scholarly journals Combinatorics of KP hierarchy structural constants

2021 ◽  
Vol 81 (12) ◽  
Author(s):  
A. Andreev ◽  
A. Popolitov ◽  
A. Sleptsov ◽  
A. Zhabin
Keyword(s):  
Initial Data ◽  
Integral Representation ◽  
Internal Structure ◽  
Generating Functions ◽  
Transport Coefficients ◽  
Mathematical Physics ◽  
Combinatorial Structure ◽  
Kp Hierarchy ◽  
Topological Recursion ◽  
Structure Research

AbstractWe investigate the structural constants of the KP hierarchy, which appear as universal coefficients in the paper of Natanzon–Zabrodin arXiv:1509.04472. It turns out that these constants have a combinatorial description in terms of transport coefficients in the theory of flow networks. Considering its properties we want to point out three novel directions of KP combinatorial structure research: connection with topological recursion, eigenvalue model for the structural constants and its deformations, possible deformations of KP hierarchy in terms of the structural constants. Firstly, in this paper we study the internal structure of these coefficients which involves: (1) construction of generating functions that have interesting properties by themselves; (2) restrictions on topological recursion initial data; (3) construction of integral representation or matrix model for these coefficients with non-trivial Ward identities. This shows that these coefficients appear in various problems of mathematical physics, which increases their value and significance. Secondly, we discuss their role in integrability of KP hierarchy considering possible deformation of these coefficients without changing the equations on $$\tau $$ τ -function. We consider several plausible deformations. While most failed even very basic checks, one deformation (involving Macdonald polynomials) passes all the simple checks and requires more thorough study.

NEW PROPERTIES OF QUASIPERIODIC SOLUTIONS OF THE COMPLEX GINZBURG-LANDAU EQUATION

1992 ◽  
Vol 02 (04) ◽  
pp. 955-972 ◽  
Author(s):  
TATIANA S. AKHROMEYEVA ◽  
GEORGE G. MALINETSKII ◽  
ALEXEY B. POTAPOV ◽  
GEORGE Z. TSERTSVADZE
Keyword(s):  
Numerical Methods ◽  
Initial Data ◽  
Landau Equation ◽  
Mathematical Physics ◽  
Ginzburg Landau ◽  
New Class ◽  
A Value ◽  
Ginzburg Landau Equation ◽  
New Type ◽  
No Fluxes

By using analytical and numerical methods the authors study one of the basic models of mathematical physics—the so-called complex Ginzburg-Landau equation [Formula: see text] with the provision that no fluxes exist at the segment boundaries. A new class of solutions is found for this equation. It is shown that among its solutions there are analogs of limiting cycles of the second kind. A value describing these analogs is introduced, and a scenario of its variation depending on the parameters of the problem is given. A new type of spontaneous appearance of symmetry is shown when we go from initial data in the general form to spatially symmetrical solutions describing quasiperiodic regimes.

VI.—Generating Functions of Certain Continuous Orthogonal Systems

1941 ◽  
Vol 61 (1) ◽  
pp. 61-70
Author(s):  
A. Erdélyi
Keyword(s):  
Generating Functions ◽  
Point Spectrum ◽  
Formal Theory ◽  
Mathematical Physics ◽  
Hypergeometric Type ◽  
Orthogonal Systems ◽  
Mixed Systems ◽  
Orthogonal Sequences

Summary10. Generating functions and bilinear generating functions (of the type of Mehler's celebrated formula) are known to be of great importance in the formal theory of orthogonal sequences. The present paper contains analogous formulae for a number of continuous orthogonal systems as well as “mixed” systems (which have a point spectrum as well as a continuous one). Four systems of the hypergeometric type have been selected as examples which are thought to be of some importance because of their presenting themselves in certain problems of Mathematical Physics.My thanks are due to the Carnegie Trust for the Universities of Scotland for grants towards the printing of this paper and my paper in Proceedings, vol. lx, no. 26, 1940.

scholarly journals Miscellaneous Properties of the Gamma Distribution Polynomials

2019 ◽  
Author(s):  
Ugur Duran ◽  
Mehmet Acikgoz
Keyword(s):  
Integral Representation ◽  
Explicit Formula ◽  
Gamma Distribution ◽  
Generating Functions ◽  
Euler Polynomials ◽  
Addition Formula ◽  
Special Polynomial

The main aim of this paper is to investigate multifarious properties and relations for the gamma distribution. The approach to reach this purpose will be introducing a special polynomial including gamma distribution. Several formulas covering addition formula, derivative property, integral representation and explicit formula are derived by means of the series manipulation method. Furthermore, two correlations including Bernoulli and Euler polynomials for gamma distribution polynomials are provided by utilizing of their generating functions.

