Symmetries of the multi-component Boussinesq hierarchy

2021 ◽  
pp. 2150111
Author(s):  
Meiyan Hu ◽  
Chuanzhong Li
Keyword(s):  
Wave Function ◽  
Generating Function ◽  
String Equation ◽  
Additional Symmetry ◽  
Infinite Dimensional ◽  
Lax Operator ◽  
Sato Theory ◽  
The Way

In this paper, we construct the Lax operator of the multi-component Boussinesq hierarchy. Based on the Sato theory and the dressing structure of the multi-component Boussinesq hierarchy, the adjoint wave function and the Orlov–Schulman’s operator are introduced, which are useful for constructing the additional symmetry of the multi-component Boussinesq hierarchy. Besides, the additional flows can commute with the original flows, and these flows form an infinite dimensional [Formula: see text] algebra. Taking the above discussion into account, we mainly study the additional symmetry flows and the generating function for both strongly and weakly multi-component of the Boussinesq hierarchies. By the way, using the [Formula: see text] constraint of the multi-component Boussinesq hierarchy, the string equation can be derived.

scholarly journals THE LAX OPERATOR APPROACH FOR THE VIRASORO AND THE W-CONSTRAINTS IN THE GENERALIZED KdV HIERARCHY

1993 ◽  
Vol 08 (20) ◽  
pp. 3457-3478 ◽  
Author(s):  
SUDHAKAR PANDA ◽  
SHIBAJI ROY
Keyword(s):  
Symmetry Operator ◽  
String Equation ◽  
Kp Hierarchy ◽  
Operator Approach ◽  
Additional Symmetry ◽  
Kdv Hierarchy ◽  
Lax Operator

We show directly in the Lax operator approach how the Virasoro and W-constraints on the τ-function arise in the p-reduced KP hierarchy or generalized KdV hierarchy. In particular, we consider the KdV and the Boussinesq hierarchy to show that the Virasoro and the W-constraints follow from the string equation by expanding the "additional symmetry" operator in terms of the Lax operator. We also mention how this method could be generalized for higher KdV hierarchies.

scholarly journals Strong Local Survival of Branching Random Walks is Not Monotone

2014 ◽  
Vol 46 (02) ◽  
pp. 400-421 ◽  
Author(s):  
Daniela Bertacchi ◽  
Fabio Zucca
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Generating Function ◽  
Continuous Time ◽  
Local Property ◽  
Branching Random Walk ◽  
Infinite Dimensional ◽  
Branching Random Walks ◽  
Local Survival ◽  
Branching Law

In this paper we study the strong local survival property for discrete-time and continuous-time branching random walks. We study this property by means of an infinite-dimensional generating functionGand a maximum principle which, we prove, is satisfied by every fixed point ofG. We give results for the existence of a strong local survival regime and we prove that, unlike local and global survival, in continuous time, strong local survival is not a monotone property in the general case (though it is monotone if the branching random walk is quasitransitive). We provide an example of an irreducible branching random walk where the strong local property depends on the starting site of the process. By means of other counterexamples, we show that the existence of a pure global phase is not equivalent to nonamenability of the process, and that even an irreducible branching random walk with the same branching law at each site may exhibit nonstrong local survival. Finally, we show that the generating function of an irreducible branching random walk can have more than two fixed points; this disproves a previously known result.

scholarly journals Strong Local Survival of Branching Random Walks is Not Monotone

2014 ◽  
Vol 46 (2) ◽  
pp. 400-421 ◽  
Author(s):  
Daniela Bertacchi ◽  
Fabio Zucca
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Generating Function ◽  
Continuous Time ◽  
Local Property ◽  
Branching Random Walk ◽  
Infinite Dimensional ◽  
Branching Random Walks ◽  
Local Survival ◽  
Branching Law

