The Fast Numerical Solution of Very Large Elliptic Difference Schemes

Let L_kv_k = g_k be a system of difference equations discretizingan elliptic boundary value problem. Assume the system to be"very large", that means that the number of unknowns exceedsthe capacity of storage. We present a method for solving theproblem with much less storage requirement. For two-dimensionalproblems the size of the needed storage decreases from O(h^–2)to (or even O(h^–5/4)). The computational work increasesonly by a factor about six. The technique can be generalizedto nonlinear problems. The algorithm is also useful for computerswith a small number of parallel processors.     

Long-Time Behavior of Solutions of the Fast Diffusion Equations with Critical Absorption Terms

This paper is devoted to the long-time behavior of solutionsto the Cauchy problem of the porous medium equation u_t = (u^m)– u^p in Rⁿ x (0,) with (1 – 2/n)₊ < m < 1and the critical exponent p = m + 2/n. For the strictly positiveinitial data u(x,0) = O(1 + |x|)^–k with n + mn(2 –n + nm)/(2[2 – m + mn(1 – m)]) k < 2/(1 –m), we prove that the solution of the above Cauchy problem convergesto a fundamental solution of u_t = (u^m) with an additional logarithmicanomalous decay exponent in time as t .     

Necessary and Sufficient Conditions for Exponential Stability and Ultimate Boundedness of Systems Governed by Stochastic Partial Differential Equations

Consider the following infinite dimensional stochastic evolutionequation over some Hilbert space H with norm |·|: It is proved that under certain mild assumptions, the strongsolution X_t(x₀)VHV*, t 0, is mean square exponentially stableif and only if there exists a Lyapunov functional (·,·):HxR₊R¹ which satisfies the following conditions: (i)c₁|x|²–k₁e^–µ₁t(x,t)c₂|x|²+k₂+k₂e^–µ₂t; (ii) L(x,t)–c₃(x,t)+k₃e^–µ₃t, xV, t0; where L is the infinitesimal generator of the Markov processX_t and c_i, k_i, µ_i, i = 1, 2, 3, are positive constants.As a by-product, the characterization of exponential ultimateboundedness of the strong solution is established as the nulldecay rates (that is, µ_i = 0) are considered.     

The Baire Method for the Prescribed Singular Values Problem

The paper investigates the vectorial Dirichlet problem definedby S_j( u(x))=1,xnO; a.e., j=1,...,n, u(x)=\(x),\,x\in|O. end{cases}Here O is an open bounded subset of Rⁿ with boundary |O, and_j(A) (j=1,...,n) denote the singular values of the gradient u(x). The existence of solutions is established under one ofthe following assumptions: : O – Rⁿ is continuous on Oand locally contractive on O, or : |O – Rⁿ is contractiveon |O. This extends a result due to Dacorogna and Marcellini.The approach is based on the Baire category method developedearlier by the authors.     

Isomorphisms between Semisimple Weighted Measure Algebras

It is shown that, if is an isomorphism between semisimple weightedmeasure algebras M(w₁) and M(w₂), then maps L¹(R⁺, w₁) ontoL¹(R⁺, w₂). This is used to describe all the automorphisms ofM(R⁺, w). A necessary and sufficient condition is given forM(w₁) and M(w₂) to be isomorphic.     

Shapiro's Cyclic Sum

Shapiro's cyclic sum is defined by , If K is the cone in Rⁿ of points withnon-negative coordinates, it is shown that the minimum of Ein K is a fixed point of T², where T is the non-linear operatordefined by (Tx)_i = x_n–i+1/(x_n–i+2 + x_n–i+3)²for i = 1,2,...,n. It is conjectured that Tx = S^kx, where Sis the shift operator in Rⁿ, and a proof is given under someadditional hypotheses. One of the consequences is a simple proofthat at the minimum point, a_i(x) = a_n–i+1–k(x) fori = 1,2,...,n.     

