Advances in Applied Mathematics最新文献

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Refining a chain theorem from matroids to internally 4-connected graphs 完善从矩阵到内部 4 连接图的链式定理

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Advances in Applied Mathematics

Pub Date : 2024-11-06 DOI: 10.1016/j.aam.2024.102802

Chanun Lewchalermvongs , Guoli Ding

Graph theory and matroid theory are interconnected with matroids providing a way to generalize and analyze the structural and independence properties within graphs. Chain theorems, vital tools in both matroid and graph theory, enable the analysis of matroid structures associated with graphs. In a significant contribution, Chun, Mayhew, and Oxley [2] established a chain theorem for internally 4-connected binary matroids, clarifying the operations involved. Our research builds upon this by specifying the matroid result to internally 4-connected graphs. The primary goal of our research is to refine this chain theorem for matroids into a chain theorem for internally 4-connected graphs, making it more accessible to individuals less acquainted with matroid theory.

图论和矩阵理论相互关联，矩阵为概括和分析图的结构和独立性提供了一种方法。链定理是矩阵和图论中的重要工具，可以分析与图相关的矩阵结构。Chun、Mayhew 和 Oxley [2] 的一项重大贡献是建立了内部 4 连接二元矩阵的链定理，阐明了其中的操作。我们的研究在此基础上将矩阵结果具体化为内部 4 连接的图。我们研究的主要目标是将这个矩阵的链式定理完善为内部 4 连接图的链式定理，使对矩阵理论不太熟悉的人更容易理解。

引用次数: 0

On the enumeration of series-parallel matroids 关于串并联矩阵的枚举

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Advances in Applied Mathematics

Pub Date : 2024-10-30 DOI: 10.1016/j.aam.2024.102801

Nicholas Proudfoot , Yuan Xu , Benjamin Young

By the work of Ferroni and Larson, Kazhdan–Lusztig polynomials and Z-polynomials of complete graphs have combinatorial interpretations in terms of quasi series-parallel matroids. We provide explicit formulas for the number of series-parallel matroids and the number of simple series-parallel matroids of a given rank and cardinality, extending results of Ferroni–Larson and Gao–Proudfoot–Yang–Zhang.

根据费罗尼和拉尔森的研究成果，完整图的卡兹丹-卢兹提格多项式和 Z 多项式可以用准数列平行矩阵来组合解释。我们提供了给定秩和心数的数列平行矩阵数和简单数列平行矩阵数的明确公式，扩展了费罗尼-拉森和高-普鲁福-杨-张的结果。

引用次数: 0

Identifiability of homoscedastic linear structural equation models using algebraic matroids 利用代数矩阵识别同源线性结构方程模型

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Advances in Applied Mathematics

Pub Date : 2024-10-15 DOI: 10.1016/j.aam.2024.102794

Mathias Drton, Benjamin Hollering, Jun Wu

We consider structural equation models (SEMs), in which every variable is a function of a subset of the other variables and a stochastic error. Each such SEM is naturally associated with a directed graph describing the relationships between variables. When the errors are homoscedastic, recent work has proposed methods for inferring the graph from observational data under the assumption that the graph is acyclic (i.e., the SEM is recursive). In this work, we study the setting of homoscedastic errors but allow the graph to be cyclic (i.e., the SEM to be non-recursive). Using an algebraic approach that compares matroids derived from the parameterizations of the models, we derive sufficient conditions for when two simple directed graphs generate different distributions generically. Based on these conditions, we exhibit subclasses of graphs that allow for directed cycles, yet are generically identifiable. We also conjecture a strengthening of our graphical criterion which can be used to distinguish many more non-complete graphs.

我们考虑结构方程模型（SEM），其中每个变量都是其他变量子集和随机误差的函数。每个这样的 SEM 自然都与描述变量间关系的有向图相关联。当误差为同方误差时，最近的研究提出了从观测数据推断图的方法，前提是图是非循环的（即 SEM 是递归的）。在这项工作中，我们研究了同方差误差的设置，但允许图是循环的（即 SEM 是非递归的）。我们使用代数方法比较从模型参数化得到的矩阵，推导出两个简单有向图产生不同分布的充分条件。基于这些条件，我们展示了允许有向循环但一般可识别的图的子类。我们还猜想我们的图形标准会得到加强，可以用来区分更多的非完整图形。

引用次数: 0

Minimal skew semistandard tableaux and the Hillman–Grassl correspondence 最小倾斜半标准表和希尔曼-格拉斯尔对应关系

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Advances in Applied Mathematics

