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1.
We prove that if WN, d is a Brownian sheet mapping to Rd and E is a set in (0, )N of Hausdorff dimensiongreater than , then for almost every rotation about a point x and translation x such that x(E) (0, )N, the set x(E) is such that almost surely W(E) containsinterior points. The techniques are adapted from Kahane andRosen and generalize to higher dimensional time and range.  相似文献   

2.
In 1985, Sárközy proved a conjecture of Erdösby showing that ) is never square-free for sufficiently large n. By applying a new estimateon exponential sums, we prove that this also holds for if dis not ‘too big’. Let 0 < < 1, p0 N. For m m0 and 1 k m satisfying |m– 2k| < m1 – , there is a prime p > p0 suchthat .  相似文献   

3.
A minimal surface of general type with pg(S) = 0 satisfies 1 K2 9, and it is known that the image of the bicanonical map is a surface for , whilst for , the bicanonical map is always a morphism. In this paper it is shown that is birationalif , and that the degree of is at most 2 if or By presenting two examples of surfaces S with and 8 and bicanonical map of degree 2, it is alsoshown that this result is sharp. The example with is, to our knowledge, a new example of a surfaceof general type with pg = 0. The degree of is also calculated for two other known surfacesof general type with pg = 0 and . In both cases, the bicanonical map turns out to be birational.  相似文献   

4.
Let (an)n0 be a sequence of complex numbers, and, for n0, let A number of results are proved relating the growth of the sequences(bn) and (cn) to that of (an). For example, given p0, if bn= O(np and for all > 0,then an=0 for all n > p. Also, given 0 < p < 1, then for all > 0 if and onlyif . It is further shown that, given rß > 1, if bn,cn=O(rßn), then an=O(n),where , thereby proving a conjecture of Chalendar, Kellay and Ransford. The principal ingredientsof the proogs are a Phragmén-Lindelöf theorem forentire functions of exponential type zero, and an estimate forthe expected value of e(X), where X is a Poisson random variable.2000 Mathematics Subject Classification 05A10 (primary), 30D15,46H05, 60E15 (secondary).  相似文献   

5.
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 = ek. 2000Mathematics Subject Classification 34K11 (primary); 34C10 (secondary).  相似文献   

6.
Let Fn be the free group of rank n with basis x1, x2, ..., xn,and let d(G) denote the minimal number of generators of thefinitely generated group G. Suppose that n d(G). There existsan exact sequence and wemay view the free abelian group as a right ZG-module by defining (rR')g = rg–1R' for allg G, where g–1 is any preimage of g under , and = (g–1)–1 r(g–1),the conjugate of r by g–1. We call the relation module of G associated with the presentation(1), and say that has ambient rank n. Furthermore, we call the group Fn/R' the free abelianizedextension of G associated with (1). 1991 Mathematics SubjectClassification 20F05, 20C07.  相似文献   

7.
Let C be a germ at O R2 of a real analytic plane curve, andCC its complexification; let Ct B be a fiber of a real smoothdeformation of C in the ball B = B(O,). The following inequalityis proved between the integrals of real curvature k of Ct andthose of Gaussian curvature K of : The sharpness of this inequality is proved in the case whereC is a real irreducible germ. Similar results are proved foran affine algebraic curve C R2 of degree d. 2000 MathematicsSubject Classification 14H20, 14H50, 53A04.  相似文献   

8.
Let K be a compact subset of Rn, 0 s n. Let , Ps denote s-dimensional packing premeasure andmeasure, respectively. We discuss in this paper the relationbetween and Ps. We prove:if , then ; and if , then for any > 0, there exists a compact subset F of K such that and Ps(F) Ps(K) – .1991 Mathematics Subject Classification 28A80, 28A78.  相似文献   

9.
Let L and be orthogonal complementary rational linear subspaces of En, and let = L Zn and $$\stackrel{\&macr;}{\Lambda}$$ = Zn be the sublatticesof the usual integer lattice Zn induced by L and . Then the determinants of and are equal. The samerelationship holds between the determinants of the lattices and obtained by orthogonal projection of Zn on to L and .  相似文献   

10.
Let K. denote the graded Koszul complex associated to the regularsequence (x0, ..., xn) in the graded polynomial ring A = k[x0,..., xn], |xi| = 1 for all i, over an arbitrary field k. Let denote the Koszul complex associated to another regular sequence of homogeneous elements(p0, ..., pn) in A. In [5] we have studied ranks of graded chaincomplex morphisms with the property f0 = id. Let k (respectively, 'k) denote the kernelof the Koszul differential d: Kk Kk–1 (respectively,), and let denote the restriction of fk. The main result wasthat Rank . 1991 MathematicsSubject Classification 13D25.  相似文献   

11.
Remarks on Maximal Operators Over Arbitrary Sets of Directions   总被引:1,自引:0,他引:1  
Throughout this paper, we shall let be a subset of [0, 1] havingcardinality N. We shall consider to be a set of slopes, andfor any s , we shall let es be the unit vector of slope s inR2. Then, following [7], we define the maximal operator on R2associated with the set by The history of the bounds obtained on is quite curious. The earliest study of relatedoperators was carried out by Cordoba [2]. He obtained a boundof C(1 + log N) on the L2 operator norm of the Kakeya maximaloperator over rectangles of length 1 and eccentricity N. Thisoperator is analogous to with However, for arbitrary sets, the best known result seems to be C(1 + log N). This followsfrom Lemma 5.1 in [1], but a point of view which produces aproof appears already in [8]. However, in this paper, we provethe following.  相似文献   

