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For general antiferromagnetic 2-spin systems, including the hardcore model and the antiferromagnetic Ising model, there is an $\mathsf{FPTAS}$ for the partition function on graphs of maximum degree $\Delta$ when the infinite regular tree lies in the uniqueness region by Li et al. (2013). Moreover, in the tree non-uniqueness region, Sly (2010) showed that there is no $\mathsf{FPRAS}$ to estimate the partition function unless $\mathsf{NP}=\mathsf{RP}$. The algorithmic results follow from the correlation decay approach due to Weitz (2006) or the polynomial interpolation approach developed by Barvinok (2016). However the running time is only polynomial for constant $\Delta$. For the hardcore model, recent work of Anari et al. (2020) establishes rapid mixing of the simple single-site Markov chain known as the Glauber dynamics in the tree uniqueness region. Our work simplifies their analysis of the Glauber dynamics by considering the total pairwise influence of a fixed vertex $v$ on other vertices, as opposed to the total influence on $v$, thereby extending their work to all 2-spin models and improving the mixing time. More importantly our proof ties together the three disparate algorithmic approaches: we show that contraction of the tree recursions with a suitable potential function, which is the primary technique for establishing efficiency of Weitz's correlation decay approach and Barvinok's polynomial interpolation approach, also establishes rapid mixing of the Glauber dynamics. We emphasize that this connection holds for all 2-spin models (both antiferromagnetic and ferromagnetic), and existing proofs for correlation decay or polynomial interpolation immediately imply rapid mixing of Glauber dynamics. Our proof utilizes that the graph partition function divides that of Weitz's self-avoiding walk trees, leading to new tools for analyzing influence of vertices., Comment: Section 7 and Appendix E are updated to fix a small error

A natural problem in the context of the coupon collector's problem is the behavior of the maximum of independent geometrically distributed random variables (with distinct parameters). This question has been addressed by Brennan et al. (British J. of Math. & CS. 8 (2015), 330-336). Here we provide explicit asymptotic expressions for the moments of that maximum, as well as of the maximum of exponential random variables with corresponding parameters. We also deal with the probability of each of the variables being the maximal one. The calculations lead to expressions involving Hurwitz's zeta function at certain special points. We find here explicitly the values of the function at these points. Also, the distribution function of the maximum we deal with is closely related to the generating function of the partition function. Thus, our results (and proofs) rely on classical results pertaining to the partition function., 32 pages, 2 tables

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.

We revisit the Riemann-Hilbert problem determined by Donaldson-Thomas invariants for the resolved conifold and for other small crepant resolutions. While this problem can be recast as a system of TBA-type equations in the conformal limit, solutions are ill-defined due to divergences in the sum over infinite trajectories in the spectrum of D2-D0-brane bound states. We explore various prescriptions to make the sum well-defined, show that one of them reproduces the existing solution in the literature, and identify an alternative solution which is better behaved in a certain limit. Furthermore, we show that a suitable asymptotic expansion of the $\tau$ function reproduces the genus expansion of the topological string partition function for any small crepant resolution. As a by-product, we conjecture new integral representations for the triple sine function, similar to Woronowicz' integral representation for Faddeev's quantum dilogarithm., Comment: 25+14 pages; added proof for integral representation of the double sine function and a similar conjecture for the triple sine; version accepted for publication in Annales Henri Poincar\'e

The partitions in which even parts come in two colours are known as cubic partitions. In this paper, we introduce and investigate the cubic partition function A(n) which is defined as the difference between the number of cubic partitions of n into an even numbers of parts and the number of cubic partitions of n into an odd numbers of parts. We present a collection of identities relating A(n) and provide analytic proofs. Among these identities we remark two Ramanujan-like congruences: for all $$n\geqslant 0$$ , $$A(9n+5) \equiv 0 \pmod 3$$ and $$A(27n+26) \equiv 0 \pmod 3$$ . Combinatorial interpretations for A(n) are given in terms of the ordinary partitions into parts congruent to $$\pm 1, 4 \pmod 8$$ when n is even and congruent to $$\pm 3, 4 \pmod 8$$ when n is odd. In this context, some infinite families of linear inequalities involving A(n) are proposed as open problems.

