Generalized Partition Function Research Articles

Approximating the chromatic polynomial is as hard as computing it exactly

We show that for any non-real algebraic number q, such that|q-1|>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|q-1|>1$$\\end{document} or ℜ(q)>32\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Re(q)>\\frac{3}{2}$$\\end{document} it is #P-hard to computea multiplicative (resp. additive) approximation to the absolutevalue (resp. argument) of the chromatic polynomial evaluated at q on planar graphs. This implies #P-hardness for allnon-real algebraic q on the family of all graphs. We, moreover,prove several hardness results for q, such that |q-1|≤1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|q-1|\\leq 1$$\\end{document}.Our hardness results are obtained by showing that a polynomial timealgorithm for approximately computing the chromaticpolynomial of a planar graph at non-real algebraic q (satisfyingsome properties) leads to a polynomial time algorithm forexactly computing it, which is known to be hard by a resultof Vertigan. Many of our results extend in fact to the more generalpartition function of the random cluster model, a well-knownreparametrization of the Tutte polynomial.

CPT-even electrodynamics in a multidimensional torus: Casimir effect at finite temperature

The energy–momentum tensor for the electromagnetism with Lorentz breaking even term of the Standard Model Extended (SME) photon sector confined in a hyper torus is determined. A generalized partition function method is used, following in parallel the thermofield dynamics formalism written in N-dimensional toroidal manifold. After considering general aspects of the SME photon sector in a toroidal manifold, the influence of the isotropic CPT-even electromagnetic sector of the SME is analysed. The approach is then applied to the Casimir effect at finite temperature, corresponding to a topology (S^{1})^{r}times R^{N-r}, where N is the dimension of the Minkowski space-time, and r is the number of compactified dimensions.influence of the isotropic CPT-even electromagnetic sector of the SME is analysed.

Classical codes and chiral CFTs at higher genus

Higher genus modular invariance of two-dimensional conformal field theories (CFTs) is a largely unexplored area. In this paper, we derive explicit expressions for the higher genus partition functions of a specific class of CFTs: code CFTs, which are constructed using classical error-correcting codes. In this setting, the Sp(2g, ℤ) modular transformations of genus g Riemann surfaces can be recast as a simple set of linear maps acting on 2g polynomial variables, which comprise an object called the code enumerator polynomial. The CFT partition function is directly related to the enumerator polynomial, meaning that solutions of the linear constraints from modular invariance immediately give a set of seemingly consistent partition functions at a given genus. We then find that higher genus constraints, plus consistency under degeneration limits of the Riemann surface, greatly reduces the number of possible code CFTs. This work provides a step towards a full understanding of the constraints from higher genus modular invariance on 2d CFTs.

Congruences modulo prime for fractional colour partition function

PurposeLet p[1,r;t] be defined by ∑n=0∞p[1,r;t](n)qn=(E1Er)t, where t is a non-zero rational number, r ≥ 1 is an integer and Er=∏n=0∞(1−qr(n+1)) for |q| < 1. The function p[1,r;t](n) is the generalisation of the two-colour partition function p[1,r;−1](n). In this paper, the authors prove some new congruences modulo odd prime ℓ by taking r = 5, 7, 11 and 13, and non-integral rational values of t.Design/methodology/approachUsing q-series expansion/identities, the authors established general congruence modulo prime number for two-colour partition function.FindingsIn the paper, the authors study congruence properties of two-colour partition function for fractional values. The authors also give some particular cases as examples.Originality/valueThe partition functions for fractional value is studied in 2019 by Chan and Wang for Ramanujan's general partition function and then extended by Xia and Zhu in 2020. In 2021, Baruah and Das also proved some congruences related to fractional partition functions previously investigated by Chan and Wang. In this sequel, some congruences are proved for two-colour partitions in this paper. The results presented in the paper are original.

