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A brief history of partitions of numbers, partition functions and their modern applications

INTERNATIONAL JOURNAL OF MATHEMATICAL EDUCATION IN SCIENCE AND TECHNOLOGY 2016年第3期47卷 329-355页

作者： Debnath, Lokenath Univ Texas Pan Amer Dept Math Edinburg TX 78538 USA

[GRAPHICS] 'Number rules the universe.' The Pythagoras [GRAPHICS] [GRAPHICS] This paper is written to commemorate the centennial anniversary of the Mathematical Association of America. It deals with a short history of different kinds of natural numbers including triangular, square, pentagonal, hexagonal and k-gonal numbers, and their simple properties and their geometrical representations. Included are Euclid's and Pythagorean's main contributions to elementary number theory with the main contents of the Euclid Elements of the 13-volume masterpiece of mathematical work. This is followed by Euler's new discovery of the additive number theory based on partitions of numbers. Special attention is given to many examples, Euler's theorems on partitions of numbers with geometrical representations of Ferrers' graphs, Young's diagrams, Lagrange's four-square theorem and the celebrated Waring problem. Included are Euler's generating functions for the partitions of numbers, Euler's pentagonal number theorem, Gauss' triangular and square number theorems and the Jacobi triple product identity. Applications of the theory of partitions of numbers to different statistics such as the Bose- Einstein, Fermi- Dirac, Gentile, and Maxwell- Boltzmann statistics are briefly discussed. Special attention is given to pedagogical information through historical approach to number theory so that students and teachers at the school, college and university levels can become familiar with the basic concepts of partitions of numbers, partition functions and their modern applications, and can pursue advanced study and research in analytical and computational number theory.

关键词： partitions of numbers generating functions partition functions asymptotic formulas of partition functions applications to statistical physics 01A45 11P83 01A50 1100 11P82

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partition functions of N = (2,2) Gauge Theories on S² and Vortices

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COMMUNICATIONS IN MATHEMATICAL PHYSICS 2015年第3期334卷 1483-1527页

作者： Benini, Francesco Cremonesi, Stefano SUNY Stony Brook Simons Ctr Geometry & Phys Stony Brook NY 11794 USA Univ London Imperial Coll Sci Technol & Med Theoret Phys Grp London SW7 2AZ England

We apply localization techniques to compute the partition function of a two-dimensional N = (2, 2) R-symmetric theory of vector and chiral multiplets on S-2. The path integral reduces to a sum over topological sectors of a matrix integral over the Cartan subalgebra of the gauge group. For gauge theories which would be completely Higgsed in the presence of a Fayet-Iliopoulos term in flat space, the path integral alternatively reduces to the product of a vortex times an antivortex partition functions, weighted by semiclassical factors and summed over isolated points on the Higgs branch. For applications, we evaluate the partition function for some U(N) gauge theories, showing equality of the path integrals for theories conjectured to be dual by Hori and Tong and deriving new expressions for vortex partition functions.

关键词： partition functions GAUGE field theory PATH integrals VORTEX methods HIGGS bosons VECTOR fields

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Vortex partition functions, Wall Crossing and Equivariant Gromov-Witten Invariants

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COMMUNICATIONS IN MATHEMATICAL PHYSICS 2015年第2期333卷 717-760页

作者： Bonelli, Giulio Sciarappa, Antonio Tanzini, Alessandro Vasko, Petr Int Sch Adv Studies SISSA I-34136 Trieste Italy Ist Nazl Fis Nucl Sez Trieste Trieste Italy

In this paper we identify the problem of equivariant vortex counting in a (2, 2) supersymmetric two dimensional quiver gauged linear sigma model with that of computing the equivariant Gromov-Witten invariants of the GIT quotient target space determined by the quiver. We provide new contour integral formulae for the I and J - functions encoding the equivariant quantum cohomology of the target space. Its chamber structure is shown to be encoded in the analytical properties of the integrand. This is explained both via general arguments and by checking several key cases. We show how several results in equivariant Gromov-Witten theory follow just by deforming the integration contour. In particular, we apply our formalism to compute Gromov-Witten invariants of the C-3/Z(n) orbifold, of the Uhlembeck (partial) compactification of the moduli space of instantons on C-2, and of A(n) and D-n singularities both in the orbifold and resolved phases. Moreover, we analyse dualities of quantum cohomology rings of holomorphic vector bundles over Grassmannians, which are relevant to BPS Wilson loop algebrae.

