Polynomial Representation Research Articles

Algebraic theory of degenerate general bivariate Appell polynomials and related interpolation hints

The algebraic study of polynomials based on determinant representations is important in many fields of mathematics, ranging from algebraic geometry to optimization. The motivation to introduce determinant expressions of special polynomials comes from the fact that they are useful in scientific computing in solving systems of equations effectively. It is critical for this application to have determinant representations not just for single valued polynomials but also for bivariate polynomials. In this article, a family of degenerate general bivariate Appell polynomials is introduced. Several different explicit representations, recurrence relations, and addition theorems are established for this family. With the aid of different recurrence relations, we establish the determinant expressions for the degenerate general bivariate Appell polynomials. We also establish determinant definitions for degenerate general polynomials. Several examples are framed as the applications of this family and their graphical representations are shown. As concluding remarks, we propose a linear interpolation problem for these polynomials and some hints are provided.

Partial isospectrality of a matrix pencil and circularity of the c-numerical range

We study when functions of the eigenvalues of the pencil(1)Re(e−itA)=cos⁡(t)ReA+sin⁡(t)ImA are constant functions of t. The results are then applied to questions regarding the numerical range, the higher rank numerical range and the c-numerical range, and we derive trace type conditions for when these numerical ranges are disks centered at 0. The theory of symmetric polynomials plays an important part in the proofs.

Fast quasi-null-filling of radiation patterns for multiple solutions generation

Here we present an improved, rapid method for filling quasi-nulls in symmetrical radiation patterns synthesized by equispaced linear arrays, leading to the generation of multiple solutions. Considering the polynomial representation of the pattern, this null-filling is achieved by displacing the roots radially off the unit circle, keeping a constant displacement. This allows analyzing how the potential solutions vary with the quasi-uniform filling and the associated directivity loss. This method is based on the Cardano-Vieta relations, which link the coefficients of a complex Schelkunoff polynomial with its roots. As examples of application, we have considered a 20/100 element Dolph-Chebyshev pattern, with a spacing between the elements λ/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda / 2$$\\end{document}, side lobe level of − 20/− 28 dB and three inner sidelobes at − 40/− 50 dB.

Method of Performing Operations on the Elements of GF(2m) Using a Sparse Table

For the implementation of error-correcting codes, cryptographic algorithms, and the construction of homomorphic methods for privacy-preserving, there is a need for methods of performing operations on elements GF(2m) that have low computational complexity. This paper analyzes the existing methods of performing operations on the elements GF(2m) and proposes a new method based on the use of a sparse table of elements of this field. The object of research is the processes of operations in information security systems. The subject of research is methods and algorithms for performing operations on elements GF(2m). The purpose of this research is to develop and improve methods and algorithms for performing operations on elements GF(2m) to reduce their computational complexity. Empirical methods and methods of mathematical and software modeling are used in the research. Existing and proposed algorithms are implemented using the C# programming language in the Visual Studio 2015 development environment. Experimental research of existing and developed algorithms was carried out according to the proposed method, which allows to level the influence of additional parameters on the results of the research. The conducted research on methods for performing operations on the elements GF(2m) shows the expediency of using a sparse table of field elements. This approach makes it possible to reduce the amount of RAM required for the software and hardware implementation of the developed method compared to the classical tabular method, which requires storage of a full table of correspondence of the polynomial and index representation of the field elements. In addition, the proposed method gives an increase in speed of more than 4 times for the operations of calculating the multiplicative inverse element and exponentiation. As a result, the proposed method allows to reduce the computational complexity of error-correcting codes, cryptographic algorithms, and the homomorphic methods for privacy-preserving.

On polynomial representation by U-numbers

On polynomial representation by U-numbers

Crystal bases of parabolic Verma modules over the quantum orthosymplectic superalgebras

We show that there exists a unique crystal base of a parabolic Verma module over a quantum orthosymplectic superalgebra, which is induced from a q-analogue of a polynomial representation of a general linear Lie superalgebra.

Products, Polynomials and Differential Equations in the Stream Calculus

We study connections among polynomials, differential equations, and streams over a field 𝕂, in terms of algebra and coalgebra. We first introduce the class of (F,G) - products on streams, those where the stream derivative of a product can be expressed as a polynomial function of the streams and their derivatives. Our first result is that, for every (F,G) -product, there is a canonical way to construct a transition function on polynomials such that the resulting unique final coalgebra morphism from polynomials into streams is the (unique) commutative 𝕂-algebra homomorphism—and vice versa. This implies that one can algebraically reason on streams via their polynomial representation. We apply this result to obtain an algebraic-geometric decision algorithm for polynomial stream equivalence, for an underlying generic (F,G) -product. Finally, we extend this algorithm to solve a more general problem: finding all valid polynomial equalities that fit in a user specified polynomial template.

