Contrary to convexity problems, the symmetry methods, such as moving plane technique works very well for free boundary value problems for systems, when there is a symmetry in the given equation, the bulk domain, and boundary values. Here, we present the most simple example, leaving several obvious generalizations towards other problems to the reader. The original Saint Venant problem, with an overdetermination of the boundary gradient condition (here expressed as system) is to show that whenever there is a solution vector $\mathbf{u}$ to the following problem \begin{equation}\tag{1} \begin{cases} \Delta \mathbf{u} = - \mathbf{k} & \text{in } \Omega , \\ \mathbf{u} = \mathbf{0} & \text{on } \partial\Omega , \\ |\nabla \mathbf{u}| = 1 & \text{on } \partial\Omega , \\ \end{cases} \end{equation} the domain $\Omega $ has to be a ball. For scalar case, James Serrin [5] gave a very nice proof of this based on moving plane technique (of A. D. Alexandrov). ...

The system case of the free boundary for general $F(x, \mathbf{p})$, even if $F$ is independent of $x$, and convex in $\mathbf{p}$-variables, is of course a non-trivial problem. Let us consider the particular case as in ...

Convexity results for the Bernoulli type problems, represented in the system of equations \begin{equation}\begin{cases} \Delta \mathbf{u} = 0 & \text{in } \Omega \setminus \mathbf{D}, \\ |\nabla \mathbf{u}| = g(x) & \text{on } \partial\Omega . \\ \end{cases} \end{equation} seems for now out of reach, at least with the methods we know of. It is also not straightforward what conditions we should impose on the domains $\mathbf{D}$ (besides being convex and maybe $D_i$'s being homothetic). ...

This is a classical problem in additive combinatorics that can be equivalently phrased as a problem in real analysis (this equivalence is due to Javier Cilleruelo, Imre Z. Ruzsa, and Carlos Vinuesa): what random variable $X$ with compact support on $\mathbb{R}$ has the property that the distribution of $X+X$ is as flat as possible? ...

Stanislaw Ulam, in a 1964 book, described what is now called the "Ulam sequence". It is defined by setting a_1 = 1, a_2 =2 and then picking a_n in a greedy manner as the smallest integer that can be uniquely written as the sum of two distinct earlier elements of the sequence. This results in the sequence

1,2,3,4,6,8,11, ... ...

The Hadamard maximal determinant problem is a fascinating combinatorial optimization problem with many concrete applications to signal processing, coding theory, experimental design theory, and cryptography . Let $M$ be an $n\times n$ matrix, all of whose entries are at most 1 in modulus. Note that $det(M)$ is equal to the volume of the $n$-dimensional parallelepiped spanned by the rows of $M$. By assumption, each row has Euclidean length at most $n^{1/2}$. Then $det(M) \leq n^{n/2}$, and equality holds if and only if every entry of $M$ is $\pm 1$ and the rows of $M$ are orthogonal. Hadamard established this upper bound which is only attainable if the order n equals $1, 2,$ or a multiple of $4.$ A matrix attaining the bound is called a Hadamard matrix. The classical and very old Hadamard conjecture asserts that a Hadamard matrix exists of every order divisible by $4.$ The smallest multiple of $4$ for which no such matrix is currently known is $668$, the value $428$ having been settled only in $2005$. Due to its extensive real life applications, it is still a very active line of research. ...

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? ...