The minimal generating space of a polynomial

Given a polynomial $\text{P(X}{}_1\text{,…,X}{}_{\mathrm{N}}\text{)∈}\text{R}\text{[}\text{X}\text{]}$, we calculate a subspace G_p of the linear space 〈X〉 generated by the indeterminates which is minimal with respect to the property $\text{P∈}\text{R}\text{[}\text{G}{}_{\mathrm{p}}\text{]}$ (the algebra generated by G_p, and prove its uniqueness. Furthermore, we use this result to characterize the pairs (P,Q) of polynomials P(X₁,…,X_n) and Q(X₁,…,X_n) for which there exists an isomorphism T:〈X〉 →〈X〉 that “separates P from Q,” i.e., such that for some k(1<k<n) we can write P and Q as $\text{P}{}^1\text{(Y}{}_1\text{,…,Y}{}_{\mathrm{k}}\text{)}$ and $\text{Q}{}^1\text{(Y}{}_{\mathrm{<mtext>k+</mtext><mtext>1</mtext>}}\text{,…,Y}{}_{\mathrm{n}}\text{)}$ respectively, where $\text{Y}\text{=T}\text{X}$.     

On the Smale theorem in weak shape theory

We modify various lemmas from Dydak's paper on the Vietoris-Smale theorem to obtain more general results. We consider closed subsets X and Y of paracompact spaces M, N respectively, and a map F:M→N such that F∥X:X→Y is closed and surjective and X=F^-1(Y) to obtain the following result.(Theorem). If $\text{F}{}^{-1}\text{(y)ϵAC}{}^{\mathrm{n}}{}_{\mathrm{M}}\text{(}\text{K}\text{)}$ for each y ϵ Y and N is LCⁿ⁺¹, then the morphism in-pro-$\text{π}{}_{\mathrm{k}}\text{[F]:π}{}_{\mathrm{k}}\text{U}{}^{\mathrm{<mtext>K</mtext>}}{}_{\mathrm{M}}\text{(X,x)→π}{}_{\mathrm{k}}\text{U}{}^{\mathrm{<mtext>K</mtext>}}{}_{\mathrm{N}}\text{(Y,y)}$ induced by the morphism $\text{[F]:U}{}^{\mathrm{<mtext>K</mtext>}}{}_{\mathrm{M}}\text{(X,x)→U}{}^{\mathrm{<mtext>K</mtext>}}{}_{\mathrm{N}}\text{(Y,y)}$ is an isomorphism of in-pro-Grp for k⩽n and an epimorphism for k=n+1.     

Strong law for the bootstrap

Let X₁, X₂, X₃, … be i.i.d. r.v. with E|X₁| < ∞, E X₁ = μ. Given a realization X = (X₁,X₂,…) and integers n and m, construct Y_n,i, i = 1, 2, …, m as i.i.d. r.v. with conditional distribution $\text{P}{}^1\text{(Y}{}_{\mathrm{n,i}}\text{= X}{}_{\mathrm{j}}\text{) =}\text{1}\text{n}$ for 1 ? j ? n. ($\text{P}{}^1$ denotes conditional distribution given X). Conditions relating the growth rate of m with n and the moments of X₁ are given to ensure the almost sure convergence of $\text{(}\text{1}\text{m}\text{)Σ}{}^{\mathrm{m}}{}_{\mathrm{i=1}}\text{Y}{}_{\mathrm{n,i}}$ toμ. This equation is of some relevance in the theory of Bootstrap as developed by Efron (1979) and Bickel and Freedman (1981).     

Banach spaces whose duals contain complemented subspaces isomorphic to C[0, 1]

We give several characterizations of those Banach spaces X such that the dual $\text{X}{}^1$ contains a complemented subspace isomorphic to $\text{C[0, 1]}{}^1$. We investigate operators on separable $\text{L}$_∞ spaces whose adjoints have nonseparable ranges and apply our results to obtain a structure theorem for $\text{L}$_∞ spaces whose duals are not isomorphic to l₁(Γ).     

An equation of Lyapunov type

A new method of finding explicit solutions of Lyapunov equations is described based on a lemma on one-dimensional perturbations of invertible operators. If Y satisfies the equation $\text{Y?CYC}{}^1\text{= bb}{}^1$ for an appropriate vector b, then X = Y^-1 satisfies $\text{X? C}{}^1\text{XC = aa}{}^1$ for a given vector a. A concrete example [with a=(1,0,…,0)^T] is given.     

