A Relation between Hausdorff Dimension and a Condition on Time Sets for the Image by the Brownian Sheet to Possess Interior Points

We prove that if W_{N, d} is a Brownian sheet mapping to R^d 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.     

On the ideals and singularities of secant varieties of Segre varieties

We find minimal generators for the ideals of secant varietiesof Segre varieties in the cases of _k(¹ x ⁿ x ^m) for all k, n,m, ₂(ⁿ x ^m x ^p x ^r) for all n, m, p, r (GSS conjecture for fourfactors), and ₃(ⁿ x ^m x ^p) for all n, m, p and prove they arenormal with rational singularities in the first case and arithmeticallyCohen–Macaulay in the second two cases.     

Relation Modules of Infinite Groups

Let F_n be the free group of rank n with basis x₁, x₂, ..., x_n,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 = r^g–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 F_n/R' the free abelianizedextension of G associated with (1). 1991 Mathematics SubjectClassification 20F05, 20C07.     

Determinants of Lattices induced by Rational Subspaces

Let L and be orthogonal complementary rational linear subspaces of Eⁿ, and let = L Zⁿ and $$\stackrel{\¯}{\Lambda}$$ = 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 .     

Approximation of Ground State Eigenvalues and Eigenfunctions of Dirichlet Laplacians

Let be a bounded connected open set in R^N, 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.     

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).     

The Bicanonical Map of Surfaces with pg = 0 and K2 >= 7

A minimal surface of general type with p_g(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 p_g = 0. The degree of is also calculated for two other known surfacesof general type with p_g = 0 and . In both cases, the bicanonical map turns out to be birational.     

Correction: Abelian Varieties defined over their Fields of Moduli, I

Bull London Math. Soc, 4 (1972), 370–372. The proof of the theorem contains an error. Before giving acorrect proof, we state two lemmas. LEMMA 1. Let K/k be a cyclic Galois extension of degree m, let generate Gal (K/k), and let (A, I, ) be defined over K. Supposethat there exists an isomorphism :(A,I,) (A, I, ) over K suchthat v^m–1 ... = 1, where v is the canonical isomorphism(A^m, I^m, ^m) (A, I, ). Then (A, I, ) has a model over k, whichbecomes isomorphic to (A, I, ) over K. Proof. This follows easily from [7], as is essentially explainedon p. 371. LEMMA 2. Let G be an abelian pro-finite group and let : G Q/Z be a continuous character of G whose image has order p.Then either: (a) there exist subgroups G' and H of G such that H is cyclicof order p^m for some m, (G') = 0, and G = G' x H, or (b) for any m > 0 there exists a continuous character m ofG such that p^m _m = . Proof. If (b) is false for a given m, then there exists an element G, of order p^r for some r m, such that () ¦ 0. (Considerthe sequence dual to 0 Ker (p^m) G ^pm G). There exists an opensubgroup G_o of G such that (G₀) = 0 and has order p^r in G/G₀.Choose H to be the subgroup of G generated by , and then aneasy application to G/G₀ of the theory of finite abelian groupsshows the existence of G' (note that () ¦ 0 implies that is not a p-th. power in G). We now prove the theorem. The proof is correct up to the statement(iv) (except that (i) should read: F' k₁ F'ab). To removea minor ambiguity in the proof of (iv), choose to be an elementof Gal (F'_ab/k₂) whose image $$\stackrel{\&macr;}{\sigma}$$ in Gal (k₁/k₂) generates this last group. The error occursin the statement that the canonical map v : A^P A acts on pointsby sending a^p a; it, of course, sends a a. The proof is correct, however, in the case that it is possibleto choose so that ^p = 1 (in Gal (F'/k₂)). By applying Lemma 2 to G = Gal (F'_ab/k₂) and the map G Gal(k₁/k₂) one sees that only the following two cases have to beconsidered. (a) It is possible to choose so that ^pm = 1, for some m, andG = G' x H where G' acts trivially on k₁ and H is generatedby . (b) For any m > 0 there exists a field K, F'ab K k₁ k₂is a cyclic Galois extension of degree p^m. In the first case, we let K F'_ab be the fixed field of G'.Then (A, I, ), regarded as being defined over K, has a modelover k₂. Indeed, if m = 1, then this was observed above, butwhen m > 1 the same argument applies. In the second case, let : (A, I, ) (A$$\stackrel{\&macr;}{\sigma}$$, I$$\stackrel{\&macr;}{\sigma }$$, $$\stackrel{\&macr;}{\sigma}$$) be an isomorphism defined over k₁ and let v ... ^p–1 = µ(R). If is replaced by for some Aut_k1((A, I, )) then is replacedby ^P. Thus, as µ(R) is finite, we may assume that ^pm–1= 1 for some m. Choose K, as in (b), to be of degree p^m overk₂. Let _m be a generator of Gal (K/k₂) whose restriction tok₁ is $$\stackrel{\&macr;}{\sigma }$$. Then : (A, I, ) (A^{$$\stackrel{\&macr;}{\sigma }$$}, I^{$$\stackrel{\&macr;}{\sigma}$$}, ^{$$\stackrel{\&macr;}{\sigma }$$} = (A^{$$\stackrel{\&macr;}{\sigma}$$m}, I^{$$\stackrel{\&macr;}{\sigma }$$m}, ^{$$\stackrel{\&macr;}{\sigma}$$m} is an isomorphism defined over K and v ^mpm–1, ... ^m =^pm–1 = 1, and so, by) Lemma 1, (A, I, ) has a model overk₂ which becomes isomorphic to (A, I, over K. The proof may now be completed as before. Addendum: Professor Shimura has pointed out to me that the claimon lines 25 and 26 of p. 371, viz that µ(R) is a puresubgroup of _R^*t, does not hold for all rings R. Thus this condition,which appears to be essential for the validity of the theorem,should be included in the hypotheses. It holds, for example,if µ(R) is a direct summand of µ(F).     

