数学代写|表示论代写Representation theory代考|MTH4107

Doug I. Jones

Doug I. Jones

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我们提供的表示论Representation theory及其相关学科的代写,服务范围广, 其中包括但不限于:

  • Statistical Inference 统计推断
  • Statistical Computing 统计计算
  • Advanced Probability Theory 高等概率论
  • Advanced Mathematical Statistics 高等数理统计学
  • (Generalized) Linear Models 广义线性模型
  • Statistical Machine Learning 统计机器学习
  • Longitudinal Data Analysis 纵向数据分析
  • Foundations of Data Science 数据科学基础
数学代写|表示论代写Representation theory代考|MTH4107

数学代写|表示论代写Representation theory代考|Notes and Exercises

The concept of topological persistence came to the fore in early 2000 with the paper by Edelsbrunner, Letscher, and Zomorodian [152] though the concept was proposed in a rudimentary form (e.g., zero-dimensional homology) in other papers by Frosini [162] and Robins [266]. The persistence algorithm as described in this chapter was presented in [152] which has become the cornerstone of topological data analysis. The original algorithm was described without any matrix reduction, which first appeared in [105]. Since then various versions of the algorithm have been presented. We have already seen that persistence for filtrations of simplicial 1-complexes (graphs) with $n$ simplices can be computed in $O(n \alpha(n))$ time. Persistence for filtrations of simplicial 2manifolds also can be computed in an $O(n \alpha(n))$ time algorithm by essentially reducing the problem to computing persistence on a dual graph. In general, for any constant $d \geq 1$, the persistence pairs between $d$ – and $(d-1)$-simplices of a simplicial $d$-manifold can be computed in $O(n \alpha(n))$ time by considering the dual graph. If the manifold has a boundary, then one has to consider a “dummy” vertex that connects to every dual vertex of a $d$-simplex adjoining a boundary $(d-1)$-simplex.

For efficient implementation, clearing and compression strategies as described in Section $3.3 .2$ were presented by Chen and Kerber [94]. We have given a proof based on matrix reduction that the same persistence pairs can be computed by considering the anti-transpose of the boundary matrix. This is termed the cohomology algorithm first introduced in [114]. The name is justified by the fact that, considering cohomology groups and the resulting persistence module that reverses the arrows (Fact 2.14), we obtain the same barcode. The anti-transpose of the boundary matrix indeed represents the coboundary matrix filtered reversely. These tricks are further used by Bauer for processing Rips filtration efficiently in the Ripser software [19]; see also [304]. Boissonnat et al. [41, 42] have suggested a technique to reduce the size of a given filtration using the strong collapse of Barmak and Minian [17]. The collapse on the complex can be efficiently achieved through only simple manipulations of the boundary matrix.

The concept of bottleneck distance for persistence diagrams was first proposed by Cohen-Steiner et al. [101] who also showed the stability of such diagrams in terms of bottleneck distances with respect to the infinity norm of the difference between functions generating them. This result was extended to Wasserstein distance, though in a weaker form in [103] which got improved recently [278]. The more general concept of interleaving distance between persistence modules and the stability of persistence diagrams with respect to them was presented by Chazal et al. [77]. The fact that the bottleneck distance between persistence diagrams is not only bounded from above by the interleaving distance but is indeed equal to it was shown by Lesnick [221] which was further studied by Bauer and Lesnick [23] later. Also, see [54] for more generalization at algebraic level.

The use of the reduced Betti numbers for the lower link of a vertex to quantify its criticality was originally introduced in [150] for a PL-function defined on a triangulation of a $d$-manifold. Our PL-criticality considers both the lower link and upper link for more general simplicial complexes. As far as we know, the relations between such PL-critical points and homology groups of sublevel sets for the PL-setting have not been stated explicitly elsewhere in the literature. The concept of homological critical values was first introduced in [101], and the more general concept of “levelset critical values” (and levelset tame functions) was originally introduced in [63].

The idea of using union-find data structure to compute the zeroth persistent homology group was already introduced in the original persistence algorithm paper [152]. In this chapter, we present a modification for the PL-function setting.

数学代写|表示论代写Representation theory代考|Stability of Towers

Just like in the previous chapter, we define the stability with respect to the perturbation of the towers themselves, forgetting the functions that generate them. This requires a definition of the distance between towers at simplicial (space) levels and homology levels.

It turns out that it is convenient and sometimes appropriate if the objects (spaces, simplicial complexes, or vector spaces) in a tower are indexed over the positive real axis instead of a discrete subset of it. This, in turn, requires one to spell out the connecting map between every pair of objects.

