Formulas for zonal polynomials

New integral and differential formulas for zonal polynomials are proved. As illustrations, zonal polynomials corresponding to partitions of two parts are computed. A method is presented, based on a certain partial differential operator, for expressing an orthogonally invariant polynomial as a linear combination of zonals. Zonal polynomials are expressed as linear combinations of well-known symmetric polynomials.     

On differential properties for bivariate orthogonal polynomials

In this paper, we consider bivariate orthogonal polynomials associated with a quasi-definite moment functional which satisfies a Pearson-type partial differential equation. For these polynomials differential properties are obtained. In particular, we deduce some structure and orthogonality relations for the successive partial derivatives of the polynomials.      

Real perturbations of complex polynomials

In this article the dynamics of generic C^r (r ≥ 3) perturbations of complex polynomials are considered. The attention is focused on the determination of the existence of large or invariant components of the complement of the basin of ∞, where the interesting dynamics occur.     

Schubert Class and Cyclotomic NilHecke Algebras

Let (l) and n be positive integers such that l ≥ n,and let Gn,l be the Grassmannian which consists of the set of n-dimensional subspaces of Cl.There is a Z-graded algebra isomorphism between the cohomology H*(Gn,l,Z) of Gn,l and a nat-ural Z-form B of the Z-graded basic algebra of the type A cyclotomic nilHecke algebra(H)(0) =<ψ1,… ψn-1,y1,…,yn>.We show that the isomorphism can be chosen such that the image of each (geometrically defined) Schubert class (a1,...,an) coincides with the basis element bλ constructed by Hu and Liang by purely algebraic method,where 0 ≤ a1 ≤ a2 ≤ … ≤ an ≤ l-n with ai ∈ Z for each i,and λ is the l-multipartition of n associated to (l + 1-(an + n),l + 1-(an-1 + n-1),...,l + 1-(a1 + 1)).A similar correspondence between the Schubert class basis of the cohomology of the Grassmanni-an Gl-n,l and the bλ's basis (λ is an l-multipartition of n with each component being either (1) or empty) of the natural Z-form B of the Z-graded basic algebra of (H)(0)l,n is also obtained.As an application,we obtain a second version of the Giambelli formula for Schubert classes.     

Bases for Cooperations in K-Theory

Gaussian polynomials are used to define bases with good multiplicative properties for the algebra K _*(K) of cooperations in K-theory and for the invariants under conjugation.     

Symbolic Computation of Newton Sum Rules for the Zeros of Polynomial Eigenfunctions of Linear Differential Operators

A symbolic algorithm based on the generalized Lucas polynomials of first kind is used in order to compute the Newton sum rules for the zeros of polynomial eigenfunctions of linear differential operators with polynomial coefficients.     

Upper bounds on a two-term exponential sum*

We obtain upper bounds for mixed exponential sums of the type where p^m is a prime power with m⩾ 2 and X is a multiplicative character (mod p^m). If X is primitive or p⫮(a, b) then we obtain |S(χ,f,p ^m)| ⩽2np ^{2/3 m}. If X is of conductor p and p⫮( a, b) then we get the stronger bound |S(χ,f,p ^m)|⩽np ^m/2. This paper is dedicated to Prof. Wang Yuan on the occasion of his 70th birthday.     

Fractional revival and association schemes

Fractional revival occurs between two vertices in a graph if a continuous-time quantum walk unitarily maps the characteristic vector of one vertex to a superposition of the characteristic vectors of the two vertices. This phenomenon is relevant in quantum information in particular for entanglement generation in spin networks. We study fractional revival in graphs whose adjacency matrices belong to the Bose–Mesner algebra of association schemes. A specific focus is a characterization of balanced fractional revival (which corresponds to maximal entanglement) in graphs that belong to the Hamming scheme. Our proofs exploit the intimate connections between algebraic combinatorics and orthogonal polynomials.     

A new algorithm for fixed design regression and denoising

In this paper, we present a new algorithm to estimate a regression function in a fixed design regression model, by piecewise (standard and trigonometric) polynomials computed with an automatic choice of the knots of the subdivision and of the degrees of the polynomials on each sub-interval. First we give the theoretical background underlying the method: the theoretical performances of our penalized least-squares estimator are based on non-asymptotic evaluations of a mean-square type risk. Then we explain how the algorithm is built and possibly accelerated (to face the case when the number of observations is great), how the penalty term is chosen and why it contains some constants requiring an empirical calibration. Lastly, a comparison with some well-known or recent wavelet methods is made: this brings out that our algorithm behaves in a very competitive way in term of denoising and of compression.