Projections on spaces of invariant functions

Let (X, ∑, μ) be a measure space and S be a semigroup of measure-preserving transformations T:X → X. In case μ(X) < ∞, Aribaud [1] proved the existence of a positive contractive projection P of L¹(μ) such that for every $\text{? ? L}{}^1\text{(μ), Pf}$ belongs to the closure $\text{C}{}_1\text{(?)}\text{in}\text{L}{}^1\text{(μ)}$ of the convex hull $\text{C(?)}$ of the set {$\text{? ○ T:T ? S}$}. In this paper we extend this result in three directions: we consider infinite measure spaces, vector-valued functions, and L^p spaces with 1 ? p < ∞, and prove that P is in fact the conditional expectation with respect to the σ-algebra Λ of sets of ∑ which are invariant with respect to all T?S.     

On convergence rates and the expectation of sums of random variables

Let X_i be iidrv's and S_n=X₁+X₂+…+X_n. When EX²₁<+∞, by the law of the iterated logarithm $\text{(S}{}_{\mathrm{n}}\text{?α}{}_{\mathrm{n}}\text{)}\text{(n log n)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{→0 a.s.}$ for some constants α_n. Thus the r.v. $\text{Y=sup}{}_{\mathrm{n?1}}\text{[|S}{}_{\mathrm{n}}\text{?α}{}_{\mathrm{n}}\text{|?(δn log n)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{]}{}^{\mathrm{+}}$ is a.s.finite when δ>0. We prove a rate of convergence theorem related to the classical results of Baum and Katz, and apply it to show, without the prior assumption EX²₁<+∞ that EY^h<+∞ if and only if $\text{E}\text{|X}{}_1\text{|}{}^{\mathrm{2+h}}\text{[log|X}{}_1\text{|]}{}^{-1}\text{<+∞}$ for 0<h<1 and δ> $\text{h}\text{E}\text{(X}{}_1\text{?}\text{E}\text{X}{}_1\text{)}{}^2$, whereas $\text{E}\text{Y}{}^{\mathrm{h}}\text{=+∞}$ whenever h>0 and $\text{0<δ<h}\text{E}\text{(X}{}_1\text{?}\text{E}\text{X}{}_1\text{)}{}^2$.     

On the exponential map of classical Lie groups and linear differential systems with periodic coefficients

Let X and Y be Banach spaces, $\text{?: X → Y}{}^1\text{and}\text{P: X → Y}$; P is said to be strongly ?-accretive if $\text{〈Px ? Py, ?(x ? y)〉 ? c ¦|x ? y¦|}{}^2$ for some c > 0 and each x,y?X. These mappings constitute a generalization simultaneously of monotone mappings ($\text{when}\text{Y = X}{}^1$) and accretive mappings (when Y = X). By applying a theorem of 1. Ekeland, it is shown that a localized class of these mappings must be surjective under appropriate geometric assumptions on $\text{Y}{}^1$ and continuity assumptions on P. The results generalize two theorems of F. E. Browder and the proofs further refine the methodology for dealing with such mappings.     

Zeta matrices of elliptic curves

Let $\text{O =}\text{lim}{}_{\mathrm{n}}\text{Z/p}{}^{\mathrm{n}}\text{Z}$, let $\text{A}\text{= O[g}{}_2\text{, g}{}_3\text{]}{}_{\mathrm{\Delta }}$, where g₂ and g₃ are coefficients of the elliptic curve: Y² = 4X³ ? g₂X ? g₃ over a finite field and Δ = g₂³ ? 27g₃² and let $\text{B}\text{=}\text{A}\text{[X, Y]}\text{(Y}{}^2\text{? 4X}{}^3\text{+ g}{}_2\text{X + g}{}_3\text{)}$. Then the p-adic cohomology theory will be applied to compute explicitly the zeta matrices of the elliptic curves, induced by the pth power map on the free $\text{A}{}^2\text{?}{}_{\mathrm{Z}}\text{Q}$-module $\text{H}{}^1\text{(X,}\text{A}{}^2\text{?}{}_{\mathrm{Z}}\text{Q)}$. Main results are; Theorem 1.1: X²dY and YdX are basis elements for $\text{H}{}^1\text{(}\text{X}\text{, Γ}{}_{\mathrm{<mtext>A</mtext>}}{}^1\text{(}\text{X}\text{)}{}^2\text{?}{}_{\mathrm{Z}}\text{Q)}$; Theorem 1.2: YdX, X²dY, Y^?1dX, Y^?2dX and XY^?2dX are basis elements for $\text{H}{}^1\text{(}\text{X}\text{? (Y = 0), Γ}{}_{\mathrm{<mtext>A</mtext>}}{}^1\text{(}\text{X}\text{)}{}^2\text{?}{}_{\mathrm{Z}}\text{Q)}$, where $\text{X}$ is a lifting of X, and all the necessary recursive formulas for this explicit computation are given.     

