scholarly journals Rank partition functions and truncated theta identities

2021 ◽  
pp. 23-23
Author(s):  
Mircea Merca
Keyword(s):  
Partition Function ◽  
Generating Functions ◽  
Linear Inequalities ◽  
Partition Functions ◽  
Integer Partition ◽  
Garden Of Eden ◽  
Freeman Dyson

In 1944, Freeman Dyson defined the concept of rank of an integer partition and introduced without definition the term of crank of an integer partition. A definition for the crank satisfying the properties hypothesized for it by Dyson was discovered in 1988 by G.E. Andrews and F.G. Garvan. In this paper, we introduce truncated forms for two theta identities involving the generating functions for partitions with non-negative rank and non-negative crank. As corollaries we derive new infinite families of linear inequalities for the partition function p(n). The number of Garden of Eden partitions are also considered in this context in order to provide other infinite families of linear inequalities for p(n).

Some generating functions and inequalities for the andrews–stanley partition functions

2021 ◽  
pp. 1-16
Author(s):  
Na Chen ◽  
Shane Chern ◽  
Yan Fan ◽  
Ernest X. W. Xia
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Partition Functions ◽  
Integer Partition

Abstract Let $\mathcal {O}(\pi )$ denote the number of odd parts in an integer partition $\pi$ . In 2005, Stanley introduced a new statistic $\operatorname {srank}(\pi )=\mathcal {O}(\pi )-\mathcal {O}(\pi ')$ , where $\pi '$ is the conjugate of $\pi$ . Let $p(r,\,m;n)$ denote the number of partitions of $n$ with srank congruent to $r$ modulo $m$ . Generating function identities, congruences and inequalities for $p(0,\,4;n)$ and $p(2,\,4;n)$ were then established by a number of mathematicians, including Stanley, Andrews, Swisher, Berkovich and Garvan. Motivated by these works, we deduce some generating functions and inequalities for $p(r,\,m;n)$ with $m=16$ and $24$ . These results are refinements of some inequalities due to Swisher.

Partition Function and Density of States in Models of a Finite Number of Ising Spins with Direct Exchange between the Minimum and Maximum Number of Nearest Neighbors

2016 ◽  
Vol 247 ◽  
pp. 142-147
Author(s):  
Petr Dmitrievich Andriushchenko ◽  
Konstantin V. Nefedev
Keyword(s):  
Partition Function ◽  
Density Of States ◽  
Generating Functions ◽  
Energy State ◽  
Energy Levels ◽  
Nearest Neighbors ◽  
Ising Models ◽  
Partition Functions ◽  
Ising Chain ◽  
Ising Spins

The results of studies of 1D Ising models and Curie-Weiss models partition functions structure are presented in this work. Exact calculation of the partition function using the authors combinatorial approach for such system is discussed. The distribution of the energy levels degeneracy was calculated. Analytical solution for density of states of 1D Ising chain were obtained. Generating functions for these models were obtained. It was suggested that in Curie-Weiss model the transition to a low-energy state occurs without the formation of separation boundaries

scholarly journals Free partition functions and an averaged holographic duality

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Nima Afkhami-Jeddi ◽  
Henry Cohn ◽  
Thomas Hartman ◽  
Amirhossein Tajdini
Keyword(s):  
Partition Function ◽  
Spectral Gap ◽  
Central Charge ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Partition Functions ◽  
General Constraints ◽  
Dimensional Gravity ◽  
Holographic Duality

Abstract We study the torus partition functions of free bosonic CFTs in two dimensions. Integrating over Narain moduli defines an ensemble-averaged free CFT. We calculate the averaged partition function and show that it can be reinterpreted as a sum over topologies in three dimensions. This result leads us to conjecture that an averaged free CFT in two dimensions is holographically dual to an exotic theory of three-dimensional gravity with U(1)c×U(1)c symmetry and a composite boundary graviton. Additionally, for small central charge c, we obtain general constraints on the spectral gap of free CFTs using the spinning modular bootstrap, construct examples of Narain compactifications with a large gap, and find an analytic bootstrap functional corresponding to a single self-dual boson.

scholarly journals Gauge theories on compact toric manifolds

2021 ◽  
Vol 111 (3) ◽  
Author(s):  
Giulio Bonelli ◽  
Francesco Fucito ◽  
Jose Francisco Morales ◽  
Massimiliano Ronzani ◽  
Ekaterina Sysoeva ◽  
...  
Keyword(s):  
Partition Function ◽  
Gauge Theories ◽  
Blow Up ◽  
Gauge Coupling ◽  
Piecewise Constant Function ◽  
Partition Functions ◽  
Piecewise Constant ◽  
Toric Manifolds ◽  
Holomorphic Anomaly ◽  
The One

AbstractWe compute the $$\mathcal{N}=2$$ N = 2 supersymmetric partition function of a gauge theory on a four-dimensional compact toric manifold via equivariant localization. The result is given by a piecewise constant function of the Kähler form with jumps along the walls where the gauge symmetry gets enhanced. The partition function on such manifolds is written as a sum over the residues of a product of partition functions on $$\mathbb {C}^2$$ C 2 . The evaluation of these residues is greatly simplified by using an “abstruse duality” that relates the residues at the poles of the one-loop and instanton parts of the $$\mathbb {C}^2$$ C 2 partition function. As particular cases, our formulae compute the SU(2) and SU(3) equivariant Donaldson invariants of $$\mathbb {P}^2$$ P 2 and $$\mathbb {F}_n$$ F n and in the non-equivariant limit reproduce the results obtained via wall-crossing and blow up methods in the SU(2) case. Finally, we show that the U(1) self-dual connections induce an anomalous dependence on the gauge coupling, which turns out to satisfy a $$\mathcal {N}=2$$ N = 2 analog of the $$\mathcal {N}=4$$ N = 4 holomorphic anomaly equations.