Definition 4.1. (Tower) A tower indexed in an ordered set $A \subseteq \mathbb{R}$ is any collection $\mathrm{T}=\left{T_a\right}_{a \in A}$ of objects $T_a, a \in A$, together with maps $t_{a, a^{\prime}}: T_a \rightarrow$ $T_{a^{\prime}}$ so that $t_{a, a}=\mathrm{id}$ and $t_{a^{\prime}, a^{\prime \prime}} \circ t_{a, a^{\prime}}=t_{a, a^{\prime \prime}}$ for all $a \leq a^{\prime} \leq a^{\prime \prime}$. Sometimes we write $\mathrm{T}=\left{T_a \stackrel{t_{a, a^{\prime}}^{\longrightarrow}}{\longrightarrow} T_{a^{\prime}}\right}_{a \leq a^{\prime}}$ to denote the collection with the maps. We say that the tower $\mathrm{T}$ has resolution $r$ if $a \geq r$ for every $a \in A$.

When $\mathrm{T}$ is a collection of topological spaces connected with continuous maps, we call it a space tower. When it is a collection of simplicial complexes connected with simplicial maps, we call it a simplicial tower, and when it is a collection of vector spaces connected with linear maps, we call it a vector space tower.

Remark 4.1. As we have already seen, in practice, it may happen that a tower needs to be defined over a discrete set or more generally an index set A that is only a subposet of $\mathbb{R}$. In such a case, one can “embed” A into $\mathbb{R}$ and convert the input to a tower according to Definition $4.1$ by assuming that for any $a<$ $a^{\prime} \in A$ with $\left(a, a^{\prime}\right) \notin A$ and for any $a \leq b<b^{\prime}<a^{\prime}, t_{b, b^{\prime}}$ is an isomorphism.
Definition 4.2. (Interleaving of simplicial (space) towers) Let $X=\left{X_a \stackrel{x_{a, a^{\prime}}}{\longrightarrow}\right.$ $\left.X_{a^{\prime}}\right}_{a \leq a^{\prime}}$ and $y=\left{Y_a \stackrel{y_{a, a^{\prime}}}{\longrightarrow} Y_{a^{\prime}}\right}_{a \leq a^{\prime}}$ be two towers of simplicial complexes (resp. spaces) indexed in $\mathbb{R}$.

数学代写|表示论代写Representation theory代考|MTH4107


数学代写|表示论代写Representation theory代考|Notes and Exercises

拓扑持久性的概念在 2000 年初随着 Edelsbrunner、 Letscher 和 Zomorodian [152] 的论文而脱颖而出,尽 管 Frosini [162] 在其他论文中以基本形式 (例如,零维 同源性) 提出了这个概念。和知更鸟 [266]。本章描述 的持久性算法在[152]中提出,它已成为拓扑数据分析的 基石。原始算法的描述没有任何矩阵缩咸,首次出现在 [105] 中。从那时起,出现了该算法的各种版本。我们 已经看到简单 1-复形 (图) 的过滤邿久性 $n$ 单纯形可以 计算在 $O(n \alpha(n))$ 时间。简单 2 流形的过滤持久性也可 以在 $O(n \alpha(n))$ 通过本质上将问题简化为计算对偶图上 的持久性来实现时间算法。一般来说,对于任何常数 $\mathrm{l}$ Igeq $1 d \geq 1$ ,之间的持久性对 $d$ – 和 $(d-1)$-单纯形的 单纯形 $d$-流形可以计算在 $O(n \alpha(n))$ 通过考虑对偶图来 计算时间。如果流形有边界,则必须考虑连接到流形的 每个双顶点的“虚拟”顶点 $d-$ 毗邻边界的单纯形 $(d-1)$ 单纯形。
如第节所述的高效实施、清除和压缩策略3.3.2由 Chen 和 Kerber [94] 提出。我们已经给出了基于矩阵约简的 证明,即可以通过考虑边界矩阵的反转置来计算相同的 持久性对。这被称为在 [114] 中首次引入的上同调算 法。这个名字是合理的,考虑到上同调群和反转箭头的 结果持久性模块 (事实 2.14),我们获得了相同的条形 码。边界矩阵的反转置确实代表了反向过滤的余边界矩 阵。Bauer 进一步使用这些技巧在 Ripser 软件 [19] 中 有效地处理 Rips 过滤; 另见 [304]。Boissonnat 等人。 $[41,42]$ 提出了一种使用 Barmak 和 Minian [17] 的强 坍塌来减小给定过滤尺寸的技术。
Cohen-Steiner 等人首先提出了持久图瓶颈距离的概 念。 [101] 他们还展示了此类图在瓶颈距离方面的稳定 性,相对于生成它们的函数之间差异的无穷范数。这个 结果被扩展到 Wasserstein 距离,尽管在 [103] 中的形 式较弱,最近得到改进 [278]。Chazal 等人提出了持久 性模块之间的交错距离和持久性图相对于它们的稳定性 的更一般概念。[77]。Lesnick [221] 展示了持久性图之 间的瓶颈距离不仅从上方受交错距离限制,而且实际上 等于交错距离,这一事实后来由 Bauer 和 Lesnick [23] 进一步研究。还,
[150] 最初在 [150] 中针对在 $a$ 的三角剖分上定义的 PL 函数引入了对顶点的下部链接使用减少的 Betti 数来量 化其临界性 $d$-歧管。我们的 PL-criticality 考虑了更一般 的单纯复形的下链㢺和上链接。据我们所知,此类 $P L$ 临界点与 PL 设置的子水平集的同源群之间的关系尚末 在其他文献中明确说明。同源临界值的概念首先在 [101] 中引入,更一般的概念”水平集临界值”(和水平集 驯服函数)最初在 [63] 中引入。
在最初的持久性算法论文[152]中已经引入了使用联合查 找数据结构来计算第零个持久性同调群的想法。在本章 中,我们将对 PL 功能设置进行修改。

