Advances in Applied Mathematics最新文献

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Polynomial perturbations of Euler's and Clausen's identities 欧拉恒等式和克劳森恒等式的多项式摄动

IF 1.3 3区数学 Q3 MATHEMATICS, APPLIED

Advances in Applied Mathematics

Pub Date : 2026-01-16 DOI: 10.1016/j.aam.2026.103042

Dmitrii Karp

A product of two hypergeometric series is generally not hypergeometric. However, there are a few cases when such a product does reduce to a single hypergeometric series. The oldest result of this type, beyond the obvious

{(1 - x)}^{a} {(1 - x)}^{b} = {(1 - x)}^{a + b}

, is Euler's transformation for the Gauss hypergeometric function

{}_{2}F_{1}

. Another important one is the celebrated Clausen's identity, dating back to 1828, which expresses the square of a suitable

{}_{2}F_{1}

function as a single

{}_{3}F_{2}

. By equating coefficients, each product identity corresponds to a special type of summation theorem for terminating series. Over the last two decades Euler's transformations and many summation theorems have been extended by introducing additional parameter pairs differing by positive integers. This amounts to multiplication of the power series coefficients by values of a fixed polynomial at nonnegative integers. The main goal of this paper is to present an extension of Clausen's identity obtained by such polynomial perturbation. To this end, we first reconsider the polynomial perturbations of Euler's transformations found by Miller and Paris around 2010. We propose new, simplified proofs of their transformations relating them to polynomial interpolation and exhibiting various new forms of the characteristic polynomials. We further introduce the notion of the Miller-Paris operators which play a prominent role in the construction of the extended Clausen's identity.

两个超几何级数的乘积一般不是超几何级数。然而，在少数情况下，这样的乘积确实简化为单个超几何级数。除了明显的（1−x）a(1−x)b=(1−x)a+b之外，这种类型最古老的结果是高斯超几何函数F12的欧拉变换。另一个重要的是著名的Clausen的身份，可以追溯到1828年，它将合适的F12函数的平方表示为单个F23。通过使系数相等，每个乘积恒等式对应于一种特殊类型的求和定理，用于终止级数。在过去的二十年里，欧拉变换和许多求和定理通过引入额外的参数对被正整数差分而得到了扩展。这相当于幂级数系数乘以非负整数上的固定多项式的值。本文的主要目的是给出由这种多项式摄动得到的克劳森恒等式的推广。为此，我们首先重新考虑Miller和Paris在2010年前后发现的欧拉变换的多项式摄动。我们提出了新的、简化的证明，证明了它们与多项式插值有关的变换，并展示了特征多项式的各种新形式。我们进一步介绍了米勒-巴黎算子的概念，它在扩展克劳森身份的构建中起着突出的作用。

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引用次数: 0

On a question about real-rooted polynomials and f-polynomials of simplicial complexes 关于简单复合体的实根多项式和f多项式的一个问题

IF 1.3 3区数学 Q3 MATHEMATICS, APPLIED

Advances in Applied Mathematics

Pub Date : 2026-01-15 DOI: 10.1016/j.aam.2026.103041

Lili Mu , Volkmar Welker

For a polynomial

f (t) = 1 + f_{0} t + \dots + f_{d - 1} t^{d}

with positive integer coefficients Bell and Skandera (2007) [2] ask if real-rootedness of

f (t)

implies that there is a simplicial complex with f-vector

(1, f_{0}, \dots, f_{d - 1})

or equivalently a simplicial complex with f-polynomial

f (t)

. In this paper we discover properties implied by the real-rootedness of

f (t)

in terms of the binomial representation

f_{i} = (\begin{matrix} x_{i + 1} \\ i + 1 \end{matrix})

i \geq 0

. We use these to partially answer the question by Bell and Skandera. We also describe two further approaches to the question and use one to verify that some well studied real-rooted classical polynomials are f-polynomials.

Finally, we provide a series of results showing that the set of f-vectors of simplicial complexes is closed under constructions also preserving real-rootedness of their generating polynomials.