scholarly journals Modular invariance in finite temperature Casimir effect

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Francesco Alessio ◽  
Glenn Barnich
Keyword(s):  
Partition Function ◽  
Chemical Potential ◽  
Casimir Effect ◽  
Temperature Inversion ◽  
Massless Scalar ◽  
Partition Functions ◽  
Parallel Plates ◽  
Modular Transformations ◽  
Real Analytic ◽  
Set Up

Abstract The temperature inversion symmetry of the partition function of the electromagnetic field in the set-up of the Casimir effect is extended to full modular transformations by turning on a purely imaginary chemical potential for adapted spin angular momentum. The extended partition function is expressed in terms of a real analytic Eisenstein series. These results become transparent after explicitly showing equivalence of the partition functions for Maxwell’s theory between perfectly conducting parallel plates and for a massless scalar with periodic boundary conditions.

A COMPACT BOSON COUPLED TO TWO-DIMENSIONAL GRAVITY

1991 ◽  
Vol 06 (15) ◽  
pp. 2743-2754 ◽  
Author(s):  
NORISUKE SAKAI ◽  
YOSHIAKI TANII
Keyword(s):  
Field Theory ◽  
Partition Function ◽  
Quantum Gravity ◽  
Matrix Models ◽  
Riemann Surfaces ◽  
Partition Functions ◽  
Two Dimensional ◽  
Dimensional Gravity ◽  
The Continuum ◽  
Continuum Field

The radius dependence of partition functions is explicitly evaluated in the continuum field theory of a compactified boson, interacting with two-dimensional quantum gravity (noncritical string) on Riemann surfaces for the first few genera. The partition function for the torus is found to be a sum of terms proportional to R and 1/R. This is in agreement with the result of a discretized version (matrix models), but is quite different from the critical string. The supersymmetric case is also explicitly evaluated.

scholarly journals Algorithms for Estimating the Partition Function of Restricted Boltzmann Machines (Extended Abstract)

2020 ◽  
Author(s):  
Oswin Krause ◽  
Asja Fischer ◽  
Christian Igel
Keyword(s):  
Partition Function ◽  
Markov Random Fields ◽  
Probabilistic Models ◽  
Ratio Method ◽  
Parallel Tempering ◽  
Partition Functions ◽  
Path Sampling ◽  
Reference Distribution ◽  
Restricted Boltzmann Machines ◽  
Boltzmann Machines

Estimating the normalization constants (partition functions) of energy-based probabilistic models (Markov random fields) with a high accuracy is required for measuring performance, monitoring the training progress of adaptive models, and conducting likelihood ratio tests. We devised a unifying theoretical framework for algorithms for estimating the partition function, including Annealed Importance Sampling (AIS) and Bennett's Acceptance Ratio method (BAR). The unification reveals conceptual similarities of and differences between different approaches and suggests new algorithms. The framework is based on a generalized form of Crooks' equality, which links the expectation over a distribution of samples generated by a transition operator to the expectation over the distribution induced by the reversed operator. Different ways of sampling, such as parallel tempering and path sampling, are covered by the framework. We performed experiments in which we estimated the partition function of restricted Boltzmann machines (RBMs) and Ising models. We found that BAR using parallel tempering worked well with a small number of bridging distributions, while path sampling based AIS performed best with many bridging distributions. The normalization constant is measured w.r.t.~a reference distribution, and the choice of this distribution turned out to be very important in our experiments. Overall, BAR gave the best empirical results, outperforming AIS.

scholarly journals Automatic contraction of unstructured tensor networks

2020 ◽  
Vol 8 (1) ◽  
Author(s):  
Adam Jermyn
Keyword(s):  
Partition Function ◽  
Statistical Physics ◽  
Discrete Models ◽  
Correlation Structure ◽  
Central Problem ◽  
Partition Functions ◽  
Lattice Systems ◽  
Tensor Networks ◽  
Tensor Network

The evaluation of partition functions is a central problem in statistical physics. For lattice systems and other discrete models the partition function may be expressed as the contraction of a tensor network. Unfortunately computing such contractions is difficult, and many methods to make this tractable require periodic or otherwise structured networks. Here I present a new algorithm for contracting unstructured tensor networks. This method makes no assumptions about the structure of the network and performs well in both structured and unstructured cases so long as the correlation structure is local.

scholarly journals Generating functions and congruences for some partition functions related to mock theta functions

2019 ◽  
Vol 16 (02) ◽  
pp. 423-446 ◽  
Author(s):  
Nayandeep Deka Baruah ◽  
Nilufar Mana Begum
Keyword(s):  
Generating Functions ◽  
Theta Functions ◽  
Partition Functions ◽  
Mock Theta Functions ◽  
Third Order

Recently, Andrews, Dixit and Yee introduced partition functions associated with Ramanujan/Watson third-order mock theta functions [Formula: see text] and [Formula: see text]. In this paper, we find several new exact generating functions for those partition functions as well as the associated smallest part functions and deduce several new congruences modulo powers of 5.

scholarly journals MODIFIED PARTITION FUNCTIONS, CONSISTENT ANOMALIES AND CONSISTENT SCHWINGER TERMS

2011 ◽  
Vol 08 (08) ◽  
pp. 1747-1762 ◽  
Author(s):  
AMIR ABBASS VARSHOVI
Keyword(s):  
Partition Function ◽  
Gauge Theories ◽  
Partition Functions ◽  
Gauge Invariant

A gauge invariant partition function is defined for gauge theories which leads to the standard quantization. It is shown that the descent equations and consequently the consistent anomalies and Schwinger terms can be extracted from this modified partition function naturally.

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