Pub Date : 2024-10-14 DOI: 10.1016/j.aam.2024.102792

Alejandro H. Morales , Greta Panova , GaYee Park

Standard tableaux of skew shape are fundamental objects in enumerative and algebraic combinatorics and no product formula for the number is known. In 2014, Naruse gave a formula (NHLF) as a positive sum over excited diagrams of products of hook-lengths. Subsequently, Morales, Pak, and Panova gave a q-analogue of this formula in terms of skew semistandard tableaux (SSYT). They also showed, partly algebraically, that the Hillman–Grassl bijection, restricted to skew semistandard tableaux, is behind their q-analogue. We study the problem of circumventing the algebraic part and proving the bijection completely combinatorially, which we do for the case of border strips. For general skew shapes, we define minimal semistandard Young tableaux, that are in correspondence with excited diagrams via a new description of the Hillman–Grassl bijection and have an analogue of excited moves. Lastly, we relate the minimal skew SSYT with the terms of the Okounkov-Olshanski formula (OOF) for counting standard tableaux of skew shape. Our construction immediately implies that the summands in the NHLF are less than the summands in the OOF and we characterize the shapes where both formulas have the same number of summands.

歪斜形状的标准表图是枚举和代数组合学中的基本对象，目前还不知道其数量的乘积公式。2014 年，Naruse 给出了一个公式（NHLF），它是钩长乘积的激发图的正和。随后，莫拉莱斯、帕克和帕诺娃用偏斜半标准表（SSYT）给出了该公式的 q 类似形式。他们还部分地从代数学角度证明，希尔曼-格拉斯尔双射公式（Hillman-Grassl bijection）局限于偏斜半标准表式，是他们的 q-analogue 背后的原因。我们研究的问题是绕过代数部分，完全以组合的方式证明偏射，我们针对边条的情况做到了这一点。对于一般偏斜图形，我们定义了最小半标准杨表，通过对希尔曼-格拉斯尔偏射的新描述，使其与激发图相对应，并具有激发移动的相似性。最后，我们将最小偏斜扬格图与奥孔科夫-奥尔尚斯基公式（OOF）中用于计算偏斜形状标准台形的项联系起来。我们的构造立即意味着 NHLF 中的求和项少于 OOF 中的求和项，我们还描述了两个公式具有相同求和项数的形状。

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引用次数: 0

Proof of a conjecture about Parrondo's paradox for two-armed slot machines 双臂老虎机帕隆多悖论猜想的证明

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Advances in Applied Mathematics

Pub Date : 2024-10-03 DOI: 10.1016/j.aam.2024.102793

Huaijin Liang , Zengjing Chen

The 1936 Mills Futurity slot machine had the feature that, if a player loses 10 times in a row, the 10 lost coins are returned. Ethier and Lee (2010) studied a generalized version of this machine, with 10 replaced by deterministic parameter J. They established the Parrondo effect for a hypothetical two-armed machine with the Futurity award. Specifically, arm A and arm B, played individually, are asymptotically fair, but when alternated randomly (the so-called random mixture strategy), the casino makes money in the long run. They also considered the nonrandom periodic pattern strategy for patterns with r As and s Bs (e.g.,

A B A B B

r = 2

and

s = 3

). They established the Parrondo effect if

r + s

divides J, and conjectured it in four other situations, including the case

J = 2

with

r \geq 1

and

s \geq 1

. We prove the conjecture in the latter case.

1936 年的米尔斯 "未来奖 "老虎机有一个特点，即如果玩家连续输掉 10 次，输掉的 10 枚硬币将被返还。Ethier 和 Lee（2010 年）研究了这种老虎机的通用版本，用确定性参数 J 代替了 10。具体来说，单独玩的 A 臂和 B 臂在近似上是公平的，但如果随机交替使用（即所谓的随机混合策略），赌场就会长期赚钱。他们还考虑了具有 r As 和 s Bs 的非随机周期模式策略（例如，如果 r=2 和 s=3，则为 ABABB）。他们确定了 r+s 除以 J 时的帕隆多效应，并猜想了其他四种情况，包括 J=2 且 r≥1 和 s≥1 的情况。我们证明了后一种情况下的猜想。

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引用次数: 0

A topological approach to mapping space signatures 映射空间特征的拓扑方法

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Advances in Applied Mathematics

Pub Date : 2024-09-27 DOI: 10.1016/j.aam.2024.102787

Chad Giusti , Darrick Lee , Vidit Nanda , Harald Oberhauser

A common approach for describing classes of functions and probability measures on a topological space

X

is to construct a suitable map Φ from

X

into a vector space, where linear methods can be applied to address both problems. The case where

X

is a space of paths

[0, 1] \to R^{n}

and Φ is the path signature map has received much attention in stochastic analysis and related fields. In this article we develop a generalized Φ for the case where

X

is a space of maps

{[0, 1]}^{d} \to R^{n}

for any

d \in N

, and show that the map Φ generalizes many of the desirable algebraic and analytic properties of the path signature to

d \geq 2

. The key ingredient to our approach is topological; in particular, our starting point is a generalization of K-T Chen's path space cochain construction to the setting of cubical mapping spaces.