Best Constants for Uncentred Maximal Functions

We precisely evaluate the operator norm of the uncentred Hardy–Littlewoodmaximal function on L^p(R¹). Consequently, we compute the operatornorm of the ‘strong’ maximal function on L^p(Rⁿ),and we observe that the operator norm of the uncentred Hardy–Littlewoodmaximal function over balls on L^p(Rⁿ) grows exponentially asn. 1991 Mathematics Subject Classification 42B25.     

On the Discreteness and Convergence in n-Dimensional Mobius Groups

Throughout this paper, we adopt the same notations as in [1,6, 8] such as the Möbius group M(Rⁿ), the Clifford algebraC_n–1, the Clifford matrix group SL(2, _n), the Cliffordnorm of ||A||=(|a|²+|b|²+|c|²+|d|²) (1) and the Clifford metric of SL(2, _n) or of the Möbius groupM(Rⁿ) d(A₁,A₂)=||A₁–A₂||(|a₁–a₂|²+|b₁–b₂|²+|c₁–c₂|²+|d₁–d₂|²)(2) where |·| is the norm of a Clifford number and represents f_i M(), i = 1,2, and so on. In addition, we adopt some notions in [6, 12]:the elementary group, the uniformly bounded torsion, and soon. For example, the definition of the uniformly bounded torsionis as follows.     

Complete Manifolds with Sectional Curvature Bounded Below and Large Volume Growth

This paper provides a proof that an n-dimensional complete openRiemannian manifold M with sectional curvature K_M –1is diffeomorphic to a Euclidean n-space Rⁿ if the volume growthof geodesic balls in M is close to that of the balls in an n-dimensionalhyperbolic space Hⁿ(–1) of sectional curvature –1.     

Counterexamples to Tischler's Strong Form of Smale's Mean Value Conjecture

Smale's mean value conjecture asserts that for every polynomial P of degree d satisfying P(0)=0,where K = (d–1)/d and the minimum is taken over all criticalpoints of P. A stronger conjecture due to Tischler assertsthat with . Tischler's conjecture is known to be true: (i) for local perturbations of the extremumP₀(z)=z^d – dz, and (ii) for all polynomials of degreed 4. In this paper, Tischler's conjecture is verified for alllocal perturbations of the extremum P₁(z)=(z – 1)^d –(–1)^d, but counterexamples to the conjecture are givenin each degree d 5. In addition, estimates for certain weightedL¹- and L²-averages of the quantities are established, which lead to the best currentlyknown value for K₁ in the case d=5. 2000 Mathematics SubjectClassification 30C15.     

Binary Quadratic Forms with Large Discriminants and Sums of Two Squareful Numbers II

Let F = (F₁, ..., F_m) be an m-tuple of primitive positive binaryquadratic forms and let U_F(x) be the number of integers notexceeding x that can be represented simultaneously by all theforms F_j, j = 1, ... , m. Sharp upper and lower bounds for U_F(x)are given uniformly in the discriminants of the quadratic forms. As an application, a problem of Erds is considered. Let V(x)be the number of integers not exceeding x that are representableas a sum of two squareful numbers. Then V(x) = x(log x)^–+o(1)with = 1 – 2^–1/3 = 0.206....     

Similarity Solutions of a Non-linear Diffusion Equation

In several physical contexts the equations for the dispersionof a buoyant contaminant can be approximated by the Erdogan-Chatwin(1967) equation {dot}c = {dot}_y{[D_o + ({dot}_yc)²D₂]{dot}_yc}. Here it is shown that in the limit of strong non-linearity (i.e.D_o = 0) there are similarity solutions for a concentration jumpand for a finite discharge. A stability analysis for the latterproblem involves a new family of orthogonal polynomials Y_n(z)where (1 – z⁴)Y – 6z³Y + n(n + 5)z² Y_n = 0 and the degree n is restricted to the values 0, 1, 4, 5, 8,9,.... A numerical solution of the Erdogan-Chatwin equationis given which describes the transition between the non-linearand linear (Gaussian) similarity solutions.     

