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.     

Right Cancellation in the LUC-Compactification of a Locally Compact Group

Write G^* = G^LUC\G where G^LUC is the largest semigroup compactificationof the locally compact group G. Then the set of points of G^*which are right cancellable in G^* = G^LUC is large; in fact ithas an interior in G^* which is dense in G^*. Corollaries aregiven about the number of left ideals in G^* = G^LUC and the sizeof right ideals in the algebra LUC(G)^*.     

On time regularity and related conditions for power-bounded operators

Let T be a bounded linear operator in a complex Banach space.Our main result gives various characterizations of the condition:T is power-bounded and an estimate ||(I – T)Tⁿ || cn^–1/2 holds for all positive integers n. In particular, this conditionholds if and only if T = β S + (1 – β)I, forsome β (0, 1) and some power-bounded operator S; or ifand only if T is power-bounded and the discrete semigroup (Tⁿ)is dominated by the continuous semigroup (e^{– t(I –T)})_{t 0} in a natural sense. As a consequence of our main results,for 1/2 < 1 we characterize the condition that T is power-boundedand ||(I – T)Tⁿ || c n^– for all n, in terms ofestimates on the semigroup e^{–t(I – T)}.     

Weakly Almost Periodic Functions on N with a Negative Base

Weakly almost periodic compactifications have been seriouslystudied for over 30 years. In the pioneering papers of de Leeuwand Glicksberg [4] and [5], the approach adopted was operator-theoretic.The current definition is more likely to be created from theperspective of universal algebra (see [1, Chapter 3]). For adiscrete group or semigroup S, the weakly almost periodic compactificationwS is the largest compact semigroup which (i) contains S asa dense subsemigroup, and (ii) has multiplication continuousin each variable separately (where largest means that any othercompact semigroup with the properties (i) and (ii) is a quotientof wS). A third viewpoint is to envisage wS as the Gelfand spaceof the C*-algebra of bounded weakly almost periodic functionson S (for the definition of such functions, see below). In this paper, we are concerned only with the simplest semigroup(N, +). The three approaches described above give three methodsof obtaining information about wN. An early striking resultabout wN, that it contains more than one idempotent, was obtainedby T. T. West using operator theory [13]. He considered theweak operator closure of the semigroup {T, T², T³, ...} of iteratesof a single operator T on the Hilbert space L²(µ) fora particular measure µ on [0, 1]. Brown and Moran, ina series of papers culminating in [2], used sophisticated techniquesfrom harmonic analysis to produce measures µ that permittedthe detection of further structure in wN; in particular, theyfound 2^cdistinct idempotents. However, for many years, no otherway of showing the existence of more than one idempotent inwN was found. The breakthrough came in 1991, and it was made by Ruppert [11].In his paper, he created a direct construction of a family ofweakly almost periodic functions which could detect 2^c differentidempotents in wN. His method was very ingenious (he used aunique variant of the p-adic expansion of integers) and rathercomplicated. Our main aim in this paper is to construct weaklyalmost periodic functions which are easy to describe and soappear more ‘natural’ than Ruppert's. We also showthat there are enough functions of our type to distinguish 2^cidempotentsin wN.     

A Generalised Skolem-Mahler-lech Theorem for Affine Varieties

The Skolem–Mahler–Lech theorem states that if f(n)is a sequence given by a linear recurrence over a field of characteristic0, then the set of m such that f(m) is equal to 0 is the unionof a finite number of arithmetic progressions in m 0 and afinite set. We prove that if X is a subvariety of an affinevariety Y over a field of characteristic 0 and q is a pointin Y, and is an automorphism of Y, then the set of m such that^m(q) lies in X is a union of a finite number of complete doubly-infinitearithmetic progressions and a finite set. We show that thisis a generalisation of the Skolem–Mahler–Lech theorem.     

