Decay property for second order hyperbolic systems of viscoelastic materials

We study a class of second order hyperbolic systems with dissipation which describes viscoelastic materials. The considered dissipation is given by the sum of the memory term and the damping term. When the dissipation is effective over the whole system, we show that the solution decays in L² at the rate t−n/4 as t→∞, provided that the corresponding initial data are in L²∩L¹, where n is the space dimension. The proof is based on the energy method in the Fourier space. Also, we discuss similar systems with weaker dissipation by introducing the operator (1−Δ)^−θ/2 with θ>0 in front of the dissipation terms and observe that the decay structure of these systems is of the regularity-loss type.     

The L resolvents of elliptic operators with uniformly continuous coefficients

We consider the strongly elliptic operator A of order 2m in the divergence form with bounded measurable coefficients and assume that the coefficients of top order are uniformly continuous. For 1<p<∞, A is a bounded linear operator from the L^p Sobolev space H^m,p into H^−m,p. We will prove that (A−λ)⁻¹ exists in H^−m,p for some λ and estimate its operator norm.     

Sharp bounds for the number of roots of univariate fewnomials

Let K be a field and t?0. Denote by Bm(t,K) the supremum of the number of roots in K^?, counted with multiplicities, that can have a non-zero polynomial in K[x] with at most t+1 monomial terms. We prove, using an unified approach based on Vandermonde determinants, that Bm(t,L)?t²Bm(t,K) for any local field L with a non-archimedean valuation v:L→R∪{∞} such that vZ≠0_|≡0 and residue field K, and that Bm(t,K)?(t²−t+1)(pf−1) for any finite extension K/Qp with residual class degree f and ramification index e, assuming that p>t+e. For any finite extension K/Qp, for p odd, we also show the lower bound Bm(t,K)?(2t−1)(pf−1), which gives the sharp estimation Bm(2,K)=3(pf−1) for trinomials when p>2+e.     

Upper bounds on the first eigenvalue for a diffusion operator via Bakry-Émery Ricci curvature

Let L=Δ−∇φ⋅∇ be a symmetric diffusion operator with an invariant measure on a complete Riemannian manifold. In this paper we give an upper bound estimate on the first eigenvalue of the diffusion operator L on the complete manifold with the m-dimensional Bakry-Émery Ricci curvature satisfying Ricm,n(L)?−(n−1), and therefore generalize a Cheng's result on the Laplacian (S.-Y. Cheng (1975) [8]) to the case of the diffusion operator.     

On the dimension of global sections of adjoint bundles for polarized 3-folds and 4-folds

Let (X,L) be a polarized manifold of dimension n defined over the field of complex numbers. In this paper, we treat the case where n=3 and 4. First we study the case of n=3 and we give an explicit lower bound for h⁰(K_X+L) if κ(X)≥0. Moreover, we show the following: if κ(K_X+L)≥0, then h⁰(K_X+L)>0 unless κ(X)=−∞ and h¹(O_X)=0. This gives us a partial answer of Effective Non-vanishing Conjecture for polarized 3-folds. Next for n=4 we investigate the dimension of H⁰(K_X+mL) for m≥2. If n=4 and κ(X)≥0, then a lower bound for h⁰(K_X+mL) is obtained. We also consider a conjecture of Beltrametti-Sommese for 4-folds and we can prove that this conjecture is true unless κ(X)=−∞ and h¹(O_X)=0. Furthermore we prove the following: if (X,L) is a polarized 4-fold with κ(X)≥0 and h¹(O_X)>0, then h⁰(K_X+L)>0.     

An example concerning the zero set of the Jacobian

Let f∈W1,1(Ω,Rn) be a homeomorphism of finite distortion K. It is known that if K1/(n−1)∈L¹(Ω), then the Jacobian Jf of f is positive almost everywhere in Ω. We will show that this integrability assumption on K is sharp in any Orlicz-scale: if α is increasing function (satisfying minor technical assumptions) such that limt→∞α(t)=∞, then there exists f such that K1/(n−1)/α(K)∈L¹(Ω) and Jf vanishes in a set of positive measure.     

On the convergence and superconvergence for a class of two-dimensional time fractional reaction-subdiffusion equations

This paper analyzes a class of two-dimensional (2-D) time fractional reaction-subdiffusion equations with variable coefficients. The high-order L2-1_σ time-stepping scheme on graded meshes is presented to deal with the weak singularity at the initial time t = 0, and the bilinear finite element method (FEM) on anisotropic meshes is used for spatial discretization. Using the modified discrete fractional Grönwall inequality, and combining the interpolation operator and the projection operator, the L²-norm error estimation and H¹-norm superclose results are rigorously proved. The superconvergence result in the H¹-norm is derived by applying the interpolation postprocessing technique. Finally, numerical examples are presented to verify the validation of our theoretical analysis.     

Regularization with differential operators. I. General theory

The method of regularization is used to obtain least squares solutions of the linear equation Kx = y, where K is a bounded linear operator from one Hilbert space into another and the regularizing operator L is a closed densely defined linear operator. Existence, uniqueness, and convergence analyses are developed. An application is given to the special case when K is a first kind integral operator and L is an nth order differential operator in the Hilbert space L²[a, b].     