Sharp Results Concerning the Expression of Functions as Sums of Finite Differences

Let * denote convolution and let _x denote the Dirac measureat a point x. A function in L²(R)) is called a difference oforder 1 if it is of the form g-_x * g for some x R and g L²(R)).Also, a difference of order 2 is a function of the form for some x R and g L²(R)). In fact,the concept of a ‘difference of order s’ may bedefined in a similar manner for each s 0. If f denotes the Fouriertransform of f, it is known that a function f in L²(R)) is afinite sum of differences of order s if and only if , and the vector space of all suchfunctions is denoted by D_s (L²(R)). Every function in D_s (L²(R))is a sum of int(2s) + 1 differences of order s, where int(t)denotes the integer part of t. Thus, every function in D₁ (L²(R))is a sum of three first order differences, but it was provedin 1994 that there is a function in D₁ (L(R)) which is neverthe sum of two first order differences. This complemented, forthe group R, the corresponding result for first order differencesobtained by Meisters and Schmidt in 1972 for the circle group.The results show that there is a function in L₂ R such that,for each s 1/2, this function is a sum of int (2s) + 1 differencesof order s but it is never the sum of int (2s) differences oforder s. The proof depends upon extending to higher dimensionsthe following result in two dimensions obtained by Schmidt in1972 in connection with Heilbronn's problem: if x₁, x__n arepoints in the unit square, Following on from the work of Meisters and Schmidt, this workfurther develops a connection between certain estimates in combinatorialgeometry and some questions of sharpness in harmonic analysis.2000 Mathematics Subject Classification 42A38 (primary), 52A40(secondary).     

Euler Characteristics of Real Varieties

In this paper we show how to associate to any real projectivealgebraic variety Z RP^n–1 a real polynomial F₁:Rⁿ,0 R, 0 with an algebraically isolated singularity, having theproperty that (Z) = (1 – deg (grad F₁), where deg (gradF₁ is the local real degree of the gradient grad F₁:Rⁿ, 0 Rⁿ,0. This degree can be computed algebraically by the method ofEisenbud and Levine, and Khimshiashvili [5]. The variety Z neednot be smooth. This leads to an expression for the Euler characteristic ofany compact algebraic subset of Rⁿ, and the link of a quasihomogeneousmapping f: Rⁿ, 0 Rⁿ, 0 again in terms of the local degree ofa gradient with algebraically isolated singularity. Similar expressions for the Euler characteristic of an arbitraryalgebraic subset of Rⁿ and the link of any polynomial map aregiven in terms of the degrees of algebraically finite gradientmaps. These maps do involve ‘sufficiently small’constants, but the degrees involved ar (theoretically, at least)algebraically computable.     

On the Solution of Some Elliptic Difference Equations

Iterative methods for the solution of some nonlinear ellipticdifference systems, approximating the first boundary value problemare considered. If h > 0 is the network step in the spaceof variables x = (x₁, x₂,..., x_p) and 2m is the order of theoriginal boundary value problem, then the iterative methodsproposed give solution of accuracy with the expenditure ofO(|In | h^–(p+m–)) and O(|In | |In h| h^–p)arithmetic operations in the case of a general region and arectangular parallelepiped respectively. In the case p = 2 theestimate O(|In | h^{–[2+ (m/2)]}) is obtained if the regionis made up of rectangles with sides parallel to the co-ordinateaxes.     

