Suppose $Y$ is a pair of pants with a hyperbolic structure and $\gamma_i; i = 1, 2, 3$ are the geodesic boundaries of length $l_i; i=1, 2, 3$ respectively. Now consider a essential …

Suppose $X$ is a CW complex and $Y$ is a subcomplex. Let $G$ be a compact Lie group that acts on $X$ and $Y$. Suppose further that the CW structures on $X$ and $Y$ are $G$-stable …

In recent work with Michael Albert and Nik Ruškuc, a family of words has arisen which is counted by the Catalan numbers. I've looked at Richard Stanley's Catalan exercises in EC2 a …

I want to slove the Laplace equation on $R^3_+$ with Neumann boundary condition. The equation reads: $-\Delta u = f$ in $R^3_+$, $\partial_3 u|_{x_3=0}=g$ on $R^2$. If $f$, $g$ sa …

My question is: usually, a partial differential equation, for example, those coming from physics, is written in a lauguage of vector calculus in a local coordinate, is there anyway …

Hello, Could you please help me in answering the following question? Initially I thought that the following problem can be solved through Monge-Ampere equation, but with Monge-Am …

EDIT: Mark Sapir pointed a reference (in the comments) giving a lower bound of $2^{1/4}$ for the minimal rate. Is this the state of art? The third question remains unanswered. If …

Hello, I am currently studying Extension 1 Mathematics. I missed two classes and I figured out that tomorrow I will have a quiz. Can you help me to solve this permutation and combi …

Question: Suppose that $f$ is an entire function (i.e. analytic in $\mathbb{C}$), and satisfies the condition $\iint_{\mathbb{C}}|f|^p dxdy<\infty$ for some $p\in (0,1)$. I gues …

The following problem is not from me, yet I find it a big challenge to give a nice (in contrast to 'heavy computation') proof. The motivation for me to post it lies in its concise …

For a homogeneous space $M = G/B$, with $G$ a (complex) semi-simple Lie group, it is very well-known that equivariant vector bundles $E$ over $M$ correspond to representations $(V_ …

Hi. Consider a a sequence of non-negative functions $(f_n)_n$, bounded in $L^1([-1,1])$ and weakly$-\star$ converging in $\mathscr{M}^1([-1,1])$ to some $f\in L^1([-1,1])$. What I …

Consider the “product” $\gamma = \alpha \times \beta$ of two binary strings $\alpha$, $\beta$ $\in \lbrace 0,1\rbrace^+$ which one gets by replacing every 1 in $\beta$ …

Let $V:=\oplus_{j\in\mathbb{Z}}V_j$ be a graded $\mathbb{F}$-vector space over the field $\mathbb{F}$. The graded tensor product of graded vector spaces is given by $V \otimes W: …

Hi, I was wondering about the following question: if you have a faithful action of a group G on the real line R by orientation-preserving homeomorphisms, it is easy to construct …