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This paper describes the spectroscopic (FT-IR and UV–Visible), electronic, structural, vibrational properties of celecoxib compound using density functional theory. The optimized geometry structure of the titled molecule is explained by the method of B3LYP with 6–31 + G (d, p) basis set. We found that the simulated optimized geometry of the titled molecule is simulated to the experimental ones. The infrared spectrum of celecoxib compound is represented in the present work. The computational information about the organic molecule can be explained by the charge transfer properties, which is a computational tool for the precise quantum chemical calculations of space charge density distribution and HOMO-LUMO states. The transition states of celecoxib compound have explained by the time-dependent density functional theory (TD-DFT) which is observed by the UV–visible spectrum. Docking research [1] is a modeling method, with the help of docking method we can find out new drug and chemical behavior of the drug which is useful in the application purpose of the drug, the interaction between protein and legends provides the information of the binding energy and physical parameter like partition function, free energy, internal energy, entropy.

In order to give overpartition analogues of Rogers-Ramanujan type theorems for the ordinary partition function, Andrews defined the so-called singular overpartitions. Singular overpartition function C ‾ k , i ( n ) counts the number of overpartitions of n in which no part is divisible by k and only parts ≡ ± i ( mod k ) may be overlined. Andrews also proved two beautiful Ramanujan type congruences modulo 3 satisfied by C ‾ 3 , 1 ( n ) . Later on, Aricheta proved that for an infinite family of l, C ‾ 3 l , l ( n ) is almost always divisible by 2. In this article, for an infinite subfamily of l considered by Aricheta, we prove that C ‾ 3 l , l ( n ) is almost always divisible by arbitrary powers of 2. We also prove that C ‾ 3 l , l ( n ) is almost always divisible by arbitrary powers of 3 when l = 3 , 6 , 12 , 24 . Proofs of our density results rely on the modularity of certain eta-quotients which arise naturally as generating functions for the Andrews' singular overpartition functions.

Counting the independent sets of a graph is a classical #P-complete problem, even in the bipartite case. We give an exponential-time approximation scheme for this problem which is faster than the best known algorithm for the exact problem. The running time of our algorithm on general graphs with error tolerance $\varepsilon$ is at most $O(2^{0.2680n})$ times a polynomial in $1/\varepsilon$. On bipartite graphs, the exponential term in the running time is improved to $O(2^{0.2372n})$. Our methods combine techniques from exact exponential algorithms with techniques from approximate counting. Along the way we generalise (to the multivariate case) the FPTAS of Sinclair, Srivastava, \v{S}tefankovi\v{c} and Yin for approximating the hard-core partition function on graphs with bounded connective constant. Also, we obtain an FPTAS for counting independent sets on graphs with no vertices with degree at least 6 whose neighbours' degrees sum to 27 or more. By a result of Sly, there is no FPTAS that applies to all graphs with maximum degree 6 unless $\mbox{P}=\mbox{NP}$., Comment: 52pp

Topological phases in non-Hermitian systems have become fascinating subjects recently. In this paper, we attempt to classify topological phases in 1D interacting non-Hermitian systems. We begin with the non-Hermitian generalization of the Su-Schrieffer-Heeger (SSH) model and discuss its many-body topological Berry phase, which is well defined for all interacting quasi-Hermitian systems (non-Hermitian systems that have real energy spectrum). We then demonstrate that the classification of topological phases for quasi-Hermitian systems is exactly the same as their Hermitian counterparts. Finally, we construct the fixed point partition function for generic 1D interacting non-Hermitian local systems and find that the fixed point partition function still has a one-to-one correspondence to their Hermitian counterparts. Thus, we conclude that the classification of topological phases for generic 1D interacting non-Hermitian systems is still exactly the same as Hermitian systems., 15 pages, 5 figures

A folklore conjecture on the partition function asserts that the density of odd values of p ( n ) is 1 2 . In general, for a positive integer t, let p t ( n ) be the t-multipartition function and δ t be the density of the odd values of p t ( n ) . It is widely believed that δ t exists. Given an odd integer a and an integer b depending on a and t, Judge and Zanello framed an infinite family of conjectural congruence relations on p t ( a n + b ) ( mod 2 ) which establishes a striking connection between δ a and δ 1 . As a special case t = 1 , it implies that δ 1 > 0 if ( 3 , a ) = 1 and δ a > 0 . This conjecture was proved for several values of a by Judge, Keith and Zanello. In this paper we prove that the conjecture is true for a = l α is a prime power with l ≥ 5 and a = 3 .