关键词： partition functions VECTOR analysis COHOMOLOGY theory QUANTUM theory GROMOV-Witten invariants SUPERSYMMETRY (Physics)

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Gluing Nekrasov partition functions

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COMMUNICATIONS IN MATHEMATICAL PHYSICS 2015年第2期337卷 785-816页

作者： Qiu, Jian Tizzano, Luigi Winding, Jacob Zabzine, Maxim Univ Luxembourg Math L-1359 Luxembourg Luxembourg Uppsala Univ Dept Phys & Astron S-75120 Uppsala Sweden

In this paper we summarise the localisation calculation of 5D super Yang-Mills on simply connected toric Sasaki-Einstein (SE) manifolds. We show how various aspects of the computation, including the equivariant index, the asymptotic behaviour and the factorisation property are governed by the combinatorial data of the toric geometry. We prove that the perturbative partition function on a simply connected SE manifold corresponding to an n-gon toric diagram factorises to n copies of perturbative part (zero instanton sector) of the Nekrasov partition function. This leads us to conjecture a prescription for the computation of the complete partition function, by gluing n copies of the full Nekrasov partition functions. This work is a generalisation of some earlier computation carried out on Y (p,q) manifolds, whose moment map cone has a quadrangle base and our result is valid for manifolds whose moment map cones have pentagon base, hexagon base, etc. The algorithm we used for dealing with general cones may also be of independent interest.

关键词： GLUING partition functions LOCALIZATION (Mathematics) YANG-Mills theory MANIFOLDS (Mathematics) FACTORIZATION (Mathematics)

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partition functions of Polychronakos like spin chains associated with polarized spin reversal operators

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NUCLEAR PHYSICS B 2014年第1期883卷 501-528页

作者： Basu-Mallick, B. Bondyopadhaya, Nilanjan Banerjee, Pratyay Saha Inst Nucl Phys Div Theory Kolkata 700064 W Bengal India Integrated Sci Educ & Res Ctr Santini Ketan 731235 W Bengal India

We construct polarized spin reversal operator (PSRO) which yields a class of representations for the BCN type of Weyl algebra, and subsequently use this PSRO to find out novel exactly solvable variants of the BCN type of spin Calogero model. The strong coupling limit of such spin Calogero models generates the BCN type of Polychronakos spin chains with PSRO. We derive the exact spectra of the BCN type of spin Calogero models with PSRO and compute the partition functions of the related spin chains by using the freezing trick. We also find out an interesting relation between the partition functions of the BCN type and A(N-1) type of Polychronakos spin chains. Finally, we study spectral properties like level density and distribution of spacing between consecutive energy levels for BCN type of Polychronakos spin chains with PSRO. (C) 2014 The Authors. Published by Elsevier B.V.

关键词： partition functions POLARIZATION (Nuclear physics) OPERATOR theory PARTICLE density (Nuclear chemistry) functions (Mathematics) MATHEMATICAL models

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Polymer quantization and the saddle point approximation of partition functions

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Physical Review D 2015年第10期92卷 104029-104029页

作者： Hugo A. Morales-Técotl Daniel H. Orozco-Borunda Saeed Rastgoo Departamento de Física Universidad Autónoma Metropolitana-- Iztapalapa San Rafael Atlixco 186 Mexico D.F. 09340 Mexico

The saddle point approximation of the path integral partition functions is an important way of deriving the thermodynamical properties of black holes. However, there are certain black hole models and some mathematically analog mechanical models for which this method cannot be applied directly. This is due to the fact that their action evaluated on a classical solution is not finite and its first variation does not vanish for all consistent boundary conditions. These problems can be dealt with by adding a counterterm to the classical action, which is a solution of the corresponding Hamilton-Jacobi equation. In this work we study the effects of polymer quantization on a mechanical model presenting the aforementioned difficulties and contrast it with the above counterterm method. This type of quantization for mechanical models is motivated by the loop quantization of gravity, which is known to play a role in the thermodynamics of black hole systems. The model we consider is a nonrelativistic particle in an inverse square potential, and we analyze two polarizations of the polymer quantization in which either the position or the momentum is discrete. In the former case, Thiemann’s regularization is applied to represent the inverse power potential, but we still need to incorporate the Hamilton-Jacobi counterterm, which is now modified by polymer corrections. In the latter, momentum discrete case, however, such regularization could not be implemented. Yet, remarkably, owing to the fact that the position is bounded, we do not need a Hamilton-Jacobi counterterm in order to have a well-defined saddle point approximation. Further developments and extensions are commented upon in the discussion.

