## 数学代写|泛函分析作业代写Functional Analysis代考|Math255A

statistics-lab™ 为您的留学生涯保驾护航 在代写泛函分析Functional Analysis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。我们的专家在代写泛函分析Functional Analysis代写方面经验极为丰富，各种代写泛函分析Functional Analysis相关的作业也就用不着说。

## 数学代写|泛函分析作业代写Functional Analysis代考|Energy Spaces

We have already introduced the energy space for a string. Let us consider other examples. In what follows, we shall employ only dimensionless variables, parameters, and functions of state of a body.
Bending of a Bar
In the Introduction we considered the problem of bending a clamped bar, which was governed by (4). The potential energy of the bar is
$$\mathcal{E}_1(y)=\frac{1}{2} \int_0^l B(x)\left(y^{\prime \prime}\right)^2 d x .$$
On the set $S$ consisting of all functions $y(x)$ that are twice continuously differentiable on $[0, l]$ and that satisfy
$$y(0)=y^{\prime}(0)=y(l)=y^{\prime}(l)=0,$$
let us consider
$$d\left(y_1, y_2\right)=\left(2 \mathcal{E}_1\left(y_1-y_2\right)\right)^{1 / 2}=\left(\int_0^l B(x)\left[y_1^{\prime \prime}(x)-y_2^{\prime \prime}(x)\right]^2 d x\right)^{1 / 2} .$$
For this, D1 and D3 obviously hold. Satisfaction of D4 follows from the fact that $\mathcal{E}_1(y)$ is quadratic in $y$. To verify D2, we need only show that $d(y, z)=0$ implies $y(x)=z(x)$. But $d(y, z)=0$ implies $(y(x)-z(x))^{\prime \prime}=0$, hence $y(x)-z(x)=a_1 x+a_2$ where $a_1, a_2$ are constants; imposing (1.3.1), we arrive at $a_1=a_2=0$. So $d\left(y_1, y_2\right)$ is indeed a metric on $S$.

The potential energy of a membrane occupying a domain $\Omega \subset \mathbb{R}^2$ is proportional to
$$\mathcal{E}2(u)=\int{\Omega}\left[\left(\frac{\partial u}{\partial x}\right)^2+\left(\frac{\partial u}{\partial y}\right)^2\right] d x d y .$$
So we can try
$$d(u, v)=\left(\mathcal{E}2(u-v)\right)^{1 / 2}$$ as a metric on the functions $u=u(x, y)$ that describe the normal displacements of the membrane. We first consider the case where the edge of the membrane is clamped, i.e., $$\left.u\right|{\partial \Omega}=0$$
where $\partial \Omega$ is the boundary of $\Omega$. The function $d(u, v)$ of (1.3.2) is a metric on the set $C^{(1)}(\Omega)$. Axioms D1 and D3 hold obviously; D2 holds by (1.3.3), and D4 holds by the quadratic nature of $\mathcal{E}2(u)$. This space is appropriate for investigating the corresponding boundary value problem $$\Delta u=-f,\left.\quad u\right|{\partial \Omega}=0,$$
called the Dirichlet problem for Poisson’s equation. This describes the behavior of the clamped membrane under a load $f=f(x, y)$.

## 数学代写|泛函分析作业代写Functional Analysis代考|A Plate

For a linear elastic plate the potential energy is
$$\mathcal{E}3(w)=\int{\Omega} \frac{D}{2}\left{(\Delta w)^2+2(1-\nu)\left[\left(\frac{\partial^2 w}{\partial x \partial y}\right)^2-\frac{\partial^2 w}{\partial x^2} \frac{\partial^2 w}{\partial y^2}\right]\right} d x d y$$
where $D$ is the bending stiffness of the plate, $\nu$ is Poisson’s ratio, and $w(x, y)$ is the normal displacement of the mid-surface of the plate, which is denoted by $\Omega$ in the $x y$-plane. If the edge of the plate is clamped we get
$$\left.w\right|{\partial \Omega}=\left.\frac{\partial w}{\partial n}\right|{\partial \Omega}=0 .$$
If $\mathcal{E}_3(w)=0$, then $w=a+b x+c y$ and, from (1.3.7), $w=0$. So D2 is fulfilled by the distance function
$$d\left(w_1, w_2\right)=\left(2 \mathcal{E}_3\left(w_1-w_2\right)\right)^{1 / 2} .$$
The remaining metric axioms are easily checked, and $d\left(w_1, w_2\right)$ is a metric on the subset of $C^{(2)}(\Omega)$ consisting of all functions satisfying (1.3.7). This is the energy space for the plate.

