorem of the above mentioned work, and we find necessary and sufficient conditions for a kind of strengthening of this statement to be true.     

CONVEX CONCENTRATION INEQUALITIES FOR CONTINUOUS GAS AND STOCHASTIC DOMINATION

In this article, we consider the continuous gas in a bounded domain ∧ of R^＋ or R^d described by a Gibbsian probability measure μη∧ associated with a pair interaction φ, the inverse temperature β, the activity z 〉 0, and the boundary condition η. Define F ∫ωf（s）wA（ds）. Applying the generalized Ito＇s formula for forward-backward martingales （see Klein et M. [5]）, we obtain convex concentration inequalities for F with respect to the Gibbs measure μη∧. On the other hand, by FKG inequality on the Poisson space, we also give a new simple argument for the stochastic domination for the Gibbs measure.     

DIRICHLET PROBLEMS FOR STATIONARY VON NEUMANN-LANDAU WAVE EQUATIONS

In this article, we are concerned with the Dirichlet problem of the stationary von Neumann-Landau wave equation：
{（-△x＋△y）φ（x,y）=0,x,y∈Ω
φ｜δΩxδΩ=f
where Ω is a bounded domain in R^n. By introducing anti-inner product spaces, we show the existence and uniqueness of the generalized solution for the above Dirichlet problem by functional-analytic methods.     

LOCAL TIME ANALYSIS OF ADDITIVE LEVY PROCESSES WITH DIFFERENT L(E)VY EXPONENTS

Let X1 XN be independent, classical Levy processes on R^d with Levy exponents ψ1,…, ψN, respectively. The corresponding additive Levy process is defined as the following N-parameter random field on R^d, X（t） △= X1（t1） ＋ ... ＋ XN（tN）, At∈N. Under mild regularity conditions on the ψi＇s, we derive estimate for the local and uniform moduli of continuity of local times of X = {X（t）; t ∈R^N}.     

Preliminary test estimators and phi-divergence measures in generalized linear models with binary data

We consider the problem of estimation of the parameters in Generalized Linear Models (GLM) with binary data when it is suspected that the parameter vector obeys some exact linear restrictions which are linearly independent with some degree of uncertainty. Based on minimum -divergence estimation (ME), we consider some estimators for the parameters of the GLM: Unrestricted ME, restricted ME, Preliminary ME, Shrinkage ME, Shrinkage preliminary ME, James–Stein ME, Positive-part of Stein-Rule ME and Modified preliminary ME. Asymptotic bias as well as risk with a quadratic loss function are studied under contiguous alternative hypotheses. Some discussion about dominance among the estimators studied is presented. Finally, a simulation study is carried out.     

A characterization of convex calibrable sets in with respect to anisotropic norms

A set is called “calibrable” if its characteristic function is an eigenvector of the subgradient of the total variation. The main purpose of this paper is to characterize the “-calibrability” of bounded convex sets in with respect to a norm (called anisotropy in the sequel) by the anisotropic mean -curvature of its boundary. It extends to the anisotropic and crystalline cases the known analogous results in the Euclidean case. As a by-product of our analysis we prove that any convex body C satisfying a -ball condition contains a convex -calibrable set K such that, for any V[|K|,|C|], the subset of C of volume V which minimizes the -perimeter is unique and convex. We also describe the anisotropic total variation flow with initial data the characteristic function of a bounded convex set.     

THE CONSTRUCTION OF DENUMERABLE q-PROCESSES IN RANDOM ENVIRONMENTS SATISFYING （F） OR （B）

This article is a continuation of[9].Based on the discussion of random Kolmogorov forward（backward）equations,for any given q-matrix in random environment, Q（θ）=（q（θ;x,y）,x,y∈X）,an infinite class of q-processes in random environments satisfying the random Kolmogorov forward（backward）equation is constructed.Moreover, under some conditions,all the q-processes in random environments satisfying the random Kolmogorov forward（backward）equation are constructed.     

Compressing Mappings on Primitive Sequences over Z/(2) and Its Galois Extension

Let f(x) be a strongly primitive polynomial of degree n over Z/(2^e), η(x₀,x₁,…,x_e−2) a Boolean function of e−1 variables and (x₀,x₁,…,x_e−1)=x_e−1+η(x₀,x₁,…,x_e−2)G (f(x),Z/(2^e)) denotes the set of all sequences over Z/(2^e) generated by f(x), F₂^∞ the set of all sequences over the binary field F₂, then the compressing mapping

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is injective, that is, for , G(f(x),Z/(2^e)), = if and only if Φ( )=Φ( ), i.e., ( ₀,…, _e−1)=( ₀,…, _e−1) mod 2. In the second part of the paper, we generalize the above result over the Galois rings.     

Relative rearrangement and Lebesgue spaces with variable exponent

We apply the techniques of monotone and relative rearrangements to the nonrearrangement invariant spaces L^p()(Ω) with variable exponent. In particular, we show that the maps uL^p()(Ω)→k(t)u_*L^p_*()(0,measΩ) and uL^p()(Ω)→u_*L^p*()(0,measΩ) are locally -Hölderian (u_* (resp. p^*) is the decreasing (resp. increasing) rearrangement of u (resp. p)). The pointwise relations for the relative rearrangement are applied to derive the Sobolev embedding with eventually discontinuous exponents.     

