L-spaces and S-spaces in P(ω)

We show that CH implies that $\text{P}\text{(ω)}$, when equipped with the Vietoris topology, has a subspace which is an L-space and a subspace which is an S-space. This is an immediate consequence of the following purely combinatorial result: CH implies the existence of an ω₁-sequence 〈x_α: α < ω₁〉 in $\text{P}\text{(ω)}$ such that (1) if α<β<ω₁, then $\text{X}{}_{\mathrm{\beta }}\text{?}{}^1\text{X}{}_{\mathrm{\alpha }}$; (2) if I ?ω₁ is unaccountable, then there are distinct α, β ∈ I with X_β ?X_α.     

Some special vapnik-chervonenkis classes

For a class $\text{C}$ of subsets of a set X, let V($\text{C}$) be the smallest n such that no n-element set F?X has all its subsets of the form A ∩ F, A ∈ $\text{C}$. The condition V($\text{C}$) <+∞ has probabilistic implications. If any two-element subset A of X satisfies both A ∩ C = Ø and A ? D for some C, D∈$\text{C}$, then $\text{V(}\text{C}\text{)=2}$ if and only if $\text{C}$ is linearly ordered by inclusion. If $\text{C}$ is of the form $\text{C}\text{={∩}{}^{\mathrm{n}}{}_{\mathrm{i=1}}\text{C}{}_{\mathrm{i}}\text{:C}{}_{\mathrm{i}}\text{∈}\text{C}{}_{\mathrm{i}}$, i=1,2,…,n}, where each $\text{C}{}_{\mathrm{i}}$ is linearly ordered by inclusion, then $\text{V(}\text{C}\text{)?n+1}$. If H is an (n-1)-dimensional affine hyperplane in an n-dimensional vector space of real functions on X, and $\text{C}$ is the collection of all sets {x: f(x)>0} for f in H, then $\text{V(}\text{C}\text{)=n}$.     

Estimation d'une rupture en dépendance faibleChange point estimation for a weakly dependent sequence

Let $\text{(X}{}_{\mathrm{n}}\text{,Y}{}_{\mathrm{n}}\text{)}{}_{\mathrm{<mtext>n\in </mtext><mtext>N</mtext>}}$ be a stationary sequence governed by the model Y_n=m(X_n)+σ(X_n)ε_n where $\text{(ε}{}_{\mathrm{n}}\text{)}{}_{\mathrm{<mtext>n\in </mtext><mtext>N</mtext>}}$ is i.i.d. and independent from $\text{(X}{}_{\mathrm{n}}\text{)}{}_{\mathrm{<mtext>n\in </mtext><mtext>N</mtext>}}\text{.}$ The latter sequence satisfy a weak dependence condition proposed by Doukhan and Louhichi in [2]. We provide a Central Limit Theorem for jumps in the regression function. Our method deals with linear local regression described in [4]. We use a variation on Lindeberg–Rio method as in [5]. To cite this article: P. Ango Nze, C. Prieur, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 267–270.     

γ-sets and other singular sets of real numbers

A family of $\text{J}$ of open subsets of the real line is called an ω-cover of a set X iff every finite subset of X is contained in an element of $\text{J}$. A set of reals X is a γ-set iff for every ω-cover $\text{J}$ of X there exists $\text{〈D}{}_{\mathrm{n}}\text{: n < ω〉?}\text{J}{}^{\mathrm{\omega }}$ such that

In this paper we show that assuming Martin's axiom there is a γ-set X of cardinality the continuum.     

