Let \[ \xi\!\left(\frac12+z\right) =\sum_{n\geq 0}\frac{\gamma(n)}{n!}z^{2n}, \qquad J^{d,n}(X) =\sum_{j=0}^{d}\binom dj\gamma(n+j)X^j . \] The Riemann hypothesis is equivalent to the hyperbolicity of $J^{d,n}$ for every $d,n\geq0$. We prove that there is an absolute constant $K>0$ such that \[ n^3\log^2(n+2)\geq Kd^5 \quad\Longrightarrow\quad J^{d,n}\ \text{is hyperbolic}. \] Along every sequence with $n,d\to\infty$ in this region, the empirical measure of the naturally centered and scaled zeros also converges to Wigner's semicircle law. This gives a simultaneous degree--derivative version of the global semicircle consequence of the fixed-degree Hermite limit of Griffin, Ono, Rolen, and Zagier.
Let $\Omega\subset\mathbb{R}^d$, $d\ge2$, be a bounded connected domain with boundary of class $C^{1,\alpha}$, where $0<\alpha<1$. For the adjoint Neumann--Poincar\'e operator $K^*_{\partial\Omega}$, normalised so that its distinguished eigenvalue is $1/2$, let $\lambda_j^+(\Omega)$ denote the upper min--max values on...
We prove that for every even integer $N\geq 4$ or $N\in\{3,5,7\}$, every axially symmetric solution to the $Q$-curvature-type problem $$ \alpha P_N u + (N-1)!(1-\frac{e^{Nu}}{\int_{\mathbb{S}^N} e^{Nu}dw})=0 \ \ \ \ \ \mbox{on} \ \mathbb{S}^N $$ is constant, provided that $ \alpha\ge\frac{1}{2}$ and $\alpha \not =1$. T...
Chang-Feng Gui, Tuo Li, Jun-Cheng Wei et al.· 0 citations
The Bernstein problem for the affine maximal type equation \[ \sum_{i,j=1}^n f^{ij} w_{ij}=0,\qquad w\equiv \left[\det\left(\frac{\partial^2 f}{\partial x_i\partial x_j}\right)\right]^a,\quad x\in\Omega\subset\mathbb R^n, \] is a central problem in affine geometry. It originates from Chern's conjecture on entire locall...
We compute the large-$N$ asymptotics of the determinant of the identity plus or minus a generalized Hilbert matrix. For $N \in \mathbb{N}$ and $\delta \in \mathbb{R} \setminus \{-1, -2, \ldots\}$, let \[ H_N^\delta := \left( \frac{\sin((1+\delta)\pi)}{\pi(j+k+1+\delta)} \right)_{j,k=0}^{N-1} \] be the generalized Hilbe...
Let $d,K,N\in \mathbb{N}$ with $K\geq 3$ and $d\geq 4K+4$. Let $\Delta\subset \mathbb{Z}^d$ be the vertex set of a nondegenerate $(K-1)$-simplex, and let $A\subseteq[N]^d$ contain no nontrivial similar copy of $\Delta$. We prove that \[ |A|\ll_{\Delta,d} N^d\exp\!\left(-c_{\Delta,d}\sqrt{\log N}\right) \] improving upo...
Andrew Lott, Á. Magyar, N. R. Ponagandla· 0 citations
Let $1 \leq q \le 2$ and denote by $2 \leq q'$ its corresponding conjugate exponent. We prove the continuity of solutions $u \in W^{1,(\frac{n}{n-1},q')}(B^n, \mathbb{R}^m)$ to the critical elliptic system $-\Delta u = \Omega \cdot \nabla u$ in dimension $n \ge 3$, where the potential $\Omega \in L^{(n,q)}(B^n, \mathfr...
Carolin Bayer· 0 citations
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