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This is false for other permutations. For example, call $(e_1,e_2,e_3)$ the canonical basis of $\mathbb{R}^3$ and let A,B,C be the $3 \times 3$ real matrices such that $Ae_1=e_2$, $Be_2=e_3$, $Ce_3=e_ …

The integral with regard do $\mathrm{d}M^*$ is a pathwise Stieltjès integral, so the question is an analysis problem. Let $f : \mathbb{R}_+ \to \mathbb{R}$ be any continuous function, $F$ its current …

Nice questions, with nice motivation. I think that $f_-(x) = x^2$ and $f_+(x) = 2x-x^2 = 1-(1-x)^2$ is a non-trivial solution. Indeed, for every $x$ and $y$ in $[0,1]$, $$0 \le x^2 \le x \le 2x-x^2 \l …

ADDENDUM. Vokram told me that I answered another question, not his question. Therefore, I give another counterexample (not so different from the previous one) disproving the equality. Choose a functio …

I think you need to assume that $K$ has a smooth boundary and is strictly convex to ensure that $g$ and $h$ are differentiable outside $0$. Anyway, I do not think that the result is true. Assume that …

The answer is negative : let $A = [-1,1]$, $U_i(a,s) = a^2$ for every $i \in [0,1]$ and $(a,s) \in A \times \mathbb{R}$, and $(\sigma_n)_n \to \sigma$ be any convergent sequence of real numbers. For e …

If the $\lambda_i$ take only two different values namely $\lambda_\max > \lambda_\min$, the equation $F(t)=s$ can be written $$At^2(\lambda_\min+t)^2 + Bt^2(\lambda_\max+t)^2 = s (\lambda_\max+t)^2 (\ …

The notations are contradictory. Once $p$ is fixed, and then it varies. Do you set $\mu_{n}:=\frac{1}{n}\sum_{i=0}^{n-1}T_{\ast}^{i}\nu$ for ALL $n \ge 1$ and assume that $T_{\ast}^{p} \nu = \nu$ for …