A canonical decomposition of the probability measure of sets of isotropic random points in Rn

The probability measure of X = (x₀,…, x_r), where x₀,…, x_r are independent isotropic random points in $\text{R}$ⁿ (1 ≤ r ≤ n ? 1) with absolutely continuous distributions is, for a certain class of distributions of X, expressed as a product measure involving as factors the joint probability measure of (ω, ?), the probability measure of p, and the probability measure of $\text{Y}{}^1\text{= (y}{}_0{}^1\text{,…, y}{}_{\mathrm{r}}{}^1\text{)}$. Here ω is the r-subspace parallel to the r-flat η determined by X, ? is a unit vector in ω^⊥ with ‘initial’ point at the origin [ω^⊥ is the (n ? r)-subspace orthocomplementary to ω], p is the norm of the vector z from the origin to the orthogonal projection of the origin on η, and $\text{y}{}_{\mathrm{i}}{}^1\text{=}\text{(x}{}_{\mathrm{i}}\text{? z)}\text{α(p}{}^2\text{)}$, where α is a scale factor determined by p. The probability measure for ω is the unique probability measure on the Grassmann manifold of r-subspaces in $\text{R}$ⁿ invariant under the group of rotations in $\text{R}$ⁿ, while the conditional probability measure of ? given ω is uniform on the boundary of the unit (n ? r)-ball in ω^⊥ with centre at the origin. The decomposition allows the evaluation of the moments, for a suitable class of distributions of X, of the r-volume of the simplicial convex hull of {x₀,…, x_r} for 1 ≤ r ≤ n.     

Positive definite distributions on K\GK

Let G be a semisimple noncompact Lie group with finite center and let K be a maximal compact subgroup. Then W. H. Barker has shown that if T is a positive definite distribution on G, then T extends to Harish-Chandra's Schwartz space $\text{C}$¹(G). We show that the corresponding property is no longer true for the space of double cosets $\text{K}\text{G}\text{K}$. If G is of real-rank 1, we construct liner functionals $\text{T}{}_{\mathrm{p}}\text{? (C}{}_{\mathrm{c}}{}^{\mathrm{\infty }}\text{(K}\text{G}\text{K}\text{))′}$ for each p, 0 < p ? 2, such that $\text{T}{}_{\mathrm{p}}\text{(f 1 f}{}^1\text{) ? 0, ?f ? C}{}_{\mathrm{c}}{}^{\mathrm{\infty }}\text{(K}\text{G}\text{K}\text{)}$ but T_p does not extend to a continuous functional on $\text{C}{}^{\mathrm{p}}\text{(K}\text{G}\text{K}\text{)}$. In particular, if p ? 1, T_v does not extend to a continuous functional on $\text{C}{}^1\text{(K}\text{G}\text{K}\text{)}$. We use this to answer a question (in the negative) raised by Barker whether for a K-bi-invariant distribution T on G to be positive definite it is enough to verify that $\text{T(f 1 f}{}^1\text{) ? 0, ?f ? C}{}_{\mathrm{c}}{}^{\mathrm{\infty }}\text{(K}\text{G}\text{K}\text{)}$. The main tool used is a theorem of Trombi-Varadarajan.     

The sum of aΩ(n) over integers n ≤ x with all prime factors between a and y

For a > 0 let $\text{ψ}{}_{\mathrm{a}}\text{(x, y) =}\text{Σ}\text{a}{}^{\mathrm{\Omega (n)}}$, the sum taken over all n, 1 ≤ n ≤ x such that if p is prime and p|n then a < p ≤ y. It is shown for u < about ($\text{log log}\text{x}\text{log log log}\text{x}$) that $\text{ψ}{}_{\mathrm{a}}\text{(x, x}{}^{\mathrm{<mtext>1</mtext><mtext>u</mtext>}}\text{) ? x(}\text{log}\text{x)}{}^{\mathrm{a?1}}\text{p}{}_{\mathrm{a}}\text{(u)}$, where p_a(u) solves a delay differential equation much like that for the Dickman function p(u), and the asymptotic behavior of p_a(u) is worked out.     

Equivariant embeddings of principal Z-bundles into complex vector bundles

Let π: E→X be a principal $\text{Z}$_n-bundle and p:V→X an m-dimensional complex vector bundle over, say, a connected CW-complex X. An equivariant embedding of π into p is an embedding h:E → V commuting with projections such that h(e · z)=zh(e) for all eεE and $\text{zε}\text{Z}{}_{\mathrm{n}}\text{?S}{}^1\text{?}\text{Z}$. We compute the primary obstruction $\text{cεH}{}^{\mathrm{2m}}\text{(X;}\text{Z}\text{)}$ to embedding π equivariantly into p. If dim X?2m, then c=0 if and only if π admits an equivariant embedding into p. If dim X>2m and π embeds equivariantly into p, then c=0. Other embedding criteria exist in case p is the trivial m-plane bundle ε^m. We use these criteria for a discussion of the classification of the equivalence classes of principal $\text{Z}$-bundles that admit equivariant embeddings into ε^m. Finally, we offer an example of a principal $\text{Z}$-bundle that admit an ordinary but not an equivariant embedding into ε¹.     

