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.     

Immanant Inequalities, Induced Characters, and Rank Two Partitions

If = {₁, ₂, ..., _s}, where _{1 2} ... _s > 0, is a partitionof n then denotes the associated irreducible character of S_n,the symmetric group on {1, 2, ..., n}, and, if cCS_n, the groupalgebra generated by C and S_n, then d_c(·) denotes thegeneralized matrix function associated with c. If c₁, c₂ CS_nthen we write c₁ c₂ in case (A) (A) for each n x n positivesemi-definite Hermitian matrix A. If cCS_n and c(e) 0, wheree denotes the identity in S_n, then or denotes (c(e))^–1 c. The main result, an estimate for the norms of tensors of a certainanti-symmetry type, implies that if = {₁, ₂, ..., _s, 1^t} isa partition of n such that s > 1 and _s = 2, and ' denotes{₁, ₂, ..., _s-1, 1^t} then (, _{2}) where denotes characterinduction from S_n–2 x S₂ to S_n. This in turn implies thatif = {₁, ₂, ..., _s, 1^t} with s > 1, _s = 2, and ßdenotes {₁ + 2, ₂, ..., _s-1, 1^t} then _ß which,in conjunction with other known results, provides many new inequalitiesamong immanants. In particular it implies that the permanentfunction dominates all normalized immanants whose associatedpartitions are of rank 2, a result which has proved elusivefor some years. We also consider the non-relationship problem for immanants– that is the problem of identifying pairs, (,ß)such that _ß and _ß are both false.     

Controllability of a system of parabolic equations with non-diagonal diffusion matrix

In this paper we give a necessary and sufficient algebraic conditionfor the approximate controllability of the following systemof parabolic equations with Dirichlet boundary condition: {z_t = D z + b₁(x)u₁ + ··· + b_m(x)u_m, t 0, z ⁿ, z = 0, on where is a sufficiently smooth bounded domain in ^N, b_i L²(;ⁿ), the control functions u_i L²(0, t₁; ); i = 1, 2, ..., mand D is an n x n non-diagonal matrix whose eigenvalues aresemi-simple with positive real part. This algebraic conditionis checkable since it is given in terms of the n_j x m matricesDP_j and P_jB, i.e. Rank [P_jBDP_jBD²P_jB··· D^n_j–1 P_jB]= n_j, where P_jBu = P_jb₁u₁ + ··· + P_jb_mu_m. Finally,this result can be applied to those systems of partial differentialequations that can be rewritten as a diffusion system (see deOliveira, 1998).     

A Necessary Condition for Transitivity of a Finite Permutation Group

Suppose the group G is generated by permutations g₁, g₂, ...,g₈ acting on a set of size n, such that g₁g₂...g₈ is the identitypermutation. If the generator g_i has exactly c_i cycles (for1 i s), and G is transitive on , then n(s–2)– is a non-negative even integer. Thisis proved using an elementary graph-theoretic argument.     

Long-Time Behavior of Solutions of the Fast Diffusion Equations with Critical Absorption Terms

This paper is devoted to the long-time behavior of solutionsto the Cauchy problem of the porous medium equation u_t = (u^m)– u^p in Rⁿ x (0,) with (1 – 2/n)₊ < m < 1and the critical exponent p = m + 2/n. For the strictly positiveinitial data u(x,0) = O(1 + |x|)^–k with n + mn(2 –n + nm)/(2[2 – m + mn(1 – m)]) k < 2/(1 –m), we prove that the solution of the above Cauchy problem convergesto a fundamental solution of u_t = (u^m) with an additional logarithmicanomalous decay exponent in time as t .     

A characterization of finite soluble groups

Let G be a finite soluble group of order m and let w(x₁, ...,x_n) be a group word. Then the probability that w(g₁, ..., g_n)= 1 (where (g₁, ..., g_n) is a random n-tuple in G) is at leastp^–(m–t), where p is the largest prime divisor ofm and t is the number of distinct primes dividing m. This contrastswith the case of a non-soluble group G, for which Abérthas shown that the corresponding probability can take arbitrarilysmall positive values as n .     

