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Factor graphs are important models for succinctly representing probability distributions in machine learning, coding theory, and statistical physics. Several computational problems, such as computing marginals and partition functions, arise naturally when working with factor graphs. Belief propagation is a widely deployed iterative method for solving these problems. However, despite its significant empirical success, several questions regarding the correctness and efficiency of belief propagation remain open. The Bethe approximation is an optimization-based method for approximating the partition functions. While it is known that the stationary points of the Bethe approximation coincide with the fixed points of belief propagation, in general, the relation between the Bethe approximation and the partition function is not well understood. It has been observed that for a few classes of factor graphs, the Bethe approximation gives a lower bound to the partition function, which distinguishes them from the general case, where neither a lower bound nor an upper bound holds universally. This has been rigorously proved for permanents and for attractive graphical models. Here, we consider bipartite factor graphs over binary alphabet and show that if the local constraints satisfy a certain analytic property, the Bethe approximation is a lower bound to the partition function, generalizing an analogous inequality between the permanent and the Bethe permanent of a matrix with non-negative entries. We arrive at this result by viewing the factor graphs through the lens of polynomials, which allows us to reformulate the Bethe approximation as an optimization problem involving polynomials. The sufficient condition for our lower bound property to hold is inspired by the recent developments in the theory of real stable polynomials. We believe that this way of viewing factor graphs and its connection to real stability might lead to a better understanding of belief propagation and factor graphs in general. [ABSTRACT FROM AUTHOR]

In the framework of Atiyah's axioms of topological quantum field theory with unitarity, we give a direct proof of the fact that symmetry protected topological phases without Hall effects are classified by cobordism invariants. We first show that the partition functions of those theories are cobordism invariants after a tuning of the Euler term. Conversely, for a given cobordism invariant, we construct a unitary topological field theory whose partition function is given by the cobordism invariant, assuming that a certain bordism group is finitely generated. Two theories having the same cobordism invariant partition functions are isomorphic. [ABSTRACT FROM AUTHOR]

W -representation realizes partition functions by an action of a cut-and-join operator on the vacuum state with a zero-mode background. We provide explicit formulas of this kind for β - and q , t -deformations of the simplest rectangular complex matrix model. In the latter case, instead of the complicated definition in terms of multiple Jackson integrals, we define partition functions as the weight-two series, made from Macdonald polynomials, which are evaluated at different loci in the space of time variables. Resulting expression for the W ˆ operator appears related to the problem of simple Hurwitz numbers (contributing are also the Young diagrams with all but one lines of length two and one). This problem is known to exhibit nice integrability properties. Still the answer for W ˆ can seem unexpectedly sophisticated and calls for improvements. Since matrix models lie at the very basis of all gauge- and string-theory constructions, our exercise provides a good illustration of the jump in complexity between β - and q , t -deformations – which is not always seen at the accidentally simple level of Calogero-Ruijsenaars Hamiltonians (where both deformations are equally straightforward). This complexity is, however, quite familiar in the theories of network models, topological vertices and knots. [ABSTRACT FROM AUTHOR]

Abstract We show that the three-dimensional map between fermions and bosons at finite temperature generalises for all odd dimensions d > 3. We further argue that such a map has a nontrivial large d limit. Evidence comes from studying the gap equations, the free energies and the partition functions of the U (N) Gross–Neveu and CP N − 1 models for odd d ≥ 3 in the presence of imaginary chemical potential. We find that the gap equations and the free energies can be written in terms of the Bloch–Wigner–Ramakrishnan D d (z) functions analysed by Zagier. Since D 2 (z) gives the volume of ideal tetrahedra in 3 d hyperbolic space our three-dimensional results are related to resent studies of complex Chern–Simons theories, while for d > 3 they yield corresponding higher dimensional generalizations. As a spinoff, we observe that particular complex saddles of the partition functions correspond to the zeros and the extrema of the Clausen functions C l d (θ) with odd and even index d respectively. These saddles lie on the unit circle at positions remarkably well approximated by a sequence of rational multiples of π. [ABSTRACT FROM AUTHOR]

The partition functions of three-dimensional N=2 supersymmetric gauge theories on different manifolds can be expressed as q-hypergeometric integrals. Comparing the partition functions of three-dimensional mirror dual theories, we derive complicated integral identities. In some cases, these identities can be written in the form of pentagon relations. Such identities are often interpreted as the Pachner 3–2 move for triangulated manifolds using the so-called 3d–3d correspondence. From the physics perspective, another important application of pentagon identities is that they can be used to construct new solutions of the quantum Yang–Baxter equation. [ABSTRACT FROM AUTHOR]

