DETERMINANTAL IDENTITIES ON INTEGRABLE MAPPINGS

1994 ◽  
Vol 08 (16) ◽  
pp. 2157-2201 ◽  
Author(s):  
S. BOUKRAA ◽  
J-M. MAILLARD ◽  
G. ROLLET
Keyword(s):  
Elliptic Curves ◽  
Common Factor ◽  
Discrete Groups ◽  
Infinite Dimensional ◽  
Lattice Statistical Mechanics ◽  
The Matrix ◽  
Q Matrix ◽  
Canonical Polynomials ◽  
Integrable Mappings ◽  
Determinantal Identities

We describe birational representations of discrete groups generated by involutions, having their origin in the theory of exactly solvable vertex-models in lattice statistical mechanics. These involutions correspond respectively to two kinds of transformations on q×q matrices: the inversion of the q×q matrix and an (involutive) permutation of the entries of the matrix. In a case where the permutation is a particular elementary transposition of two entries, it is shown that the iteration of this group of birational transformations yield algebraic elliptic curves in the parameter space associated with the (homogeneous) entries of the matrix. It is also shown that the successive iterated matrices do have remarkable factorization properties which yield introducing a series of canonical polynomials corresponding to the greatest common factor in the entries. These polynomials do satisfy a simple nonlinear recurrence which also yields algebraic elliptic curves, associated with biquadratic relations. In fact, these polynomials not only satisfy one recurrence but a whole hierarchy of recurrences. Remarkably these recurrences are universal: they are independent of q, the size of the matrices. This study provides examples of infinite dimensional integrable mappings.

scholarly journals Certain unitary representations of the infinite symmetric group, II

1987 ◽  
Vol 106 ◽  
pp. 143-162 ◽  
Author(s):  
Nobuaki Obata
Keyword(s):  
Symmetric Group ◽  
Discrete Group ◽  
Inductive Limit ◽  
Symmetric Groups ◽  
Discrete Groups ◽  
Type I ◽  
Infinite Symmetric Group ◽  
Locally Finite ◽  
Distinctive Features ◽  
Infinite Dimensional

The infinite symmetric group is the discrete group of all finite permutations of the set X of all natural numbers. Among discrete groups, it has distinctive features from the viewpoint of representation theory and harmonic analysis. First, it is one of the most typical ICC-groups as well as free groups and known to be a group of non-type I. Secondly, it is a locally finite group, namely, the inductive limit of usual symmetric groups . Furthermore it is contained in infinite dimensional classical groups GL(ξ), O(ξ) and U(ξ) and their representation theories are related each other.

scholarly journals A note on the Ritz method with an application to overtone stellar pulsation theory

1968 ◽  
Vol 8 (2) ◽  
pp. 275-286 ◽  
Author(s):  
A. L. Andrew
Keyword(s):  
Numerical Solution ◽  
Eigenvalue Problems ◽  
Ritz Method ◽  
Linear Operators ◽  
Matrix Equations ◽  
Infinite Dimensional ◽  
Stellar Pulsation ◽  
Finite Matrix ◽  
The Matrix ◽  
Matrix Eigenvalue Problems

The Ritz method reduces eigenvalue problems involving linear operators on infinite dimensional spaces to finite matrix eigenvalue problems. This paper shows that for a certain class of linear operators it is possible to choose the coordinate functions so that numerical solution of the matrix equations is considerably simplified, especially when the matrices are large. The method is applied to the problem of overtone pulsations of stars.

scholarly journals Small-Deviation Inequalities for Sums of Random Matrices

Symmetry ◽  
2019 ◽  
Vol 11 (5) ◽  
pp. 638
Author(s):  
Xianjie Gao ◽  
Chao Zhang ◽  
Hongwei Zhang
Keyword(s):  
Random Matrices ◽  
Large Deviation ◽  
Small Deviation ◽  
High Dimensional ◽  
Main Research ◽  
Infinite Dimensional ◽  
The Matrix ◽  
Largest Eigenvalues ◽  
Deviation Inequalities ◽  
Eigenvalues Of Random Matrices

Random matrices have played an important role in many fields including machine learning, quantum information theory, and optimization. One of the main research focuses is on the deviation inequalities for eigenvalues of random matrices. Although there are intensive studies on the large-deviation inequalities for random matrices, only a few works discuss the small-deviation behavior of random matrices. In this paper, we present the small-deviation inequalities for the largest eigenvalues of sums of random matrices. Since the resulting inequalities are independent of the matrix dimension, they are applicable to high-dimensional and even the infinite-dimensional cases.

