数学代写代考|Recursion 离散数学代写 - 代考代写：100%准时可靠您的作业代写专家

Some functions (or objects) used in mathematics (e.g. the Fibonacci sequence) are
difficult to define explicitly, and are best defined by a recurrence relation: i.e. an
equation that recursively defines a sequence of values, once one or more initial
values are defined. Recursion may be employed to define functions, sequences and
sets.
There are two parts to a recursive definition, namely the base case and the
recursive (inductive) step. The base case usually defines the value of the function at
$n=0$ or $n=1$, whereas the recursive step specifies how the application of the
function to a number may be obtained from its application to one or more smaller
numbers. It is important that care is taken with the recursive definition, to ensure that that it
is not circular, and does not lead to an infinite regress. The argument of the function
on the right-hand side of the definition in the recursive step is usually smaller than
the argument on the left-hand side to ensure termination (there are some unusual
recursively defined functions such as the McCarthy 91 function where this is not the
case). It is natural to ask when presented with a recursive definition whether it means
anything at all, and in some cases the answer is negative. The fixed-point theory
provides the mathematical foundations for recursion, and ensures that the
functions/objects are well defined.
Chapter 12 (Sect. 12.6) discusses various mathematical structures such as partial
orders, complete partial orders and lattices, which may be employed to give a
secure foundation for recursion. A precise mathematical meaning is given to
recursively defined functions in terms of domains and fixed-point theory, and it is
essential that the conditions in which recursion may be used
The reader is referred to [1] for more detailed information.
A recursive definition will include at least one non-recursive branch with every recursive branch occurring in a context that is different from the original, and brings
it closer to the non-recursive case. Recursive definitions are a powerful and elegant
way of giving the denotational semantics of language constructs.
Next, we present examples of the recursive definition of the factorial function $\mathrm{~ N e x t i v i n g ~ d e n t i o n t}$
and Fibonacci numbers.
Example 4.4 (Recursive Definition of Functions) The factorial function $n !$ is very
common in mathematics and its well-known definition is $n !=$
$n(n-1)(n-2) \ldots 3.2 .1$ and $0 !=1$. The formal definition in terms of a base case
and inductive step is given as follows:
$\begin{array}{ll}\text { Base Step } & \text { fac }(0)=1 \ \text { Recursive Step } & \text { fac }(n)=n * \operatorname{fac}(n-1)\end{array}$
Some functions (or objects) used in mathematics (e.g. the Fibonacci sequence) are
difficult to define explicitly, and are best defined by a recurrence relation: i.e. an
equation that recursively defines a sequence of values, once one or more initial
values are defined. Recursion may be employed to define functions, sequences and
sets.
There are two parts to a recursive definition, namely the base case and the
recursive (inductive) step. The base case usually defines the value of the function at
$n=0$ or $n=1$, whereas the recursive step specifies how the application of the
function to a number may be obtained from its application to one or more smaller
numbers.
It is important that care is taken with the recursive definition, to ensure that that it
is not circular, and does not lead to an infinite regress. The argument of the function
on the right-hand side of the definition in the recursive step is usually smaller than
the argument on the left-hand side to ensure termination (there are some unusual
recursively defined functions such as the $M c$ Carthy 91 function where this is not the
case).
It is natural to ask when presented with a recursive definition whether it means
anything at all, and in some cases the answer is negative. The fixed-point theory
provides the mathematical foundations for recursion, and ensures that the
functions/objects are well defined.
Chapter 12 (Sect. $12.6$ ) discusses various mathematical structures such as partial
orders, complete partial orders and lattices, which may be employed to give a
secure foundation for recursion. A precise mathematical meaning is given to
recursively defined functions in terms of domains and fixed-point theory, and it is
essential that the conditions in which recursion may be used safely be understood.
$\mathrm{~ T h e ~ r e a d e r ~ i s ~ r e f e r}$
This recursive definition defines the procedure by which the factorial of a number is
determined from the base case, or by the product of the number by the factorial of

$5.2$ 序列和系列
一个序列 $a_{1}, a_{2}, \ldots 。 a_{n} \ldots$ 是任何连续的项（通常是数字），我们在前面的章节中讨论了斐波那契数列。 4. 斐波那契数列中的每一项（除了前两项）都是从序列中前两项之和得到的：
$$
1,1,2,3,5,8,13,21, \ldots
$$
89
(C) 作者，获得 Springer Nature Switzerland AG 2021 的独家许可
G. O’Regan，离散数学指南，计算机科学文本，
https://doi.org/10.1007/978-3-030-81588-2_5
90
5 序列、系列、排列和组合
序列可能是有限的（具有固定数量的项）或无限的。斐波那契数列是无限的，而数列 $2,4,6,8,10$ 是有限的。我们区分收敛序列和发散序列，其中收敛序列随着 $n$ 变得越来越大而接近某个值（技术上，我们说 $_{n \rightarrow \infty} \lim a_{n}$ 存在（即存在 $a_{n}$ 的限制））。否则，称该序列是发散的。