Recently, Andrews defined a partition function $\mathcal{EO}(n)$ which counts the number of partitions of $n$ in which every even part is less than each odd part. He also defined a partition function $\overline{\mathcal{EO}}(n)$ which counts the number of partitions of $n$ enumerated by $\mathcal{EO}(n)$ in which only the largest even part appears an odd number of times. Andrews proposed to undertake a more extensive investigation of the properties of $\overline{\mathcal{EO}}(n)$. In this article, we prove infinite families of congruences for $\overline{\mathcal{EO}}(n)$. We next study parity properties of $\overline{\mathcal{EO}}(n)$. We prove that there are infinitely many integers $N$ in every arithmetic progression for which $\overline{\mathcal{EO}}(N)$ is even; and that there are infinitely many integers $M$ in every arithmetic progression for which $\overline{\mathcal{EO}}(M)$ is odd so long as there is at least one. Very recently, Uncu has treated a different subset of the partitions enumerated by $\mathcal{EO}(n)$. We prove that Uncu's partition function is divisible by $2^k$ for almost all $k$. We use arithmetic properties of modular forms and Hecke eigenforms to prove our results., Accepted in Journal of Number Theory

We consider the limit behavior of partition function of directed polymers in random environment, which is represented by a linear model instead of a family of i.i.d.variables in 1 + 1 dimensions. Under the assumption on the environment that its spatial correlation decays algebraically, using the method developed in Alberts et al. (2014), we show that the scaled partition function, as a process defined on [ 0 , 1 ] × R , converges weakly to the solution to some stochastic heat equations driven by fractional Brownian field. The fractional Hurst parameter is determined by the correlation exponent of the random environment. Here multiple Ito integral with respect to fractional Gaussian field and spectral representation of stationary process are heavily involved.

Promoter is an important functional elements of DNA sequences, which is in charge of gene transcription initiation. Recognizing promoter have important help for understanding the relative life phenomena. Based on the concept that promoter is mainly determined by its sequence and structure, a novel statistical physics model for predicting promoter in Escherichia coli K-12 is proposed. The total energies of DNA local structure of sequence segments in the three benchmark promoter sequence datasets, the sole prediction parameter, are calculated by using principles from statistical physics and information theory. The better results are obtained. And a web-server PhysMPrePro for predicting promoter is established at http://202.207.14.87:8032/bioinformation/PhysMPrePro/index.asp, so that other scientists can easily get their desired results by our web-server.

The Turan inequality and its higher order analog arise in the study of Maclaurin coefficients of an entire function in the Laguerre-Polya class. It is well known that if a real entire function ψ ( x ) is in the LP class, the Maclaurin coefficients satisfy both the Turan inequality and the higher order Turan inequality. Chen, Jia and Wang proved that for n ≥ 95 , the higher order Turan inequality holds for the partition function p ( n ) and the 3-rd associated Jensen polynomials p ( n ) + 3 p ( n + 1 ) x + 3 p ( n + 2 ) x 2 + p ( n + 3 ) x 3 have only real zeros. Recently, Griffin, Ono, Rolen and Zagier showed that Jensen polynomials for a large family of functions, including those associated to ξ ( s ) and the partition function, are hyperbolic for sufficiently large n. This result gave evidence for Riemann hypothesis. In this paper, we give a unified approach to investigate the higher order Turan inequality for the sequences { a n / n ! } n ≥ 0 , where a n satisfy a three-term recurrence relation. In particular, we prove higher order Turan inequality for the sequences { a n / n ! } n ≥ 0 , where a n are the Motzkin numbers, the Fine number, the Franel numbers of order 3 and the Domb numbers. As a consequence, for these combinatorial sequences, the 3-rd associated Jensen polynomials a n n ! + 3 a n + 1 ( n + 1 ) ! x + 3 a n + 2 ( n + 2 ) ! x 2 + a n + 3 ( n + 3 ) ! x 3 have only real zeros. Furthermore, for these combinatorial sequences we conjecture that for any given integer m ≥ 4 , there exists an integer N ( m ) such that for n > N ( m ) , the m-th associated Jensen polynomials ∑ i = 0 m ( m i ) a n + i ( n + i ) ! x i have only real zeros.

