Jairo Bochi
  • Member for 12 years, 2 months
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Applications of the Cayley-Hamilton theorem
17 votes

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 $\rho(A)$ ...

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Geometric/combinatorial depiction of algebraic identity?
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9 votes

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 ...

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A question on invariant measures
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7 votes

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 ...

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A metric for Grassmannians
6 votes

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: $$ d(V,W):= \inf_J \| J - i_V\| $$ ...

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Approximating a convex disk by an ellipse
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5 votes

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, ...

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Generalizations and relative applications of Fekete's subadditive lemma
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5 votes

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 ...

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What is the defining formula for Sectional Curvature
5 votes

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 (.....

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Algebraic characterization of transitive spaces of matrices
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4 votes

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 ...

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Maximizing the expectation of a polynomial function of iid random variables
3 votes

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)...

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Approximating a convex disk by an ellipse
3 votes

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 ...

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Number of disjoint simple closed geodesics
3 votes

The flat torus has infinitely many disjoint simple closed geodesics.

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Counterexamples against all odds
2 votes

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} ...

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Existence of algebraic integers with certain properties
2 votes

@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 ...

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Growth rate of bounded Lipschitz functions on compact finite-dimensional space
2 votes

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 $\epsilon$-...

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Quantitative approximation of invariant measures by periodic ones
2 votes

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 ...

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Sets of matrices which are irreducible but not strongly irreducible
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2 votes

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{...

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Maximum cost optimal transport
1 votes

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 $X$...

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Maximizing the expectation of a polynomial function of iid random variables
1 votes

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 ...

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Relationship between Multiplicative Ergodic Theorems
1 votes

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 ...

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Intersection of subvarieties in grassmannian space
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1 votes

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 ...

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