6 problems

Consider the axially symmetric steady-state ansatz … with $l>-1$, $0<k<l+\frac{3}{2}$, and weighted norms involving … where $r(x)=\sqrt{x_1^2+x_2^2}$. Weighted-estimate conjecture.…

Let $X$ be a random vector, and for $h\in\mathbb{S}^{d-1}$ define … The main theorem proves the corresponding characterization in dimension $d=2$ using a random sample of two proje…

Let $X$ be a random vector in $\mathbb{R}^d$, and let $d$ independent random projection directions be sampled. In dimension two, agreement between the projections of a distribution…

Let $\Sigma$ be a closed genus-zero surface and let $X:\Sigma\longrightarrow\mathbb{R}^3$ be an axially symmetric immersion with non-constant mean curvature. Let $\mathcal{H}$ deno…

An axially symmetric Helfrich topological sphere is a closed genus-zero axially symmetric surface that is critical for the Helfrich energy. The plane of symmetry is the horizontal…

Constant-solution conjecture. If $\frac{1}{2}<\alpha<1$, then the equation admits only constant solutions, and consequently