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

George Andrews and Ae Ja Yee recently established beautiful results involving bivariate generalizations of the third-order mock theta functions $$\omega (q)$$ and $$\nu (q)$$ , thereby extending their earlier results with the second author. Generalizing the Andrews–Yee identities for trivariate generalizations of these mock theta functions remained a mystery, as pointed out by Li and Yang in their recent work. We partially solve this problem and generalize these identities. Several new as well as well-known results are derived. For example, one of our two main theorems gives, as a corollary, a special case of Soon-Yi Kang’s three-variable reciprocity theorem. A relation between a new restricted overpartition function $$p^{*}(n)$$ and a weighted partition function $$p_*(n)$$ is obtained from one of the special cases of our second theorem.

We look for the origins of the single equation, which is a peculiar combination of W-constrains, which provides the non-abelian W-representation for generalized Kontsevich model (GKM), i.e. is enough to fix the partition function unambiguously. Namely we compare it with the scalar projection of the matrix Ward identity. It turns out that, though similar, the two equations do not coincide, moreover, the latter one is non-polynomial in time-variables. This discrepancy disappears for the cubic model if partition function is reduced to depend on odd times (belong to KdV sub-hierarchy of KP), but in general such reduction is not enough. We consider the failure of such direct interpretation of the "single equation" as a new kind of anomaly, which should be explained and eliminated in the future analysis of GKM., to Andrei Mironov on occasion of his 60's birthday; 12 pages

For fixed m and R⊆{0,1,…,m−1} , take A to be the set of positive integers congruent modulo m to one of the elements of R, and let pA(n) be the number of ways to write n as a sum of elements of A. N...

In this work, the atomic parameters of Ag XLIV (Be-like Ag) are examined and evaluated by implementing the GRASP2K package with the multi-configuration Dirac–Hartree–Fock (MCDHF) method for the calculation of wave-functions. We have listed fine structure energy levels of the lowest 170 levels with radiative data for multipole moments, such as electric dipole (E1), electric quadrupole (E2), magnetic dipole (M1), and magnetic quadrupole (M2) transitions, that lie in the region of extreme ultraviolet (EUV) and soft X-ray (SXR) for Ag XLIV from the ground state within the lowest 170 levels. We have compared our GRASP2K and FAC results with theoretical results available in the literature for some levels. Additionally, we have also calculated partition function and thermodynamic quantities for temperature ranges from 104 to 107 K. We believe that our presented details and data may be beneficial not only in plasma modeling but also in imaging of nanostructure as well as in medicine, semiconductors, and EUV and SXR laser applications.

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

The partition crank is a statistic on partitions introduced by Freeman Dyson to explain Ramanujan's congruences. In this paper, we prove that the crank is asymptotically equidistributed modulo Q, for any odd number Q. To prove this, we obtain effective bounds on the error term from Zapata Rolon's asymptotic estimate for the crank function. We then use those bounds to prove the surjectivity and strict log-subadditivity of the crank function., 14 pages 1) Some clarifications and corrections have been made. 2) This paper will appear in Ramanujan Journal

Using the relationship between Siegel cusp forms of degree $2$ and cuspidal automorphic representations of $\mathrm{GSp}(4,\mathbb{A}_{\mathbb{Q}})$, we derive some congruences involving dimensions of spaces of Siegel cusp forms of degree $2$ and the class number of $\mathbb{Q}(\sqrt{-p})$. We also obtain some congruences between the $4$-core partition function $c_4(n)$ and dimensions of spaces of Siegel cusp forms of degree $2$., Comment: Some minor changes. Published online in The Ramanujan Journal

We present a comprehensive and elaborate study of W LVI (K-like W55+) by using multi-configuration Dirac–Fock method (MCDF). We have included relativistic corrections, QED (quantum electrodynamics) and Breit corrections in our computation. We have reported energy levels and radiative data for multipole transitions (i.e., electric dipole (E1), electric quadrupole (E2), magnetic dipole (M1), and magnetic quadrupole (M2)) within the lowest 142 fine-structure levels and predicted soft X-ray transition (SXR) and extreme ultraviolet transitions (EUV) from higher excited states to the ground state. We have compared our calculated data with energy levels compiled by NIST and other available results in the literature and the small discrepancies found with them are discussed. Because only a few of the lowest levels are available in the literature, for checking excitation energies of higher excited states we have performed the same calculations with the distorted wave method. Furthermore, we have also provided relative population for the first five excited states, with both the partition function and thermodynamic quantities for W LVI and studied their variations with temperature. We believe that our reported new atomic data of W LVI may be useful in identification and analysis of spectral lines from various astrophysical and fusion plasma sources and may also be beneficial in plasma modeling.

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 .

