Le théorème ergodique pour les opérateurs positifs sur les espaces Lp (1<p<∞) revisitéErgodic theorem for positive operators on Lp-spaces Lp (1<p<∞) revisited

We consider positive linear operators on L_p-spaces (1<p<∞), (A(L_p⁺)?L_p⁺), satisfying the inequality A^m+n<A^m+Aⁿ for all $\text{m,n∈}\text{N}\text{.}$ We describe the structure of these operators (Theorem 1). As a consequence we obtain for all f∈L^p,Aⁿf converges a.e. The last statement contains the theorem of a.e. convergence of Cesaro averages for positive mean bounded operators. To cite this article: A. Brunel, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 205–207.     

An Improved Interval Linearization for Solving Nonlinear Problems

Let f(x) denote a system of n nonlinear functions in m variables, m≥n. Recently, a linearization of f(x) in a box x has been suggested in the form L(x)=Ax+b where A is a real n×m matrix and b is an interval n-dimensional vector. Here, an improved linearization L(x,y)=Ax+By+b, x∈x, y∈y is proposed where y is a p-dimensional vector belonging to the interval vector y while A and B are real matrices of appropriate dimensions and b is a real vector. The new linearization can be employed in solving various nonlinear problems: global solution of nonlinear systems, bounding the solution set of underdetermined systems of equations or systems of equalities and inequalities, global optimization. Numerical examples illustrating the superiority of L(x,y)=Ax+By+b over L(x)=Ax+b have been solved for the case where the problem is the global solution of a system of nonlinear equations (n=m).     

Enumeration of matrices of given rank with submatrices of given rank

The authors determine the number of (n+m)×t matrices A¹ of rank r+v, over a finite field GF(q), whose last m rows are those of a given matrix A of rank r+v over GF(q) and whose first n rows have a given rank u.     

On the sequence of powers of a stochastic matrix with large exponent

We consider the class of primitive stochastic n×n matrices A, whose exponent is at least (n²−2n+2)/2+2. It is known that for such an A, the associated directed graph has cycles of just two different lengths, say k and j with k>j, and that there is an α between 0 and 1 such that the characteristic polynomial of A is λⁿ−αλ^n−j−(1−α)λ^n−k. In this paper, we prove that for any mn, if α1/2, then A^m+k−A^m_∞A^m−1w^T_∞, where 1 is the all-ones vector and w^T is the left-Perron vector for A, normalized so that w^T1=1. We also prove that if jn/2, n31 and , then A^m+j−A^m_∞A^m−1w^T_∞ for all sufficiently large m. Both of these results lead to lower bounds on the rate of convergence of the sequence A^m.     

Nonparametric estimation of mixed partial derivatives of a multivariate density

On the basis of a random sample of size n on an m-dimensional random vector X, this note proposes a class of estimators f_n^(p) of f^(p), where f is a density of X w.r.t. a σ-finite measure dominated by the Lebesgue measure on R^m, p = (p₁,…,p_m), p_j ≥ 0, fixed integers, and for x = (x₁,…,x_m) in R^m, f^(p)(x) = ?^p₁+…+p_m f(x)/(?^p₁x₁ … ?^p_mx_m). Asymptotic unbiasedness as well as both almost sure and mean square consistencies of f_n^(p) are examined. Further, a necessary and sufficient condition for uniform asymptotic unbisedness or for uniform mean square consistency of f_n^(p) is given. Finally, applications of estimators of this note to certain statistical problems are pointed out.     

Order-constrained uniform approximations to the exponential based on restricted rationals

LetR _n/m(z∶γ)=P _n(z∶γ)/(1?γz) ^m be a rational approximation to exp (z),z ∈C, of ordern for all real positiveγ. In this paper we show there exists exactly one value ofγ in each of min(n+1,m) interpolation intervals such that the uniform error overR ^? is at a local minimum.     