描述拓扑空间 X 上的函数类和概率度量的常用方法是构建一个合适的映射 Φ，从 X 映射到一个向量空间，其中线性方法可用于解决这两个问题。X 是路径空间 [0,1]→Rn，Φ 是路径签名图，这种情况在随机分析和相关领域受到广泛关注。在本文中，我们针对 X 是任意 d∈N 的映射空间 [0,1]d→Rn 的情况，开发了广义的 Φ，并证明该映射 Φ 将路径签名的许多理想代数和分析性质推广到了 d≥2。我们的方法的关键要素是拓扑；特别是，我们的出发点是将陈康泰的路径空间共链构造推广到立方映射空间的设置中。

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引用次数: 0

Permanent identities, combinatorial sequences, and permutation statistics 永久同一性、组合序列和置换统计

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Advances in Applied Mathematics

Pub Date : 2024-09-25 DOI: 10.1016/j.aam.2024.102789

Shishuo Fu , Zhicong Lin , Zhi-Wei Sun

<div><div>In this paper, we confirm six conjectures on the exact values of some permanents, relating them to the Genocchi numbers of the first and second kinds as well as the Euler numbers. For example, we prove that<span><span><span><math><mrow><mi>per</mi></mrow><msub><mrow><mo>[</mo><mrow><mo>⌊</mo><mfrac><mrow><mn>2</mn><mi>j</mi><mo>−</mo><mi>k</mi></mrow><mrow><mi>n</mi></mrow></mfrac><mo>⌋</mo></mrow><mo>]</mo></mrow><mrow><mn>1</mn><mo>≤</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo>≤</mo><mi>n</mi></mrow></msub><mo>=</mo><mn>2</mn><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>1</mn><mo>)</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>,</mo></math></span></span></span> where <span><math><msub><mrow><mi>B</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>B</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo></math></span> are the Bernoulli numbers. We also show that<span><span><span><math><mrow><mi>per</mi></mrow><msub><mrow><mo>[</mo><mrow><mi>sgn</mi></mrow><mrow><mo>(</mo><mi>cos</mi><mo>⁡</mo><mi>π</mi><mfrac><mrow><mi>i</mi><mo>+</mo><mi>j</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>)</mo></mrow><mo>]</mo></mrow><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>≤</mo><mi>n</mi></mrow></msub><mspace></mspace><mspace></mspace><mo>=</mo><mrow><mo>{</mo><mtable><mtr><mtd><mo>−</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>m</mi></mrow></msubsup><mrow><mo>(</mo><mtable><mtr><mtd><mi>m</mi></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></mrow><msub><mrow><mi>E</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mtd><mtd><mspace></mspace><mrow><mtext>if </mtext><mi>n</mi><mo>=</mo><mn>2</mn><mi>m</mi><mo>+</mo><mn>1</mn></mrow><mo>,</mo></mtd></mtr><mtr><mtd><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>m</mi></mrow></msubsup><mrow><mo>(</mo><mtable><mtr><mtd><mi>m</mi></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></mrow><msub><mrow><mi>E</mi></mrow><mrow><mn>2</mn><mi>k</mi></mrow></msub></mtd><mtd><mspace></mspace><mrow><mtext>if </mtext><mi>n</mi><mo>=</mo><mn>2</mn><mi>m</mi></mrow><mo>,</mo></mtd></mtr></mtable></mrow></math></span></span></span> where <span><math><mrow><mi>sgn</mi></mrow><mo>(</mo><mi>x</mi><mo>)</mo></math></span> is the sign function, and <span><math><msub><mrow><mi>E</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>E</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>E</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo></math></span> are the Euler (zigzag) numbers.</div><div>In the course of linking the evaluation of these permanents to the aforementioned combinatorial sequences, the classical permutation statistic – the exceda

