scholarly journals AUTOISOMETRIES AND AUTOSIMILARITIES OF LIE ALGEBRA 𝒜(1) ⊕ ℛ2

2020 ◽  
pp. 25-36
Author(s):  
M. N. Podoksenov ◽  
V. V. Chernykh
Keyword(s):  
Lie Algebra ◽  
Scalar Product ◽  
Parameter Group ◽  
Isotropic Cone ◽  
The One ◽  
Groups Of Isometries

We consider four-dimensional Lie algebra 𝒜(1) ⊕ ℛ2 endowed with Lorentzian scalar product. We find all the one-parameter groups of isometries and similarities, which are simultaneously automorphisms of Lie algebra, and also we find the conditions of existence of such one-parameter group. Conditions of existence are associated with the location of ideals with respect to isotropic cone.

scholarly journals SYMMETRY ANALYSIS OF TELEGRAPH EQUATION

2011 ◽  
Vol 04 (01) ◽  
pp. 117-126
Author(s):  
Mehdi Nadjafikhah ◽  
Seyed-Reza Hejazi
Keyword(s):  
Symmetry Group ◽  
Lie Algebra ◽  
Telegraph Equation ◽  
Lie Symmetry ◽  
Symmetry Analysis ◽  
Optimal System ◽  
Group Method ◽  
Parameter Group ◽  
One Dimensional

Lie symmetry group method is applied to study the telegraph equation. The symmetry group and one-parameter group associated to the symmetries with the structure of the Lie algebra symmetries are determined. The reduced version of equation and its one-dimensional optimal system are given.

QUANTUM DEFORMATIONS OF QUANTUM MECHANICS

1993 ◽  
Vol 08 (01) ◽  
pp. 89-96 ◽  
Author(s):  
MARCELO R. UBRIACO
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Time Evolution ◽  
Function Space ◽  
Scalar Product ◽  
Hypergeometric Functions ◽  
Quantum Mechanical ◽  
Basic Hypergeometric Functions ◽  
Quantum Deformations ◽  
The One

Based on a deformation of the quantum mechanical phase space we study q-deformations of quantum mechanics for qk=1 and 0<q<1. After defining a q-analog of the scalar product on the function space we discuss and compare the time evolution of operators in both cases. A formulation of quantum mechanics for qk=1 is given and the dynamics for the free Hamiltonian is studied. For 0<q<1 we develop a deformation of quantum mechanics and the cases of the free Hamiltonian and the one with a x2-potential are solved in terms of basic hypergeometric functions.

scholarly journals Approximate Symmetries of the Harry Dym Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Mehdi Nadjafikhah ◽  
Parastoo Kabi-Nejad
Keyword(s):  
Lie Algebra ◽  
Optimal System ◽  
Differential Invariants ◽  
Approximate Symmetries ◽  
One Dimensional ◽  
First Order ◽  
Optimal System Of Subalgebras ◽  
Dym Equation ◽  
The One ◽  
Harry Dym Equation

We derive the first-order approximate symmetries for the Harry Dym equation by the method of approximate transformation groups proposed by Baikov et al. (1989, 1996). Moreover, we investigate the structure of the Lie algebra of symmetries of the perturbed Harry Dym equation. We compute the one-dimensional optimal system of subalgebras as well as point out some approximately differential invariants with respect to the generators of Lie algebra and optimal system.

Pointwise bounds for the solutions of certain boundary-value problems

1951 ◽  
Vol 208 (1093) ◽  
pp. 170-175 ◽  
Keyword(s):  
Boundary Value Problems ◽  
Function Space ◽  
Scalar Product ◽  
Boundary Value ◽  
The Other ◽  
Green Functions ◽  
Regular Functions ◽  
Cutting Out ◽  
Neumann Type ◽  
The One

The boundary-value problems considered are of the Dirichlet-Neumann type. The method now given for obtaining pointwise bounds of the solution and its derivatives is a compromise between the methods of Diaz and Greenberg on the one hand and Maple and Synge on the other; it appears to be simpler than either. The solution having been located on a hypercircle in function space, the pointwise bounds are obtained by taking the scalar product of the solution by certain vectors (Green’s vectors). Divergence of integrals due to the poles of the Green functions is avoided by the use of regular functions matching the Green functions on the boundary (Diaz-Greenberg device) instead of by cutting out spheres from the domain (Maple-Synge device).

scholarly journals Lie symmetries and singularity analysis for generalized shallow-water equations

2020 ◽  
Vol 21 (7-8) ◽  
pp. 739-747
Author(s):  
Andronikos Paliathanasis
Keyword(s):  
Lie Algebra ◽  
Three Dimensional ◽  
Singularity Analysis ◽  
Complete Classification ◽  
Similarity Solutions ◽  
Lie Symmetries ◽  
Reduced Equations ◽  
Complete Study ◽  
The One

AbstractWe perform a complete study by using the theory of invariant point transformations and the singularity analysis for the generalized Camassa-Holm (CH) equation and the generalized Benjamin-Bono-Mahoney (BBM) equation. From the Lie theory we find that the two equations are invariant under the same three-dimensional Lie algebra which is the same Lie algebra admitted by the CH equation. We determine the one-dimensional optimal system for the admitted Lie symmetries and we perform a complete classification of the similarity solutions for the two equations of our study. The reduced equations are studied by using the point symmetries or the singularity analysis. Finally, the singularity analysis is directly applied on the partial differential equations from where we infer that the generalized equations of our study pass the singularity test and are integrable.