Lorentz Spaces Of Vector-Valued Measures

Given a non-atomic, finite and complete measure space (,,µ)and a Banach space X, the modulus of continuity for a vectormeasure F is defined as the function _F(t) = sup_µ(E)t |F|(E)and the space V^p,q(X) of vector measures such that t^–1/p'_F(t) L^q((0,µ()],dt/t) is introduced. It is shown thatV^p,q(X) contains isometrically L^p,q(X) and that L^p,q(X) = V^p,q(X)if and only if X has the Radon–Nikodym property. It isalso proved that V^p,q(X) coincides with the space of cone absolutelysumming operators from L^p',q^' into X and the duality V^p,q(X^*)=(Lp^',q^'(X))^*where 1/p+1/p^'= 1/q+1/q^' = 1. Finally, V^p,q(X) is identifiedwith the interpolation space obtained by the real method (V¹(X),V(X))_1/p^',q. Spaces where the variation of F is replaced bythe semivariation are also considered.     

On the Local and Superlinear Convergence of Quasi-Newton Methods

This paper presents a local convergence analysis for severalwell-known quasi-Newton methods when used, without line searches,in an iteration of the form to solve for x^* such that Fx^* = 0. The basic idea behind theproofs is that under certain reasonable conditions on x_o, Fand x_o, the errors in the sequence of approximations {H_k} toF'(x^*)^–1 can be shown to be of bounded deterioration inthat these errors, while not ensured to decrease, can increaseonly in a controlled way. Despite the fact that H_k is not shownto approach F'(x^*)^–1, the methods considered, includingthose based on the single-rank Broyden and double-rank Davidon-Fletcher-Powellformulae, generate locally Q-superlinearly convergent sequences{x_k}.     

On Positive Multipeak Solutions of a Nonlinear Elliptic Problem

In this paper we continue our investigation in [5, 7, 8] onmultipeak solutions to the problem –²u+u=Q(x)|u|^q–2u, xR^N, uH¹(R^N) (1.1) where = ^N_i=1²/x²_i is the Laplace operator in R^N, 2 < q < for N = 1, 2, 2 < q < 2N/(N–2) for N3, and Q(x)is a bounded positive continuous function on R^N satisfying thefollowing conditions. (Q₁) Q has a strict local minimum at some point x₀R^N, that is,for some > 0 Q(x)>Q(x₀) for all 0 < |x–x₀| < . (Q₂) There are constants C, > 0 such that |Q(x)–Q(y)|C|x–y| for all |x–x₀| , |y–y₀| . Our aim here is to show that corresponding to each strict localminimum point x₀ of Q(x) in R^N, and for each positive integerk, (1.1) has a positive solution with k-peaks concentratingnear x₀, provided is sufficiently small, that is, a solutionwith k-maximum points converging to x₀, while vanishing as 0 everywhere else in R^N.     

Reduction of CalderON Commutators

Let A₁,..., A_n be Lipschitz functions on R such that A'₁,...,A'_nVMO. We show that on any bounded interval, the Calderóncommutator associated with the kernel (A₁(x)–A₁(y)) ...(A_n(x) – A_n(y))/(x–y) ⁿ¹ is a compact perturbationof , where H is the Hilberttransform. 1991 Mathematics Subject Classification 47B38, 47B47,47G10, 45E99.     

A Class of Infinite Dimensional Simple Lie Algebras

Let A be an abelian group, F be a field of characteristic 0,and , ß be linearly independent additive maps fromA to F, and let ker()\{0}. Then there is a Lie algebra L = L(A,, ß, ) = _xA Fe_x under the product [e_x, e_y]]=(x–y)e_x+y+(ß) (x, y) e_x+y–. If, further, ß() = 1, and ß(A) = Z, thereis a subalgebra L⁺:=L(A⁺, , ß, ) = _xA⁺ Fe_x, whereA⁺ = {xA|ß(x)0}. The necessary and sufficient conditionsare given for L' = [L, L] and L⁺ to be simple, and all semi-simpleelements in L' and L⁺ are determined. It is shown that L' andL⁺ cannot be isomorphic to any other known Lie algebras andL' is not isomorphic to any L⁺, and all isomorphisms betweentwo L' and all isomorphisms between two L⁺ are explicitly described.