scholarly journals Matrix 3-Lie superalgebras and BRST supersymmetry

2017 ◽  
Vol 14 (11) ◽  
pp. 1750160 ◽  
Author(s):  
Viktor Abramov
Keyword(s):  
Clifford Algebra ◽  
Lie Algebra ◽  
Vector Fields ◽  
Jacobi Identity ◽  
Lie Superalgebras ◽  
Infinite Dimensional ◽  
Algebra Approach ◽  
The Matrix ◽  
Infinite Dimensional Lie Algebra

Given a matrix Lie algebra one can construct the 3-Lie algebra by means of the trace of a matrix. In the present paper, we show that this approach can be extended to the infinite-dimensional Lie algebra of vector fields on a manifold if instead of the trace of a matrix we consider a differential 1-form which satisfies certain conditions. Then we show that the same approach can be extended to matrix Lie superalgebras [Formula: see text] if instead of the trace of a matrix we make use of the supertrace of a matrix. It is proved that a graded triple commutator of matrices constructed with the help of the graded commutator and the supertrace satisfies a graded ternary Filippov–Jacobi identity. In two particular cases of [Formula: see text] and [Formula: see text], we show that the Pauli and Dirac matrices generate the matrix 3-Lie superalgebras, and we find the non-trivial graded triple commutators of these algebras. We propose a Clifford algebra approach to 3-Lie superalgebras induced by Lie superalgebras. We also discuss an application of matrix 3-Lie superalgebras in BRST-formalism.

scholarly journals Corrections to Wigner–Eckart Relations by Spontaneous Symmetry Breaking

Symmetry ◽  
2020 ◽  
Vol 12 (7) ◽  
pp. 1120
Author(s):  
Carlo Heissenberg ◽  
Franco Strocchi
Keyword(s):  
Symmetry Breaking ◽  
Spontaneous Symmetry Breaking ◽  
Irreducible Representations ◽  
Matrix Elements ◽  
Infinite Dimensional ◽  
Infinite Dimensional Systems ◽  
Goldstone Bosons ◽  
The Matrix ◽  
Symmetry Breaking Term ◽  
Breaking Term

The matrix elements of operators transforming as irreducible representations of an unbroken symmetry group G are governed by the well-known Wigner–Eckart relations. In the case of infinite-dimensional systems, with G spontaneously broken, we prove that the corrections to such relations are provided by symmetry breaking Ward identities, and simply reduce to a tadpole term involving Goldstone bosons. The analysis extends to the case in which an explicit symmetry breaking term is present in the Hamiltonian, with the tadpole term now involving pseudo Goldstone bosons. An explicit example is discussed, illustrating the two cases.

ALMOST INTEGRABLE MAPPINGS

1994 ◽  
Vol 08 (01n02) ◽  
pp. 137-174 ◽  
Author(s):  
S. BOUKRAA ◽  
J.-M. MAILLARD ◽  
G. ROLLET
Keyword(s):  
Elliptic Curves ◽  
Parameter Space ◽  
Single Variable ◽  
Algebraic Varieties ◽  
Linear Recurrences ◽  
Birational Transformations ◽  
Higher Dimensional ◽  
Non Linear ◽  
Integrable Mappings

We analyze birational transformations obtained from very simple algebraic calculations, namely taking the inverse of q × q matrices and permuting some of the entries of these matrices. We concentrate on 4 × 4 matrices and elementary transpositions of two entries. This analysis brings out six classes of birational transformations. Three classes correspond to integrable mappings, their iteration yielding elliptic curves. Generically, the iterations corresponding to the three other classes are included in higher dimensional non-trivial algebraic varieties. Nevertheless some orbits of the parameter space lie on (transcendental) curves. These transformations act on fifteen (or q2 − 1) variables, however one can associate to them remarkably simple non-linear recurrences bearing on a single variable. The study of these last recurrences gives a complementary understanding of these amazingly regular non-integrable mappings, which could provide interesting tools to analyze weak chaos.

scholarly journals A Diffie-Hellman Encryption Scheme over Elliptic Curvesusing Golden Matrices

2019 ◽  
Vol 9 (2S3) ◽  
pp. 160-163
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Encryption Scheme ◽  
Diffie Hellman ◽  
The Matrix ◽  
Bijective Function

In this paper, we proposed Diffie-Hellman encryption scheme based on golden matrices over the elliptic curves. This algorithm works with a bijective function defined as characters of ASCII from the elliptic curve points and the matrix developed the additional personal key, which was obtained from the golden matrices.

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