Weighted estimates for maximal multilinear commutators

Let T be a Calderón–Zygmund operator with regular kernel K and T * _b be the maximal multilinear commutator defined by . In this paper, the following weighted estimates for T * _b are discussed. Precisely, for 0 < p < ∞, ω ∈ A _∞ and b _j ∈ Osc equation/tex2gif-inf-5.gif, r_j ≥ 1, j = 1, … , m , there exists a positive constant C such that . For p = 1 and ω ∈ A ₁, the weighted weak L (log L )^1/r ‐type estimates are also established. Our theorems are parallel to the ones of the multilinear commutators of Calderón–Zygmund operators obtained in [18] and extend the main result in [14] essentially. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A note on a third‐order multi‐point boundary value problem at resonance

Based on the coincidence degree theory of Mawhin, we prove some existence results for the following third‐order multi‐point boundary value problem at resonance where f: [0, 1] × R³ → R is a continuous function, 0 < ξ₁ < ??? < ξ_m < 1, α_i ∈ R, i = 1, …, m, m ≥ 1 and 0 < η₁ < η₂ < ??? < η_n < 1, β_j ∈ R, j = 1, 2, …, n, n ≥ 2. In this paper, the dimension of the linear space Ker L (linear operator L is defined by Lx = x′^′′) is equal to 2. Since all the existence results for third‐order differential equations obtained in previous papers are for the case dim Ker L = 1, our work is new.     

Behavior of the constant in Korenblum's maximum principle

Let A^p (??) (p ≥ 1) be the Bergman space over the open unit disk ?? in the complex plane. Korenblum's maximum principle states that there is an absolute constant c ∈ (0, 1) (may depend on p), such that whenever |f (z)| ≤ |g (z)| (f, g ∈ A^p (??)) in the annulus c < |z | < 1, then ∥f ≤ ∥g ∥. For p ≥ 1, let c_p be the largest value of c for which Korenblum's maximum principle holds. In this note we prove that c_p → 1 as p → ∞. Thus we give a positive answer of a question of Hinkkanen. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the number of positive solutions of elliptic systems

The paper deals with the existence, multiplicity and nonexistence of positive radial solutions for the elliptic system div(|?|^{p –2}?) + λk_i (|x |) fⁱ (u₁, …,u_n) = 0, p > 1, R₁ < |x | < R₂, u_i (x) = 0, on |x | = R₁ and R₂, i = 1, …, n, x ∈ ?^N , where k_i and fⁱ, i = 1, …, n, are continuous and nonnegative functions. Let u = (u₁, …, u_n), φ (t) = |t |^{p –2}t, fⁱ₀ = lim_{‖ u ‖→0}((fⁱ ( u ))/(φ (‖ u ‖))), fⁱ_∞= lim_{‖ u ‖→∞}((fⁱ ( u ))/(φ (‖ u ‖))), i = 1, …, n, f = (f¹, …, fⁿ), f ₀ = ∑ⁿ _{i =1} fⁱ ₀ and f _∞ = ∑ⁿ _{i =1} fⁱ _∞. We prove that either f ₀ = 0 and f _∞ = ∞ (superlinear), or f ₀ = ∞and f _∞ = 0 (sublinear), guarantee existence for all λ > 0. In addition, if fⁱ ( u ) > 0 for ‖ u ‖ > 0, i = 1, …, n, then either f ₀ = f _∞ = 0, or f ₀ = f _∞ = ∞, guarantee multiplicity for sufficiently large, or small λ, respectively. On the other hand, either f₀ and f_∞ > 0, or f₀ and f_∞ < ∞ imply nonexistence for sufficiently large, or small λ, respectively. Furthermore, all the results are valid for Dirichlet/Neumann boundary conditions. We shall use fixed point theorems in a cone. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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     