In this paper we study the strong local survival property for discrete-time and continuous-time branching random walks. We study this property by means of an infinite-dimensional generating function G and a maximum principle which, we prove, is satisfied by every fixed point of G. We give results for the existence of a strong local survival regime and we prove that, unlike local and global survival, in continuous time, strong local survival is not a monotone property in the general case (though it is monotone if the branching random walk is quasitransitive). We provide an example of an irreducible branching random walk where the strong local property depends on the starting site of the process. By means of other counterexamples, we show that the existence of a pure global phase is not equivalent to nonamenability of the process, and that even an irreducible branching random walk with the same branching law at each site may exhibit nonstrong local survival. Finally, we show that the generating function of an irreducible branching random walk can have more than two fixed points; this disproves a previously known result.

scholarly journals Obtaining Easily Sums of Powers on Arithmetic Progressions and Properties of Bernoulli Polynomials by Operator Calculus

2017 ◽  
Vol 9 (5) ◽  
pp. 73
Author(s):  
Do Tan Si
Keyword(s):  
Differential Operator ◽  
Generating Function ◽  
Arithmetic Progression ◽  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Arithmetic Progressions ◽  
Operator Calculus ◽  
Sums Of Powers ◽  
The Way

We show that a sum of powers on an arithmetic progression is the transform of a monomial by a differential operator and that its generating function is simply related to that of the Bernoulli polynomials from which consequently it may be calculated. Besides, we show that it is obtainable also from the sums of powers of integers, i.e. from the Bernoulli numbers which in turn may be calculated by a simple algorithm.By the way, for didactic purpose, operator calculus is utilized for proving in a concise manner the main properties of the Bernoulli polynomials. 

The q-deformed mKP hierarchy with two parameters

2018 ◽  
Vol 32 (16) ◽  
pp. 1850170
Author(s):  
Kelei Tian ◽  
Yanyan Ge ◽  
Xiaoming Zhu
Keyword(s):  
Additional Symmetry ◽  
Lax Equation ◽  
Quantum Calculus ◽  
Two Parameters ◽  
Sato Theory

In this paper, with the help of the biparametric quantum calculus we construct the Sato theory on the q-deformation modified Kadomtsev–Petviashvili hierarchy with two parameters (qp-mKP), which is a new deformation of classical mKP hierarchy. The Lax equation and dressing operator of qp-mKP hierarchy are derived. By considering the M operator and [Formula: see text] operator, the additional symmetry of qp-mKP hierarchy is obtained.

Sato–Bäcklund transformations and string equations of the mKP hierarchy

2019 ◽  
Vol 34 (25) ◽  
pp. 1950142 ◽  
Author(s):  
Huizhan Chen ◽  
Lumin Geng ◽  
Jipeng Cheng
Keyword(s):  
Bäcklund Transformations ◽  
Kp Hierarchy ◽  
Time And Space ◽  
Additional Symmetry ◽  
Tau Functions ◽  
Virasoro Constraint ◽  
Backlund Transformations ◽  
Lax Operator ◽  
Important Kind

Additional symmetry is an important kind of symmetries depending explicitly on the time and space variables, which can be expressed through Sato–Bäcklund transformations. In this paper, we construct Sato–Bäcklund transformations of the modified KP hierarchy and its constrained cases. Then the string equations of the [Formula: see text]-reduced modified KP hierarchy are established by requiring the system independent on some additional symmetry flows, which are expressed by the Lax operator [Formula: see text] and the Orlov–Shulman’s operator [Formula: see text]. At last, we obtain the negative Virasoro constraint on the two tau functions of the 2-reduced modified KP hierarchy satisfying the string equations.

scholarly journals Infinite Systems of Functional Equations and Gaussian Limiting Distributions

2012 ◽  
Vol DMTCS Proceedings vol. AQ,... (Proceedings) ◽  
Author(s):  
Michael Drmota ◽  
Bernhard Gittenberger ◽  
Johannes F. Morgenbesser
Keyword(s):  
Generating Function ◽  
Functional Equations ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Combinatorial Enumeration ◽  
Limiting Distributions ◽  
Infinite Dimensional ◽  
Infinite Systems ◽  
Enumeration Problems ◽  
International Audience

International audience In this paper infinite systems of functional equations in finitely or infinitely many random variables arising in combinatorial enumeration problems are studied. We prove sufficient conditions under which the combinatorial random variables encoded in the generating function of the system tend to a finite or infinite dimensional limiting distribution.

