Varignon’s Theorem is a statement in Euclidean geometry by Pierre Varignon that was first published in 1731. It deals with the construction of a particular parallelogram (Varignon parallelogram) from an arbitrary quadrangle (another name for a quadrilateral). The midpoints of the sides of an arbitrary quadrangle form a parallelogram. If the quadrangle is convex, then the area of the parallelogram is half as large as the area of the quadrangle. You will prove the first part of the theorem by solving Problem 35 Problem 35. Prove that if consecutive midpoints of all sides of a quadrilateral are connected, that they form a parallelogram.
Solution. I have watched many students trying to solve this problem. Many of them draw a nice, accurate picture of the problem. From the picture, they make a conclusion: “It is always a parallelogram.” Yes, an accurate picture is $50 \%$ of a successful solution in geometry, but it is not the solution itself.
数学代写|复几何代写Complex Geometry代考|Trapezoids
A trapezoid is a convex quadrilateral with at least one pair of parallel sides. Any pair of angles adjacent to a parallel side sums to $180^{\circ}$ since they are supplementary. The parallel sides are bases of the trapezoid. A midline segment (midsegment) of a trapezoid connects the midpoints of nonparallel sides and is parallel to the bases.
Trapezoid Midsegment Length Theorem. The length of a midsegment is equal to $\frac{1}{2}(a+b)$, where $a$ and $b$ are the bases of a trapezoid.
So, for a trapezoid with bases 4 and 10, irrespective of the lengths of other sides, the midline segment equals $(4+10) / 2=7$.
We will give the proof of the Trapezoid Midsegment Length Theorem in Problem 41.
Problem 41. A trapezoid $A B C D$ with bases $A D=a$ and $B C=b$ is given. $M$ is the midpoint of $A B, N$ is the midpoint of $C D$. Prove that $M N=(a+b) / 2$.
现代博弈论始于约翰-冯-诺伊曼（John von Neumann）提出的两人零和博弈中的混合策略均衡的观点及其证明。冯-诺依曼的原始证明使用了关于连续映射到紧凑凸集的布劳威尔定点定理，这成为博弈论和数学经济学的标准方法。在他的论文之后，1944年，他与奥斯卡-莫根斯特恩（Oskar Morgenstern）共同撰写了《游戏和经济行为理论》一书，该书考虑了几个参与者的合作游戏。这本书的第二版提供了预期效用的公理理论，使数理统计学家和经济学家能够处理不确定性下的决策。
平面几何代写plane geometry代考|MATH113 Angle Bisector and its Properties
数学代写|复几何代写Complex Geometry代考|Angle Bisector and its Properties
Theorem of the Three Angle Bisectors. In a triangle all angle bisectors intersect at one point. This point is the center of the circle inscribed into the triangle. One of the proofs of this statement can be done easily with the use of Ceva’s Theorem and knowledge of some properties of angle bisectors that are given below. However, just for demonstration, we sketch Figure $1.45$ with the use of Geometry Sketch Pad and obtain the following:
Triangle Angle Bisector Theorem. A bisector of an angle of a triangle divides the opposite side of the triangle into segments, which are proportional to the adjacent sides of the triangle. The following formulas are valid. $$ \frac{b}{c}=\frac{a_b}{a_c}, \quad \frac{a}{b}=\frac{c_a}{c_b}, \quad \frac{c}{a}=\frac{b_c}{b_a} $$
数学代写|复几何代写Complex Geometry代考|Median, Bisector and Height from a Vertex
Here are formulas for the bisector, $l_a$, median, $m_a$, and height, $h_a$ dropped from the same vertex to the opposite side, $a$, of triangle $A B C$ (Fig. 1.48): $$ \begin{aligned} l_a &=\frac{2 b c \cdot \cos \left(\frac{\angle A}{2}\right)}{b+c} \ l_a^2 &=b c-a_b \cdot a_c \ m_a &=\frac{1}{2} \sqrt{2 b^2+2 c^2-a^2} \ h_a &=\frac{b c}{a} \cdot \sin (\angle A) \end{aligned} $$ Note. By analogy, you can rewrite these with respect to other sides.