We study functional clones, which are sets of non-negative pseudo-Boolean functions (functions { 0 , 1 } k → R ≥ 0 ) closed under (essentially) multiplication, summation and limits. Functional clones naturally form a lattice under set inclusion and are closely related to counting Constraint Satisfaction Problems (CSPs). We identify a sublattice of interesting functional clones and investigate the relationships and properties of the functional clones in this sublattice. [ABSTRACT FROM AUTHOR]

Abstract: Character formulae for positive energy unitary representations of the superconformal group are obtained through use of reduced Verma modules and Weyl group symmetry. Expansions of these are given which determine the particular representations present and results such as dimensions of superconformal multiplets. By restriction of variables various ‘blind’ characters are also obtained. Limits, corresponding to reduction to particular subgroups, in the characters isolate contributions from particular subsets of multiplets and in many cases simplify the results considerably. As a special case, the index counting short and semi-short multiplets which do not form long multiplets found recently is shown to be related to particular cases of reduced characters. Partition functions of super-Yang–Mills are investigated. Through analysis of these, exact formulae are obtained for counting - and some -BPS operators in the free case. Similarly, partial results for the counting of semi-short operators are given. It is also shown in particular examples how certain short operators which one might combine to form long multiplets due to group theoretic considerations may be protected dynamically. [Copyright &y& Elsevier]

In this paper, we investigate the stability of the configurations of harmonic oscillator potential that are directly proportional to the square of the displacement. We derive expressions for fluctuations in partition function due to variations of the parameters, viz. the mass, temperature and the frequency of oscillators. Here, we introduce the Hessian matrix of the partition function as the model embedding function from the space of parameters to the set of real numbers. In this framework, we classify the regions in the parameter space of the harmonic oscillator fluctuations where they yield a stable statistical configuration. The mechanism of stability follows from the notion of the fluctuation theory. In Secs. 7 and 8, we provide the nature of local and global correlations and stability regions where the system yields a stable or unstable statistical basis, or it undergoes into geometric phase transitions. Finally, in Sec. 9, the comparison of results is provided with reference to other existing research. [ABSTRACT FROM AUTHOR]

We consider the problem of computing the partition function $\sum _x e^{f(x)}$ , where $f: \{-1, 1\}^n \longrightarrow {\mathbb R}$ is a quadratic or cubic polynomial on the Boolean cube $\{-1, 1\}^n$. In the case of a quadratic polynomial f, we show that the partition function can be approximated within relative error $0 in quasi-polynomial $n^{O(\ln n - \ln \epsilon)}$ time if the Lipschitz constant of the non-linear part of f with respect to the $\ell ^1$ metric on the Boolean cube does not exceed $1-\delta $ , for any $\delta>0$ , fixed in advance. For a cubic polynomial f, we get the same result under a somewhat stronger condition. We apply the method of polynomial interpolation, for which we prove that $\sum _x e^{\tilde {f}(x)} \ne 0$ for complex-valued polynomials $\tilde {f}$ in a neighborhood of a real-valued f satisfying the above mentioned conditions. The bounds are asymptotically optimal. Results on the zero-free region are interpreted as the absence of a phase transition in the Lee–Yang sense in the corresponding Ising model. The novel feature of the bounds is that they control the total interaction of each vertex but not every single interaction of sets of vertices. [ABSTRACT FROM AUTHOR]

In this paper, a new localized radial basis function (RBF) method based on partition of unity (PU) is proposed for solving boundary and initial-boundary value problems. The new method benefits from a direct discretization approach and is called the "direct RBF partition of unity (D-RBF-PU)" method. Thanks to avoiding all derivatives of PU weight functions as well as all lower derivatives of local approximants, the new method is faster and simpler than the standard RBF-PU method. Besides, the discontinuous PU weight functions can now be utilized to develop the method in a more efficient and less expensive way. Alternatively, the new method is an RBF-generated finite difference (RBF-FD) method in a PU setting which is much faster and in some situations more accurate than the original RBF-FD. The polyharmonic splines are used for local approximations, and the error and stability issues are considered. Some numerical experiments on irregular two- and three-dimensional domains, as well as cost comparison tests, are performed to support the theoretical analysis and to show the efficiency of the new method. [ABSTRACT FROM AUTHOR]