On the Fourier transforms of retarded Lorentz-invariant functions

In this article we evaluate the Fourier transforms of retarded Lorentz-invariant functions (and distributions) as limits of Laplace transforms. Our method works generally for any retarded Lorentz-invariant functions φ(t) (t?Rⁿ) which is, besides, a continuous function of slow growth. We give, among others, the Fourier transform of G_R(t, α, m², n) and G_A(t, α, m², n), which, in the particular case α = 1, are the characteristic functions of the volume bounded by the forward and the backward sheets of the hyperboloid u = m² and by putting α = ?k are the derivatives of k-order of the retarded and the advanced-delta on the hyperboloid u = m². We also obtain the Fourier transform of the function W(t, α, m², n) introduced by M. Riesz (Comm. Sem. Mat. Univ. Lund4 (1939)). We finish by evaluating the Fourier transforms of the distributional functions G_R(t, α, m², n), G_A(t, α, m², n) and W(t, α, m², n) in their singular points.     

Recursion relations for the extended Krylov subspace method

The evaluation of matrix functions of the form f(A)v, where A is a large sparse or structured symmetric matrix, f is a nonlinear function, and v is a vector, is frequently subdivided into two steps: first an orthonormal basis of an extended Krylov subspace of fairly small dimension is determined, and then a projection onto this subspace is evaluated by a method designed for small problems. This paper derives short recursion relations for orthonormal bases of extended Krylov subspaces of the type K^m,mi+1(A)=span{A^-m+1v,…,A^-1v,v,Av,…,A^miv}, m=1,2,3,…, with i a positive integer, and describes applications to the evaluation of matrix functions and the computation of rational Gauss quadrature rules.     

A New Algorithm for the Symmetric Tridiagonal Eigenvalue Problem

We apply a novel approach to approximate within ϵ to all the eigenvalues of an n × n symmetric tridiagonal matrix A using at most n²([3 log₂(625n⁶)] + (83n − 34)[log₂ (log₂((λ₁ − λ_n)/(2ϵ))/log₂(25_n))]) arithmetic operations where λ₁ and λ_n denote the extremal eigenvalues of A. The algorithm can be modified to compute any fixed numbers of the largest and the smallest eigenvalues of A and may also be applied to the band symmetric matrices without their reduction to the tridiagonal form.     

Albanese and Picard 1-motives

We define, in a purely algebraic way, 1-motives Alb⁺(X), Alb⁻(X), Pic⁺(X), and Pic⁻(X) associated with any algebraic scheme X over an algebraically closed field of characteristic zero. For X over C of dimension n, the Hodge realizations are, respectively, H^{2n − 1} (X, Z(n))/(torsion), H₁ (X, Z)/(torsion), H¹ (X, Z(1)), and H_{2n − 1} (X, Z(1 n))/(torsion).     

Some lower bounds for the spectral radius of matrices using traces

Let A be an n×n matrix with eigenvalues λ₁,λ₂,…,λ_n, and let m be an integer satisfying rank(A)?m?n. If A is real, the best possible lower bound for its spectral radius in terms of m, trA and trA² is obtained. If A is any complex matrix, two lower bounds for are compared, and furthermore a new lower bound for the spectral radius is given only in terms of trA,trA²,‖A‖,‖A^∗A-AA^∗‖,n and m.     

On the convex hull of the multiform numerical range

Let A ₁,…,A_m be nxn hermitian matrices. Definine

W(A ₁,…,A_m )={(xA₁x ^?,…xA_mx ^?):x?C ⁿ ,xx ^?=1}. We will show that every point in the convex hull of W(A ₁,…,A_m ) can be represented as a convex combination of not more than k(m,n) points in W(A ₁,…,A_m ) where k(m,n)=min{n,[√m]+δ _n ² _m+1}.     

On the multiplicity and the Cohen-Macaulayness of fiber cones of graded algebras

Let A be a Noetherian local ring with the maximal ideal m and an m-primary ideal J. Let S=?_n≥0S_n be a finitely generated standard graded algebra over A. Set S⁺=?_n>0S_n. Denote by F_J(S)=?_n≥0→(S_n/JS_n) the fiber cone of S with respect to J. The paper characterizes the multiplicity and the Cohen-Macaulayness of F_J(S) in terms of minimal reductions of S⁺.     

The Zygmund Morse-Sard Theorem

The classical Morse-Sard Theorem says that the set of critical values off:R ^n+k →R ⁿ has Lebesgue measure zero iff ∈C ^k+1. We show theC ^k+1 smoothness requirement can be weakened toC ^k+Zygmund. This is corollary to the following theorem: For integersn >m >r > 0, lets = (n ?r)/(m ?r); iff:R ⁿ →R ^m belongs to the Lipschitz class Λ _s andE is a set of rankr forf, thenf(E) has measure zero.     

