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.     

On the spectral synthesis problem for hypersurfaces of RN

Let F¹(Rⁿ) denote the Fourier algebra on $\text{R}$ⁿ, and $\text{D}$($\text{R}$ⁿ) the space of test functions on $\text{R}$ⁿ. A closed subset E of $\text{R}$ⁿ is said to be of spectral synthesis if the only closed ideal J in F¹(Rⁿ) which has E as its hull

$\text{h(J)={x ? R}{}^{\mathrm{n}}\text{:f(x)=0}\text{for all}\text{f ? J}}$

is the ideal

$\text{k(E)={f?F}{}^1\text{(R}{}^{\mathrm{n}}\text{):f(E)=0}}$

. We consider sufficiently regular compact subsets of smooth submanifolds of $\text{R}$ⁿ with constant relative nullity. For such sets E we give an estimate of the degree of nilpotency of the algebra $\text{(k(E)∩D(R}{}^{\mathrm{n}}\text{))}{}^{\mathrm{?}}\text{j(E)}$, where j(E) denotes the smallest closed ideal in F¹(Rⁿ) with hull E. Especially in the case of hypersurfaces this estimate turns out to be exact. Moreover for this case we prove that k(E)∩D(Rⁿ) is dense in k(E). Together this solves the synthesis problem for such sets.     

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.     

Transcendence order over Qp in Cp

Let (K, ∥ · ∥) be a valued transcendence degree 1 extension of $\text{Q}$_p. An element x ∈ K transcendental over $\text{Q}$_p is said to have order ≤a (a > 0) if there exists C_x > 0 such that every polynomial P(X) ∈ $\text{Q}$_p [X] satisfies

$\text{?}\text{log;}\text{(P(x))? ?}\text{log}\text{∥P∥+c}{}_{\mathrm{x}}\text{(}\text{deg}\text{P)}{}^{\mathrm{a}}$

when ∥ · ∥ is the Gauss norm on $\text{Q}$_p[X]. No x ∈ $\text{C}$_p can have order ≤α if α < 1 but we construct some x ∈ $\text{C}$_p with order ≤ 1. Furthermore, we prove order ≤α is stable by algebraic extension.     

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.     

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.     

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.     

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 ∞.     

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.     

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.     

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.     

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.     

Large time behaviour of the solutions of a semilinear parabolic equation in RN

The asymptotic behaviour as t tends to +∞ of the solution of $\text{(?u}\text{?t)}\text{? Δu + u¦u¦}{}^{\mathrm{p\; ?\; 1}}\text{= 0}$ in $\text{R}$^N × $\text{R}$⁺, p > 1, was studied. It was proved that the behaviour depends strongly on the sign of $\text{(N + 2)}\text{N ? p}$ and also on the rate of decay of the admissible initial data u(0, x) as $\text{¦x¦}$ tends to +∞.     

Spectre de A(\̄gW) pour des domaines bornés faiblement pseudoconvexes réguliers

If Ω is a weakly pseudoconvex domain in a Stein manifold, then the spectrum of the Frechet Algebra $\text{A}{}^{\mathrm{\infty }}\text{(}\text{}\text{?}\text{gW)}$ is isomorphic to $\text{}\text{?}\text{gW, A}{}^{\mathrm{\infty }}\text{(}\text{}\text{?}\text{gW)}$ denotes the space of holomorphic functions smooth up to the boundary. The spectrum of the uniform algebra $\text{A(}\text{}\text{?}\text{gW)}$ is also isomorphic to -Ω. As a corollary we prove an approximation theorem for plurisubharmonic functions in $\text{Ω}$ continuous in -Ω.     

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).     

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.     

Frontière de Martin des marches aléatoires sur certains hypergroupes d-dimensionnelsMartin boundary of random walks on certain d-dimensional hypergroups

We give an integral representation formula for harmonic functions of Markov chains on $\text{N}{}^{\mathrm{d}}$ and $\text{R}{}_{\mathrm{+}}{}^{\mathrm{d}}$ which transition probability is invariant by translations of a hypergroup, product of polynomial hypergroups for $\text{N}{}^{\mathrm{d}}$ and product of Sturm–Liouville hypergroups for $\text{R}{}_{\mathrm{+}}{}^{\mathrm{d}}$. To cite this article: L. Godefroy, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 1029–1034.     

Lifting commuting pairs of C1 algebras

Given a commuting pair $\text{A}$₁, $\text{A}$₂ of abelian $\text{C}{}^1$ subalgebras of the Calkin algebra, we look for a commuting pair $\text{B}$₁,$\text{B}$₂ of $\text{C}{}^1$ subalgebras of $\text{B(H)}$ which project onto $\text{A}$₁ and $\text{A}$₂. We do not insist that $\text{B}$_i, be abelian, so $\text{B}$_i, may contain nontrivial compact operators. If X is the joint spectrum σ($\text{A}$₁, $\text{A}$₂), it is shown that the existence of a pair $\text{B}$₁, $\text{B}$₂ depends only on the element τ in Ext(X) determined by $\text{A}$₁, $\text{A}$₂. The set L(X) of those τ in Ext(X) which “lift” in this sense is shown to be a subgroup of Ext(X) when Ext(X) is Hausdorff, and also when $\text{A}$_i are singly generated. In this latter case, L(X) can be explicitly calculated for large classes of joint spectra. These results are applied to lift certain pairs of commuting elements of the Calkin algebra to pairs of commuting operators.     

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.