## 数学代写|数论作业代写number theory代考|PMATH441

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## 数学代写|数论作业代写number theory代考|SIMULTANEOUS EQUATIONS WITH INTEGRAL COEFFICIENTS

Let $a, b, c, d$ and $p, q$ be integers. If $a d-b c=\pm 1$, then the simultaneous equations
$$\left{\begin{array}{l} a x+b y=p \ c x+d y=q \end{array}\right.$$
have an integral solution $x, y$. Conversely, if the system has an integral solution for all integers $p, q$, then $a d-b c=\pm 1$.
Proof. The solution can be written
$$\left(\begin{array}{l} x \ y \end{array}\right)=\left(\begin{array}{ll} a & b \ c & d \end{array}\right)^{-1}\left(\begin{array}{l} p \ q \end{array}\right)=\frac{1}{a d-b c}\left(\begin{array}{cc} d & -b \ -c & a \end{array}\right)\left(\begin{array}{l} p \ q \end{array}\right)=\frac{1}{a d-b c}\left(\begin{array}{l} d p-b q \ a q-c p \end{array}\right)$$
provided that $a d-b c \neq 0$. It is clear that if $a d-b c=\pm 1$, then $x$ and $y$ are integers. Conversely, suppose that $|a d-b c|>1$ and consider the solutions when $p=1, q=0$ and when $p=0, q=1$. If these solutions are to be integers, then $a d-b c$ must be a factor of $a, b, c$ and $d$. But this leads to
$$(a d-b c)^{2} \mid a d-b c$$
which is impossible. Finally note that if $a d-b c=0$, then there exist $p, q$ for which the system has no solution at all, and therefore certainly no integral solution.

Exercise. Let $A$ be an $n \times n$ matrix with integral entries. Show that the linear equations $A \mathbf{x}=\mathbf{b}$ have a solution $\mathbf{x}$ with integral components for all integer vectors $\mathbf{b}$, if and only if $\operatorname{det}(A)=\pm 1$.

## 数学代写|数论作业代写number theory代考|BASIC MUSICAL TERMINOLOGY

Some musical terminology, for readers who may not already be familiar with it. Consider a piano keyboard, part of which is shown in figure $4.3$. The white keys are labelled with the first seven letters of the alphabet: $A, B, C, D, E$, $\mathrm{F}$, G. After using all of these we start again. Note that $A$ follows $G$, and that $\mathrm{C}$ appears at both the left and right-hand ends of the diagram. A black key is given the name of the white key just below it, with a sharp ( $#$ ) added; or of the white key just above, with a flat (b) added. Thus the leftmost black key in the diagram is called $\mathrm{C} \sharp$ or $\mathrm{D}$ b, pronounced “C sharp”, ” $\mathrm{D}$ flat”.

The interval from any key to the next is called a semitone. This is the smallest interval used in the majority of traditional Western music. For example, $\mathrm{C}-\mathrm{C} \sharp \mathrm{G}-\mathrm{A}, \mathrm{B}-\mathrm{C}$ are all semitones. The interval from any key to the next key of the same name (for example $\mathrm{C}-\mathrm{C}, \mathrm{E} b-\mathrm{E}$ ) ) is called an octave. Counting five steps up a scale (including the first and last notes) gives the interval of a fifth, sometimes, for emphasis, called a perfect fifth. (Musical readers will know that there are other kinds of fifths, but we shall not be concerned with them here.) Instances of perfect fifths are $C-G, B-F \sharp$ and Gb-Db.

Musical sounds are caused by regular vibrations in the air (or in other media). The number of vibrations per second causing any particular note is the frequency of that note in units of $\operatorname{Hertz}(\mathrm{Hz})$. For example, the modern standard of orchestral pitch is established by defining the note A above middle C to have a frequency of exactly $440 \mathrm{~Hz}$ – that is, 440 vibrations per second. It is found by observation (and backed up by psychological and physiological theories) that when two notes of different pitches are sounded simultaneously or consecutively, the result is most pleasing to the ear if the ratio of the frequencies of the pitches is a simple fraction. The simplest possible fractions are $\frac{2}{1}$ and $\frac{3}{2}$, and these correspond to the intervals of the octave and the perfect fifth respectively. Middle $\mathrm{C}$, for example, has a frequency of $262 \mathrm{~Hz}$; a perfect fifth above is $\mathrm{G}$ with a frequency of $393 \mathrm{~Hz}$; the octave above middle $\mathrm{C}$ is the $\mathrm{C}$ with frequency $524 \mathrm{~Hz}$.

The above is adapted, with permission, from the present author’s article [3]. For accessible reading on acoustical aspects of music, the classic text is that by Sir James Jeans [34]. Very much more detailed information may be found in $[12]$.

## 数学代写|数论作业代写number theory代考|SIMULTANEOUS EQUATIONS WITH INTEGRAL COEFFICIENTS

$$\left{ 一个X+b是=p CX+d是=q\正确的。 H一个在和一个n一世n吨和Gr一个ls○l在吨一世○nX,是.C○n在和rs和l是,一世F吨H和s是s吨和米H一个s一个n一世n吨和Gr一个ls○l在吨一世○nF○r一个ll一世n吨和G和rsp,q,吨H和n一个d−bC=±1.磷r○○F.吨H和s○l在吨一世○nC一个nb和在r一世吨吨和n \剩下（ X 是\右）=\左（ 一个b Cd\right)^{-1}\left( p q\right)=\frac{1}{a db c}\left( d−b −C一个\右左（ p q\right)=\frac{1}{a db c}\left( dp−bq 一个q−Cp\正确的） pr○在一世d和d吨H一个吨一个d−bC≠0.我吨一世sCl和一个r吨H一个吨一世F一个d−bC=±1,吨H和nX一个nd是一个r和一世n吨和G和rs.C○n在和rs和l是,s在pp○s和吨H一个吨|一个d−bC|>1一个ndC○ns一世d和r吨H和s○l在吨一世○ns在H和np=1,q=0一个nd在H和np=0,q=1.我F吨H和s和s○l在吨一世○ns一个r和吨○b和一世n吨和G和rs,吨H和n一个d−bC米在s吨b和一个F一个C吨○r○F一个,b,C一个ndd.乙在吨吨H一世sl和一个ds吨○ (a db c)^{2} \mid a db c$$

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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。