Hatcher 1.3 Problem 20 Give examples of a covering space $Y$ of a Klein bottle $K$ where [*]$Y$ is a torus and the cover is normal [*]$Y$ is a torus and the cover is not a normal cover [*]$Y$ is a Klein bottle and the cover is not a normal cover
Let $X$ be a path connected cell complex and suppose that there is a map $X\to S^1$ that is not null-homotopic. Let $x$ be a zero-cell of $X$. Prove that $X\vee_x X$ has an irregular cover of degree 3. Does $X\vee_x X$ admit a retract onto a wedge of 2 circles?
Exercise 1.3.25. Exercise 1.3.25 Consider the action on $X=\mathbb{R}^2 \backslash\{0\}$ sending $(x, y) \mapsto(2 x, y / 2)$. Show that this generates a covering space action of $\mathbb{Z}$ on $X$, and that $X / \mathbb{Z}$ is not Hausdorff. Show that $X / \mathbb{Z}$ contains four subspaces homeo
\begin{align*} \theta_{1}&=\arcsin\left(\frac{\sin(\varphi)\cos(\psi)\sqrt{1-k^{2}\sin\!^{2}\>\!\psi}+\sin(\psi)\cos(\varphi)\sqrt{1-k^{2}\sin\!^{2}\>\!\varphi}} {1-k^{2}\sin\!^{2}(\varphi)\sin\!^{2}(\psi)}\right)\\ P&=\sqrt{\big(1-k^{2}\sin\!^{2}\>\!\theta_{1}\big)\big(1-k^{2}\sin\!^{2}\>\!\xi\,\b
https://gr.inc/question/prove-that-any-fat-cantor-set-is-homeomorphic-to-the-regular-cantor-set-then-d/ [*]any fat Cantor set is homeomorphic to the regular Cantor set. [*]∄ a diffeomorphism from [0,1] to itself such that a fat Cantor set is mapped to the ternary Cantor set.
1.$\lim_{n \to \infty}{\frac{1}{\ln n}\sum_{k=1}^n\abs{\frac{\sin{kx}}{k}}}$是否存在？存在的话极限是多少？如果把正弦换为余弦又会怎样？ 2.$\lim_{n \to \infty}{\frac{1}{n}\sum_{k=1}^n\abs{\sin{kx}}}$是否存在？存在的话极限是多少？如果把正弦换为余弦又会怎样？ 谢了！
To determine the divergence of the series \(\sum_{n=1}^{\infty} \frac{e^{-n} n^n}{n!}\), we can use Stirling's approximation for factorials. Stirling's approximation states that for large \(n\), \[ n! \approx \sqrt{2\pi n} \left(\frac{n}{e}\right)^n. \] Using this approximation, we can rewrite the g
设函数 $f(x)$ 满足：$\forall x, y \inR, f(x+y)=f(x)+f(y)$．若 $f(x)$ 有唯一解 $f(x)=f(1) \cdot x$，则 $f(x)$ 需要满足什么条件？ （换言之，加什么样的条件才能保证没有另外的解，比如不连续函数、病态函数？）
Hyperbolic metric geodesically complete Consider the upper half plane model of the hyperbolic space ($\mathbb{H}$ with the riemannian metric $g=\frac{dx^2+dy^2}{y^2}$). It is known that $(\mathbb{H},g)$ is geodesically complete, which means that no geodesic can reach the border $\partial \mathbb{H}$
用分离变量法求解如下受外部阻尼力的弦振动方程的初边值问题 \[ \begin{cases}\partial_t^2 u-\partial_x^2 u+\partial_t u=0, & (t, x) \in(0,+\infty) \times(0, L), \\ u(0, x)=\varphi(x), \partial_t u(0, x)=\psi(x), & x \in(0, L), \\ u(t, 0)=u(t, L)=0, & t \in(0,+\infty) .\end{cases} \] 并说明当 $t \rightar
[*]To show there is no point of \(\mathcal{E}(\mathbb{Q}_p)\) reducing to the singular point \((0,0) \in \widetilde{\mathcal{E}}_p(\mathbb{F}_p)\), consider the reduction map: \[ \mathrm{red}: \mathcal{E}(\mathbb{Q}_p) \to \widetilde{\mathcal{E}}_p(\mathbb{F}_p). \] A point \((x, y) \in \mathcal{E}(
A hyperbolic element in $\text{SL}_2(\Bbb Z)$ is a matrix with trace absolute value greater than 2 https://gr.inc/question/given-sl_2mathbbz-what-interpretation-is-available-for-the-hyperbolic-e They correspond to hyperbolic isometries of the upper half-plane, acting as translations along geodesic a
I remember that modular forms of integral weight are defined on congruence subgroups like Γ₀(N) or Γ₁(N), but for half-integral weights, things are different. The primary reason is to ensure the existence of a consistent set of square roots https://gr.inc/question/explain-why-half-integral-weigh
Let $ U $ be an arbitrary neighborhood of the identity $ e \in G $. Define $ S = \langle U \rangle $, the subgroup generated by $ U $. Explicitly, $ S $ consists of all finite products of elements from $ U $ and their inverses: $$S = \{ u_1 u_2 \cdots u_n \mid u_i \in U \cup U^{-1}, \, n \in \mathbb
Lie's Third Theorem: For every finite-dimensional real or complex Lie algebra $\mathfrak{g}$, there exists a unique (up to isomorphism) simply connected Lie group $G$ whose Lie algebra is isomorphic to $\mathfrak{g}$. Proof:[*]Apply Ado's Theorem: By Ado's Theorem, there exists a finite-dimension