对于系数为正整数的多项式f(t)=1+f0t+⋯+fd−1td, Bell and Skandera(2007)[2]，问f(t)的实数性是否意味着存在一个具有f-向量（1,f0，…，fd−1）的简单复形，或者等价地具有f-多项式f(t)的简单复形。本文发现了f(t)在二项表示形式fi=(xi+1i+1)， i≥0下的实数根所蕴涵的性质。我们用这些来部分回答Bell和Skandera提出的问题。我们还描述了两种进一步的方法来解决这个问题，并使用一种方法来验证一些研究得很好的实根经典多项式是f多项式。最后，我们给出了一系列结果，证明了简单复形的f向量集在构造下是封闭的，并且保持了它们的生成多项式的实根性。

引用次数: 0

Graphs missing a connected partition 缺少连通分区的图

IF 1.3 3区数学 Q3 MATHEMATICS, APPLIED

Advances in Applied Mathematics

Pub Date : 2026-01-15 DOI: 10.1016/j.aam.2026.103044

Foster Tom

We prove that a graph with a cut vertex whose deletion produces at least five connected components must be missing a connected partition of some type. We prove that this also holds if there are four connected components that each have at least two vertices. In particular, the chromatic symmetric function of such a graph cannot be e-positive. This brings us very close to the conjecture by Dahlberg, She, and van Willigenburg of non-e-positivity for all trees with a vertex of degree at least four. We also prove that spiders with four legs cannot have an e-positive chromatic symmetric function.

证明了具有切顶点的图，其删除至少产生5个连通分量，必然缺少某种类型的连通分区。我们证明，如果有四个连接的组件，每个组件至少有两个顶点，这也成立。特别地，这种图的色对称函数不可能是e正的。这使我们非常接近Dahlberg， She和van Willigenburg的猜想，即顶点度至少为4的所有树都是非正性的。我们还证明了四条腿的蜘蛛不可能具有e正的色对称函数。

引用次数: 0

Capacity bounds on integral flows and the Kostant partition function 积分流的容量界与Kostant配分函数

IF 1.3 3区数学 Q3 MATHEMATICS, APPLIED

Advances in Applied Mathematics

Pub Date : 2026-01-12 DOI: 10.1016/j.aam.2025.103002

Jonathan Leake , Alejandro H. Morales

The type A Kostant partition function is an important combinatorial object with various applications: it counts integer flows on the complete directed graph, computes Hilbert series of spaces of diagonal harmonics, and can be used to compute weight and tensor product multiplicities of representations. In this paper we study asymptotics of the Kostant partition function, improving on various previously known lower bounds and settling conjectures of O'Neill and Yip. Our methods build upon recent results and techniques of Brändén-Leake-Pak, who used Lorentzian polynomials and Gurvits' capacity method to bound the number of lattice points of transportation and flow polytopes. Finally, we also give new two-sided bounds using the Lidskii formulas from subdivisions of flow polytopes.

A型科斯坦配分函数是一个重要的组合对象，具有多种应用：它对完全有向图上的整数流进行计数，计算对角谐波空间的希尔伯特级数，并可用于计算表示的权重和张量积多重。本文研究了Kostant配分函数的渐近性，改进了以前已知的各种下界，并解决了O'Neill和Yip的猜想。我们的方法建立在Brändén-Leake-Pak最近的结果和技术的基础上，他们使用洛伦兹多项式和Gurvits的容量方法来限制运输和流动多面体的晶格点数量。最后，我们还利用Lidskii公式，从流多面体的细分中给出了新的双面边界。

引用次数: 0

A vector bundle approach to Nash equilibria 纳什均衡的向量束方法

IF 1.3 3区数学 Q3 MATHEMATICS, APPLIED

Advances in Applied Mathematics

Pub Date : 2026-01-08 DOI: 10.1016/j.aam.2025.103028

Hirotachi Abo , Irem Portakal , Luca Sodomaco

We use vector bundles to study the locus of totally mixed Nash equilibria of an n-player game in normal form, which we call the Nash equilibrium scheme. When the payoff tensor format is balanced, we study the Nash discriminant variety, i.e., the algebraic variety of games whose Nash equilibrium scheme is nonreduced or has a positive dimensional component. We prove that this variety has codimension one. We classify all possible components of the Nash equilibrium scheme for a binary three-player game. We prove that if the payoff tensor is of boundary format, then the Nash discriminant variety has two components: an irreducible hypersurface and a larger-codimensional component. A generic game with an unbalanced payoff tensor format does not admit totally mixed Nash equilibria. We define the Nash resultant variety of games admitting a positive number of totally mixed Nash equilibria. We prove that it is irreducible and determine its codimension and degree.