We discuss some properties of generative models for word embeddings. Namely, (Arora et al., 2016) proposed a latent discourse model implying the concentration of the partition function of the word vectors. This concentration phenomenon led to an asymptotic linear relation between the pointwise mutual information (PMI) of pairs of words and the scalar product of their vectors. Here, we first revisit this concentration phenomenon and prove it under slightly weaker assumptions, for a set of random vectors symmetrically distributed around the origin. Second, we empirically evaluate the relation between PMI and scalar products of word vectors satisfying the concentration property. Our empirical results indicate that, in practice, this relation does not hold with arbitrarily small error. This observation is further supported by two theoretical results: (i) the error cannot be exactly zero because the corresponding shifted PMI matrix cannot be positive semidefinite; (ii) under mild assumptions, there exist pairs of words for which the error cannot be close to zero. We deduce that either natural language does not follow the assumptions of the considered generative model, or the current word vector generation methods do not allow the construction of the hypothesized word embeddings. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a closed formula for the Kostant's partition function for the Lie algebras sl4(C) and sp6(C): Using this function, one can compute the weight multiplicity of irreducible representations of the Lie algebras sl4(C) and sp6(C). [ABSTRACT FROM AUTHOR]

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). [ABSTRACT FROM AUTHOR]

Nicolas [8] and DeSalvo and Pak [3] proved that the partition function p(n) is log concave for n ≥ 25 . Chen et al. [2] proved that p(n) satisfies the third order Turán inequality, and that the associated degree 3 Jensen polynomials are hyperbolic for n ≥ 95 . Recently, Griffin et al. [5] proved more generally that for all d, the degree d Jensen polynomials associated to p(n) are hyperbolic for sufficiently large n. In this paper, we prove that the same result holds for the k-regular partition function p k (n) for k ≥ 2 . In particular, for any positive integers d and k, the order d Turán inequalities hold for p k (n) for sufficiently large n. The case when d = k = 2 proves a conjecture by Neil Sloane that p 2 (n) is log concave. [ABSTRACT FROM AUTHOR]

Approximating the partition function of the ferromagnetic Ising model with general external fields is known to be\#BIS-hard in the worst case, even for bounded-degree graphs, and it is widely believed that no polynomial-time approximation scheme exists. This motivates an averagecase question: are there classes of instances for which polynomial-time approximation schemes exist? We investigate this question for the random field Ising model on graphs with maximum degree\Delta. We establish the existence of fully polynomial-time approximation schemes and samplers with high probability over the random fields if the external fields are independent and identically distributed Gaussians with variance larger than a constant depending only on the inverse temperature and\Delta. Our methods work more generally for external fields that are large in absolute value with high probability. The main challenge is that such a hypothesis does not rule out the existence of a positive density of vertices at which the external field is small. These regions, which may have connected components of size\Theta (log n), are a barrier to algorithms based on establishing zero-free regions of partition functions and cause worst-case analyses of Glauber dynamics to fail. The analysis of our algorithm is based on percolation on a self-avoiding walk tree. [ABSTRACT FROM AUTHOR]

We continue the investigation of the connection between the genus expansion of matrix models and the ħ expansion of integrable hierarchies. In this paper, we focus on the BKP hierarchy, which corresponds to the infinite-dimensional Lie algebra of type B. We consider the genus expansion of such important solutions as Brézin–Gross–Witten (BGW) model, Kontsevich model, and generating functions for spin Hurwitz numbers with completed cycles. We show that these partition functions with inserted parameter ħ , which controls the genus expansion, are solutions of the ħ -BKP hierarchy with good quasi-classical behavior. ħ -BKP language implies the algorithmic prescription for ħ -deformation of the mentioned models in terms of hypergeometric BKP τ -functions and gives insight into the similarities and differences between the models. Firstly, the insertion of ħ into the Kontsevich model is similar to the one in the BGW model, though the Kontsevich model seems to be a very specific example of hypergeometric τ -function. Secondly, generating functions for spin Hurwitz numbers appear to possess a different prescription for genus expansion. This property of spin Hurwitz numbers is not the unique feature of BKP: already in the KP hierarchy, one can observe that generating functions for ordinary Hurwitz numbers with completed cycles are deformed differently from the standard matrix model examples. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a generalization of the Bessenrodt–Ono inequality by following Gian–Carlo Rota's advice in studying problems in combinatorics and number theory in terms of roots of polynomials. We consider the number of k-colored partitions of n as special values of polynomials P n (x) . We prove for all real numbers x > 2 and a , b ∈ N with a + b > 2 the inequality: P a (x) · P b (x) > P a + b (x). We show that P n (x) < P n + 1 (x) for x ≥ 1 , which generalizes p (n) < p (n + 1) , where p(n) denotes the partition function. Finally, we observe for small values, the opposite can be true, since, for example: P 2 (- 3 + 10) = P 3 (- 3 + 10) . [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We propose a field-theoretic interpretation of Ruelle zeta function and show how it can be seen as the partition function for BF theory 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. [ABSTRACT FROM AUTHOR]