Andrews, Lewis, and Lovejoy introduced the partition function PD(n) as the number of partitions of n with designated summands. A bipartition of n is an ordered pair of partitions $$(\pi _1, \pi _2)$$ with the sum of all of the parts being n. In this paper, we introduce a generalized crank named the pd-crank for bipartitions with designated summands and give some inequalities for the pd-crank of bipartitions with designated summands modulo 2 and 3. We also define the pd-crank moments weighted by the parity of pd-cranks $$\mu _{2k,bd}(-1,n)$$ and show the positivity of $$(-1)^n\mu _{2k,bd}(-1,n)$$ . Let $$M_{bd}(m,n)$$ denote the number of bipartitions of n with designated summands with pd-crank m. We prove a monotonicity property of pd-cranks of bipartitions with designated summands and find that the sequence $$\{M_{bd}(m,n)\}_{|m|\le n}$$ is unimodal for $$n\not = 1,5,7$$ .

The sliding electrical contact interface will generate a huge total heat source including Joule heat and frictional heat when an armature slides along a pair of rails driven by Lorentz forces. The total heat is partitioned at the contact interface flowing into armature and rail respectively. The heat partition at the rail-armature interface and the resulting temperature rise have important effects on the melt wear characteristics of the contact material. A computational method was proposed for determining heat partition at the rail-armature interface based on least squares regression. Temperatures equations of rail and armature were developed in terms of an unknown heat partition function. Solution equations of heat partition function were derived by matching the average temperature of rail and armature at the contact surfaces. The heat partition function was assumed to be a polynomial form and an estimate for the heat partition function was obtained by minimizing the error term in the least squares sense. Then, effects of polynomial order on the heat partition function are analyzed. Furthermore, the heat partition solution from the least squares regression and previous analytical method are compared. Variation characteristics of heat partition coefficient were also analyzed. In the end, the variation mechanism and influencing factors of heat partition coefficient were discussed.

We generalize the asymptotic estimates by Bubboloni, Luca and Spiga (2012) on the number of $k$-compositions of $n$ satisfying some coprimality conditions. We substantially refine the error term concerning the number of $k$-compositions of $n$ with pairwise relatively prime summands. We use a different approach, based on properties of multiplicative arithmetic functions of $k$ variables and on an asymptotic formula for the restricted partition function., Comment: revised

We study resurgence for some 3-manifold invariants when $G_{\mathbb{C}}=SL(2, \mathbb{C})$. We discuss the case of an infinite family of Seifert manifolds for general roots of unity and the case of the torus knot complement in $S^3$. Via resurgent analysis, we see that the contribution from the abelian flat connections to the analytically continued Chern-Simons partition function contains the information of all non-abelian flat connections, so it can be regarded as a full partition function of the analytically continued Chern-Simons theory on 3-manifolds $M_3$. In particular, this directly indicates that the homological block for the torus knot complement in $S^3$ is an analytic continuation of the full $G=SU(2)$ partition function, i.e. the colored Jones polynomial., Comment: 42 pages, 4 figures

Linear inequalities involving Euler’s partition function p(n) have been the subject of recent studies. In this article, we consider the partition function Q(n) counting the partitions of n into distinct parts. Using truncated theta series, we provide four infinite families of linear inequalities for Q(n) and partition theoretic interpretations for these results.

For any positive integer [Formula: see text], let [Formula: see text] be the number of solutions of the equation [Formula: see text] with integers [Formula: see text], where [Formula: see text] is the integral part of real number [Formula: see text]. Recently, Luca and Ralaivaosaona gave an asymptotic formula for [Formula: see text]. In this paper, we give an asymptotic development of [Formula: see text] for all [Formula: see text]. Moreover, we prove that the number of such partitions is even (respectively, odd) infinitely often.

We have carried out DFT-based first-principles calculations of hydrogen isotope fractionation among serpentine, talc, brucite, and water as a function of temperature. The results for the fractionat...

We study the divisibility properties of the partition function associated with the eighth-order mock theta function $$V_0(q)$$ , introduced by Gordon and McIntosh. We obtain congruences modulo powers of 2 for certain coefficients of the partition function, akin to Ramanujan’s partition congruences. Further, we also present several infinite families of congruences modulo 13, 25, and 27.

Andrews introduced the partition function $\overline {C}_{k, i}(n)$ , called the singular overpartition function, which counts the number of overpartitions of n in which no part is divisible by k and only parts $\equiv \pm i\pmod {k}$ may be overlined. We prove that $\overline {C}_{6, 2}(n)$ is almost always divisible by $2^k$ for any positive integer k. We also prove that $\overline {C}_{6, 2}(n)$ and $\overline {C}_{12, 4}(n)$ are almost always divisible by $3^k$ . Using a result of Ono and Taguchi on nilpotency of Hecke operators, we find infinite families of congruences modulo arbitrary powers of $2$ satisfied by $\overline {C}_{6, 2}(n)$ .

