## 数学代写|概率论代写Probability theory代考|MATHS7103

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• 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|概率论代写Probability theory代考|Set, Operation, and Function

Set. In general, a set is a collection of objects equipped with an equality relation. To define a set is to specify how to construct an element of the set, and how to prove that two elements are equal. A set is also called a family.

A member $\omega$ in the collection $\Omega$ is called an element of the latter, or, in symbols, $\omega \in \Omega$

The usual set-theoretic notations are used. Let two subsets $A$ and $B$ of a set $\Omega$ be given. We will write $A \cup B$ for the union, and $A \cap B$ or $A B$ for the intersection. We write $A \subset B$ if each member $\omega$ of $A$ is a member of $B$. We write $A \supset B$ for $B \subset A$. The set-theoretic complement of a subset $A$ of the set $\Omega$ is defined as the set ${\omega \in \Omega: \omega \in A$ implies a contradiction $}$. We write $\omega \notin A$ if $\omega \in A$ implies a contradiction.

Nonempty set. A set $\Omega$ is said to be nonempty if we can construct some element $\omega \in \Omega$.

Empty set. A set $\Omega$ is said to be empty if it is impossible to construct an element $\omega \in \Omega$. We will let $\phi$ denote an empty set.

Operation. Suppose $A, B$ are sets. A finite, step-by-step, method $X$ that produces an element $X(x) \in B$ given any $x \in A$ is called an operation from $A$ to $B$. The element $X(x)$ need not be unique. Two different applications of the operation $X$ with the same input element $x$ can produce different outputs. An example of an operation is [. $]_1$, which assigns to each $a \in R$ an integer $[a]_1 \in$ $(a, a+2)$. This operation is a substitute of the classical operation [-] and will be used frequently in the present work.

Function. Suppose $\Omega, \Omega^{\prime}$ are sets. Suppose $X$ is an operation that, for each $\omega$ in some nonempty subset $A$ of $\Omega$, constructs a unique member $X(\omega)$ in $\Omega^{\prime}$. Then the operation $X$ is called a function from $\Omega$ to $\Omega^{\prime}$, or simply a function on $\Omega$. The subset $A$ is called the domain of $X$. We then write $X: \Omega \rightarrow \Omega^{\prime}$, and write domain $(X)$ for the set $A$. Thus a function $X$ is an operation that has the additional property that if $\omega_1=\omega_2$ in $\operatorname{domain}(X)$, then $X\left(\omega_1\right)=X\left(\omega_2\right)$ in $\Omega^{\prime}$. To specify a function $X$, we need to specify its domain as well as the operation that produces the image $X(\omega)$ from each given member $\omega$ of $\operatorname{domain}(X)$.
Two functions $X, Y$ are considered equal, $X=Y$ in symbols, if
$\operatorname{domain}(X)=\operatorname{domain}(Y)$,
and if $X(\omega)=Y(\omega)$ for each $\omega \in \operatorname{domain}(X)$. When emphasis is needed, this equality will be referred to as the set-theoretic equality, in contradistinction to almost everywhere equality, to be defined later.

## 数学代写|概率论代写Probability theory代考|Metric Space

The definitions and notations related to metric spaces in [Bishop and Bridges 1985], with few exceptions, are familiar to readers of classical texts. A summary of these definitions and notations follows.

Metric complement. Let $(S, d)$ be a metric space. If $J$ is a subset of $S$, its metric complement is the set ${x \in S: d(x, y)>0$ for all $y \in J}$. Unless otherwise specified, $J_c$ will denote the metric complement of $J$.

Condition valid for all but countably many points in metric space. A condition is said to hold for all but countably many members of $S$ if it holds for each member in the metric complement $J_c$ of some countable subset $J$ of $S$.

Inequality in a metric space. We will say that two elements $x, y \in S$ are unequal, and write $x \neq y$, if $d(x, y)>0$.

Metrically discrete subset of a metric space. We will call a subset $A$ of $S$ metrically discrete if, for each $x, y \in A$ we have $x=y$ or $d(x, y)>0$. Classically, each subset $A$ of $S$ is metrically discrete.

Limit of a sequence of functions with values in a metric space. Let $\left(f_n\right){n=1,2, \ldots .}$ be a sequence of functions from a set $\Omega$ to $S$ such that the set $$D \equiv\left{\omega \in \bigcap{i=1}^{\infty} \operatorname{domain}\left(f_i\right): \lim {i \rightarrow \infty} f_i(\omega) \text { exists in } S\right}$$ is nonempty. Then $\lim {i \rightarrow \infty} f_i$ is defined as the function with domain $\left(\lim {i \rightarrow \infty} f_i\right) \equiv D$ and with value $$\left(\lim {i \rightarrow \infty} f_i\right)(\omega) \equiv \lim {i \rightarrow \infty} f_i(\omega)$$ for each $\omega \in D$. We emphasize that $\lim {i \rightarrow \infty} f_i$ is well defined only if it can be shown that $D$ is nonempty.

## 数学代写|概率论代写Probability theory代考|Set, Operation, and Function

. Inotherwords, unlessotherwisespecified, convergenceofaseriesofrealnumbersmeansabsolute $2.18$

## 数学代写|概率论代写Probability theory代考|Summary

This chapter reviewed the uncertainty in geotechnical engineering, and mainly discussed the uncertainties involved in the estimation of soil properties and geotechnical models. The influencing factors on the uncertainties and relevant studies were summarized. It was pointed out that the uncertainty of soil parameters should be considered and analyzed in estimating a certain soil property. Bayesian probabilistic approach as a useful tool was outlined from two application aspects, that is, parametric identification and model class selection. In view of the complex updated PDF, the chapter reviewed several available numerical simulation methods.

The chapter then reviewed previous studies on two problems of geotechnical engineering, that is, soil water retention property of unsaturated soil and creep behavior of soft soil. In the sections of soil water retention of unsaturated soil, the soil suction as an important factor for the development of the unsaturated soil mechanics was explained first, and its contribution on the soil shear strength, permeability and compressibility was discussed by reviewing the existing studies. The soil-water characteristic curve was then explained, and its influencing factors were summarized. The four commonly used methods for estimating SWCC were finally presented. The objective in the study of SWCC was mentioned, and a new model, which can consider the effect of initial void ratio on the SWCC of same textured soil sample, was required to be constructed and the relevant uncertainty analysis should also be conducted.

In the sections of creep behavior of soft soil, the mechanism of soil creep deformation was presented briefly, and the existing studies on the time-dependent models for describing the creep behavior were reviewed. As the basis of this study, the conceptual time line model proposed by Bjerrum [114] was illustrated, and the 1-D elastic viscoplastic model developed by Yin and Graham [122, 123, 149] based on the Bjerrum’s model and the development of EVP models were reviewed. Several methods for determining the parameters of EVP model were summarized, and the objectives in the study of creep behavior were proposed, that is, to analyze the model parameters by using the Bayesian probabilistic method and to select the suitable model for the predictions of creep behavior of soft soil.

## 数学代写|概率论代写Probability theory代考|Shear Behavior of Granular Soils

：排水条件下空隙率降低，不排水条件下平均有效应力降低。
(b) 高超固结黏土和致密砂在剪切过程中表现出体积膨胀，即膨胀特征（排水条件下孔隙比变大，不排水条件下平均有效应力变大），峰值应力比高于临界应力比。

## 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。