scholarly journals On a class of biorthogonal polynomials on the unit circle

2016 ◽  
Vol 438 (1) ◽  
pp. 465-473 ◽  
Author(s):  
J. Borrego-Morell ◽  
Fernando Rodrigo Rafaeli
Keyword(s):  
Unit Circle ◽  
Biorthogonal Polynomials

scholarly journals An algebraic treatment of the Askey biorthogonal polynomials on the unit circle

2021 ◽  
Vol 9 ◽  
Author(s):  
Luc Vinet ◽  
Alexei Zhedanov
Keyword(s):  
Unit Circle ◽  
Jacobi Polynomials ◽  
Algebraic Interpretation ◽  
Algebraic Treatment ◽  
Biorthogonal Polynomials

Abstract A joint algebraic interpretation of the biorthogonal Askey polynomials on the unit circle and of the orthogonal Jacobi polynomials is offered. It ties their bispectral properties to an algebra called the meta-Jacobi algebra $m\mathfrak {J}$ .

scholarly journals Some results on biorthogonal polynomials

1994 ◽  
Vol 17 (4) ◽  
pp. 625-636 ◽  
Author(s):  
Richard W. Ruedemann
Keyword(s):  
Analytic Continuation ◽  
Unit Circle ◽  
Lebesgue Measure ◽  
Laurent Polynomials ◽  
Biorthogonal Polynomials

Some biorthogonal polynomials of Hahn and Pastro are derived using a polynomial modification of the Lebesgue measuredθcombined with analytic continuation. A result is given for changing the measures of biorthogonal polynomials on the unit circle by the multiplication of their measures by certain Laurent polynomials.

The Restricted Arc-Width of a Graph

10.37236/1734 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
David Arthur
Keyword(s):  
Unit Circle ◽  
Single Point ◽  
Minimum Width ◽  
Graph Operations ◽  
Function Mapping

An arc-representation of a graph is a function mapping each vertex in the graph to an arc on the unit circle in such a way that adjacent vertices are mapped to intersecting arcs. The width of such a representation is the maximum number of arcs passing through a single point. The arc-width of a graph is defined to be the minimum width over all of its arc-representations. We extend the work of Barát and Hajnal on this subject and develop a generalization we call restricted arc-width. Our main results revolve around using this to bound arc-width from below and to examine the effect of several graph operations on arc-width. In particular, we completely describe the effect of disjoint unions and wedge sums while providing tight bounds on the effect of cones.

scholarly journals A Discontinuity of the Energy of Quantum Walk in Impurities

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1134
Author(s):  
Kenta Higuchi ◽  
Takashi Komatsu ◽  
Norio Konno ◽  
Hisashi Morioka ◽  
Etsuo Segawa
Keyword(s):  
Unit Circle ◽  
Stationary State ◽  
Discrete Time ◽  
Quantum Walk ◽  
Time Limit ◽  
The Other ◽  
Unitary Matrix ◽  
Initial State ◽  
Time Quantum ◽  
Long Time

We consider the discrete-time quantum walk whose local dynamics is denoted by a common unitary matrix C at the perturbed region {0,1,⋯,M−1} and free at the other positions. We obtain the stationary state with a bounded initial state. The initial state is set so that the perturbed region receives the inflow ωn at time n(|ω|=1). From this expression, we compute the scattering on the surface of −1 and M and also compute the quantity how quantum walker accumulates in the perturbed region; namely, the energy of the quantum walk, in the long time limit. The frequency of the initial state of the influence to the energy is symmetric on the unit circle in the complex plain. We find a discontinuity of the energy with respect to the frequency of the inflow.

scholarly journals Testing Stability of Digital Filters Using Optimization Methods with Phase Analysis

Energies ◽  
2021 ◽  
Vol 14 (5) ◽  
pp. 1488
Author(s):  
Damian Trofimowicz ◽  
Tomasz P. Stefański
Keyword(s):  
Unit Circle ◽  
Phase Analysis ◽  
Characteristic Equation ◽  
Digital Filter ◽  
Digital Filters ◽  
Computing Time ◽  
Complex Function ◽  
Stochastic Methods ◽  
Final Population ◽  
Candidate Regions

In this paper, novel methods for the evaluation of digital-filter stability are investigated. The methods are based on phase analysis of a complex function in the characteristic equation of a digital filter. It allows for evaluating stability when a characteristic equation is not based on a polynomial. The operation of these methods relies on sampling the unit circle on the complex plane and extracting the phase quadrant of a function value for each sample. By calculating function-phase quadrants, regions in the immediate vicinity of unstable roots (i.e., zeros), called candidate regions, are determined. In these regions, both real and imaginary parts of complex-function values change signs. Then, the candidate regions are explored. When the sizes of the candidate regions are reduced below an assumed accuracy, then filter instability is verified with the use of discrete Cauchy’s argument principle. Three different algorithms of the unit-circle sampling are benchmarked, i.e., global complex roots and poles finding (GRPF) algorithm, multimodal genetic algorithm with phase analysis (MGA-WPA), and multimodal particle swarm optimization with phase analysis (MPSO-WPA). The algorithms are compared in four benchmarks for integer- and fractional-order digital filters and systems. Each algorithm demonstrates slightly different properties. GRPF is very fast and efficient; however, it requires an initial number of nodes large enough to detect all the roots. MPSO-WPA prevents missing roots due to the usage of stochastic space exploration by subsequent swarms. MGA-WPA converges very effectively by generating a small number of individuals and by limiting the final population size. The conducted research leads to the conclusion that stochastic methods such as MGA-WPA and MPSO-WPA are more likely to detect system instability, especially when they are run multiple times. If the computing time is not vitally important for a user, MPSO-WPA is the right choice, because it significantly prevents missing roots.

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