Recently, a new partition function associated with Ramanujan's third order mock theta function ω ( q ) was discovered, and subsequently its overpartition analogue was introduced. In this paper, we prove an intriguing identity arising from the study of that analogue, which involves a double series of Hecke-Rogers type. In addition, two further identities will be given. Our proofs heavily rely on formulas from the work of Liu [12] .

The inverse Langevin-like function in the presence of an orienting field is calculated via inversion of the exact, numerically calculated Langevin-like function and/or inversion of its Taylor representation. This representation is obtained using the cumulant method for the natural logarithm of the chain partition function. As a result, a power series representation of the inverse function is obtained. A simple algorithm is formulated for the calculation of this series. The chain free energy and the deformation work and force are calculated. For various strengths of the field and relative orientations of the chain end-to-end distance and the orienting field direction, the convergence of results obtained by using of the exact inverse Langevin-like function or its Taylor representation is discussed.

Recent trends in the numbers of minority language speakers have given cause for concern as to the effectiveness of traditional language-acquisition and survival strategies. This paper addresses the establishment of a recognized survival scenario, that of diglossia, the allocation of different languages to complementary social domains. The method is the novel application of an analogue of the partition function of statistical mechanics, embodied in a cellular automaton, to a population of subsets of social domains in which either of two languages may be spoken. The case of modern Wales is discussed in detail. Conclusions are drawn regarding the role of diglossia in the preservation of endangered languages.

Cheng et al. gave the microscopic expression of entransy in the ideal gas system for monatomic molecules in an approximate non-interaction particle system. Diatomic molecular gas or polyatomic molecular gas is more widely used than monatomic molecules gas in industry. So this paper deduces the microscopic expression of entransy in the ideal gas system for diatomic molecules by using the approximate calculation of partition function.

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

In his unpublished manuscript on the partition and tau functions, Ramanujan obtained several striking congruences for the partition function p ( n ) , the number of unrestricted partitions of n. The most notable of them are p ( 5 n + 4 ) ≡ 0 ( mod 5 ) and p ( 7 n + 5 ) ≡ 0 ( mod 7 ) which holds for all positive integers n. More surprisingly, Ramanujan obtained certain identities between q-series from which the above congruences follow as consequences. In this paper, we adopt Ramanujan's approach and prove an identity which witnesses another famous Ramanujan congruence, namely, p ( 11 n + 6 ) ≡ 0 ( mod 11 ) and also establish some new identities for the generating functions for p ( 17 n + 5 ) , p ( 19 n + 7 ) and p ( 23 n + 1 ) . We also find explicit evaluations for F p ( q ) in the cases p = 17 , 19 , 23 where F p is the function appearing in Ramanujan's circular summation formula.

Text For any positive real number r, let p r ( n ) be the number of solutions of the equation n = ⌊ a 1 r ⌋ + ⋯ + ⌊ a k r ⌋ with integers 1 ≤ a 1 ≤ ⋯ ≤ a k . Recently, Luca and Ralaivaosaona gave an asymptotic formula for p 2 ( n ) . In Part I, it is proved that, for any real number r > 1 , we have exp ⁡ ( τ 1 n r / ( r + 1 ) ) ≤ p r ( n ) ≤ exp ⁡ ( τ 2 n r / ( r + 1 ) ) for two positive constants τ 1 and τ 2 (depending only on r). In this paper, we prove that, for any real number r > 1 , p r ( n ) = exp ⁡ ( c 1 n r / ( r + 1 ) + c 2 n ( r − 1 ) / ( r + 1 ) + ⋯ + c l n ( r − l + 1 ) / ( r + 1 ) + O ( n 1 / ( r + 1 ) ) ) , where l is the integer with l r ≤ l + 1 and c 1 , c 2 , … , c l are computable constants depending only on r. In particular, c 1 = ( 1 + r − 1 ) ( r ζ ( r + 1 ) Γ ( r + 1 ) ) 1 / ( r + 1 ) . For 0 r 1 , we pose a conjecture for the asymptotic formula for p r ( n ) . Video For a video summary of this paper, please visit https://youtu.be/Z9HiSbr9eJY .

We characterize the Shapley value using (together with standard conditions of efficiency and equal gains in two-player games) a condition of ‘undominated merge-externalities’. Similar to the well-known ‘balanced contributions’ characterization, our characterization corresponds intuitively to ‘threat points’ present in bargaining. It derives from the observation that all semivalues satisfy ‘balanced merge-externalities’. Our characterization is applicable to useful, narrow sub-classes of games (including monotonic simple games), and also extends naturally to encompass games in partition function form.

