## 数学代写|交换代数代写commutative algebra代考|MAST90025

statistics-lab™ 为您的留学生涯保驾护航 在代写交换代数commutative algebra方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写交换代数commutative algebra代写方面经验极为丰富，各种代写交换代数commutative algebra相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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|交换代数代写commutative algebra代考|Introduction to Boolean Algebras

A lattice is a set $\mathbf{T}$ equipped with an order relation $\leqslant$ for which there exist a minimum element, denoted by $0_{\mathbf{T}}$, a maximum element, denoted by $1_{\mathbf{T}}$, and every pair of elements $(a, b)$ admits an upper bound, denoted by $a \vee b$, and a lower bound, denoted by $a \wedge b$. A mapping from one lattice to another is called a lattice homomorphism if it respects the operations $\vee$ and $\wedge$ as well as the constants 0 and 1 . The lattice is called a distributive lattice when each of the two operations $\vee$ and $\wedge$ is distributive with respect to the other.

We will give a succinct study of the structure of distributive lattices and of structures that relate back to them in Chap. XI.
3.1 Proposition and definition (Boolean algebras)

1. By definition a ring $\mathbf{B}$ is $a$ Boolean algebra if and only if every element is idempotent. Consequently $2=\mathbf{B} 0$ (because $2=\mathrm{B} 4$ ).
2. We can define over $\mathrm{B}$ an order relation $x \preccurlyeq y$ by: $x$ is a multiple of $y$, i.e. $\langle x\rangle \subseteq\langle y\rangle$. Then, two arbitrary elements admit a lower bound, their lcm $x \wedge y=x y$, and an upper bound, their gcd $x \vee y=x+y+x y$. We thus obtain a distributive lattice with 0 as its minimum element and 1 as its maximum element.
3. For every $x \in \mathbf{B}$, the element $x^{\prime}=1+x$ is the unique element that satisfies the equalities $x \wedge x^{\prime}=0$ and $x \vee x^{\prime}=1$, we call it the complement of $x$.

Notation conflict Here we find ourselves with a conflict of notation. Indeed, divisibility in a ring leads to a notion of the gcd, which is commonly denoted by $a \wedge b$, because it is taken as a lower bound ( $a$ divides $b$ being understood as ” $a$ smaller than $b$ ” in the sense of the divisibility). This conflicts with the gcd of the elements in a Boolean algebra, which is an upper bound. This is due to the fact that the order relation has been reversed, so that the elements 0 and 1 of the Boolean algebra are indeed the minimum and the maximum in the lattice. This inevitable conflict will appear in an even stronger sense when we will consider the Boolean algebra of the idempotents of a ring $\mathbf{A}$.

Even though all the elements of a Boolean algebra are idempotents we will keep the terminology “fundamental system of orthogonal idempotents” ” for a finite family $\left(x_i\right)$ of pairwise orthogonal elements (i.e. $x_i x_j=0$ for $i \neq j$ ) with sum 1. This convention is all the more justified in that we will mainly preoccupy ourselves with the Boolean algebra that naturally appears in commutative algebra: that of the idempotents of a ring $\mathbf{A}$.

## 数学代写|交换代数代写commutative algebra代考|Discrete Boolean Algebras

3.2 Proposition (Every discrete Boolean algebra behaves in computations as the algebra of the detachable subsets of a finite set) Let $\left(r_1, \ldots, r_m\right)$ be a finite family in a Boolean algebra $\mathbf{B}$.

Let $s_i=1-r_i$ and, for a finite subset $I$ of ${1, \ldots, m}$, let $r_I=\prod_{i \in I} r_i \prod_{j \notin I} s_j$.

1. The $r_I$ ‘s form a fundamental system of orthogonal idempotents and they generate the same Boolean algebra as the $r_i$ ‘s.
2. Suppose that $\mathbf{B}$ is discrete. Then, if there are exactly $N$ nonzero elements $r_I$, the Boolean subalgebra generated by the $r_i$ ‘s is isomorphic to the algebra of finite subsets of a set with $N$ elements.

