On Lp boundedness of wave operators for 4-dimensional Schrodinger operators with threshold singularities

Let H=–+V(x) be a Schrödinger operator on L²(R⁴),H₀=–. Assume that |V(x)|+| V(x)|C x ^– for some>8. Let be the wave operators. It is known that W_± extend to bounded operators in L^p(R⁴)for all 1p, if 0 is neither an eigenvalue nor a resonance ofH. We show that if 0 is an eigenvalue, but not a resonance ofH, then the W_± are still bounded in L^p(R⁴) for all psuch that 4/3<p<4.     

On the Existence of Extremals for the Sobolev Trace Embedding Theorem with Critical Exponent

In this paper, the existence problem is studied for extremalsof the Sobolev trace inequality W^1,p()L^p*(), where is a boundedsmooth domain in R^N, p*=p(N–1)/(N–p), is the criticalSobolev exponent, and 1 < p < N. 2000 Mathematics SubjectClassification 35J65 (primary), 35B33 (secondary).     

The Hardy Operator and the Gap Between L{infty} and BMO

We study boundedness and compactness properties of the Hardyintegral operator from a weighted Banach function space X(v) into L and BMO. We give a new simplecharacterization of compactness of T from X(v) into BMO. Weconstruct examples of spaces X(v) such that T(X(v)) is (a) boundedin L but not compact in BMO; (b) compact in BMO but not boundedin L; (c) bounded in BMO but neither bounded in L nor compactin BMO; (d) bounded in L, compact in BMO and yet not compactin L. In order to obtain the last of the counterexamples weconstruct a new weighted Banach function space.     

Noether Numbers for Subrepresentations of Cyclic Groups of Prime Order

Let W be a finite-dimensional Z/p-module over a field, k, ofcharacteristic p. The maximum degree of an indecomposable elementof the algebra of invariants, k[W]^Z/p, is called the Noethernumber of the representation, and is denoted by rß(W).A lower bound for rß(W) is derived, and it is shownthat if U is a Z/p submodule of W, then rß(U) rß(W).Aset of generators, in fact a SAGBI basis, is constructed fork[V₂ V₃]^Z/p, where V_n is the indecomposable Z/p-module of dimensionn. 2000 Mathematics Subject Classification 13A50, 20J06.     

On a p-Local Geometry Associated with a Complex of Non-Trivial p-Subgroups

Let D be a complex of non-trivial p-subgroups of a finite groupG, closed under G-conjugation. In this paper, a p-local geometry^p(G; D) is introduced for G associated with the complex D. Thehomotopy equivalence between ^p(G; D) and D is also studied.2000 Mathematics Subject Classification 20E15, 51E24.     

A Criterion for Finite Topological Determinacy of Map-Germs

Let f, g: (Rⁿ, 0) (R^p, 0) be two C map-germs. Then f and gare C⁰-equivalent if there exist homeomorphism-germs h and lof (Rⁿ, 0) and (R^p, 0) respectively such that g = l f h^–1.Let k be a positive integer. A germ f is k-C⁰-determined ifevery germ g with j^k g(0) = j^k f(0) is C⁰-equivalent to f. Moreover,we say that f is finitely topologically determined if f is k-C⁰-determinedfor some finite k. We prove a theorem giving a sufficient conditionfor a germ to be finitely topologically determined. We explainthis condition below. Let N and P be two C manifolds. Consider the jet bundle J^k(N,P) with fiber J^k(n, p). Let z in J^k(n, p) and let f be suchthat z = j^kf(0). Define Whether (f) < k depends only on z, not on f. We can thereforedefine the set Let W^k(N, P) be the subbundle of J^k(N, P) with fiber W^k(n, p).Mather has constructed a finite Whitney (b)-regular stratificationS^k(n, p) of J^k(n, p) – W^k(n, p) such that all strata aresemialgebraic and K-invariant, having the property that if S^k(N,P) denotes the corresponding stratification of J^k(N, P) –W^k(N, P) and f C(N, P) is a C map such that j^kf is multitransverseto S^k(N, P), j^kf(N) W^k(N, P) = and N is compact (or f is proper),then f is topologically stable. For a map-germ f: (Rⁿ, 0) (R^p, 0), we define a certain ojasiewiczinequality. The inequality implies that there exists a representativef: U R^p such that j^kf(U – 0) W^k (Rⁿ, R^p = and suchthat j^kf is multitransverse to S^k (Rⁿ, R^p) at any finite setof points S U – 0. Moreover, the inequality controlsthe rate j^kf becomes non-transverse as we approach 0. We showthat if f satisfies this inequality, then f is finitely topologicallydetermined. 1991 Mathematics Subject Classification: 58C27.     