For any positive integers $n$ and $r$, let $p_r(n)$ denotes the number of partitions of $n$ where each part has $r$ distinct colours. Many authors studied the partition function $p_r(n)$ for particular values of $r$. In this paper, we prove some general congruences modulo $5$ and $7$ for the colour partition function $p_r(n)$ by considering some general values of $r$. To prove the congruences we employ some $q$-series identities which is also in the spirit of Ramanujan., 10 Pages

The purpose of this paper is to study which coalition structures have stable distributions. We employ the projective core as a stability concept. Although the projective core is often defined only for the grand coalition, we define it for every coalition structure. We apply the core notion to a variety of economic models including the public goods game, the Cournot and Bertrand competition, and the common pool resource game. We use a partition function to formulate these models. We argue that symmetry is a common property of these models in terms of a partition function. We offer some general results that hold for all symmetric partition function form games and discuss their implications in the economic models. We also provide necessary and sufficient conditions for the projective core of the models to be nonempty. In addition, we show that our results hold even in the presence of small perturbations of symmetry.

We propose a field-theoretic interpretation of Ruelle zeta function and show how it can be seen as the partition function forBFtheory when an unusual gauge-fixing condition on contact manifolds is imposed. This suggests an alternative rephrasing of a conjecture due to Fried on the equivalence between Ruelle zeta function and analytic torsion, in terms of homotopies of Lagrangian submanifolds., Annales Henri Poincaré, 21 (12), ISSN:1424-0661, ISSN:1424-0637

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

In this paper, we investigated the influence of energy-dependent potentials on the thermodynamic properties of the Klein-Gordon oscillator(KGO): in this way all thermal properties have been determinate via the well-know Euler-Maclaurin method. After this, we extend our study to the case of superstatistical properties of our problem in question. The probability densityf(β)followsχ2− superstatistics (=Tsallis statistics or Gamma distribution). Under the approximation of the low-energy asymptotics of superstatistics, the partition function, at first, has been calculated. This approximation leads to a universal parameterqfor any superstatistics, not only for Tsallis statistics. By using the desired partition function, all thermal properties have been obtained in terms of the parameterq. Also, the influence of the this type of potentials on these properties, via the parameterγ, are well discussed.

In this work we build the foundations of a quantum Monte Carlo as a stochastic numerical method to solve lattice many-body quantum systems with nearest-neighbor interactions at most. As motivation, we briefly describe the bilinear-biquadratic Heisenberg model with an external field, for spin-1 particles, as an effective Hamiltonian of the Bose-Hubbard model with an external quadratic Zeeman field in the Mott insulator phase at unit filling. Then, we discuss how to implement the world line Monte Carlo with local updates to circumvent the difficulties that arise on these type of systems by mapping the quantum partition function into the one of an effective classical model, in one additional dimension, given by the imaginary time evolution of the system. Such a mapping is performed by means of the Suzuki-Trotter decomposition, which transforms the original partition function into a summation of weights given by the classical configurations. Later, we present a set of observables that can be measured through this method and show how to use a Metropolis update scheme to accomplish the measurements. At last, we present the maximization of the configuration weights for three parameter sets as the first and relevant step to perform future measurements.

Let $$(m_1,\ldots ,m_J)$$ and $$(r_1,\ldots ,r_J)$$ be two sequences of J positive integers satisfying $$1\le r_j< m_j$$ for all $$j=1,\ldots ,J$$ . Let $$(\delta _1,\ldots ,\delta _J)$$ be a sequence of J nonzero integers. In this paper, we study the asymptotic behavior of the Taylor coefficients of the infinite product $$\begin{aligned} \prod _{j=1}^J\Bigg (\prod _{k\ge 1}\big (1-q^{r_j+m_j(k-1)}\big )\big (1-q^{-r_j+m_jk}\big )\Bigg )^{\delta _j}. \end{aligned}$$ Our work generalizes many known results, including an asymptotic formula due to Lehner for the partition function arising from the first Rogers–Ramanujan identity. The main technique used here is based on Rademacher’s circle method.

We give a combinatorial proof of a generalization of an identity involving the sum of divisors function [Formula: see text] and the partition function [Formula: see text] which is a companion of Euler’s recurrence formula for [Formula: see text]

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).