We analyze the mock modular behavior of P ¯ ω ( q ) , a partition function introduced by Andrews, Dixit, Schultz, and Yee. This function arose in a study of smallest parts functions related to classical third order mock theta functions, one of which is ω ( q ) . We find that the modular completion of P ¯ ω ( q ) is not simply a harmonic Maass form, but is instead the derivative of a linear combination of products of various harmonic Maass forms and theta functions. We precisely describe its behavior under modular transformations and find that the image under the Maass lowering operator lies in a relatively simpler space.

We provide partition-theoretic interpretation of two truncated identities of Gauss solving a problem by Guo and Zeng. We also reveal that these results, together with our previous truncation of Euler's pentagonal number theorem, are essentially corollaries of the RogersFine identity. Finally we examine further positivity questions related to the partition function.

Andrews, Lewis and Lovejoy introduced the partition function $PD(n)$ as the number of partitions of $n$ with designated summands. In a recent work, Lin studied a partition function $PD_{t}(n)$ which counts the number of tagged parts over all the partitions of $n$ with designated summands. He proved that $PD_{t}(3n+2)$ is divisible by $3$. In this paper, we first introduce a structure named partitions with overline designated summands, which is counted by $PD_t(n)$. We then define a generalized rank of partitions with overline designated summands and give a combinatorial interpretation of the congruence for $PD_t(3n+2)$., 14 pages

The study reports an accurate and new closed-form representation of the ro-vibrational partition function, using a small amount of spectroscopic constants of the ground electronic state. A rigorous factorization method is proposed for the vibrational and rotational contributions to the partition function using an average vibrational quantum number.

The factorization formula in the non-interacting quantum field theories that relates the fermionic partition function to the bosonic partition function considered recently by Chair (2013) [3] is obtained for the harmonic oscillator using the path integral formulation. By using the latter, the fermionic partition function turns out to be the ratio of two determinants of the same operator ( ∂ τ + ω ) , whose eigenmodes being both periodic on the imaginary time intervals [ 0 , 2 β ] , [ 0 , β ] . The natural generalization of the factorization formula when β → 2 m β is derived, such a factorization implies that the bosonic oscillator at temperature β can be seen as a non-interacting mixture of a bosonic oscillator at temperature 2 m β and m-fermionic oscillators at different temperatures 2 m − k β , k = 1 , 2 , … , m . As a consequence, a general relationship between the bosonic and fermionic thermal zeta functions is deduced.

There are a number of situations in which several signals are simultaneously recorded in complex systems, which exhibit long-term power-law cross-correlations. This paper presents two-dimensional multifractal cross-correlation analysis based on the partition function (2D-MFXPF), two-dimensional multifractal cross-correlation analysis based on the detrended fluctuation analysis (2D-MFXDFA) and two-dimensional multifractal cross-correlation analysis based on the detrended moving average analysis (2D-MFXDMA). We apply these methods to pairs of two-dimensional multiplicative cascades (2D-MC) to do a comparative study. Then, we apply the two-dimensional multifractal cross-correlation analysis based on the detrended fluctuation analysis (2D-MFXDFA) to real images and unveil intriguing multifractality in the cross correlations of the material structures. At last, we give the main conclusions and provide a valuable reference on how to choose the multifractal algorithms in the potential applications in the field of SAR image classification and detection.

The unsuccessful astronomical searches for some important astrophysical and astrobiological molecules have been linked to the large partition function of these molecules. This letter reports an extensive investigation of the relationship between partition function and astronomical observation of interstellar isomers using high level quantum chemical calculations. 120 molecules from 30 different isomeric groups have been considered. Partition function and thermodynamic stabilities are determined for each set of isomeric species. From the results, there is no direct correlation between partition function and astronomical observation of the same isomeric species. Though interstellar formations processes are generally controlled by factors like kinetics, thermodynamics, formation and destruction pathways. However, the observation of the isomers seems to correlate well with thermodynamics. For instance, in all the groups considered, the astronomically detected isomers are the thermodynamically most stable molecules in their respective isomeric groups. The implications of these results in accounting for the limited number of known cyclic interstellar molecules, unsuccessful searches for amino acid and the possible molecules for astronomical observations are discussed. (C) 2016 COSPAR. Published by Elsevier Ltd. All rights reserved.