As a corollary we obtain the following fact and the fundamental structure theorem that summarizes it. Recall that we denote by $\mathrm{P}_{\mathrm{f}}(S)$ the set of finite subsets of a set $S$.
In a discrete Boolean algebra an element $e$ is called an atom if it satisfies one of the following equivalent properties.

• $e$ is minimal among the nonzero elements.
• $e \neq 0$ and for every $f, f$ is orthogonal or greater than $e$.
• $e \neq 0$ and for every $f, e f=0$ or $e$, or $e f=0$ or $e(1-f)=0$.
• $e \neq 0$ and the equality $e=e_1+e_2$ with $e_1 e_2=0$ implies $e_1=0$ or $e_2=0$.
We also say that $e$ is indecomposable. It is clear that an automorphism of a discrete Boolean algebra preserves the set of atoms and that for two atoms $e$ and $f$, we have $e=f$ or $e f=0$.
3.3 Theorem (Structure theorem)
1. Every finite Boolean algebra is isomorphic to the algebra of the detachable subsets of a finite set.
2. More precisely, for a Boolean algebra $C$ the following properties are equivalent.
a. $C$ is finite.
b. $C$ is discrete and finitely generated.
c. The set $S$ of atoms is finite, and $1_C$ is the sum of this set.
In such a case $C$ is isomorphic to the Boolean algebra $\mathrm{P}_{\mathrm{f}}(S)$.

# 交换代数代考

## 数学代写|交换代数代写commutative algebra代考|Introduction to Boolean Algebras

3.1 命题与定义 (布尔代数)

1. 根据定义环 $\mathrm{B}$ 是 $a$ 布尔代数当且仅当每个元素都是幂等的。最后 $2=B 0$ (因为 $2=B 4$ ).
2. 我们可以定义B顺序关系 $x \preccurlyeq y$ 经过: $x$ 是的倍数 $y ， \mathrm{IE}\langle x\rangle \subseteq\langle y\rangle$. 然后，两个任意元素承认下 界，他们的 $\operatorname{lcm} x \wedge y=x y$ ，以及一个上限，他们的 $\operatorname{gcd} x \vee y=x+y+x y$. 这样我们就得到 了一个以0为最小元素，1为最大元素的分配格。
3. 对于每一个 $x \in \mathbf{B}$ ，元素 $x^{\prime}=1+x$ 是满足等式的唯一元素 $x \wedge x^{\prime}=0$ 和 $x \vee x^{\prime}=1$ ，我们称之 为补码 $x$.
符号冲突 在这里，我们发现自己遇到了符号冲突。实际上，环中的可分性导致了 gcd 的概念，通常表示 为 $a \wedge b$, 因为它被视为下界 ( $a$ 分裂 $b$ 被理解为“ $a$ 小于 $b$ ” 在可分性的意义上) 。这与布尔代数中作为上限的 元素的 gcd 冲突。这是由于顺序关系被颠倒了，使得布尔代数的元素0和1确实是格中的最小值和最大 值。当我们考虑环的幂等项的布尔代数时，这种不可避免的冲突将以更强烈的意义出现 $\mathbf{A}$.
即使布尔代数的所有元素都是幂等的，我们仍会为有限族保留术语“正交幂等的基本系统” $\left(x_i\right)$ 成对正交元 素 (即 $x_i x_j=0$ 为了 $i \neq j$ ) 总和为 1 。这个约定更加合理，因为我们将主要关注自然出现在交换代数中 的布尔代数：环的幂等元 $\mathbf{A}$.

## 数学代写|交换代数代写commutative algebra代考|Discrete Boolean Algebras

3.2 命题（每个离散布尔代数在计算中表现为有限集的可分离子集的代数）令 $\left(r_1, \ldots, r_m\right)$ 是布尔代数中 的有限族 $\mathbf{B}$.