Consider further the triangle $A B C$ of Fig. $1.48$ where $A B=c, C B=a$, and $A C=b . M$ is the midpoint of $B C$ and $m_a=A M$ is a median. If $m \angle B A D=m \angle D A C$, then $l_a=A D$ is the bisector of $\angle B A C$. Let $B D=a_c, D C=a_b, A G \perp B C$ and $h_a=A G$ is the height of triangle $A B C$.
平面几何代写plane geometry代考|MATH113 Angle Bisector and its Properties
现代博弈论始于约翰-冯-诺伊曼（John von Neumann）提出的两人零和博弈中的混合策略均衡的观点及其证明。冯-诺依曼的原始证明使用了关于连续映射到紧凑凸集的布劳威尔定点定理，这成为博弈论和数学经济学的标准方法。在他的论文之后，1944年，他与奥斯卡-莫根斯特恩（Oskar Morgenstern）共同撰写了《游戏和经济行为理论》一书，该书考虑了几个参与者的合作游戏。这本书的第二版提供了预期效用的公理理论，使数理统计学家和经济学家能够处理不确定性下的决策。
平面几何代写plane geometry代考|MAT116 Law of Cosines and Law of Sines
数学代写|复几何代写Complex Geometry代考|Law of Cosines and Law of Sines
The Law of Cosines is the relationship between the sides and the angles of a triangle: $a^2=b^2+c^2-2 b c \cdot \cos (\angle A)$, where $\angle A$ is opposite of side $a$. Students usually learn this formula at the end of a geometry course but don’t understand how to apply it until taking trigonometry. Nevertheless, this formula is very powerful. In addition, the Pythagorean Theorem is a particular case of the Law of Cosines. Let us now solve Problem 8 :
Problem 8. A triangle with sides of length 5, 12, and 13 is given. Find the angle that is opposite the biggest side.
Solution. First, we will draw an accurate picture and put all known information on it so it will help us to calculate the measure of angle $C$ (Fig. 1.15).
Let us apply the Law of Cosines to the triangle: $$ \begin{aligned} &13^2=5^2+12^2-2 \cdot 5 \cdot 12 \cdot \cos (\angle C) \ &0=\cos (\angle C), \quad \text { then } \quad m \angle C=90^{\circ} \end{aligned} $$ We found that triangle $A B C$ is a right triangle. Let us prove that the Pythagorean Theorem is a particular case of the Law of Cosines. For this purpose we will write the Law of Cosines for the cosine of angle $C$ using the same picture: $$ \begin{aligned} &c^2=a^2+b^2-2 a b \cos (\angle C) \ &\cos (\angle C)=\frac{a^2+b^2-c^2}{2 a b} \end{aligned} $$
数学代写|复几何代写Complex Geometry代考|Similar Triangles
Two triangles are called similar if their corresponding angles are equal and the ratio of corresponding sides is the same. Similar triangles or, in general, any similar figures have a similar shape. You probably remember that two triangles are similar to each other by two angles $(A A)$, by two sides and the included angle $(S A S)$, and by three sides (SSS). Especially important is the fact that in similar triangles the ratio of corresponding sides, medians, heights, and bisectors equals $k$, the coefficient of similitude. The ratio of the areas of similar triangles equals $k^2$, the square of the coefficient of similitude.
The following picture will illustrate one way of constructing similar triangles by hand, with $k>1$ (magnification) and $k<1$ (making the image smaller). This is based on Homothetic Transformation or such transformation at which any point of the original (pre-image), its image and the center of homothety lie on the same line. Using homothety, different similar figures can be constructed, so all corresponding angles are equal.
