scholarly journals Sklyanin-like algebras for (q-)linear grids and (q-)para-Krawtchouk polynomials

2021 ◽  
Vol 62 (1) ◽  
pp. 013505
Author(s):  
Geoffroy Bergeron ◽  
Julien Gaboriaud ◽  
Luc Vinet ◽  
Alexei Zhedanov
Keyword(s):  
Krawtchouk Polynomials

New Set of Invariant Quaternion Krawtchouk Moments for Color Image Representation and Recognition

2021 ◽  
pp. 2250037
Author(s):  
Gaber Hassan ◽  
Khalid M. Hosny ◽  
R. M. Farouk ◽  
Ahmed M. Alzohairy
Keyword(s):  
Quaternion Algebra ◽  
Color Image ◽  
Image Representation ◽  
Color Images ◽  
Color Channel ◽  
Krawtchouk Polynomials ◽  
Krawtchouk Moments ◽  
Similarity Transformations ◽  
The Stability ◽  
Discrete Moments

One of the most often used techniques to represent color images is quaternion algebra. This study introduces the quaternion Krawtchouk moments, QKrMs, as a new set of moments to represent color images. Krawtchouk moments (KrMs) represent one type of discrete moments. QKrMs use traditional Krawtchouk moments of each color channel to describe color images. This new set of moments is defined by using orthogonal polynomials called the Krawtchouk polynomials. The stability against the translation, rotation, and scaling transformations for QKrMs is discussed. The performance of the proposed QKrMs is evaluated against other discrete quaternion moments for image reconstruction capability, toughness against various types of noise, invariance to similarity transformations, color face image recognition, and CPU elapsed times.

scholarly journals Quantum walks on graphs of the ordered Hamming scheme and spin networks

2019 ◽  
Vol 7 (1) ◽  
Author(s):  
Hiroshi Miki ◽  
Satoshi Tsujimoto ◽  
Luc Vinet
Keyword(s):  
Magnetic Fields ◽  
Quantum Walks ◽  
Perfect State ◽  
Spin Networks ◽  
Krawtchouk Polynomials ◽  
Energy Eigenstates ◽  
State Transfer ◽  
Perfect State Transfer ◽  
Local Magnetic Fields ◽  
Basis Vectors

It is shown that the hopping of a single excitation on certain triangular spin lattices with non-uniform couplings and local magnetic fields can be described as the projections of quantum walks on graphs of the ordered Hamming scheme of depth 2. For some values of the parameters the models exhibit perfect state transfer between two summits of the lattice. Fractional revival is also observed in some instances. The bivariate Krawtchouk polynomials of the Tratnik type that form the eigenvalue matrices of the ordered Hamming scheme of depth 2 give the overlaps between the energy eigenstates and the occupational basis vectors.

scholarly journals Reliable Recurrence Algorithm for High-Order Krawtchouk Polynomials

Entropy ◽  
2021 ◽  
Vol 23 (9) ◽  
pp. 1162
Author(s):  
Khaled A. AL-Utaibi ◽  
Sadiq H. Abdulhussain ◽  
Basheera M. Mahmmod ◽  
Marwah Abdulrazzaq Naser ◽  
Muntadher Alsabah ◽  
...  
Keyword(s):  
Recurrence Relation ◽  
Coding Theory ◽  
State Of The Art ◽  
Reconstruction Error ◽  
Krawtchouk Polynomials ◽  
Polynomial Size ◽  
Main Challenge ◽  
Recurrence Algorithm ◽  
Symmetry Relations ◽  
Parameter Values

Krawtchouk polynomials (KPs) and their moments are promising techniques for applications of information theory, coding theory, and signal processing. This is due to the special capabilities of KPs in feature extraction and classification processes. The main challenge in existing KPs recurrence algorithms is that of numerical errors, which occur during the computation of the coefficients in large polynomial sizes, particularly when the KP parameter (p) values deviate away from 0.5 to 0 and 1. To this end, this paper proposes a new recurrence relation in order to compute the coefficients of KPs in high orders. In particular, this paper discusses the development of a new algorithm and presents a new mathematical model for computing the initial value of the KP parameter. In addition, a new diagonal recurrence relation is introduced and used in the proposed algorithm. The diagonal recurrence algorithm was derived from the existing n direction and x direction recurrence algorithms. The diagonal and existing recurrence algorithms were subsequently exploited to compute the KP coefficients. First, the KP coefficients were computed for one partition after dividing the KP plane into four. To compute the KP coefficients in the other partitions, the symmetry relations were exploited. The performance evaluation of the proposed recurrence algorithm was determined through different comparisons which were carried out in state-of-the-art works in terms of reconstruction error, polynomial size, and computation cost. The obtained results indicate that the proposed algorithm is reliable and computes lesser coefficients when compared to the existing algorithms across wide ranges of parameter values of p and polynomial sizes N. The results also show that the improvement ratio of the computed coefficients ranges from 18.64% to 81.55% in comparison to the existing algorithms. Besides this, the proposed algorithm can generate polynomials of an order ∼8.5 times larger than those generated using state-of-the-art algorithms.