An Example on Sobolev Space Approximation

For each d2 we construct a connected open set R^d such that = int (clos()), and for each k 1 and each p [1, ), the subsetW^k, () fails to be dense in the Sobolev space W^{k, p}(), in thenorm of W^{k, p}(). 1991 Mathematics Subject Classification 46E35,46F05.     

A qualocation method for parabolic partial differential equations

In this paper a qualocation method is analysed for parabolicpartial differential equations in one space dimension. Thismethod may be described as a discrete H¹-Galerkin method inwhich the discretization is achieved by approximating the integralsby a composite Gauss quadrature rule. An O (h^4-i) rate of convergencein the W^i.p norm for i = 0, 1 and 1 p is derived for a semidiscretescheme without any quasi-uniformity assumption on the finiteelement mesh. Further, an optimal error estimate in the H² normis also proved. Finally, the linearized backward Euler methodand extrapolated Crank-Nicolson scheme are examined and analysed.     

A fourth-order cubic spline method for linear second-order two-point boundary-value problems

A cubic spline method for linear second-order two-point boundary-valueproblems is analysed. The method is a Petrov-Galerkin methodusing a cubic spline trial space, a piecewise-linear test space,and a simple quadrature rule for the integration, and may beconsidered a discrete version of the H¹-Galerkin method. Themethod is fully discrete, allows an arbitrary mesh, yields alinear system with bandwidth five, and under suitable conditionsis shown to have an 0(h^4– rate of convergence in the W_p¹norm for i = 0, 1, 2, 1p. The H¹-Galerkin method and orthogonalspline collocation with Hermite cubics are also discussed.     

Kato Class Potentials for Higher Order Elliptic Operators

Our goal in this paper is to determine conditions on a potentialV which ensure that an operator such as H:=(–)^m+V (1) acting on L²(R^N) defines a semigroup in L^p(R^N) for various valuesof p including p=1. The operator is defined as a quadratic formsum. That is, we put for (all integrals are on R^N and are with respect to Lebesgue measure), and note thatthe closure of the form is non-negative and has domain equalto the Sobolev space W^m,2. We then assume that the potentialhas quadratic form bound less than 1 with respect to Q₀, anddefine This form is closed and is associated with a semibounded self-adjointoperator H in L² (see [17, p. 348; 5, Theorem 4.23]). One canthen ask whether the semigroup e^–Ht defined on L² fort0 is extendable to a strongly continuous one-parameter semigroupon L^p for other values of p, and if so whether one can describethe domain and spectrum of its generator.     

A Critical Phenomenon for Sublinear Elliptic Equations in Cone-Like Domains

The authors of this paper study positive supersolutions to theelliptic equation -u = c|x|^–su^p in Cone-like domains ofR^N (N 2), where p, s R and c > 0. They prove that in thesublinear case p < 1 there exists a critical exponent p_*> 1 such that the equation has a positive supersolution ifand only if – < p < p_*. The value of p_* is determinedexplicitly by s and the geometry of the cone. 2000 MathematicsSubject Classification 35J60 (primary), 35B05, 35R45 (secondary).     

Traces and extensions of matrix-weighted Besov spaces

Let V be a matrix weight on ⁿ⁺¹ and let W be a matrix weighton ⁿ, satisfying, for example, the matrix A_p condition. Definethe trace, or restriction, operator Tr by Tr (f)(x^')=f(x^', 0),where x^'n and f is a function on ⁿ⁺¹. If –1/p>n (1/p–1)₊+(β–n)/p,where β is the doubling exponent of W, then the trace operatoris bounded from into (matrix-weighted Besov spaces) if and only ifthe weights V and W uniformly satisfy an estimate controllingthe average of on anydyadic cube I ⁿ by the average of on Q(I)=Ix[0, (I)], for all . If V and W satisfy the converse inequality, then there existsa continuous linear map .If both inequalities hold, then Tr Ext is the identity on .     

Exponential stability and maximal attractors for a one-dimensional nonlinear thermoviscoelasticity

This paper is concerned with the global existence, exponentialstability of solutions and associated nonlinear C₀-semigroupas well as the existence of maximal attractors in Hⁱ (i = 1,2, 4) for a nonlinear one-dimensional thermoviscoelasticitydescribing a kind of solid-like material. Some new ideas andmore delicated estimates are employed to prove the global existenceand exponential stability of solutions. The important featurefor the existence of maximal attractors in Hⁱ₊ (i = 1, 2, 4)is that the metric spaces H¹₊, H²₊ and H⁴₊ we work with arethree incomplete metric spaces, as can be seen from the physicalconstraints, i.e. > 0 and u > 0, with and u being absolutetemperature and deformation gradient (strain). For any positiveparameters ₁, ₂, ..., ₅ verifying some conditions, a sequenceof closed subspaces Hⁱ Hⁱ+ (i = 1, 2, 4) is found, and theexistence of maximal attractors in Hⁱ (i = 1, 2, 4) is established.     

