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• 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
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## 数学代写|拓扑学代写Topology代考|Notes and Exercises

Simplicial complexes are a fundamental structure in algebraic topology. A good source for the subject is Munkres [242].

The concept of nerve is credited to Aleksandroff [7]. The Nerve Theorem has different versions. It holds for open covers for topological spaces with some mild conditions [300]. Borsuk proved it for closed covers again with some conditions on the space and covers [45]. The assumptions of both are satisfied by metric spaces and finite covers with which we state the theorem in Section 2.2. A version of the theorem is also credited to Leray [220].

Čech and Vietoris-Rips complexes have turned out to be very effective data structures in topological data analysis. Cech complexes were introduced to define Čech homology. Leonid Vietoris [293] introduced the Vietoris complex to extend the homology theory from simplicial complexes to metric spaces. Later, Eliyahu Rips used it in hyperbolic group theory [176]. Jean-Claude Hausmann named it as Vietoris-Rips complex and showed that it is homotopy equivalent to a compact Riemannian manifold when the vertex set spans all points of the manifold and the parameter to build it is sufficiently small [187].

## 数学代写|拓扑学代写Topology代考|Complexes and Homology Groups

This result was further improved by Latschev [218] who showed that the homotopy equivalence holds even when the vertex set is finite.

Delaunay complexes are a very well known and useful data structure for various geometric applications in two and three dimensions. They enjoy various optimal properties. For example, for a given point set $P \subset \mathbb{R}^{2}$, among all simplicial complexes linearly embedded in $\mathbb{R}^{2}$ with vertex set $P$, the Delaunay complex maximizes the minimum angle over all triangles as stated in Fact 2.5. Many such properties and algorithms for computing Delaunay complexes are described in the books by Edelsbrunner [148] and Cheng et al. [96]. The alpha complex was proposed in [151] and further developed in [153]. The first author of this book can attest to the historic fact that the development of the persistence algorithm was motivated by the study of alpha complexes and their Betti numbers. The book by Edelsbrunner and Harer [149] confirms this. Witness complexes were proposed by de Silva and Carlsson [113] in an attempt to build a sparser complex out of a dense point sample. The graph induced complex is also another such construction proposed in [123].

Homology groups and their associated concepts are main algebraic tools used in topological data analysis. Many associated structures and results about them exist in algebraic topology. We only cover the main necessary concepts that are used in this book and leave others. Interested readers can familiarize themselves with these omitted topics by reading Munkres [242], Hatcher [186], or Ghrist [170], among many other excellent sources.

## 数学代写|拓扑学代写Topology代考|Topological Persistence

Suppose we have point cloud data $P$ sampled from a 3D model. A quantified summary of the topological features of the model that can be computed from this sampled representation helps in further processing such as shape analysis in geometric modeling. Persistent homology offers this avenue, as Figure $3.1$ illustrates. For further explanation, consider $P$ sampled from a curve in $\mathbb{R}^{2}$ as in Figure 3.3 later. Our goal is to get the information that the sampled space had two loops, one bigger and more prominent than the other. The notion of persistence captures this information. Consider the distance function $r: \mathbb{R}^{2} \rightarrow \mathbb{R}$ defined over $\mathbb{R}^{2}$ where $r(x)$ equals $\mathrm{d}(x, P)$, that is, the minimum distance of $x$ to the points in $P$. Now let us look at the sublevel sets of $r$, that is, $r^{-1}[-\infty, a]$ for some $a \in \mathbb{R}^{+} \cup{0}$. These sublevel sets are the union of closed balls of radius $a$ centering the points. We can observe from Figure $3.3$ that if we increase $a$ starting from zero, we come across different holes surrounded by the union of these balls which ultimately get filled up at different times. However, the two holes corresponding to the original two loops persist longer than the others. We can abstract out this observation by looking at how long a feature (homological class) survives when we scan over the increasing sublevel sets. This weeds out the “false” features (noise) from the true ones. The notion of persistent homology formalizes and discretizes this idea: It takes a function defined on a topological space (simplicial complex) and quantifies the changes in homology classes as the sublevel sets (subcomplexes) grow with increasing value of the funetion.

There are two predominant scenarios where persistence appears, though in slightly different contexts. One is when the function is defined on a topological space which requires considering singular homology groups of the sublevel sets. The other is when the function is defined on a simplicial complex and the sequence of sublevel sets is implicitly given by a nested sequence of subcomplexes called a filtration. This involves simplicial homology.

## 数学代写|拓扑学代写Topology代考|Complexes and Homology Groups

Latschev [218] 进一步改进了这一结果，他表明即使顶点集是有限的，同伦等价也是成立的。
Delaunay 复形是一种众所周知且有用的数据结构，适用于二维和三维的各种几何应用。他们享有各种最佳性能。例如，对于给定的点集 $P \subset \mathbb{R}^{2}$ ，在所有线性嵌入的单纯复形中 $\mathbb{R}^{2}$ 有顶点集 $P$ ，如事实 $2.5$ 所述，Delaunay 复形使所有三角形的最小角度最大化。Edelsbrunner [148] 和 Cheng 等人的书中描述了许多用于计算 Delaunay 复形的此类属性和算法。[96]。 $\alpha$ 复合物在 [151] 中提出并在 [153] 中进一步发 展。本书的第一作者可以证明一个历史事实，即持久性算法的发展是由对 alpha 复形及其 Betti 数的研究推动的。Edelsbrunner 和 Harer [149] 的书证实了这一点。de Silva 和 Carlsson [113] 提出了见证复合体，试图从密集的点样本中构建一个更稀疏的复合体。图诱导复合体 也是 [123] 中提出的另一种此类构造。

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## MATLAB代写

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