The partition function of the exactly solvable four-vertex model on a square grid in the presence of a linearly growing external field is calculated. Namely, we study a system when the external field is applied to one of the columns of the grid. The model is considered both for the fixed and periodic boundary conditions.

Dyson’s famous conjectures (proved by Atkin and Swinnerton-Dyer) gave a combinatorial interpretation of Ramanujan’s congruences for the partition function. The proofs of these results center on one of the universal mock theta functions that generate partitions according to Dyson’s rank. George Beck has generalized the study of partition function congruences related to rank by considering the total number of parts in the partitions of [Formula: see text]. The related generating functions are no longer part of the world of mock theta functions. However, George Beck has conjectured that certain linear combinations of the related enumeration functions do satisfy congruences modulo 5 and 7. The conjectures are proved here.

We derive a general formula giving a representation of the partition function of the one-dimensional Ising model of a system of N particles in the form of an explicitly defined functional of the spectral invariants of finite submatrices of a certain infinite Toeplitz matrix. We obtain an asymptotic representation of the partition function for large N, which can be a base for explicitly calculating some thermodynamic averages, for example, the specific free energy, in the case of a general translation-invariant spin interaction (not necessarily only between nearest neighbors). We estimate the partition function from above and below in the plane of the complex variable β (β is the inverse temperature) and consider the conditions under which these estimates are asymptotically equivalent as N → ∞

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.

We consider a univariate beta integral composed from general modular quantum dilogarithm functions and prove its exact evaluation formula. It represents the partition function of a particular $3d$ supersymmetric field theory on the general squashed lens space. Its possible applications to $2d$ conformal field theory are briefly discussed as well., Comment: typos corrected, 20 pages

Supersymmetry allows a $D^4 R^4$ interaction in M-theory, but such an interaction is inconsistent with string theory dualities and so is known to be absent. We provide a novel proof of the absence of the $D^4 R^4$ M-theory interaction by calculating 4-point scattering amplitudes of 11d supergravitons from ABJM theory. This calculation extends a previous calculation performed to the order corresponding to the $R^4$ interaction. The new ingredient in this extension is the interpretation of the fourth derivative of the mass deformed $S^3$ partition function of ABJM theory, which can be determined using supersymmetric localization, as a constraint on the Mellin amplitude associated with the stress tensor multiplet 4-point function. As part of this computation, we relate the 4-point function of the superconformal primary of the stress tensor multiplet of any 3d ${\cal N} = 8$ SCFT to some of the 4-point functions of its superconformal descendants. We also provide a concise formula for a general integrated 4-point function on $S^d$ for any $d$., Comment: 54 pages, Mathematica notebook attached; v2 typos fixed

The 3D-RISM-KH molecular solvation theory is applied to liquid long chain unsaturated oleyl alcohol to explore the structure of the liquid and to calculate alcohol/water partition coefficients. The united atom TraPPE force field parameters resulted in comparable structural information between RISM and MD simulations. The RISM theory and the MD simulation point to a relatively ordered liquid structure of oleyl alcohol. The molecular partition coefficients were computed with a root mean square deviation of 0.48 unit via a predictive quantitative structure activity relationship.

We show that the fermionic matrix model can be realized by $W$-representation. We construct the Virasoro constraints with higher algebraic structures, where the constraint operators obey the Witt algebra and null 3-algebra. The remarkable feature is that the character expansion of the partition function can be easily derived from such Virasoro constraints. It is a $\tau$-function of the KP hierarchy. We construct the fermionic Aristotelian tensor model and give its $W$-representation. Moreover, we analyze the fermionic red tensor model and present the $W$-representation and character expansion of the partition function., Comment: 20 pages. Revised version accepted for publication in Nuclear Physics B

In 2012, Andrews and Merca established a truncated form of Euler’s pentagonal number theory. In this note, using the Rogers–Fine identity, we present a partition theoretic interpretation of truncated sums on the partition function which solve a problem given by Merca. Furthermore, we prove an inequality on ordinary partition which implies the positive result on truncated sums of partition function due to Andrews and Merca.

In Alexandrov's work [4,5] it has been shown that the extended partition function exp⁡(Fo,ext+Fc) introduced by Buryak in [14,15] is a tau-function of the KP hierarchy. In this work, we compute the affine coordinates of this tau-function on the Sato Grassmannian, and rewrite the Virasoro constraints as recursions for the affine coordinates in the fermionic picture. As applications we derive some formulas for the extended partition function and the connected n-point functions using methods developed by Zhou in [48] based on the boson-fermion correspondence.