The Approximation Functional and Interpolation with Perturbed Continuity

The continuity conditions at the endpoints of interpolation theorems, Ta_BjM_j a_Aj for j=0, 1, can be written with the help of the approximation functional: E(t, Ta; B₁, B₀)_L^∞M₀ a_A0 and E(t, Ta; B₀, B₁)_L^∞M₁ a_A1. As a special case of the results we present here we show that in the hypotheses of the interpolation theorem the L^∞ norms can be replaced by BMO( ₊) norms. This leads to a strong version of the Stein-Weiss theorem on interpolation with change of measure. Another application of our results is that the condition fL⁰, i.e., f*L^∞, where f*(γ)=μ{|f|>γ} is the distribution function of f, can be replaced in interpolation with L(p, q) spaces by the weaker f*BMO( ₊).     

Widths and shape-preserving widths of Sobolev-type classes of s-monotone functions

Let I be a finite interval, , and 1p∞. Given a set M, of functions defined on I, denote by the subset of all functions yM such that the s-difference is nonnegative on I, τ>0. Further, denote by the Sobolev class of functions x on I with the seminorm x^(r)_Lp1. We obtain the exact orders of the Kolmogorov and the linear widths, and of the shape-preserving widths of the classes in L_q for s>r+1 and (r,p,q)≠(1,1,∞). We show that while the widths of the classes depend in an essential way on the parameter s, which characterizes the shape of functions, the shape-preserving widths of these classes remain asymptotically ≈n^-2.     

On sharp Burkholder–Rosenthal-type inequalities for infinite-degree U-statistics

In this paper, we present a method that allows one to obtain a number of sharp inequalities for expectations of functions of infinite-degree U-statistics. Using the approach, we prove, in particular, the following result: Let D be the class of functions f :R₊→R₊ such that the function f(x+z)−f(x) is concave in xR₊ for all zR₊. Then the following estimate holds: for all fD and all U-statistics ∑_1i1<<ilnY_i1,…,il(X_i1,…,X_il) with nonnegative kernels Y_i1,…,il :R^l→R₊, 1i_kn; i_r≠i_s, r≠s; k,r,s=1,…,l; l=0,…,m, in independent r.v.'s X₁,…,X_n. Similar inequality holds for sums of decoupled U-statistics. The class D is quite wide and includes all nonnegative twice differentiable functions f such that the function f″(x) is nonincreasing in x>0, and, in particular, the power functions f(x)=x^t, 1<t2; the power functions multiplied by logarithm f(x)= (x+x₀)^t ln(x+x₀), 1<t<2, x₀max(e^{(3t2−6t+2)/(t(t−1)(2−t))},1); and the entropy-type functions f(x)=(x+x₀)ln(x+x₀), x₀1. As an application of the results, we determine the best constants in Burkholder–Rosenthal-type inequalities for sums of U-statistics and prove new decoupling inequalities for those objects. The results obtained in the paper are, to our knowledge, the first known results on the best constants in sharp moment estimates for U-statistics of a general type.     

Interlacing and spacing properties of zeros of polynomials, in particular of orthogonal and -minimal polynomials,

Let be a sequence of polynomials with real coefficients such that uniformly for [α-δ,β+δ] with G(eⁱ)≠0 on [α,β], where 0α<βπ and δ>0. First it is shown that the zeros of are dense in [α,β], have spacing of precise order π/n and are interlacing with the zeros of p_n+1(cos) on [α,β] for every nn₀. Let be another sequence of real polynomials with uniformly on [α-δ,β+δ] and on [α,β]. It is demonstrated that for all sufficiently large n the zeros of p_n(cos) and strictly interlace on [α,β] if on [α,β]. If the last expression is zero then a weaker kind of interlacing holds. These interlacing properties of the zeros are new for orthogonal polynomials also. For instance, for large n a simple criteria for interlacing of zeros of Jacobi polynomials on [-1+,1-], >0, is obtained. Finally it is shown that the results hold for wide classes of weighted L_q-minimal polynomials, q[1,∞], linear combinations and products of orthogonal polynomials, etc.     

The supremum of Brownian local times on Hölder curves

For , we consider L^f_t, the local time of space-time Brownian motion on the curve f. Let be the class of all functions whose Hölder norm of order α is less than or equal to 1. We show that the supremum of L^f₁ over f in is finite if α>1/2 and infinite if α<1/2.     

Weak Copositive and Intertwining Approximation

It is known that shape preserving approximation has lower rates than unconstrained approximation. This is especially true for copositive and intertwining approximations. ForfL_p, 1p<∞, the former only has rateω(f, n⁻¹)_p, and the latter cannot even be bounded byC f_p. In this paper, we discuss various ways to relax the restrictions in these approximations and conclude that the most sensible way is the so-calledalmostcopositive/intertwining approximation in which one relaxes the restriction on the approximants in a neighborhood of radiusΔ_n(y_j) of each sign changey_j.     

On Ulyanov inequalities in Banach spaces and semigroups of linear operators

Let X,Y be Banach spaces and {T(t):t≥0} be a consistent, equibounded semigroup of linear operators on X as well as on Y; it is assumed that {T(t)} satisfies a Nikolskii type inequality with respect to X and Y:T(2t)f_Y(t)T(t)f_X. Then an abstract Ulyanov type inequality is derived between the (modified) K-functionals with respect to (X,D_X((-A)^α)) and (Y,D_Y((-A)^α)),α>0, where A is the infinitesimal generator of {T(t)}. Particular choices of X,Y are Lorentz–Zygmund spaces, of {T(t)} are those connected with orthogonal expansions such as Fourier, spherical harmonics, Jacobi, Laguerre, Hermite series. Known characterizations of the K-functionals lead to concrete Ulyanov type inequalities. In particular, results of Ditzian and Tikhonov in the case , are partly covered.