Total relative displacement of vertex permutations of K

Let α denote a permutation of the n vertices of a connected graph G. Define δ_α(G) to be the number , where the sum is over all the unordered pairs of distinct vertices of G. The number δ_α(G) is called the total relative displacement of α (in G). So, permutation α is an automorphism of G if and only if δ_α(G) = 0. Let π(G) denote the smallest positive value of δ_α(G) among the n! permutations α of the vertices of G. A permutation α for which π(G) = δ_α(G) has been called a near‐automorphism of G [ 2 ]. We determine π(K) and describe permutations α of K for which π(K) = δ_α(K). This is done by transforming the problem into the combinatorial optimization problem of maximizing the sums of the squares of the entries in certain t by t matrices with non–negative integer entries in which the sum of the entries in the ith row and the sum of the entries in the ith column each equal to n_i,1≤i≤t. We prove that for positive integers, n₁≤n₂≤…≤n_t, where t≥2 and n_t≥2, where k₀ is the smallest index for which n = n+1. As a special case, we correct the value of π(K_m,n), for all m and n at least 2, given by Chartrand, Gavlas, and VanderJagt [ 2 ]. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 85–100, 2002     

The existence of positive solutions to a non‐local singular boundary value problem

We consider the non‐local singular boundary value problem (1) where q ∈ C⁰([0,1]) and f, h ∈ C⁰((0,∞)), limf(x)=?∞, limh(x)=∞. We present conditions guaranteeing the existence of a solution x ∈ C¹([0,1]) ∩ C²((0,1]) which is positive on (0,1]. The proof of the existence result is based on regularization and sequential techniques and on a non‐linear alternative of Leray–Schauder type. Copyright © 2005 John Wiley & Sons, Ltd.     

Detecting and creating oscillations using multifractal methods

By comparing the Hausdorff multifractal spectrum with the large deviations spectrum of a given continuous function f, we find sufficient conditions ensuring that f possesses oscillating singularities. Using a similar approach, we study the nonlinear wavelet threshold operator which associates with any function f = ∑_j ∑_k d_j,k ψ _j,k ∈ L ²(?) the function series f^t whose wavelet coefficients are d ^t _j,k = d_j,k 1 , for some fixed real number γ > 0. This operator creates a context propitious to have oscillating singularities. As a consequence, we prove that the series f^t may have a multifractal spectrum with a support larger than the one of f . We exhibit an example of function f ∈ L ²(?) such that the associated thresholded function series f^t effectively possesses oscillating singularities which were not present in the initial function f . This series f^t is a typical example of function with homogeneous non‐concave multifractal spectrum and which does not satisfy the classical multifractal formalisms. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Covering arrays with mixed alphabet sizes

Covering arrays with mixed alphabet sizes, or simply mixed covering arrays, are natural generalizations of covering arrays that are motivated by applications in software and network testing. A (mixed) covering array A of type is a k × N array with the cells of row i filled with elements from ? and having the property that for every two rows i and j and every ordered pair of elements (e,f) ∈ ? × ?, there exists at least one column c, 1 ≤ c ≤ N, such that A_i,c = e and A_j,c = f. The (mixed) covering array number, denoted by , is the minimum N for which a covering array of type with N columns exists. In this paper, several constructions for mixed covering arrays are presented, and the mixed covering array numbers are determined for nearly all cases with k = 4 and for a number of cases with k = 5. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 413–432, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10059     

On the form of homomorphisms into the differential group Ls 1 and their extensibility