通常，序列中的第 $n$ 项有一个数学表达式（例如，对于偶数序列 $2,4,6,8，\ldots$，$a_{n}$ 的一般表达式由 $ a_{n}=2 n$)。显然，偶数的序列是发散的，因为它不接近特定值，因为 $n$ 变得越来越大。考虑以下顺序：
$$
1,-1,1,-1,1,-1
$$
然后这个序列是发散的，因为它没有接近某个值，因为 $n$ 变得越来越大，因为它继续在 1 和 $-1$ 之间交替。序列中第 $n$ 项的公式可以由下式给出。
$$
(-1)^{n+1}
$$
序列 $1,1 / 2,1 / 3,1 / 4, \ldots 1 / n \ldots$ 是收敛的并且收敛到 0 。序列中的第 $n$ 项由 $1 /_{n}$ 给出，并且随着 $n$ 越来越大，它越来越接近 0 。

一个系列是一个序列中各项的总和，序列 $a_{1}, a_{2}, \ldots a_{n} \ldots$ 的前 $n$ 项的总和由 $a_ 给出{1}+a_{2}+\cdots+a_{n}$，记为
$$
\sum_{k=1}^{n} a_{k}
$$
如果随着 $n$ 越来越大，一个级数的总和接近某个值 $S$，则该级数是收敛的，这正式写为
$$
\lim {n \rightarrow \infty} \sum{k=1}^{n} a_{k}=\mathrm{S} 。
$$
否则，该系列被称为发散的。

图论代考

数学中使用的一些函数（或对象）（例如斐波那契数列）是
难以明确定义，最好由递归关系定义：即
递归定义一系列值的方程，一旦一个或多个初始值
值被定义。递归可用于定义函数、序列和
套。
递归定义有两个部分，即基本情况和
递归（归纳）步骤。基本情况通常定义函数的值
$n=0$ 或 $n=1$，而递归步骤指定如何应用
一个数的函数可以从其应用到一个或多个更小的函数中获得
数字。重要的是要注意递归定义，以确保它
不是循环的，也不会导致无限回归。函数的参数
递归步骤中定义的右侧通常小于
左侧的参数以确保终止（有一些不寻常的
递归定义的函数，例如 McCarthy 91 函数，其中 this 不是
案子）。当出现递归定义时，很自然地会问它是否意味着
任何东西，在某些情况下答案是否定的。不动点理论
为递归提供数学基础，并确保
功能/对象定义明确。
第 12 章（第 12.6 节）讨论了各种数学结构，例如偏
阶、完全偏阶和格，可用于给出
为递归奠定坚实的基础。给出了精确的数学含义
根据域和不动点理论递归定义的函数，它是
重要的是可以使用递归的条件
更多详细信息请参阅 [1]。
递归定义将包括至少一个非递归分支，每个递归分支都发生在与原始分支不同的上下文中，并带来
它更接近非递归情况。递归定义是一种强大而优雅的
给出语言结构的指称语义的方法。
接下来，我们给出阶乘函数 $\mathrm{~ N e x t i v i n g ~ d e n t i o n t}$ 的递归定义的例子
和斐波那契数。
例 4.4（函数的递归定义）阶乘函数 $n !$ 非常
在数学中很常见，其著名的定义是 $n !=$
$n(n-1)(n-2) \ldots 3.2 .1$ 和 $0 !=1$。根据基本情况的正式定义
归纳步骤如下：
$\begin{array}{ll}\text { 基本步骤 } & \text { fac }(0)=1 \ \text { 递归步骤 } & \text { fac }(n)=n * \operatorname{fac }(n-1)\end{数组}$
数学中使用的一些函数（或对象）（例如斐波那契数列）是
难以明确定义，最好由递归关系定义：即
递归定义一系列值的方程，一旦一个或多个初始值
值被定义。递归可用于定义函数、序列和
套。
递归定义有两个部分，即基本情况和
递归（归纳）步骤。基本情况通常定义函数的值
$n=0$ 或 $n=1$，而递归步骤指定如何应用
一个数的函数可以从其应用到一个或多个更小的函数中获得
数字。
重要的是要注意递归定义，以确保它
不是循环的，也不会导致无限回归。函数的参数
递归步骤中定义的右侧通常小于
左侧的参数以确保终止（有一些不寻常的
递归定义的函数，例如 $M c$ Carthy 91 函数，其中 this 不是
案子）。
当出现递归定义时，很自然地会问它是否意味着
任何东西，在某些情况下答案是否定的。不动点理论
为递归提供数学基础，并确保
功能/对象定义明确。
第 12 章（Sect. $12.6$）讨论了各种数学结构，例如偏
阶、完全偏阶和格，可用于给出
为递归奠定坚实的基础。给出了精确的数学含义
根据域和不动点理论递归定义的函数，它是
理解可以安全使用递归的条件至关重要。
$\mathrm{~ T h e ~ r e a d e r ~ i s ~ r e f e r}$
这个递归定义定义了一个数字的阶乘的过程
由基本情况确定，或由数字乘以的阶乘确定

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Cryptosystem
A system that describes how to encrypt or decrypt messages
Plaintext
Message in its original form
Ciphertext
Message in its encrypted form
Cryptographer
Invents encryption algorithms
Cryptanalyst
Breaks encryption algorithms or implementations

编码理论代写

编码理论（英语：Coding theory）是研究编码的性质以及它们在具体应用中的性能的理论。编码用于数据压缩、加密、纠错，最近也用于网络编码中。不同学科（如信息论、电机工程学、数学、语言学以及计算机科学）都研究编码是为了设计出高效、可靠的数据传输方法。这通常需要去除冗余并校正（或检测）数据传输中的错误。

编码共分四类：^[1]