Related papers: Hyperelliptic Theta-Functions and Spectral Methods
In this paper we complement the program concerning the application of symmetrization methods to nonlocal PDEs by providing new estimates, in the sense of mass concentration comparison, for solutions to linear fractional elliptic and…
Riemannian optimization is concerned with problems, where the independent variable lies on a smooth manifold. There is a number of problems from numerical linear algebra that fall into this category, where the manifold is usually specified…
A numerical scheme is presented for approximating fractional order Poisson problems in two and three dimensions. The scheme is based on reformulating the original problem posed over $\Omega$ on the extruded domain…
We consider a finite element method for elliptic equation with heterogeneous and possibly high-contrast coefficients based on primal hybrid formulation. A space decomposition as in FETI and BDCC allows a sequential computations of the…
The stationary, axisymmetric reduction of the vacuum Einstein equations, the so-called Ernst equation, is an integrable nonlinear PDE in two dimensions. There now exists a general method for analyzing boundary value problems for integrable…
Exact expressions for the partition functions of the rigid string and membrane at any temperature are obtained in terms of hypergeometric functions. By using zeta function regularization methods, the results are analytically continued and…
We consider the numerical solution of high-frequency scattering problems modeled by the Helmholtz equation with a bounded obstacle. Although the analysis of this problem dates back at least 50 years, over the past decade or so, tools and…
Some computations made about the Riemann Hypothesis and in particular, the verification that zeroes of zeta belong on the critical line and the extension of zero-free region are useful to get better effective estimates of number theory…
We prove sharp upper and lower bounds for the nodal length of Steklov eigenfunctions on real-analytic Riemannian surfaces with boundary. The argument involves frequency function methods for harmonic functions in the interior of the surface…
Boundary value problems for integrable nonlinear evolution PDEs formulated on the finite interval can be analyzed by the unified method introduced by one of the authors and used extensively in the literature. The implementation of this…
It is shown that eigenvalues of Laplace-Beltrami operators on compact Riemannian manifolds can be determined as limits of eigenvalues of certain finite-dimensional operators in spaces of polyharmonic functions with singularities. In…
Elliptic partial differential equations arise in many fields of science and engineering such as steady state distribution of heat, fluid dynamics, structural/mechanical engineering, aerospace engineering and seismology etc. In three…
We present a new formulation of Einstein's equations for an axisymmetric spacetime with vanishing twist in vacuum. We propose a fully constrained scheme and use spherical polar coordinates. A general problem for this choice is the…
We study the space of periodic solutions of the elliptic $\sinh$-Gordon equation by means of spectral data consisting of a Riemann surface $Y$ and a divisor $D$. We show that the space $M_g^{\mathbf{p}}$ of real periodic finite type…
A computer-algebra aided method is carried out, for determining geometric objects associated to differential operators that satisfy the elliptic ansatz. This results in examples of Lame curves with double reduction and in the explicit…
This paper, commissioned as a survey of the Riemann Hypothesis, provides a comprehensive overview of 165 years of mathematical approaches to this fundamental problem, while introducing a new perspective that emerged during its preparation.…
We consider numerical approximations of spectral fractional Laplace-Beltrami problems on closed surfaces. The proposed numerical algorithms rely on their Balakrishnan integral representation and consist of a sinc quadrature coupled with…
The theory of error-correcting codes is concerned with constructing codes that optimize simultaneously transmission rate and relative minimum distance. These conflicting requirements determine an asymptotic bound, which is a continuous…
Phase space cuts are implemented by inserting Heaviside theta functions in the integrands of momentum-space Feynman integrals. By directly parametrizing theta functions and constructing integration-by-parts (IBP) identities in the…
We review recent developments in the method of algebro-geometric integration of integrable systems related to deformations of algebraic curves. In particular, we discuss the theta-functional solutions of Schlesinger system, Ernst equation…