A Mean Value Property of Poly-Temperatures on a Strip Domain

We consider the iterates of the heat operator on Rⁿ⁺¹={(X, t); X=(x₁, x₂, ..., x_n)Rⁿ, tR}. Let Rⁿ⁺¹ be a domain,and let m1 be an integer. A lower semi-continuous and locallyintegrable function u on is called a poly-supertemperatureof degree m if (–H)^mu0 on (in the sense of distribution). If u and –u are both poly-supertemperatures of degreem, then u is called a poly-temperature of degree m. Since His hypoelliptic, every poly-temperature belongs to C(), andhence (–H)^m u(X, t)=0 (X, t). For the case m=1, we simply call the functions the supertemperatureand the temperature. In this paper, we characterise a poly-temperature and a poly-supertemperatureon a strip D={(X, t);XRⁿ, 0<t<T} by an integral mean on a hyperplane. To state our result precisely,we define a mean A[·, ·]. This plays an essentialrole in our argument.     

The Singular Homology of the Hawaiian Earring

The singular homology groups of compact CW-complexes are finitelygenerated, but the groups of compact metric spaces in generalare very easy to generate infinitely and our understanding ofthese groups is far from complete even for the following compactsubset of the plane, called the Hawaiian earring: Griffiths [11] gave a presentation of the fundamental groupof H and the proof was completed by Morgan and Morrison [15].The same group is presented as the free -product of integers Z in [4, Appendix]. Hence the firstintegral singular homology group H₁(H) is the abelianizationof the group . These results have been generalized to non-metrizable counterparts H_I of H(see Section 3). In Section 2 we prove that H₁(X) is torsion-free and H_i(X) =0 for each one-dimensional normal space X and for each i 2.The result for i 2 is a slight generalization of [2, Theorem5]. In Section 3 we provide an explicit presentation of H₁(H)and also H₁(H_I) by using results of [4]. Throughout this paper, a continuum means a compact connectedmetric space and all maps are assumed to be continuous. Allhomology groups have the integers Z as the coefficients. Thebouquet with n circles is denoted by B_n. The base point (0, 0) of B_n is denoted by o forsimplicity.     

Serre's Theorem on the Cohomology Algebra of a p-Group

The purpose of this note is to give a proof of a theorem ofSerre, which states that if G is a p-group which is not elementaryabelian, then there exist an integer m and non-zero elementsx₁, ..., x_m H¹ (G, Z/p) such that with ß the Bockstein homomorphism. Denote by m_G thesmallest integer m satisfying the above property. The theoremwas originally proved by Serre [5], without any bound on m_G.Later, in [2], Kroll showed that m_G p^k – 1, with k =dim_Z/pH¹ (G, Z/p). Serre, in [6], also showed that m_G (p^k –1)/(p – 1). In [3], using the Evens norm map, Okuyamaand Sasaki gave a proof with a slight improvement on Serre'sbound; it follows from their proof (see, for example, [1, Theorem4.7.3]) that m_G (p + 1)p^k–2. However, m_G can be sharpenedfurther, as we see below. For convenience, write H^*(G, Z/p) = H^*(G). For every x_i H¹(G),set 1991 Mathematics SubjectClassification 20J06.     

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.     

Limits Along Parallel Lines and the Classical Fine Topology

The fine topology on Rⁿ (n2) is the coarsest topology for whichall superharmonic functions on Rⁿ are continuous. We refer toDoob [11, 1.XI] for its basic properties and its relationshipto the notion of thinness. This paper presents several theoremsrelating the fine topology to limits of functions along parallellines. (Results of this nature for the minimal fine topologyhave been given by Doob – see [10, Theorem 3.1] or [11,1.XII.23] – and the second author [15].) In particular,we will establish improvements and generalizations of resultsof Lusin and Privalov [18], Evans [12], Rudin [20], Bagemihland Seidel [6], Schneider [21], Berman [7], and Armitage andNelson [4], and will also solve a problem posed by the latterauthors. An early version of our first result is due to Evans [12, p.234], who proved that, if u is a superharmonic function on R³,then there is a set ER²x{0}, of two-dimensional measure 0, suchthat u(x, y,·) is continuous on R whenever (x, y, 0)E.We denote a typical point of Rⁿ by X=(X' x), where X'R^n–1and xR. Let :RⁿR^n–1x{0} denote the projection map givenby (X', x) = (X', 0). For any function f:Rⁿ[–, +] andpoint X we define the vertical and fine cluster sets of f atX respectively by C_V(f;X)={l[–, +]: there is a sequence (t_m) of numbersin R\{x} such that t_mx and f(X', t_m)l}| and C_F(f;X)={l[–, +]: for each neighbourhood N of l in [–,+], the set f^–1(N) is non-thin at X}. Sets which are open in the fine topology will be called finelyopen, and functions which are continuous with respect to thefine topology will be called finely continuous. Corollary 1(ii)below is an improvement of Evans' result.     

