Cayley-Hamilton theorem can be used to prove Gelfand's formula (whose usual proofs rely either on complex analysis or normal forms of matrices). Let $A$ be a $d\times d$ complex matrix, let $\rho(A)$ ...

The identity can be rewritten as $(a+b+c)^3=a^3+b^3+c^3+3(a+b)(a+c)(b+c)$ by means of a linear change of variables $a:=(−x+y+z)/2$, etc. Let $T$ be a circle of length $a+b+c$, and let's chop it into ...

Your set consists exactly of weak coboundaries plus constants. A function $f\in C(X)$ is called a coboundary if $f = h \circ T - h$ for some $h\in C(X)$. A function is called a weak coboundary if it ...

My "answer" is just a (still another) reformulation of Ian's. Given two subspaces $V$, $W \subset \mathbb{R}^n$ with the same dimension, define their distance as: $$ d(V,W):= \inf_J \| J - i_V\| $$ ...

Question (a) has a positive answer in the centrally symmetric case. The proof is involved and I will only summarize the strategy here. Full details can be found in this ArXiv paper. Comments, ...

Since you mentioned Kingman's subadditive ergodic theorem, you may find interesting the following semi-uniform subadditive ergodic theorem: Let $T \colon X \to X$ be a continuous map of a compact ...

For a discussion about the sign choice, see Lang, Fundamentals of Diff. Geom., p. 235-237. I quote him: Classically, starting with surface theory, people wanted some formulas such as Gauss-Bonnet (.....

Here is the real answer: A space of matrices is transitive iff its "orthogonal complement" contains no matrix of rank one. The idea was not mine; it is I found in Sec. 4 from the paper below (See ...

Suppose $\mu = \mu_{\max}$ is a (Borel) probability measure on $[0,1]$ for which the following integral is maximized: \begin{equation}\tag{1} \mathcal{I}(\mu) := \iiint |(y-x)(z-y)(z-x)| d\mu(x)d\mu(y)...

Update: I previously claimed that I had a negative answer to question (a) in the centrally symmetric case, but thanks to Matt F. I found that there was an error in my calculation. I corrected the ...

Let $\mathsf{A} = (A_1,\dots,A_m)$ be a tuple of $d \times d$ matrices. The joint spectral radius (JSR) of $\mathsf{A}$ is $\mathrm{JSR}(\mathsf{A}) := \lim_{n\to\infty} \sup_{i_1,\dots,i_n} \|A_{i_1} ...

@orthodontist has already given a clean and educated answer to my question. Nevertheless let me include an elementary answer, tweaking Vijayaraghavan's construction. The calculations are not terribly ...

Fine estimates of covering and packing numbers of several metric spaces of functions have been obtained in the paper: A.N. Kolmogorov, V.M. Tihomirov. $\epsilon$-entropy and $\epsilon$-...

For a result of a somewhat similar type, check the paper "Rate of approximation of minimizing measures" by Bressaud and Quas, 2007, especially Theorem 4. In their case the dynamics is a subshift of ...

The answer to your second question is no. Let $E_1$, $E_2$, $E_3$ be pairwise transverse $2$-dimensional subspaces of $\mathbb{R}^4$. Consider the following semigroup: $$ \Sigma := \{M \in \mathrm{...

Consider the following variant of the problem. Let $(X,d)$ be a compact metric space, and let $\delta>0$. Take the cost function $c_\delta(x,y) := d(x,y)^{-\delta}$. Given a probability $\nu$ on $X$...

Here is a complete and elementary solution for the Baby-squared problem, that is, the Baby Problem with the integrand squared. The idea is to reformulate the problem in terms of moments, and ...

The question has already been answered, so let me just complement what has been said. The book which I find most accessible is Lang's Fundamentals of Diff. Geom. (it was recommended to me by ...

Well, after a crash course on cohomology of the grassmannians, Schubert calculus, etc, I think I've found the answer to the second question: If the codimension of $X$ in $G_m(\mathbb{C}^n)$ is less ...