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# 经济代写| Axioms: convexity, monotonicity, and continuity微观经济学代写

## 经济代写

3.1. Convex preferences. We will often assume convexity of preferences and monotonicity. We will first need some math.
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IV. ORDINAL PREFERENCE THEORY
FIGURE IV.7. Convex sets?
DefinItiON IV 17 (CONVEX COMBINATION). Let $x$ and $y$ be elements of $\mathbb{R}^{\ell}$. Then,
$$k x+(1-k) y, k \in[0,1]$$
is called the convex combination of $x$ and $y$.
We have seen a convex combination before, in exercise IV.1 (p. 59). The convex combination of $x$ and $y$ lies on the line connecting $x$ and $y$. The smaller $k$, the closer the convex combination to $y$, which is also clear from
$$k x+(1-k) y=y+k(x-y) \text {. }$$
DEFINITION IV.18 (CONVEX SET). A set $M \subseteq \mathbb{R}^{\ell}$ is called convex if for any two points $x$ and $y$ from $M$, their convex combinations are also contained in $M$.

Here, convexity is a property of sets and is not to be confused with the convexity of functions. In figure IV.7, the left-hand example shows a set that is not convex, while the other two sets exhibit convexity.

EXERCISE IV.12. Show that the intersection of two convex sets is also convex.

DEFINITION IV.19 (STRICTLY CONVEX SET). A set $M$ is called strictly convex if for any two points $x$ and $y$ from $M, x \neq y$,
$$k x+(1-k) y$$
is an interior point of $M$ for any $k \in(0,1)$.
The right-most set in figure IV.7 is strictly convex, while the middle set is convex but not strictly so. Convince yourself that open $\varepsilon-b a l l s ~ i n ~ \mathbb{R}^{2}$ are strictly convex for any norm. How about closed $\varepsilon-b a l l s ?$

EXERCISE IV. 13. Are the intervals $(0, \infty),[0,3]$, or $[0, \infty)$ convex or strictly convex?

1. AXIOMS: CONVEXITY, MONOTONICITY, AND CONTINUITY
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FIGURE IV.8. Two interior points
DEFINITION IV.20 (CONVEX PREFERENCE RELATION). A preference relation $\succ$ on $\mathbb{R}_{+}^{\ell}$ is
• convex if all its better sets $B_{y}$ are convex,
• strictly convex if all its better sets $B_{y}$ are strictly convex,
• concave if all its worse sets $W_{y}$ are convex,
• strictly concave if all its worse sets $W_{y}$ are strictly convex.
A rough description of convexity is “mixtures are prefered to extremes”.
I, for example, would rather have 1 glass of milk and one donut than two

3.1。凸偏好。我们通常会假设偏好的凸性和单调性。我们首先需要一些数学知识。
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$$k x+(1-k) y, k \in[0,1]$$

$$k x+(1-k) y=y+k(x-y) \text {. }$$

$$k x+(1-k) y$$

1. 公理：凸性、单调性和连续性
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图 IV.8。两个内点
定义 IV.20（凸偏好关系）。 $\mathbb{R}_{+}^{\ell}$ 上的偏好关系 $\succ$ 是
• 如果所有更好的集合 $B_{y}$ 都是凸的，则为凸的，
• 如果所有更好的集合 $B_{y}$ 都是严格凸的，则严格凸的，
• 如果所有较差的集合 $W_{y}$ 都是凸的，则为凹的，
• 如果所有较差的集合 $W_{y}$ 都是严格凸的，则严格凹。
对凸性的粗略描述是“混合物优于极端”。
例如，我宁愿喝 1 杯牛奶和一个甜甜圈，也不愿两个

## 经济代考

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## 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

## 复分析代考

(1) 提到复变函数 ，首先需要了解复数的基本性左和四则运算规则。怎么样计算复数的平方根， 极坐标与 $x y$ 坐标的转换，复数的模之类的。这些在高中的时候囸本上都会学过。
(2) 复变函数自然是在复平面上来研究问题，此时数学分析里面的求导数之尖的运算就会很自然的 引入到复平面里面，从而引出解析函数的定义。那/研究解析函数的性贡就是关楗所在。最关键的 地方就是所谓的Cauchy一Riemann公式，这个是判断一个函数是否是解析函数的关键所在。
(3) 明白解析函数的定义以及性质之后，就会把数学分析里面的曲线积分 $a$ 的概念引入复分析中， 定义几乎是一致的。在引入了闭曲线和曲线积分之后，就会有出现复分析中的重要的定理: Cauchy 积分公式。 这个是易分析的第一个重要定理。