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.     

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.     

On Extensions of Myers' Theorem

Let M be a compact Riemannian manifold, and let h be a smoothfunction on M. Let p^h(x) = inf||–₁(Ric_x(,)–2Hess(h_x(,)).Here Ric_x denotes the Ricci curvature at x and Hess(h) is theHessian of h. Then M has finite fundamental group if ^h–p^h<0. Here ^h =:+2L_h is the Bismut-Witten Laplacian. This leadsto a quick proof of recent results on extension of Myers' theoremto manifolds with mostly positive curvature. There is also asimilar result for noncompact manifolds.     

Quasi-Affinity in certain Classes of Operators

The family of operators S + V (, C, Re > 0), where V isan injective S-Volterra operator (that is, [S, V[ = V²) and— A — V^–1 generates a uniformly bounded C₀-semigroup,is studied in the context of similarity and of the weaker quasi-affinityrelation. It is shown that S is similar to S + V for all , C,Re > 1, and is a quasi-affine transform of S + tV for allt 0 and 0 < < 1.     

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).     

Cardinal Invariants and Eventually Different Functions

In this paper we study several kinds of maximal almost disjointfamilies. In the main result of this paper we show that forsuccessor cardinals , there is an unexpected connection betweeninvariants a_e(), b() and a certain cardinal invariant m_d(⁺)on ⁺. As a corollary we get for example the following result.For a successor cardinal , even assuming that ^< = and 2= ⁺, the following is not provable in Zermelo–Fraenkelset theory. There is a ⁺-cc poset which does not collapse andwhich forces a() = ⁺ < a_e() = ⁺⁺ = 2. We also apply the ideasfrom the proofs of these results to study a = a() and non(M).2000 Mathematics Subject Classification 03E17 (primary), 03E05(secondary).     

Metric Entropy of Convex Hulls in Hilbert Spaces

We show in this note the following statement which is an improvementover a result of R. M. Dudley and which is also of independentinterest. Let X be a set of a Hilbert space with the propertythat there are constants , >0, and for each n N, the setX can be covered by at most n balls of radius n^–. Then,for each n N, the convex hull of X can be covered by 2ⁿ ballsof radius . The estimate is best possible for all n N, apart from the value c=c(, , X).In other words, let N(, X), >0, be the minimal number ofballs of radius covering the set X. Then the above result isequivalent to saying that if N(, X)=O(^–1/) as 0, thenfor the convex hull conv (X) of X, N(, conv (X)) =O(exp(^–2/(12))). Moreover, we give an interplay between several coveringparameters based on coverings by balls (entropy numbers) andcoverings by cylindrical sets (Kolmogorov numbers). 1991 MathematicsSubject Classification 41A46.     

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.     

On A Minimization Problem for Vector Fields in L1

This paper treats the problem of minimizing the norm of vectorfields in L¹ with prescribed divergence. The ridge of . playsan important role in the analysis, and in the case where R²is a polygonal domain, the ridge is thoroughly analysed andsome examples are presented. In the case where Rⁿ is a Lipschitzdomain and the divergence is a finite positive Borel measure,the infimum is calculated, and it is shown that if an extremalexists, then it is of the form ₁ = –Fd, where F is a nonnegativefunction and d(x) is the distance from x to the boundary .Finally, if R² is a polygonal domain and the measure is representedby a nonnegative continuous function, then an explicit expressionfor the extremal is given, and it is proven that this extremalis unique.     

Existence of Periodic Solutions in Nonlinear Asymmetric Oscillations

The existence of periodic solutions for the nonlinear asymmetricoscillator x' + x⁺ – rßx^– = h(t),(' =d/dt (is discussed, where , rß are positive constantssatisfying for some positive integer n N and h(t) L (0,2) is 2-periodic with x^±= max {±x,0}. 2000 Mathematics Subject Classification34C10, 34C25.     