Given a finite set of agents, an embedded coalition consists of a coalition and a partition of the rest of agents. We study a partial order on the set of embedded coalitions of a finite set of agents. An embedded coalition precedes another one if the first coalition is contained in the second and the second partition equals the first one after removing the agents in the second coalition. This poset is not a lattice. We describe the maximal lower bounds and minimal upper bounds of a finite subset, whenever they exist. It is a graded poset and we are able to count the number of elements at a given level as well as the total number of chains. The study of this structure allows us to derive results for games with externalities. In particular, we introduce a new concept of convexity and show that it is equivalent to having non-decreasing contributions to embedded coalitions of increasing size.

In coalitional games with uncertain payoffs, a deviating coalition can only form expectations regarding its post-deviation payoff. Classical approaches address the problem from the side of conservatism, expecting the worst, or by explicit assumptions of the emerging state of the world. We borrow the idea of risk from the finance literature and compare the payoff of staying with the original outcome with the risk of deviating. Employing this idea to the core leads to a new concept that we call the risk-based core. We introduce this concept and discuss its properties. We find an inclusion relation between cores of games with increasingly conservative players. The model is also suitable to study cooperative games in partition function form where the value of a coalition depends on the entire partition. For the cores of such games our main result yields many of the familiar inclusion relations as corollaries, while the inclusion of the optimistic core in the optimistic recursive core turns out to be non-robust.

We obtain a finite analogue of a recent generalization of an identity in Ramanujan's Notebooks. Differentiating it with respect to one of the parameters leads to a result whose limiting case gives a finite analogue of Andrews' famous identity for $\textup{spt}(n)$. The latter motivates us to extend the theory of the restricted partition function $p(n, N)$, namely, the number of partitions of $n$ with largest parts less than or equal to $N$, by obtaining the finite analogues of rank and crank for vector partitions as well as of the rank and crank moments. As an application of the identity for our finite analogue of the spt-function, namely $\textup{spt}(n, N)$, we prove an inequality between the finite second rank and crank moments. The other results obtained include finite analogues of a recent identity of Garvan, an identity relating $d(n, N)$ and lpt$(n, N)$, namely the finite analogues of the divisor and largest parts functions respectively, and a finite analogue of the Beck-Chern theorem. We also conjecture an inequality between the finite analogues of $k^{\textup{th}}$ rank and crank moments., 46 pages, submitted for publication

The rovibrational partition function is expressed in terms of analytic functions using summation formula. It is derived for a double series, which represents the rovibrational partition function in quantum statistical mechanics. With the summation formula obtained, the rovibrational partition function of isotonic oscillator is calculated in an analytic form and its temperature dependence is analyzed carefully.

In this paper, the author provides a method to compute the values of Euler's partition function p ( n ) that requires only the values of p ( k ) with k ⩽ n / 2 . This method is combined with Ewell's recurrence relation for the partition function p ( n ) to obtain a simple and fast computation of the value of p ( n ) .

A method to differentiate posed and spontaneous expressions by capturing their global spatial patterns using RBM is proposed.Partition function of RBM is estimated by extending AIS to RBM with continuous visible units.Gender and expression category applied as privileged information for posed and spontaneous expression distinction.Experimental results on benchmark databases demonstrate the effectiveness of the proposed approach. In this paper, we introduce methods to differentiate posed expressions from spontaneous ones by capturing global spatial patterns embedded in posed and spontaneous expressions, and by incorporating gender and expression categories as privileged information during spatial pattern modeling. Specifically, we construct multiple restricted Boltzmann machines (RBMs) with continuous visible units to model spatial patterns from facial geometric features given expression-related factors, i.e., gender and expression categories. During testing, only facial geometric features are provided, and the samples are classified into posed or spontaneous expressions according to the RBM with the largest likelihood. Furthermore, we propose efficient inference algorithm by extending annealing importance sampling to RBM with continuous visible units for calculating partition function of RBMs. Experimental results on benchmark databases demonstrate the effectiveness of the proposed approach in modelling global spatial patterns as well as its superior posed and spontaneous expression distinction performance over existing approaches.

The objective of this paper is to develop a solution concept for stability of coalitional games with externalities. The existing solution concepts for this class of coalitional games can be empty. Using the partition function form representation, we propose a new solution concept called equivalence nucleolus, which is shown to be unique and always non-empty.