If the edge of the plate is free from geometrical fixing (clamping), the situation is similar to the Neumann problem of membrane theory: we must eliminate “rigid” motions of the plate. We shall consider this in detail later.

# 泛函分析代写

## 数学代写|泛函分析作业代写Functional Analysis代考|Energy Spaces

$$\mathcal{E}_1(y)=\frac{1}{2} \int_0^l B(x)\left(y^{\prime \prime}\right)^2 d x .$$

$$y(0)=y^{\prime}(0)=y(l)=y^{\prime}(l)=0,$$

$$d\left(y_1, y_2\right)=\left(2 \mathcal{E}_1\left(y_1-y_2\right)\right)^{1 / 2}=\left(\int_0^l B(x)\left[y_1^{\prime \prime}(x)-y_2^{\prime \prime}(x)\right]^2 d x\right)^{1 / 2} .$$

$$\mathcal{E}2(u)=\int{\Omega}\left[\left(\frac{\partial u}{\partial x}\right)^2+\left(\frac{\partial u}{\partial y}\right)^2\right] d x d y .$$

$$d(u, v)=\left(\mathcal{E}2(u-v)\right)^{1 / 2}$$作为描述膜的正常位移的函数$u=u(x, y)$的度量。我们首先考虑膜的边缘被夹住的情况，即$$\left.u\right|{\partial \Omega}=0$$

## 数学代写|泛函分析作业代写Functional Analysis代考|A Plate

$$\mathcal{E}3(w)=\int{\Omega} \frac{D}{2}\left{(\Delta w)^2+2(1-\nu)\left[\left(\frac{\partial^2 w}{\partial x \partial y}\right)^2-\frac{\partial^2 w}{\partial x^2} \frac{\partial^2 w}{\partial y^2}\right]\right} d x d y$$

$$\left.w\right|{\partial \Omega}=\left.\frac{\partial w}{\partial n}\right|{\partial \Omega}=0 .$$

$$d\left(w_1, w_2\right)=\left(2 \mathcal{E}_3\left(w_1-w_2\right)\right)^{1 / 2} .$$

## 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|泛函分析作业代写Functional Analysis代考|MATH784

statistics-lab™ 为您的留学生涯保驾护航 在代写泛函分析Functional Analysis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。我们的专家在代写泛函分析Functional Analysis代写方面经验极为丰富，各种代写泛函分析Functional Analysis相关的作业也就用不着说。

## 数学代写|泛函分析作业代写Functional Analysis代考|Preliminaries

Consider a set of particles $P_i, i=1, \ldots, n$. To locate these particles in the space $\mathbb{R}^3$, we need a reference system. Let the Cartesian coordinates of $P_i$ be $\left(\xi_i, \eta_i, \zeta_i\right)$ for each $i$. Identifying $\left(\xi_1, \eta_1, \zeta_1\right)$ with $\left(x_1, x_2, x_3\right),\left(\xi_2, \eta_2, \zeta_2\right)$ with $\left(x_4, x_5, x_6\right)$, and so on, we obtain a vector $\mathbf{x}$ of the Euclidean space $\mathbb{R}^{3 n}$ with coordinates $\left(x_1, x_2, \ldots, x_{3 n}\right)$. This vector determines the positions of all particles in the set.