We consider the dimer model on a hexagonal lattice. This model can be represented as a "pile of cubes in a box." The energy of a configuration is given by the volume of the pile. The partition function is computed by the classical MacMahon formula or as the determinant of the Kasteleyn matrix. We use the MacMahon formula to derive the scaling behavior of free energy in the limit as the lattice spacing goes to zero and temperature goes to infinity. We consider the case of a finite hexagonal domain, the case where one side of the hexagonal box is infinite, and the case of inhomogeneous Boltzmann weights. We obtain an asymptotic expansion of free energy, which is called finite-size corrections, and discuss the universality and physical meaning of the expansion coefficients. [ABSTRACT FROM AUTHOR]

The generating function for restricted partitions is a finite product with a Laurent expansion at each root of unity. The question of the behavior of these Laurent coefficients as the size of the product increases goes back to Rademacher and his work on partitions. Building on the methods of Drmota, Gerhold and previous results of the author, we complete this description and give the full asymptotic expansion of each coefficient at every root of unity. These techniques are also shown to give the asymptotics of Sylvester waves. [ABSTRACT FROM AUTHOR]

Nuclear processes such as DNA replication, transcription, and RNA processing each depend on the concerted action of many different protein and RNA molecules. How biomolecules with shared functions find their way to specific locations has been assumed to occur largely by diffusion-mediated collisions. Recent studies have shown that many nuclear processes occur within condensates that compartmentalize and concentrate the protein and RNA molecules required for each process, typically at specific genomic loci. These condensates have common features and emergent properties that provide the cell with regulatory capabilities beyond canonical molecular regulatory mechanisms. We describe here the shared features of nuclear condensates, the components that produce locus-specific condensates, elements of specificity, and the emerging understanding of mechanisms regulating these compartments. Most nuclear regulatory processes are compartmentalized in condensates. Components with shared functions partition selectively into specific condensates. Bifunctional proteins with both structured and condensate-promoting domains localize condensates to specific genomic loci. Diverse RNA species and RNA-binding proteins promote formation of specific condensates. Further understanding of condensates may provide new therapeutic opportunities for diseases. [ABSTRACT FROM AUTHOR]

We study generating functions of ordinary and plane partitions coloured by the action of a finite subgroup of the corresponding special linear group. After reviewing known results for the case of ordinary partitions, we formulate a conjecture concerning a basic factorisation property of the generating function of coloured plane partitions that can be thought of as an orbifold analogue of a conjecture of Maulik et al., now a theorem, in three-dimensional Donaldson–Thomas theory. We study natural quantisations of the generating functions arising from geometry, discuss a quantised version of our conjecture, and prove a positivity result for the quantised coloured plane partition function under a geometric assumption. [ABSTRACT FROM AUTHOR]

We present several results concerning the free energy of odd-dimensional conformal field theories (CFTs) on squashed spheres. First, we propose a formula which computes this quantity for holographic CFTs dual to higher-curvature gravities with second-order linearized equations of motion. As opposed to standard on-shell action methods for Taub geometries, our formula only involves a simple evaluation of the corresponding bulk Lagrangian on an auxiliary pure anti-de Sitter (AdS) space. The expression is closely related to the function determining the possible AdS vacua of the bulk theory in question, which we argue to act as a generating functional from which correlation functions of the boundary stress tensor can be easily characterized. Finally, based on holographic results and free-field numerical calculations, we conjecture that the subleading term in the squashing-parameter free-energy expansion is universally controlled by the stress-tensor three-point function charge t4 for general (2+1)-dimensional CFTs. [ABSTRACT FROM AUTHOR]

We propose a non-perturbative definition for refined topological strings. This can be used to compute the partition function of superconformal theories in 5 dimensions on squashed S5 and the superconformal index of a large number of 6 dimensional (2, 0) and (1, 0) theories, including that of N coincident M5 branes. The result can be expressed as an integral over the product of three combinations of topological string amplitudes. SL(3,Z) modular transformations acting by inverting the coupling constants of the refined topological string play a key role. [ABSTRACT FROM AUTHOR]

We propose a set of novel expansions of Nekrasov’s instanton partition functions. Focusing on 5d supersymmetric pure Yang-Mills theory with unitary gauge group on ℂq,t−12×S1, we show that the instanton partition function admits expansions in terms of partition functions of unitary gauge theories living on the 3d subspaces ℂq×S1, ℂt−1×S1 and their intersection along S1. These new expansions are natural from the BPS/CFT viewpoint, as they can be matched with Wq,t correlators involving an arbitrary number of screening charges of two kinds. Our constructions generalize and interpolate existing results in the literature. [ABSTRACT FROM AUTHOR]

