The third kind of generating functions of stochastic symplectic integrators

2012 ◽  
Author(s):  
Lijin Wang ◽  
Tong Zhao
Keyword(s):  
Generating Functions ◽  
Symplectic Integrators ◽  
The Third

Poisson Brackets & Angular Momentum

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Generating Functions ◽  
First Integrals ◽  
Lagrangian Mechanics ◽  
Poisson Brackets ◽  
Coordinate Transformations ◽  
Canonical Transformations ◽  
Gauge Transformations ◽  
The Third ◽  
Point Transformations ◽  
Canonical Coordinate

This chapter discusses canonical transformations and gauge transformations and is divided into three sections. In the first section, canonical coordinate transformations are introduced to the reader through generating functions as the extension of point transformations used in Lagrangian mechanics, with the harmonic oscillator being used as an example of a canonical transformation. In the second section, gauge theory is discussed in the canonical framework and compared to the Lagrangian case. Action-angle variables, direct conditions, symplectomorphisms, holomorphic variables, integrable systems and first integrals are examined. The third section looks at infinitesimal canonical transformations resulting from functions on phase space. Ostrogradsky equations in the canonical setting are also detailed.

scholarly journals Generating Functions for Orthogonal Polynomials of A2, C2 and G2

Symmetry ◽  
2018 ◽  
Vol 10 (8) ◽  
pp. 354 ◽  
Author(s):  
Tomasz Czyżycki ◽  
Jiří Hrivnák ◽  
Jiří Patera
Keyword(s):  
Orthogonal Polynomials ◽  
Lie Algebras ◽  
Chebyshev Polynomials ◽  
Generating Functions ◽  
Bell Polynomials ◽  
The Third ◽  
Weight Lattice ◽  
Hybrid Character ◽  
Polynomial Family

The generating functions of fourteen families of generalized Chebyshev polynomials related to rank two Lie algebras A 2 , C 2 and G 2 are explicitly developed. There exist two classes of the orthogonal polynomials corresponding to the symmetric and antisymmetric orbit functions of each rank two algebra. The Lie algebras G 2 and C 2 admit two additional polynomial collections arising from their hybrid character functions. The admissible shift of the weight lattice permits the construction of a further four shifted polynomial classes of C 2 and directly generalizes formation of the classical univariate Chebyshev polynomials of the third and fourth kinds. Explicit evaluating formulas for each polynomial family are derived and linked to the incomplete exponential Bell polynomials.

Anatomy of the canonical transformation

1993 ◽  
Vol 345 (1677) ◽  
pp. 577-598 ◽  
Keyword(s):  
Fluid Dynamics ◽  
Canonical Transformation ◽  
Generating Functions ◽  
Hamiltonian Structure ◽  
Rigid Bodies ◽  
Symplectic Integrators ◽  
Classical Dynamics ◽  
Canonical Transformations ◽  
Geophysical Fluid Dynamics ◽  
Dynamics Of Particles

A concise account of the structure of the canonical transformation is given, in the lowest dimensional case. This case is chosen because it offers a special clarity in several respects. In particular, the diversity of possible generating functions is illustrated by m any examples which are not available elsewhere. Many of these are of physical interest, and some of them are multivalued. These examples are used to inform a comparative study of the several different definitions of a canonical transformation to be found in the literature. The paper is pertinent to all those branches of mechanics which can be given a hamiltonian representation. These include not only the classical dynamics of particles and rigid bodies, but also some more recent studies in continuum mechanics, including geophysical fluid dynamics. An area of particular modern interest is that of symplectic integrators. These are numerical integrating algorithms which generate a solution to Hamilton’s equations via a sequence of canonical transformations, which preserve the hamiltonian structure in the numerical solution.

scholarly journals Some results on dual third-order Jacobsthal quaternions

Filomat ◽  
2019 ◽  
Vol 33 (7) ◽  
pp. 1865-1876 ◽  
Author(s):  
Gamaliel Cerda-Morales
Keyword(s):  
Generating Functions ◽  
Lucas Numbers ◽  
Third Order ◽  
The Third

Dual Fibonacci and dual Lucas numbers are defined with dual Fibonacci and Lucas quaternions in Nurkan and G?ven [14]. In this study, we define the dual third-order Jacobsthal quaternion and the dual third-order Jacobsthal-Lucas quaternion. We derive the relations between the dual third-order Jacobsthal quaternion and dual third-order Jacobsthal-Lucas quaternion which connected the third-order Jacobsthal and third-order Jacobsthal-Lucas numbers. In addition, we give the generating functions, the Binet and Cassini formulas for these new types of quaternions.

scholarly journals Determinant Expressions of Constraint Polynomials and the Spectrum of the Asymmetric Quantum Rabi Model

2020 ◽  
Author(s):  
Kazufumi Kimoto ◽  
Cid Reyes-Bustos ◽  
Masato Wakayama
Keyword(s):  
Generating Functions ◽  
The Third ◽  
Asymmetric Quantum ◽  
Pauli Matrix ◽  
Rabi Model