1. 这 $r_I$ 形成了一个基本的正交幂等系统，并且它们生成了与 $r_i$ 的。
2. 假设B是离散的。那么，如果恰好有 $N$ 非零元素 $r_I$ ，由生成的布尔子代数 $r_i$ 的同构于一个集合的有 限子集的代数 $N$ 元素。
作为推论，我们得到以下事实和总结它的基本结构定理。回想一下，我们用 $\mathrm{P}_{\mathrm{f}}(S)$ 集合的有限子集 $S$.
在离散布尔代数中，一个元素 $e$ 如果满足以下等效属性之一，则称为原子。
• $e$ 在非零元素中是最小的。
• $e \neq 0$ 对于每一个 $f, f$ 正交或大于 $e$.
• $e \neq 0$ 对于每一个 $f, e f=0$ 或者 $e$ ，或者 $e f=0$ 或者 $e(1-f)=0$.
• $e \neq 0$ 和平等 $e=e_1+e_2$ 和 $e_1 e_2=0$ 暗示 $e_1=0$ 或者 $e_2=0$.
我们还说 $e$ 是不可分解的。很明显，离散布尔代数的自同构保留了原子集和两个原子的 $e$ 和 $f$ ，我们 有 $e=f$ 或者 $e f=0$.
3.3 定理 (结构定理)
1. 每个有限布尔代数都同构于有限集的可分离子集的代数。
2. 更准确地说，对于布尔代数 $C$ 以下属性是等效的。
A。 $C$ 是有限的。
b. $C$ 是离散且有限生成的。
C。套装 $S$ 原子的数量是有限的，并且 $1_C$ 是这个集合的总和。
在这种情况下 $C$ 与布尔代数同构 $\mathrm{P}_{\mathrm{f}}(S)$.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|交换代数代写commutative algebra代考|MATH4312

statistics-lab™ 为您的留学生涯保驾护航 在代写交换代数commutative algebra方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写交换代数commutative algebra代写方面经验极为丰富，各种代写交换代数commutative algebra相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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|交换代数代写commutative algebra代考|The Dynamic Method

In classical mathematics proofs of existence are rarely explicit. Two essential obstacles appear each time that we try to render such a proof explicit.

The first obstacle is the application of LEM. For instance, if you consider the proof that every univariate polynomial over a field $\mathbf{K}$ admits a decomposition into prime factors, you have a kind of algorithm whose key ingredient is: if $P$ is irreducible all is well, if $P$ can be decomposed into a product of two factors of degree $\geqslant 1$, all is still well, by induction hypothesis. Unfortunately the disjunction used to make the proof work ” $P$ is irreducible or $P$ can be decomposed into a product of two factors of degree $\geqslant 1$ ” is not explicit in general. In other words, even if a field is defined constructively, we cannot be sure that this disjunction can be made explicit by an algorithm. Here we find ourselves in the presence of a typical case where LEM “is an issue,” because the existence of an irreducible factor cannot be the object of a general algorithm.

The second obstacle is the application of Zorn’s lemma, which allows us to generalize to the uncountable case the usual proofs by induction in the countable case.
For example in Modern Algebra by van der Waerden the second pitfall is avoided by limiting ourselves to the countable algebraic structures.
However, we have two facts that are now well established from experience:

• The universal concrete results proven by the dubious abstract methods above have never been contradicted. We have even very often successfully extracted unquestionable constructive proofs from them. This would suggest that even if the abstract methods are in some way incorrect or contradictory, they have until now only been used with a sufficient amount of discernment.
• The key concrete results proven by the dubious abstract methods have not been invalidated either. On the contrary, they have often been validated by algorithms proven constructively. 1

Faced with this slightly paradoxical situation: the abstract methods are a priori dubious, but they do not fundamentally deceive us when they give us a result of a concrete nature. There are two possible reactions.

Either we believe that the abstract methods are fundamentally correct because they reflect a “truth,” some sort of “ideal Cantor universe” in which exists the true semantic of mathematics. This is the stance taken by Platonic realism, defended for instance by Gödel.