Infinitesimal Kinematics Methods in the Mobility Determination of Kinematic Chains

2009 ◽  
Author(s):  
J. M. Rico ◽  
J. J. Cervantes ◽  
A. Tadeo ◽  
J. Gallardo ◽  
L. D. Aguilera ◽  
...  
Keyword(s):  
Lie Algebra ◽  
Good Deal ◽  
Higher Order ◽  
The Other ◽  
Velocity Analysis ◽  
Euclidean Group ◽  
Kinematic Chains ◽  
Order Analysis ◽  
The One

In recent years, there has been a good deal of controversy about the application of infinitesimal kinematics to the mobility determination of kinematic chains. On the one hand, there has been several publications that promote the use of the velocity analysis, without any additional results, for the determination of the mobility of kinematic chains. On the other hand, the authors of this contribution have received several reviews of researchers who have the strong belief that no infinitesimal method can be used to correctly determine the mobility of kinematic chains. In this contributions, it is attempted to show that velocity analysis by itself can not correctly determine the mobility of kinematic chains. However, velocity and higher order analysis coupled with some recent results about the Lie algebra, se(3), of the Euclidean group, SE(3), can correctly determine the mobility of kinematic chains.

THE GENERALIZED SPHERICALLY-SYMMETRIC METRIC

2020 ◽  
Vol 22 (4) ◽  
pp. 223-226
Author(s):  
M.M. Khashaev
Keyword(s):  
Lie Algebra ◽  
Spherical Symmetry ◽  
Vector Fields ◽  
Killing Vector ◽  
Space Time ◽  
Spherically Symmetric ◽  
Killing Vector Fields ◽  
Parameter Group ◽  
Killing Vectors ◽  
Spherically Symmetric Metric

Four parameter group of transformations containing rotations and time translations is consi[1]dered due to spherical symmetry and stationarity of the space-time metric. It is found that there exists such a quartet of Killing vector fields which constitute the Lie algebra of the transforma[1]tion group and in which space-like vectors are not orthogonal to the time-like one. The metric corresponding to the Lie algebra of Killing vectors is composed. It is shown that the metric is non-static.

scholarly journals Weyl n-algebras and the Kontsevich integral of the unknot

2016 ◽  
Vol 25 (12) ◽  
pp. 1642008
Author(s):  
Nikita Markarian
Keyword(s):  
Lie Algebra ◽  
Wilson Loop ◽  
Scalar Product ◽  
Symplectic Structure ◽  
Loop Invariant ◽  
First Order ◽  
Order Deformation ◽  
Definition Of ◽  
Hochschild Complex

Given a Lie algebra with a scalar product, one may consider the latter as a symplectic structure on a [Formula: see text]-scheme, which is the spectrum of the Chevalley–Eilenberg algebra. In Sec. 1 we explicitly calculate the first-order deformation of the differential on the Hochschild complex of the Chevalley–Eilenberg algebra. The answer contains the Duflo character. This calculation is used in the last section. There we sketch the definition of the Wilson loop invariant of knots, which is, hopefully, equal to the Kontsevich integral, and show that for unknot they coincide. As a byproduct, we get a new proof of the Duflo isomorphism for a Lie algebra with a scalar product.

scholarly journals The Finsler geometry of groups of isometries of Hilbert Space

1987 ◽  
Vol 42 (2) ◽  
pp. 196-222 ◽  
Author(s):  
C. J. Atkin
Keyword(s):  
Hilbert Space ◽  
Lie Algebra ◽  
Orthogonal Group ◽  
Finsler Geometry ◽  
Operator Norm ◽  
Full Description ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Finsler Structure ◽  
Groups Of Isometries

AbstractThe paper deals with six groups: the unitary, orthogonal, symplectic, Fredholm unitary, special Fredholm orthogonal, and Fredholm symplectic groups of an infinite-dimensional Hilbert space. When each is furnished with the invariant Finsler structure induced by the operator-norm on the Lie algebra, it is shown that, between any two points of the group, there exists a geodesic realising this distance (often, indeed, a unique geodesic), except in the full orthogonal group, in which there are pairs of points that cannot be joined by minimising geodesics, and also pairs that cannot even be joined by minimising paths. A full description is given of each of these possibilities.

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