Internal Structure Research of Shungite by Broadband Ultrasonic Spectroscopy

2017 ◽  
Vol 755 ◽  
pp. 242-247 ◽  
Author(s):  
Elena B. Cherepetskaya ◽  
Alexander A. Karabutov ◽  
Vladimir A. Makarov ◽  
Elena A. Mironova ◽  
Ivan A. Shibaev ◽  
...  
Keyword(s):  
Internal Structure ◽  
Wave Velocity ◽  
Elastic Waves ◽  
Elastic Moduli ◽  
Young Modulus ◽  
Ultrasonic Waves ◽  
Parallel Plates ◽  
Ultrasonic Spectroscopy ◽  
Accuracy Of Measurements ◽  
Structure Research

The internal structure of plane-parallel plates of shungite is studied. The broadband ultrasonic pulses are used to measure the velocities of longitudinal and shear elastic ultrasonic waves. The accuracy of measurements is 0.3% in the case of longitudinal wave velocity and 0.5% in the case of shear wave velocity (scanning pitch over the surface of specimens was 0.5 mm). Local elastic moduli of shungite (Young modulus, shear modulus and Poisson's ratio) are uniquely determined from the velocities of elastic waves.

scholarly journals On Bers Generating Functions for First Order Systems of Mathematical Physics

2010 ◽  
Vol 21 (3) ◽  
pp. 547-559 ◽  
Author(s):  
Vladislav V. Kravchenko ◽  
Marco P. Ramirez T.
Keyword(s):  
Generating Functions ◽  
Mathematical Physics ◽  
First Order

scholarly journals Counting Markov Types

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Philippe Jacquet ◽  
Charles Knessl ◽  
Wojciech Szpankowski
Keyword(s):  
Information Theory ◽  
Integral Representation ◽  
Generating Functions ◽  
Diophantine Equations ◽  
Analytical Techniques ◽  
Saddle Point Method ◽  
Empirical Distributions ◽  
Linear Diophantine Equations ◽  
Markov Source ◽  
International Audience

International audience The method of types is one of the most popular techniques in information theory and combinatorics. Two sequences of equal length have the same type if they have identical empirical distributions. In this paper, we focus on Markov types, that is, sequences generated by a Markov source (of order one). We note that sequences having the same Markov type share the same so called $\textit{balanced frequency matrix}$ that counts the number of distinct pairs of symbols. We enumerate the number of Markov types for sequences of length $n$ over an alphabet of size $m$. This turns out to coincide with the number of the balanced frequency matrices as well as with the number of special $\textit{linear diophantine equations}$, and also balanced directed multigraphs. For fixed $m$ we prove that the number of Markov types is asymptotically equal to $d(m) \frac{n^{m^{2-m}}}{(m^2-m)!}$, where $d(m)$ is a constant for which we give an integral representation. For $m \to \infty$ we conclude that asymptotically the number of types is equivalent to $\frac{\sqrt{2}m^{3m/2} e^{m^2}}{m^{2m^2} 2^m \pi^{m/2}} n^{m^2-m}$ provided that $m=o(n^{1/4})$ (however, our techniques work for $m=o(\sqrt{n})$). These findings are derived by analytical techniques ranging from multidimensional generating functions to the saddle point method.

scholarly journals The Translated Dowling Polynomials and Numbers

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Mahid M. Mangontarum ◽  
Amila P. Macodi-Ringia ◽  
Normalah S. Abdulcarim
Keyword(s):  
Integral Representation ◽  
Explicit Formula ◽  
Generating Function ◽  
Generating Functions ◽  
Hankel Transform ◽  
Bell Polynomials ◽  
Basic Properties ◽  
Whitney Numbers ◽  
Exponential Generating Function

More properties for the translated Whitney numbers of the second kind such as horizontal generating function, explicit formula, and exponential generating function are proposed. Using the translated Whitney numbers of the second kind, we will define the translated Dowling polynomials and numbers. Basic properties such as exponential generating functions and explicit formula for the translated Dowling polynomials and numbers are obtained. Convexity, integral representation, and other interesting identities are also investigated and presented. We show that the properties obtained are generalizations of some of the known results involving the classical Bell polynomials and numbers. Lastly, we established the Hankel transform of the translated Dowling numbers.

scholarly journals Gottlieb Polynomials and Their q-Extensions

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1499
Author(s):  
Esra ErkuŞ-Duman ◽  
Junesang Choi
Keyword(s):  
Power Series ◽  
Integral Representation ◽  
Orthogonal Polynomials ◽  
Generating Functions ◽  
Discrete Orthogonal Polynomials ◽  
Discrete Orthogonal ◽  
Finite Power

Since Gottlieb introduced and investigated the so-called Gottlieb polynomials in 1938, which are discrete orthogonal polynomials, many researchers have investigated these polynomials from diverse angles. In this paper, we aimed to investigate the q-extensions of these polynomials to provide certain q-generating functions for three sequences associated with a finite power series whose coefficients are products of the known q-extended multivariable and multiparameter Gottlieb polynomials and another non-vanishing multivariable function. Furthermore, numerous possible particular cases of our main identities are considered. Finally, we return to Khan and Asif’s q-Gottlieb polynomials to highlight certain connections with several other known q-polynomials, and provide its q-integral representation. Furthermore, we conclude this paper by disclosing our future investigation plan.

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