Relative locality of derivations

Let H and K be symmetric linear operators on a C¹-algebra $\text{U}$ with domains D(H) and D(K). H is defined to be strongly K-local if $\text{ω(K(A)}{}^1\text{K(A)) = 0}$ implies $\text{ω(H(A)}{}^1\text{H(A)) = 0}$ for A?D(H) ∩ D(K) and ω in the state space of $\text{U}$, and H is completely strongly K-local if $\text{Ω(K(A)}{}^1\text{K(A))=0}$ implies $\text{Ω(H(A)}{}^1\text{H(A))=0}$ for A ∈ D(H) ∩ D(K) and Ω in the state of $\text{U}$, and H is cpmpletely strongly K-local if $\text{H??}{}_{\mathrm{n}}$ is $\text{K??}{}_{\mathrm{n}}$-local on U?M_n for all n ? 1, where $\text{1}$_n is the identity on the n × n matrices M_n. If $\text{U}$ is abelian then strong locality and complete strong locality are equivalent. The main result states that if τ is a strongly continuous one-parameter group of ¹-automorphisms of $\text{U}$ with generator δ₀ and δ is a derivation which commutes with τ and is completely strongly δ₀-local then δ generates a group α of ¹-automorphisms of $\text{U}$. Various characterizations of α are given and the particular case of periodic τ is discussed.     

On projections and limit mappings of inverse systems of compact spaces

A compactificaton αX of a completely regular space X is “determined” by a subset F of C¹(X) if αX is the smallest compactificaton of X to which each element of F extends, and is “generated” by F if the evaluation map $\text{e}{}_{\mathrm{F}}\text{:X →}\text{R}{}^{\mathrm{n}}\text{,n = |F|}$, is an embedding and $\text{αX =}\text{e}{}_{\mathrm{F}}\text{(X)}$. Evidently, if F either determines or generates αX, then every elements of F has an extension to αX; whenever F satisfies this latter condition, the set of all such extensions is denoted F^α.A major results of our previous paper is that F determines αX if and only if F^α separates points of αX ? X. A major result of the present paper is that F generates αX if and only if F^α separates points of αX.     

Poisson limits for a hard-core clustering model

Let X_1n,…,X_>nn denote the locations of n points in a bounded, γ-dimensional, Euclidean region D_n which has positive γ-dimensional Lebesgue measure μ(D_n). Let {Y_n(r): r > 0} be the interpoint distance process for these points where Y_n(r) is the number of pairs of points(X_in, X_in) which with i < j have Euclidean distance 6X_in ? X_>in6 < r. In this article we study the limiting distribution of Y_n(r) when n → ∞ and μ(D_n) → ∞, and the joint density of X_1n,…,X_nnis of the form

where r₀ is a positive constant and C_n is a normalizing constant. These joint densities modify the Strauss [11] clustering model densities by introducing a hard-core component (no two points can have 6X_in ? X_in6 < r₀) found in the Matérn [4] models. In our main result we show that the interpoint distance process converges to a non-homogeneous Poisson process for r values in a bounded interval 0 < r₀ < r < r₀₀ provided sparseness conditions discussed by Saunders and Funk [9] hold. The sparseness conditions which require $\text{μ(D}{}_{\mathrm{n}}\text{)}\text{n}{}^2$ converges to a positive constant and the boundary of D_n is negligible are essentially equivalent to requiring that although the number of points n is large the region is large enough so that the points are sparse in this region. That is, it is rare for a point to have another point close to it. These results extend results for v ? 0 given by Saunders and Funk [9] where it is shown that without the hard core component such results do not hold for v > 0. Statistical applications are discussed.     

On Berry-Esséen rates,a law of the iterated logarithm and an invariance principle for the proportion of the sample below the sample mean

Let F_n(x) be the empirical distribution function based on n independent random variables X₁,…,X_n from a common distribution function F(x), and let $\text{X}\text{= Σ}{}_{\mathrm{i=1}}{}^{\mathrm{n}}\text{X}{}_{\mathrm{i}}\text{n}$ be the sample mean. We derive the rate of convergence of $\text{F}{}_{\mathrm{n}}\text{(}\text{X}\text{)}$ to normality (for the regular as well as nonregular cases), a law of iterated logarithm, and an invariance principle for $\text{F}{}_{\mathrm{n}}\text{(}\text{X}\text{)}$.     