Or we think that the abstract methods truly are questionable. But then, unless we believe that mathematics falls within the domain of magic or of miracles, it must be explained why classical mathematics makes such few mistakes. If we believe in neither Cantor, nor miracles, we are led to believe that the abstract proofs of concrete results necessarily contain sufficient “hidden ingredients” to construct the corresponding concrete proofs.

## 数学代写|交换代数代写commutative algebra代考|Splitting Fields and Galois Theory in Classical Mathematics

In this subsection we will offer a possible presentation of the splitting field of an arbitrary polynomial and of the Galois theory of a separable polynomial in classical mathematics. This allows us to understand the “detours” that we will be obligated to take to have an entirely constructive theory.

If $f$ is a monic polynomial, we work with the universal splitting algebra of $f$, $\mathbf{A}=\mathrm{Adu}_{\mathbf{K}, f}$ in which $f(T)=\prod_i\left(T-x_i\right)$, with $\mathrm{S}_n$ as a group of automorphisms (see Sect. III-4).

This algebra being a finite dimensional $\mathbf{K}$-vector space, all the ideals are themselves finite dimensional $\mathbf{K}$-vector spaces and we have the right to consider a strict ideal $\mathfrak{m}$ of maximum dimension as a $\mathbf{K}$-vector space (all of this by applying LEM). This ideal is automatically a maximal ideal. The quotient algebra $\mathbf{L}=\mathbf{A} / \mathrm{m}$ is then a splitting field for $f$. The group $G=\operatorname{St}(\mathfrak{m})$ operates on $\mathbf{L}$ and the fixed field of $G$, $\mathbf{L}^G=\mathbf{K}_1$, possesses the two following properties:

• $\mathbf{L} / \mathbf{K}_1$ is a Galois extension with $\operatorname{Gal}\left(\mathbf{L} / \mathbf{K}_1\right) \simeq G$.
• $\mathbf{K}_1 / \mathbf{K}$ is an extension obtained by successive additions of $p^{\text {th }}$ roots, where $p=$ char $(\mathbf{K})$.

Moreover, if $\mathbf{L}^{\prime}$ is another splitting field for $f$ with $f=\prod_i\left(T-\xi_i\right)$ in $\mathbf{L}^{\prime}[T]$, we have a unique homomorphism of $\mathbf{K}$-algebras $\varphi: \mathbf{A} \rightarrow \mathbf{L}^{\prime}$ satisfying the equalities $\varphi\left(x_i\right)=\xi_i$ for $i \in \llbracket 1 . . n \rrbracket$. We can then show that $\operatorname{Ker} \varphi$, which is a maximal ideal of $A$, is necessarily a conjugate of $\mathfrak{m}$ under the action of $S_n$. Thus the splitting field is unique, up to isomorphism (this isomorphism is not unique if $G \neq{\mathrm{Id}}$ ).

Finally, when $f$ is separable, the situation is simplified because the universal splitting algebra is étale, and $\mathbf{K}_1=\mathbf{K}$.

The previous approach is possible from a constructive point of view if the field $\mathbf{K}$ is separably factorial and if the polynomial $f$ is separable, because then, since the universal splitting algebra $\mathbf{A}$ is étale, it can be decomposed into a finite product of étale fields over $\mathbf{K}$ (Corollary VI-1.13).

But when the field is not separably factorial, we face an a priori insurmountable obstacle, and we cannot hope to systematically and algorithmically obtain a splitting field that is strictly finite over $\mathbf{K}$.

If the characteristic is finite and if the polynomial is not separable, we need stronger factorization properties to construct a splitting field (the question is delicate, and very well presented in [MRR]).