平面几何代写plane geometry代考|MAT116 Law of Cosines and Law of Sines
现代博弈论始于约翰-冯-诺伊曼（John von Neumann）提出的两人零和博弈中的混合策略均衡的观点及其证明。冯-诺依曼的原始证明使用了关于连续映射到紧凑凸集的布劳威尔定点定理，这成为博弈论和数学经济学的标准方法。在他的论文之后，1944年，他与奥斯卡-莫根斯特恩（Oskar Morgenstern）共同撰写了《游戏和经济行为理论》一书，该书考虑了几个参与者的合作游戏。这本书的第二版提供了预期效用的公理理论，使数理统计学家和经济学家能够处理不确定性下的决策。
数学代写|复几何代写Complex Geometry代考|COMP5045 Direct and Inverse Images
数学代写|复几何代写Complex Geometry代考|Direct and Inverse Images
It is useful to transfer a sheaf from one topological space to another via a continuous map $f: X \rightarrow Y$. In fact, we have already done this in special cases. We start by explaining how to push a sheaf on $X$ down to $Y$.
Definition 3.7.1. Given a presheaf $\mathscr{F}$ on $X$, the direct image $f_* \mathscr{F}$ is a presheaf on $Y$ given by $f_* \mathscr{F}(U)=\mathscr{F}\left(f^{-1} U\right)$ with restrictions given by $$ \rho_{f^{-1} U f^{-1} V}: \mathscr{F}\left(f^{-1} U\right) \rightarrow \mathscr{F}\left(f^{-1} V\right) . $$ Lemma 3.7.2. Direct images of sheaves are sheaves. Proof. Suppose that $f: X \rightarrow Y$ is a continuous map and $\mathscr{F}$ is a sheaf on $X$. Let $\left{U_i\right}$ be an open cover of $U \subseteq Y$, and $s_i \in f_* \mathscr{F}\left(U_i\right)$ a collection of sections that agree on the intersections. Then $\left{f^{-1} U_i\right}$ is an open cover of $f^{-1} U$, and we can regard $s_i \in \mathscr{F}\left(f^{-1} U_i\right)$ as a compatible collection of sections for it. Thus we can patch $s_i$ to get a uniquely defined $s \in f_* \mathscr{F}(U)=\mathscr{F}\left(f^{-1} U\right)$ such that $\left.s\right|{U_i}=s_i$. This proves that $f* \mathscr{F}$ is a sheaf.
Now we want to consider the opposite direction. Suppose that $\mathscr{G}$ is a sheaf on $Y$. We would like to pull it back to $X$. We will denote this by $f^{-1} \mathscr{G}$, since $f^*$ is reserved for something related to be defined later on. Naively, we can simply try to define $$ f^{-1} \mathscr{G}(U)=\mathscr{G}(f(U)) . $$ However, this does not yet make sense unless $f(U)$ is open. So as a first step, given any subset $S \subset Y$ of a topological space and a presheaf $\mathscr{G}$, define $$ \mathscr{G}(S)=\lim _{\rightarrow} \mathscr{G}(V) $$ as $V$ ranges over all open neighborhoods of $S$. When $S$ is a point, $\mathscr{G}(S)$ is just the stalk. An element of $\mathscr{G}(S)$ can be viewed as germ of a section defined in a neighborhood of $S$, where two sections define the same germ if their restrictions agree in a common neighborhood. If $S^{\prime} \subset S$, there is a natural restriction map $\mathscr{G}(S) \rightarrow \mathscr{G}\left(S^{\prime}\right)$ given by restriction of germs. So our naive attempt can now be made precise.