Sufficient conditions for ergodicity and recurrence of Markov chains on a general state space

Let {X_n} be a ?-irreducible Markov chain on an arbitrary space. Sufficient conditions are given under which the chain is ergodic or recurrent. These extend known results for chains on a countable state space. In particular, it is shown that if the space is a normed topological space, then under some continuity conditions on the transition probabilities of {X_n} the conditions for ergodicity will be met if there is a compact set K and an ? > 0 such that $\text{E}\text{{6X}{}_{\mathrm{n+1}}\text{6 — 6X}{}_{\mathrm{n}}\text{6 ∣ X}{}_{\mathrm{n}}\text{= x} ? ??}$ whenever x lies outside K and $\text{E}\text{{6X}{}_{\mathrm{n+1}}\text{6 ∣ X}{}_{\mathrm{n}}\text{=x}}$ is bounded, x ∈ K; whilst the conditions for recurrence will be met if there exists a compact K with $\text{E}\text{{6X}{}_{\mathrm{n+1}}\text{6 ? 6X}{}_{\mathrm{n}}\text{6 ∣ X}{}_{\mathrm{n}}\text{= x} ? 0}$ for all x outside K. An application to queueing theory is given.     

A Feynman-Kac gauge for solvability of the Schrödinger equation

Let {X_t, t ≥ 0} be Brownian motion in $\text{R}$^d (d ≥ 1). Let D be a bounded domain in $\text{R}$^d with C² boundary, ?D, and let q be a continuous (if d = 1), Hölder continuous (if d ≥ 2) function in D?. If the Feynman-Kac “gauge” E_x{exp(∝₀^τ_Dq(X_t)dt)1_A(X_τD)}, where τ_D is the first exit time from D, is finite for some non-empty open set A on ?D and some x?D, then for any $\text{? ? C}{}^0\text{(?D}$), is the unique solution in $\text{C}{}^2\text{(D) ∩ C}{}^0\text{(}\text{D}\text{?}\text{)}$ of the Schrödinger boundary value problem $\text{(}\text{1}\text{2}\text{Δ}\text{+ q)φ = 0}\text{in}\text{D, φ = ?}\text{on}\text{?D}$.     

Linear operators preserving the (p,q)-numerical range

Let $\text{C}{}_{\mathrm{n\times n}}$ and $\text{H}{}_{\mathrm{n}}$ denote respectively the space of n×n complex matrices and the real space of n×n hermitian matrices. Let p,q,n be positive integers such that p?q?n. For $\text{A?}\text{C}{}_{\mathrm{n\times n}}$, the (p,q)-numerical range of A is the set

$\text{W}{}_{\mathrm{p,q}}\text{(A)={trC}{}_{\mathrm{p}}\text{(J}{}_{\mathrm{q}}\text{UAU}{}^1\text{):U}\text{unitary}\text{}}$

, where C_p(X) is the pth compound matrix of X, and Jq is the matrix I_q?O_n-q. Let $\text{L}$ denote $\text{H}$_n or $\text{C}{}_{\mathrm{n\times n}}$. The problem of determining all linear operators T: $\text{L}$→$\text{L}$ such that

$\text{W}{}_{\mathrm{p,q}}\text{(T(A))=W}{}_{\mathrm{p,q}}\text{(A)}\text{for all}\text{A?}\text{L}$

is treated in this paper.     