On the grading numbers of direct products of chains

For a finite lattice L, denote by $\text{l}{}^1\text{(L)}$ and $\text{l}{}_1\text{(L)}$ respectively the upper length and lower length of L. The grading number g(L) of L is defined as $\text{g(L) = l}{}^1\text{(}\text{Sub}\text{(L))-l}{}_1\text{(}\text{Sub}\text{(L))}$ where Sub(L) is the sublattice-lattice of L. We show that if K is a proper homomorphic image of a distributive lattice L, then $\text{l}{}_1\text{(}\text{Sub}\text{(K)) < l}{}_1\text{(}\text{Sub}\text{(L))}$; and derive from this result, formulae for $\text{l}{}_1\text{(}\text{Sub}\text{(L))}$ and g(L) where L is a product of chains.     

Trace formulas for a class of Toeplitz-like operators II

Let P_T denote the orthogonal projection of L²(R¹, dΔ) onto the space of entire functions of exponential type ? T which are square summable on the line with respect to the measure $\text{dΔ(γ) = ¦ h(γ)¦}{}^2\text{dγ}$, and let G denote the operator of multiplication by a suitably restricted complex valued function g. It is shown that if $\text{(γ}{}^2\text{+ 1)}{}^{\mathrm{?1}}\text{log}\text{¦ h(γ)¦}$ is summable, if $\text{¦ h ¦}{}^{\mathrm{?2}}$ is locally summable, and if $\text{h}\text{h}{}^{\mathrm{\#}}$ belongs to the span in L^∞ of e^?iyTH^∞:T ? 0, in which h is chosen to be an outer function and h^#(γ) agrees with the complex conjugate of h(γ) on the line, then

$\text{lim trace}\text{T↑∞}\text{{(P}{}_{\mathrm{T}}\text{GP}{}_{\mathrm{T}}\text{)}{}^{\mathrm{n}}\text{? P}{}_{\mathrm{T}}\text{G}{}^{\mathrm{n}}\text{P}{}_{\mathrm{T}}\text{}}$

exists and is independent of h for every positive integer n. This extends the range of validity of a formula due to Mark Kac who evaluated this limit in the special case h = 1 using a different formalism. It also extends earlier results of the author which were established under more stringent conditions on h. The conclusions are based in part upon a preliminary study of a more general class of projections.     

Sufficiency and invariance

Suppose that a statistical decision problem is invariant under a group of transformations g?G. T (X) is equivariant if there exists $\text{g}{}^1\text{? G}{}^1$ such that $\text{T(g(X)) = g}{}^1\text{(T((X))}$. We show that the minimal sufficient statistic is equivalent and that if T(X) is an equivariant sufficient statistics and d(X) is invariant under G, then $\text{d}{}^1\text{(T) = Ed(X)∥T}$ is invariant under $\text{G}{}^1$.     

A generalization of doubly stochastic matrices over a field

Let X and Y be m×n matrices over a field F such that Y^TX is nonsingular, and let Λ and Λ′ be sets of n-square matrices over F. Solutions A to the simultaneous equations AX = XK and $\text{Y}{}^{\mathrm{T}}\text{A =}\text{K}\text{?}\text{Y}{}^{\mathrm{T}}$ where K?Λ and $\text{K}\text{?}\text{? Λ′}$ are considered. It is shown that many properties of doubly stochastic matrices over a field have a natural generalization in terms of the set Δ(Λ,Λ′) of all such solutions.     

