# All Questions

19 views

### Wave equation. Perturbation methods

The problem I want to solve is, \begin{equation*} u_{tt} = u_{xx}, \, t >0, \end{equation*} \begin{equation*} u_x(0,t) = - \epsilon, \, u_x(1,t) = \epsilon, \, t > 0, \end{equation*} ...
27 views

### Union of compact sets

Let $K_1$,$K_2$,...,$K_N$ be compact subsets of the metric space $(X,d)$. Show that a) $K_1\cap K_2\,\cap\,...\cap\,K_N$ is compact. b) $K_1\cup K_2\,\cup\,...\cup\,K_N$ is compact. My own idea is ...
16 views

### Absolute convergence of an infinite complex series

Let $(z_n)$ be a sequence of complex numbers. If $\sum_{n=1}^\infty \vert z_n\vert$ converges then $\sum_{n=1}^\infty z_n$ does too. Show that this statement is true assuming we know this is true for ...
22 views

### Can a graph be a subgraph if each component is not connected?

Say you two connected components of G. Can they be "combined" as one subgraph with no edges connecting them?
27 views

### Which $y$ does verify: $y \pm \sqrt{y^2-4} > 0$? [on hold]

How should one analyze this kind of inequalities?
17 views

### Characteristics of $\vec u$ in equation of the reflection of $\vec x$ about the line $N$

The linear transformation of the reflection of $\vec x$ about line $N$ is $$\vec x = 2 \text{proj}_N(\vec x)\vec x - \vec x= 2(\vec x \cdot \vec u) \vec x- \vec x.$$ Is the unit vector, $\vec u$, ...
31 views

### Why is the infimum of the following relation 1?

The question was: What is the infimum from $K=\{x ~ |~0 \leq x \lt 1\}$ for the partial order $\{ (x,y) ~ | ~ ( \exists z \in \mathbb{R}_{\geq 0})[x-z=y] \}$ on set $\mathbb{R}$. This was a question ...
15 views

### For the ring $\mathbb{Z}[\sqrt{d}]$

For the ring $\mathbb{Z}[\sqrt{d}]$, we define the norm function $f:\mathbb{Z}[\sqrt {d}] \rightarrow \mathbb{N}\cup \{0\}$ by: $f(a+b\sqrt d)=|a^2-db^2|$. How can I prove that $f(x)=0$ if and only ...
11 views

### How to find arbitrage for forward rate of exchange?

Assume that $S(0)$ is the current rate of exchange for foreign currency. Assume that and $K_h$ and $K_f$ are rates of return on home and foreign currency if it is invested over a period $T$. (A) ...
17 views

### Every set of pairwise disjoint sets has a choice function implies AC

Suppose that for every collection $A$ of non empty pairwise disjoint sets has a choice function. I need to prove that this implies the axiom of choice. Let $S$ be a collection of sets. For every $B$ ...
13 views

### Sign of a permutation including a trivial cycle

This may be a rather basic question, but I can't see mention of this anywhere. Suppose I have a permutation $p\in S_5$ (say). Suppose further that $p$ decomposes as $p=(1 2)(3)(4 5)$. What is the ...