Absence of Critical Points of Solutions to the Helmholtz Equation in 3D

The focus of this paper is to show the absence of critical points for the solutions to the Helmholtz equation in a bounded domain $\Omega\subset\mathbb{R}^{3}$, given by $$\left\{ \begin{array}{l} -\rm{div}(a\,\nabla u_{\omega}^{g})-\omega qu_{\omega}^{g}=0\quad\text{in $\Omega$,}\\ u_{\omega}^{g}=g\quad\text{on $\partial\Omega$.} \end{array}\right.$$ We prove that for an admissible $g$ there exists a finite set of frequencies $K$ in a given interval and an open cover $\overline{\Omega}=\cup_{\omega\in K}\Omega_{\omega}$ such that $|\nabla u_{\omega}^{g}(x)|>0$ for every $\omega\in K$ and $x\in\Omega_{\omega}$. The set $K$ is explicitly constructed. If the spectrum of the above problem is simple, which is true for a generic domain $\Omega$, the admissibility condition on $g$ is a generic property.Comment: 14 page

Spectral analysis of non-self-adjoint Jacobi operator associated with Jacobian elliptic functions

We perform the spectral analysis of a family of Jacobi operators $J(\alpha)$ depending on a complex parameter $\alpha$. If $|\alpha|\neq1$ the spectrum of $J(\alpha)$ is discrete and formulas for eigenvalues and eigenvectors are established in terms of elliptic integrals and Jacobian elliptic functions. If $|\alpha|=1$, $\alpha \neq \pm 1$, the essential spectrum of $J(\alpha)$ covers the entire complex plane. In addition, a formula for the Weyl $m$-function as well as the asymptotic expansions of solutions of the difference equation corresponding to $J(\alpha)$ are obtained. Finally, the completeness of eigenvectors and Rodriguez-like formulas for orthogonal polynomials, studied previously by Carlitz, are proved.Comment: published version, 2 figures added; 21 pages, 3 figure

Approximate computations with modular curves

This article gives an introduction for mathematicians interested in numerical computations in algebraic geometry and number theory to some recent progress in algorithmic number theory, emphasising the key role of approximate computations with modular curves and their Jacobians. These approximations are done in polynomial time in the dimension and the required number of significant digits. We explain the main ideas of how the approximations are done, illustrating them with examples, and we sketch some applications in number theory

Pendulum Integration and Elliptic Functions

Revisiting canonical integration of the classical pendulum around its unstable equilibrium, normal hyperbolic canonical coordinates are constructe

Linearizing torsion classes in the Picard group of algebraic curves over finite fields

We address the problem of computing in the group of $\ell^k$-torsion rational points of the jacobian variety of algebraic curves over finite fields, with a view toward computing modular representations.Comment: To appear in Journal of Algebr

The arithmetic of hyperelliptic curves

We summarise recent advances in techniques for solving Diophantine problems on hyperelliptic curves; in particular, those for finding the rank of the Jacobian, and the set of rational points on the curve