Inner functions in the unit ball of Cn

Let B be the open unit ball of $\text{C}$ⁿ, n > 1. Let I (for “inner”) be the set of all u ? H °(B) that have $\text{¦u¦ = 1}$ a.e. on the boundary S of B. Aleksandrov proved recently that there exist nonconstant u ? I. This paper strengthens his basic theorem and provides further information about I and the algebra Q generated by I. Let XY be the finite linear span of products xy, x ? X, y ? Y, and let $\text{¦X¦}$ be the norm closure, in L^∞ = L^∞(S), of X. Some results: set I is dense in the unit ball of H^∞(B) in the compact-open topology. On $\text{S,}\text{Q}\text{?}\text{Q}$ is weak¹-dense in $\text{L}{}^{\mathrm{\infty }}\text{, ¦}\text{Q}\text{?}\text{Q¦}$ does not contain $\text{H}{}^{\mathrm{\infty }}\text{, C(S) ?¦}\text{Q}\text{?}\text{H}{}^{\mathrm{\infty }}\text{¦ ≠ ¦}\text{H}\text{?}{}^{\mathrm{\infty }}\text{H}{}^{\mathrm{\infty }}\text{¦ ≠ L}{}^{\mathrm{\infty }}$. (When $\text{n = 1, ¦Q¦ = H}{}^{\mathrm{\infty }}\text{and}\text{¦}\text{Q}\text{?}\text{Q¦ = L}{}^{\mathrm{\infty }}$.) Every unimodular $\text{?}\text{?}\text{L}{}^{\mathrm{\infty }}$ is a pointwise limit a.e. of products $\text{u}\text{v}\text{?}\text{, u}\text{?}\text{I, ν}\text{?}\text{I}$. The zeros of every $\text{? ? 0}$ in the ball algebra (but not of every H^∞-function) can be matched by those of some u ? I, as can any finite number of derivatives at 0 if $\text{∥?∥}{}_{\mathrm{\infty }}\text{< 1}$. However, $\text{?}\text{u}$ cannot be bounded in B if u ? I is non-constant.     

Properties of certain algebras between L∞and H∞

The main result of this paper is that if F is a closed subset of the unit circle, then $\text{(H}{}^{\mathrm{\infty }}\text{+ L}{}_{\mathrm{F}}{}^{\mathrm{\infty }}\text{)}\text{H}{}^{\mathrm{\infty }}$ is an M-ideal of $\text{L}{}^{\mathrm{\infty }}\text{H}{}^{\mathrm{\infty }}$. Consequently, if $\text{? ∈ L}{}^{\mathrm{\infty }}$ then ? has a closest element in H^∞ + L_F^∞. Furthermore, if $\text{¦F¦ >0}\text{then}\text{L}{}^{\mathrm{\infty }}\text{(H}{}^{\mathrm{\infty }}\text{+ L}{}_{\mathrm{F}}{}^{\mathrm{\infty }}\text{)}$ is not the dual of any Banach space.     

Cohérence différentielle des F-isocristaux unités

Let $\text{V}$ be a mixed characteristic complete discrete valuation ring, $\text{P}$ a smooth formal scheme over $\text{V}$, P its special fiber, X a smooth subscheme of P, T a divisor in P such that T_X=T∩X is a divisor in X and $\text{D}{}^2{}_{\mathrm{<mtext>P</mtext>}}$ the weak completion of the sheaf of differential operators on $\text{P}$. We prove that the unit-root F-isocrystals on X?T_X overconvergent along T_X are coherent over $\text{D}{}^2{}_{\mathrm{<mtext>P</mtext><mtext>,</mtext><mtext>Q</mtext>}}$. To cite this article: D. Caro, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

On the exact asymptotic behaviour of the distribution of ladder epochs

Let T₊ denote the first increasing ladder epoch in a random walk with a typical step-length X. It is known that for a large class of random walks with E(X)=0,E(X²)=∞, and the right-hand tail of the distribution function of X asymptotically larger than the left-hand tail, $\text{P}\text{T}{}_{\mathrm{+}}\text{?n∽n}{}^{\mathrm{<mtext>1</mtext><mtext>\beta ?1</mtext>}}\text{L}{}_{\mathrm{+}}\text{(n)}$ as n→∞, with 1<β<2 and L₊ slowly varying, if and only if$\text{P}\text{{X?x}∽ 1/{x}{}^{\mathrm{\beta }}\text{L(x)}}$ as x→+∞, with L slowly varying. In this paper it is shown how the asymptotic behaviour of L determines the asymptotic behaviour of L₊ and vice versa. As a by-product, it follows that a certain class of random walks which are in the domain of attraction of one-sided stable laws is such that the down-going ladder height distribution has finite mean.     

Algebras of bounded functions invariant under the restricted backward shift

Let T be the unit circle in the complex plane and let A be a vector space of bounded Lebesgue measurable functions on T. A is said to be invariant under the restricted backward shift if, whenever ? is in A and the 0-th Fourier coefficient of ? vanishes, then $\text{e}{}^{\mathrm{?i\theta }}\text{?(e}{}^{\mathrm{i\theta }}\text{)}$ is also in A. The theorems of this paper provide a characterization of the uniformly closed subalgebras of C(T) which contain the constants and which are invariant under the restricted backward shift and, a similar characterization of the weak-¹ closed subalgebras of L^∞(T, dθ) which contain the constants and which are invariant under the restricted backward shift.