Generating functions for a class ofq-polynomials

Summary Some simple ideas are used here to prove a theorem on generating functions for a certain class of q-polynomials. This general theorem is then applied to derive a fairly large number of known as well as new generating functions for the familiar q-analogues of various polynomial systems including, for example, the classical orthogonal polynomials of Hermite, Jacobi, and Laguerre. A number of other interesting consequences of the theorem are also discussed.     

Approximation of Matrices and a Family of Gander Methods for Polar Decomposition

Two matrix approximation problems are considered: approximation of a rectangular complex matrix by subunitary matrices with respect to unitarily invariant norms and a minimal rank approximation with respect to the spectral norm. A characterization of a subunitary approximant of a square matrix with respect to the Schatten norms, given by Maher, is extended to the case of rectangular matrices and arbitrary unitarily invariant norms. Iterative methods, based on the family of Gander methods and on Higham’s scaled method for polar decomposition of a matrix, are proposed for computing subunitary and minimal rank approximants. Properties of Gander methods are investigated in details. AMS subject classification (2000) 65F30, 15A18     

Self-assembly of a linear mesoscale tetramer with electrical connectivity

Capillary interaction-based self-assembly of block-shaped mesoscale components into an electrically interconnected 1-D tetramer is reported. Low melting point solder droplets, selectively patterned on the faces of the blocks, were employed to drive the sequential alignment, registration, linking and electrical interconnection of each block. The solder patterns were designed so that successful assembly would only occur when the solder patterns on one block face were correctly aligned with those on the face of an adjacent block. For assembly, the blocks were agitated in a flask containing KBr solution. At 60 °C the solder was molten, and collisions between blocks enabled the solder menisci to easily interact. To minimize interfacial free energy, the menisci coalesced and quickly drove the interacting blocks to form a stable, registered and aligned assembly. When agitation was terminated and the solution cooled, the self-aligned, linear tetrameric arrangement of blocks was permanently captured by solder solidification, a process that provided good mechanical bonding and electrical interconnection between each block. PACS 81.16.Dn; 68.03.Cd; 85.40.-e     

A Genus-3 Topological Recursion Relation

In this paper, we give a new genus-3 topological recursion relation for Gromov-Witten invariants of compact symplectic manifolds. This formula also applies to intersection numbers on moduli spaces of spin curves. A by-product of the proof of this formula is a new relation in the tautological ring of the moduli space of 1-pointed genus-3 stable curves. Research of the first author was partially supported by NSF grant DMS-0204824 Research of the second author was partially supported by NSF grant DMS-0505835     

Asymptotic expansions in sequential estimation for the first-order random coefficient autoregressive model: Regenerative approach

Edgeworth expansions for the distribution of a sequential least squares estimator in the random coefficient autoregressive (RCA) model are derived. The regenerative approach to second-order asymptotic analysis of Markov-type statistical models is developed.     

Nonlinear magneto-optical diffraction by two-dimensional magnetic superstructures

Nonlinear (at the second-harmonic frequency of the incident light) optical reflection by two-dimensional magnetic superstructures is theoretically studied. A square lattice of magnetic dots and a hexagonal lattice of magnetic bubbles (cylindrical magnetic domains) are considered. Because the periods of these structures are comparable with the wavelengths of the fundamental and the second-harmonic radiation, it would be possible to observe diffraction at the second-harmonic frequency. A polarization analysis of nonlinearly diffracted radiation is performed and the numbers of observable diffraction orders for the above structures are estimated. Received: 10 January 2002 / Published online: 11 June 2002     

Nuclear effects in the hadroproduction and photoproduction ofJ/ψ

Hadroprduction ofJ/ψ in π^? nucleus collisions is studied in the context of the colour singlet model using nuclear structure functions from three different models of the EMC effect. We conclude that it is possible to understand the data on the large transverse momentum (p _T ) and small Feynmanx (x _F ) of theJ/ψ in terms of the nuclear dependence of structure functions alone. We further show that these data can be used to distinguish between different models of the EMC effect, and that a study of distributions in various kinematic variables in photoproduction ofJ/ψ should be able to very effectively supplement the information obtained from hadroproduction experiments.     

Mechanism of detachment of particles from the surface of high-temperature superconducting targets absorbing laser pulses

A cluster mechanism is considered for the detachment of particles when the surface of a high-temperature superconducting target absorbs laser pulses. By proposing a fractal nature for the surface, one can model it as an assembly of a large number of nonlinear oscillators (clusters). Utilizing the method of sudden perturbations, a calculation was made of the probability of dissociation of these oscillators, i.e., of the detachment of clusters from the surface. The calculated probability decreases rapidly as the power density of the laser radiation is reduced to some limiting value. This is in agreement with experiments. State Institute of Sensory Microelectronics. Siberian Branch of the Russian Academy of Sciences. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 73–77, June, 1997.     

Orthonormal polynomial bases in function spaces

We construct polynomial orthonormal bases in various function spaces. Our bases have linear order of growth of degrees of polynomials. We show that this order is optimal.