The indeterminate moment problem for the qq-Meixner polynomials

For a class of orthogonal polynomials related to the $q$-Meixner polynomials corresponding to an indeterminate moment problem we give a one-parameter family of orthogonality measures. For these measures we complement the orthogonal polynomials to an orthogonal basis for the corresponding weighted $L^2$-space explicitly. The result is proved in two ways; by a spectral decomposition of a suitable operator and by direct series manipulation. We discuss extensions to explicit non-positive measures and the relation to other indeterminate moment problems for the continuous $q^{-1}$-Hahn and $q$-Laguerre polynomials.Comment: 26 page

Geometry of invariant domains in complex semi-simple Lie groups

We investigate the joint action of two real forms of a semi-simple complex Lie group S by left and right multiplication. After analyzing the orbit structure, we study the CR structure of closed orbits. The main results are an explicit formula of the Levi form of closed orbits and the determination of the Levi cone of generic orbits. Finally, we apply these results to prove q-completeness of certain invariant domains in S.Comment: 20 page

Crossings, Motzkin paths and Moments

Kasraoui, Stanton and Zeng, and Kim, Stanton and Zeng introduced certain $q$-analogues of Laguerre and Charlier polynomials. The moments of these orthogonal polynomials have combinatorial models in terms of crossings in permutations and set partitions. The aim of this article is to prove simple formulas for the moments of the $q$-Laguerre and the $q$-Charlier polynomials, in the style of the Touchard-Riordan formula (which gives the moments of some $q$-Hermite polynomials, and also the distribution of crossings in matchings). Our method mainly consists in the enumeration of weighted Motzkin paths, which are naturally associated with the moments. Some steps are bijective, in particular we describe a decomposition of paths which generalises a previous construction of Penaud for the case of the Touchard-Riordan formula. There are also some non-bijective steps using basic hypergeometric series, and continued fractions or, alternatively, functional equations.Comment: 21 page

Dynamic Factors in the Presence of Block Structure

Macroeconometric data often come under the form of large panels of time series, themselves decomposing into smaller but still quite large subpanels or blocks. We show how the dynamic factor analysis method proposed in Forni et al (2000), combined with the identification method of Hallin and Liska (2007), allows for identifying and estimating joint and block-specific common factors. This leads to a more sophisticated analysis of the structures of dynamic interrelations within and between the blocks in such datasets, along with an informative decomposition of explained variances. The method is illustrated with an analysis of the Industrial Production Index data for France, Germany, and Italy.Panel data; Time series; High dimensional data; Dynamic factor model; Business cycle; Block specific factors; Dynamic principal components; Information criterion.

Hierarchical Orthogonal Matrix Generation and Matrix-Vector Multiplications in Rigid Body Simulations

In this paper, we apply the hierarchical modeling technique and study some numerical linear algebra problems arising from the Brownian dynamics simulations of biomolecular systems where molecules are modeled as ensembles of rigid bodies. Given a rigid body $p$ consisting of $n$ beads, the $6 \times 3n$ transformation matrix $Z$ that maps the force on each bead to $p$'s translational and rotational forces (a $6\times 1$ vector), and $V$ the row space of $Z$, we show how to explicitly construct the $(3n-6) \times 3n$ matrix $\tilde{Q}$ consisting of $(3n-6)$ orthonormal basis vectors of $V^{\perp}$ (orthogonal complement of $V$) using only $\mathcal{O}(n \log n)$ operations and storage. For applications where only the matrix-vector multiplications $\tilde{Q}{\bf v}$ and $\tilde{Q}^T {\bf v}$ are needed, we introduce asymptotically optimal $\mathcal{O}(n)$ hierarchical algorithms without explicitly forming $\tilde{Q}$. Preliminary numerical results are presented to demonstrate the performance and accuracy of the numerical algorithms

A Householder-based algorithm for Hessenberg-triangular reduction

The QZ algorithm for computing eigenvalues and eigenvectors of a matrix pencil $A - \lambda B$ requires that the matrices first be reduced to Hessenberg-triangular (HT) form. The current method of choice for HT reduction relies entirely on Givens rotations regrouped and accumulated into small dense matrices which are subsequently applied using matrix multiplication routines. A non-vanishing fraction of the total flop count must nevertheless still be performed as sequences of overlapping Givens rotations alternately applied from the left and from the right. The many data dependencies associated with this computational pattern leads to inefficient use of the processor and poor scalability. In this paper, we therefore introduce a fundamentally different approach that relies entirely on (large) Householder reflectors partially accumulated into block reflectors, by using (compact) WY representations. Even though the new algorithm requires more floating point operations than the state of the art algorithm, extensive experiments on both real and synthetic data indicate that it is still competitive, even in a sequential setting. The new algorithm is conjectured to have better parallel scalability, an idea which is partially supported by early small-scale experiments using multi-threaded BLAS. The design and evaluation of a parallel formulation is future work

Global classical solutions for partially dissipative hyperbolic system of balance laws

This work is concerned with ($N$-component) hyperbolic system of balance laws in arbitrary space dimensions. Under entropy dissipative assumption and the Shizuta-Kawashima algebraic condition, a general theory on the well-posedness of classical solutions in the framework of Chemin-Lerner's spaces with critical regularity is established. To do this, we first explore the functional space theory and develop an elementary fact that indicates the relation between homogeneous and inhomogeneous Chemin-Lerner's spaces. Then this fact allows to prove the local well-posedness for general data and global well-posedness for small data by using the Fourier frequency-localization argument. Finally, we apply the new existence theory to a specific fluid model-the compressible Euler equations with damping, and obtain the corresponding results in critical spaces.Comment: 39 page