数学代写|复几何代写Complex Geometry代考|Differentials
With basic sheaf theory in hand, we can now construct sheaves of differential forms on manifolds and varieties in a unified way. In order to motivate things, let us start with a calculation. Suppose that $X=\mathbb{R}^n$ with coordinates $x_1, \ldots, x_n$. Given a $C^{\infty}$ function $f$ on $X$, we can develop a Taylor expansion about $\left(y_1, \ldots, y_n\right)$ : $$ f\left(x_1, \ldots, x_n\right)=f\left(y_1, \ldots, y_n\right)+\sum \frac{\partial f}{\partial x_i}\left(y_1, \ldots, y_n\right)\left(x_i-y_i\right)+O\left(\left(x_i-y_i\right)^2\right) $$ Thus the differential is given by $$ d f=f\left(x_1, \ldots, x_n\right)-f\left(y_1, \ldots, y_n\right) \bmod \left(x_i-y_i\right)^2 . $$ We can view $x_1, \ldots, x_n, y_1, \ldots, y_n$ as coordinates on $X \times X=\mathbb{R}^{2 n}$, so that $x_i-y_i=0$ defines the diagonal $\Delta$. Then $d f$ lies in the ideal of $\Delta$ modulo its square.
Let $X$ be a $C^{\infty}$ or complex manifold or an algebraic variety over a field $k$. We take $k=\mathbb{R}$ or $\mathbb{C}$ in the first two cases. We have a diagonal map $\delta: X \rightarrow X \times X$ given by $x \mapsto(x, x)$, and projections $p_i: X \times X \rightarrow X$. Let $\mathscr{I}{\Delta}$ be the ideal sheaf of the image of $\delta$, and let $\mathscr{I}{\Delta}^2 \subseteq \mathscr{I}{\Delta}$ be the sub-ideal sheaf locally generated by products of pairs of sections of $\mathscr{I}{\Delta}$. Then we define the sheaf of 1 -forms by $$ \Omega_X^1=\left.\left(\mathscr{I}{\Delta} / \mathscr{I}{\Delta}^2\right)\right|_{\Delta} . $$
数学代写|复几何代写Complex Geometry代考|COMP5045 Direct and Inverse Images
现代博弈论始于约翰-冯-诺伊曼（John von Neumann）提出的两人零和博弈中的混合策略均衡的观点及其证明。冯-诺依曼的原始证明使用了关于连续映射到紧凑凸集的布劳威尔定点定理，这成为博弈论和数学经济学的标准方法。在他的论文之后，1944年，他与奥斯卡-莫根斯特恩（Oskar Morgenstern）共同撰写了《游戏和经济行为理论》一书，该书考虑了几个参与者的合作游戏。这本书的第二版提供了预期效用的公理理论，使数理统计学家和经济学家能够处理不确定性下的决策。
数学代写|复几何代写Complex Geometry代考|MATH3405 The Category of Sheaves
数学代写|复几何代写Complex Geometry代考|The Category of Sheaves
It will be convenient to define presheaves of things other than functions. For instance, one might consider sheaves of equivalence classes of functions, distributions, and so on. For this more general notion of presheaf, the restriction maps have to be included as part of the data:
Definition 3.1.1. A presheaf $\mathscr{P}$ of sets (respectively groups or rings) on a topological space $X$ consists of a set (respectively group or ring) $\mathscr{P}(U)$ for each open set $U$, and maps (respectively homomorphisms) $\rho_{U V}: \mathscr{P}(U) \rightarrow \mathscr{P}(V)$ for each inclusion $V \subseteq U$ such that:
$\rho_{U U}=\mathrm{id}_{\mathscr{P}(U)}$;
$\rho_{V W} \circ \rho_{U V}=\rho_{U W}$ if $W \subseteq V \subseteq U$. We will usually write $\left.f\right|V=\rho{U V}(f)$. Here is a simple example of a presheaf given abstractly.
Example 3.1.2. Let $X$ be topological space. Then $$ \mathscr{P}(U)= \begin{cases}\mathbb{Z} & \text { if } U=X, \ 0 & \text { otherwise }\end{cases} $$ with all $\rho_{U V}=0$, is a presheaf. A more natural class of examples, which arises frequently, is given by the quotient construction.