Two-variable q-Laguerre polynomials from the context of quasi-monomiality

We introduce the theory of two-variable q-Laguerre polynomials by means of a generating function involving zeroth-order q-Bessel Tricomi functions and we establish two-variable q-Laguerre polynomials from the context of quasi-monomiality. Then we obtain the operational representations and q-integrodifferential equations for two-variable q-Laguerre polynomials. Moreover, we introduce mth-order two-variable q-Laguerre polynomials, we deduce the q-partial differential equations and q-integrodifferential equations for mth-order two-variable q-Laguerre polynomials and we also examine the quasi-monomiality characteristics of these new q-special polynomials. Finally, we give some graphical representations of q-Laguerre polynomials.

Representation of positive polynomials on a generalized strip and its application to polynomial optimization

Representation of positive polynomials on a generalized strip and its application to polynomial optimization

Hypergraphs with Polynomial Representation: Introducing $r$-splits

Inspired by the split decomposition of graphs and rank-width, we introduce the notion of $r$-splits. We focus on the family of $r$-splits of a graph of order $n$, and we prove that it forms a hypergraph with several properties. We prove that such hypergraphs can be represented using only $\mathcal O(n^{r+1})$ of its hyperedges, despite its potentially exponential number of hyperedges. We also prove that there exist hypergraphs that need at least $\Omega(n^r)$ hyperedges to be represented, using a generalization of set orthogonality.

A note on the post quantum-Sheffer polynomial sequences

Abstract In this article, the post quantum analogue of Sheffer polynomial sequences is introduced using concepts of post quantum calculus. The series representation, recurrence relations, determinant expression and certain other properties of this class are established. Further, the 2D-post quantum-Sheffer polynomials are introduced via generating function and their properties are established. Certain identities and integral representations for the 2D-post quantum-Hermite polynomials, 2D-post quantum-Laguerre polynomials, and 2D-post quantum-Bessel polynomials are also considered.

Representations of quadratic Heisenberg-Weyl algebras and polynomials in the fourth Painlevé transcendent

<p>We provide new insights into the solvability property of a Hamiltonian involving the fourth Painlevé transcendent and its derivatives. This Hamiltonian is third-order shape invariant and can also be interpreted within the context of second supersymmetric quantum mechanics. In addition, this Hamiltonian admits third-order lowering and raising operators. We have considered the case when this Hamiltonian is irreducible, i.e., when no special solutions exist for given parameters $ \alpha $ and $ \beta $ of the fourth Painlevé transcendent $ P_{IV}(x, \alpha, \beta) $. This means that the Hamiltonian does not admit a potential in terms of rational functions (or the hypergeometric type of special functions) for those parameters. In such irreducible cases, the ladder operators are as well involving the fourth Painlevé transcendent and its derivative. An important case for which this occurs is when the second parameter (i.e., $ \beta $) of the fourth Painlevé transcendent $ P_{IV}(x, \alpha, \beta) $ is strictly positive, $ \beta > 0 $. This Hamiltonian was studied for all hierarchies of rational solutions that come in three families connected to the generalized Hermite and Okamoto polynomials. The explicit form of ladder, the associated wavefunctions involving exceptional orthogonal polynomials, and recurrence relations were also completed described. Much less is known for the irreducible case, in particular the excited states. Here, we developed a description of the induced representations based on various commutator identities for the highest and lowest weight type representations. We also provided for such representations a new formula concerning the explicit form of the related excited states from the point of view of the Schrödinger equation as two-variables polynomials that involve the fourth Painlevé transcendent and its derivative.</p>

Jan Stochel, a stellar mathematician

The occasion for this survey article was the 70th birthday of Jan Stochel, professor at Jagiellonian University, former head of the Chair of Functional Analysis and a prominent member of the Krak�w school of operator theory. In the course of his mathematical career, he has dealt, among other things, with various aspects of functional analysis, single and multivariable operator theory, the theory of moments, the theory of orthogonal polynomials, the theory of reproducing kernel Hilbert spaces, and mathematical aspects of quantum mechanics.

