Schur Q-polynomials, multiple hypergeometric series and enumeration of marked shifted tableaux

We study Schur Q-polynomials evaluated on a geometric progression, or equivalently q-enumeration of marked shifted tableaux, seeking explicit formulas that remain regular at q=1. We obtain several such expressions as multiple basic hypergeometric series, and as determinants and pfaffians of continuous q-ultraspherical or continuous q-Jacobi polynomials. As special cases, we obtain simple closed formulas for staircase-type partitions.     

A new polynomial sequence as a guide to numerical solutions for applied‐physics‐related partial differential equations under Dirichlet‐Newman‐type exogenous boundary conditions

In this article, a new set of polynomials is proposed as a new tool for solving bivaried partial differential equation related with some appropriated applied physics problems. A concrete formulation of canonical expressions, with respect to more common boundary conditions, is yielded out of the recently established Boubaker polynomials. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Direct amplitude detection of Zernike modes by computer-generated holographic wavefront sensor: Modeling and simulation

A fast holographic wavefront sensor is proposed using a computer-generated hologram (CGH). This CGH is a multiplexed hologram of different Zernike mode–amplitude combinations, and is designed in such a manner as to get the corresponding spots on the detector according to the presence and strength of a particular aberration. Interference between the aberrated wavefront (with a single mode–amplitude combination) and the Fourier transform of an image with single bright pixel (defined as dot image) is numerically calculated for one hologram. Different mode–amplitude combination and corresponding different positions of bright pixels (dots) are taken to compute various holograms and then all the holograms are multiplexed to get the final hologram. When the aberrated wavefront with a particular mode–amplitude combination is incident onto the multiplexed hologram, the corresponding dot is generated in the Fourier plane. A lens performs the Fourier transform in optical domain and provides the instant detection of amplitude of the respective Zernike mode. The main advantage of the scheme is to avoid the need of any computations, which makes it really fast. The simulation results are presented with the cross-talk analysis for few Zernike terms.     

Algebras Generated by Linearly Dependent Elements with Prescribed Spectra

We consider associative algebras presented by a finite set of generators and relations of special form: each generator is annihilated by some polynomial, and the sum of generators is zero. The growth of this algebra in dependence on the degrees of the polynomials annihilating the generators is studied. The tuples of degrees for which the algebras are finite-dimensional, have polynomial growth, or have exponential growth are indicated. To the tuple of degrees, we assign a graph, and the above-mentioned cases correspond to Dynkin diagrams, extended Dynkin diagrams, and the other graphs, respectively. For extended Dynkin diagrams, we indicate the hyperplane in the space of parameters (roots of the polynomials) on which the corresponding algebras satisfy polynomial identities.     

Digital computation of transient characteristics of an induction motor on switched supply

A novel method of computing the nonlinear differential equations relating to the transient behaviour of an induction motor using a polynomial series, is presented. The stator and rotor currents and the angle of rotation are expressed as polynomial series dependent on time. These are then substituted into the differential equations of the induction motor to compute the polynomial coefficients and, consequently, the transient quantities. Since the equations are nonlinear, the computations are carried out using the step-by-step method. The stator and rotor currents, speed and torque are calculated for acceleration and braking conditions. The results are compared both analytically and experimentally.     

Finite-Time Optimal Control of Polynomial Systems Using Successive Suboptimal Approximations

It is shown in this paper that the finite-time optimal control of polynomial systems can be obtained by solving a sequence of optimal control problems for the linearized problem. The paper provides proof of convergence as well as illustration of the procedure by two examples.     

Mathematical Relationships Between Bond-Bending Force Fields

Molecular mechanics softwares adopt various set of force field functions. In some cases, reliable data from one set of force field parameters cannot be used in a software that adopts another set of force field. Using mathematical approach, exact relationships between parameters from three bond-bending force fields, namely the (i) harmonic cosine angle, (ii) polynomial series, and (iii) Fourier series, are herein developed. Parameters from these three potential functions are further related to the approximate form, the harmonic angle function, which is valid for small change in chemical bond angle.     

THE UNIQUENESS OF LIMIT CYCLE AND THE STRUCTURE OF CRITICAL POINT AT INFINITY FOR A CLASS OF CUBIC SYSTEM

A class of cubic system, which is an accompany system of a quadratic differential one, is studied. It is proved that the system has at most one limit cycle, and the critical point at infinity is a higher order one. The structure and algebraic character of the critical point at infinity are obtained.     

Supersolvable LL-lattices of binary trees

Some posets of binary leaf-labeled trees are shown to be supersolvable lattices and explicit EL-labelings are given. Their characteristic polynomials are computed, recovering their known factorization in a different way.     

Tutte uniqueness of line graphs

We prove that if a graph H has the same Tutte polynomial as the line graph of a d-regular, d-edge-connected graph, then H is the line graph of a d-regular graph. Using this result, we prove that the line graph of a regular complete t-partite graph is uniquely determined by its Tutte polynomial. We prove the same result for the line graph of any complete bipartite graph.