Superconvergence of Legendre projection methods for the eigenvalue problem of a compact integral operator

In this paper, we consider the Galerkin and collocation methods for the eigenvalue problem of a compact integral operator with a smooth kernel using the Legendre polynomials of degree ≤n. We prove that the error bounds for eigenvalues are of the order O(n^−2r) and the gap between the spectral subspaces are of the orders O(n^−r) in L₂-norm and O(n^1/2−r) in the infinity norm, where r denotes the smoothness of the kernel. By iterating the eigenvectors we show that the iterated eigenvectors converge with the orders of convergence O(n^−2r) in both L₂-norm and infinity norm. We illustrate our results with numerical examples.     

Noncommutative localization in algebraic L-theory

Given a noncommutative (Cohn) localization A→σ⁻¹A which is injective and stably flat we obtain a lifting theorem for induced f.g. projective σ⁻¹A-module chain complexes and localization exact sequences in algebraic L-theory, matching the algebraic K-theory localization exact sequence of Neeman-Ranicki [Amnon Neeman, Andrew Ranicki, Noncommutative localisation in algebraic K-theory I, Geom. Topol. 8 (2004) 1385-1425] and Neeman [Amnon Neeman, Noncommutative localisation in algebraic K-theory II, Adv. Math. 213 (2007) 785-819].     

A local smoothing estimate for the Schrödinger equation

We show that the Schrödinger operator eitΔ is bounded from Wα,q(Rn) to Lq(Rn×[0,1]) for all α>2n(1/2−1/q)−2/q and q?2+4/(n+1). This is almost sharp with respect to the Sobolev index. We also show that the Schrödinger maximal operator sup0<t<1|eitΔf| is bounded from Hs(Rn) to when s>s₀ if and only if it is bounded from Hs(Rn) to L²(Rn) when s>2s₀. A corollary is that sup0<t<1|eitΔf| is bounded from Hs(R²) to L²(R²) when s>3/4.     

The Wente problem associated to the modified Helmholtz operator

In this paper, we consider the solution to Wente's problem with the modified Helmholtz operator −Δ+αI, where α is a positive constant. We study the best constant in the so-called Wente's inequality. At first, we consider the best constant associated to the L^∞ norm. Next, We study the case of the L² norm.     

Asymmetry of twisted convolution operators

Let K be a distribution on $\text{R}$². We denote by λ(K) the twisted convolution operator f → K × f defined by the formula K × f(x, y) = ∝∝ dudvK(x ? u, y ? v) f(u, v) exp(ixv ? iyu). We show that there exists K such that the operator λ(K) is bounded on L^p(R)² for every p in (1, 2¦, but is unbounded on L^q(R)² for every q > 2.     

Equivalence of <Emphasis Type="Italic">K</Emphasis>-functionals and modulus of smoothness constructed by generalized Dunkl translations

In a Hilbert space L _2,α := L ₂(?, |x|^2α+1 dx), α > ? 1/2, we study the generalized Dunkl translations constructed by the Dunkl differential-difference operator. Using the generalized Dunkl translations, we define generalized modulus of smoothness in the space L _2,α . Based on the Dunkl operator we define Sobolev-type spaces and K-functionals. The main result of the paper is the proof of the equivalence theorem for a K-functional and a modulus of smoothness.     

A series of Siamese twin designs

A {0,±1}-matrix S is called a Siamese twin design sharing the entries of I if S=I+K−L, where I,K,L are nonzero {0,1}-matrices and both I+K and I+L are incidence matrices of symmetric designs with the same parameters. Let p and 2p+3 be prime powers and . We construct a Siamese twin design with parameters (4(p+1)²,2p²+3p+1,p²+p).     

On the spectra of a Hardy kernel

A class of integral operators, K, on L^p spaces is studied. The operator I ? K is invertible on L^rand L^s but not on L^p for some p between r and s. This gives an example of an operator T densely defined and bounded in L^rand L^s norms but which is not bounded in L^p norm for some intermediate p.     

Effective non-vanishing of global sections of multiple adjoint bundles for polarized 3-folds

Let X be a smooth complex projective variety of dimension 3 and let L be an ample line bundle on X. In this paper, we provide a lower bound for h⁰(m(K_X+L)) under the assumption that κ(K_X+L)≥0. In particular, we get the following: (1) if 0≤κ(K_X+L)≤2, then h⁰(K_X+L)>0 holds. (2) If κ(K_X+L)=3, then h⁰(2(K_X+L))≥3 holds. Moreover we get a classification of (X,L) with κ(K_X+L)=3 and h⁰(2(K_X+L))=3 or 4.     

Trace formulas for non-self-adjoint periodic Schrödinger operators and some applications

Recently, a trace formula for non-self-adjoint periodic Schrödinger operators in L²(R) associated with Dirichlet eigenvalues was proved in [Differential Integral Equations 14 (2001) 671-700]. Here we prove a corresponding trace formula associated with Neumann eigenvalues. In addition we investigate Dirichlet and Neumann eigenvalues of such operators. In particular, using the Dirichlet and Neumann trace formulas, we provide detailed information on location of the Dirichlet and Neumann eigenvalues for the model operator with the potential Ke2ix, where K∈C.