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.     

Packing, Tiling, Orthogonality and Completeness

Let R^d be an open set of measure 1. An open set DR^d is calleda ‘tight orthogonal packing region’ for if D–Ddoes not intersect the zeros of the Fourier transform of theindicator function of , and D has measure 1. Suppose that isa discrete subset of R^d. The main contribution of this paperis a new way of proving the following result: D tiles R^d whentranslated at the locations if and only if the set of exponentialsE = {exp 2i, x: } is an orthonormal basis for L²(). (This resulthas been proved by different methods by Lagarias, Reeds andWang [9] and, in the case of being the cube, by Iosevich andPedersen [3]. When is the unit cube in R^d, it is a tight orthogonalpacking region of itself.) In our approach, orthogonality ofE is viewed as a statement about ‘packing’ R^d withtranslates of a certain non-negative function and, additionally,we have completeness of E in L²() if and only if the above-mentionedpacking is in fact a tiling. We then formulate the tiling conditionin Fourier analytic language, and use this to prove our result.2000 Mathematics Subject Classification 52C22, 42B99, 11K70.     

Analytic Discs Attached From the Outside Have Knotted Boundaries

Let be a pseudoconvex domain in C² with smooth boundary, andlet be a smooth embedded analytic disc intersecting transversally along the curve A. Then A isknotted in . 2000 Mathematics Subject Classification 32U99.     

Correction: on the Intersection Matrices of Curves lying on Irregular Algebraic Surfaces

Professor W. F. Hammond has kindly drawn my attention to a blunderin 4 of the above paper. He referred to the ( – 2r) xß submatrix D of the skew-symmetric matrix displayednear the top of page 181, of which it is asserted that it issquare and non-singular, and pointed out that, from the factthat the matrix of which D forms part is regular, it may onlybe deduced that the columns of D are linearly independent; thatis, it only follows that – 2r ß. The validity of the equation – 2r = ß is essentialto the succeeding argument and, fortunately, may be establishedby alternative means. Using the nomenclature of the paper, wehave on F the set ₁^*, ..., _2r^*, ₁^*, ..., _ß^* of independent3-cycles (independent because they cut independent 1-cycleson the curve C), which may be completed, to form a basis forsuch cycles on F, by a further set ₁', ..., _2q–2r–pof independent 3-cycles, each of which meets C in a cycle homologousto zero on C. The cycles ₁^*, ..., ^* are invariant cycles andare independent on F so that, if > 2r + ß, thereis a non-trivial linear combination ^* of these having zero intersectionon C with each of the cycles ₁^*, ..., _2r^*, ₁^*, ..., _ß^*.Thus we have. (^* ._k^*)c = 0 = (^* ._i^*)c i.e. (^* ._k^*) = 0 = (^* ._i^* on F (1 k 2r; 1 i ß). Furthermore, (_j . C) 0 on C and we have (^* ._j .C)_C = 0 i.e. (^* ._j) = 0 on F (1 j 2q – 2r – ß). It now follows that ^* 0 on F (for it has zero intersectionwith every member of a basic set of 3-cycles on F). But thiscondradicts the assumption that ^* is a non-trivial linear combinationof the independent cycles ₁^*, ...,^*; and hence < 2r + ß.     

Realisations of Non v-Sufficient Jets with value in Rp, p > 1

The purpose of this paper is to show that if a jet cue oJ^r(n,p), n p, p > 1, is not v-sufficient in C^r+1, there existsan infinite sequence (f_i)_iN^* of realisations of o with mutuallynon-homeomorphic germs of varieties . Bochnak and Kuo [2, 5] showed it when p = 1 and thought thatthe same argument slightly modified can be used in the casep 2 [7, p. 225]. But when n p + 2, p > 1, we have to proceeddifferently. Moreover, it is necessary to prove separately theresult when n = p and n = p + 1. About C⁰-sufriciency and p> 1, Brodersen [3, p. 168] showed a similar theorem.     

Singular Integrals and Maximal Functions Associated to Surfaces of Revolution

We obtain L^p estimates for singular integrals and maximal functionsassociated to hypersurfaces in Rⁿ⁺¹, n 2, which are obtainedby rotating a curve around one of the coordinate axes.     

Anzai Skew Products with Lebesgue Component of Infinite Multiplicity

Let f be a 1-periodic C¹-function whose Fourier coefficientssatisfy the condition _n|n|³|f(n|² < . For every R\Q andm Z\{0}, we consider the Anzai skew product T(x, y) = (x +, y + mx + f(x)) acting on the 2-torus. It is shown that T hasinfinite Lebesgue spectrum on the orthocomplement L²(dx) ofthe space of functions depending only on the first variable.This extends some earlier results of Kushnirenko, Choe, Lemaczyk,Rudolph, and the author. 1991 Mathematics Subject Classification28D05.     

Finite groups in Stone-Cech compactifications

Let be an infinite cardinal and let G = ₂. Now let β Gbe the Stone–ech compactification of G as a discrete semigroup,and let =_<c_{β G} {xG\{0}:minsupp (x)}. We show that thesemigroup contains no nontrivial finite group.     

Some Relations Between Packing Premeasure and Packing Measure

Let K be a compact subset of Rⁿ, 0 s n. Let , P^s denote s-dimensional packing premeasure andmeasure, respectively. We discuss in this paper the relationbetween and P^s. We prove:if , then ; and if , then for any > 0, there exists a compact subset F of K such that and P^s(F) P^s(K) – .1991 Mathematics Subject Classification 28A80, 28A78.     

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.     

Normal families and omitted functions II

Let k 2 be an integer and let be a family of functions meromorphicon a domain D in , all of whose poles are multiple and whosezeros all have multiplicity at least k + 1. Let h be a functionmeromorphic on D, h 0, . Suppose that for each f , f^(k)(z) h(z) for z D. Then is a normal family on D.     