scholarly journals Composition Markov chains of multinomial type

2009 ◽  
Vol 41 (01) ◽  
pp. 270-291 ◽  
Author(s):  
Hua Zhou ◽  
Kenneth Lange
Keyword(s):  
Markov Chain ◽  
Random Walks ◽  
Continuous Time ◽  
Spectral Decomposition ◽  
Equilibrium Distribution ◽  
Convergence To Equilibrium ◽  
Light Bulb ◽  
Krawtchouk Polynomials ◽  
Particle Counts ◽  
Coalescence Times

Suppose that n identical particles evolve according to the same marginal Markov chain. In this setting we study chains such as the Ehrenfest chain that move a prescribed number of randomly chosen particles at each epoch. The product chain constructed by this device inherits its eigenstructure from the marginal chain. There is a further chain derived from the product chain called the composition chain that ignores particle labels and tracks the numbers of particles in the various states. The composition chain in turn inherits its eigenstructure and various properties such as reversibility from the product chain. The equilibrium distribution of the composition chain is multinomial. The current paper proves these facts in the well-known framework of state lumping and identifies the column eigenvectors of the composition chain with the multivariate Krawtchouk polynomials of Griffiths. The advantages of knowing the full spectral decomposition of the composition chain include (a) detailed estimates of the rate of convergence to equilibrium, (b) construction of martingales that allow calculation of the moments of the particle counts, and (c) explicit expressions for mean coalescence times in multi-person random walks. These possibilities are illustrated by applications to Ehrenfest chains, the Hoare and Rahman chain, Kimura's continuous-time chain for DNA evolution, a light bulb chain, and random walks on some specific graphs.

scholarly journals Spin lattices, state transfer, and bivariate Krawtchouk polynomials

2015 ◽  
Vol 93 (9) ◽  
pp. 979-984 ◽  
Author(s):  
Vincent X. Genest ◽  
Hiroshi Miki ◽  
Luc Vinet ◽  
Alexei Zhedanov
Keyword(s):  
Three Dimensional ◽  
Rotation Group ◽  
Matrix Elements ◽  
Quantum State Transfer ◽  
Krawtchouk Polynomials ◽  
Triangular Domain ◽  
State Transfer ◽  
Exactly Solvable ◽  
Spin Lattices ◽  
Transfer Properties

The quantum state transfer properties of a class of two-dimensional spin lattices on a triangular domain are investigated. Systems for which the 1-excitation dynamics is exactly solvable are identified. The exact solutions are expressed in terms of the bivariate Krawtchouk polynomials that arise as matrix elements of the unitary representations of the rotation group on the states of the three-dimensional harmonic oscillator.

scholarly journals REPRESENTATIONS OF sl(2) IN THE BOOLEAN LATTICE, AND THE HAMMING AND JOHNSON SCHEMES

2012 ◽  
Vol 15 (03) ◽  
pp. 1250019
Author(s):  
PHILIP FEINSILVER
Keyword(s):  
Special Functions ◽  
Regular Representation ◽  
Boolean Lattice ◽  
Irreducible Representations ◽  
Leibniz Rule ◽  
Krawtchouk Polynomials ◽  
Lattice Representation ◽  
Boolean Poset ◽  
Group Law

Starting with the zero-square "zeon algebra," the regular representation gives rise to a Boolean lattice representation of sl(2). We detail the su(2) content of the Boolean lattice, providing the irreducible representations carried by the algebra generated by the subsets of an n-set. The group elements are found, exhibiting the "special functions" in this context. The corresponding Leibniz rule and group law are shown. Krawtchouk polynomials, the Hamming and the Johnson schemes appear naturally. Applications to the Boolean poset and the structure of Hadamard–Sylvester matrices are shown as well.

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