Geometric and Asymptotic Properties of Brillouin Zones in Lattices

Each lattice in R^d determines a sequence of Brillouin zonesB_n, 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 B_n (called Landsberg subzones). Fermi surfacesare also briefly described.     

Logarithmic Convexity for Supremum Norms of Harmonic Functions

We prove the following convexity property for supremum normsof harmonic functions. Let be a domain in Rⁿ, ₀ and E a subdomainand a compact sebset of ,respectively. Then there exists a constant = (E, ₀, ) (0, 1) such that for all harmonic functions u on, the inequality is valid.The case of concentric balls E plays a key role in the proof.For positive harmonic funcitons ono osuch balls, we determinethe sharp constant in the inequlity.     

On Prime Divisors of Binomial Coefficients

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| < m^{1 –} , there is a prime p > p₀ suchthat .     

A problem of Hirst on continued fractions with sequences of partial quotients

Let B denote an infinite sequence of positive integers b₁ <b₂ < ..., and let denote the exponent of convergence ofthe series _{n = 1} 1/b_n; that is, = inf {s 0 : _{n = 1} 1/b_n^s <}. Define E(B) = {x [0, 1]: a_n(x) B (n 1) and a_n(x) asn }. K. E. Hirst [Proc. Amer. Math. Soc. 38 (1973) 221–227]proved the inequality dim_H E(B) /2 and conjectured (see ibid.,p. 225 and [T. W. Cusick, Quart. J. Math. Oxford (2) 41 (1990)p. 278]) that equality holds. In this paper, we give a positiveanswer to this conjecture.     

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     

On the 'pits effect' of Littlewood and Offord

Asymptotic behaviour of the entire functions , with real _n is studied. It turns out that the Phragmén–Lindelöfindicator of such a function is always non-negative, unlessf(z)=e^az. For a special choice of _n= n² with irrational , theindicator is constant and f has completely regular growth inthe sense of Levin and Pfluger. Similar functions of arbitraryorder are also considered.     

A Radial Uniqueness Theorem for Sobolev Functions

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 + .     

Singular Cardinal Problem: Shelah's Theorem on 2N{omega}

This is an expository paper giving a complete proof of a theoremof Saharon Shelah: if 2 < for all n < , then 2 < ₄.     

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.     

Local Limitations of the Ext Functor Do Not Exist

In this note it is shown that for k a field, and for the four-dimensionalalgebra = kx, y/x², y², xy + qyx when qⁿ 1, 0 for all n, thereexist a two-dimensional module M and a family of two-dimensionalmodules M_i, i = 1, 2, ..., such that for i equal to 0, j and j + 1, and otherwise. This is probably the most straightforward examplegiving a negative answer to a question raised by Maurice Auslander.2000 Mathematics Subject Classification 16D10, 16E10, 16E30,16G10, 16G20.     

Outer Automorphisms on Cuntz Algebras

We prove that if an automorphism on the Cuntz algebra O_n satisfies(S₁) = S₁ for one isometry S₁ of generators S₁, ..., S_n suchthat , , l i n and a complex number with modulus one,then it is outer. In particular, any non trivial automorphismon O_n leaving one generator fixed is outer.     

On The Curvature of the Real Milnor Fiber

Let C be a germ at O R² of a real analytic plane curve, andC^C its complexification; let C_t 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 C_t 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.