Icosahedral viruses are under a micrometer in diameter, their infectious genome encapsulated by a shell assembled by a multiscale process, starting from an integer multiple of 60 viral capsid or coat protein (VP) monomers. We predict and validate inter-atomic hotspot interactions between VP monomers that are important for the assembly of 3 types of icosahedral viral capsids: Adeno Associated Virus serotype 2 (AAV2) and Minute Virus of Mice (MVM), both T = 1 single stranded DNA viruses, and Bromo Mosaic Virus (BMV), a T = 3 single stranded RNA virus. Experimental validation is by in-vitro, site-directed mutagenesis data found in literature. We combine ab-initio predictions at two scales: at the interface-scale, we predict the importance (cruciality) of an interaction for successful subassembly across each interface between symmetry-related VP monomers; and at the capsid-scale, we predict the cruciality of an interface for successful capsid assembly. At the interface-scale, we measure cruciality by changes in the capsid free-energy landscape partition function when an interaction is removed. The partition function computation uses atlases of interface subassembly landscapes, rapidly generated by a novel geometric method and curated opensource software EASAL (efficient atlasing and search of assembly landscapes). At the capsid-scale, cruciality of an interface for successful assembly of the capsid is based on combinatorial entropy. Our study goes all the way from resource-light, multiscale computational predictions of crucial hotspot inter-atomic interactions to validation using data on site directed mutagenesis’ effect on capsid assembly. By reliably and rapidly narrowing down target interactions, (no more than 1.5 hours per interface on a laptop with Intel Core i5-2500K@3.2 Ghz CPU and 8GB of RAM) our predictions can inform and reduce time consuming in-vitro and in-vivo experiments, or more computationally intensive in-silico analyses. [ABSTRACT FROM AUTHOR]

We prove the conjectural relationship recently proposed in Dubrovin and Yang (Commun Number Theory Phys 11:311–336, 2017) between certain special cubic Hodge integrals of the Gopakumar–Mariño–Vafa type (Gopakumar and Vafa in Adv Theor Math Phys 5:1415–1443, 1999, Mariño and Vafa in Contemp Math 310:185–204, 2002) and GUE correlators, and the conjecture proposed in Dubrovin et al. (Adv Math 293:382–435, 2016) that the partition function of these Hodge integrals is a tau function of the discrete KdV hierarchy. [ABSTRACT FROM AUTHOR]

In 2010, Andrews investigated a variety of parity questions in the classical partition identities of Euler, Rogers, Ramanujan and Gordon. In particular, he considered the Rogers-Ramanujan-Gordon partitions with some constraints on even and odd parts. At the end of this paper, he left fifteen open questions, of which the eleventh is to extend his parity consideration to overpartitions. The main purpose of this paper is to undertake that question. In 2013, Chen, Sang and Shi derived an overpartition analogue of the Rogers–Ramanujan–Gordon theorem. Motivated by their work, we define two kinds of Rogers–Ramanujan–Gordon type overpartitions with some parity constraints on even and odd parts. We then provide the generating functions for such partitions in some cases. [ABSTRACT FROM AUTHOR]

Recently, Andrews defined a partition function 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 EO ‾ (n) which counts the number of partitions of n enumerated by 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 EO ‾ (n). In this article, we prove infinite families of congruences for EO ‾ (n). We next study distribution of EO ‾ (n). We prove that there are infinitely many integers N in every arithmetic progression for which EO ‾ (2 N) is even; and that there are infinitely many integers M in every arithmetic progression for which EO ‾ (2 M) is odd so long as there is at least one. We further prove that EO ‾ (n) is even for almost all n. Very recently, Uncu has treated a different subset of the partitions enumerated by 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. [ABSTRACT FROM AUTHOR]

We consider the Laguerre partition function and derive explicit generating functions for connected correlators with arbitrary integer powers of traces in terms of products of Hahn polynomials. It was recently proven in Cunden et al. (Ann. Inst. Henri Poincaré D, to appear) that correlators have a topological expansion in terms of weakly or strictly monotone Hurwitz numbers that can be explicitly computed from our formulae. As a second result, we identify the Laguerre partition function with only positive couplings and a special value of the parameter α = - 1 / 2 with the modified GUE partition function, which has recently been introduced in Dubrovin et al. (Hodge-GUE correspondence and the discrete KdV equation. arXiv:1612.02333) as a generating function for Hodge integrals. This identification provides a direct and new link between monotone Hurwitz numbers and Hodge integrals. [ABSTRACT FROM AUTHOR]