In the paper the author uses the quintuple product identity to derive a combinatorial interpretation for a bisection of the pentagonal number recurrence for the partition function.

In this paper we extend previous work of Galleas and the author to elliptic SOS models. We demonstrate that the dynamical reflection algebra can be exploited to obtain a functional equation characterizing the partition function of an elliptic SOS model with domain-wall boundaries and one reflecting end. Special attention is paid to the structure of the functional equation. Through this approach we find a novel multiple-integral formula for that partition function., Comment: 31 pages, 3 figures; v2: minor improvements, reference added

Extending the definition of phase-space (Wigner) quantum projectors to thermodynamic ensembles usually results into an efficient platform for quantifying their elementary information content. Given the spectral decomposition profile of a thermalized quantum system, general expressions for the quantum purity quantifier, $\mathcal{P}(\beta)$, and for phase-space projectors related to the quantum fidelity, $\mathcal{F}(\beta)$, are explicitly derived in terms of an explicit correspondence with the related partition function, $\mathcal{Z}(\beta)$. Besides quantifying the storage of information capacity of thermodynamical ensembles, the tools here introduced extend the role of the partition function in expressing the quantum behavior of thermodynamic ensembles., Comment: 12 pages, 03 figures

We present the partition function of Chern-Simons theory with the exceptional gauge group on three-sphere in the form of a partition function of the refined closed topological string with relation $2\tau=g_s(1-b) $ between single K\"ahler parameter $\tau$, string coupling constant $g_s$ and refinement parameter $b$, where $b=\frac{5}{3},\frac{5}{2},3,4,6$ for $G_2, F_4, E_6, E_7, E_8$, respectively. The non-zero BPS invariants $N^d_{J_L,J_R}$ ($d$ - degree) are $N^2_{0,\frac{1}{2}}=1, N^{11}_{0,1}=1$. Besides these terms, partition function of Chern-Simons theory contains term corresponding to the refined constant maps of string theory. Derivation is based on the universal (in Vogel's sense) form of a Chern-Simons partition function on three-sphere, restricted to exceptional line $Exc$ with Vogel's parameters satisfying $\gamma=2(\alpha+\beta)$. This line contains points, corresponding to the all exceptional groups. The same results are obtained for $F$ line $\gamma=\alpha+\beta$ (containing $SU(4), SO(10)$ and $E_6$ groups), with the non-zero $N^2_{0,\frac{1}{2}}=1, N^{7}_{0,1}=1$. In both cases refinement parameter $b$ ($=-\epsilon_2/\epsilon_1$ in terms of Nekrasov's parameters) is given in terms of universal parameters, restricted to the line, by $b=-\beta/\alpha$., Comment: 8 pages

In this paper, we consider the thermal properties of one-dimensional Dirac in the framework of the theory of superstatistics where the probability density f ( β ) follows χ 2 − superstatistics (=Tsallis statistics or Gamma distribution). Under the approximation of the low-energy asymptotics of superstatistics, the partition function, at first, has been calculated by using both Mellin Transformation and Zeta function. This approximation leads to a universal parameter q for any superstatistics, not only for Tsallis statistics. By using the desired partition function, all thermal properties have been obtained in terms of the parameter q . As an application, we extend this concept to the case of Graphene: the reason of this choice is due the existence of an exact mapping about the Dirac oscillator and the compound in question.

The partition function plays an important role in determining column densities from emission lines in molecular clouds, since under typical conditions the two are linearly related. Here, we focus on successively applying quantum and centrifugal distortion correction terms to the classical rotational partition function for asymmetric top molecules, in an effort to uncover the relative importance of each term. We do this for example asymmetric top molecules that contain from one to four heavy atoms (water, methanimine, vinyl alcohol, and glycolaldehyde), at temperatures relevant to interstellar molecular clouds. Particular attention is paid to methanimine, for which it was recently attested that the value of the classical partition function significantly differs from the numerically summed (exact) value at several hundred Kelvin [Sharma et al., Astron. Nachrichten. 338, 125 (2017)]. We show that the classical values are in fact within 1% of the exact values from ~50 K to well above 400 K; the addition of three simple correction terms reduces the error to ≲0.001% (down to 10 K).