数学代写|表示论代写Representation theory代考|Stability of Towers

就像在上一章中一样,我们根据塔本身的扰动来定义稳 定性,而忘记了产生它们的函数。这需要在单纯(空 间)水平和同源水平上定义塔之间的距离。
事实证明,如果塔中的对象(空间、单纯复形或向量空 间)在正实轴而不是它的离散子集上进行索引,这很方 便,有时也是合适的。反过来,这需要一个人拼出每对 对象之间的连接图。
定义 4.1。 (塔) 在有序集合A Isubseteq $\backslash m a t h b b{R}$ 中索引的塔 $A \subseteq \mathbb{R}$ 是任何集合
Imathrm ${T}=V$ left{T_alright $}_{-}{a$ lin $A}$
Imathrm{T}=\left{T_a\right } } { a \backslash \text { in A } } \text { 对象的 } T _ { a } , a \in A \text { , }
连同地图 $t_{a, a^{\prime}}: T_a \rightarrow T_{a^{\prime}}$ 以便 $t_{a, a}=\mathrm{id}$ 和
$t_{a^{\prime}, a^{\prime \prime}} \circ t_{a, a^{\prime}}=t_{a, a^{\prime \prime}}$ 对所有人 $a \leq a^{\prime} \leq a^{\prime \prime}$. 有时我们
Imathrm ${T}=| l$ eft $\left{T_{-} a\right.$ Istackrel $\left{t_{-}{a, a \wedge{1 p r i m e}} \wedge{\right.$ Vlongrigh
用地图表示集合。我们说塔Imathrm{T}T分辨率为 $r r$ 如 果 $a \geq r$ 每一个 $a \in A$.
什么时候T是用连续映射连接的拓扑空间的集合,我们 称它为空间塔。当它是单纯形复形的集合与单纯映射相 连时,我们称它为单纯塔,当它是向量空间与线性映射 相连的集合时,我们称它为向量空间塔。
备注 4.1。正如我们已经看到的,在实践中,可能需要 在离散集或更一般的索引集 $\mathrm{A}$ 上定义一个塔,它只是 $\mathbb{R}$. 换为塔4.1通过假设对于任何 $a<a^{\prime} \in A$ 和 $\left(a, a^{\prime}\right) \notin A$ 对于任何 $a \leq b<b^{\prime}<a^{\prime}, t_{b, b^{\prime}}$ 是一个同 构。
定义 4.2。 (简单 (空间) 塔的交错) 让 $X=\backslash l$ eft $\left{X\right.$ a Istackrel $\left{X_{-}{a, a \wedge{\right.$ a prime $\left.}}\right} \backslash$ Vongrightarrow $} \backslash$ rig

$\mathrm{y}=$ Veft $\left{Y_{-} a\right.$ Istackrel $\left{y_{-}\left{a, a^{\wedge}{{\right.\right.$ prime $\left.}}\right}{$ Vlongrightarrow $} Y_{-}$
是索引在 $\mathbb{R}$.

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术语 广义线性模型(GLM)通常是指给定连续和/或分类预测因素的连续响应变量的常规线性回归模型。它包括多元线性回归,以及方差分析和方差分析(仅含固定效应)。



有限元是一种通用的数值方法,用于解决两个或三个空间变量的偏微分方程(即一些边界值问题)。为了解决一个问题,有限元将一个大系统细分为更小、更简单的部分,称为有限元。这是通过在空间维度上的特定空间离散化来实现的,它是通过构建对象的网格来实现的:用于求解的数值域,它有有限数量的点。边界值问题的有限元方法表述最终导致一个代数方程组。该方法在域上对未知函数进行逼近。[1] 然后将模拟这些有限元的简单方程组合成一个更大的方程系统,以模拟整个问题。然后,有限元通过变化微积分使相关的误差函数最小化来逼近一个解决方案。





随机过程,是依赖于参数的一组随机变量的全体,参数通常是时间。 随机变量是随机现象的数量表现,其时间序列是一组按照时间发生先后顺序进行排列的数据点序列。通常一组时间序列的时间间隔为一恒定值(如1秒,5分钟,12小时,7天,1年),因此时间序列可以作为离散时间数据进行分析处理。研究时间序列数据的意义在于现实中,往往需要研究某个事物其随时间发展变化的规律。这就需要通过研究该事物过去发展的历史记录,以得到其自身发展的规律。


多元回归分析渐进(Multiple Regression Analysis Asymptotics)属于计量经济学领域,主要是一种数学上的统计分析方法,可以分析复杂情况下各影响因素的数学关系,在自然科学、社会和经济学等多个领域内应用广泛。


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