# 交换代数代考

## 数学代写|交换代数代写commutative algebra代考|The Dynamic Method

• 上述可疑的抽象方法所证明的普遍具体结果从未被反驳过。我们甚至经常从他们那里成功地提取出无可置疑的建设性证据。这表明即使抽象方法在某种程度上是不正确的或自相矛盾的，它们直到现在也只是在足够的辨别力下被使用。
• 可疑的抽象方法证明的关键具体结果也没有作废。相反，它们经常被建设性证明的算法所验证。1个

## 数学代写|交换代数代写commutative algebra代考|Splitting Fields and Galois Theory in Classical Mathematics

• $\mathbf{L} / \mathbf{K}_1$ 是一个伽罗华扩展 $\mathrm{Gal}\left(\mathbf{L} / \mathbf{K}_1\right) \simeq G$.
• $\mathbf{K}_1 / \mathbf{K}$ 是通过连续添加获得的扩展 $p^{\text {th }}$ 根，在哪里 $p=$ 字符 $(\mathbf{K})$.
此外，如果 $\mathbf{L}^{\prime}$ 是另一个分裂领域 $f$ 和 $f=\prod_i\left(T-\xi_i\right)$ 在 $\mathbf{L}^{\prime}[T]$ ，我们有一个唯一的同态 $\mathbf{K}$-代数 $\varphi: \mathbf{A} \rightarrow \mathbf{L}^{\prime}$ 满足等式 $\varphi\left(x_i\right)=\xi_i$ 为了 $i \in \backslash$ llbracket1.. $n \backslash$ rrbracket. 然后我们可以证明Ker $\varphi$ ， 这是一个极大的理想 $A$, 必然是的共轭 $m$ 的作用下 $S_n$. 因此，分裂场是唯一的，直到同构（这种同构不是 唯一的，如果 $G \neq \mathrm{Id}$ ).
最后，当 $f$ 是可分离的，情况被简化了，因为泛分裂代数是 étale，并且 $\mathbf{K}_1=\mathbf{K}$.
从建设性的角度来看，如果该领域K是可分离的阶乘，如果多项式 $f$ 是可分的，因为那时，自从通用分裂 代数 $\mathbf{A}$ 是 étale，它可以分解为 étale 域的有限乘积 $\mathbf{K}$ (推论 VI-1.13)。
但是当场不是可分阶乘时，我们面临着一个先验不可逾越的障碍，我们不能希望系统地和算法地获得一个 严格有限的分裂场K.
如果特征是有限的并且多项式不可分，我们需要更强的因式分解性质来构造分裂域（这个问题很微妙，在 [MRR] 中有很好的介绍)。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|凸优化作业代写Convex Optimization代考|CPD131

statistics-lab™ 为您的留学生涯保驾护航 在代写凸优化Convex Optimization方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写凸优化Convex Optimization代写方面经验极为丰富，各种代写凸优化Convex Optimization相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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|凸优化作业代写Convex Optimization代考|Sparse descriptions and basis pursuit

In basis pursuit, there is a very large number of basis functions, and the goal is to find a good fit of the given data as a linear combination of a small number of the basis functions. (In this context the function family is linearly dependent, and is sometimes referred to as an over-complete basis or dictionary.) This is called basis pursuit since we are selecting a much smaller basis, from the given over-complete basis, to model the data.

Thus we seek a function $f \in \mathcal{F}$ that fits the data well,
$$f\left(u_i\right) \approx y_i, \quad i=1, \ldots, m$$
with a sparse coefficient vector $x$, i.e., $\operatorname{card}(x)$ small. In this case we refer to
$$f=x_1 f_1+\cdots+x_n f_n=\sum_{i \in \mathcal{B}} x_i f_i$$
where $\mathcal{B}=\left{i \mid x_i \neq 0\right}$ is the set of indices of the chosen basis elements, as a sparse description of the data. Mathematically, basis pursuit is the same as the regressor selection problem (see $\S 6.4$ ), but the interpretation (and scale) of the optimization problem are different.

Sparse descriptions and basis pursuit have many uses. They can be used for de-noising or smoothing, or data compression for efficient transmission or storage of a signal. In data compression, the sender and receiver both know the dictionary, or basis elements. To send a signal to the receiver, the sender first finds a sparse representation of the signal, and then sends to the receiver only the nonzero coefficients (to some precision). Using these coefficients, the receiver can reconstruct (an approximation of) the original signal.

