### 数学代写|组合学代写Combinatorics代考|MATH393

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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
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• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|组合学代写Combinatorics代考|Recurrence

A sequence is an infinite collection of numbers ordered following the example of the natural series, which is itself a prime and benchmark sequence. It has its beginning (the number 1) and has no end. When we count: one, two, three, four, …, we spell out natural numbers in the order, in which they form the most fundamental of all sequences. The ability to make a further step at any time during the counting evidence the infinite nature of this sequence.
The structure of the sequence of natural numbers (natural series) can be completely described by several definitive properties, which we have been familiar with since the first years of study of arithmetic. These properties are outlined below.

1. The first natural number is 1 . This is the only natural number which has no predecessor.
2. For every natural number, there is a successive one, and the successor is unique.
3. Every natural number, except for 1, has a preceding one, and the predecessor is unique.
4. Starting with the number 1, then moving to the next number (2), and to the next (3) and so on, after finite (though possibly large) amount of steps we will get to any natural number.

The last property might be hard to understand but it is extremely important. Actually, it means that although the natural series is infinite, every natural number has finite place in it, if one begins the count at 1 .

Now, assume that under every number of the natural series we write another number following some rule (denote these numbers by $a_i$, and let the index $i$ coincide with the corresponding natural number):

## 数学代写|组合学代写Combinatorics代考|Definition of a Sequence by a Recurrence Relation

The fact that a sequence is a function of natural argument, and its members are ordered as a natural series, there is another opportunity to define it, which is essentially different from the previous. In the above discussion, we have considered the direct rule of dependence of the members of a sequence on their numbers. A direct formula explicitly expresses this dependence establishing the correspondence between natural numbers (the numbers of the members of a sequence) and the elements of a sequence.

Another approach is to define the value of each following member of a sequence through values of several previous members and not only with its number. A formula establishing the required relation is called a recurrence relation. An elementary example of such a formula is
$$a_n=a_{n-1}+a_{n-2} .$$
What is the sense of this expression? It tells us about the sequence $\left(a_n\right)$, the members of which follow the rule: each of them (as $n$ is an arbitrary natural number) is the sum of two previous members (because $a_{n-1}$ and $a_{n-2}$ immediately precede $a_n$ ). Is this information about the sequence sufficient to reproduce it? For instance, are we able to determine a few of its starting elements? Clearly, the answer is no. In particular, it is impossible to determine the first member of the sequence. As well as the second one. The formula $a_n=a_{n-1}+a_{n-2}$ can not be applied to the first two members of the sequence, since neither of them has two preceding elements. Therefore, the formula fails from the very beginning. In order to make it work, it is necessary to define two starting members of the sequence. Given this preliminary information, the formula begins operation, tirelessly and relentlessly expanding the sequence: the third term is the sum of the first and second, the fourth term is the sum of the second and third, etc., to infinity.

Obviously, a recurrence relation defines a class of sequences and not the exact sequence. The class comprises all the sequences following this recurrence relation. To distinguish one of the sequences of the class one needs to define a certain amount of its starting members.

## 数学代写|组合学代写Combinatorics代考|Recurrence

1. 第一个自然数是 1 。这是唯一没有前身的自然数。
2. 对于每一个自然数，都有一个后继数，并且后继数是唯一的。
3. 每个自然数，除 1 外，都有一个前导数，且前导数是唯一的。
4. 从数字 1 开始，然后移动到下一个数字 (2)，然后移动到下一个数字 (3)，依此类推，经过有限（尽管可能很大）的步骤后，我们将得到任何自然数。

## 数学代写|组合学代写Combinatorics代考|Definition of a Sequence by a Recurrence Relation

$$a_n=a_{n-1}+a_{n-2} .$$

## 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。