Derangement model of ligand-receptor binding

Abstract We introduce a derangement model of ligand-receptor binding that allows us to quantitatively frame the question “How can ligands seek out and bind to their optimal receptor sites in a sea of other competing ligands and suboptimal receptor sites?” To answer the question, we first derive a formula to count the number of partial generalized derangements in a list, thus extending the derangement result of Gillis and Even. We then compute the general partition function for the ligand-receptor system and derive the equilibrium expressions for the average number of bound ligands and the average number of optimally bound ligands. A visual model of squares assembling onto a grid allows us to easily identify fully optimal bound states. Equilibrium simulations of the system reveal its extremes to be one of two types, qualitatively distinguished by whether optimal ligand-receptor binding is the dominant form of binding at all temperatures and quantitatively distinguished by the relative values of two critical temperatures. One of those system types (termed “search-limited,” as it was in previous work) does not exhibit kinetic traps and we thus infer that biomolecular systems where optimal ligand-receptor binding is functionally important are likely to be search-limited.

PDE/Statistical Mechanics Duality: Relation Between Guerra\u2019s Interpolated p-Spin Ferromagnets and the Burgers Hierarchy

We examine the duality relating the equilibrium dynamics of the mean-field p-spin ferromagnets at finite size in the Guerra’s interpolation scheme and the Burgers hierarchy. In particular, we prove that—for fixed p—the expectation value of the order parameter on the first side w.r.t. the generalized partition function satisfies the p-1-th element in the aforementioned class of nonlinear equations. In the light of this duality, we interpret the phase transitions in the thermodynamic limit of the statistical mechanics model with the development of shock waves in the PDE side. We also obtain the solutions for the p-spin ferromagnets at fixed N, allowing us to easily generate specific solutions of the corresponding equation in the Burgers hierarchy. Finally, we obtain an effective description of the finite N equilibrium dynamics of the p=2 model with some standard tools in PDE side.

Arithmetic properties of Ramanujan’s general partition function for modulo 11

In the present work, for the general partition function pr(n), we establish five new infinite families of congruences modulo 11. Our emphasis throughout this paper is to exhibit the use of q-identities to generate congruences of pr(n).

Approximate minimization of weighted tree automata

This paper studies the following approximate minimization problem: given a minimal weighted tree automaton A with n states recognizing a weighted tree language f, can we construct a smaller automaton Aˆ with nˆ<n states recognizing a language fˆ that is a good approximation of f? The corresponding problem for weighted automata on strings was recently studied by Balle et al. [16,15], where the authors introduced a new canonical form for weighted automata called singular value automata inspired by spectral methods, and showed that truncating such canonical form yields a solution for the problem satisfying a certain approximation criteria. In this paper we take a similar approach and show that in the tree case one can obtain an analogous canonical form that we call singular value tree automata, and use it to study the approximate minimization problem for weighted tree automata. We first establish the existence of this canonical form for weighted tree automata and then provide bounds on the quality of the resulting approximation method based on truncation. We also study the problem of computing the canonical form given a minimal weighted tree automaton and show that in the tree case this task is considerably more complicated than in the string case. In particular, computing the canonical form reduces to solving a system of polynomial equations. By further reducing this problem to the computation of generalized partition functions for weighted tree automata, we propose and analyze two methods for computing the canonical form based on iterative methods: fixed point iteration and Newton's method. Our analysis of Newton's method unveils a connection between iterative methods and sequences of sets of trees satisfying a certain technical condition that might be of independent interest.

Fractality and Multifractality in a Stock Market’s Nonstationary Financial Time Series

A financial time series, such as a stock market index, foreign exchange rate, or a commodity price, fluctuates heavily and shows scaling behaviors. Scaling and multi-scaling behaviors are measured for a nonstationary time series, such as stock market indices, high-frequency stock prices of individual stocks, or the volatility time series of a stock index. We review the fractality, multi-scaling, and multifractality of the financial time series of a stock market. We introduce a detrended fluctuation analysis of the financial time series to extract fluctuation patterns. Multifractality is measured using various methods, such as generalized Hurst exponents, the generalized partition function method, a detrended fluctuation analysis, the detrended moving average method, and a wavelet transformation.