Let L_s ¹ (s ∈ ?) be the s-th differential group, that is the set {(x1,…,x_s): x₁ ≠ 0, x_n ∈ K, n =1,2,…,s} (K ∈ {?,?}) together with the group operation which describes the chain rules (up to order s) for C^s-functions with fixed point 0. We consider homomorphisms Φ_s, Φ_s = (f₁,…,f_s) from an abelian group (G,+) into L_s ¹ such that f₁ = 1, f₂ = … = f_p+2 = 0, 0_p+2 ≠ 0 for a fixed, but arbitrary p ≥ 0 such that p + 2 ≤ s (then f_p+2 is necessarily a homomorphism from (G, +) to (K, +). Let l ∈ ? or l = ∞. We present a criterion for the extensibility of Φ_s to a homomorphism Φ_s+l from (G, +) to L_s+1 ¹ (L_∞ ¹, if l = ∞), by proving that such an extension (continuation) exists iff the component functions f_n of Φ_s with s - p ≤ n ≤ min(s - p + l - 1,s) are certain polynomials in f_P+2 (see Theorem 1). We also formulate the problem in the language of truncated formal power series in one indeterminate X over K. The somewhat easier situation f ₁ ≠ 1 will be studied in a separate paper.     

Ramsey numbers for multiple copies of complete graphs

The Ramsey numbers r(m₁K_p, …, m_cK) are calculated to within bounds which are independent of m₁, …, m_c when c > 2 and p₁, …, p_c > 2.     

Universal robustness of scale‐free networks against cascading edge failures

Considering the effect of the local topology structure of an edge on cascading failures, we investigate the cascading reaction behaviors on scale‐free networks with respect to small edge‐based initial attacks. Adopt the initial load of an edge ij in a network to be L_ij = (k_ik_j)^α[(∑k_a)(∑k_b)]^β with k_i and k_j being the degrees of the nodes connected by the edge ij, where α and β are tunable parameters, governing the strength of the edge initial load, and Γ_i and Γ_j are the sets of neighboring nodes of i and j, respectively. Our aim is to explore the relationship between some parameters and universal robustness characteristics against cascading failures on scale‐free networks. We find by the theoretical analysis that the Baraba'si‐Albert (BA) scale‐free networks can reach the strongest robustness level against cascading failures when α + β = 1, where the robustness is quantified by a transition from normal state to collapse. And the network robustness has a positive correlation with the average degree. We furthermore confirm by the numerical simulations these results.     

Coin flipping from a cosmic source: On error correction of truly random bits

We study a problem related to coin flipping, coding theory, and noise sensitivity. Consider a source of truly random bits x ∈ {0, 1}ⁿ, and k parties, who have noisy version of the source bits yⁱ ∈ {0, 1}ⁿ, when for all i and j, it holds that P [y = x_j] = 1 ? ?, independently for all i and j. That is, each party sees each bit correctly with probability 1 ? ?, and incorrectly (flipped) with probability ?, independently for all bits and all parties. The parties, who cannot communicate, wish to agree beforehand on balanced functions f_i: {0, 1}ⁿ → {0, 1} such that P [f₁(y¹) = … = f_k(y^k)] is maximized. In other words, each party wants to toss a fair coin so that the probability that all parties have the same coin is maximized. The function f_i may be thought of as an error correcting procedure for the source x. When k = 2,3, no error correction is possible, as the optimal protocol is given by f_i(yⁱ) = y. On the other hand, for large values of k, better protocols exist. We study general properties of the optimal protocols and the asymptotic behavior of the problem with respect to k, n, and ?. Our analysis uses tools from probability, discrete Fourier analysis, convexity, and discrete symmetrization. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2005     

Convergence of boundary optimal controls in mixed elliptic problems

We consider a steady-state heat conduction problem P_α withmixed boundary conditions for the Poisson equation in a bounded multidimensional domain Ω depending of a positive parameter α which represents the heat transfer coefficient on a portion Γ₁ of the boundary of Ω. We consider, for each α > 0, a cost function J_α and we formulate boundary optimal control problems with restrictions over the heat flux q on a complementary portion Γ₂ of the boundary of Ω. We obtain that the optimality conditions are given by a complementary free boundary problem in Γ₂ in terms of the adjoint state. We prove that the optimal control q and its corresponding system state u and adjoint state p for each α are strongly convergent to q_op, u and p in L²(Γ₂), H¹(Ω), and H¹(Ω) respectively when α → ∞. We also prove that these limit functions are respectively the optimal control, the system state and the adjoint state corresponding to another boundary optimal control problem with restrictions for the same Poisson equation with a different boundary condition on the portion Γ₁. We use the elliptic variational inequality theory in order to prove all the strong convergences. In this paper, we generalize the convergence result obtained in Ben Belgacem-El Fekih-Metoui, ESAIM:M2AN, 37 (2003), 833-850 by considering boundary optimal control problems with restrictions on the heat flux q defined on Γ₂ and the parameter α (which goes to infinity) is defined on Γ₁. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A compact formulation of a mixed-integer set

