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In this paper, we provide generating functions and counting formulas for spanning trees and spanning forests in hypergraphs in two different ways: (1) We represent spanning trees and spanning forests in hypergraphs through Berezin-Grassmann integrals on Zeon algebra and hyper-Hafnians (orders and signs are not considered); (2) We establish a Hyper-Pfaffian-Cactus Spanning Forest Theorem through Berezin-Grassmann integrals on Grassmann algebra (orders and signs are considered), which generalizes the Hyper-Pfaffian-Cactus Theorem by Abdesselam (2004) [1] and Pfaffian matrix tree theorem by Masbaum and Vaintrob (2002) [15]. • We provide generating functions and counting formulas for spanning trees and spanning forests in hypergraphs through Berezin-Grassmann integrals on Zeon algebra and hyper- Hafnians when orders and signs are not considered. • We establish a Hyper-Pfaffian-Cactus Spanning Forest Theorem through Berezin-Grassmann integrals on Grassmann algebra (orders and signs are considered), which generalizes the Hyper-Pfaffian-Cactus Theorem by Abdesselam [Advances in Applied Mathematics (2004) Vol. 33: 51-70] and Pfaffian matrix tree theorem by Masbaum and Vaintrob [Internat. Math. Res. Notices (2002) Vol. 27: 1397-1426]. [ABSTRACT FROM AUTHOR]

In this paper, we study the desingularization problem in the first q -Weyl algebra. We give an order bound for desingularized operators, and thus derive an algorithm for computing desingularized operators in the first q -Weyl algebra. Moreover, an algorithm is presented for computing a generating set of the first q -Weyl closure of a given q -difference operator. As an application, we certify that several instances of the colored Jones polynomial are Laurent polynomial sequences by computing the corresponding desingularized operator. [ABSTRACT FROM AUTHOR]

The Morris constant term identity is known to be equivalent to the famous Selberg integral. In this paper, we regard Morris type constant terms as polynomials of a certain parameter. Thus, we can take the leading coefficients and obtain new identities. These identities happen to be crucial in finding lower bounds for cardinalities of some restricted sumsets, and in calculating the volume of a Tesler polytope. [ABSTRACT FROM AUTHOR]

This paper presents a geometric model of the Auslander–Reiten quiver of a type A quiver together with a stability function for which all indecomposable modules are stable. We also introduce a new Catalan object which we call a maximal almost rigid representation. We show that its endomorphism algebra is a tilted algebra of type A. We define a partial order on the set of maximal almost rigid representations and use our new geometric model to show that this partial order is a Cambrian lattice. [ABSTRACT FROM AUTHOR]

Frieze patterns form a nexus between algebra, combinatorics, and geometry. T -paths with respect to triangulations of surfaces have been used to obtain expansion formulae for cluster variables. This paper will introduce the concepts of weak friezes and T -paths with respect to dissections of polygons. Our main result is that weak friezes are characterised by satisfying an expansion formula which we call the T -path formula. We also show that weak friezes can be glued together, and that the resulting weak frieze is a frieze if and only if so was each of the weak friezes being glued. [ABSTRACT FROM AUTHOR]

In his 1998 Fubini Lectures, Rota discusses twelve problems in probability that "no one likes to bring up". The first problem calls for a revision of the notion of a sample space, guided by the belief that mention of sample points in a probabilistic argument is bad form and that a "pointless" foundation of probability should be provided by algebras of random variables. In 1958 Chang introduced MV-algebras to prove the completeness theorem of Łukasiewicz logic Ł ∞. The aim of this paper is to show that MV-algebras provide a solution of Rota's first problem. The adjunction between MV-algebras and unital commutative C*-algebras equips every MV-algebra A with a natural ring structure, as advocated by Nelson for algebras of random variables. The closed compact set S (A) ⊆ [ 0 , 1 ] A of finitely additive probability measures on A (the states of A) coincides with the set of [ 0 , 1 ] -valued functions on A whose finite restrictions are consistent in de Finetti's sense. MV-algebras and Ł ∞ thus provide the framework for a generalization (known as ŁIPSAT) of Boole's probabilistic inference problem, and its modern reformulation known as probabilistic satisfiability, PSAT. We construct an affine homeomorphism γ A of S (A) onto the weakly compact space of regular Borel probability measures on the maximal spectral space μ (A). The latter is the most general compact Hausdorff space. As a consequence, for every Kolmogorov probability space (Ω , F Ω , P) , with F Ω the sigma-algebra of Borel sets of a compact Hausdorff space Ω, and P a regular probability measure on F Ω , there is an MV-algebra A and a state σ of A such that (Ω , F Ω , P) ≅ (μ (A) , F μ (A) , γ A (σ)). [ABSTRACT FROM AUTHOR]