关键词： RESEARCH Quantization (Physics) Saddlepoint approximations partition functions

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Rovibrational energies, partition functions and equilibrium fractionation of the CO₂ isotopologues

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JOURNAL OF QUANTITATIVE SPECTROSCOPY & RADIATIVE TRANSFER 2014年 147卷 233-251页

作者： Cerezo, J. Bastida, A. Requena, A. Zuniga, J. Univ Murcia Dept Quim Fis E-30100 Murcia Spain

Rovibrational energy levels, partition functions and relative abundances of the stable isotopologues of CO2 in gas phase at equilibrium are calculated using an empirical Morsecosine potential energy surface (PES) refined by fitting to the updated pure (l(2) = 0) vibrational frequencies observed for the main (CO2)-C-12-O-16 isotopologue. The rovibrational energy levels are calculated variationally using a system of optimized hyperspherical normal coordinates, and from these the vibrational terms G(v) and rotational constants B of the isotopologues are determined. The refined potential surface is shown to be clearly superior to the original potential surface, with the former reproducing the observed values of the spectroscopic constants G(v) and B-v with accuracies of about 0.1 cm(-1) and 0.00020 cm(-1), respectively, for levels with l(2) >= 0 up to 10,000 cm(-1) above the ground state. The internal partition functions of the isotopologues are calculated by approximated direct summation over the rovibrational energies and compared with both previous partition sums and values obtained from analytical expressions based on the harmonic oscillator and rigid rotor models. The partition functions calculated by approximated direct summation are then used to determine the abundances of the CO2 isotopologues at thermodynamic equilibrium using the method developed by Wang et al. [74]. Significant variations in the relative abundances of some of the CO2 multiple substituted isotopologues at terrestrial temperatures with respect to those provided by the classical harmonicbased Urey theory are found, which may be of relevance in geochemical processes. (C) 2014 Elsevier Ltd. All rights reserved.

关键词： CO2 isotopologues Potential energy surface Vibrational terms and rotational constants partition functions Relative abundances

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RAMANUJAN-TYPE CONGRUENCES FOR SEVERAL partition functions

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INTERNATIONAL JOURNAL OF NUMBER THEORY 2014年第3期10卷 641-652页

作者： Zhang, Wenlong Wang, Chun Dalian Univ Technol Sch Math Sci Dalian 116024 Peoples R China

Inspired by the recent work of Baruah and Ojah, we investigate several partition functions and establish some interesting Ramanujan-type congruences.

关键词： Ramanujan-type congruences partition functions Jacobi's identity

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Theoretical study on the thermodynamic and transition properties of the interstellar molecule: FeC

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THERMOCHIMICA ACTA 2025年 745卷

作者： Fang, Nan Zhang, Chuan-Yu Wan, Ming-Jie Huang, Xiao-Peng Chengdu Univ Technol Sch Math & Phys Chengdu 610059 Peoples R China Yibin Univ Computat Phys Key Lab Sichuan Prov Yibin 644000 Peoples R China

As an interstellar molecule, the thermodynamic properties and transition characteristics of FeC hold significant importance. In this study, high-precision ab initio methods were employed to obtain the potential energy functions of the five Lambda-S states of FeC. Based on these results, the partition functions, proportions of each molecular and thermodynamic properties were calculated for temperatures from 50 K to 10,000 K. In addition, the spin orbit coupling effects on the ground state (X 3 Delta ) split it into three Omega states (Omega = 3, 2, 1), with Omega = 3 being the lowest. The rotational spectrum for Omega = 3 at 298.15 K shows significantly higher transition intensity for the v ' = 0 and v '' = 0 band. Moreover, this paper presents the ultraviolet spectrum of two transitions: X 3 Delta <-> 13 Pi and 13 Pi <-> 1 3 Sigma- , at a temperature of 298.15 K, with the 13 Pi <-> 1 3 Sigma- transition having stronger absolute intensity.

关键词： partition functions Thermodynamic properties Ultraviolet spectrum Rotational spectrum

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Truncated theta series from the Bailey lattice

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ADVANCES IN APPLIED MATHEMATICS 2025年 167卷

作者： Ding, Xiangyu Sun, Lisa Hui Nankai Univ Ctr Combinator LPMC Tianjin 300071 Peoples R China

In 2012, Andrews and Merca obtained a truncated version of Euler's pentagonal number theorem and showed the non-negativity related to partition functions. Meanwhile, Andrews and Merca, Guo and Zeng independently conjectured that the truncated Jacobi triple product series has nonnegative coefficients, which has been confirmed analytically and also combinatorially. In 2022, Merca proposed a stronger version for this conjecture. In this paper, by applying Agarwal, Andrews and Bressoud's identity derived from the Bailey lattice, we obtain a truncated version for the Jacobi triple product series with odd basis, which reduces to the Andrews-Gordon identity as a special instance. As consequences, we obtain new truncated forms for Euler's pentagonal number theorem, Gauss' theta series on triangular numbers and square numbers, which lead to inequalities for certain partition functions. Moreover, by considering a truncated theta series involving B-regular partitions, we confirm a conjecture proposed by Ballantine and Merca about 6-regular partitions and show that Merca's stronger conjecture on truncated Jacobi triple product series holds when R = 3S for S >= 1. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

关键词： partition functions Truncated theta series The Bailey lattice Euler's pentagonal number theorem

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