Integrals over Grassmannians and random permutations

Testing the independence of two Gaussian populations involves the distribution of the sample canonical correlation coefficients, given that the actual correlation is zero. The “Laplace transform” (as a function of x) of this distribution is not only an integral over the Grassmannian of p-dimensional planes in real, complex or quaternion n-space , but is also related to a generalized hypergeometric function. Such integrals are solutions of Painlevé-like equations; in the complex case, they are solutions to genuine Painlevé equations. These integrals over have remarkable expansions in x, related to random words of length ? formed with an alphabet of p letters 1,…,p. The coefficients of these expansions are given by the probability that a word (i) contains a subsequence of letters p,p−1,…,1 in that order and (ii) that the maximal length of the disjoint union of p−1 increasing subsequences of the word is ?k, where k refers to the power of x. Note that, if each letter appears in the word, then the maximal length of the disjoint union of p increasing subsequences of the word is automatically =? and is thus trivial.     

Painlevé V and a Pollaczek–Jacobi type orthogonal polynomials

We study a sequence of polynomials orthogonal with respect to a one-parameter family of weights defined for x∈[0,1]. If t=0, this reduces to a shifted Jacobi weight. Our ladder operator formalism and the associated compatibility conditions give an easy determination of the recurrence coefficients.For t>0, the factor induces an infinitely strong zero at x=0. With the aid of the compatibility conditions, the recurrence coefficients are expressed in terms of a set of auxiliary quantities that satisfy a system of difference equations. These, when suitably combined with a pair of Toda-like equations derived from the orthogonality principle, show that the auxiliary quantities are particular Painlevé V and/or allied functions.It is also shown that the logarithmic derivative of the Hankel determinant, satisfies the Jimbo–Miwa–Okamoto σ-form of the Painlevé V equation and that the same quantity satisfies a second-order non-linear difference equation which we believe to be new.     

The space of scalarly integrable functions for a Fréchet-space-valued measure

The space of all scalarly integrable functions with respect to a Fréchet-space-valued vector measure ν is shown to be a complete Fréchet lattice with the σ-Fatou property which contains the (traditional) space L¹(ν), of all ν-integrable functions. Indeed, L¹(ν) is the σ-order continuous part of . Every Fréchet lattice with the σ-Fatou property and containing a weak unit in its σ-order continuous part is Fréchet lattice isomorphic to a space of the kind .     

On duality between étale groupoids and Hopf algebroids

For any étale Lie groupoid G over a smooth manifold M, the groupoid convolution algebra of smooth functions with compact support on G has a natural coalgebra structure over the commutative algebra which makes it into a Hopf algebroid. Conversely, for any Hopf algebroid A over we construct the associated spectral étale Lie groupoid over M such that is naturally isomorphic to G. Both these constructions are functorial, and is fully faithful left adjoint to . We give explicit conditions under which a Hopf algebroid is isomorphic to the Hopf algebroid of an étale Lie groupoid G.     

Characteristic classes and transversality

Let ξ be a smooth vector bundle over a differentiable manifold M. Let be a generic bundle morphism from the trivial bundle of rank n−i+1 to ξ. We give a geometric construction of the Stiefel-Whitney classes when ξ is a real vector bundle, and of the Chern classes when ξ is a complex vector bundle. Using h we define a differentiable closed manifold and a map whose image is the singular set of h. The ith characteristic class of ξ is the Poincaré dual of the image, under the homomorphism induced in homology by ?, of the fundamental class of the manifold . We extend this definition for vector bundles over a paracompact space, using that the universal bundle is filtered by smooth vector bundles.     

On a theorem of Chowla

Let be a finite field with q=p^felements, where p is a prime number and f is a positive integer. For a nonprincipal multiplicative character χ and a nontrivial additive character ψ on , it is well known that Gauss sum has absolute value . In this paper, we investigate when is a root of unity.     