To distinguish different configurations $\mathbf{x}$ and $\mathbf{y}$ of the system, we can introduce a distance from $\mathbf{x}$ to $\mathbf{y}$ :
$$d_E(\mathbf{x}, \mathbf{y})=\left(\sum_{i=1}^{3 n}\left(x_i-y_i\right)^2\right)^{1 / 2} .$$
This is the Euclidean distance (or metric) of $\mathbb{R}^{3 n}$. Alternatively, we could characterize the distance from $\mathbf{x}$ to $\mathbf{y}$ using the function
$$d_S(\mathbf{x}, \mathbf{y})=\max \left{\left|x_1-y_1\right|,\left|x_2-y_2\right|, \ldots,\left|x_{3 n}-y_{3 n}\right|\right} .$$
It is easily seen that each of the metrics $d_E$ and $d_S$ satisfy the following properties, known as the metric axioms:
D1. $d(\mathbf{x}, \mathbf{y}) \geq 0$;
D2. $d(\mathbf{x}, \mathbf{y})=0$ if and only if $\mathbf{x}=\mathbf{y}$;

D3. $d(\mathbf{x}, \mathbf{y})=d(\mathbf{y}, \mathbf{x})$;
D4. $d(\mathbf{x}, \mathbf{y}) \leq d(\mathbf{x}, \mathbf{z})+d(\mathbf{z}, \mathbf{y})$.
Any real valued function $d(\mathbf{x}, \mathbf{y})$ defined for all $\mathbf{x}, \mathbf{y} \in \mathbb{R}^{3 n}$ is called a metric on $\mathbb{R}^{3 n}$ if it satisfies properties D1-D4. Property D1 is called the axiom of positiveness, property D3 is called the axiom of symmetry, and property D4 is called the triangle inequality.

## 数学代写|泛函分析作业代写Functional Analysis代考|Some Metric Spaces of Functions

To describe the behavior or change in state of a body in space, we use functions of one or more variables. Displacements, velocities, loads, and temperatures are all functions of position. So we must learn how to distinguish different states of a body; the appropriate tool for this is, of course, the notion of metric space. In mechanics of materials, we deal mostly with real-valued continuous or differentiable functions.

Let $\Omega$ be a closed and bounded domain in $\mathbb{R}^n$. A natural measure of the deviation between two continuous functions $f(\mathbf{x})$ and $g(\mathbf{x}), \mathbf{x} \in \Omega$, is
$$d(f, g)=\max {\mathbf{x} \in \Omega}|f(\mathbf{x})-g(\mathbf{x})| .$$ It is obvious that $d(f, g)$ satisfies axioms D1-D3. Let us verify D4. Since $|f(\mathbf{x})-g(\mathbf{x})|$ is a continuous function on $\Omega$, there exists a point $\mathbf{x}_0 \in \Omega$ such that $$d(f, g)=\max {\mathbf{x} \in \Omega}|f(\mathbf{x})-g(\mathbf{x})|=\left|f\left(\mathbf{x}_0\right)-g\left(\mathbf{x}_0\right)\right| .$$
For any function $h(\mathbf{x})$ which is continuous on $\Omega$, we get
\begin{aligned} d(f, g) & =\left|f\left(\mathbf{x}_0\right)-g\left(\mathbf{x}_0\right)\right| \ & \leq\left|f\left(\mathbf{x}_0\right)-h\left(\mathbf{x}_0\right)\right|+\left|h\left(\mathbf{x}_0\right)-g\left(\mathbf{x}_0\right)\right| \ & \leq d(f, h)+d(h, g) . \end{aligned}
(Here we use the Weierstrass theorem that on a compact set a continuous function attains its maximum and minimum values.) Thus $d(f, g)$ in (1.2.1) is a metric.

