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 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.

scholarly journals PRESBURGER ARITHMETIC, RATIONAL GENERATING FUNCTIONS, AND QUASI-POLYNOMIALS

2015 ◽  
Vol 80 (2) ◽  
pp. 433-449 ◽  
Author(s):  
KEVIN WOODS
Keyword(s):  
Generating Functions ◽  
Counting Function ◽  
Order Theory ◽  
Characteristic Functions ◽  
Presburger Arithmetic ◽  
First Order ◽  
First Order Theory ◽  
Rational Generating Function ◽  
Rational Generating Functions

AbstractPresburger arithmetic is the first-order theory of the natural numbers with addition (but no multiplication). We characterize sets that can be defined by a Presburger formula as exactly the sets whose characteristic functions can be represented by rational generating functions; a geometric characterization of such sets is also given. In addition, ifp= (p1, . . . ,pn) are a subset of the free variables in a Presburger formula, we can define a counting functiong(p) to be the number of solutions to the formula, for a givenp. We show that every counting function obtained in this way may be represented as, equivalently, either a piecewise quasi-polynomial or a rational generating function. Finally, we translate known computational complexity results into this setting and discuss open directions.

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 On modelling harmonic waves in linear micropolar elastic media by four screw fields

2020 ◽  
pp. 27-36
Author(s):  
Юрий Николаевич Радаев
Keyword(s):  
Differential Equations ◽  
Linear Theory ◽  
Harmonic Wave ◽  
Mathematical Physics ◽  
Harmonic Waves ◽  
Micropolar Elasticity ◽  
Alternative Analysis ◽  
First Order ◽  
Physical Fields ◽  
Vector Differential Equation

В статье рассматриваются дифференциальные уравнения для потенциалов, обеспечивающие выполнение связанных векторных дифференциальных уравнений линейной теории микрополярной упругости в случае гармонической зависимости поля перемещений и микровращений от времени. Предложена альтернативная схема расщепления связанных векторных дифференциальных уравнений микрополярной теории упругости для потенциалов на несвязанные уравнения первого порядка. Она основана на пропорциональности с разными масштабными факторами вихревых составляющих перемещений и микровращений одному вихревому винтовому полю. Найдено представление векторов перемещений и микровращений с помощью четырех винтовых векторов, обеспечивающее выполнимость связанных векторных дифференциальных уравнений линейной теории микрополярной упругости. В результате проблема нахождения вихревых составляющих перемещений и микровращений сводится к решению четырех несвязанных между собой векторных винтовых дифференциальных уравнений первого порядка с частными производными. Полученные результаты могут быть использованы в прикладных задачах механики, связанных с распространением гармонических (монохроматических) волн перемещений и микровращений вдоль длинных волноводов. The paper is devoted to study of the coupled vector differential equations of the linear theory of micropolar elasticity formulated in terms of displacements and microrotations in the case of a harmonic dependence of the physical fields on time. An alternative analysis aimed at splitting the coupled vector differential equation of the linear theory of micropolar elasticity into uncoupled equations is given. It is based on a notion of proportionality of the vortex parts of the displacements and microrotations to the single vector, which satisfies the screw equation well known from the mathematical physics. As a result, the problem of finding the vortex parts of the displacements and microrotations fields is reduced to solution of four uncoupled screw differential equations of the first order with partial derivatives. Obtained results are to be used in applied problems of the micropolar elasticity and in particular in studies of harmonic wave propagation along waveguides

Hamiltonian Mechanics for Functionals Involving Second-Order Derivatives

2002 ◽  
Vol 69 (6) ◽  
pp. 749-754 ◽  
Author(s):  
B. Tabarrok ◽  
C. M. Leech
Keyword(s):  
Differential Equations ◽  
Generating Functions ◽  
Jacobi Equation ◽  
Equations Of Motion ◽  
Second Order ◽  
Hamiltonian Mechanics ◽  
Energy Functionals ◽  
First Order ◽  
Independent Variable ◽  
Hamilton Jacobi Equation

Hamilton’s principle was developed for the modeling of dynamic systems in which time is the principal independent variable and the resulting equations of motion are second-order differential equations. This principle uses kinetic energy which is functionally dependent on first-order time derivatives, and potential energy, and has been extended to include virtual work. In this paper, a variant of Hamiltonian mechanics for systems whose motion is governed by fourth-order differential equations is developed and is illustrated by an example invoking the flexural analysis of beams. The variational formulations previously associated with Newton’s second-order equations of motion have been generalized to encompass problems governed by energy functionals involving second-order derivatives. The canonical equations associated with functionals with second order derivatives emerge as four first-order equations in each variable. The transformations of these equations to a new system wherein the generalized variables and momenta appear as constants, can be obtained through several different forms of generating functions. The generating functions are obtained as solutions of the Hamilton-Jacobi equation. This theory is illustrated by application to an example from beam theory the solution recovered using a technique for solving nonseparable forms of the Hamilton-Jacobi equation. Finally whereas classical variational mechanics uses time as the primary independent variable, here the theory is extended to include static mechanics problems in which the primary independent variable is spatial.

Generating functions and semi-classical orthogonal polynomials

1994 ◽  
Vol 124 (5) ◽  
pp. 1003-1011 ◽  
Author(s):  
Pascal Maroni ◽  
Jeannette Van Iseghem
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Orthogonal Polynomials ◽  
Recurrence Relation ◽  
Generating Function ◽  
Generating Functions ◽  
Special Form ◽  
Classical Orthogonal Polynomials ◽  
Order Linear ◽  
First Order

An orthogonal family of polynomials is given, and a link is made between the special form of the coefficients of their recurrence relation and a first-order linear homogenous partial differential equation satisfied by the associated generating function. A study is also made of the semiclassical character of such families.

scholarly journals Generating solutions for charged stellar models in general relativity

2021 ◽  
Vol 81 (3) ◽  
Author(s):  
B. V. Ivanov
Keyword(s):  
Differential Equations ◽  
Generating Functions ◽  
Maxwell Equations ◽  
Radial Pressure ◽  
Tangential Pressure ◽  
First Order ◽  
Embedding Class One ◽  
Stellar Models ◽  
Linear Differential ◽  
First Order Differential Equations

AbstractIt is shown that the expressions for the tangential pressure, the anisotropy factor and the radial pressure in the Einstein–Maxwell equations may serve as generating functions for charged stellar models. The latter can incorporate an equation of state when the expression for the energy density is also used. Other generating functions are based on the condition for the existence of conformal motion (conformal flatness in particular) and the Karmarkar condition for embedding class one metrics, which do not depend on charge. In all these cases the equations are linear first order differential equations for one of the metric components and Riccati equations for the other. The latter may be always transformed into second order homogenous linear differential equations. These conclusions are illustrated by numerous particular examples from the study of charged stellar models.

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