### 计算机代写|计算机图形学作业代写computer graphics代考|Algebra

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等楖率论
• 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

## 计算机代写|计算机图形学作业代写computer graphics代考|Introduction

Some people, including me, find learning a foreign language a real challenge; one of the reasons being the inconsistent rules associated with its syntax. For example, why is a table feminine in French, ‘la table’, and a bed masculine, ‘le lit’? They both have four legs! The rules governing natural language are continuously being changed by each generation, whereas mathematics appears to be logical and consistent. The reason for this consistency is due to the rules associated with numbers and the way they are combined together and manipulated at an abstract level. Such rules, or axioms, generally make our life easy, however, as we saw with the invention of negative numbers, extra rules have to be introduced, such as ‘two negatives make a positive’, which is easily remembered. However, as we explore mathematics, we discover all sorts of inconsistencies, such as there is no real value associated with the square-root of a negative number. It’s forbidden to divide a number by zero. Zero divided by zero gives inconsistent results. Nevertheless, such conditions are easy to recognise and avoided. At least in mathematics, we don’t have to worry about masculine and feminine numbers!

As a student, I discovered Principia Mathematica [1], a three-volume work written by the British philosopher, logician, mathematician and historian Bertrand Russell (1872-1970), and the British mathematician and philosopher Alfred North Whitehead (1861-1947), in which the authors attempt to deduce all of mathematics using the axiomatic system developed by the Italian mathematician Giuseppe Peano (18581932). The first volume established type theory, the second was devoted to numbers, and the third to higher mathematics. The authors did intend a fourth volume on geometry, but it was too much effort to complete. It made extremely intense reading. In fact, I never managed to get pass the first page! It took the authors almost 100 pages of deep logical analysis in the second volume to prove that $1+1=2$ !

## 计算机代写|计算机图形学作业代写computer graphics代考|Background

Modern algebraic notation has evolved over thousands of years where different civilisations developed ways of annotating mathematical and logical problems. The word ‘algebra’ comes from the Arabic ‘al-jabr w’al-muqabal’ meaning ‘restoration and reduction’. In retrospect, it does seem strange that centuries passed before the ‘equals’ sign (=) was invented, and concepts such as ‘zero’ (CE 876) were introduced, especially as they now seem so important. But we are not at the end of this evolution, because new forms of annotation and manipulation will continue to emerge as new mathematical objects are invented.

One fundamental concept of algebra is the idea of giving a name to an unknown quantity. For example, $m$ is often used to represent the slope of a $2 \mathrm{D}$ line, and $c$ is the line’s $y$-coordinate where it intersects the $y$-axis. René Descartes formalised the idea of using letters from the beginning of the alphabet $(a, b, c, \ldots$ ) to represent arbitrary quantities, and letters at the end of the alphabet $(p, q, r, s, t, \ldots, x, y, z)$ to represent quantities such as pressure $(p)$, time $(t)$ and coordinates $(x, y, z)$.

With the aid of the basic arithmetic operators: $+,-, \times, /$ we can develop expressions that describe the behaviour of a physical process or a logical computation. For example, the expression $a x+b y-d$ equals zero for a straight line. The variables $x$ and $y$ are the coordinates of any point on the line and the values of $a, b$ and $d$ determine the position and orientation of the line. The $=$ sign permits the line equation to be expressed as a self-evident statement:
$$0=a x+b y-d$$
Such a statement implies that the expressions on the left- and right-hand sides of the = sign are ‘equal’ or ‘balanced’, and in order to maintain equality or balance,

whatever is done to one side, must also be done to the other. For example, adding $d$ to both sides, the straight-line equation becomes
$$d=a x+b y .$$
Similarly, we could double or treble both expressions, divide them by 4 , or add 6 , without disturbing the underlying relationship. When we are first taught algebra, we are often given the task of rearranging a statement to make different variables the subject. For example, $(3.1)$ can be rearranged such that $x$ is the subject:
\begin{aligned} y &=\frac{x+4}{2-\frac{1}{z}} \ y\left(2-\frac{1}{z}\right) &=x+4 \ x &=y\left(2-\frac{1}{z}\right)-4 . \end{aligned}
Making $z$ the subject requires more effort:
\begin{aligned} y &=\frac{x+4}{2-\frac{1}{z}} \ y\left(2-\frac{1}{z}\right) &=x+4 \ 2 y-\frac{y}{z} &=x+4 \ 2 y-x-4 &=\frac{y}{z} \ z &=\frac{y}{2 y-x-4} \end{aligned}
Parentheses are used to isolate part of an expression in order to select a subexpression that is manipulated in a particular way. For example, the parentheses in $c(a+b)+d$ ensure that the variables $a$ and $b$ are added together before being multiplied by $c$, and finally added to $d$.

## 计算机代写|计算机图形学作业代写computer graphics代考|Solving the Roots of a Quadratic Equation

Problem solving is greatly simplified if one has solved it before, and having a good memory is always an advantage. In mathematics, we keep coming across problems that have been encountered before, apart from different numbers. For example,

$(a+b)(a-b)$ always equals $a^{2}-b^{2}$, therefore factorising the following is a trivial exercise:
\begin{aligned} a^{2}-16 &=(a+4)(a-4) \ x^{2}-49 &=(x+7)(x-7) \ x^{2}-2 &=(x+\sqrt{2})(x-\sqrt{2}) . \end{aligned}
A perfect square has the form:
$$a^{2}+2 a b+b^{2}=(a+b)^{2}$$
Consequently, factorising the following is also a trivial exercise:
\begin{aligned} a^{2}+4 a b+4 b^{2} &=(a+2 b)^{2} \ x^{2}+14 x+49 &=(x+7)^{2} \ x^{2}-20 x+100 &=(x-10)^{2} \end{aligned}
Now let’s solve the roots of the quadratic equation $a x^{2}+b x+c=0$, i.e. those values of $x$ that make the equation equal zero. As the equation involves an $x^{2}$ term, we will exploit any opportunity to factorise it. We begin with the quadratic where $a \neq 0$ :
$$a x^{2}+b x+c=0 .$$
Step 1: Subtract $c$ from both sides to begin the process of creating a perfect square:
$$a x^{2}+b x=-c$$
Step 2: Divide both sides by $a$ to create an $x^{2}$ term:
$$x^{2}+\frac{b}{a} x=-\frac{c}{a} .$$
Step 3: Add $b^{2} / 4 a^{2}$ to both sides to create a perfect square on the left side:
$$x^{2}+\frac{b}{a} x+\frac{b^{2}}{4 a^{2}}=\frac{b^{2}}{4 a^{2}}-\frac{c}{a}$$
Step 4: Factorise the left side:
$$\left(x+\frac{b}{2 a}\right)^{2}=\frac{b^{2}}{4 a^{2}}-\frac{c}{a}$$

0=一种X+b是−d

d=一种X+b是.

## 计算机代写|计算机图形学作业代写computer graphics代考|Solving the Roots of a Quadratic Equation

(一种+b)(一种−b)总是等于一种2−b2，因此分解以下是一个简单的练习：

X2+b一种X=−C一种.

X2+b一种X+b24一种2=b24一种2−C一种

(X+b2一种)2=b24一种2−C一种

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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。