Prove that every nilpotent Lie algebra in characteristic zero can be represented by strictly upper triangular matrices. You may use Ado's theorem and consider the case where the field is algebraically closed and the case where the field is arbitrary. https://gr.inc/question/prove-that-every-nilpote
Consider the unit $n$-ball $B^n$ and unit $(n+1)$-sphere $S^{n+1}$: $$ \begin{aligned} B^n & =\left\{x=\left(x_1, \ldots, x_n\right) \in \mathbb{R}^n: x_1^2+\cdots+x_n^2 \leqslant 1\right\} \\ S^{n+1} & =\left\{x=\left(x_1, \ldots, x_{n+2}\right) \in \mathbb{R}^{n+2}: x_1^2+\cdots+x_{n+2}^2=1\right\
The closed form of the sum $\sum_{n=1}^\infty q^{-n^2} z^n$ for $|z|<1$ is given by: \[ \frac{z e^{-z/\sqrt{q}}}{1 - e^{-z/\sqrt{q}}} \] https://gr.inc/question/find-the-closed-form-of-the-summation-sum_n1infty-q-n2-zn-where/
\begin{gather*} \left\{\begin{split} p&=18x^{6}-9(4+\sqrt{3})x^{4}+9(2+\sqrt{3})x^{2}-3-2\sqrt{3}\\ q&=3x(6x^{4}-(9+2\sqrt{3})x^{2}+2+\sqrt{3})\\ r&=3x^{2}(6x^{2}-\sqrt{3}-6)\\ s&=9x(2x^{2}-1)\\ y&=((x^{2}-1)(x^{2}-\frac{3+2\sqrt{3}}{3}))^{1/4} \end{split}\right.\\ \\ 9(97+56\sqrt{3})=\prod_{k=1}^{
https://gr.inc/question/show-that-there-is-no-closed-minimal-surface-in-r3/ Suppose for contradiction that there exists a closed (i.e. compact without boundary) minimal surface $\Sigma$ immersed in $\mathbb{R}^3$. Let $x = (x_1, x_2, x_3)$ be the position vector of the immersion. A key fact about mi
https://mathematica.stackexchange.com/q/232830
Let $G$ be a compact Lie group, and $\rho_1, \rho_2$ be finite-dimensional complex representations with $\chi_{\rho_1} = \chi_{\rho_2}$. By the complete reducibility theorem, every finite-dimensional representation of $G$ decomposes into a direct sum of irreducible representations. Write: $$\rho_1 \
在 Math StackExchange 上讨论了一个特定的算子乘积展开式的问题。 提问者考虑如下的算子乘积： $$ \prod_{j=0}^{n-1} \sum_{i=0}^{s}(n-j-s+i)_i E^i $$ 其中，$(x)_n$ 表示下降阶乘（falling factorial），$E$ 是定义为 $Ef(s) = f(s-1)$ 的移位算子。ChatGPT 展开该表达式，得到： $$ \sum_{m=0}^{ns}\sum_{\substack{i
https://math.stackexchange.com/questions/295758 The Lie bracket of a left-invariant vector field $X_L$ and a right-invariant vector field $X_R$ on a Lie group $G$ vanishes. Proof: The flow $\phi_t^{X_L}$ of $X_L$ is given by right multiplication: $\phi_t^{X_L}(g) = g \exp(tA)$, where $A = X_L(e) \in
设有定义在$[0,1]$内的函数$f(x)$能展开为三角级数： $$f(x)=\sum_{k=0}^{\infty} a_k cos(2k+1)\pi x,$$ 其中$a_k\geq 0。$ 求证 $$\int_{0}^{1} (1-|x-t|)f(t) dt\leq \frac{1}{4}\underset{n\in \mathbb{N}}{max}|a_n|.$$
来自https://news.ycombinator.com/item?id=42913867 The set of natural transformations between two functors $F,G\colon\mathcal{C}\to\mathcal{D}$ can be expressed as the end $$\mathrm{Nat}(F,G)\cong\int_{A}\mathrm{Hom}_{\mathcal{D}}(F(A),G(A)).$$Define set of natural cotransformations from $F$ to $G$
\begin{align*} N\left(u+v\sqrt[3]{t}+w\sqrt[3]{t^2}\,\right)&=u^3+tv^3+t^2w^3-3tuvw\\ N\left(U+V\sqrt[3]{t}+W\sqrt[3]{t^2}\,\right)&=U^3+tV^3+t^2W^3-3tUVW\\ N(\alpha)N(\beta)&=N(\alpha\beta) \end{align*} The Euclidean Condition in Pure Cubic and Complex Quartic Fields \begin{gather*} \begin{vmatrix
Mock Exam [*]Prove each of the following statements for a compact manifold $M$ of dimension $n \geqslant 2$. [*]If $M$ admits a Riemannian metric $g$ with sectional curvature $K \leqslant 0$ then $M$ has an infinite fundamental group. [*]If $M$ admits a Riemannian metric with sectional curvature $K>
Mock Exam Let $\gamma:[0, L]\to(M, g)$ be a normalised geodesic in a Riemannian manifold $(M, g)$ with Riemann curvature $R$. [*]Define what is meant by a Jacobi field along $\gamma$. [*]Let $J$ be a Jacobi field along $\gamma$ with $J(0)=0,|J'(0)|=1$ and $g(J'(0), \gamma'(0))=0$. [*]Show that \[ g(
Mock Exam Let $(M_j, g_j)$ be complete Riemannian manifolds of dimension $n_j \geqslant 2$ for $j=1,2$ and let $M=M_1 \times M_2$. Given that $T_{(p_1, p_2)}(M_1 \times M_2) \cong T_{p_1} M_1 \times T_{p_2} M_2$ for all $(p_1, p_2) \in M_1 \times M_2$, we define a section $g$ of $S^2 T^*(M_1 \times
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