We further the study of the Donaldson–Thomas theory of the banana 3-folds which were recently discovered and studied by Bryan [ 3 ]. These are smooth proper Calabi–Yau 3-folds which are fibred by Abelian surfaces such that the singular locus of a singular fibre is a non-normal toric curve known as a 'banana configuration'. In [ 3 ], the Donaldson–Thomas partition function for the rank 3 sub-lattice generated by the banana configurations is calculated. In this article, we provide calculations with a view towards the rank 4 sub-lattice generated by a section and the banana configurations. We relate the findings to the Pandharipande–Thomas theory for a rational elliptic surface and present new Gopakumar–Vafa invariants for the banana 3-fold. [ABSTRACT FROM AUTHOR]

We consider the quantum Sherrington–Kirkpatrick (SK) spin-glass model with a transverse field and provide a formula for its free energy in the thermodynamic limit, valid for all inverse temperatures β > 0. To characterize the free energy, we use the path integral representation of the partition function and approximate the model by a sequence of finite-dimensional vector-spin glasses with R d -valued spins. This enables us to use the results of Panchenko who generalized [Ann. Probab. 46(2), 829–864 (2018); ibid, 46(2), 865–896 (2018)] the Parisi formula to classical vector-spin glasses. As a consequence, we can express the thermodynamic limit of the free energy of the quantum SK model as the d → ∞ limit of the free energies of the d-dimensional approximations of the model. [ABSTRACT FROM AUTHOR]

3d $${\mathcal{N}=2}$$ partition functions on the squashed three-sphere $${S^{3}_{b}}$$ and on the twisted product $${S^{2} \times S^{1}}$$ have been shown to factorize into sums of squares of solid tori partition functions, the so-called holomorphic blocks. The same set of holomorphic blocks realizes $${S^{3}_{b}}$$ and $${S^{2} \times S^{1}}$$ partition functions but the two cases involve different inner products, the S-pairing and the id-pairing, respectively. We define a class of q-deformed CFT correlators where chiral blocks are controlled by a deformation of Virasoro symmetry and are paired by S-pairing and id-pairing, respectively. Applying the bootstrap approach to a class of degenerate correlators we are able to derive three-point functions. We show that degenerate correlators can be mapped to 3d partition functions while the crossing symmetry of correlators corresponds to the flop symmetry of 3d gauge theories. We explore how non-degenerate q-deformed correlators are related to 5d partition functions. We argue that id-pairing correlators are associated with the superconformal index on $${S^{4} \times S^{1}}$$ while S-pairing three-point function factors capture the one-loop part of S partition functions. This is consistent with the interpretation of $${S^{2} \times S^{1}}$$ and $${S^{3}_{b}}$$ gauge theories as codimension two defect theories inside $${S^{4} \times S^{1}}$$ and S, respectively. [ABSTRACT FROM AUTHOR]

We study the spectrum of permutation orbifolds of 2d CFTs. We find examples where the light spectrum grows faster than Hagedorn, which is different from known cases such as symmetric orbifolds. We also describe how to compute their partition functions using a generalization of Hecke operators. [ABSTRACT FROM AUTHOR]

The aim of this article is twofold: first, we show how to recover the smooth Deligne-Beilinson cohomology groups from a Heegaard splitting of a closed oriented smooth 3-manifold by extending the usual C ̌ ech-de Rham construction; second, thanks to the above and still relying on a Heegaard splitting, we explain how to compute the partition functions of the U(1) Chern-Simons and BF theories. [ABSTRACT FROM AUTHOR]

We prove a Givental type decomposition for partition functions that arise out of topological recursion applied to spectral curves. Copies of the Konstevich–Witten KdV tau function arise out of regular spectral curves and copies of the Brezin–Gross–Witten KdV tau function arise out of irregular spectral curves. We present the example of this decomposition for the matrix model with two hard edges and spectral curve (x 2 − 4) y 2 = 1. [ABSTRACT FROM AUTHOR]

The partition function of a three-dimensional N=2 theory on the manifold ℳg,p, an S1 bundle of degree p over a closed Riemann surface Σg, was recently computed via supersymmetric localization. In this paper, we compute these partition functions at large N in a class of quiver gauge theories with holographic M-theory duals. We provide the supergravity bulk dual having as conformal boundary such three-dimensional circle bundles. These configurations are solutions to N=2 minimal gauged supergravity and pertain to the class of Taub-NUT-AdS and Taub-Bolt-AdS preserving 1/4 of the supersymmetries. We discuss the conditions for the uplift of these solutions to M-theory, and compute the on-shell action via holographic renormalization. We show that the uplift condition and on-shell action for the Bolt solutions are correctly reproduced by the large N limit of the partition function of the dual superconformal field theory. In particular, the Σg × S1 = ℳg,0 partition function, which was recently shown to match the entropy of AdS4 black holes, and the S3 ≅ ℳ0,1 free energy, occur as special cases of our formalism, and we comment on relations between them. [ABSTRACT FROM AUTHOR]