# 泛函分析代写

## 数学代写|泛函分析作业代写Functional Analysis代考|Preliminaries

$$d_E(\mathbf{x}, \mathbf{y})=\left(\sum_{i=1}^{3 n}\left(x_i-y_i\right)^2\right)^{1 / 2} .$$

$$d_S(\mathbf{x}, \mathbf{y})=\max \left{\left|x_1-y_1\right|,\left|x_2-y_2\right|, \ldots,\left|x_{3 n}-y_{3 n}\right|\right} .$$

d。$d(\mathbf{x}, \mathbf{y}) \geq 0$;
d。$d(\mathbf{x}, \mathbf{y})=0$当且仅当$\mathbf{x}=\mathbf{y}$;

d3。$d(\mathbf{x}, \mathbf{y})=d(\mathbf{y}, \mathbf{x})$;
d4;$d(\mathbf{x}, \mathbf{y}) \leq d(\mathbf{x}, \mathbf{z})+d(\mathbf{z}, \mathbf{y})$。

## 数学代写|泛函分析作业代写Functional Analysis代考|Some Metric Spaces of Functions

$$d(f, g)=\max {\mathbf{x} \in \Omega}|f(\mathbf{x})-g(\mathbf{x})| .$$很明显，$d(f, g)$满足公理D1-D3。让我们验证D4。由于$|f(\mathbf{x})-g(\mathbf{x})|$是$\Omega$上的连续函数，因此存在一个点$\mathbf{x}_0 \in \Omega$，使得$$d(f, g)=\max {\mathbf{x} \in \Omega}|f(\mathbf{x})-g(\mathbf{x})|=\left|f\left(\mathbf{x}_0\right)-g\left(\mathbf{x}_0\right)\right| .$$

\begin{aligned} d(f, g) & =\left|f\left(\mathbf{x}_0\right)-g\left(\mathbf{x}_0\right)\right| \ & \leq\left|f\left(\mathbf{x}_0\right)-h\left(\mathbf{x}_0\right)\right|+\left|h\left(\mathbf{x}_0\right)-g\left(\mathbf{x}_0\right)\right| \ & \leq d(f, h)+d(h, g) . \end{aligned}
(这里我们使用Weierstrass定理，即在紧集合上连续函数达到最大值和最小值。)因此(1.2.1)中的$d(f, g)$是一个度量。

## 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|泛函分析作业代写Functional Analysis代考|MAT4450

statistics-lab™ 为您的留学生涯保驾护航 在代写泛函分析Functional Analysis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。我们的专家在代写泛函分析Functional Analysis代写方面经验极为丰富，各种代写泛函分析Functional Analysis相关的作业也就用不着说。

## 数学代写|泛函分析作业代写Functional Analysis代考|Inductive spectra of locally convex spaces

Given an inductive spectrum $\left(X_n\right)n$ there is a canonical algebraically exact sequence $$0 \longrightarrow \bigoplus{n \in \mathbb{N}} X_n \stackrel{d}{\longrightarrow} \bigoplus_{n \in \mathbb{N}} X_n \stackrel{\sigma}{\longrightarrow} X \longrightarrow 0$$
where $d\left(\left(x_n\right){n \in \mathbb{N}}\right)=\left(x_n-x{n-1}\right){n \in \mathbb{N}}, x_0=0$, and $\sigma\left(\left(x_n\right){n \in \mathbb{N}}\right)=\sum_{n \in \mathbb{N}} x_n$.
$d$ and $\sigma$ are continuous, and $\sigma$ is also open. The spectrum is called (weakly) acyclic if $d$ is (weakly) open onto its range.

This definition is related to the question whether a subspace $L$ of an inductive limit ind $X_n$ is topologically the inductive limit of the spaces $L_n=X \cap L_n$ (in this case $L$ is called a limit subspace) or if it has at least the same topological dual as the inductive limit (then $L$ is called well-located). Using either the duality explained below or a diagram chase on gets:
A subspace $L$ of the limit $X=\operatorname{ind} X_n$ of a (weakly) acyclic inductive spectrum is (well-located) a limit subspace if and only if the spectrum $\left(X_n / L_n\right)_{n \in \mathbb{N}}$ is (weakly) acyclic.