我们用向量束研究了n人博弈的标准形式的完全混合纳什均衡的轨迹，我们称之为纳什均衡方案。当支付张量格式平衡时，我们研究了纳什判别变量，即纳什均衡格式非约简或具有正维分量的对策的代数变量。我们证明了这个变体的余维为1。我们对一个二元三人博弈的纳什均衡方案的所有可能组成部分进行分类。证明了如果支付张量是边界格式的，则纳什判别式有两个分量：一个不可约的超曲面和一个较大的协维分量。具有非平衡收益张量格式的一般博弈不允许完全混合纳什均衡。我们定义了含有正数个完全混合纳什均衡的纳什结果博弈。证明了它是不可约的，并确定了它的余维数和次。

引用次数: 0

Distribution of new statistics of parking functions and their generalizations 停放函数新统计量的分布及其推广

IF 1.3 3区数学 Q3 MATHEMATICS, APPLIED

Advances in Applied Mathematics

Pub Date : 2026-01-07 DOI: 10.1016/j.aam.2025.103026

Stephan Wagner , Catherine H. Yan , Mei Yin

In this paper we present new results on the enumeration of parking functions and labeled forests. We introduce new statistics on parking functions, which are then extended to labeled forests via bijective correspondences. We determine the joint distribution of two statistics on parking functions and their counterparts on labeled forests. Our results on labeled forests also serve to explain the mysterious equidistribution between two seemingly unrelated statistics in parking functions recently identified by Stanley and Yin and give an explicit bijection between the two statistics. Extensions of our techniques are discussed, including joint distribution on further refinement of these new statistics.

本文给出了停车函数和标记森林枚举的新结果。我们引入了停车函数的新统计量，然后通过双目标对应将其扩展到标记森林。我们确定了停车函数的两个统计量的联合分布及其对应的标记森林。我们对标记森林的研究结果也有助于解释Stanley和Yin最近发现的停车函数中两个看似不相关的统计数据之间的神秘均衡分布，并给出两个统计数据之间的显式双射。讨论了我们的技术的扩展，包括联合分布对这些新统计的进一步改进。

引用次数: 0

Toric multivariate Gaussian models from symmetries in a tree 树对称的环面多元高斯模型

IF 1.3 3区数学 Q3 MATHEMATICS, APPLIED

Advances in Applied Mathematics

Pub Date : 2026-01-06 DOI: 10.1016/j.aam.2025.103024

Emma Cardwell , Aida Maraj , Álvaro Ribot

Given a rooted tree

T

on n non-root leaves with colored and zeroed nodes, we construct a linear space

L_{T}

n \times n

symmetric matrices with constraints determined by the combinatorics of the tree. When

L_{T}

represents the covariance matrices of a Gaussian model, it provides natural generalizations of Brownian motion tree (BMT) models in phylogenetics and a step toward a more accurate model for phylogenetic networks with symmetries for species hybridization. When

L_{T}

represents a space of concentration matrices of a Gaussian model, it gives certain colored Gaussian graphical models, which we refer to as BMT derived models. We investigate conditions under which the reciprocal variety

L_{T}^{- 1}

is toric. Relying on the birational isomorphism of the inverse matrix map, we show that if the BMT derived graph of

T

is vertex-regular and a block graph, under the derived Laplacian transformation, which we introduce,

L_{T}^{- 1}

is the vanishing locus of a toric ideal. This ideal is given by the sum of the toric ideal of the Gaussian graphical model on the block graph, the toric ideal of the original BMT model, and binomial linear conditions coming from vertex-regularity. To this end, we provide monomial parametrizations for these toric models realized through paths among leaves in

T

给定一棵有根树T，它有n个非根叶，结点为有色和零，我们构造了一个由n×n对称矩阵组成的线性空间LT，其约束由树的组合决定。当LT表示高斯模型的协方差矩阵时，它提供了系统发育中布朗运动树（BMT）模型的自然推广，并向具有物种杂交对称性的系统发育网络更准确的模型迈出了一步。当LT表示一个高斯模型的浓度矩阵空间时，它给出了一定的彩色高斯图形模型，我们称之为BMT衍生模型。我们研究了倒数变化LT−1是环面的条件。利用逆矩阵映射的二分同构性，我们证明了如果T的BMT导出图是顶点正则图和块图，在我们引入的推导拉普拉斯变换下，LT−1是一个环理想的消失轨迹。该理想由块图上高斯图形模型的环向理想、原BMT模型的环向理想以及由顶点正则性产生的二项式线性条件的和给出。为此，我们为这些通过T中的叶间路径实现的环形模型提供单项参数化。

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引用次数: 0

A central limit theorem on two-sided descents of Mallows distributed elements of finite Coxeter groups 有限Coxeter群的Mallows分布元的双侧下降的中心极限定理

IF 1.3 3区数学 Q3 MATHEMATICS, APPLIED

Advances in Applied Mathematics

Pub Date : 2026-01-05 DOI: 10.1016/j.aam.2025.103025

Maxwell Sun

The Mallows distribution is a non-uniform distribution, first introduced over permutations to study non-ranked data, in which permutations are weighted according to their length. It can be generalized to any Coxeter group, and we study the distribution of

des (w) + des (w^{- 1})

where w is a Mallows distributed element of a finite irreducible Coxeter group. We show that the asymptotic behavior of this statistic is Gaussian. The proof uses a size-bias coupling with Stein's method.

