作者：
Chen, Shi-ChaoHenan Univ
Inst Contemporary Math Dept Math & Stat Sci Kaifeng 475001 Peoples R China
Let k >= 2 be an integer, B a prime and Fe the finite field with B elements. A sequence (a(n)),,>= 0 is called k-automatic if there exists a deterministic finite automaton with output that reads the canonical ba...
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Let k >= 2 be an integer, B a prime and Fe the finite field with B elements. A sequence (a(n)),,>= 0 is called k-automatic if there exists a deterministic finite automaton with output that reads the canonical base-k representation of nand the outputs a(n). We apply the properties of automatic sequences to prove the transcendence of a formal power series over Fe(X) related to infinite products. As applications, the parity results of various partition functions are obtained, including the root partition function and the prime parts partition function. We also establish the transcendence of the power series associated with holomorphic modular forms with integer coefficients. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar
In physics, the continuum limit consists of transforming discretely-spaced degrees of freedom on a lattice into smoothly-varying fields on a continuous domain. In these transformations, the degrees of freedom are typi...
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In physics, the continuum limit consists of transforming discretely-spaced degrees of freedom on a lattice into smoothly-varying fields on a continuous domain. In these transformations, the degrees of freedom are typically dynamical variables like position or probability. However, in statistical physics, partition functions themselves can be seen as existing on a discrete lattice whose individual sites are identified by the number of degrees of freedom in the system. In this work, we pursue this interpretation and show that the continuum limit of a certain class of mean-field theory partition functions yields a partial differential equation whose solution provides large N approximations to the partition function of the original system. The equation is obtained by promoting the number of degrees of freedom to a continuous variable and then re-interpreting (as finite differences in a continuous space) the recursive relation that defines the partition function. For some example systems, we show that solutions to this equation yield transition temperatures with the same parameter-scaling behavior as those of the original system. We conclude by discussing how this formalism can motivate 'diffusion in degree-of-freedom space' interpretations of how interacting partition functions vary with temperature.
The computation of classical Ising partition functions, coming from statistical physics, is a natural generalization of binary optimization. This is a notoriously hard problem in general, which makes it an especially ...
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ISBN:
(纸本)9798400717987
The computation of classical Ising partition functions, coming from statistical physics, is a natural generalization of binary optimization. This is a notoriously hard problem in general, which makes it an especially interesting task to consider in the search for practical quantum advantage in near term quantum computers. In this work we view classical Ising models (on certain graphs) as quantum imaginary time evolution, which is enabled by the use of the transfer matrix mapping. We study this mapping from two points of view: (1) following Onsager and Kaufman's original solution of the 2D Ising model, which serves as a starting point, we consider more general models and the possibility of a similar Lie-theoretic solution;(2) we consider quantum algorithms for the computation of partition functions and thermal averages via transfer matrices, which can be implemented either with block encodings inside larger unitaries or by approximating the state trajectories with unitary operators.
In this work, using quotients of Dedekind eta functions of weight 12, we express certain Eisenstein series associated with congruence subgroups Gamma(0)(p), p= 5, 7, 13, of arbitrary weight. We then obtain new Ramanuj...
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In this work, using quotients of Dedekind eta functions of weight 12, we express certain Eisenstein series associated with congruence subgroups Gamma(0)(p), p= 5, 7, 13, of arbitrary weight. We then obtain new Ramanujan type identities on partition functions associated with certain eta quotients. The method we apply in this work is inspired by the work of Fine, Kolberg, and Garvan on Eisenstein series. It is different from those used by Ramanujan, Zuckerman, Carlitz, and others.
We show factorization formulas for a class of partition functions of rational six vertex model. First we show factorization formulas for partition functions under triangular boundary. Further, by combining the factori...
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We show factorization formulas for a class of partition functions of rational six vertex model. First we show factorization formulas for partition functions under triangular boundary. Further, by combining the factorization formulas with the explicit forms of the generalized domain wall boundary partition functions by Belliard-Pimenta-Slavnov, we derive factorization formulas for partition functions under trapezoid boundary which can be viewed as a generalization of triangular boundary. We also discuss an application to emptiness formation probabilities under trapezoid boundary which admit determinant representations.
We implement a version of conformal field theory in a doubly connected domain with numerous conformal types to connect it to the theory of annulus SLE of various types, including the standard annulus SLE, the reversib...
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We implement a version of conformal field theory in a doubly connected domain with numerous conformal types to connect it to the theory of annulus SLE of various types, including the standard annulus SLE, the reversible annulus SLE, and the annulus SLE with several force points. This implementation considers the statistical fields generated under the OPE multiplication by the Gaussian free field and its central/background charge modifications with a weighted combination of Dirichlet and excursion-reflected boundary conditions. We derive the Eguchi-Ooguri version of Ward's equations and Belavin-Polyakov-Zamolodchikov equations for those statistical fields and use them to show that the correlations of fields in the OPE family under the insertion of the one-leg operators are martingale-observables for various annulus SLEs. We find Coulomb gas (Dotsenko-Fateev integral) solutions to the parabolic partial differential equations for partition functions of conformal field theory for the reversible annulus SLE.
We will revisit Gupta's result regarding properties of a formula for restricted partitions and generalize this. We will then use this result to prove an infinite family of congruences for a certain restricted part...
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We will revisit Gupta's result regarding properties of a formula for restricted partitions and generalize this. We will then use this result to prove an infinite family of congruences for a certain restricted partition function. We find and prove combinatorial witnesses, also known as cranks, for the congruences using polyhedral geometry.
We focus on a set of r-valued n-variable functions that are defined by a partition P on the set of r(n) input vectors. Specifically, each block of P specifies input vectors, all of which map to the same function value...
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ISBN:
(纸本)9781728154060
We focus on a set of r-valued n-variable functions that are defined by a partition P on the set of r(n) input vectors. Specifically, each block of P specifies input vectors, all of which map to the same function value. For example, a symmetric function is defined by a partition where input vectors in the same block are permutations of each other. Given the partition P and the set S of functions associated with P, we analyze the set S' of functions that are a maximal distance from S. Such functions hold promise for use in crypto-systems. In this paper, we characterize functions in S'. From this, we compute the distance to their corresponding partition functions. We show that, when r and n increase without bound, this distance approaches the maximum possible, r(n). Bent functions achieve only half the maximum possible distance when n is large. We show that functions a maximal distance from partition functions tend to have a uniform distribution across the r possible function values. Such functions tend to be immune to statistics-based attacks. Finally, we show that, if the set S' of functions is maximally distant from a set S of partition functions, then the converse is true;that is, S is maximally distant from S'.
This article gives a proof of the Langlands-Shelstad fundamental lemma for the spherical Hecke algebra for every unramified p-adic reductive group G in large positive characteristic. The proof is based on the transfer...
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This article gives a proof of the Langlands-Shelstad fundamental lemma for the spherical Hecke algebra for every unramified p-adic reductive group G in large positive characteristic. The proof is based on the transfer principle for constructible motivic integration. To carry this out, we introduce a general family of partition functions attached to the complex L-group of the unramified p-adic group G. Our partition functions specialize to Kostant's q-partition function for complex connected groups and also specialize to the Langlands L-function of a spherical representation. These partition functions are used to extend numerous results that were previously known only when the L-group is connected (that is, when the p-adic group is split). We give explicit formulas for branching rules, the inverse of the weight multiplicity matrix, the Kato-Lusztig formula for the inverse Satake transform, the Plancherel measure, and Macdonald's formula for the spherical Hecke algebra on a nonconnected complex group (that is, nonsplit unramified p-adic group).
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