Spectral theory of finite volume domains in Rn

Let Ω be an arbitrary open subset of $\text{R}$ⁿ of finite positive measure, and assume the existence of a subset Λ ? $\text{R}$ⁿ such that the exponential functions e_λ = exp i(λ₁x₁ + … + λ_nx_n), λ = (λ₁,…, λ_n) ∈ Λ, form an orthonormal basis for $\text{L}{}_2\text{(Ω)}$ with normalized measure. Assume 0 ∈ Λ and define subgroups K and A of ($\text{R}$ⁿ, +) by K = Λ⁰ = {γ ∈ $\text{R}$ⁿ:γ·λ ∈ 2π$\text{Z}$}, A = {a ∈ $\text{R}$ⁿ:U_a$\text{m}$ U1_a = $\text{m}$}, where U_t is the unitary representation of $\text{R}$ⁿ on $\text{L}{}_2\text{(Ω)}$ given by U_te = e^itλe_λ, t ∈ $\text{R}$ⁿ, λ ∈ Λ, and where $\text{m}$ is the multiplication algebra of $\text{L}{}_{\mathrm{\infty }}\text{(Ω)}$ on L₂. Assume that A is discrete. Then there is a discrete subgroup D ? A of dimension n, a fundamental domain $\text{D}$ for D, and finite sets of representers R_Λ, R_Γ, $\text{R}{}_{\mathrm{\Omega }}$, each containing 0, R_Λ for $\text{A}\text{K}$ in K⁰, and $\text{R}{}_{\mathrm{\Omega }}$ for $\text{A}\text{K}$ in A such that Ω is disjoint union of translates of $\text{D}$: Ω = ∪_a∈RΩ (a + $\text{D}$), neglecting null sets, and Λ = R_Λ ⊕ D⁰. If R_Γ is a set of representers for $\text{D}\text{A}$ in D, then Γ = R_Γ ⊕ K is a translation set for Ω, i.e., Ω ⊕ Γ = $\text{R}$ⁿ, direct sum, (neglecting null sets). The case A = $\text{R}$ⁿ corresponds to Ω = $\text{D}$, Λ = D⁰ and Γ = K. This last case corresponds in turn to a function theoretic assumption of Forelli.     

Multiplicativity factors for C-numerical radii

For matrices $\text{A, C?}\text{C}{}_{\mathrm{n}}\text{x}{}_{\mathrm{n}}$ , the C-numerical radius of A is the nonnegative quantity

$\text{r}{}_{\mathrm{c}}\text{(A)=max{|tr(CU}{}^1\text{AU)|:Uunitary}}$

. This generalizes the classical numerical radius r(A). It is known that r_c constitutes a norm on $\text{C}{}_{\mathrm{n}}\text{x}{}_{\mathrm{n}}$ if and only if C is nonscalar and trC≠0. For all such C we obtain multiplicativity factors for r_c, i.e., constant μ>0 for which μr_c is submultiplicative on $\text{C}{}_{\mathrm{n}}\text{x}{}_{\mathrm{n}}$.     

On a property related to convergence in probability and some applications to branching processes

It is shown that any real-valued sequence of random variables {X_n} converging in probability to a non-degenerate, not necessarily a.s. finite limit X possesses the following property: for any c with P(X? (c ? δ, c + δ)) > 0 for all δ > 0, there exists a sequence {c_n} with lim_n→∞ c_n = c such that for any ε > 0, lim_n→∞ P(Xδ (c ? ε, c + ε) |X_n = c_n) = 1. This property is applied to various types of branching processes where $\text{X}{}_{\mathrm{n}}\text{=}\text{Z}{}_{\mathrm{n}}\text{C}{}_{\mathrm{n}}$ or $\text{X}{}_{\mathrm{n}}\text{=}\text{U(Z}{}_{\mathrm{n}}\text{)}\text{C}{}_{\mathrm{n}}\text{{C}{}_{\mathrm{n}}\text{}}$ being a sequence of constants or random variables and U a slowly varying function. If {Z_n} is a supercritical branching process in varying or random environment, X is shown to have a continuous and strictly increasing distribution function on (0, ∞). Characterizations of the tail of the liniting distribution of the finite mean and the infinite mean supercritical Galton-Watson processes are also obtained.     