malos分布是一种非均匀分布，首先引入排列来研究非排序数据，其中排列根据其长度进行加权。它可以推广到任何Coxeter群，我们研究了des(w)+des（w−1）的分布，其中w是有限不可约Coxeter群的Mallows分布元。我们证明了这个统计量的渐近性质是高斯的。该证明使用了Stein方法的尺寸偏差耦合。

引用次数: 0

Mixed Berndt-type integrals and generalized Barnes multiple zeta functions 混合berndt型积分与广义Barnes多重zeta函数

IF 1.3 3区数学 Q3 MATHEMATICS, APPLIED

Advances in Applied Mathematics

Pub Date : 2026-01-05 DOI: 10.1016/j.aam.2025.103027

Jianing Zhou

In this paper, we define and study four families of Berndt-type integrals, called mixed Berndt-type integrals, whose integrands contain (hyperbolic) sine and cosine functions. Using contour integration, these integrals are first converted to hyperbolic (infinite) sums of Ramanujan type, all of which can be calculated in closed form by comparing both the Fourier and the Maclaurin series expansions of certain Jacobi elliptic functions. These sums can be expressed as rational polynomials in

Γ (1 / 4)

and

π^{- 1}

which give rise to the closed formulas of the mixed Berndt-type integrals we are interested in. Moreover, we also present some interesting consequences and illustrative examples. Additionally, we define a generalized Barnes multiple zeta function, and obtain an integral representation of it. Furthermore, we give an alternative evaluation of the mixed Berndt-type integrals in terms of the generalized Barnes multiple zeta function. Finally, we obtain some direct evaluations of rational linear combinations of the generalized Barnes multiple zeta function.

本文定义并研究了四类称为混合berndt型积分的berndt型积分族，其积分族包含（双曲）正弦和余弦函数。利用轮廓积分，首先将这些积分转换为拉马努金型的双曲（无限）和，所有这些积分都可以通过比较某些Jacobi椭圆函数的傅里叶级数展开和麦克劳林级数展开以封闭形式计算。这些和可以表示为Γ（1/4）和π−1中的有理多项式，从而得到我们感兴趣的混合伯恩特型积分的封闭公式。此外，我们还提出了一些有趣的结果和说明性的例子。此外，我们定义了广义Barnes多重zeta函数，并得到了它的积分表示。此外，我们给出了用广义Barnes多重zeta函数表示混合berndt型积分的另一种评价方法。最后，我们得到了广义Barnes多重zeta函数的有理线性组合的一些直接评价。

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引用次数: 0

Characterizations of certain matroids by maximizing valuative invariants 用最大值不变量刻画某些拟阵

IF 1.3 3区数学 Q3 MATHEMATICS, APPLIED

Advances in Applied Mathematics

Pub Date : 2026-01-02 DOI: 10.1016/j.aam.2025.103022

Joseph E. Bonin

Luis Ferroni and Alex Fink recently introduced a polytope of all unlabeled matroids of rank r on n elements, and they showed that the vertices of this polytope come from matroids that can be characterized by maximizing a sequence of valuative invariants. We prove that a number of the matroids that they conjectured to yield vertices indeed do (these include cycle matroids of complete graphs, projective geometries, and Dowling geometries), and we give additional examples (including truncations of cycle matroids of complete graphs, Bose-Burton geometries, and binary and free spikes with tips). We prove a special case of a conjecture of Ferroni and Fink by showing that direct sums of uniform matroids yield vertices of their polytope, and we prove a similar result for direct sums whose components are in certain restricted classes of matroids.

Luis Ferroni和Alex Fink最近引入了一个由n个元素上秩为r的所有未标记的拟阵组成的多面体，他们证明了这个多面体的顶点来自于可以通过最大化一系列赋值不变量来表征的拟阵。我们证明了他们推测的一些能产生顶点的拟阵（包括完全图的环拟阵、射影几何和道林几何）确实能产生顶点，并给出了额外的例子（包括完全图的环拟阵的截断、玻色-伯顿几何、带尖的二进制和自由尖峰）。我们证明了Ferroni和Fink猜想的一种特殊情况，证明了一致拟阵的直接和产生其多面体的顶点，并证明了其分量在某些限制类拟阵中的直接和的类似结果。

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全部化学•材料生命科学医学物理工程技术环境•农林材料科学地球科学法学管理学化学环境科学与生态学计算机科学教育学经济学农林科学人文科学生物学数学物理与天体物理心理学综合性期刊其他工业工程理学历史学农学文学信息工程

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