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.     

An Order of Convergence for Some Radial Basis Functions

The quasi-interpolant to a function f : RⁿR on an infinite regulargrid of spacing h can be defined by where : RⁿR is a function which decays quickly for large argument.In the case of radial basis functions has the form where : R⁺R is known as a radial basis function and, in general,?_j R (j = 1,...,m) and x^j Rⁿ (j = 1,...,m), though here onlythe particular case x^j Zⁿ (j = 1,..., m) is considered. Thispaper concentrates on the case (r) = r, a generalization oflinear interpolation, although some of the analysis is moregeneral. It is proved that, if n is odd, then there is a function such that the maximum difference between a sufficiently smoothfunction and its quasi-interpolant is bounded by a constantmultiple of hⁿ⁺¹. This is done by first showing that such aquasi-interpolation formula can reproduce polynomials of degreen.     

On hearing the shape of a bounded domain with Robin boundary conditions

The asymptotic expansions of the trace of the heat kernel (t)= [sum ]_j=1 exp(-t_j) for small positive t, where {_j} _j=1 arethe eigenvalues of the negative Laplacian -_n = -[sum ]ⁿ_k=1 (/x^k)²in Rⁿ (n = 2 or 3), are studied for a general multiply connectedbounded domain which is surrounded by simply connected boundeddomains _i with smooth boundaries _i (i = 1,...,m), where smoothfunctions Y_i (i = 1,...,m) are assuming the Robin boundary conditions(n_i + Y_i) = 0 on _i. Here /n_i denote differentiations along theinward-pointing normals to _i (i = 1,...,m). Some applicationsof an ideal gas enclosed in the multiply connected bounded containerwith Neumann or Robin boundary conditions are given.     

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 $$\stackrel{\¯}{\sigma}$$ 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$$\stackrel{\¯}{\sigma}$$, I$$\stackrel{\¯}{\sigma }$$, $$\stackrel{\¯}{\sigma}$$) 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 $$\stackrel{\¯}{\sigma }$$. Then : (A, I, ) (A^{$$\stackrel{\¯}{\sigma }$$}, I^{$$\stackrel{\¯}{\sigma}$$}, ^{$$\stackrel{\¯}{\sigma }$$} = (A^{$$\stackrel{\¯}{\sigma}$$m}, I^{$$\stackrel{\¯}{\sigma }$$m}, ^{$$\stackrel{\¯}{\sigma}$$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).     

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.     

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.     

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.     

Congruences De Sommes De Chiffres De Valeurs Polynomiales

Let m, g, q N with q 2 and (m, q – 1) = 1. For n N,denote by s_n(n) the sum of digits of n in the q-ary digitalexpansion. Given a polynomial f with integer coefficients, degreed 1, and such that f(N) N, it is shown that there exists C= C(f, m, q) > 0 such that for any g Z, and all large N, In the special case m = q = 2 and f(n)= n², the value C = 1/20 is admissible. 2000 Mathematics SubjectClassification 11B85 (primary), 11N37, 11N69 (secondary).     

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.     

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.     

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.     

The Baire Method for the Prescribed Singular Values Problem

The paper investigates the vectorial Dirichlet problem definedby S_j( u(x))=1,xnO; a.e., j=1,...,n, u(x)=\(x),\,x\in|O. end{cases}Here O is an open bounded subset of Rⁿ with boundary |O, and_j(A) (j=1,...,n) denote the singular values of the gradient u(x). The existence of solutions is established under one ofthe following assumptions: : O – Rⁿ is continuous on Oand locally contractive on O, or : |O – Rⁿ is contractiveon |O. This extends a result due to Dacorogna and Marcellini.The approach is based on the Baire category method developedearlier by the authors.     

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.     

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.     

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.     

Extremal Properties of Contraction Semigroups on Hilbert and Banach Spaces

Let f be a unit vector and T = {T(t) = e^tA: t 0} be a (C₀)contraction semigroup generated by A on a complex Hilbert spaceX. If |T(t)f,f| 1 as t then f is an eigenvector of A correspondingto a purely imaginary eigenvalue. If one allows X to be a Banachspace, the same situation can be considered by replacing T(t)f,fby (T(t)f) where is a unit vector in X^* dual to f. If |(T(t)f)| 1, as t , is f an eigenvector of A? The answer is sometimesyes and sometimes no.     