Operator properties of Sobolev imbeddings over unbounded domains

For an open set Ω ? $\text{R}$^N, 1 ? p ? ∞ and λ ∈ $\text{R}$⁺, let $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω)}$ denote the Sobolev-Slobodetzkij space obtained by completing $\text{C}{}_0{}^{\mathrm{\infty }}\text{(Ω)}$ in the usual Sobolev-Slobodetzkij norm (cf. A. Pietsch, “r-nukleare Sobol. Einbett. Oper., Ellipt. Dgln. II,” Akademie-Verlag, Berlin, 1971, pp. 203–215). Choose a Banach ideal of operators $\text{U}$, 1 ? p, q ? ∞ and a quasibounded domain Ω ? $\text{R}$^N. Theorem 1 of the note gives sufficient conditions on λ such that the Sobolev-imbedding map $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω) λ L}{}_{\mathrm{q}}\text{(Ω)}$ exists and belongs to the given Banach ideal $\text{U}$: Assume the quasibounded domain fulfills condition C_k^l for some l > 0 and 1 ? k ? N. Roughly this means that the distance of any $\text{x ? Ω}$ to the boundary ?Ω tends to zero as $\text{O(¦ x ¦}{}^{\mathrm{?l}}\text{)}$ for $\text{¦ x ¦ → ∞}$, and that the boundary consists of sufficiently smooth ?(N ? k)-dimensional manifolds. Take, furthermore, 1 ? p, q ? ∞, p > k. Then, if μ, ν are real positive numbers with λ = μ + v ∈ $\text{N}$, μ > λ _S($\text{U}$; p,q:N) and v > N/l · λ_D($\text{U}$;p,q), one has that $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω) λ L}{}_{\mathrm{q}}\text{(Ω)}$ belongs to the Banach ideal $\text{U}$. Here λ_D($\text{U}$;p,q;N)∈$\text{R}$⁺ and λ_S($\text{U}$;p,q;N)∈$\text{R}$⁺ are the D-limit order and S-limit order of the ideal $\text{U}$, introduced by Pietsch in the above mentioned paper. These limit orders may be computed by estimating the ideal norms of the identity mappings l_pⁿ → l_qⁿ for n → ∞. Theorem 1 in this way generalizes results of R. A. Adams and C. Clark for the ideals of compact resp. Hilbert-Schmidt operators (p = q = 2) as well as results on imbeddings over bounded domains.Similar results over general unbounded domains are indicated for weighted Sobolev spaces.As an application, in Theorem 2 an estimate is given for the rate of growth of the eigenvalues of formally selfadjoint, uniformly strongly elliptic differential operators with Dirichlet boundary conditions in $\text{L}{}_2\text{(Ω)}$, where Ω fulfills condition C₁^l.For an open set Ω in $\text{R}$^N, let $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω)}$ denote the Sobolev-Slobodetzkij space obtained by completing $\text{C}{}_0{}^{\mathrm{\infty }}\text{(Ω)}$ in the usual Sobolev-Slobodetzkij norm, see below. Taking a fixed Banach ideal of operators and 1 ? p, q ? ∞, we consider quasibounded domains Ω in $\text{R}$^N and give sufficient conditions on λ such that the Sobolev imbedding operator $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω) λ L}{}_{\mathrm{q}}\text{(Ω)}$ exists and belongs to the Banach ideal. This generalizes results of C. Clark and R. A. Adams for compact, respectively, Hilbert-Schmidt operators (p = q = 2) to general Banach ideals of operators, as well as results on imbeddings over bounded domains. Similar results over general unbounded domains may be proved for weighted Sobolev spaces. As an application, we give an estimate for the rate of growth of the eigenvalues of formally selfadjoint, uniformly strongly elliptic differential operators with Dirichlet boundary conditions in $\text{L}{}_2\text{(Ω)}$, where Ω is a quasibounded open set in $\text{R}$^N.     