The Characterization of the Regularity of the Jacobian Determinant in the Framework of Bessel Potential Spaces on Domains

Let 2 m n. The paper gives necessary and sufficient conditionson the parameters s1, s2, ..., sm, p1, p2, ..., pm such thatthe Jacobian determinant extends to a bounded operator fromHs1p1 x Hs2p2 x ... x Hsmpm into S'. Here all spaces are definedon Rn or on domains Rn. In almost all cases the regularity ofthe Jacobian determinant is calculated exactly.     

On the Algorithmic Complexity of Minkowski's Reconstruction Theorem

In 1903 Minkowski showed that, given pairwise different unitvectors µ₁, ..., µ_m in Euclidean n-space Rⁿ whichspan Rⁿ, and positive reals µ₁, ..., µ_m such that^m_i=1µ_iµ_i = 0, there exists a polytope P in Rⁿ, uniqueup to translation, with outer unit facet normals µ₁, ...,µ_m and corresponding facet volumes µ₁, ..., µ_m.This paper deals with the computational complexity of the underlyingreconstruction problem, to determine a presentation of P asthe intersection of its facet halfspaces. After a natural reformulationthat reflects the fact that the binary Turing-machine modelof computation is employed, it is shown that this reconstructionproblem can be solved in polynomial time when the dimensionis fixed but is #P-hard when the dimension is part of the input. The problem of ‘Minkowski reconstruction’ has variousapplications in image processing, and the underlying data structureis relevant for other algorithmic questions in computationalconvexity.     

The Characterisation of the Regularity of the Jacobian Determinant in the Framework of Potential Spaces

We give necessary and sufficient conditions on the parameterss₁, s₂, ..., s_m, p₁, p₂, ..., p_m such that the Jacobian determinantextends to a bounded operator from into Z'. Here all spaces are defined on Rⁿ and 2mn. In almostall cases the regularity of the Jacobian determinant is calculatedexactly.     

SIMULTANEOUS DIOPHANTINE APPROXIMATION BY SQUARE-FREE NUMBERS

Let ₁, ...,_r R be ‘not very well approximable’,for example, Q-linearly independent real algebraic numbers.Then there are infinitely many positive square-free integersn such that ||n_i|| << n^–(2/3r)+(1 i r), where||·|| denotes distance to the nearest integer.     

The Direct Limits Of The Banach-mazur Compacta

Let 1 p . For each n-dimensional Banach space E = (E, || ·||), we define a norm || · ||_p on E x R as follows: [formula] It is shown that the correspondence (E, || · ||) (Ex R, || · ||_p) defines a topological embedding of oneBanach–Mazur compactum, BM(n), into another, BM(n 1),and hence we obtain a tower of Banach–Mazur compacta:BM(1) BM(2) BM(3) ···. Let BM_p be thedirect limit of this tower. We prove that BM_p is homeomorphicto Q = dir lim Qⁿ, where Q = [0, 1] is the Hilbert cube. 1991Mathematics Subject Classification 46B04, 46B20, 52A21, 57N20,54H15.     

Universal Operator Algebras Associated to Contractive Sequences of Non-Commuting Operators

We characterize the minimal isometric dilation of a non-commutativecontractive sequence of operators as a universal object forcertain diagrams of completely positive maps. A non-spatialconstruction of the minimal isometric dilation is also given,using Hilbert modules over C*-algebras. It is shown that the non-commutative disc algebras A_n (n2) arethe universal algebras generated by contractive sequences ofoperators and the identity, and C*(S₁, ..., S_n) (n2), the extensionthrough compact operators of the Cuntz algebra O_n, is the universalC*-algebra generated by a contractive sequence of isometries.It is also shown that the algebras A_n and C*(S₁, ..., S_n) arecompletely isometrically isomorphic to some free operator algebrasconsidered by D. Blecher. In particular, the universal operatoralgebra of a row (respectively column) contraction is identifiedwith a subalgebra of C*(S₁, ..., S_n). The internal characterizationof the matrix norm on a universal algebra leads to some factorizationtheorems.     