Hypersurface Singularities with 2-Dimensional Critical Locus

Consider an analytic germ f:(C^m, 0)(C, 0) (m3) whose criticallocus is a 2-dimensional complete intersection with an isolatedsingularity (icis). We prove that the homotopy type of the Milnorfiber of f is a bouquet of spheres, provided that the extendedcodimension of the germ f is finite. This result generalizesthe cases when the dimension of the critical locus is zero [8],respectively one [12]. Notice that if the critical locus isnot an icis, then the Milnor fiber, in general, is not homotopicallyequivalent to a wedge of spheres. For example, the Milnor fiberof the germ f:(C⁴, 0)(C, 0), defined by f(x₁, x₂, x₃, x₄) =x₁x₂x₃x₄ has the homotopy type of S¹xS¹xS¹. On the other hand,the finiteness of the extended codimension seems to be the rightgeneralization of the isolated singularity condition; see forexample [9–12, 17, 18]. In the last few years different types of ‘bouquet theorems’have appeared. Some of them deal with germs f:(X, x)(C, 0) wheref defines an isolated singularity. In some cases, similarlyto the Milnor case [8], F has the homotopy type of a bouquetof (dim X–1)-spheres, for example when X is an icis [2],or X is a complete intersection [5]. Moreover, in [13] Siersmaproved that F has a bouquet decomposition FF₀Sⁿ...Sⁿ (whereF₀ is the complex link of (X, x)), provided that both (X, x)and f have an isolated singularity. Actually, Siersma conjecturedand Tibr proved [16] a more general bouquet theorem for thecase when (X, x) is a stratified space and f defines an isolatedsingularity (in the sense of the stratified spaces). In thiscase F_iF_i, where the F_i are repeated suspensions of complexlinks of strata of X. (If (X, x) has the ‘Milnor property’,then the result has been proved by Lê; for details see[6].) In our situation, the space-germ (X, x) is smooth, but f hasbig singular locus. Surprisingly, for dim Sing f^–1(0)2,the Milnor fiber is again a bouquet (actually, a bouquet ofspheres, maybe of different dimensions). This result is in thespirit of Siersma's paper [12], where dim Sing f^–1(0)= 1. In that case, there is only a rather small topologicalobstruction for the Milnor fiber to be homotopically equivalentto a bouquet of spheres (as explained in Corollary 2.4). Inthe present paper, we attack the dim Sing f^–1(0) = 2 case.In our investigation some results of Zaharia are crucial [17,18].     

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 Artin's Conjecture, I: Systems of Diagonal Forms

As a special case of a well-known conjecture of Artin, it isexpected that a system of R additive forms of degree k, say [formula] with integer coefficients a_ij, has a non-trivial solution inQ_p for all primes p whenever [formula] Here we adopt the convention that a solution of (1) is non-trivialif not all the x_i are 0. To date, this has been verified onlywhen R=1, by Davenport and Lewis [4], and for odd k when R=2,by Davenport and Lewis [7]. For larger values of R, and in particularwhen k is even, more severe conditions on N are required toassure the existence of p-adic solutions of (1) for all primesp. In another important contribution, Davenport and Lewis [6]showed that the conditions [formula] are sufficient. There have been a number of refinements of theseresults. Schmidt [13] obtained N>>R²k³ log k, and Low,Pitman and Wolff [10] improved the work of Davenport and Lewisby showing the weaker constraints [formula] to be sufficient for p-adic solubility of (1). A noticeable feature of these results is that for even k, onealways encounters a factor k³ log k, in spite of the expectedk² in (2). In this paper we show that one can reach the expectedorder of magnitude k². 1991 Mathematics Subject Classification11D72, 11D79.     

Bounding the number of stable homotopy types of a parametrized family of semi-algebraic sets defined by quadratic inequalities

We prove a nearly optimal bound on the number of stable homotopytypes occurring in a k-parameter semi-algebraic family of setsin R, each defined in terms of m quadratic inequalities. Ourbound is exponential in k and m, but polynomial in . More precisely,we prove the following. Let R be a real closed field and let = {P₁, ... , P_m} R[Y₁, ... ,Y,X₁, ... ,X_k], with deg_Y(P_i) 2, deg_X(P_i) d, 1 i m. Let S R^+k be a semi-algebraic set,defined by a Boolean formula without negations, with atoms ofthe form P 0, P 0, P . Let : R^+k R^k be the projection onthe last k coordinates. Then the number of stable homotopy typesamongst the fibers S_x = ^–1(x) S is bounded by (2^mkd)^O(mk).     