On the Phenomenon of Low-Frequency, Large-Amplitude Oscillations in a High-Dimensional Linear Dynamical System

This article provides an explanation of the observed phenomenon on the basis of a complete analytical solution of the system. The solution is constructed in a standard way by reducing the dynamic problem to the problem of eigenvalues and eigenvectors for the system matrix. When solving, we use methods from the theory of orthogonal polynomials. In addition, we discuss two physical interpretations of the system. The connection between these interpretations and the system is established through the Schrödinger variables.

Local theory of stable polynomials and bounded rational functions of several variables

Local theory of stable polynomials and bounded rational functions of several variables

Polynomial Representation of Biclique Neighborhood in Graphs

Polynomial Representation of Biclique Neighborhood in Graphs

NN2Poly: A Polynomial Representation for Deep Feed-Forward Artificial Neural Networks.

Interpretability of neural networks (NNs) and their underlying theoretical behavior remain an open field of study even after the great success of their practical applications, particularly with the emergence of deep learning. In this work, NN2Poly is proposed: a theoretical approach to obtain an explicit polynomial model that provides an accurate representation of an already trained fully connected feed-forward artificial NN a multilayer perceptron (MLP). This approach extends a previous idea proposed in the literature, which was limited to single hidden layer networks, to work with arbitrarily deep MLPs in both regression and classification tasks. NN2Poly uses a Taylor expansion on the activation function, at each layer, and then applies several combinatorial properties to calculate the coefficients of the desired polynomials. Discussion is presented on the main computational challenges of this method, and the way to overcome them by imposing certain constraints during the training phase. Finally, simulation experiments as well as applications to real tabular datasets are presented to demonstrate the effectiveness of the proposed method.

Analyzing the effects of suction and injection Reynolds number on the transport process in a hydromagnetic flow through a channel of reactive porous walls

The present paper analyzes the effects of suction and injection Reynolds number on the dispersion process of solute in an incompressible magneto-hydrodynamics flow through a channel. The walls of the channel are permeable and a first order heterogeneous reaction is applied on it. Also, an inclined magnetic field is employed at both walls of the channel. The unsteady convection-diffusion equation is solved semi-analytically to find out the coefficient of dispersion, skewness and kurtosis of the solute distribution for all time period. Using Hermite polynomial representation, the distribution of mean concentration is obtained. The effects of various parameters such as suction Reynolds number (R), injection Reynolds number (R′), boundary absorption parameter (β), inclination angle of the magnetic field (α), Hartmann number (M) and dispersion time (t) on dispersion process of the solute are examined. It is shown that with the increment of suction Reynolds number, the peak of the mean concentration of solute enhances prominently. But, when injection Reynolds number increases, the amplitude of the mean concentration decreases. The current research gives an insight to understand the complex behavior of transport process of the solute in a hydro-magnetic channel flow with reactive porous walls.

Random and mean Lyapunov exponents for

AbstractWe consider orthogonally invariant probability measures on $\operatorname {\mathrm {GL}}_n(\mathbb {R})$ and compare the mean of the logs of the moduli of eigenvalues of the matrices with the Lyapunov exponents of random matrix products independently drawn with respect to the measure. We give a lower bound for the former in terms of the latter. The results are motivated by Dedieu and Shub [On random and mean exponents for unitarily invariant probability measures on $\operatorname {\mathrm {GL}}_n(\mathbb {C})$ . Astérisque287 (2003), xvii, 1–18]. A novel feature of our treatment is the use of the theory of spherical polynomials in the proof of our main result.

Systematic methods for the synthesis of equidistant MIMO arrays

Abstract. Finding a feasible antenna arrangement for multiple input multiple output (MIMO) arrays to serve a specific purpose is a first crucial step towards a successful MIMO radar system design. Design methods to synthesize uniformly weighted and equidistant MIMO arrays are proposed and investigated. The methods can be used to gain a design foundation for 1D or 2D arrays without software tools or programming effort. Since the presented approach does not consider electromagnetic fields, electromagnetic full-wave simulations might be required additionally. The method is based on sequentially copying and displacing antenna groups with the help of a number scheme. A nomenclature is proposed to classify the degrees of freedom in the design procedure. If the antennas are aligned to a uniform grid, a polynomial representation of the array can be chosen alternatively. This method is beneficial when redundancies of a produced array and where they appear must be analyzed. A new design problem arises when an array is to consist of only transceiving antennas, which can be analyzed with polynomial multiplication. One strategy to find a suitable MIMO array consisting of transceiver elements is given and evaluated.