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.     

Optimal Stopping in the L log L-Inequality of Hardy and Littlewood

Let B = (B_t)_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/e^c(c – 1) and = (c – 1)/c for c >1, with X_t = |B_t| and S_t = max_0rt|B_r|. 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.     

On the Structure of Specht Modules

Most of our notation is taken from James [7], where furtherdetails of the representation theory of the symmetric groupsmay be found; note, however, that we write functions on theleft. Let n be a non-negative integer, and a partition of n. Saythat two -tableaux are row equivalent if one can be obtainedfrom the other by permuting the entries within each row, anddefine column equivalence similarly. Let _row and _col denotethese relations.     

Minimal vanishing sums of roots of unity with large coefficients

A vanishing sum , where_n is a primitive nth root of unity and the a_is are non-negativeintegers is called minimal if the coefficient vector (a₀, ..., a_n–1) does not properly dominate the coefficient vectorof any other such non-zero sum. We show that for every c thereis a minimal vanishing sum of nth roots of unity with its greatestcoefficient equal to c, where n is of the form 3pq for odd primesp, q. This solves an open problem posed by Lenstra Jr.     

Central L-Values and Toric Periods for GL(2)

Let be a cusp form on GL(2) over a number field F and let Ebe a quadratic extension of F. Denote by _E the base change of to E and by a unitary character of A^x_E/ E^x. We use the relativetrace formula to give an explicit formula for L(1/2, _E ) interms of period integrals of Gross–Prasad test vectors.We give an application of this formula to equidistribution ofgeodesics on a hyperbolic 3-fold.     

Reaping Numbers of Boolean Algebras

A subset A of a Boolean algebra B is said to be (n,m)-reapedif there is a partition of unity p B of size n such that |{b p:b a 0}| m for all a A. The reaping number r_n,m (B) ofa Boolean algebra B is the minimum cardinality of a set A B\{0}which cannot be (n,m)-reaped. It is shown that for each n, thereis a Boolean algebra B such that r_n+1,2(B) r_n,2(B). Also, {r_n,m(B):mn } consists of at most two consecutive cardinals. The existenceof a Boolean algebra B such that r_n,m (B) r_n',m' (B) is equivalentto a statement in finite combinatorics which is also discussed.     

Rado Partition Theorem for Random Subsets of Integers

For an l x k matrix A = (a_ij) of integers, denote by L(A) thesystem of homogenous linear equations a_i1x₁ + ... + a_ikx_k =0, 1 i l. We say that A is density regular if every subsetof N with positive density, contains a solution to L(A). Fora density regular l x k matrix A, an integer r and a set ofintegers F, we write if for any partition F = F₁ ... F_r there exists i {1, 2,..., r} and a column vector x such that Ax = 0 and all entriesof x belong to F_i. Let [n]_N be a random N-element subset of{1, 2, ..., n} chosen uniformly from among all such subsets.In this paper we determine for every density regular matrixA a parameter = (A) such that lim_n P([n]_N (A)_r)=0 if N =O(n) and 1 if N = (n). 1991 Mathematics Subject Classification:05D10, 11B25, 60C05     

Properties of the {ell}1-solutions of the Linear Matrix Equation AX+YB=C

The properties of the ₁ of the linear matrix equation AX+YB=Care investigated, where A, B, and C are given real matricesof dimensions m x r, s x n, and m x n, respectively, with m> r and n > s. An algorithm which is an ₁version of thegeneral alternating method is developed. This algorithm utilizesa special form of the equation AX+YB=C and adjusts X and Y alternately.It gives an ₁-solution of AX+YB=C under additional assumptions.Some numerical examples are given.