We study Type IIB supergravity solutions with spacetime of the form AdS6×S2 warped over a Riemann surface Σ, where Σ includes punctures around which the super-gravity fields have non-trivial SL(2, R) monodromy. Solutions without monodromy have a compelling interpretation as near-horizon limits of (p, q) 5-brane webs, and the punctures have been interpreted as additional 7-branes in the web. In this work we provide further support for this interpretation and clarify several aspects of the identification of the supergravity solutions with brane webs. To further support the identification of the punctures with 7-branes, we show that punctures with infinitesimal monodromy match a probe 7-brane analysis using κ-symmetry. We then construct families of solutions with fixed 5-brane charges and punctures with finite monodromy, corresponding to fully back-reacted 7-branes. We compute the sphere partition functions of the dual 5d SCFTs and use the results to discuss concrete brane web interpretations of the supergravity solutions. [ABSTRACT FROM AUTHOR]

We consider 4d supersymmetric (special) unitary Γ quiver gauge theories on compact manifolds which are fibrations over . We show that their partition functions are correlators of vertex operators and screening charges of the modular double version of elliptic algebras. We also consider a generating function of BPS surface defects supported on and show that it can be identified with a particular coherent state in the Fock module over the elliptic Heisenberg algebra. [ABSTRACT FROM AUTHOR]

Dubrovin establishes a certain relationship between the GUE partition function and the partition function of Gromov–Witten invariants of the complex projective line. In this paper, we give a direct proof of Dubrovin's result. We also present in a diagram the recent progress on topological gravity and matrix gravity. [ABSTRACT FROM AUTHOR]

where in the thermodynamic limit critical scaling is described by mean-field theory. Recent works show that the correlation length is not bound by the system's physical size, a belief that long held sway. Instead, two scaling regimes can be observed -- at the critical and pseudo-critical temperatures. We demonstrate that both are manifest for free boundaries. We use numerical simulations of the d = 5 Ising model to analyse the magnetization, susceptibility, magnetization Fourier modes and the partition function zeros. While some of the response functions hide the dual finite-size scaling, the precision enabled by the analysis of Lee--Yang zeros allows this be brought to the fore. In particular, finite-size scaling of leading zeros at the pseudo-critical point confirms recent predictions coming from correlations exceeding the system size. This paper is dedicated to Jaroslav Ilnytskyi on the occasion of his 60th birthday. [ABSTRACT FROM AUTHOR]

In this paper we continue the study of the double ramification hierarchy of Buryak (Commun Math Phys 336(3):1085-1107, 2015). After showing that the DR hierarchy satisfies tau-symmetry we define its partition function as the (logarithm of the) tau-function of the string solution and show that it satisfies various properties (string, dilaton, and divisor equations plus some important degree constraints). We then formulate a stronger version of the conjecture from Buryak (2015): for any semisimple cohomological field theory, the Dubrovin-Zhang and double ramification hierarchies are related by a normal [i.e. preserving the tau-structure (Dubrovin et al. in Adv Math 293:382-435, 2016)] Miura transformation which we completely identify in terms of the partition function of the CohFT. In fact, using only the partition functions, the conjecture can be formulated even in the non-semisimple case (where the Dubrovin-Zhang hierarchy is not defined). We then prove this conjecture for various CohFTs (trivial CohFT, Hodge class, Gromov-Witten theory of CP1, 3-, 4- and 5-spin classes) and in genus 1 for any semisimple CohFT. Finally we prove that the higher genus part of the DR hierarchy is basically trivial for the Gromov-Witten theory of smooth varieties with non-positive first Chern class and their analogue in Fan-Jarvis-Ruan-Witten quantum singularity theory (Fan et al. in Ann Math 178(1):1-106, 2013). [ABSTRACT FROM AUTHOR]

Underground 3D density variation can be obtained via the inversion of gravity data, which is a very important basis for structural division, oil and gas structure definition, and mineral resource evaluation. A depth-weighting function is usually introduced as a structural constraint in density inversion to solve the skin effect. We propose an adaptive space–location-weighting (ASW) function for gravity field data to improve the resolution of the inversion, which adds the position and depth information provided by the DEXP method to form a new weighting function. The weighting function is partitioned according to the horizontal distribution of the source and can effectively improve the resolution of field sources with different positions and different depths. The results of model tests have shown that the ASW function method can significantly improve the precision and resolution of density inversion results and has good noise immunity. The ASW method was applied to interpret the real gravity data of a mining area in Shandong, and we speculated potential mineralization based on the inversion results, which corresponded well with the logging results. [ABSTRACT FROM AUTHOR]

