# All Questions

21 views

### What is the dimension of a certain vector subspace of invariant matrices?

$\newcommand{\Sig}{\Sigma}$ Let $\Sig$ be a diagonal matrix with strictly positive entries on the diagonal. Define $V=\{B \in M_n\mid B\Sig +\Sig B^T=\Sig B +B^T \Sig \}$ (where $M_n$ is the vector ...
846 views

### Metric assuming the value infinity

If we instead define a metric as $d:X\times X \rightarrow [0,\infty]$, do we lose any nice properties of metric spaces? The reason I ask is that I saw this theorem: Given a finite measure space ...
4 views

### Proof of identity relating composition of permutations and sign in alternating tensors.

I was having some confusion trying to prove the following identity: Let $A: \mathcal{L}^{k} \to \mathcal{A}^{k}$, where $\mathcal{L}^{k}$ denotes the set of all $k$-tensors and $\mathcal{A}^{k}$ ...
18 views

### joint infima involved in relation to distance between sets

Consider a metric space $(X,d)$, and let $A,K \subseteq X$ such that A is closed and K is compact. I have to show that there exists an element $k_0\in K$ which achieves the minimum between the sets, ...
17 views

### Prove by induction: A tree on n≥2 vertices has ≥2 leaves

this is what I have. I'm pretty sure this is quite incorrect, but am I at least headed in the right direction? Base Case: P(2): Tree on 2 vertices can only have one edge, the edge connecting the 2 ...
55 views

### The sequence $(-1)^n\binom{\alpha-1}{n}$ converges.

I need to show that for $n \in \mathbb N_0$ and $\alpha \ge 0$ the sequence $(-1)^n\binom{\alpha-1}{n}$ converges. It can be shown that the sequence convereges to zero using a theorem claiming that ...
11 views

### Proof that dim(L(V,W))=dim V *dim W, when V, W are finite dimensional.

I've been looking a few proofs of this, and I have a problem with the standard one, which uses the concept of isomorphism. Suppose dim V=n and dim W=M.The proof uses the function M from L(V,W)(the ...
14 views

### Bounds on a quadratic form

I am currently in the middle of a proof where it would be nice to have some estimates on the size of a quadratic form. In particular, I am looking at $$x^TAx$$ where $A$ is "small" (in the analyst's ...
9 views

### Sum of sum of binomial coefficients

I know there is no simple way to solve the sum: $$\sum_{k=0}^{j}{{n}\choose{k}}$$ But what about the equation: $$\sum_{j=1}^{n}{\sum_{k=0}^{j}{{n}\choose{k}}}$$ Are there any simplifications or ...
20 views

### Why is the Fundamental Theorem of Arithmetic so important?

I've recently read about the Fundamental Theorem of Arithmetic and I think that I have just about understood the proof. What I found quite interesting at first was the "Fundamental" part in the name. ...
I am trying to understand the following: If $X$ is normally distributed and $Y$ is distributed according to the Poisson distribution, how to find out which distribution $Z=X/Y^2$ has? Is there any ...