scholarly journals Congruence properties ofS-partition functions

2011 ◽  
Vol 4 (4) ◽  
pp. 411-416
Author(s):  
Andrew Gruet ◽  
Linzhi Wang ◽  
Katherine Yu ◽  
Jiangang Zeng
Keyword(s):  
Partition Functions ◽  
Congruence Properties

scholarly journals Congruence properties for a certain kind of partition functions

2016 ◽  
Vol 290 ◽  
pp. 739-772 ◽  
Author(s):  
Su-Ping Cui ◽  
Nancy S.S. Gu ◽  
Anthony X. Huang
Keyword(s):  
Partition Functions ◽  
Congruence Properties

scholarly journals Congruence Properties of Binary Partition Functions

2013 ◽  
Vol 17 (1) ◽  
pp. 15-26 ◽  
Author(s):  
Katherine Anders ◽  
Melissa Dennison ◽  
Jennifer Weber Lansing ◽  
Bruce Reznick
Keyword(s):  
Partition Functions ◽  
Binary Partition ◽  
Congruence Properties

On congruence properties of the restricted partition functions $$p_{\omega }(n,k)$$ p ω ( n , k ) and $$p_{\nu }(n,k)$$ p ν ( n , k )

2018 ◽  
Vol 49 (1) ◽  
pp. 105-113
Author(s):  
Robson da Silva ◽  
Kelvin Souza de Oliveira ◽  
Almir Cunha da Graça Neto
Keyword(s):  
Partition Functions ◽  
Restricted Partition ◽  
Congruence Properties

scholarly journals Some restricted partition functions: Congruences modulo 3

1970 ◽  
Vol 11 (1) ◽  
pp. 82-90 ◽  
Author(s):  
D. B. Lahiri
Keyword(s):  
Partition Functions ◽  
General Category ◽  
Previous Communication ◽  
Restricted Partition ◽  
Image Position ◽  
Congruence Properties

In a previous communication [5] the author has dealt with the congruence properties of some restricted partition functions. The general category of such functions may be denoted.

ON THE BROKEN 1-DIAMOND PARTITION

2008 ◽  
Vol 04 (02) ◽  
pp. 199-218 ◽  
Author(s):  
ERIC MORTENSON
Keyword(s):  
Modular Forms ◽  
Generating Functions ◽  
Infinite Family ◽  
Partition Functions ◽  
Congruence Properties

We introduce a crank-like statistic for a different class of partitions. In [4], Andrews and Paule initiated the study of broken k-diamond partitions. Their study of the respective generating functions led to an infinite family of modular forms, about which they were able to produce interesting arithmetic theorems and conjectures for the related partition functions. Here we establish a crank-like statistic for the broken 1-diamond partition and discuss its role in congruence properties.

Congruence properties of pk(n)

2019 ◽  
Vol 15 (06) ◽  
pp. 1267-1290 ◽  
Author(s):  
Julia Q. D. Du ◽  
Edward Y. S. Liu ◽  
Jack C. D. Zhao
Keyword(s):  
Positive Integer ◽  
Product Formula ◽  
Partition Functions ◽  
Euler Product ◽  
Modular Equations ◽  
Arbitrary Positive Integer ◽  
Modular Functions ◽  
Unified Approach ◽  
Linear Recurring Sequences ◽  
Congruence Properties

We present a unified approach to establish infinite families of congruences for [Formula: see text] for arbitrary positive integer [Formula: see text], where [Formula: see text] is given by the [Formula: see text]th power of the Euler product [Formula: see text]. For [Formula: see text], define [Formula: see text] to be the least positive integer such that [Formula: see text] and [Formula: see text] the least non-negative integer satisfying [Formula: see text]. Using the Atkin [Formula: see text]-operator, we find that the generating function of [Formula: see text] (respectively, [Formula: see text]) can be expressed as the product of an integral linear combination of modular functions on [Formula: see text] and [Formula: see text] (respectively, [Formula: see text]) for any [Formula: see text] and [Formula: see text]. By investigating the properties of the modular equations of the [Formula: see text]th order under the Atkin [Formula: see text]-operator, we obtain that these generating functions are determined by some linear recurring sequences. Utilizing the periodicity of these linear recurring sequences modulo [Formula: see text], we are led to infinite families of congruences for [Formula: see text] modulo any [Formula: see text] with [Formula: see text] and periodic relations between the values of [Formula: see text] modulo powers of [Formula: see text]. As applications, infinite families of congruences for many partition functions such as [Formula: see text]-core partition functions, the partition function and Andrews’ spt-function are easily obtained.

Bimolecular Reactions, Transition-State Theory

2018 ◽  
Author(s):  
Niels Engholm Henriksen ◽  
Flemming Yssing Hansen
Keyword(s):  
Transition State ◽  
Saddle Point ◽  
Transition State Theory ◽  
Classical Mechanics ◽  
Small Degree ◽  
State Theory ◽  
Reaction Coordinate ◽  
Partition Functions ◽  
Bimolecular Reactions ◽  
Activated Complex

This chapter discusses an approximate approach—transition-state theory—to the calculation of rate constants for bimolecular reactions. A reaction coordinate is identified from a normal-mode coordinate analysis of the activated complex, that is, the supermolecule on the saddle-point of the potential energy surface. Motion along this coordinate is treated by classical mechanics and recrossings of the saddle point from the product to the reactant side are neglected, leading to the result of conventional transition-state theory expressed in terms of relevant partition functions. Various alternative derivations are presented. Corrections that incorporate quantum mechanical tunnelling along the reaction coordinate are described. Tunnelling through an Eckart barrier is discussed and the approximate Wigner tunnelling correction factor is derived in the limit of a small degree of tunnelling. It concludes with applications of transition-state theory to, for example, the F + H2 reaction, and comparisons with results based on quasi-classical mechanics as well as exact quantum mechanics.

Hamilton-Jacobi Theory

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Phase Space ◽  
Bbgky Hierarchy ◽  
Distribution Functions ◽  
Symplectic Integrators ◽  
Partition Functions ◽  
Von Neumann ◽  
Equipartition Theorem ◽  
Recurrence Theorem ◽  
Phase Amplitude ◽  
Canonical Ensembles

This chapter focuses on Liouville’s theorem and classical statistical mechanics, deriving the classical propagator. The terms ‘phase space volume element’ and ‘Liouville operator’ are defined and an n-particle phase space probability density function is constructed to derive the Liouville equation. This is deconstructed into the BBGKY hierarchy, and radial distribution functions are used to develop n-body correlation functions. Koopman–von Neumann theory is investigated as a classical wavefunction approach. The chapter develops an operatorial mechanics based on classical Hilbert space, and discusses the de Broglie–Bohm formulation of quantum mechanics. Partition functions, ensemble averages and the virial theorem of Clausius are defined and Poincaré’s recurrence theorem, the Gibbs H-theorem and the Gibbs paradox are discussed. The chapter also discusses commuting observables, phase–amplitude decoupling, microcanonical ensembles, canonical ensembles, grand canonical ensembles, the Boltzmann factor, Mayer–Montroll cluster expansion and the equipartition theorem and investigates symplectic integrators, focusing on molecular dynamics.

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