Ken Dykema, Catherine Yan
We prove a recursion formula for generating functions of certain renormalizations of *-moments of the DT(δ_0,1)-operator T, involving an operation \odot on formal power series and a transformation E that converts \odot to usual multiplication. This recursion formula is used to prove that all of these generating functions are rational functions, and to find a few of them explicitly.
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We study a three-player variation of the impartial avoidance game introduced by Anderson and Harary. Three players take turns selecting previously-unselected elements of a finite group. The losing player is the one who selects an element that causes the set of jointly-selected elements to be a generating set for the group, with the previous player winning and the remaining player coming in second place. We describe the winning strategy for these games on cyclic, dihedral, and nilpotent groups.
Tanisaki introduced generating sets for the defining ideals of the schematic intersections of the closure of conjugacy classes of nilpotent matrices with the set of diagonal matrices. These ideals are naturally labeled by integer partitions. Given such a partition $λ$, we define several methods to produce a reduced generating set for the associated ideal $I_λ$. For particular shapes we find nice generating sets. By comparing our sets with some generating sets of $I_λ$ arising from a work of Weyman, we find a counterexample to a related conjecture of Weyman.
Impartial achievement games for generating nilpotent groups
研究Anderson和Harary提出的公平博弈，确定形如$T \ imes H$（$T$为$2$-群，$H$为奇数阶群）的有限群的nim数，涵盖所有幂零群和阿贝尔群。
We study representation zeta functions of finitely generated, torsion-free nilpotent groups which are rational points of unipotent group schemes over rings of integers of number fields. Using the Kirillov orbit method and p-adic integration, we prove rationality and functional equations for almost all local factors of the Euler products of these zeta functions. We further give explicit formulae, in terms of Dedekind zeta functions, for the zeta functions of class-2-nilpotent groups obtained from three infinite families of group schemes, generalising the integral Heisenberg group. As an immedia…
In [13] and [14], we found so-called the reccurence diagram. This paper is the application of such recurrence diagram. As a first application for the reccurence diagram, we will re-compute the well-known moment series of the generating opeator of the free group factor L(F_N). Also, we will compute the trivial L(F_1)-valued moment series of the generating operator of L(F_2). This result can be extended for the general case when we have L(F_N), instead of L(F_2). To study operator-valued moment series is to study the operator-valued distribution. By doing that, we can get some algebraic and comb…
We give a generating function for the number of pairs of $n\times n$ matrices $(A, B)$ over a finite field that are mutually annihilating, namely, $AB=BA=0$. This generating function can be viewed as a singular analogue of a series considered by Cohen and Lenstra. We show that this generating function has a factorization that allows it to be meromorphically extended to the entire complex plane. We also use it to count pairs of mutually annihilating nilpotent matrices. This work is essentially a study of the motivic aspects about the variety of modules over $\mathbb C[u,v]/(uv)$ as well as the …
Feit and Fine derived a generating function for the number of ordered pairs of commuting n by n matrices over the finite field F_q. This has been reproved and studied by Bryan and Morrison from the viewpoint of motivic Donaldson-Thomas theory. In this note we give a new proof of the Feit-Fine result, and generalize it to the Lie algebra of finite unitary groups and to the Lie algebra of odd characteristic finite symplectic groups. We extract some asymptotic information from these generating functions. Finally, we derive generating functions for the number of commuting nilpotent elements for th…
In this paper, we investigate certain combinatorial numbers, the \textit{moment generating Stirling numbers}. They are a special case of Hsu's generalized Stirling numbers and satisfy many more properties and combinatorial identities than are known in the general case. As application, we provide the computation of the moments and central moments of the phase type distribution, the recurrence time in Markov chains, the geometric distribution, the negative binomial distribution and of a class of distributions generalizing the negative binomial distribution. All computations can be performed in c…
Moment Generating Stirling Numbers of the first kind and Applications
本文引入第一类矩生成Stirling数MSN1，作为MSN2的逆元使统计分布的矩可用闭式公式表示，并证明其与r-Stirling数的关系及在任意实数参数下的转换性质。
We define two-parameter generalizations of two combinatorial constructions of Andrews: the kth symmetrized rank moment and the k-marked Durfee symbol. We prove that three specializations of the associated generating functions are so-called quasimock theta functions, while a fourth specialization gives quasimodular forms. We then define a two-parameter generalization of Andrews' smallest parts function and note that this leads to quasimock theta functions as well. The automorphic properties are deduced using q-series identities relating the relevant generating functions to known mock theta func…
Electronic transport through chaotic quantum dots exhibits universal behaviour which can be understood through the semiclassical approximation. Within the approximation, transport moments reduce to codifying classical correlations between scattering trajectories. These can be represented as ribbon graphs and we develop an algorithmic combinatorial method to generate all such graphs with a given genus. This provides an expansion of the linear transport moments for systems both with and without time reversal symmetry. The computational implementation is then able to progress several orders highe…
Quantification of the Fourth Moment Theorem for Cyclotomic Generating Functions
本文针对概率生成函数根为实数或位于单位圆上的随机变量序列，利用Stein-Tikhomirov方法统一建立了定量正态逼近界，其中分圆情形需额外引入第四累积量或矩。
Canonical forms and moment-generating functions of plane polypols
本文研究了与由有理代数弧边界限定的平面域相关的两个密切相关对象：正几何意义下的规范形式和归一化矩生成函数（或Fantappie变换）。对于多边形，这两个对象通过极性相关联：多边形的归一化Fantappie变换是极化多边形的规范形式。对于真正的弯曲多边形，这种双几何机制仍然存在，但变换不再是理性的对数规范形式；而是成为一种全息的、通常分支的周期，其奇点由顶点超平面和非线性边界组件的射影对偶曲线控制。本文给出了具体的例子，包括扇形和半圆，并解释了谐波矩生成函数如何作为相同Fantappie变换的一维限制出现。
On generating functions of Hausdorff moment sequences
本文刻画了[0,1]上有限正测度矩（完全单调序列）生成函数为(-∞,1)上解析且正的Pick函数，推导其推论，刻画凸/凹概率分布矩的生成函数，并证明特定条件下Fuss-Catalan等数为概率分布矩。
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