An easy calculation shows that the transposed of $d$ is
$$\Psi: \prod_{n \in \mathbb{N}} X_n^{\prime} \longrightarrow \prod_{n \in \mathbb{N}} X_n^{\prime},\left(f_n\right){n \in \mathbb{N}} \longmapsto\left(f_n-\varrho{n+1}^n\left(f_{n+1}\right)\right){n \in \mathbb{N}}$$ where $\varrho{n+1}^n: X_{n+1}^{\prime} \longrightarrow X_n^{\prime}$ is the restriction (i.e. the transposed of the inclusion). Thus, there is a close connection between (weakly) acyclic inductive spectra and the derived projective limit functor. In particular, an inductive spectrum $\left(X_n\right)_{n \in \mathbb{N}}$ is weakly acyclic if and only if $\operatorname{Proj}^1 \mathscr{Y}=0$ where $\mathscr{Y}=\left(X_n^{\prime}, \varrho_m^n\right)$ with the restrictions as spectral maps. Thus, the results of chapter 3 have immediate counterparts for weakly acyclic spectra which we do not state explicitely.

There is also a direct relation between acyclic inductive spectra and properties of projective limits. Using theorem 2.2 .2 we obtain that $\left(X_n\right){n \in \mathbb{N}}$ is acyclic if and only if for every set $I$ we have $\operatorname{Proj}^1\left(L\left(X_n, \ell_I^{\infty}\right), R_m^n\right)=0$ where $R{n+1}^n(T)=T \circ \varrho_{n+1}^n$. The spaces $L\left(X_n, \ell_I^{\infty}\right)$ are covered by the system $\left{\left(U^{\circ}\right)^I: U \in \mathscr{U}0\left(X_n\right)\right}$ (here, we identify $T \in L\left(X_n, \ell_I^{\infty}\right)$ with a family $\left.\left(T_i\right){i \in I}\right)$ and using theorem 3.2 .14 and the remark following its proof we obtain a general sufficient condition for acyclicity (a direct proof of this can be obtained as in [69]).

## 数学代写|泛函分析作业代写Functional Analysis代考|The duality functor

This final chapter is concerned with the problem when the transposed map of a homomorphism in the category of locally convex spaces is again a homomorphism. Let
$$0 \longrightarrow X \stackrel{f}{\longrightarrow} Y \stackrel{g}{\longrightarrow} Z \longrightarrow 0$$
be an exact sequence in the category of locally convex spaces. The HahnBanach theorem implies that the dual sequence
$$0 \longrightarrow Z^{\prime} \stackrel{g^t}{\longrightarrow} Y^{\prime} \stackrel{f^t}{\longrightarrow} X^{\prime} \longrightarrow 0$$
is exact as a sequence of vector spaces, but if all duals are endowed with the strong topology neither $f^t$ nor $g^t$ must be a homomorphism. Let $D$ be the contravariant functor assigning to a locally convex space $X$ its strong dual $X_\beta^{\prime}$ and to $f: X \longrightarrow Y$ the transposed map. Then an exact complex
$$0 \longrightarrow X \longrightarrow Y \longrightarrow Z \longrightarrow 0$$
is transformed into an acyclic complex
$$0 \longrightarrow D(Z) \longrightarrow D(Y) \longrightarrow D(X) \longrightarrow 0$$
To measure the exactness of this complex Palamodov used the functors $H_M$ introduced in section 2.2. For any non-empty set $M$ we define the covariant functor $D_M=H_M \circ D$ from $\mathcal{L C S}$ to the category of vector spaces. Explicitely, to a locally convex space $X$ we $\operatorname{assign} D_M(X)=\operatorname{Hom}\left(X_\beta^{\prime}, \ell_M^{\infty}\right)$, and for a morphism $f: X \longrightarrow Y$ the linear map
$$f^*=D_M(f): \operatorname{Hom}\left(X_\beta^{\prime}, \ell_M^{\infty}\right) \longrightarrow \operatorname{Hom}\left(Y_\beta^{\prime}, \ell_M^{\infty}\right)$$
is defined by $T \mapsto T \circ f^t$. From theorem 2.2.2 we deduce that for an exact sequence
$$0 \longrightarrow X \stackrel{f}{\longrightarrow} Y \stackrel{g}{\longrightarrow} Z \longrightarrow 0$$