Additional symmetries of the dispersionless extended noncommutative Gelfand–Dickey hierarchy

2021 ◽  
pp. 2150145
Author(s):  
Meiyan Hu ◽  
Chuanzhong Li
Keyword(s):  
Algebraic Structure ◽  
String Equation ◽  
Additional Symmetry ◽  
Quantum Torus ◽  
Additional Symmetries

This paper aims at additional symmetries of the unextended and extended, commutative and noncommutative dispersionless Gelfand–Dickey (dGD) hierarchies. Being similar to the Lax formalism of the Gelfand–Dickey (GD) hierarchy, we construct the function [Formula: see text] and Orlov–Schulman function [Formula: see text] of the hierarchies. Meanwhile, the additional symmetry will be studied with the infinite flows of [Formula: see text] and [Formula: see text] function of the dGD hierarchy and one can find that only a part of additional flows can survive under the GD constraints with the corresponding string equation. Furthermore, we pay attention to the additional symmetries of the dispersionless extended Gelfand–Dickey (dEGD) hierarchy which has a quantum torus algebraic structure and show the flows in detail. The additional symmetry of dispersionless noncommutative Gelfand–Dickey (dNCGD) hierarchy and dispersionless extended noncommutative Gelfand–Dickey (dENCGD) hierarchy are studied.

scholarly journals WEIGHTED TRACES ON ALGEBRAS OF PSEUDO-DIFFERENTIAL OPERATORS AND GEOMETRY ON LOOP GROUPS

2002 ◽  
Vol 05 (04) ◽  
pp. 503-540 ◽  
Author(s):  
A. CARDONA ◽  
C. DUCOURTIOUX ◽  
J. P. MAGNOT ◽  
S. PAYCHA
Keyword(s):  
Elliptic Operator ◽  
Differential Operators ◽  
Loop Groups ◽  
Linear Functionals ◽  
Geometric Framework ◽  
Infinite Dimensional ◽  
Type Formula ◽  
Pseudo Differential Operators ◽  
Set Up ◽  
The Way

Using weighted traces which are linear functionals of the type [Formula: see text] defined on the whole algebra of (classical) pseudo-differential operators (P.D.Os) and where Q is some admissible invertible elliptic operator, we investigate the geometry of loop groups in the light of the cohomology of pseudo-differential operators. We set up a geometric framework to study a class of infinite dimensional manifolds in which we recover some results on the geometry of loop groups, using again weighted traces. Along the way, we investigate properties of extensions of the Radul and Schwinger cocycles defined with the help of weighted traces.

scholarly journals From Quark to Infinite Dimensionality

2021 ◽  
pp. 1-3
Author(s):  
Joseph E Brierly ◽  
Keyword(s):  
Particle Physics ◽  
Physical Reality ◽  
Nanometer Scale ◽  
Epr Paradox ◽  
Infinite Dimensional ◽  
Exact Number ◽  
Stable Means ◽  
Infinite Dimensionality ◽  
The Universe ◽  
The Way

This article gives a overall picture of how the universe works from the likelihood that our universe is infinite dimensional at the nanometer scale of an indestructible quark. The article explains that we only can perceive for sure up to 4 dimensions of physical reality. However, the speculation in this article seems very clear that likely we are seeing activity in the 5th dimension in particle physics experimentation explaining the EPR paradox and other mysteries seen in particle physics. Finally, the article shows why the Mendeleev Chart has historically listed possible stable atoms without giving the exact number possible. The way protons and other hadrons are composed of six quarks and six antiquarks held together by gluons leads to the inevitable conclusion that only 108 stable atoms can exist. Being stable means the protons in an atom are composed of 3 quarks/antiquarks having charge 1. Recent discoveries in particle physics research demonstrates that there exists a particle named the pentaquark composed of five quarks. The article explains that pentaquarks have been identified in recent particle research. It is not known yet whether the pentaquark leads to a different proton that leads in turn to a pentaquark atom. New particle research will likely answer this question

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