Using entirely algebraic or elementary methods, partly inspired by recent ideas of P. Monsky on quadratic representations, we determine a new asymptotic lower bound for the number of odd values of a fundamental modular function in number theory, the Klein j-function. Namely, the number of n≤x such that the Klein j-function — or equivalently, the cubic partition function — is odd has order at least xlog⁡log⁡xlog⁡x, for x→∞. This improves recent results of Berndt, Yee and Zaharescu and Chen and Lin, approaching significantly the best bound currently known for p(n), obtained using modular forms. Then, in the final section, we show how to employ modular forms to slightly refine our bound. In fact, our brief argument, combining a recent result of Nicolas–Serre with a classical theorem of Serre on level 1 modular forms, more generally provides a lower bound for the odd values of any positive power of the generating function of p(n).

The well-known partition function p ( n ) , which is the number of solutions of the equation n = a 1 + ⋯ + a k with integers 1 ≤ a 1 ≤ ⋯ ≤ a k , has a long research history. In this note, we investigate a new partition function. Let q ( n ) be the number of solutions of the equation n = [ a 1 ] + ⋯ + [ a k ] with integers 1 ≤ a 1 ≤ ⋯ ≤ a k , where [ x ] denotes the integral part of x. We prove that exp ⁡ ( c 1 n 2 / 3 ) ≤ q ( n ) ≤ exp ⁡ ( c 2 n 2 / 3 ) for two explicit positive constants c 1 and c 2 .

In his 1984 AMS Memoir, Andrews introduced the k-colored generalized Frobenius partition function c ϕ k ( n ) , which denotes the number of generalized Frobenius partitions of n with k colors. Recently, Baruah and Sarmah found the generating function for c ϕ 6 ( n ) . They also established 2- and 3-dissections of the generating function for c ϕ 6 ( n ) and proved that c ϕ 6 ( 2 n + 1 ) ≡ 0 ( mod 4 ) , c ϕ 6 ( 3 n + 1 ) ≡ c ϕ 6 ( 3 n + 2 ) ≡ 0 ( mod 9 ) . Furthermore, they conjectured that c ϕ 6 ( 3 n + 2 ) ≡ 0 ( mod 27 ) for n ≥ 0 . In this paper, we confirm this conjecture by employing the generating function for c ϕ 6 ( 3 n + 2 ) given by Baruah and Sarmah and the ( p , k ) -parametrization of theta functions due to Alaca, Alaca and Williams. Moreover, we conjecture that for n ≥ 0 , c ϕ 6 ( 9 n + 7 ) ≡ 0 ( mod 27 ) and c ϕ 6 ( 27 n + 16 ) ≡ 0 ( mod 3 5 ) .

Multifractality of a time series can be analyzed using the partition function method based on empirical moments of the process. In this paper we analyze the method when the underlying process has heavy-tailed increments. A nonlinear estimated scaling function and non-trivial spectrum are usually considered as signs of a multifractal property in the data. We show that a large class of processes can produce these effects and that this behavior can be attributed to heavy tails of the process increments. Examples are provided indicating that multifractal features considered can be reproduced by simple heavy-tailed Lévy process.

The constants of Landau and Lebesgue are defined, for all integers n⩾0, in order, by Gn=∑k=0n116k(2kk)2andLn=12π∫−ππ|sin((n+12)t)sin(12t)|dt, which play important roles in the theories of complex analysis and Fourier series, respectively. Certain inequalities and asymptotic expansions for the constants Gn and Ln have been investigated by many authors. Here we aim at establishing new asymptotic expansions for the constants Gn and Ln of Landau and Lebesgue, respectively, by mainly using Bell polynomials and the partition function.

Properties of four quintic theta functions are developed in parallel with those of the classical Jacobi null theta functions. The quintic theta functions are shown to satisfy analogues of Jacobiʼs quartic theta function identity and counterparts of Jacobiʼs Principles of Duplication, Dimidiation and Change of Sign Formulas. The resulting library of quintic transformation formulas is used to describe series multisections for modular forms in terms of simple matrix operations. These efforts culminate in a formal technique for deducing congruences modulo powers of five for a variety of combinatorial generating functions, including the partition function. Further analysis of the quintic theta functions is undertaken by exploring their modular properties and their connection to Eisenstein series. The resulting relations lead to a coupled system of differential equations for the quintic theta functions.