the dual sequence
$$0 \longrightarrow X^{\prime} \stackrel{g^t}{\longrightarrow} Y^{\prime} \stackrel{f^t}{\longrightarrow} X^{\prime} \longrightarrow 0$$
is left exact at $Y^{\prime}$ (i.e. $g^t$ is open onto its range) or right exact at $Y^{\prime}$ (i.e. $f^t$ is open) respectively if and only if for every set $M \neq \emptyset$ the complex
$$0 \longrightarrow D_M(X) \stackrel{f^}{\longrightarrow} D_M(Y) \stackrel{g^}{\longrightarrow} D_M(Z) \longrightarrow 0$$
is exact at $D_M(Z)$ or exact at $D_M(Y)$, respectively. (We note that the complex is always exact at $D_M(X)$, i.e. $D_M$ is a semi-injective functor.) The exactness at $D_M(Z)$ and $D_M(Y)$ are measured by $D_M^1(X)$ and $D_M^{+}(X)$, respectively. Let us recall the definitions of $D_M^1$ and $D_M^{+}$. If
$$0 \longrightarrow X \stackrel{i}{\longrightarrow} I_0 \stackrel{i_1}{\longrightarrow} I_1 \stackrel{i_2}{\longrightarrow} \ldots$$
is any injective resolution of $X$ we have
$$\begin{gathered} D_M^{+}(X)=\operatorname{ker} D_M\left(i_0\right) / \operatorname{im} D_M(i) \text { and } \ D_M^1(X)=\operatorname{ker} D_M\left(i_1\right) / \operatorname{im} D_M\left(i_0\right) . \end{gathered}$$

# 泛函分析代写

## 数学代写|泛函分析作业代写Functional Analysis代考|Inductive spectra of locally convex spaces

$d$和$\sigma$是连续的，$\sigma$也是开放的。如果$d$在其范围内(弱)开放，则称为(弱)无环谱。

$$\Psi: \prod_{n \in \mathbb{N}} X_n^{\prime} \longrightarrow \prod_{n \in \mathbb{N}} X_n^{\prime},\left(f_n\right){n \in \mathbb{N}} \longmapsto\left(f_n-\varrho{n+1}^n\left(f_{n+1}\right)\right){n \in \mathbb{N}}$$其中$\varrho{n+1}^n: X_{n+1}^{\prime} \longrightarrow X_n^{\prime}$是限制(即包含的转置)。因此，在(弱)无环感应谱和推导出的射影极限函子之间存在着密切的联系。特别地，一个感应谱$\left(X_n\right)_{n \in \mathbb{N}}$是弱无环的当且仅当$\operatorname{Proj}^1 \mathscr{Y}=0$其中$\mathscr{Y}=\left(X_n^{\prime}, \varrho_m^n\right)$具有谱映射的限制条件。因此，第3章的结果对我们没有明确说明的弱无环谱有直接对应。

## 数学代写|泛函分析作业代写Functional Analysis代考|The duality functor

$$0 \longrightarrow X \stackrel{f}{\longrightarrow} Y \stackrel{g}{\longrightarrow} Z \longrightarrow 0$$

$$0 \longrightarrow Z^{\prime} \stackrel{g^t}{\longrightarrow} Y^{\prime} \stackrel{f^t}{\longrightarrow} X^{\prime} \longrightarrow 0$$

$$0 \longrightarrow X \longrightarrow Y \longrightarrow Z \longrightarrow 0$$

$$0 \longrightarrow D(Z) \longrightarrow D(Y) \longrightarrow D(X) \longrightarrow 0$$

$$f^*=D_M(f): \operatorname{Hom}\left(X_\beta^{\prime}, \ell_M^{\infty}\right) \longrightarrow \operatorname{Hom}\left(Y_\beta^{\prime}, \ell_M^{\infty}\right)$$

$$0 \longrightarrow X \stackrel{f}{\longrightarrow} Y \stackrel{g}{\longrightarrow} Z \longrightarrow 0$$

## 有限元方法代写

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## 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。