On Finite and InfiniteCk-A-Determinacy

Let f: (Rⁿ,0) (R^p,0) be a C map-germ. We define f to be finitely,or -, A-determined, if there exists an integer m such that allgerms g with j^mg(0) = j^mf(0), or if all germs g with the sameinfinite Taylor series as f, respectively, are A-equivalentto f. For any integer k, 0 k < , we can consider A' sC^kcounterpart (consisting of C^k diffeomorphisms) A^(k), and wecan define the notion of finite, or -,A^(k)-determinacy in asimilar manner. Consider the following conditions for a C germf: (a_k) f is -A^(k)-determined, (b_k) f is finitely A^(k)-determined,(t) , (g) there exists a representative f : U R^p defined on some neighbourhood U of 0 in Rⁿ such thatthe multigerm of f is stable at every finite set , and (g') every f' with j f'(0)=j f(0) satisfiescondition (g). We also define a technical condition which willimply condition (g) above. This condition is a collection ofp+1 Lojasiewicz inequalities which express that the multigermof f is stable at any finite set of points outside 0 and onlybecomes unstable at a finite rate when we approach 0. We willdenote this condition by (e). With this notation we prove thefollowing. For any C map germ f:(Rⁿ,0) (R^p,0) the conditions(e), (t), (g') and (a) are equivalent conditions. Moreover,each of these conditions is equivalent to any of (a_k) (p+1 k < , (b_k) (p+1 k < ). 1991 Mathematics Subject Classification:58C27.     

A Krylov subspace algorithm for multiquadric interpolation in many dimensions

We consider the version of multiquadric interpolation wherethe interpolation conditions are the equations s(x_i) = f_i, i= 1,2,..., n, and where the interpolant has the form s(x) =_j=1ⁿ _j (||x – x_j ||² + c²)^1/2 + x R^d, subject to theconstraint _j=1ⁿ _j = 0. The points x_i R^d, the right-hand sidesf_i, i = 1,2,...,n, and the constant c are data. The equationsand the constraint define the parameters _j, j = 1,2,...,n, and. The resultant approximation s f is useful in many applications,but the calculation of the parameters by direct methods requiresO (n³) operations, and n may be large. Therefore iterative proceduresfor this calculation have been studied at Cambridge since 1993,the main task of each iteration being the computation of s(x_i),i = 1,2,...,n, for trial values of the required parameters.These procedures are based on approximations to Lagrange functions,and often they perform very well. For example, ten iterationsusually provide enough accuracy in the case d = 2 and c = 0,for general positions of the data points, but the efficiencydeteriorates if d and c are increased. Convergence can be guaranteedby the inclusion of a Krylov subspace technique that employsthe native semi-norm of multiquadric functions. An algorithmof this kind is specified, its convergence is proved, and carefulattention is given to the choice of the operator that definesthe Krylov subspace, which is analogous to pre-conditioningin the conjugate gradient method. Finally, some numerical resultsare presented and discussed, for values of d and n from theintervals [2,40] and [200,10 000], respectively.     

The Set of {omega}-Limit Points of A Map of the Circle is Closed

Let f be a continuous self-map of the unit circle, S¹. The -limitpoints (x) of a point x are the set of all limit points of thesequence of iterates of f acting on x. We shall show that theset of all -limit points _xS¹(x) a closed set in S¹.     

Invariant Subspaces of C1.-Contractions with Non-Reductive Unitary Extensions

Let T be a contraction acting on the Hilbert space H such thatlim_n||Tⁿh||0, for every nonzero h;H. It is proved that if theunitary operator attached to T in a canonic way contains thebilateral shift, then T has a non-trivial invariant subspace.Furthermore, if in addition lim_n||T^*nh||0 holds for every nonzeroh H, then T is shown to be reflexive.