Example 3.1.3. Let $\mathscr{P}$ be a presheaf of abelian groups. Then a subpresheaf $\mathscr{P}^{\prime} \subseteq \mathscr{P}$ is a collection of subgroups $\mathscr{P}^{\prime}(U) \subseteq \mathscr{P}(U)$ stable under the restriction maps $\rho_{U V}$. The presheaf quotient is given by $$ \left(\mathscr{P} / \mathscr{P}^{\prime}\right)^P(U)=\mathscr{P}(U) / \mathscr{P}(U)^{\prime} $$ with the induced restrictions. (This somewhat clumsy notation is used to distinguish this from the quotient sheaf to be defined later on.) The definition of a sheaf carries over verbatim.
数学代写|复几何代写Complex Geometry代考|Exact Sequences
The categories $\operatorname{PAb}(X)$ and $\operatorname{Ab}(X)$ are additive, which means among other things that $\operatorname{Hom}(A, B)$ has an abelian group structure such that composition is bilinear.
Actually, more is true. These categories are abelian $[44,82,118]$, which implies that they possess many of the basic constructions and properties of the category of abelian groups. In particular, given a morphism, we can form kernels, cokernels, and images, characterized by the appropriate universal properties. This is spelled out more fully in the exercises. Here we just define these operations. Given a morphism of presheaves $f: \mathscr{A} \rightarrow \mathscr{B}$, we define the presheaf kernel, image, and cokernel by $$ \begin{aligned} (\operatorname{pker} f)(U) &=\operatorname{ker} f_U:[\mathscr{A}(U) \rightarrow \mathscr{B}(U)], \ (\operatorname{pim} f)(U) &=\operatorname{im} f_U:[\mathscr{A}(U) \rightarrow \mathscr{B}(U)], \ (\operatorname{pcoker} f)(U) &=\operatorname{coker} f_U:[\mathscr{A}(U) \rightarrow \mathscr{B}(U)] . \end{aligned} $$ This is an isomorphism if $f_U$ is an isomorphism for every $U$, or equivalently if pker $f=\operatorname{pcoker} f=0$. For a morphism of sheaves $f: \mathscr{A} \rightarrow \mathscr{B}$, the sheaf kernel, etc. is given by $$ \operatorname{ker} f=(\operatorname{pker} f)^{+}, \quad \operatorname{im} f=(\operatorname{pim} f)^{+}, \quad \operatorname{coker} f=(\operatorname{pcoker} f)^{+} . $$ We may get a better sense of these by looking at the stalks: $$ \begin{aligned} (\operatorname{ker} f)_x &=(\operatorname{pker} f)_x=\operatorname{ker} f_x:\left[\mathscr{A}_x \rightarrow \mathscr{B}_x\right] \ (\operatorname{im} f)_x &=(\operatorname{pim} f)_x=\operatorname{im} f_x:\left[\mathscr{A}_x \rightarrow \mathscr{B}_x\right] \ (\operatorname{coker} f)_x &=(\operatorname{pcoker} f)_x=\operatorname{coker} f_x:\left[\mathscr{A}_x \rightarrow \mathscr{B}_x\right] . \end{aligned} $$
数学代写|复几何代写Complex Geometry代考|MATH3405 The Category of Sheaves
现代博弈论始于约翰-冯-诺伊曼（John von Neumann）提出的两人零和博弈中的混合策略均衡的观点及其证明。冯-诺依曼的原始证明使用了关于连续映射到紧凑凸集的布劳威尔定点定理，这成为博弈论和数学经济学的标准方法。在他的论文之后，1944年，他与奥斯卡-莫根斯特恩（Oskar Morgenstern）共同撰写了《游戏和经济行为理论》一书，该书考虑了几个参与者的合作游戏。这本书的第二版提供了预期效用的公理理论，使数理统计学家和经济学家能够处理不确定性下的决策。
数学代写|复几何代写Complex Geometry代考|MATH3033 Manifolds and Varieties via Sheaves
数学代写|复几何代写Complex Geometry代考|Sheaves of Functions
As we said above, we need to define sheaves in order eventually to define manifolds and varieties. We start with a more primitive notion. In many parts of mathematics, we encounter topological spaces with distinguished classes of functions on them: continuous functions on topological spaces, $C^{\infty}$-functions on $\mathbb{R}^n$, holomorphic functions on $\mathbb{C}^n$, and so on. These functions may have singularities, so they may be defined only over subsets of the space; we will be interested primarily in the case that these subsets are open. We say that such a collection of functions is a presheaf if it is closed under restriction. Given sets $X$ and $T$, let $\operatorname{Map}_T(X)$ denote the set of maps from $X$ to $T$. Here is the precise definition of a presheaf, or rather of the kind of presheaf we need at the moment.