Compactifications of rational maps, and the implicit equations of their images

In this paper, we give different compactifications for the domain and the codomain of an affine rational map f which parameterizes a hypersurface. We show that the closure of the image of this map (with possibly some other extra hypersurfaces) can be represented by a matrix of linear syzygies. We compactify into an (n−1)-dimensional projective arithmetically Cohen-Macaulay subscheme of some . One particular interesting compactification of is the toric variety associated to the Newton polytope of the polynomials defining f. We consider two different compactifications for the codomain of f: and . In both cases we give sufficient conditions, in terms of the nature of the base locus of the map, for getting a matrix representation of its closed image, without involving extra hypersurfaces. This constitutes a direct generalization of the corresponding results established by Laurent Busé and Jean-Pierre Jouanolou (2003) [12], Laurent Busé et al. (2009) [9], Laurent Busé and Marc Dohm (2007) [11], Nicolás Botbol et al. (2009) [5] and Nicolás Botbol (2009) [4].     

Globalization of the partial isometries of metric spaces and local approximation of the group of isometries of Urysohn space

We prove the equivalence of the two important facts about finite metric spaces and universal Urysohn metric spaces U, namely Theorems A and G: Theorem A (Approximation): The group of isometry ISO(U) contains everywhere dense locally finite subgroup; Theorem G (Globalization): For each finite metric space F there exists another finite metric space and isometric imbedding j of F to such that isometry j induces the imbedding of the group monomorphism of the group of isometries of the space F to the group of isometries of space and each partial isometry of F can be extended up to global isometry in . The fact that Theorem G, is true was announced in 2005 by author without proof, and was proved by S. Solecki in [S. Solecki, Extending partial isometries, Israel J. Math. 150 (2005) 315-332] (see also [V. Pestov, The isometry group of the Urysohn space as a Lévy group, Topology Appl. 154 (10) (2007) 2173-2184; V. Pestov, A theorem of Hrushevski-Solecki-Vershik applied to uniform and coarse embeddings of the Urysohn metric space, math/0702207]) based on the previous complicate results of other authors. The theorem is generalization of the Hrushevski's theorem about the globalization of the partial isomorphisms of finite graphs. We intend to give a constructive proof in the same spirit for metric spaces elsewhere. We also give the strengthening of homogeneity of Urysohn space and in the last paragraph we gave a short survey of the various constructions of Urysohn space including the new proof of the construction of shift invariant universal distance matrix from [P. Cameron, A. Vershik, Some isometry groups of Urysohn spaces, Ann. Pure Appl. Logic 143 (1-3) (2006) 70-78].     

Notes sur la fonction ζ de Riemann, 4

Opération fondamentale de l'arithmétique, familière depuis des millénaires, la division euclidienne n'a pas livré tous ses secrets. Ainsi, notons pour k et a entiers positifs, le reste de la division euclidienne de k par a, et imaginons un instant que, par un choix convenable d'un entier n et de réels c₂,…,c_n, nous sachions rendre arbitrairement petite la quantité

     

Minimum number of ideal generators for a linear center perturbed by homogeneous polynomials

Using the algorithm presented in [J. Giné, X. Santallusia, On the Poincaré-Liapunov constants and the Poincaré series, Appl. Math. (Warsaw) 28 (1) (2001) 17-30] the Poincaré-Liapunov constants are calculated for polynomial systems of the form , , where P_n and Q_n are homogeneous polynomials of degree n. The objective of this work is to calculate the minimum number of ideal generators i.e., the number of functionally independent Poincaré-Liapunov constants, through the study of the highest fine focus order for n=4 and n=5 and compare it with the results that give the conjecture presented in [J. Giné, On the number of algebraically independent Poincaré-Liapunov constants, Appl. Math. Comput. 188 (2) (2007) 1870-1877]. Moreover, the computational problems which appear in the computation of the Poincaré-Liapunov constants and the determination of the number of functionally independent ones are also discussed.     

Almost periodic solutions of forced vectorial Liénard equations

We describe the set of bounded or almost periodic solutions of the following Liénard system: , where is almost periodic, is a symmetric and nonsingular linear operator, and ∇F denotes the gradient of the convex function F on R^N. Then, we state a result of existence and uniqueness of almost periodic solution.