An Euler-Type Version of the Local Steiner Formula for Convex Bodies

The purpose of this note is to establish a new version of thelocal Steiner formula and to give an application to convex bodiesof constant width. This variant of the Steiner formula generalizesresults of Hann [3] and Hug [6], who use much less elementarytechniques than the methods of this paper. In fact, Hann askedfor a simpler proof of these results [4, Problem 2, p. 900].We remark that our formula can be considered as a Euclideananalogue of a spherical result proved in [2, p. 46], and thatour method can also be applied in hyperbolic space. For some remarks on related formulas in certain two-dimensionalMinkowski spaces, see Hann [5, p. 363]. For further information about the notions used below, we referto Schneider's book [9]. Let Kⁿ be the set of all convex bodiesin Euclidean space Rⁿ, that is, the set of all compact, convex,non-empty subsets of Rⁿ. Let S^n–1 be the unit sphere.For KKⁿ, let NorK be the set of all support elements of K, thatis, the pairs (x, u)RⁿxS^n–1 such that x is a boundarypoint of K and u is an outer unit normal vector of K at thepoint x. The support measures (or generalized curvature measures)of K, denoted by ₀(K.), ..., _n–1(K.), are the unique Borelmeasures on RⁿxS^n–1 that are concentrated on NorK andsatisfy [formula] for all integrable functions f:RⁿR; here denotes the Lebesguemeasure on Rⁿ. Equation (1), which is a consequence and a slightgeneralization of Theorem 4.2.1 in Schneider [9], is calledthe local Steiner formula. Our main result is the following.1991 Mathematics Subject Classification 52A20, 52A38, 52A55.     

Spaces of Harmonic Functions

It is important and interesting to study harmonic functionson a Riemannian manifold. In an earlier work of Li and Tam [21]it was demonstrated that the dimensions of various spaces ofbounded and positive harmonic functions are closely relatedto the number of ends of a manifold. For the linear space consistingof all harmonic functions of polynomial growth of degree atmost d on a complete Riemannian manifold Mⁿ of dimension n,denoted by H_d(Mⁿ), it was proved by Li and Tam [20] that thedimension of the space H₁(M) always satisfies dimH₁(M) dimH₁(Rⁿ)when M has non-negative Ricci curvature. They went on to askas a refinement of a conjecture of Yau [32] whether in generaldim H_d(Mⁿ) dimH_d(Rⁿ)for all d. Colding and Minicozzi made animportant contribution to this question in a sequence of papers[5–11] by showing among other things that dimH_d(M) isfinite when M has non-negative Ricci curvature. On the otherhand, in a very remarkable paper [16], Li produced an elegantand powerful argument to prove the following. Recall that Msatisfies a weak volume growth condition if, for some constantA and , (1.1) for all x M and r R, where V_x(r) is the volume of the geodesicball B_x(r) in M; M has mean value property if there exists aconstant B such that, for any non-negative subharmonic functionf on M, (1.2) for all p M and r > 0.     

Some Remarks on p-Adic Analytic Groups

A theorem of Maranda [1, Section 30] states that if F is a finitegroup, p is a prime and p^e exactly divides |F|, then a Z_pF-latticeM is determined up to isomorphism by its finite quotient M/p^e+1M.If M is a free Z_p-module of rank d, this is equivalent to sayingthat representations of F in GL_d(Z_p) are determined up to equivalenceby their images modulo p^e+1. 1991 Mathematics Subject Classification20E18, 22E20.     

Linear Independence of Hecke Operators in the Homology of X0(N)

In Merel's recent proof [7] of the uniform boundedness conjecturefor the torsion of elliptic curves over number fields, a keystep is to show that for sufficiently large primes N, the Heckeoperators T₁, T₂, ..., T_D are linearly independent in theiractions on the cycle e from 0 to i in H₁(X₀(N) (C), Q). In particular,he shows independence when max(D⁸, 400D⁴) < N/(log N)⁴. Inthis paper we use analytic techniques to show that one can chooseD considerably larger than this, provided that N is large.     