Counting points in hypercubes and convolution measure algebras

It is shown that ifA andB are non-empty subsets of {0, 1} ⁿ (for somenεN) then |A+B|≧(|A||B|)^α where α=(1/2) log₂ 3 here and in what follows. In particular if |A|=2 ^n-1 then |A+A|≧3 ^n-1 which anwers a question of Brown and Moran. It is also shown that if |A| = 2 ^n-1 then |A+A|=3 ^n-1 if and only if the points ofA lie on a hyperplane inn-dimensions. Necessary and sufficient conditions are also given for |A +B|=(|A||B|)^α. The above results imply the following improvement of a result of Talagrand [7]: ifX andY are compact subsets ofK (the Cantor set) withm(X),m(Y)>0 then λ(X+Y)≧2(m(X)m(Y))^α wherem is the usual measure onK and λ is Lebesgue measure. This also answers a question of Moran (in more precise terms) showing thatm is not concentrated on any proper Raikov system.     

On the Cohomology Sequence in a Semiabelian Category

In a semiabelian category, a strictly exact sequence 0→A→B→C→0 of cochain complexes gives rise to the cohomology sequence ...→H ⁿ(A) →H ⁿ(B)→ H ⁿ(C)→ H ⁿ⁺¹ (A) →.... We study conditions for exactness of the homology sequence at a given term.     

Irreducibility of associated matrices

For an n by n matrix A, let K(A) be the associated matrix corresponding to a permutation group (of degree m) and one of its characters. Let D_r(A) be the coefficient of x^m?r in K(A+xI). If A is reducible, then D_r(A) is reducible. If A is irreducible and the character is identically one, then D₁(A) is irreducible. If A is row stochastic and the character is identically one, then D_r(A) is essentially row stochastic. Finally, the results motivate the definition of group induced diagraphs.     

PCR algorithm for parallel computing the solution of the general restricted linear equations

In this paper we present a parallel algorithm for parallel computing the solution of the general restricted linear equations Ax=b,x∈T, where T is a subspace of ? ⁿ and b∈AT. By this algorithm the solution x=A _T,S ⁽²⁾ b is obtained in n(log?₂ m+log?₂(n?s+1)+7)+log?₂ m+1 steps with P=mn processors when m≥2(n?1) and with P=2n(n?1) processors otherwise.     

Function algebras on which homomorphisms are point evaluations on sequences

In the study of the spectrum of a subalgebraA ofC(X), whereX is a completely regular Hausdorff space, a key question is, whether each homomorphism ?:A→R has the point evaluation property for sequences inA, that is whether, for each sequence (f _n ) inA, there exists a pointa inX such that ?(f _n )=f _n (a) for alln. In this paper it is proved that all algebras, which are closed under composition with functions inC ^∞ (R) and have a certain local property, have the point evaluation property for sequences. Such algebras are, for instance, the spaceC ^m (E) (m=0,1,...,∞) ofC ^m -functions on any real locally convex spaceE. This result yields in a trivial manner that each homomorphism ? onA is a point evaluation, ifX is Lindelöf or ifA contains a sequence which separates points inX. Further, also a well known result as well as some new ones are obtained as a consequence of the main theorem.     

Characterization and Connectivity of Generalized Filters in L 2(ℝ d )

Let A be a d×d real expansive integer matrix (i.e., a matrix with real entries whose eigenvalues are all of modules greater than one) with |det?A|=2, and let m (which is called A-dilation generalized filter) be a 2π? ^d periodic function with the property that |m(s)|²+|m(s+2π h ₂)|²=1, where h ₂∈(A ^τ )^?1? ^d ?? ^d . In this paper, we characterize the set of all A-dilation generalized filters and show that this set is path-connected in $L^{2}({\mathbb{T}}^{d})$ -norm by using the technique of filter multipliers. We also obtain an equivalent condition for an A-dilation generalized filter to be an A-dilation low pass filter. These extend the results of Manos Papadakis et al. from one dimensional case to high dimensions and matrix dilations cases.