scholarly journals Note on Dual Symmetric Functions

1931 ◽  
Vol 2 (3) ◽  
pp. 164-167 ◽  
Author(s):  
A. C. Aitken
Keyword(s):  
Generating Function ◽  
Symmetric Functions ◽  
General Theorem ◽  
Special Cases ◽  
Dual Forms

In an earlier paper, which this note is intended to supplement and in some respects improve, the writer gave a general theorem of duality relating to isobaric determinants with elements Cr and Hr, the elementary and the complete homogeneous symmetric functions of a set of variables. The result was shewn to include as special cases the dual forms of “bi-alternant” symmetric functions given by Jacobi and Naegelsbach, as well as two equivalent forms of isobaric determinant used by MacMahon as a generating function in an important problem of permutations.

ON TWO THEOREMS OF QUINZII AND RENT CONTROLLED HOUSING ALLOCATION IN SWEDEN

2007 ◽  
Vol 09 (03) ◽  
pp. 515-525
Author(s):  
KIMMO ERIKSSON ◽  
JONAS SJÖSTRAND
Keyword(s):  
Control System ◽  
Empirical Data ◽  
General Theorem ◽  
Black Market ◽  
Utility Functions ◽  
Rent Control ◽  
Black And White ◽  
The Core ◽  
Special Cases ◽  
Housing Allocation

The Swedish rent control system creates a white market for swapping rental contracts and a black market for selling rental contracts. Empirical data suggests that in this black-and-white market some people act according to utility functions that are both discontinuous and locally decreasing in money. We discuss Quinzii's theorem for the nonemptiness of the core of generalized house-swapping games, and show how it can be extended to cover the Swedish game. In a second part, we show how this theorem of Quinzii and her second theorem on nonemptiness of the core in two-sided models are both special cases of a more general theorem.

On operators which depend on some parameter

1967 ◽  
Vol 63 (2) ◽  
pp. 359-366
Author(s):  
G. O. Okikiolu
Keyword(s):  
General Theorem ◽  
Variable Parameter ◽  
Poisson Operator ◽  
Special Cases ◽  
New Inequalities

AbstractThe purpose of this paper is to study the mapping properties of certain operators which depend on a variable parameter. We prove a general theorem which is applied to some special cases. Among results obtained are new inequalities involving the Poisson operator and its conjugate.

A general procedure for rational inequalities

2015 ◽  
Vol 08 (01) ◽  
pp. 1550014
Author(s):  
B. E. Rhoades
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
General Procedure ◽  
General Theorem ◽  
Type Inequality ◽  
Special Cases ◽  
Rational Type

Recently Bhatt, Chaukiyal and Dimri proved a fixed point theorem for a pair of maps satisfying a rational type inequality. It is the purpose of this paper to show that this result, along with a number of others, are all special cases of a general theorem of Sehie Park.

scholarly journals Constructions of L∞ Algebras and Their Field Theory Realizations

2018 ◽  
Vol 2018 ◽  
pp. 1-11 ◽  
Author(s):  
Olaf Hohm ◽  
Vladislav Kupriyanov ◽  
Dieter Lüst ◽  
Matthias Traube
Keyword(s):  
Field Theory ◽  
Initial Data ◽  
Vector Space ◽  
Jacobi Identity ◽  
General Theorem ◽  
Commutator Algebra ◽  
Courant Algebroid ◽  
Linear Map ◽  
General Initial Data ◽  
Special Cases

We construct L∞ algebras for general “initial data” given by a vector space equipped with an antisymmetric bracket not necessarily satisfying the Jacobi identity. We prove that any such bracket can be extended to a 2-term L∞ algebra on a graded vector space of twice the dimension, with the 3-bracket being related to the Jacobiator. While these L∞ algebras always exist, they generally do not realize a nontrivial symmetry in a field theory. In order to define L∞ algebras with genuine field theory realizations, we prove the significantly more general theorem that if the Jacobiator takes values in the image of any linear map that defines an ideal there is a 3-term L∞ algebra with a generally nontrivial 4-bracket. We discuss special cases such as the commutator algebra of octonions, its contraction to the “R-flux algebra,” and the Courant algebroid.

A generalization of the Whitney rank generating function

1993 ◽  
Vol 113 (2) ◽  
pp. 267-280 ◽  
Author(s):  
G. E. Farr
Keyword(s):  
Generating Function ◽  
Linear Code ◽  
Tutte Polynomial ◽  
Natural Generalization ◽  
Chromatic Polynomial ◽  
Weight Enumerator ◽  
Hadamard Transform ◽  
Percolation Probability ◽  
Special Cases

AbstractThe Whitney quasi-rank generating function, which generalizes the Whitney rank generating function (or Tutte polynomial) of a graph, is introduced. It is found to include as special cases the weight enumerator of a (not necessarily linear) code, the percolation probability of an arbitrary clutter and a natural generalization of the chromatic polynomial. The crucial construction, essentially equivalent to one of Kung, is a means of associating, to any function, a rank-like function with suitable properties. Some of these properties, including connections with the Hadamard transform, are discussed.

