A closed Riemann surface X which can be realised as a p-sheeted covering of the Riemann sphere is called p-gonal, and such a covering is called a p-gonal morphism. A p-gonal Riemann surface is called real p-gonal if there is an anticonformal involution (symmetry) σ of X commuting with the p-gonal morphism. If the p-gonal morphism is a cyclic regular covering the Riemann surface is called real c...

In the present paper we obtain a necessary and sufficient condition to prove the Riemann hypothesis in terms of certain properties of local extrema of the function Ξ(t) = ξ( 1 2 + it). First, we prove that positivity of all local maxima and negativity of all local minima of Ξ(t) form a necessary condition for the Riemann hypothesis to be true. After showing that any extremum point of Ξ(t) is a ...

We are solving the classical Riemann-Hilbert problem of rank N > 1 on the extended complex plane punctured in 2m + 2 points, for N × N quasi-permutation monodromy matrices with 2m(N − 1) parameters. Our approach is based on the finite gap integration method applied to study the Riemann-Hilbert by Deift, Its, Kapaev and Zhou [1] and Kitaev and Korotkin [2, 3]. This permits us to solve the Rieman...

is called Fuchsian if the N ×N coefficient matrix A(λ) is a rational function of λ whose singularities are simple poles. Each Fuchsian system generates, via analytic continuation of its fundamental solution Ψ (λ) along closed curves, a representation of the fundamental group of the punctured Riemann sphere (punctured at the poles of A(λ)) in the group of N ×N invertible matrices. This represent...

All this seems to be started by Georg Friedrich Bernhard Riemann’s original paper whose English translation was included at least in [12], without, however forgetting the enormous influence of Leonhard Euler and his studies on the function now known as the Riemann zeta-function. After this, a huge number of papers is available in the internet on the Riemann hypothesis and the zeta-function. Fur...

We prove a general uniformization theorem for N=2 superconformal and N=1 superanalytic DeWitt super-Riemann surfaces, showing that in general an N=2 superconformal (resp. N=1 superanalytic) DeWitt superRiemann surface is N=2 superconformally (resp., N=1 superanalytically) equivalent to a manifold with transition functions containing no odd functions of the even variable if and only if the first...

We address the problem of surface inpainting, which aims to fill in holes or missing regions on a Riemann surface based on its surface geometry. In practical situation, surfaces obtained from range scanners often have holes or missing regions where the 3D models are incomplete. In order to analyze the 3D shapes effectively, restoring the incomplete shape by filling in the surface holes is neces...