12.
A Radial Uniqueness Theorem for Sobolev Functions   总被引:1,自引:0,他引:1  
We show that continuous functions u in the Sobolev space , 1 < p n, which have the limitzero in a certain weak sense in a set of positive p-capacityon B with where B is the open unit ball of Rn and for 0 > > , are identically zero. Conversely, we produce for each 1 > p n and each positive a non-constant function u in , continuous in , and a compact set EB of positive p-capacity such that u = 0 in E and the aboveinequality holds with exponent p – l + .  相似文献   

13.
Let be a bounded connected open set in RN, N 2, and let –0be the Dirichlet Laplacian defined in L2(). Let > 0 be thesmallest eigenvalue of –, and let > 0 be its correspondingeigenfunction, normalized by ||||2 = 1. For sufficiently small>0 we let R() be a connected open subset of satisfying Let – 0 be the Dirichlet Laplacian on R(), and let >0and >0 be its ground state eigenvalue and ground state eigenfunction,respectively, normalized by ||||2=1. For functions f definedon , we let Sf denote the restriction of f to R(). For functionsg defined on R(), we let Tg be the extension of g to satisfying 1991 Mathematics SubjectClassification 47F05.  相似文献   

14.
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  相似文献   

15.
Let B = (Bt)t0 be standard Brownian motion started at zero.We prove for all c > 1and all stopping times for B satisfying E(r) < for somer > 1/2. This inequality is sharp, and equality is attainedat the stopping time whereu* = 1 + 1/ec(c – 1) and = (c – 1)/c for c >1, with Xt = |Bt| and St = max0rt|Br|. Likewise, we prove for all c > 1 and all stopping times for B satisfying E(r < for some r > 1/2. This inequalityis sharp, and equality is attained at the stopping time where v* = c/e(c – 1) and =(c – 1)/c for c > 1. These results contain and refinethe results on the L log L-inequality of Gilat [6] which areobtained by analytic methods. The method of proof used hereis probabilistic and is based upon solving the optimal stoppingproblem with the payoff whereF(x) equals either xlog+ x or x log x. This optimal stoppingproblem has some new interesting features, but in essence issolved by applying the principle of smooth fit and the maximalityprinciple. The results extend to the case when B starts at anygiven point (as well as to all non-negative submartingales).1991 Mathematics Subject Classification 60G40, 60J65, 60E15.  相似文献   

16.
Each lattice in Rd determines a sequence of Brillouin zonesBn, fundamental regions for bounded by Bragg hyperplanes; forexample B1 is the Dirichlet region. Basic geometric and topologicalproperties of these zones are established, and we obtain asymptoticestimates (valid for almost all ) for (n) = , where L(n) is the number of connected componentsof the interior of Bn (called Landsberg subzones). Fermi surfacesare also briefly described.  相似文献   

17.
In this paper, perturbations of the left and right essentialspectra of 2 x 2 upper triangular operator matrix MC are studied,where is an operator acting on the Hilbert space H K. For given operators A and B, thesets and are determined, where le(T) and re(T) denote, respectively,the left essential spectrum and the right essential spectrumof an operator T. 2000 Mathematics Subject Classification 47A10,47A55.  相似文献   

18.
For 1 k < and 1 p q , the problem of finding the bestconstant Cpq in the weighted inequality involving the Riemann-Liouville integrals of theform is considered.  相似文献   

19.
Generalized Catalan Numbers, Weyl Groups and Arrangements of Hyperplanes   总被引:1,自引:0,他引:1  
For an irreducible, crystallographic root system in a Euclideanspace V and a positive integer m, the arrangement of hyperplanesin V given by the affine equations (, x) = k, for and k =0, 1, ..., m, is denoted here by . The characteristic polynomial of is related in the paper to that of the Coxeter arrangement A(corresponding to m = 0), and the number of regions into whichthe fundamental chamber of A is dissected by the hyperplanesof is deduced to be equal to the product , where e1,e2, ..., el are the exponents of and h is the Coxeter number.A similar formula for the number of bounded regions follows.Applications to the enumeration of antichains in the root posetof are included. 2000 Mathematics Subject Classification 20F55(primary), 05A15, 52C35 (secondary).  相似文献   

20.
Soient F un corps commutatif localement compact non archimédienet un caractère additif non trivial de F. Soient unereprésentation du groupe de Weil–Deligne de F,et sa contragrédiente. Nous calculons le facteur (, , ). De manière analogue, nous calculons le facteur (x, , ) pour toute représentationadmissible irréductible de GLn(F). En conséquence,si F est de caractéristique nulle et si et se correspondentpar la correspondance de Langlands construite par M. Harris,ou celle construite par les auteurs, alors les facteurs (, , s) et (x, , s) sont égaux pour tout nombre complexe s. Let F be a non-Archimedean local field and a non-trivial additivecharacter of F. Let be a representation of the Weil–Delignegroup of F and its contragredient representation. We compute (, , ). Analogously, we compute (x, , ) for all irreducible admissible representations of GLn(F).Consequently, if F has characteristic zero, and , correspondvia the Langlands correspondence established by M. Harris orthe correspondence constructed by the authors, then we have(, , s) = (x, , s) for all sC. 1991 Mathematics Subject Classification22E50.  相似文献   

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