The family of D-weights for a two-rooted graph

Let Π(G) be the set of paths of a particular class Π from the initial to the terminal root of a two-rooted (possibly directed) graph G. We consider the family of $\text{D}$-weights defined by

where Π_x(G) is the family of subsets of Π(G) which cover x(G), the vertex set or the edge (arc) set of G.A number of the common properties and interrelations of these weights are discussed. Some of the weights have been considered previously, [1, 2], in the context of percolation theory but here only combinatorial arguments are used.     

An analogue of the thom isomorphism for crossed products of a C1 algebra by an action of R

In this paper we show the existence and uniqueness of a natural isomorphism ø^j_α of K_j(A) with K_j+1(A ?_α$\text{R}$), j ? $\text{Z}$/2 where (A, $\text{R}$, α) is a $\text{C1}$ dynamical $\text{R}$-system, K is the functor of topological K theory and A ?_α$\text{R}$ is the crossed product of A by the action of $\text{R}$. The Pimsner-Voiculescu exact sequence is obtained as a corollary. We show that given an α-invariant trace τ on A, with dual trace $\text{}\text{?}$gt, one has $\text{}\text{?}\text{gtø}{}^1{}_{\mathrm{\alpha }}\text{[u] = (}\text{1}\text{2}\text{iπ) τ(δ(u)u1)}$ for any unitary u in the domain of the derivation δ of A associated to the action α. Finally, we show that the crossed product of C(S³) (continuous functions on the 3 sphere) by a minimal diffeomorphism is a simple $\text{C1}$ algebra with no nontrivial idempotent.     

Spaces N ∪ R and their dimensions

About spaces N∪$\text{R}$ (see [2, Exercise 5I]), the following are proved: (1) dim $\text{N∪}\text{R}\text{= dim β(N∪}\text{R}\text{)?N∪}\text{R}\text{,(2)if|β(N∪}\text{R}\text{)?N∪}\text{R}\text{|<2}{}^{\mathrm{?}}{}^{\mathrm{o}}$, then no real-valued continuous fu ction on N∪$\text{R}$ is onto (and hence, dim $\text{N∪}\text{R}\text{=0}$), (3) any compact metric space without isolated points is homeomorphic to some $\text{β(N∪}\text{R}\text{)?N∪}\text{R}$ and (4)there are spaces X,X₁ and X₂ of the form N∪$\text{R}$ such that X=X₁∪X₂,X₂andX₂ are zero sets of X, and dim X=n, dimX₁=dimX₂=0, where n=1,2,… or ∞.     

Resolvable decomposition of Kn1

In this paper, we prove that $\text{K}{}_{\mathrm{n}}{}^1$ admits a resolvable decomposition into TT₃ or C₃ if and only if n ≡ 0 (mod. 3), n ≠ 6.     

The covering problem for the fields Q(−1)12 and Q(−3)12

Let k be a positive square free integer, $\text{N(?k)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ the ring of algebraic integers in $\text{Q(?k)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ and S the unit sphere in C_n, complex n-space. If A₁,…, A_n are n linearly independent points of C_n then L = {u₁Au + … + u_nA_n} with $\text{u}{}_{\mathrm{r}}\text{∈ N(?k)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ is called a k-lattice. The determinant of L is denoted by d(L). If L is a covering lattice for S, then $\text{θ(S, L) =}\text{V(S)}\text{d(L)}$ is the covering density. L is called locally (absolutely) extreme if θ(S, L) is a local (absolute) minimum. In this paper we determine unique classes of extreme lattices for k = 1 and k = 3.     

On the expansion of Cϱ1(V + I) as a sum of zonal polynomials

The coefficients a_τ?, sometimes called “generalized binomial coefficients” in the expansion $\text{C}{}_{\mathrm{?}}{}^1\text{(V +I) = Σ}{}_{\mathrm{\tau }}\text{a}{}_{\mathrm{?\tau }}\text{C}{}_{\mathrm{\tau }}{}^1\text{(V)}$, are computed explicitly when t = r + 1, where ? is a partition of r and τ a partition of t. A recursion formula permits the calculation of the general a_τ?. Several properties of a_τ? are proved. A connection between the a_τ? and other coefficients is established. The main tools used are Bingham's identity, results from the theory of invariant differential operators, and a lemma concerning zonal polynomials.     

On R∞ and Q∞-manifolds

We give a characterization of manifolds modeled on $\text{R}{}^{\mathrm{\infty }}\text{= dir lim}$ or $\text{R}{}^{\mathrm{n}}\text{Q}{}^{\mathrm{\infty }}\text{=dir lim Q}{}^{\mathrm{n}}$, where Q is the Hilbert cube, and elementary short proofs of the Open Embedding Theorem for these manifolds and the following theorem generalizing the Stability Theorem: Each fine homotopy equivalence between these manifolds is a near homeomorphism. Moreover we establish the Open Embedding Approximation Theorem.     

Almost resolvable decomposition of Kn1

In this paper, we show that the complete symmetric directed graph with n vertices $\text{K}{}_{\mathrm{n}}{}^1$ admits an almost resolvable decomposition into TT₃ (the transitive tournament on 3 vertices) or C₃ (the directed cycle of length 3) if and only if n ≡ 1(mod 3).     