On the number of homotopy types of fibres of a definable map

In this paper we prove a single exponential upper bound on thenumber of possible homotopy types of the fibres of a Pfaffianmap in terms of the format of its graph. In particular, we showthat if a semi-algebraic set SR^m+n, where R is a real closedfield, is defined by a Boolean formula with s polynomials ofdegree less than d, and : R^m+nRⁿ is the projection on a subspace,then the number of different homotopy types of fibres of doesnot exceed s^2(m+1)n(2^m nd)^O(nm). As applications of our mainresults we prove single exponential bounds on the number ofhomotopy types of semi-algebraic sets defined by fewnomials,and by polynomials with bounded additive complexity. We alsoprove single exponential upper bounds on the radii of ballsguaranteeing local contractibility for semi-algebraic sets definedby polynomials with integer coefficients.     

A Characterization of Fredholm Pseudo-Differential Operators

We give a necessary and sufficient condition on an ellipticsymbol of order m to ensure that the unique closed extensionin L^p(Rⁿ) for 1 < p < , of the pseudo-differential operatorT, initially defined on the Schwartz space, is a Fredholm operatorfrom L^p(Rⁿ) into L^p(Rⁿ) with domain H^{m, p}, where H^{m, p} is theL^p Sobolev space of order m.     

Intersections of Symbolic Powers of Prime Ideals

Let (R,m) be a local ring with prime ideals p and q such that. If R is regular and containsa field, and dim(R/p)+dim(R/q)=dim(R), then it is proved thatp^(m) q⁽ⁿ⁾ m^m+n for all positive integers m and n. This isproved using a generalization of Serre's Intersection Theoremwhich is applied to a hypersurface R/fR. The generalizationgives conditions that guarantee that Serre's bound on the intersectiondimension (R/p)+(R/q)dim(R) holds when R is nonregular.     

Extensions of Simple Modules for Sp4(2n) and Suz(2m)

The group of extensions between any two irreducible 2-modularrepresentations of the groups Sp₄ (2ⁿ) and Suz (2^m) is determined.     

Asymptotic simplification for a reaction-diffusion problem with a nonlinear boundary condition

We study non-negative solutions of the porous medium equationwith a source and a nonlinear flux boundary condition, u_t =(u^m)_xx + u^p in (0, ), x (0, T); – (u^m)_x (0, t) = u^q (0,t) for t (0, T); u (x, 0) = u₀ (x) in (0, ), where m > 1,p, q > 0 are parameters. For every fixed m we prove thatthere are two critical curves in the (p, q-plane: (i) the criticalexistence curve, separating the region where every solutionis global from the region where there exist blowing-up solutions,and (ii) the Fujita curve, separating a region of parametersin which all solutions blow up from a region where both globalin time solutions and blowing-up solutions exist. In the caseof blow up we find the blow-up rates, the blow-up sets and theblow-up profiles, showing that there is a phenomenon of asymptoticsimplification. If 2q < p + m the asymptotics are governedby the source term. On the other hand, if 2q > p + m theevolution close to blow up is ruled by the boundary flux. If2q = p + m both terms are of the same order.     

The Existence of Periodic and Subharmonic Solutions of Subquadratic Second Order Difference Equations

By critical point theory, a new approach is provided to studythe existence of periodic and subharmonic solutions of the secondorder difference equation where f C(R x R^m, R^m), f(t+M,z)+f(t,z) for any (t, z)R x R^mand M is a positive integer. This is probably the first timecritical point theory has been applied to deal with the existenceof periodic solutions of difference systems.     

A Classification of Horizontally Homothetic Submersions from Space Forms of Nonnegative Curvature

In this paper, it is proved that for n 2, any horizontallyhomothetic submersion : Rⁿ⁺¹ (Nⁿ, h) is a Riemannian submersionup to a homothety. It is also shown that if : Sⁿ⁺¹ (Nⁿ, h)is a horizontally homothetic submersion, then n = 2m, (Nⁿ, h)is isometric to CP^m and, up to a homothety, is a standard Hopffibration S^2m+1 CP^m. 2000 Mathematics Subject Classification53C20, 53C12.     

On the Regular Stratifications and Conormal Structure of Subanalytic Sets

In this paper we define (b^*)-regularity for incident strataof a stratified set, a notion which refines Whitney (b)-regularity.We give three proofs that every subanalytic set admits a (b^*)-regularstratification and we discuss some related properties.     