在本文中，我们证实了关于一些永恒项精确值的六个猜想，这些猜想与第一种和第二种基诺奇数以及欧拉数有关。例如，我们证明了 per[⌊2j-kn⌋]1≤j,k≤n=2(2n+1-1)Bn+1，其中 B0,B1,B2,... 是伯努利数。我们还证明，per[sgn(cosπi+jn+1)]1≤i,j≤n={-∑k=0m(mk)E2k+1（如果 n=2m+1），∑k=0m(mk)E2k（如果 n=2m），其中 sgn(x) 是符号函数，E0,E1,E2,... 是欧拉（之字）数。在将这些永久数的评估与上述组合序列联系起来的过程中，经典的置换统计量--切除数，以及它的几种变体，起着核心作用。我们的方法以递推关系、双射以及矩阵的某些基本运算为特色，这些运算保留了矩阵的永久性。此外，我们对第二个恒等式的证明导致了对巴拉猜想的续分公式的证明，以及对 2-Eulerian 多项式的 γ 系数的意想不到的置换解释。

引用次数: 0

Continued fractions for q-deformed real numbers, {−1,0,1}-Hankel determinants, and Somos-Gale-Robinson sequences q 个变形实数的连续分数、{-1,0,1}-汉克尔行列式和索莫斯-盖尔-罗宾逊序列

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Advances in Applied Mathematics

Pub Date : 2024-09-24 DOI: 10.1016/j.aam.2024.102788

Valentin Ovsienko , Emmanuel Pedon

q-deformed real numbers are power series with integer coefficients. We study Stieltjes and Jacobi type continued fraction expansions of q-deformed real numbers and find many new examples of such continued fractions. We also investigate the corresponding sequences of Hankel determinants and find an infinite family of power series for which several of the first sequences of Hankel determinants consist of

- 1, 0

and 1 only. These Hankel sequences satisfy Somos and Gale-Robinson recurrences.

q 变形实数是具有整数系数的幂级数。我们研究了 q 变形实数的 Stieltjes 和 Jacobi 型续分数展开式，发现了许多此类续分数的新实例。我们还研究了相应的汉克尔行列式序列，并发现了一个无穷的幂级数族，其中几个汉克尔行列式的第一序列仅由-1、0 和 1 组成。这些汉克尔序列满足索莫斯和盖尔-罗宾逊递推规律。

引用次数: 0

Summing the “exactly one 42” and similar subsums of the harmonic series 求谐波数列的 "恰好一个 42 "和类似子和

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Advances in Applied Mathematics

Pub Date : 2024-09-20 DOI: 10.1016/j.aam.2024.102791

Jean-François Burnol

For $b > 1$ and αβ a string of two digits in base b, let $K_{1}$ be the subsum of the harmonic series with only those integers having exactly one occurrence of αβ. We obtain a theoretical representation of such $K_{1}$ series which, say for $b = 10$ , allows computing them all to thousands of digits. This is based on certain specific measures on the unit interval and the use of their Stieltjes transforms at negative integers. Integral identities of a combinatorial nature both explain the relation to the $K_{1}$ sums and lead to recurrence formulas for the measure moments allowing in the end the straightforward numerical implementation.

对于 b>1，αβ 是一个以 b 为底数的两位数字符串，让 K1 成为谐数列的子集，其中只包含那些αβ 恰好出现一次的整数。我们可以从理论上表示这样的 K1 数列，比如对于 b=10 的数列，可以将它们计算到数千位。这是基于单位区间上的某些特定度量，以及在负整数处使用它们的斯蒂尔杰斯变换。组合性质的积分等式既解释了与 K1 和的关系，又引出了度量矩的递推公式，最终可以直接用数字实现。

引用次数: 0

Betti numbers and torsions in homology groups of double coverings 双覆盖同调群中的贝蒂数和扭转

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Advances in Applied Mathematics

Pub Date : 2024-09-19 DOI: 10.1016/j.aam.2024.102790

Suguru Ishibashi , Sakumi Sugawara , Masahiko Yoshinaga

Papadima and Suciu proved an inequality between the ranks of the cohomology groups of the Aomoto complex with finite field coefficients and the twisted cohomology groups, and conjectured that they are actually equal for certain cases associated with the Milnor fiber of the arrangement. Recently, an arrangement (the icosidodecahedral arrangement) with the following two peculiar properties was found: (i) the strict version of Papadima-Suciu's inequality holds, and (ii) the first integral homology of the Milnor fiber has a non-trivial 2-torsion. In this paper, we investigate the relationship between these two properties for double covering spaces. We prove that (i) and (ii) are actually equivalent.

帕帕季马和苏修证明了具有有限场系数的青本复数同调群的秩与扭曲同调群之间的不等式，并猜想在与排列的米尔诺纤维相关的某些情况下，它们实际上是相等的。最近，我们发现了一种具有以下两个奇特性质的排列（icosidodecahedral arrangement）：(i) Papadima-Suciu 不等式的严格版本成立；(ii) Milnor 纤维的第一积分同调具有非三维 2 扭。在本文中，我们研究了双覆盖空间这两个性质之间的关系。我们证明(i)和(ii)实际上是等价的。

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