Finitary quasi-varieties

A pair ($\text{C}$,U) consisting of a category $\text{C}$ with coequalizers and a functor U: $\text{C}$ → Set is a weak quasi-variety if U has a left adjoint and U preserves and reflects regular epis. It is known that every weak quasi-variety is equivalent to a concrete quasi-variety, i.e. a category of Σ-algebras which has all free algebras and which is closed with respect to products and subalgebras. It is also known that if U preserves monic direct limits, $\text{C}$ is equivalent to a concrete quasi-variety of Σ-algebras in which Σ contains no function symbols of infinite rank; and if U preserves all direct limits, $\text{C}$ is equivalent to a concrete quasi-variety of Σ-algebras definable by a set of implications of the form

$\text{(t}{}_1\text{= s}{}_1\text{∧?∧t}{}_{\mathrm{m}}\text{= s}{}_{\mathrm{m}}\text{) ? t}{}_{\mathrm{m+1}}\text{= s}{}_{\mathrm{m+1}}$

where t_i and s_i are Σ-terms and m is a nonnegative integer. This paper concerns several definitions of ‘finiteness’ in a category theoretic setting and some theorems on weak quasi-varieties. Two main theorems characterize those weak quasi-varieties ($\text{C}$, U) such that U preserves all direct limits.     

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 $\text{Y}\text{=T}\text{X}$.     

The approximation of flows

Let U, V be two strongly continuous one-parameter groups of bounded operators on a Banach space $\text{X}$ with corresponding infinitesimal generators S, T. We prove the following: ∥U_t, ? V_t ∥ = O(t), t → 0, if and only if U = V; ∥U_t ? V_t∥ = O(t^α), t → 0; with 0 ? α ? 1, if and only if $\text{S = Ω(T + P)Ω}{}^{\mathrm{?1}}$, where Ω, P, are bounded operators on $\text{X}$ such that $\text{∥U}{}_{\mathrm{t}}\text{Ω ? ΩU}{}_{\mathrm{t}}\text{∥ = O(t}{}^{\mathrm{\alpha }}\text{), ∥U}{}_{\mathrm{t}}\text{P ? PU}{}_{\mathrm{t}}\text{∥ = ?O(t}{}^{\mathrm{\alpha }}\text{), t → 0; ∥U}{}_{\mathrm{t}}\text{? V}{}_{\mathrm{t}}\text{∥ = O(t)}$ if and only if $\text{S}{}^1\text{? T}{}^1$ has a bounded extension to $\text{X}$¹. Further results of this nature are inferred for semigroups, reflexive spaces, Hilbert spaces, and von Neumann algebras.     

Three problems on the lengths of increasing runs

Let U₁, U₂,… be a sequence of independent, uniform (0, 1) r.v.'s and let R₁, R₂,… be the lengths of increasing runs of {U_i}, i.e., X₁=R₁=inf{i:U_i+1<U_i},…, X_n=R₁+R₂+?+R_n=inf{i:i>X_n?1,U_i+1<U_i}. The first theorem states that the sequence $\text{(}\text{3}\text{2}\text{n)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(X}{}_{\mathrm{n}}\text{?2n)}$ can be approximated by a Wiener process in strong sense.Let τ(n) be the largest integer for which R₁+R₂+?+R_τ(n)?n, $\text{R}{}^1{}_{\mathrm{n}}\text{=n?(R}{}_1\text{+R}{}_2\text{+?+R}{}_{\mathrm{\tau (n)}}\text{)}$ and $\text{M}{}_{\mathrm{n}}\text{=}\text{max}\text{{R}{}_1\text{,R}{}_2\text{,…,R}{}_{\mathrm{\tau (n)}}\text{,R}{}^1{}_{\mathrm{n}}\text{}}$. Here M_n is the length of the longest increasing block. A strong theorem is given to characterize the limit behaviour of M_n.The limit distribution of the lengths of increasing runs is our third problem.