Abstract The purpose of this paper is to study the exceptional eigenvalues of the asymmetric quantum Rabi models (AQRMs), specifically, to determine the degeneracy of their eigenstates. Here, the Hamiltonian $H_{\textrm{Rabi}}^{\varepsilon }$ of the AQRM is defined by adding the fluctuation term $\varepsilon \sigma _x$, with $\sigma _x$ being the Pauli matrix, to the Hamiltonian of the quantum Rabi model, breaking its $\mathbb{Z}_{2}$-symmetry. The spectrum of $H_{\textrm{Rabi}}^{\varepsilon }$ contains a set of exceptional eigenvalues, considered to be remains of the eigenvalues of the uncoupled bosonic mode, which are further classified in two types: Juddian, associated with polynomial eigensolutions, and non-Juddian exceptional. We explicitly describe the constraint relations for allowing the model to have exceptional eigenvalues. By studying these relations we obtain the proof of the conjecture on constraint polynomials previously proposed by the third author. In fact we prove that the spectrum of the AQRM possesses degeneracies if and only if the parameter $\varepsilon $ is a halfinteger. Moreover, we show that non-Juddian exceptional eigenvalues do not contribute any degeneracy and we characterize exceptional eigenvalues by representations of $\mathfrak{s}\mathfrak{l}_2$. Upon these results, we draw the whole picture of the spectrum of the AQRM. Furthermore, generating functions of constraint polynomials from the viewpoint of confluent Heun equations are also discussed.

scholarly journals A discrete-time system with service control and repairs

2014 ◽  
Vol 24 (3) ◽  
pp. 471-484 ◽  
Author(s):  
Ivan Atencia
Keyword(s):  
Performance Measures ◽  
Discrete Time ◽  
Generating Functions ◽  
Continuous Time ◽  
Queueing System ◽  
Discrete Time System ◽  
Time System ◽  
The Third ◽  
Special Cases ◽  
Number Of Customers

Abstract This paper discusses a discrete-time queueing system with starting failures in which an arriving customer follows three different strategies. Two of them correspond to the LCFS (Last Come First Served) discipline, in which displacements or expulsions of customers occur. The third strategy acts as a signal, that is, it becomes a negative customer. Also examined is the possibility of failures at each service commencement epoch. We carry out a thorough study of the model, deriving analytical results for the stationary distribution. We obtain the generating functions of the number of customers in the queue and in the system. The generating functions of the busy period as well as the sojourn times of a customer at the server, in the queue and in the system, are also provided. We present the main performance measures of the model. The versatility of this model allows us to mention several special cases of interest. Finally, we prove the convergence to the continuous-time counterpart and give some numerical results that show the behavior of some performance measures with respect to the most significant parameters of the system

scholarly journals On Third-Order Bronze Fibonacci Numbers

Mathematics ◽  
2021 ◽  
Vol 9 (20) ◽  
pp. 2606
Author(s):  
Mücahit Akbiyik ◽  
Jeta Alo
Keyword(s):  
Generating Functions ◽  
Fibonacci Numbers ◽  
Fibonacci Sequence ◽  
Second Order ◽  
Matrix Representations ◽  
Third Order ◽  
The Third ◽  
Encryption And Decryption ◽  
Fibonacci Sequences

In this study, we firstly obtain De Moivre-type identities for the second-order Bronze Fibonacci sequences. Next, we construct and define the third-order Bronze Fibonacci, third-order Bronze Lucas and modified third-order Bronze Fibonacci sequences. Then, we define the generalized third-order Bronze Fibonacci sequence and calculate the De Moivre-type identities for these sequences. Moreover, we find the generating functions, Binet’s formulas, Cassini’s identities and matrix representations of these sequences and examine some interesting identities related to the third-order Bronze Fibonacci sequences. Finally, we present an encryption and decryption application that uses our obtained results and we present an illustrative example.

Progress report on 21-cm observations at low latitude between longitudes (l 11) 0° and 66°

1967 ◽  
Vol 31 ◽  
pp. 177-179
Author(s):  
W. W. Shane
Keyword(s):  
Preliminary Report ◽  
Progress Report ◽  
Galactic Centre ◽  
Low Latitude ◽  
The Third ◽  
Introductory Lecture ◽  
Centre Region

In the course of several 21-cm observing programmes being carried out by the Leiden Observatory with the 25-meter telescope at Dwingeloo, a fairly complete, though inhomogeneous, survey of the regionl11= 0° to 66° at low galactic latitudes is becoming available. The essential data on this survey are presented in Table 1. Oort (1967) has given a preliminary report on the first and third investigations. The third is discussed briefly by Kerr in his introductory lecture on the galactic centre region (Paper 42). Burton (1966) has published provisional results of the fifth investigation, and I have discussed the sixth in Paper 19. All of the observations listed in the table have been completed, but we plan to extend investigation 3 to a much finer grid of positions.

The orbit of Pluto over a long interval of time

1966 ◽  
Vol 25 ◽  
pp. 227-229 ◽  
Author(s):  
D. Brouwer
Keyword(s):  
Periodic Orbit ◽  
Numerical Integration ◽  
Restricted Problem ◽  
Three Dimensional ◽  
The Third ◽  
Circular Restricted Problem ◽  
Resonance Relation

The paper presents a summary of the results obtained by C. J. Cohen and E. C. Hubbard, who established by numerical integration that a resonance relation exists between the orbits of Neptune and Pluto. The problem may be explored further by approximating the motion of Pluto by that of a particle with negligible mass in the three-dimensional (circular) restricted problem. The mass of Pluto and the eccentricity of Neptune's orbit are ignored in this approximation. Significant features of the problem appear to be the presence of two critical arguments and the possibility that the orbit may be related to a periodic orbit of the third kind.

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