Multistep approximation of linear sectorial evolution equations

This paper concerns the linear multistep approximation of alinear sectorial evolution equation u_t = Au on a complex Banachspace X. Given a strictly A()-stable q-step method of orderp whose stability region includes a sectorial region containingthe spectrum of the operator A, the corresponding evolutionsemigroup for the method is Cⁿ(hA), n 0, defined on X^q, whereC(z) L (C^q) denotes the one-step map associated with the method.It is shown that for appropriately chosen V, Y: C C^q, basedon the principal right and left eigenvectors of C(z), Cⁿ(hA)approximates the semigroup V(hA)e^nhAY^H(hA) with optimal orderp.     

Improving the Stability of Predictor-Corrector Methods by Residue Smoothing

Residue smoothing is usually applied in order to acceleratethe convergence of iteration processes. Here, we show that residuesmoothing can also be used in order to increase the stabilityregion of predictor-corrector methods. We shall concentrateon increasing the real stability boundary. The iteration parametersand the smoothing operators are chosen such that the stabilityboundary becomes as large as c(m, q)m²4^g where m is the numberof right-hand side evaluations per step, q the number of smoothingoperations applied to each right-hand side evaluation, and c(m,q) a slowly varying function of m and q, of magnitude 1.3 ina typical case. Numerical results show that, for a variety oflinear and nonlinear parabolic equations in one and two spatialdimensions, these smoothed predictor-corrector methods are atleast competitive with conventional implicit methods.     

Triangle group representations and constructions of regular maps

A regular map of type {m,n} is a 2-cell embedding of a graphin an orientable surface, with the property that for any twodirected edges e and e' there exists an orientation-preservingautomorphism of the embedding that takes e onto e', and in whichthe face length and the vertex valence are m and n, respectively.Such maps are known to be in a one-to-one correspondence withtorsion-free normal subgroups of the triangle groups T(2,m,n).We first show that some of the known existence results aboutregular maps follow from residual finiteness of triangle groups.With the help of representations of triangle groups in speciallinear groups over algebraic extensions of Z we then constructivelydescribe homomorphisms from T(2,m,n)=y,z|y^m=zⁿ=(yz)²=1 intofinite groups of order at most c^r where c=c(m,n), such thatno non-identity word of length at most r in x,y is mapped ontothe identity. As an application, for any hyperbolic pair {m,n}and any r we construct a finite regular map of type {m,n} ofsize at most C^r, such that every non-contractible closed curveon the supporting surface of the map intersects the embeddedgraph in more than r points. We also show that this result isthe best possible up to determining C=C(m,n). For r>m thegraphs of the above regular maps are arc-transitive, of valencen, and of girth m; moreover, if each prime divisor of m is largerthan 2n then these graphs are non-Cayley. 2000 Mathematics SubjectClassification: 05C10, 05C25, 20F99, 20H25.     

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 .     

Optimum Runge-Kutta-Fehlberg Methods for Second-order Differential Equations

^* Presently at Deparment of Mathematics, Indian Institute of Technology, Madras, India. The optimum Runge-Kutta method of a particular order is theone whose truncation error is minimum. In this paper, we havederived optimum Runge-Kutta mehtods of 0(h^m+4), 0(h^m+5) and0(h^m+6) for m = 0(1)8, which can be directly used for solvingthe second order differential equation yⁿ = f(x, y, y'). Thesemethods are based on a transformation similar to that of Fehlbergand require two, three and four evaluations of f(x, y, y') respectively,for each step. The numercial solutions of one example obtainedwith these methods are given. It has been assumed that f(x,y, y')is sufficiently differentiable in the entire region ofintegration.     

Systems of cubic forms

We discuss the existence of rational and p-adic zeros of systemsof cubic forms. In particular, we prove that for p2 any systemof r cubic forms over Q_p in more than 125r³+705r²+210r variablesadmits a non-trivial p-adic zero, and that any system of r rationalcubic forms in more than O(r⁴ m⁶+r⁶ m⁵) variables admits a rationallinear space of zeros of dimension at least m.