Let $X$ be a matrix-valued Ito process $$dX_{t}=u_{t}dt+v_{t}dW_{t}$$ where $u$ and $v$ are $d\times d$-matrices, with $d\ge 2$, and $W$ is a real Brownian motion. Consider the exponential $$Q_{t}=\text{Exp}(X_{t}).$$ Under what conditions on $u,v$ is $Q$ a local martingale? ...

Let $A$ be a bounded linear operator on Hilbert space $\mathcal{H}$. For some fixed $x\in \mathcal{H}$, consider a system of vectors $$\left\{\frac{\|A^n x\|}{\|x\|} \right\}_{n=0,1,2,\dots}. $$ There are bounded (not-normal) operators for which a system of above form is an orthonormal basis (e.g. shift operator on $\mathcal{l}(\mathbb{N})$). ...

Let $$ \begin{cases} -\Delta u_f=f & \text{in }D,\\ u_f=0 &\text{on }\partial D. \end{cases} $$ It is well-known that the maximization of the functional $$ \Phi(f)=\int_D|\nabla u_f|^2dx, $$ over the set $$ \bar{\mathcal{R}}_\beta=\{f\,\,|\,\, 0\leq f\leq 1,\,\,\text{and}\,\, \int_Dfdx=\beta\} $$ has a solution $$ \hat{f}=\chi_{\{\hat{u}>\alpha\}}\in \mathcal{R}_\beta=\{\chi_E\,\,|\,\, |E|=\beta\}, $$ for some $\alpha>0$, and $\hat{u}=u_{\hat{f}}$. ...

Let $v$ be a solution to the obstacle problem $$ \Delta v = h\chi_{\{v > 0\}}, \qquad v \geq 0 \qquad \hbox{in } B_1. $$ We assume $h \geq c_0 >0 $ is Lipschitz, and a Dirichlet data on $\partial B_1$ has been prescribed. ...

Consider the probability measure $\mathbb{P}_N^n(\lambda)=\frac{\dim E_\lambda}{N^n}$ on the set of Young diagrams with $n$ cells and at most $N$ rows, where $\dim E_\lambda$ is the dimension of the isotypic component corresponding to the Young diagram $\lambda$ in the tensor representation $(\mathbb{C}^N)^{\otimes n}$ of the group $GL(N,\mathbb{C})$. ...

If $E$ is a compact subset of ${\Bbb R}^d$, of Hausdorff dimension greater than $\frac{d+1}{2}$, then there exists a non-empty open interval $I$ such that, for any $t \in I$, there exist $x^1, x^2, \dots, x^{k+1} \in E$ such that $|x^{j+1}-x^j|=t$, $1 \leq j \leq k$. ...

The following result is known for a repelling system of $N$ particles in a plane, at inverse temperature $\beta$: if we zoom at a suitable point at a natural microscopic scale, and if $\beta$ is at least proportional to $\log N$, then the particles become (almost-certainly) separated from each other by a certain fixed distance, as $N->\infty$. ...