scholarly journals Some relations of interpolated multiple zeta values

2017 ◽  
Vol 28 (05) ◽  
pp. 1750033 ◽  
Author(s):  
Zhonghua Li ◽  
Chen Qin
Keyword(s):  
Generating Function ◽  
Hypergeometric Functions ◽  
Multiple Zeta Values ◽  
Fixed Weight ◽  
Multiple Zeta ◽  
Special Cases ◽  
Zeta Values

In this paper, the extended double shuffle relations for interpolated multiple zeta values (MZVs) are established. As an application, Hoffman’s relations for interpolated MZVs are proved. Furthermore, a generating function for sums of interpolated MZVs of fixed weight, depth and height is represented by hypergeometric functions, and we discuss some special cases.

scholarly journals Algorithms for Some Euler-Type Identities for Multiple Zeta Values

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Shifeng Ding ◽  
Weijun Liu
Keyword(s):  
Generating Function ◽  
Symmetric Functions ◽  
Convergent Series ◽  
Multiple Zeta Values ◽  
Double Product ◽  
Multiple Zeta ◽  
Zeta Values ◽  
Positive Integers

Multiple zeta values are the numbers defined by the convergent seriesζ(s1,s2,…,sk)=∑n1>n2>⋯>nk>0(1/n1s1 n2s2⋯nksk), wheres1,s2,…,skare positive integers withs1>1. Fork≤n, letE(2n,k)be the sum of all multiple zeta values with even arguments whose weight is2nand whose depth isk. The well-known resultE(2n,2)=3ζ(2n)/4was extended toE(2n,3)andE(2n,4)by Z. Shen and T. Cai. Applying the theory of symmetric functions, Hoffman gave an explicit generating function for the numbersE(2n,k)and then gave a direct formula forE(2n,k)for arbitraryk≤n. In this paper we apply a technique introduced by Granville to present an algorithm to calculateE(2n,k)and prove that the direct formula can also be deduced from Eisenstein's double product.

scholarly journals GIX/MY/1 systems with resident server and generally distributed arrival and service groups

1996 ◽  
Vol 9 (2) ◽  
pp. 159-170
Author(s):  
Alexander Dukhovny
Keyword(s):  
Group Size ◽  
Generating Function ◽  
Riemann Boundary Value Problem ◽  
Riemann Boundary ◽  
Queueing Process ◽  
Imbedded Markov Chain ◽  
Special Cases ◽  
Necessary And Sufficient ◽  
Service Groups ◽  
Steady State Probabilities

Considered are bulk systems of GI/M/1 type in which the server stands by when it is idle, waits for the first group to arrive if the queue is empty, takes customers up to its capacity and is not available when busy. Distributions of arrival group size and server's capacity are not restricted. The queueing process is analyzed via an augmented imbedded Markov chain. In the general case, the generating function of the steady-state probabilities of the chain is found as a solution of a Riemann boundary value problem. This function is proven to be rational when the generating function of the arrival group size is rational, in which case the solution is given in terms of roots of a characteristic equation. A necessary and sufficient condition of ergodicity is proven in the general case. Several special cases are studied in detail: single arrivals, geometric arrivals, bounded arrivals, and an arrival group with a geometric tail.

scholarly journals On Cherednik and Nazarov-Sklyanin large N limit construction for integrable many-body systems with elliptic dependence on momenta

2021 ◽  
Vol 2021 (12) ◽  
Author(s):  
A. Grekov ◽  
A. Zotov
Keyword(s):  
Explicit Expression ◽  
Generating Function ◽  
Infinite Number ◽  
Symmetric Functions ◽  
Many Body ◽  
Quantum Double ◽  
Large N ◽  
Elliptic Model ◽  
Many Body Systems ◽  
Number Of Particles

Abstract The infinite number of particles limit in the dual to elliptic Ruijsenaars model (coordinate trigonometric degeneration of quantum double elliptic model) is proposed using the Nazarov-Sklyanin approach. For this purpose we describe double-elliptization of the Cherednik construction. Namely, we derive explicit expression in terms of the Cherednik operators, which reduces to the generating function of Dell commuting Hamiltonians on the space of symmetric functions. Although the double elliptic Cherednik operators do not commute, they can be used for construction of the N → ∞ limit.

scholarly journals Minimal Overlapping Patterns in Colored Permutations

10.37236/2021 ◽  
2011 ◽  
Vol 18 (2) ◽  
Author(s):  
Adrian Duane ◽  
Jeffrey Remmel
Keyword(s):  
General Formula ◽  
Generating Function ◽  
Minimal Length ◽  
Special Cases ◽  
Maximum Packings ◽  
Colored Permutations

A pattern $P$ of length $j$ has the minimal overlapping property if two consecutive occurrences of the pattern can overlap in at most one place, namely, at the end of the first consecutive occurrence of the pattern and at the start of the second consecutive occurrence of the pattern. For patterns $P$ which have the minimal overlapping property, we derive a general formula for the generating function for the number of consecutive occurrences of $P$ in words, permutations and $k$-colored permutations in terms of the number of maximum packings of $P$ which are patterns of minimal length which has $n$ consecutive occurrences of the pattern $P$. Our results have as special cases several results which have appeared in the literature. Another consequence of our results is to prove a conjecture of Elizalde that two permutations $\alpha$ and $\beta$ of size $j$ which have the minimal overlapping property are strongly $c$-Wilf equivalent if $\alpha$ and $\beta$ have the same first and last elements.

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