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• Statistical Inference 统计推断
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• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 电子工程代写|信号处理与线性系统作业代写Signal Processing and Linear Systems代考|electrical networks

For the reader’s convenience, we briefly recall several principal facts and definitions from the theory of weighted networks.

Let $G=(V, E)$ denote a connected undirected locally finite graph with single edges between vertices such that the vertex set $V$ is (countably) infinite, and the edge set $E$ has no loops. The set $E(x):={y \in V: y \sim x}$ of all neighbors of $x$ is finite for any vertex $x$. The edge $e \in E$ connecting vertices $x$ and $y$ is denoted by $(x y)$. The connectedness of $G$ means that, for any two vertices $x, y \in V$, there exists a finite path $\gamma=\left(x_{0}, x_{1}, \ldots, x_{n}\right)$ such that $x_{0}=x, x_{n}=y$ and $\left(x_{i} x_{i+1}\right) \in E$ for all $i$.

Definition 2.1. An weighted network $(G, c)$ is a weighted graph $G$ with a symmetric conductance function $c: V \times V \rightarrow[0, \infty)$, i.e., $c_{x y}=c_{y x}$ for any $(x y) \in E$. Moreover, $c_{x y}>0$ if and only if $(x y) \in E$. For any $x \in V$, the total conductance at $x$ is defined as
$$c(x):=\sum_{y \sim x} c_{x y} .$$
Given a weighted network $(G, c)=(V, E, c)$ with fixed conductance function $c$, we associate the following three Hilbert spaces of functions on $V$ :
\begin{aligned} l^{2}(V):=\left{u: V \in \mathbb{R}:|u|_{l^{2}}^{2}=\sum_{x \in V} u(x)^{2}<\infty\right} \ l^{2}(V, c) &=\left{u: V \in \mathbb{R}:|u|_{l^{2}(V, c)}^{2}=\sum_{x \in V} c(x) u(x)^{2}<\infty\right}, \end{aligned}
and $\mathcal{H}{E}:=$ equivalence classes of functions on $V$ such that $$|u|{\mathcal{H}{E}}^{2}=\frac{1}{2} \sum{(x y) \in E} c_{x y}(u(x)-u(y))^{2}<\infty,$$
where $u_{1}$ and $u_{2}$ are equivalent if $u_{1}-u_{2}=$ constant. The Hilbert space $\mathcal{H}_{E}$ is called the finite energy space

We note that in this paper we focus on real-valued functions. The case of complex-valued functions is considered with obvious changes.

## 电子工程代写|信号处理与线性系统作业代写Signal Processing and Linear Systems代考|From discrete to measurable setting

We recall that, for every network $(V, E, c)$, an atomic measure space $(V, m)$ is given where $m$ is the counting measure. The conductance function $c$ defines another atomic measure $\rho$ on $E \subset V \times V$ by setting $\rho(x, y)=c_{x y}$. In what follows, we define, in terms of measure spaces, similar objects which can be regarded as analogues to the basic notions for weighted networks.

Measure space. Let $V$ be a separable completely metrizable topological space (a Polish space, for short), and let $\mathcal{B}$ be the $\sigma$-algebra of Borel subsets of $V$. Then $(V, \mathcal{B})$ is called a standard Borel space. We recall that all uncountable standard Borel spaces are Borel isomorphic, so that one can use any convenient realization of the space $V$. If $\mu$ is a continuous (i.e., non-atomic) Borel measure on $(V, \mathcal{B})$, then $(V, \mathcal{B}, \mu)$ is called a standard measure space. Wé use this name for both finite and $\sigma$-finite measure spaces. Also the same notation, $\mathcal{B}$ is applied for the $\sigma$-algebras of Borel sets and measurable sets of a standard measure space. In the context of measure spaces, we always assume that $\mathcal{B}$ is complete with respect to the measure $\mu$. By $\mathcal{F}(V, \mathcal{B})$ we denote the space of real-valued Borel functions on $(V, \mathcal{B})$. For $f \in \mathcal{F}(V, \mathcal{B})$ and a Borel measure $\mu$ on $(V, \mathcal{B})$, we write
$$\mu(f)=\int_{V} f d \mu .$$
As a rule, we will deal only with continuous $\sigma$-finite measures on $(V, \mathcal{B})$ (unless the opposite is clearly indicated). This choice of measures is motivated by the discrete case where the counting measure plays the role of a $\sigma$-finite Borel measure on a measure space.

All objects, considered in the context of measure spaces (such as sets, functions, transformations, etc), are determined by modulo sets of zero measure (they are also called null sets). In most cases, we will implicitly use this mod 0 convention not mentioning the sets of zero measure explicitly.

Suppose now that a $\sigma$-finite continuous measure $\mu$ is chosen and fixed on $(V, \mathcal{B})$, so that $(V, \mathcal{B}, \mu)$ is a standard measure space. We denote by
$$\mathcal{B}{\text {fin }}=\mathcal{B}{\text {fin }}(\mu)={A \in \mathcal{B}: \mu(A)<\infty}$$
the algebra of Borel sets of finite measure $\mu$. Clearly, the set $V$ can be partitioned into a disjoint countable union of sets $A_{i}$ from $\mathcal{B}_{\text {fin }}$.

# 信号处理与线性系统代考

## 电子工程代写|信号处理与线性系统作业代写Signal Processing and Linear Systems代考|electrical networks

$$c(x):=\sum_{y \sim x} c_{x y} .$$

$\backslash 1$ ef $t$ 的分隔符缺失或无法识别

$$|u| \mathcal{H} E^{2}=\frac{1}{2} \sum(x y) \in E c_{x y}(u(x)-u(y))^{2}<\infty$$

## 电子工程代写|信号处理与线性系统作业代写Signal Processing and Linear Systems代考|From discrete to measurable setting

$$\mu(f)=\int_{V} f d \mu .$$

$$\mathcal{B} \text { fin }=\mathcal{B} \text { fin }(\mu)=A \in \mathcal{B}: \mu(A)<\infty$$

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