Advances in Applied Mathematics最新文献

We use the Poisson kernel of the continuous q-Hermite polynomials to introduce families of integral operators. One of them is semigroups of linear operators. We describe the eigenvalues and eigenfunctions of one family of operators. The action of the semigroups of operators on the Askey–Wilson polynomials is shown to only change the parameters but preserves the degrees, hence we produce transmutation relations for the Askey–Wilson polynomials. The transmutation relations are then used to derive bilinear generating functions involving the Askey–Wilson polynomials.

The entropy of a semiring ideal of tropical polynomials is introduced. The radical of a semiring ideal consists of all tropical polynomials vanishing on the tropical prevariety determined by the ideal. We prove that the entropy of the radical of a tropical bivariate polynomial with zero coefficients vanishes. Also, we prove that the entropy of a zero-dimensional tropical prevariety vanishes. An example of a non-radical semiring ideal with a positive entropy is exhibited.

We introduce a sheaf-theoretic stability condition for finite acyclic quivers. Our main result establishes that for representations of affine type $\tilde{A}$ quivers, there is a precise relationship between the associated Harder-Narasimhan filtration and the barcode of the periodic zigzag persistence module obtained by unwinding the underlying quiver.

Phylogenetic networks model evolutionary phenomena that trees fail to capture such as horizontal gene transfer and hybridization. The same Markov models used for sequence evolution on trees can also be extended to networks and similar problems, such as determining if the network parameter is identifiable or finding the invariants of the model, can be studied. This paper focuses on finding the invariants of the Cavendar-Farris-Neyman (CFN) model on level-1 phylogenetic networks by reducing the problem to finding invariants of sunlet networks, which are level-1 networks consisting of a single cycle with leaves at each vertex. We determine all quadratic invariants for sunlet networks, and conjecture these generate the full sunlet ideal.

It is well known that in q-matroids, axioms for independent spaces, bases, and spanning spaces differ from the classical case of matroids, since the straightforward q-analogue of the classical axioms does not give a q-matroid. For this reason, a fourth axiom has been proposed. In this paper we show how we can describe these spaces with only three axioms, providing two alternative ways to do that. As an application, we show direct cryptomorphisms between independent spaces and circuits and between independent spaces and bases.

A graph is chordal if every cycle of length at least four has a chord. In 1961, Dirac characterized chordal graphs as those graphs that can be built from complete graphs by repeated clique-sums. Generalizing this, we consider the class of simple $GF\left(q\right)$-representable matroids that can be built from projective geometries over $GF\left(q\right)$ by repeated generalized parallel connections across projective geometries. We show that this class of matroids is closed under induced minors. We characterize the class by its forbidden induced minors; the case when $q=2$ is distinctive.

We use the correlation functions of vertex operators to give a proof of Cauchy's formula$\prod _{i=1}^K\prod _{j=1}^N(1-x_i{y}_j)=\sum _{\mu \subseteq [K\times N]}{(-1)}^{\left|\mu \right|}{s}_{\mu }\left\{x\right\}{s}_{{\mu }^{\prime }}\left\{y\right\}.$ As an application of the interpretation, we obtain an expansion of ${\prod }_{i=1}^{\infty }{(1-q^i)}^{i-1}$ in terms of half plane partitions.

A positive integer d is a floor quotient of n if there is a positive integer k such that $d=\lfloor n/k\rfloor$. The floor quotient relation defines a partial order on the positive integers. This paper studies the internal structure of this partial order and its Möbius function.

In 1975, Don Zagier obtained a new version of the Kronecker limit formula for a real quadratic field which involved an interesting function $F\left(x\right)$ which is now known as the Herglotz function. As demonstrated by Zagier, and very recently by Radchenko and Zagier, $F\left(x\right)$ satisfies beautiful properties which are of interest in both algebraic number theory as well as in analytic number theory. In this paper, we study $F_{k,N}\left(x\right)$, an extension of the Herglotz function which also subsumes higher Herglotz function of Vlasenko and Zagier. We call it the extended higher Herglotz function. It is intimately connected with a certain generalized Lambert series as well as with a generalized cotangent Dirichlet series inspired from Krätzel's work. We derive two different kinds of functional equations satisfied by $F_{k,N}\left(x\right)$. Radchenko and Zagier gave a beautiful relation between the integral $\underset{0}{\overset{1}{\int }}\frac{\log (1+t^x)}{1+t}dt$ and $F\left(x\right)$ and used it to evaluate this integral at various rational as well as irrational arguments. We obtain a relation between $F_{k,N}\left(x\right)$ and a generalization of the above integral involving polylogarithm. The asymptotic expansions of $F_{k,N}\left(x\right)$ and some generalized Lambert series are also obtained along with other supplementary results.

Using the representation theory of $s{l}_2$ and an appropriate model for tensor product of lowest weight Verma modules, we give a realisation first of the Hahn algebra, and then of the Racah algebra, using Jacobi differential operators. While doing so we recover some known convolution formulas for Jacobi polynomials involving Hahn and Racah polynomials. Similarly, we produce realisations of the higher rank Racah algebras in which maximal commutative subalgebras are realised in terms of Jacobi differential operators.