Density of non-residues in Burgess-type intervals and applications

We show that for any fixed > 0, there are numbers >0 and p₀ 2 with the following property: for every prime p p₀ and every integer N such that p^1/(4e )+ N p, the sequence1, 2, ..., N contains at least N quadratic non-residues modulop. We use this result to obtain strong upper bounds on the sizesof the least quadratic non-residues in Beatty and Piatetski-Shapirosequences.     

Newton's method on Riemannian manifolds: Smale's point estimate theory under the {gamma}-condition

^** Email: cli{at}zju.edu.cn^*** Email: wjh{at}zjut.edu.cn The -conditions for vector fields on Riemannian manifolds areintroduced. The -theory and the -theory for Newton's methodon Riemannian manifolds are established under the -conditions.Applications to analytic vector fields are provided and theresults due to Dedieu et al. (2003, IMA J. Numer. Anal., 23,395–419) are improved.     

The largest prime factor of X3+2

The largest prime factor of X³+2 was investigated in 1978 byHooley, who gave a conditional proo that it is infinitely oftenat least as large as X¹+, with a certain positive constant .It is trivial to obtain such a result with =0. One may thinkof Hooley's result as an approximation to the conjecture thatX³+2 is infinitely often prime. The condition required by Hooley,his R^* conjecture, gives a non-trivial bound for short Ramanujan–Kloostermansums. The present paper gives an unconditional proof that thelargest prime factor of X³+2 is infinitely often at least aslarge as X¹⁺, though with a much smaller constant than thatobtained by Hooley. In order to do this we prove a non-trivialbound for short Ramanujan–Kloosterman sums with smoothmodulus. It is also necessary to modify the Chebychev method,as used by Hooley, so as to ensure that the sums that occurdo indeed have a sufficiently smooth modulus. 2000 MathematicsSubject Classification: 11N32.     

REAL HYPERSURFACES IN A EUCLIDEAN COMPLEX SPACE FORM

Let M be an orientable connected and compact real hypersurfaceof the complex space form C^{(n + 1)/2}. If the mean curvature and the function f = g(A, ) of hypersurface M satisfy the inequalityn²² (n² – 1) + f², where is the characteristic vectorfield, A is the shape operator and (n – 1) is the infimumof the Ricci curvatures of hypersurface M, then it is shownthat is a constant and M is the sphere Sⁿ(²) in C^{(n + 1)/2}.     

Conservative upwind finite-element method for a simplified Keller-Segel system modelling chemotaxis

^** Email: saito{at}infsup.jp Finite-element approximation for a non-linear parabolic–ellipticsystem is considered. The system describes the aggregation ofslime moulds resulting from their chemotactic features and iscalled a simplified Keller–Segel system. Applying an upwindtechnique, first we present a finite-element scheme that satisfiesboth positivity and mass conservation properties. Consequently,if the triangulation is of acute type, our finite-element approximationpreserves the L¹ norm, which is an important property of theoriginal system. Then, under some assumptions on the regularityof a solution and on the triangulation, we establish error estimatesin L^p x W^1, with a suitable p > d, where d is the dimensionof a spatial domain. Our scheme is well suited for practicalcomputations. Some numerical examples that validate our theoreticalresults are also presented.     

Total Flux Estimates for a Finite-Element Approximation of Parabolic Equations

An initial-boundary-value problem for a parabolic equation ina domain x (0, T) with prescribed Dirichlet data on is approximatedusing a continuous-time Galerkin finite-element scheme. It isshown that the total flux across ₁= can be approximated withan error of O(h^k) when is a curved domain in Rⁿ (n = 2 or 3)and isoparametric elements having approximation power h^k inthe L² norm are used.     

Nodal Solutions of a p-Laplacian Equation

We prove that the p-Laplacian problem –_p u = f(x, u),with u on a bounded domain R^N, with p > 1 arbitrary, has a nodal solution providedthat f : x R R is subcritical, and f(x, t) / |t|^p –² is superlinear. Infinitely many nodal solutions are obtainedif, in addition, f(x, –t) = –f(x, t). 2000 MathematicsSubject Classification 35J20, 35J65, 58E05.