Coincidence and The Colouring of Maps

In [8, 6] it was shown that for each k and n such that 2k >n, there exists a contractible k-dimensional complex Y and acontinuous map : Sⁿ Y without the antipodal coincidence property,that is, (x)(–x) for all x Sⁿ. In this paper it is shownthat for each k and n such that 2k > n, and for each fixed-pointfree homeomorphism f of an n-dimensional paracompact Hausdorffspace X onto itself, there is a contractible k-dimensional complexY and a continuous map :X Y such that (x)(f(x)) for all xX.Various results along these lines are obtained. 1991 MathematicsSubject Classication 55M10, 54C05.     

Trivial centralizers for codimension-one attractors

We show that if is a codimension-one hyperbolic attractor fora C^r diffeomorphism f, where 2 r , and f is not Anosov, thenthere is a neighborhood of f in Diff^r(M) and an open and denseset of such that any g has a trivial centralizer on thebasin of attraction for .     

Cluster Sets of Harmonic Functions at the Boundary of a Half-Space

The purpose of this paper is to answer some questions posedby Doob [2] in 1965 concerning the boundary cluster sets ofharmonic and superharmonic functions on the half-space D givenby D = R^n–1 x (0, + ), where n 2. Let f: D [–,+] and let Z D. Following Doob, we write B_Z (respectively C_Z)for the non-tangential (respectively minimal fine) cluster setof f at Z. Thus l B_Z if and only if there is a sequence (X_m)of points in D which approaches Z non-tangentially and satisfiesf(X_m) l. Also, l C_Z if and only if there is a subset E ofD which is not minimally thin at Z with respect to D, and whichsatisfies f(X) l as X Z along E. (We refer to the book byDoob [3, 1.XII] for an account of the minimal fine topology.In particular, the latter equivalence may be found in [3, 1.XII.16].)If f is superharmonic on D, then (see [2, 6]) both sets B_Z andC_Z are subintervals of [–, +]. Let denote (n –1)-dimensional measure on D. The following results are due toDoob [2, Theorem 6.1 and p. 123]. 1991 Mathematics Subject Classification31B25.     

A Combinatorial Application of the Maximal Ergodic Theorem

Let X be a compact space,µ a Borel probability measureon X, T: X X a measure preserving continuous transformationand g: X R a continuous function. Then for some yX, This Lemma is used to give an alternative proof of a resultby Ruzsa [6], which implies the following extension of a resultof Bergelson [1]. If E N satisfies then there exists a set N such that n–1|[1,n]| (E) for all, n 1, and any finite subset{₁, ... _k} satisfies Ø. 7 Moria St., Ramat Hasharon, Israel     

The Representation of Some Integers as a Subset Sum

Let A N. The cardinality (the sum of the elements) of A willbe denoted by |A| ((A)). Let m N and p be a prime. Let A {1, 2,...,p}. We prove thefollowing results. If |A| [(p+m–2)/m]+m, then for every integer x such that0 x p – 1, there is B A such that |B| = m and (B) x mod p. Moreover, the bound is attained. If |A| [(p+m–2)/m]+m!, then there is B A such that |B| 0 mod m and (B) = (m – 1)!p. If |A| [(p + 1)/3]+29, then for every even integer x such that4p s x p(p + 170)/48, there is S A such that x = (S). In particular,for every even integer a 2 such that p 192a – 170, thereare an integer j 0 and S A such that (S) = a^j+1.     

Reduction modulo p of cuspidal representations and weights in Serre's conjecture

Let be the ring of integers of a p-adic field and its maximalideal. This paper computes the Jordan–Hölder decompositionof the reduction modulo p of the cuspidal representations ofGL₂(/^e) for e 1. An alternative formulation of Serre's conjecturefor Hilbert modular forms is then provided.     

Choquet Integrals, Hausdorff Content and the Hardy-Littlewood Maximal Operator

Using the BMO-H¹ duality (among other things), D. R. Adams provedin [1] the strong type inequality whereC is some positive constant independent of f. Here M is theHardy–Littlewood maximal operator in Rⁿ, H is the -dimensionalHausdorff content, and the integrals are taken in the Choquetsense. The Choquet integral of 0 with respect to a set functionC is defined by Precise definitionsof M and H will be given below. For an application of (1) tothe Sobolev space W^{1, 1} (Rⁿ), see [1, p. 114]. The purpose of this note is to provide a self-contained, directproof of a result more general than (1). 1991 Mathematics SubjectClassification 28A12, 28A25, 42B25.     

An Upper Bound for the Hyperbolic Metric of a Convex Domain

In [1], Beardon introduced the Apollonian metric defined forany domain D in Rⁿ by This metric is Möbius invariant, and for simply connectedplane domains it satisfies the inequality _D2_D, where _D denotesthe hyperbolic distance in D, and so gives a lower bound onthe hyperbolic distance. Furthermore, it is shown in [1, Theorem6.1] that for convex plane domains, the Apollonian metric satisfies, and, by considering the example of the infinite strip {x + iy:|y|<1}, that the best possibleconstant in this inequality is at least . In this paper we makethe following improvements.     

Oscillation Criteria for First-Order Delay Equations

This paper is concerned with the oscillatory behaviour of first-orderdelay differential equations of the form (1) where is non-decreasing, (t)< t for t t₀ and . Let the numbers k andL be defined by It is proved here that when L < 1 and 0 < k 1/e all solutionsof equation (1) oscillate in several cases in which the condition holds, where ₁ is the smaller root of the equation = e^k. 2000Mathematics Subject Classification 34K11 (primary); 34C10 (secondary).