scholarly journals Comultiplication Rules for the Double Schur Functions and Cauchy Identities

10.37236/102 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
A. I. Molev
Keyword(s):  
Symmetric Functions ◽  
Schur Functions ◽  
Primitive Elements ◽  
Expansion Formula ◽  
New Family ◽  
Dual Object ◽  
Multiplication Rule ◽  
Distinguished Basis ◽  
Natural Way

The double Schur functions form a distinguished basis of the ring $\Lambda(x\!\parallel\!a)$ which is a multiparameter generalization of the ring of symmetric functions $\Lambda(x)$. The canonical comultiplication on $\Lambda(x)$ is extended to $\Lambda(x\!\parallel\!a)$ in a natural way so that the double power sums symmetric functions are primitive elements. We calculate the dual Littlewood–Richardson coefficients in two different ways thus providing comultiplication rules for the double Schur functions. We also prove multiparameter analogues of the Cauchy identity. A new family of Schur type functions plays the role of a dual object in the identities. We describe some properties of these dual Schur functions including a combinatorial presentation and an expansion formula in terms of the ordinary Schur functions. The dual Littlewood–Richardson coefficients provide a multiplication rule for the dual Schur functions.

scholarly journals κ-DEFORMED STATISTICS AND CLASSICAL FOUR-MOMENTUM ADDITION LAW

2008 ◽  
Vol 23 (09) ◽  
pp. 653-665 ◽  
Author(s):  
MARCIN DASZKIEWICZ ◽  
JERZY LUKIERSKI ◽  
MARIUSZ WORONOWICZ
Keyword(s):  
Minkowski Space ◽  
Fock Space ◽  
Scalar Fields ◽  
Cross Product ◽  
Creation And Annihilation Operators ◽  
Multiplication Rule ◽  
Symmetry Properties ◽  
Deformed Algebra ◽  
Free Scalar

We consider κ-deformed relativistic symmetries described algebraically by modified Majid–Ruegg bi-cross-product basis and investigate the quantization of field oscillators for the κ-deformed free scalar fields on κ-Minkowski space. By modification of standard multiplication rule, we postulate the κ-deformed algebra of bosonic creation and annihilation operators. Our algebra permits one to define the n-particle states with classical addition law for the four-momentum in a way which is not in contradiction with the nonsymmetric quantum four-momentum co-product. We introduce κ-deformed Fock space generated by our κ-deformed oscillators which satisfy the standard algebraic relations with modified κ-multiplication rule. We show that such a κ-deformed bosonic Fock space is endowed with the conventional bosonic symmetry properties. Finally we discuss the role of κ-deformed algebra of oscillators in field-theoretic noncommutative framework.

scholarly journals Auotomodeling rational mnemofunctions and their link to an analytical representation of distributions

2019 ◽  
Vol 55 (3) ◽  
pp. 288-298
Author(s):  
T. R. Shahava
Keyword(s):  
Asymptotic Expansion ◽  
Asymptotic Expansions ◽  
Real Line ◽  
Analytical Representation ◽  
Partial Fraction ◽  
Boundary Values ◽  
Multiplication Rule ◽  
Space Of Distributions ◽  
Partial Fraction Decomposition ◽  
Proper Rational Function

Mnemofunctions of the form f(x/ε), where f is the proper rational function without singularities on the real line, are considered in this article. Such mnemofunctions are called automodeling rational mnemofunctions. They possess the following fine properties: asymptotic expansions in the space of distributions can be written in explicit form and the asymptotic expansion of the product of such mnemofunctions is uniquely determined by the expansions of multiplicands.Partial fraction decomposition of automodeling rational mnemofunctions generates the so-called sloped analytical representation of a distribution, i.e. the representation of a distribution by a jump of the boundary values of the functions analytical in upper and lower half-planes. Sloped analytical representation is similar to the classical Cauchy analytical representation, but its structure is more complicated. The multiplication rule of such representations is described in this article.

scholarly journals Divergence & Probability Density : A Hypothesis

2020 ◽  
Author(s):  
JAYDIP DATTA
Keyword(s):  
Probability Density ◽  
Multiplication Rule ◽  
Multiplicative Rule ◽  
Algebraic Properties

The earlier work by Datta et al ( 1 ) the note has been made with the algebraic properties of Probability ie P (A) +P(B )= P ( A*B ) . This is termed as multiplicative rule of Probability. By multiplication rule any number say a can be first diverged through a series of a ( pow ) x , Where x is an integer like 0 to n . REF :PROBABILITY & WAVEFUNCTION : A NOTE November 2019 DOI: 10.31219/osf.io/awk96

scholarly journals On Complex Numbers in Higher Dimensions

Axioms ◽  
2022 ◽  
Vol 11 (1) ◽  
pp. 22
Author(s):  
Wolf-Dieter Richter
Keyword(s):  
Probability Distributions ◽  
Three Dimensional ◽  
Geometric Approach ◽  
Higher Dimensions ◽  
Space Structure ◽  
Complex Numbers ◽  
Vector Product ◽  
Multiplication Rule ◽  
Generalized Complex ◽  
General Vector

The geometric approach to generalized complex and three-dimensional hyper-complex numbers and more general algebraic structures being based upon a general vector space structure and a geometric multiplication rule which was only recently developed is continued here in dimension four and above. To this end, the notions of geometric vector product and geometric exponential function are extended to arbitrary finite dimensions and some usual algebraic rules known from usual complex numbers are replaced with new ones. An application for the construction of directional probability distributions is presented.

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