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# 数学代写代考|Recursion 离散数学

## 数学代写| Recursion 离散代考

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## 离散数学代写

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

$$1,1,2,3,5,8,13,21, \ldots$$
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

$$1,-1,1,-1,1,-1$$

$$(-1)^{n+1}$$

$$\sum_{k=1}^{n} a_{k}$$

$$\lim {n \rightarrow \infty} \sum{k=1}^{n} a_{k}=\mathrm{S} 。$$

## 图论代考

$n=0$ 或 $n=1$，而递归步骤指定如何应用

$n(n-1)(n-2) \ldots 3.2 .1$ 和 $0 !=1$。根据基本情况的正式定义

$\begin{array}{ll}\text { 基本步骤 } & \text { fac }(0)=1 \ \text { 递归步骤 } & \text { fac }(n)=n * \operatorname{fac }(n-1)\end{数组}$

$n=0$ 或 $n=1$，而递归步骤指定如何应用

$\mathrm{~ T h e ~ r e a d e r ~ i s ~ r e f e r}$

## 密码学代考

• 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

1. 数据压缩（或信源编码
2. 前向错误更正（或信道编码
3. 加密编码
4. 线路码