One common approach to basis pursuit is the same as the method for regressor selection described in $\S 6.4$, and based on $\ell_1$-norm regularization as a heuristic for finding sparse descriptions. We first solve the convex problem
$$\operatorname{minimize} \sum_{i=1}^m\left(f\left(u_i\right)-y_i\right)^2+\gamma|x|_1,$$
where $\gamma>0$ is a parameter used to trade off the quality of the fit to the data, and the sparsity of the coefficient vector. The solution of this problem can be used directly, or followed by a refinement step, in which the best fit is found, using the sparsity pattern of the solution of (6.18). In other words, we first solve (6.18), to obtain $\hat{x}$. We then set $\mathcal{B}=\left{i \mid \hat{x}i \neq 0\right}$, i.e., the set of indices corresponding to nonzero coefficients. Then we solve the least-squares problem $$\operatorname{minimize} \sum{i=1}^m\left(f\left(u_i\right)-y_i\right)^2$$
with variables $x_i, i \in \mathcal{B}$, and $x_i=0$ for $i \notin \mathcal{B}$.
In basis pursuit and sparse description applications it is not uncommon to have a very large dictionary, with $n$ on the order of $10^4$ or much more. To be effective, algorithms for solving (6.18) must exploit problem structure, which derives from the structure of the dictionary signals.

## 数学代写|凸优化作业代写Convex Optimization代考|Interpolation with convex functions

In some special cases we can solve interpolation problems involving an infinitedimensional set of functions, using finite-dimensional convex optimization. In this section we describe an example.

We start with the following question: When does there exist a convex function $f: \mathbf{R}^k \rightarrow \mathbf{R}$, with $\operatorname{dom} f=\mathbf{R}^k$, that satisfies the interpolation conditions
$$f\left(u_i\right)=y_i, \quad i=1, \ldots, m$$ at given points $u_i \in \mathbf{R}^k$ ? (Here we do not restrict $f$ to lie in any finite-dimensional subspace of functions.) The answer is: if and only if there exist $g_1, \ldots, g_m$ such that
$$y_j \geq y_i+g_i^T\left(u_j-u_i\right), \quad i, j=1, \ldots, m$$
To see this, first suppose that $f$ is convex, $\operatorname{dom} f=\mathbf{R}^k$, and $f\left(u_i\right)=y_i$, $i=1, \ldots, m$. At each $u_i$ we can find a vector $g_i$ such that
$$f(z) \geq f\left(u_i\right)+g_i^T\left(z-u_i\right)$$
for all $z$. If $f$ is differentiable, we can take $g_i=\nabla f\left(u_i\right)$; in the more general case, we can construct $g_i$ by finding a supporting hyperplane to epi $f$ at $\left(u_i, y_i\right)$. (The vectors $g_i$ are called subgradients.) By applying (6.20) to $z=u_j$, we obtain (6.19).
Conversely, suppose $g_1, \ldots, g_m$ satisfy (6.19). Define $f$ as
$$f(z)=\max _{i=1, \ldots, m}\left(y_i+g_i^T\left(z-u_i\right)\right)$$
for all $z \in \mathbf{R}^k$. Clearly, $f$ is a (piecewise-linear) convex function. The inequalities $(6.19)$ imply that $f\left(u_i\right)=y_i$, for $i=1, \ldots, m$.

We can use this result to solve several problems involving interpolation, approximation, or bounding, with convex functions.

# 凸优化代写

## 数学代写|凸优化作业代写Convex Optimization代考|Sparse descriptions and basis pursuit

$$f\left(u_i\right) \approx y_i, \quad i=1, \ldots, m$$

$$f=x_1 f_1+\cdots+x_n f_n=\sum_{i \in \mathcal{B}} x_i f_i$$

\frac{2+\log x}{(1+\log x)^{\frac{3}{2}}} \leq 2(1+logx)23​2+logx​≤2

Squaring both sides and simplifying, we get:

## Textbooks

• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.

Statistics-lab™可以为您提供harvard.edu AM221 Convex optimization现代代数课程的代写代考辅导服务！ 请认准Statistics-lab™. Statistics-lab™为您的留学生涯保驾护航。