The isometries of LP(Ω, X)

Let (Ω, ∑, μ) be a finite measure space and X a separable Banach space. We characterize the linear isometries of $\text{L}{}^{\mathrm{p}}\text{(Ω, X)}$ onto itself for 1 ? p < ∞, p ≠ 2 under the condition that X is not the l^p-direct sum of two nonzero spaces (for the same p). It is shown that T is such an isometry if and only if (Tf)(·) = S(·)h(·)(Φ(f))(·), where Φ is a set isomorphism of ∑ onto itself, S is a strongly measurable operator-valued map such that S(t) is a.e. an isometry of X onto itself, and h is a scalar function which is related to Φ. It is further shown that for a big class of measure spaces (perhaps all nontrivial ones) the condition on X is also a necessary condition for the above conclusion to hold. In the case when X is a Hilbert space the injective isometries of $\text{L}{}^{\mathrm{p}}\text{(Ω, X)}$ are also characterized. They have the same form as above, except that Φ and S(t) are not necessarily onto.     

On the square root of a positive B(X,X1)-valued function

Let (Ω, $\text{B}$, μ) be a measure space, $\text{X}$ a separable Banach space, and $\text{X}$¹ the space of all bounded conjugate linear functionals on $\text{X}$. Let f be a weak¹ summable positive B($\text{X, X}$¹)-valued function defined on Ω. The existence of a separable Hilbert space $\text{K}$, a weakly measurable B($\text{X, K}$)-valued function Q satisfying the relation $\text{Q}{}^1\text{(ω)Q(ω) = f(ω)}$ is proved. This result is used to define the Hilbert space L_2,f of square integrable operator-valued functions with respect to f. It is shown that for B⁺($\text{X, X}$¹)-valued measures, the concepts of weak¹, weak, and strong countable additivity are all the same. Connections with stochastic processes are explained.     

Extensions of factorial states of C1-algebras

Let A be a $\text{C}{}^1$-algebra, B be a $\text{C}{}^1$-subalgebra of A, and φ be a factorial state of B. Sometimes, φ may be extended to a factorial state of A by a tensor product method of Sakai (“$\text{C}{}^1$-algebras and $\text{W}{}^1$-algebras, Springer-Verlag, Berlin/Heidelberg/ New York 1971”). Sometimes, there is a weak expectation of A into $\text{π}{}_{\mathrm{\phi }}\text{(B)}$, and then factorial extensions may be found by a method of Sakai and Tsui (Yokohama Math. J.29 (1981), 157–160). These two methods are shown to have the same effect, and the factorial extensions produced by them are analysed.     

Approximation pour la distance de WassersteinApproximation for the Wasserstein distance

Let $\text{C}$ be a set of probability-measures and P a probability on $\text{R}{}^{\mathrm{d}}$. Under some conditions, we show that we have a solution to the approximation problem of P by $\text{C}\text{.}$ There exists a probability $\text{Q}{}_0\text{∈}\text{C}$, such that $\text{l(P,Q}{}_0\text{)?l(P,Q),}\text{?Q∈}\text{C}$ where l is the square of the Wasserstein distance. To cite this article: N. Belili, H. Heinich, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 537–540.     

Convergence of convex integrals in Lp spaces

We consider functionals of the form: I_f(u) = ∝_Tf[t, u(t)]μ(dt), which are defined on spaces L^p(T, R^k), and we study for these functionals the properties of a convergence for which the conjugacy $\text{I}{}_{\mathrm{f}}\text{→ I}{}_{\mathrm{f1}}$ is a continuous operator.     

Zéros des fonctions L et formes toriquesZeroes of L-functions and toric forms

An algebraic number field K defines a maximal torus T of the linear group G=GL_n. Let χ be a character of the idele class group of K, satisfying suitable assumptions. The χ-toric form are the functions defined on $\text{G}{}_{\mathrm{<mtext>Q</mtext>}}\text{Z}{}_{\mathrm{<mtext>A</mtext>}}\text{?G}{}_{\mathrm{<mtext>A</mtext>}}$ such that the Fourier coefficient corresponding to χ with respect to the subgroup induced by T is zero. The Riemann hypothesis is equivalent to certain conditions concerning some spaces of toric forms, constructed from Eisenstein series. Furthermore, we define a Hilbert space and a self-adjoint operator on this space, whose spectrum equals the set of zeroes of L(s,χ) on the critical line. To cite this article: G. Lachaud, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 219–222.     

A Hamiltonian decomposition of K2m1, 2m ≥ 8

It is shown that $\text{K}{}_{\mathrm{2m}}{}^1$, 2m ≥ 8, can be decomposed into Hamiltonian circuits. A direct construction utilizing difference methods is given for 2m ≡ 0 (mod 4). The case 2m ≡ 2 (mod 4) is handled inductively by means of a construction which shows that $\text{K}{}_{\mathrm{4m\; ?\; 2}}{}^1$ admits such a decomposition if $\text{K}{}_{\mathrm{2m}}{}^1$ does.