Definition 2.1.1. Suppose that $X$ is a topological space and $T$ a nonempty set. A presheaf of $T$-valued functions on $X$ is a collection of subsets $\mathscr{P}(U) \subseteq \operatorname{Map}_T(U)$, for each open $U \subseteq X$, such that the restriction $\left.f\right|_V$ belongs to $\mathscr{P}(V)$ whenever $f \in \mathscr{P}(U)$ and $V \subset U$
The collection of all functions $\operatorname{Map}T(U)$ is of course a presheaf. Less trivially: Example 2.1.2. Let $T$ be a topological space. Then the set of continuous functions Cont ${X, T}(U)$ from $U \subseteq X$ to $T$ is a presheaf.
Example 2.1.3. Let $X$ be a topological space and let $T$ be a set. The set $T^P(U)$ of constant functions from $U$ to $T$ is a presheaf called the constant presheaf.
Example 2.1.4. Let $X=\mathbb{R}^n$. The sets $C^{\infty}(U)$ of $C^{\infty}$ real-valued functions form a presheaf.
Example 2.1.5. Let $X=\mathbb{C}^n$. The sets $\mathscr{O}(U)$ of holomorphic functions on $U$ form a presheaf. (A function of several variables is holomorphic if it is $C^{\infty}$ and holomorphic in each variable.)
Example 2.1.6. Let $L$ be a linear differential operator on $\mathbb{R}^n$ with $C^{\infty}$ coefficients (e.g., $\Sigma \partial^2 / \partial x_i^2$ ). Let $S(U)$ denote the space of $C^{\infty}$ solutions to $L f=0$ in $U$. This is a presheaf with values in $\mathbb{R}$.
Example 2.1.7. Let $X=\mathbb{R}^n$. The sets $L^p(U)$ of measurable functions $f: U \rightarrow \mathbb{R}$ satisfying $\int_U|f|^p<\infty$ form a presheaf.
数学代写|复几何代写Complex Geometry代考|Manifolds
As explained in the introduction, a manifold consists of a topological space with a distinguished class of functions that looks locally like $\mathbb{R}^n$. We now set up the language necessary to give a precise definition. Let $k$ be a field. Then $\operatorname{Map}_k(X)$ is a commutative $k$-algebra with pointwise addition and multplication.
Definition 2.2.1. Let $\mathscr{R}$ be a sheaf of $k$-valued functions on $X$. We say that $\mathscr{R}$ is a sheaf of algebras if each $R(U) \subseteq \operatorname{Map}_k(U)$ is a subalgebra when $U$ is nonempty. We call the pair $(X, \mathscr{R})$ a concrete ringed space over $k$ or simply a concrete $k$-space. We will sometimes refer to elements of $\mathscr{R}(U)$ as distinguished functions.
The sheaf $\mathscr{R}$ is called the structure sheaf of $X$. In this chapter, we usually omit the modifier “concrete,” but we will use it later on after we introduce a more general notion. Basic examples of $\mathbb{R}$-spaces are $\left(\mathbb{R}^n\right.$, Cont $\left._{\mathbb{R}^n, \mathbb{R}}\right)$ and $\left(\mathbb{R}^n, C^{\infty}\right)$, while $\left(\mathbb{C}^n, \mathscr{O}\right)$ is an example of a $\mathbb{C}$-space.