We consider mixed-integer sets of the form X = {(s, y) ∈ ?₊ × ? ⁿ : s + a _j y _j ≥ b _j , ?j ∈ N}. A polyhedral characterization for the case where X is defined by two inequalities is provided. From that characterization we give a compact formulation for the particular case where the coefficients of the integer variables can take only two possible integer values a ₁, a ₂ ∈ ?₊ : X _n,m = {(s, y) ∈ ?₊ × ? ^n+m : s + a ₁ y _j ≥ b _j , ?j ∈ N ₁, s + a ₂ y _j ≥ b _j , j ∈ N ₂} where N ₁ = {1, …, n}, N ₂ = {n + 1, …, n + m}. In addition, we discuss families of facet-defining inequalities for the convex hull of X _n,m .     

Absolutely L – Summing Norms of Diagonal Operators in lr and Limit Orders of Lexp – Summing Operators

We compute the absolutely L – summing norms of the diagonal operators acting on l_r (1 ≤ q, r < ∞) and determine the limit orders of the absolutely L_exp – summing operators.     

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.     

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.     

The analytic spread of codimension two monomial varieties

Let ${\rm} A=k[{u_{1}^{a_{1}}},{u_{2}^{a_{2}}},\dots,{u_{n}^{a_{n}}},{u_{1}^{c_{1}}} \dots {u_{n}^{c_{n}}},{u_{1}^{b_{1}}} \dots {u_{n}^{b_{n}}}]\ \subset k[{u_{1}}, \dots {u_{n}}],$ where, a_j, b_j, C_j ∈ ?, a_j > 0, (b_j, C_j) ≠ (0,0) for 1 ≤ j ≤ n, and, further ${\underline b}:=\ ({b_{1}}, \dots,{b_{n}})\ \not=\ 0 $ and ${\underline c}:=\ ({c_{1}}, \dots,{c_{n}})\ \not=\ 0 $ . The main result says that the defining ideal I ? m = (x₁,…, x_n, y, z) ? k[x₁,…, x_n, y, z] of the semigroup ring A has analytic spread ?(I_m) at most three.     

Several Dimensional θ-Summability and Hardy Spaces

The d-dimensional Hardy spaces H_p ( T × … × T ) (d = d₁ + … + d_kand a general summability method of Fourier series and Fourier transforms are introduced with the help of integrable functions θ_j having integrable Fourier transforms. Under some conditions on θ_j we show that the maximal operator of the θ-means of a distribution is bounded from H_p ( T × … × T ) to L_p ( T ^d) where p₀ < p < ∞ and p₀ < 1 is depending only on the functions θ_j. By an interpolation theorem we get that the maximal operator is also of weak type ( L₁) (i = 1, …, k) where the Hardy space is defined by a hybrid maximal function and if k = 1. As a consequence we obtain that the θ-means of a function (log L)^k–1 converge a.e. to the function in question. If k = 1 then we get this convergence result for all f ∈ L₁. Moreover, we prove that the θ-means are uniformly bounded on the spaces H_p ( T × … × T ) whenever p₀ <p < ∞, thus the θ-means converge to f in ( T × … × T ) norm. The same results are proved for the conjugate θ-means and for d-dimensional Fourier transforms, too. Some special cases of the θ-summation are considered, such as the Weierstrass, Picar, Bessel, Fejér, Riemann, de La Vallée-Poussin, Rogosinski and Riesz summations.