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Definition 1.1. (Presheaf) Let $X$ be a topological space, where we denote $\mathfrak{Top}(X)$ be the category with: $\text{Ob}(\mathfrak{Top}(X)) = \{\text{open sets of }X\}$. $\text{Mor}_{\mathfrak{Top}(X)}(U,V) = \begin{cases} \text{inclusion} &,U \subset V\\ \varnothing &, \text{otherwise}\end{cases}$ A presheaf of abelien group on $X$ is a contravariant functor $\mathscr{F} \colon \mathfrak{Top}(X) \to \mathfrak{Ab}$, where $\mathfrak{Ab}$ is the category of abelien groups. Precisely, that is: $\mathscr{F}$ assigns an abelien group $\mathscr{F}(U)$ to every open set $U \in \mathfrak{Top}(X)$. $\mathscr{F}(\varnothing) = \text{Trivial group}$. For each $U \subset V$, $\mathscr{F}$ assigns a group homomorphism $\rho_{VU} \colon \mathscr{F}(V) \to \mathscr{F}(U)$. $\rho_{UU} = \text{Id}_{U}$. For each $U \subset V \subset W$, we have $\rho_{WU} = \rho_{WV} \circ \rho_{VU}$ We called $\mathscr{F}(U) = \Gamma(U, \mathscr{F})$ the sections of the presheaf ...

Note 1. Varieties (1) Part 1: Definitions Definition 1.1. (Variety) Let $k$ be an algebraically closed field, then an affine $n-$space  is $\mathbb{A}_k^n = k^n$. For an ideal $I \triangleleft R = k[x_i \mid 1 \leqslant i \leqslant n]$, an affine algebraic set  is defined by: $$V(I) = \{a \in \mathbb{A}_k^n \mid f(a) = 0 \text{ for all } f \in I\}$$ Proposition 1.2. Affine algebraic set satisfy the following: $V(R) = \varnothing$, $V((0)) = \mathbb{A}_k^n$. $V(I) \cup V(J) = V(IJ)$ For aribitrary index set $A$, we have $\displaystyle \bigcap_{\alpha \in A}V(I_\alpha) = V \left(\sum_{\alpha \in A}I_{\alpha}\right)$ These fact shows that by defining $\{V(I) \mid I \triangleleft R\}$ be exactly the closed set of $\mathbb{A}_k^n$, we may induce a topological space, namely, Zariski topology . Definition 1.2. A toplogical space $X$ is said to be irreducible if for any proper pair of closed set $C_1$, $C_2$, $X \neq C_1 \cup C_2$. ($X$ is not a union o...

Homework 6. Date: 108/10/31 Place: 共同教學館 102 [J] 代表微積分用書 Calculus: Early Transcendetals (8E), James Stewart. Problem 1. WeBWorK: homework 4.3, 4.4, 4.5, 4.7 Problem 2. ([J], Section 4.3, Problem 33., Page 301) Suppose $f$ is a continuous function where $f(x)>0$ for all $x$, $f(0) = 4$, $f'(x)>0$ if $x<0$ or $x>2$, $f'(x)<0$ is $0<x<2$, $f''(-1) = f''(1) = 0$, $f''(x)>0$ if $x<-1$ or $x>1$, $f''(x)<0$ if $-1<x<1$. Can $f$ have an absolute maximum? If so, sketch a possible graph of $f$. If not, explain why. Can $f$ have an absolute minimum? If so, sketch a possible graph of $f$. If not, explain why. Sketch a possible graph for $f$ that does not achieve an absolute minimum. Solution. Impossible, since $f'(x)>0$, $f''(x)>0$ for $x>>0$. (In particular, this is true for $x>2$), so $\displaystyle \lim_{x \to \infty }f(x) = \infty$, there is no absolute ma...

Homework 4. Date: 108/10/14 (Week 06) Place: 新生教學館 301 Problem 1. Find the value of $a$ that makes the following function differentiable everywhere. $$f(x) = \begin{cases} ax &,\text{ if } x<0\\ x^2-3x &, \text{ if } x \geqslant 0\end{cases}$$ Solution. Obviously, $f(x)$ is differentiable anywhere except $x = 0$. Note that we have $$\lim_{x \to 0^-}\dfrac{f(x)-f(0)}{x-0} = a$$, and, $$\lim_{x \to 0^+}\dfrac{f(x)-f(0)}{x-0} = -3$$ In order to make $f(x)$ differentiable at $x = 0$, we have $a = -3$. Problem 2. Let $f(x)$ be a strictly increasing, continuous function on $(a,b)$. Then there exists an inverse function $g(y)$ defined on $(f(a), f(b))$. Suppose furthermore that $f(x)$ is differentiable on $(a,b)$. Prove that $g(y)$ is differentiable on $(f(a), f(b))$ and $g'(y_0) = \dfrac{1}{f'(x_0)}$ if $y_0 = f(x_0)$. Solution. We assume the following fact: Since $f \colon (a, b) \to (f(a), f(b))$ is strictly increasing, there is an inverse fu...

Homework 3. Date: 108/10/03 (Week 04) Place: 新生教學館 301 Problem 1. Let $f(x)$ be a strictly increasing continuous function on $[a, b]$. That is, for any $x_1 < x_2 \in [a, b]$ we have $f(x_1) < f(x_2)$. Show that $f(b)$ is the maximum and $f(a)$ is the minimum. Show that for any $y_0 \in [f(a), f(b)]$, there exists $x_0$ such that $f(x_0) = y_0$. Show that the inverse function $f^{−1}$ is defined everywhere on $[f(a), f(b)]$. Show that $f^{-1}$ is continuous on $[f(a), f(b)]$. Solution. Since $x < b$ for all $x \in [a,b)$, we have $f(x) < f(b)$ for all $x \in [a,b)$, so $f(b)$ is the maximum, similarly, $f(a)$ is the minimum. Since $y_0 \in [f(a),f(b)]$ and $f(x)$ is continuous, by Intermediate value theorem, there exist $x_0 \in [a,b]$ such that $f(x_0) = y_0$. The previous problem gives that $f \colon [a,b] \to [f(a),f(b)]$ is surjective, and it is injective by the property strictly increasing (Why?). Hence, the function $f \colon [a,b] \to [f(a),f(b...

Note 4: Review Date: 108/10/28 (Week 08) Place: 新生教學館 301 Part 1: Graphing Problem Reference. 微積分用書 Calculus: Early Transcendetals (8E), James Stewart , Section 2.6, Limits at infinity; Horizontal Asymptotes (P.126 - P.140) 微積分用書 Calculus: Early Transcendetals (8E), James Stewart , Section 4.5, Summary of Curve Sketching (P.315 - P.323) Problem 1.1. (95.上微積分甲統一教學一組.第5題) Study the function $f(x) = \dfrac{(x-2)^2}{x+1}$, and answer the following questions. The domain of $y = f(x)$. $f'(x)$. $y = f(x)$ has critical point(s) at $x = $? $y = f(x)$ is increasing / decreasing on intervals? $f''(x)$. $y = f(x)$ is concave upward / down on intervals? Find the $(x,y)$ coordinates of the following points if exist. local maximum point(s) local minimum point(s) inflection point(s) Find the asymptotes of the graph $y = f(x)$ if exist. Vertical asymptotes(s) Horizontal asymptotes(s) Slanted asymptotes(s) Sketch the graph of $y = f(x)$ below. Solut...