Some New Congruences Modulo 5 for the General Partition Function

In the present work, we discover some new congruences modulo 5 for pr(n), the general partition function by restricting r to some sequence of negative integers. Our emphasis throughout this paper is to exhibit the use of q-identities to generate the congruences for pr(n).

Stability of the merger-to-monopoly and a core concept for partition function games

This paper is concerned with an old question: Will oligopolistic firms have incentives to merge to monopoly and will the monopoly, if the firms indeed merge, be stable? To answer this question, I motivate and introduce a new core concept for a general partition function game and prove stability of the merger-to-monopoly by applying the new core concept, labelled the strong-core, to Cournot oligopoly modelled as a partition function game. The paper shows that the Cournot oligopoly with any finite number of homogeneous firms without capacity constraints admits a non-empty strong-core and so does the Cournot oligopoly of not necessarily homogeneous firms with capacity constraints that are equal to their “historical” outputs. These results imply that oligopolistic firms will have incentives to merge to monopoly both in the long- and short-run and if the firms indeed merge to monopoly, the merger-to-monopoly will be stable.

KdV-charged black holes

We construct black hole geometries in AdS3 with non-trivial values of KdV charges. The black holes are holographically dual to quantum KdV Generalized Gibbs Ensemble in 2d CFT. They satisfy thermodynamic identity and thus are saddle point configurations of the Euclidean gravity path integral. We discuss holographic calculation of the KdV generalized partition function and show that for a certain value of chemical potentials new geometries, not the conventional BTZ ones, are the leading saddles.

Inference of Retention Time From Tracer Tests in Crystalline Rock

AbstractA statistical parametrization of transport combined with a new, general partition function for diffusive mass transfer (Cvetkovic, 2017, https://doi.org/10.1002/2017WR021187) is here developed into a practical tool for evaluating tracer tests in crystalline rock. The research question of this study is how to separate the characteristic times of retention and advection, using tracer test information alone; this decoupling is critical for upscaling of transport. Three regimes are identified based on the unconditional mean number of trapping events. Analytical expressions are derived for inferring transport‐retention parameters; these are first tested on a series of generic examples and then using two sets of tracer test data. Our results indicate that the key transport‐retention parameters can be inferred separately with reasonable accuracy by a few simple steps, provided that the macrodispersion is not too large and retention not too strong. Of particular interest is inference of the retention time from the breakthrough curve peak that avoids costly asymptotic monitoring. Finally, we summarize the retention times as inferred from a series of nonsorbing tracer tests in the Swedish granite, demonstrating the uncertainties when estimating retention based on material and structural properties from samples. Possible strategies for reducing these uncertainties that combine improved understanding of crystalline rock evolution with numerical simulations are noted as topics for future research.

Dynamical quantum phase transitions in extended toric-code models

We study the nonequilibrium dynamics of the extended toric code model (both ordered and disordered) to probe the existence of the dynamical quantum phase transitions (DQPTs). We show that in the case of the ordered toric code model, the zeros of Loschmidt overlap (generalized partition function) occur at critical times when DQPTs occur, which is confirmed by the nonanalyticities in the dynamical counter-part of the free-energy density. Moreover, we show that DQPTs occur for any non-zero field strength if the initial state is the excited state of the toric code model. In the disordered case, we show that it is imperative to study the behavior of the first time derivative of the dynamical free-energy density averaged over all the possible configurations, to characterize the occurrence of a DQPTs in the disordered toric code model since the disorder parameter itself acts as a new artificial dimension. We also show that for the case where anyonic excitations are present in the initial state, the conditions for a DQPTs to occur are the same as what happens in the absence of any excitation.