We also need to consider maps $F: X \rightarrow Y$ between such spaces. We will certainly insist on continuity, but in addition we require that when a distinguished function is precomposed with $F$, or “pulled back” along $F$, it remain distinguished. Definition 2.2.2. A morphism of $k$-spaces $(X, \mathscr{R}) \rightarrow(Y, \mathscr{S})$ is a continuous map $F: X \rightarrow Y$ such that if $f \in \mathscr{S}(U)$, then $F^* f \in \mathscr{R}\left(F^{-1} U\right)$, where $F^* f=\left.f \circ F\right|_{f^{-1} U}$. It is worthwhile noting that this completely captures the notion of a $C^{\infty}$, or holomorphic, map between Euclidean spaces. Example 2.2.3. A $C^{\infty}$ map $F: \mathbb{R}^n \rightarrow \mathbb{R}^m$ induces a morphism $\left(\mathbb{R}^n, C^{\infty}\right) \rightarrow\left(\mathbb{R}^m, C^{\infty}\right)$ of $\mathbb{R}$-spaces, since $C^{\infty}$ functions are closed under composition. Conversely, if $F$ is a morphism, then the coordinate functions on $\mathbb{R}^m$ are expressible as $C^{\infty}$ functions of the coordinates of $\mathbb{R}^n$, which implies that $F$ is $C^{\infty}$.
Example 2.2.4. Similarly, a continuous map $F: \mathbb{C}^n \rightarrow \mathbb{C}^m$ induces a morphism of $\mathbb{C}$-spaces if and only if it is holomorphic.
This is a good place to introduce, or perhaps remind the reader of, the notion of a category [82]. A category $\mathscr{C}$ consists of a set (or class) of objects $\mathrm{Obj} \mathscr{C}$ and for each pair $A, B \in \mathscr{C}$, a set $\operatorname{Hom}{\mathscr{C}}(A, B)$ of morphisms from $A$ to $B$. There is a composition law $$ \text { o: } \operatorname{Hom}{\mathscr{C}}(B, C) \times \operatorname{Hom}{\mathscr{C}}(A, B) \rightarrow \operatorname{Hom}{\mathscr{C}}(A, C), $$
and distinguished elements $i d_A \in \operatorname{Hom}{\mathscr{C}}(A, A)$ that satisfy (C1) associativity: $f \circ(g \circ h)=(f \circ g) \circ h$, (C2) identity: $f \circ \mathrm{id}_A=f$ and $\mathrm{id}_A \circ g=g$, whenever these are defined. Categories abound in mathematics. A basic example is the category of Sets. The objects are sets, $\operatorname{Hom}{\text {Sets }}(A, B)$ is just the set of maps from $A$ to $B$, and composition and $\mathrm{id}_A$ have the usual meanings. Similarly, we can form the category of groups and group homomorphisms, the category of rings and ring homomorphisms, and the category of topological spaces and continuous maps. We have essentially constructed another example. We can take the class of objects to be $k$-spaces, and morphisms as above. These can be seen to constitute a category once we observe that the identity is a morphism and the composition of morphisms is a morphism.
The notion of an isomorphism makes sense in any category. We will spell this out for $k$-spaces.
数学代写|复几何代写Complex Geometry代考|MATH3033 Manifolds and Varieties via Sheaves
现代博弈论始于约翰-冯-诺伊曼（John von Neumann）提出的两人零和博弈中的混合策略均衡的观点及其证明。冯-诺依曼的原始证明使用了关于连续映射到紧凑凸集的布劳威尔定点定理，这成为博弈论和数学经济学的标准方法。在他的论文之后，1944年，他与奥斯卡-莫根斯特恩（Oskar Morgenstern）共同撰写了《游戏和经济行为理论》一书，该书考虑了几个参与者的合作游戏。这本书的第二版提供了预期效用的公理理论，使数理统计学家和经济学家能够处理不确定性下的决策。
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