Geometric Approximation Of The Fibre Of The Freudenthal Suspension

Let Rat_k(CPⁿ) denote the space of based holomorphic maps ofdegree k from the Riemannian sphere S² to the complex projectivespace CPⁿ. The basepoint condition we assume is that f()=[1,..., 1]. Such holomorphic maps are given by rational functions: Rat_k(CPⁿ) ={(p₀(z), ..., p_n(z)):each p_i(z) is a monic, degree-kpolynomial and such that there are no roots common to all p_i(z)}.(1.1) The study of the topology of Rat_k(CPⁿ) originated in [10]. Later,the stable homotopy type of Rat_k(CPⁿ) was described in [3] interms of configuration spaces and Artin's braid groups. LetW(S²ⁿ) denote the homotopy theoretic fibre of the Freudenthalsuspension E:S²ⁿ S²ⁿ⁺¹. Then we have the following sequenceof fibrations: ²S²ⁿ⁺¹ W(S²ⁿ)S²ⁿ S²ⁿ⁺¹. A theorem in [10] tellsus that the inclusion Rat_k(CPⁿ) ²_kCP^{n 2}S²ⁿ⁺¹ is a homotopy equivalenceup to dimension k(2n–1). Thus if we form the direct limitRat(CPⁿ)= lim_k Rat_k(CPⁿ), we have, in particular, that Rat(CPⁿ)is homotopy equivalent to ²S²ⁿ⁺¹. If we take the results of [3] and [10] into account, we naturallyencounter the following problem: how to construct spaces X_k(CPⁿ),which are natural generalizations of Rat_k(CPⁿ), so that X(CPⁿ)approximates W(S²ⁿ). Moreover, we study the stable homotopytype of X_k(CPⁿ). The purpose of this paper is to give an answer to this problem.The results are stated after the following definition. 1991Mathematics Subject Classification 55P35.     

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.     

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.     

The Characterization of the Regularity of the Jacobian Determinant in the Framework of Bessel Potential Spaces on Domains

Let 2 m n. The paper gives necessary and sufficient conditionson the parameters s1, s2, ..., sm, p1, p2, ..., pm such thatthe Jacobian determinant extends to a bounded operator fromHs1p1 x Hs2p2 x ... x Hsmpm into S'. Here all spaces are definedon Rn or on domains Rn. In almost all cases the regularity ofthe Jacobian determinant is calculated exactly.     

Generalisation of the Bogomolov-Miyaoka-Yau Inequality to Singular Surfaces

The paper considers pairs (X, B) where X is a normal projectivesurface over C, and B is a Q-divisor whose coefficients are1 or 1–1/m for some natural number m. A log canonicalsingularity on such a pair is a quotient by a finite or infinitegroup, so if (X, B) has log canonical singularities, the orbifoldEuler number e_orb(X, B) can be defined. The main result is aBogomolov-Miyaoka-Yau-type inequality which implies that if(X, B) has log canonical singularities and (X, K_X + B) 0 then(K_X+B)² 3e_orb(X, B). The actual inequality proved is somewhatstronger and it also implies all the previously published versionsof the Bogomolov-Miyaoka-Yau inequality. The proof involvesthe Log Minimal Model Program, Q-sheaves when K_X+B is nef, anda study of the changes in the two sides of the inequality undera contraction. The paper also contains a further generalisationwhere the coefficients of B can be arbitrary rational numbersin [0, 1], a different condition is imposed on the singularitiesand K_X+B is required to be nef. Some applications of the inequalitiesare also given, for example, estimating the number of singularitiesor certain kinds of configurations of curves on surfaces. 1991Mathematics Subject Classification: 14J17, 14J60, 14C17.     

Embedding Spaces of Finite Length in R3

I refine a theorem from [3] to show that if (X, ) is any metricspace of finite length, it can be embedded in a compact connectedsubset of R³ of finite length in such a way as to preserve themeasure µ     

Harmonic Analogues of G. R. Maclane's Universal Functions

Let E denote the space of all entire functions, equipped withthe topology of local uniform convergence (the compact-opentopology). MacLane [15] constructed an entire function f whosesequence of derivatives (f, f', f', ...) is dense in E; hisconstruction is succinctly presented in a much later note byBlair and Rubel [2], who unwittingly rederived it (see also[3]). We shall call such a function f a universal entire function.In this note we show that analogous universal functions existin the space H_N of functions harmonic on R^N, where N2. We alsostudy the permissible growth rates of universal functions inH_N and show that the set of all such functions is very large. For purposes of comparison, we first review relevant facts aboutuniversal entire functions. The function constructed by MacLaneis of exponential type 1. Duyos Ruiz [7] observed that a universalentire function cannot be of exponential type less than 1. G.Herzog [11] refined MacLane's growth estimate by proving theexistence of a universal entire function f such that |f(z)|=O(re^r)as |z|=r. Finally, Grosse–Erdmann [10] proved the followingsharp result.