In this paper, we study a holographic description of weak measurements in conformal field theories (CFTs). Weak measurements can be viewed as a soft projection that interpolates between an identity operator and a projection operator, and can induce an effective central charge distinct from the unmeasured CFT. We model the weak measurement by an interface brane, separating different geometries dual to the post-measurement state and the unmeasured CFT, respectively. In an infinite system, the weak measurement is related to ICFT via a spacetime rotation. We find that the holographic entanglement entropy with twist operators located on the defect is consistent in both calculations for ICFT and weak measurements. We additionally calculate the boundary entropy via holographic entanglement as well as partition function. In a finite system, the weak measurement can lead to a rich phase diagram: for marginal measurements the emergent brane separates two AdS geometries, while for irrelevant measurements the post-measurement geometry features an AdS spacetime and a black hole spacetime that are separated by the brane. Although the measurement is irrelevant in the later phase, the post-measurement geometry can realize a Python's lunch. Finally, we discuss the thick brane construction for measurement and higher-dimension extensions of our model. For these general cases, our results above are still shown to be valid. [ABSTRACT FROM AUTHOR]

This paper presents the influence of microstructural defects on the vibration characteristics of the cracked functionally graded plate in a thermal environment. Extended finite element formulation is implemented to model the cracked functionally graded plate based on higher-order shear deformation theory (HSDT). The level set function is implemented to track the crack in the functionally graded plate domain. The crack in the functionally graded plate domain is addressed by enriching the primary variable with additional functions using the partition of unity technique. The power-law distribution is implemented for the gradation of the material properties along the thickness direction to form functionally graded material (FGM). The microstructural defects are assumed as porosities in the plate. The porosity index determines the porosity in the FGM plate. The convergence and comparative study have been performed to verify the current formulation's efficacy, accuracy, and reliability. Further, various numerical examples are presented. The effects of different influential parameters like porosity, volume fraction index, crack length, thickness ratio, and boundary condition on the natural frequencies have been discussed in detail. The presented computational approach is accurate, efficient, and robust enough to investigate the vibration response of the cracked FGM porous plates, which can provide a reference for future research by researchers. [ABSTRACT FROM AUTHOR]

We review and extend some recent work on testing AdS/CFT correspondence between U(N)k × U(N)–k Chern-Simons-matter 3d gauge theory and M-theory in AdS4 × S7/Zk background. We demonstrate that 1-loop term in the quantum M2 brane partition function expanded near a classical solution correctly matches localization predictions on the gauge theory side in the case of BPS Wilson loop expectation value and instanton corrections to free energy. [ABSTRACT FROM AUTHOR]

Non-Gaussian distributions in cosmology are commonly evaluated with Monte Carlo Markov chain methods, as the Fisher matrix formalism is restricted to the Gaussian case. The Metropolis–Hastings algorithm will provide samples from the posterior distribution after a burn-in period, and the corresponding convergence is usually quantified with the Gelman–Rubin criterion. In this paper, we investigate the convergence of the Metropolis–Hastings algorithm by drawing analogies to statistical Hamiltonian systems in thermal equilibrium for which a canonical partition sum exists. Specifically, we quantify virialization, equipartition, and thermalization of Hamiltonian Monte Carlo Markov chains for a toy model and for the likelihood evaluation for a simple dark energy model constructed from supernova data. We follow the convergence of these criteria to the values expected in thermal equilibrium, in comparison to the Gelman–Rubin criterion. We find that there is a much larger class of physically motivated convergence criteria with clearly defined target values indicating convergence. As a numerical tool, we employ physics-informed neural networks for speeding up the sampling process. [ABSTRACT FROM AUTHOR]

Large-scale, rovibrational variational calculations are performed for ethylene, using the potential energy surface published by Avila and Carrington [J. Chem. Phys. 135, 064101 (2011)]. Energies for J = 0 are in very good agreement with their benchmark results. Corresponding energies for J = 1 and J = 2 are also given. Calculations with a slightly reduced basis permit energies to J = 40, allowing a reliable determination of the partition function at 296 K. Using a new ab initio dipole moment surface, reported here, the infrared spectra of five dipole-allowed fundamentals are calculated. Both the partition function and infrared spectra are shown to be in excellent agreement with those in the experimental HITRAN database, with the exception of one band, which we believe is partially mis-assigned in HITRAN. [ABSTRACT FROM AUTHOR]