A Comparison Estimate for the Heat Equation with an Application to the Heat Content of the S-Adic Von Koch Snowflake

Let D be an open set in Euclidean space R^m with boundary D,and let :D[0, ) be a bounded, measurable function. Let u:DDx[0,)[0, ) be the unique weak solution of the heat equation [formula] with initial condition [formula] and with inhomogeneous Dirichlet boundary condition [formula] Then u(x; t) represents the temperature at a point xD at timet if D has initial temperature 0, while the temperature at apoint xD is kept fixed at (x) for all t>0. We define thetotal heat content (or energy) in D at time t by [formula] In this paper we wish to examine the effect of imposing additionalcooling on some subset C on both u and E_D. 1991 MathematicsSubject Classification 35K05, 60J65, 28A80.     

On Approximately Midconvex Functions

A real-valued function f defined on an open, convex set D ofa real normed space is called (, )-midconvex if it satisfies The main result of the paper states that if f is locally boundedfrom above at a point of D and is (, )-midconvex, then it satisfiesthe convexity-type inequality where : [0, 1] R is a continuous function satisfying The particular case = 0 of this result is due to Ng and Nikodem(Proc. Amer. Math. Soc. 118 (1993) 103–108), while thespecialization = = 0 yields the theorem of Bernstein and Doetsch(Math. Ann. 76 (1915) 514–526). 2000 Mathematics SubjectClassification 26A51, 26B25.     

The Dirichlet problem by variational methods

Let ^N be a bounded open set and C( ). Assume that has an extensionC() such that H^–1().Then by the Riesz representation theorem there exists a unique

We show that u+ coincides with the Perron solutionof the Dirichlet problem

This extends recent results by Hildebrandt [Math. Nachr. 278(2005), 141–144] and Simader [Math. Nachr. 279 (2006),415–430], and also gives a possible answer to Hadamard'sobjection against Dirichlet's principle.     

Are All Uniform Algebras Amnm?

A Banach algebra a is AMNM if whenever a linear functional on a and a positive number satisfy |(ab)–(a)(b)|||a||·||b||for all a, b a, there is a multiplicative linear functional on a such that ||–||=o(1) as 0. K. Jarosz [1] asked whetherevery Banach algebra, or every uniform algebra, is AMNM. B.E. Johnson [3] studied the AMNM property and constructed a commutativesemisimple Banach algebra that is not AMNM. In this note weconstruct uniform algebras that are not AMNM. 1991 MathematicsSubject Classification 46J10.     

Partial Difference Sets with Paley Parameters

Partial difference sets with parameters (,k,,µ) = (,(– 1)/2, ( – 5)/4,( – 1)/4) are called Paleypartial difference sets. By using finite local rings, we constructa family of Paley PDSs for abelian p-groups with any given exponent.Furthermore, we prove some non-existence results on Paley PDSs.Using these results, we prove that Paley PDSs exist in a rank2 abelian group if and only if the group is isomorphic to Z_p^r x Z_{p ^r} where p is an odd prime.     

On the Existence of Extremals for the Sobolev Trace Embedding Theorem with Critical Exponent

In this paper, the existence problem is studied for extremalsof the Sobolev trace inequality W^1,p()L^p*(), where is a boundedsmooth domain in R^N, p*=p(N–1)/(N–p), is the criticalSobolev exponent, and 1 < p < N. 2000 Mathematics SubjectClassification 35J65 (primary), 35B33 (secondary).     

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.     

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.     

Majorations Uniformes de Normes D'Inverses Dans Les Algebres de Beurling

The Beurling algebras l¹(D,)(D=N,Z) that are semi-simple, withcompact Gelfand transform, are considered. The paper gives anecessary and sufficient condition (on ) such that l¹(D,) possessesa uniform quantitative version of Wiener's theorem in the sensethat there exists a function :]0,+[]0,+ such that, for everyinvertible element x in the unit ball of l¹(D,), we have ||x^–1||(r(x^–1)) r(x^–1) is the spectral radiusof x^–1.     

Cocroissance Des Groupes A Petite Simplification

Let be a group presented by e₁,...,e_m|r₁,...,r_k, L the freegroup generated by e₁,...,e_m, and N = Ker(L). Let c_n be thenumber of elements of length n in N. We know that c = lim sup(c_n)^1/n exists and that (2m–1) < c 2m – 1. ifN {1}. We prove that if the group satisfies a condition slightlyweaker than the small cancellation condition C^'() with <1/6, then c(2m–1) when the lengths of the relations r_itend to infinity. A consequence of this result is a theoremof Grigorchuk.