su(2) -expansion of the Lorentz algebra so(3,1 )

2013 ◽  
Vol 91 (8) ◽  
pp. 589-598 ◽  
Author(s):  
Rutwig Campoamor-Stursberg ◽  
Hubert de Guise ◽  
Marc de Montigny
Keyword(s):  
Coherent State ◽  
Unitary Representations ◽  
Iwasawa Decomposition ◽  
Matrix Formalism ◽  
Matrix Elements ◽  
Infinite Dimensional ◽  
Lorentz Algebra ◽  
Finite Dimensional

We exploit the Iwasawa decomposition to construct coherent state representations of [Formula: see text], the Lorentz algebra in 3 + 1 dimensions, expanded on representations of the maximal compact subalgebra [Formula: see text]. Examples of matrix elements computation for finite dimensional and infinite-dimensional unitary representations are given. We also discuss different base vectors and the equivalence between these different choices. The use of the [Formula: see text]-matrix formalism to truncate the representation or to enforce unitarity is discussed.

scholarly journals To the 150th anniversary of the creation of the Periodic system of elements

2019 ◽  
Vol 55 (2) ◽  
pp. 242-254
Author(s):  
A. L. Gurskii ◽  
L. I. Hursky
Keyword(s):  
Periodic System ◽  
Mathematical Description ◽  
Modern Science ◽  
Unitary Representations ◽  
Symmetry Groups ◽  
Dynamic Symmetry ◽  
Infinite Dimensional ◽  
Quantum Numbers ◽  
Finite Dimensional ◽  
Periodic Law

The results of the discovery of the Periodic law by D. I. Mendeleev are considered, and the actual formulation of this law is given. Some examples of the use of symmetry groups in modern science are given. It is shown that the SO(4,2) group allows presenting the contents of the Periodic system of elements in full coincidence with the experimentally established structure of electronic shells of corresponding atoms without involving any additional quantum numbers characterizing the properties of atoms. adynamic substantiation of the use of representations of the dynamic symmetry group of the quantum system, isovalent to hydrogen, for a mathematical description of the properties of the symmetry of the Periodic system of elements is proposed. Using it, the splitting of the infinite-dimensional unitary representations of the group SO(4,2) into the finite-dimensional multiplets, determined by the quantum numbers describing the states of electrons, was implemented. A problem of inclusion of isotopes of elements in the Periodic system of elements is discussed.

scholarly journals An FDA-Based Approach for Clustering Elicited Expert Knowledge

Stats ◽  
2021 ◽  
Vol 4 (1) ◽  
pp. 184-204
Author(s):  
Carlos Barrera-Causil ◽  
Juan Carlos Correa ◽  
Andrew Zamecnik ◽  
Francisco Torres-Avilés ◽  
Fernando Marmolejo-Ramos
Keyword(s):  
Functional Data ◽  
Expert Knowledge ◽  
Probability Distributions ◽  
Knowledge Elicitation ◽  
Parallel Analysis ◽  
Data Sets ◽  
Dimensional Problem ◽  
Prior Distributions ◽  
Infinite Dimensional ◽  
Finite Dimensional

Expert knowledge elicitation (EKE) aims at obtaining individual representations of experts’ beliefs and render them in the form of probability distributions or functions. In many cases the elicited distributions differ and the challenge in Bayesian inference is then to find ways to reconcile discrepant elicited prior distributions. This paper proposes the parallel analysis of clusters of prior distributions through a hierarchical method for clustering distributions and that can be readily extended to functional data. The proposed method consists of (i) transforming the infinite-dimensional problem into a finite-dimensional one, (ii) using the Hellinger distance to compute the distances between curves and thus (iii) obtaining a hierarchical clustering structure. In a simulation study the proposed method was compared to k-means and agglomerative nesting algorithms and the results showed that the proposed method outperformed those algorithms. Finally, the proposed method is illustrated through an EKE experiment and other functional data sets.

scholarly journals The Farkas lemma of Shimizu, Aiyoshi and Katayama

1985 ◽  
Vol 31 (3) ◽  
pp. 445-450 ◽  
Author(s):  
Charles Swartz
Keyword(s):  
Convex Programming ◽  
Infinite Dimensional ◽  
Farkas Lemma ◽  
Finite Dimensional

Shimizu, Aiyoshi and Katayama have recently given a finite dimensional generalization of the classical Farkas Lemma. In this note we show that a result of Pshenichnyi on convex programming can be used to give a generalization of the result of Shimizu, Aiyoshi and Katayama to infinite dimensional spaces. A generalized Farkas Lemma of Glover is also obtained.

DUAL FORMULATION OF OPTIMAL PROBLEM FOR CONTINUOUS-TIME SYSTEMS

2005 ◽  
Vol 02 (03) ◽  
pp. 251-258
Author(s):  
HANLIN HE ◽  
QIAN WANG ◽  
XIAOXIN LIAO
Keyword(s):  
Objective Function ◽  
Continuous Time ◽  
Dimensional Space ◽  
Finite Dimensional Space ◽  
Error Response ◽  
Infinite Dimensional ◽  
Dual Formulation ◽  
Finite Dimensional ◽  
Continuous Time Systems ◽  
Time Systems

The dual formulation of the maximal-minimal problem for an objective function of the error response to a fixed input in the continuous-time systems is given by a result of Fenchel dual. This formulation probably changes the original problem in the infinite dimensional space into the maximal problem with some restrained conditions in the finite dimensional space, which can be researched by finite dimensional space theory. When the objective function is given by the norm of the error response, the maximum of the error response or minimum of the error response, the dual formulation for the problems of L1-optimal control, the minimum of maximal error response, and the minimal overshoot etc. can be obtained, which gives a method for studying these problems.

scholarly journals Free non-associative principal train algebras

1984 ◽  
Vol 27 (3) ◽  
pp. 313-319 ◽  
Author(s):  
P. Holgate
Keyword(s):  
Finite Number ◽  
Ground Field ◽  
Linear Forms ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
The Senses ◽  
Special Train ◽  
Complex Coefficients ◽  
Infinite Linear

The definitions of finite dimensional baric, train, and special train algebras, and of genetic algebras in the senses of Schafer and Gonshor (which coincide when the ground field is algebraically closed, and which I call special triangular) are given in Worz-Busekros's monograph [8]. In [6] I introduced applications requiring infinite dimensional generalisations. The elements of these algebras were infinite linear forms in basis elements a0, a1,… and complex coefficients such that In this paper I consider only algebras whose elements are forms which only a finite number of the xi are non zero.

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