We propose and demonstrate a limiting procedure in which, starting from the q-lifted version (or K-theoretic five dimensional version) of the (W)AGT conjecture to be assumed in this paper, the Virasoro/W block is generated in the r-th root of unity limit in q in the 2d side, while the same limit automatically generates the projection of the five dimensional instanton partition function onto that on the ALE space R^4/Z_r. This circumvents case-by-case conjectures to be made in a wealth of examples found so far. In the 2d side, we successfully generate the super-Virasoro algebra and the proper screening charge in the q -> -1, t -> -1 limit, from the defining relation of the q-Virasoro algebra and the q-deformed Heisenberg algebra. The central charge obtained coincides with that of the minimal series carrying odd integers of the N=1 superconformal algebra. In the r-th root of unity limit in q in the 2d side, we give some evidence of the appearance of the parafermion-like currents. Exploiting the q-analysis literatures, q-deformed su(n) block is readily generated both at generic q, t and the r-th root of unity limit. In the 4d side, we derive the proper normalization function for general (n, r) that accomplishes the automatic projection through the limit., 38 pages, 4 figures; v2: minor changes, references added; v3: minor changes, a comment (Note added) and three references added, published version

In this paper we present a sampling framework for RNA structures of fixed topological genus. We introduce a novel, linear time, uniform sampling algorithm for RNA structures of fixed topological genus g, for arbitrary g>0. Furthermore we develop a linear time sampling algorithm for RNA structures of fixed topological genus g that are weighted by a simplified, loop-based energy functional. For this process the partition function of the energy functional has to be computed once, which has O(n 2) time complexity.

Andrews, Lewis and Lovejoy introduced the partition function PD ( n ) as the number of partitions of n with designated summands, where we assume that among parts with equal size, exactly one is designated. They proved that PD ( 3 n + 2 ) is divisible by 3 and showed that the generating function of PD ( 3 n ) can be expressed as an infinite product of powers of ( 1 − q 2 n + 1 ) times a function F ( q 2 ) . We obtain a Ramanujan type identity which implies the congruence for PD ( 3 n + 2 ) . We also find an explicit formula for F ( q 2 ) , which leads to a formula for the generating function of PD ( 3 n ) . A formula for the generating function of PD ( 3 n + 1 ) is also obtained. Our proofs rely on Chanʼs identity on Ramanujanʼs cubic continued fraction and identities on cubic theta functions. By introducing a rank for partitions with designated summands, we give a combinatorial interpretation of the congruence for PD ( 3 n + 2 ) .

In this manuscript, we study the logarithmic Penner type nonlinear interaction in the random matrix model for interacting RNA folding and structure combinatorics. The Penner interaction originally appeared in the studies of moduli space of punctured surfaces and has been applied here in the context of RNA folding for the first time. An exact analytic formula for the generating function is derived using the orthogonal polynomial method. The partition function for a given length L of the RNA chain, derived from the generating function enumerates all possible topological structures as well as the pairings. The partition function and the asymptotic large length distribution functions are found and show a change in the critical exponent of the secondary structure contribution from L − 3 / 2 for large N (size of matrix, N > L ) to L − 1 / 2 for small N ( N ≪ L ). The exact analytic results calculated in the proposed model allow evaluation of the specific heat versus T curve for large interaction strength. In particular, the second derivative of specific heat shows a striking behavior, changing from single peaked function for large N to a double peak for small N.

Two new expansions for partial sums of Gauss' triangular and square numbers series are given. As a consequence, we derive a family of inequalities for the overpartition function $\bar{p}(n)$ and for the partition function $p_1(n)$ counting the partitions of $n$ with distinct odd parts. Some further inequalities for variations of partition function are proposed as conjectures., 9 pages, final version

We calculate zeros of the $q$-state Potts model partition function on $m$'th-iterate Sierpinski graphs, $S_m$, in the variable $q$ and in a temperature-like variable, $y$. We infer some asymptotic properties of the loci of zeros in the limit $m \to \infty$ and relate these to thermodynamic properties of the $q$-state Potts ferromagnet and antiferromagnet on the Sierpinski gasket fractal, $S_\infty$., Comment: 6 pages, 8 figures

We propose the “average approach,” where the worth of a coalition is a weighted average of its worth for different partitions of the players' set, as a unifying method to extend values for characteristic function form games. Our method allows us to extend the equal division value, the equal surplus value, the consensus value, the λ -egalitarian Shapley value, and the family of least-square values. For each of the first three extensions, we also provide an axiomatic characterization of a particular value for partition function form games. And for each of the last two extensions, we find a family of values that satisfy the properties.