Exact generalized partition function of 2D CFTs at large central charge

We discuss generalized partition function of 2d CFTs on thermal cylinder decorated by higher qKdV charges. We propose that in the large central charge limit qKdV charges factorize such that generalized partition function can be rewritten in terms of auxiliary non-interacting bosons. The explicit expression for the generalized free energy is readily available in terms of the boson spectrum, which can be deduced from the conventional thermal expectation values of qKdV charges. In other words, the picture of the auxiliary non-interacting bosons allows extending thermal one-point functions to the full non-perturbative generalized partition function. We verify this conjecture for the first seven qKdV charges using recently obtained pertrubative results and find corresponding contributions to the auxiliary boson masses. We further extend these results by conjecturing the full spectrum of bosons and find an exact expression for the generalized partition function as a function of infinite tower of chemical potentials in the limit of large central charge.

An asymptotic formula for the logarithm of generalized partition functions

Let $$\Lambda =\{ \lambda _j\}_{j=1}^\infty $$ be a set of positive integers, and let $$p_\Lambda (n)$$ be the number of ways of writing n as a sum of positive integers $$\lambda \in \Lambda $$ such that $$\chi (\lambda )=1$$ and $$f(\lambda ) \equiv j \,\,(\mathrm {mod}\,\, m)$$ , where $$m\ge 1$$ and $$j\ge 0$$ are fixed integers. Here, $$\chi $$ and f are a certain multiplicative function and an additive function, respectively. In this paper, we obtain the asymptotic formula for $$\ln \left( \sum _{n\le x} p_\Lambda (n) \right) \sim \ln p_\Lambda ([x])$$ as $$x\rightarrow \infty $$ .

Generalized Gibbs Ensemble of 2d CFTs at large central charge in the thermodynamic limit

We discuss partition function of 2d CFTs decorated by higher qKdV charges in the thermodynamic limit when the size of the spatial circle goes to infinity. In this limit the saddle point approximation is exact and at infinite central charge generalized partition function can be calculated explicitly. We show that leading 1/c corrections to free energy can be reformulated as a sum over Young tableaux which we calculate for the first two qKdV charges. Next, we compare generalized ensemble with the “eigenstate ensemble” that consists of a single primary state. At infinite central charge the ensembles match at the level of expectation values of local operators for any values of qKdV fugacities. When the central charge is large but finite, for any values of the fugacities the aforementioned ensembles are distinguishable.

Klaus Friedrich Roth, 1925-2015

Klaus Friedrich Roth, 1925-2015

New entropic inequalities for qubit and unimodal Gaussian states

The Tsallis relative entropy Sq(ρˆ,σˆ) measures the distance between two arbitrary density matrices ρˆ and σˆ. In this work the approximation to this quantity when q=1+δ (δ≪1) is obtained. It is shown that the resulting series is equal to the von Neumann relative entropy when δ=0. Analyzing the von Neumann relative entropy for an arbitrary ρˆ and a thermal equilibrium state σˆ=e−βHˆ∕Tr(e−βHˆ) is possible to define a new inequality relating the energy, the entropy, and the partition function of the system. From this inequality, a parameter that measures the distance between the two states is defined. This distance is calculated for a general qubit system and for an arbitrary unimodal Gaussian state. In the qubit case, the dependence on the purity of the system is studied for T≥0 and also for T<0. In the Gaussian case the general partition function, given a unimodal quadratic Hamiltonian, is calculated and the comparison of the thermal light state as a thermal equilibrium state of the parametric amplifier is presented.

Elliptic genus of E-strings

We study a family of 2d mathcal{N}=left(0, 4right) gauge theories which describes at low energy the dynamics of E-strings, the M2-branes suspended between a pair of M5 and M9 branes. The gauge theory is engineered using a duality with type IIA theory, leading to the D2-branes suspended between an NS5-brane and 8 D8-branes on an O8-plane. We compute the elliptic genus of this family of theories, and find agreement with the known results for single and two E-strings. The partition function can in principle be computed for arbitrary number of E-strings, and we compute them explicitly for low numbers. We test our predictions against the partially known results from topological strings, as well as from the instanton calculus of 5d Sp(1) gauge theory. Given the relation to topological strings, our computation provides the all genus partition function of the refined topological strings on the canonical bundle over frac{1}{2}K3 .