Each closed Riemann surface S of genus g ≥ 1 has associated a principally polarized Abelian variety J(S), called the Jacobian variety of S. Classical Torelli’s theorem states that S is uniquely determined, up to conformal equivalence, by J(S). On the other hand, if S is either a non-compact analytically finite Riemann surfaces or an analytically finite Riemann orbifold, then it seems that there...

1. Assume everything you know from the notes on Riemann-Stieltjes integration. (a) Show that if g : [0, 1]→ R is Riemann-integrable on [0, 1] and φ : [a, b]→ R is continuous on an interval [a, b] ⊇ g[[0, 1]], then φ ◦ g is Riemann-integrable on [0, 1]. (b) Assume that f : [0, 1]→ R is a function for which f ′(t) exists at each t ∈ [0, 1] (one-sidedly at endpoints) and that the Riemann integral ...

*Correspondence: [email protected] School of Mathematics and Physics, Anhui Jianzhu University, Hefei, 230601, P.R. China Abstract The Riemann problem for a one-dimensional nonlinear wave system with different gamma laws is considered. By the properties of wave curves, we observe that this system does not contain the composite wave compared to the barotropic models of gas dynamics with differ...

Abstract. The complex zeros of the Riemannn zeta-function are identical to the zeros of the Riemann xi-function, ξ(s). Thus, if the Riemann Hypothesis is true for the zetafunction, it is true for ξ(s). Since ξ(s) is entire, the zeros of ξ(s), its derivative, would then also satisfy a Riemann Hypothesis. We investigate the pair correlation function of the zeros of ξ(s) under the assumption that ...

A further step in the strategy for proving Riemann hypothesis proposed earlier(M. Pitkänen, math.GM/0102031) is suggested. The vanishing of Rieman Zeta reduces to an orthogonality condition for the eigenfunctions of a non-Hermitian operator D having the zeros of Riemann Zeta as its eigenvalues. The construction of D is inspired by the conviction that Riemann Zeta is associated with a physical s...

— Recently, Baker and Norine (Advances in Mathematics, 215(2): 766-788, 2007) found new analogies between graphs and Riemann surfaces by developing a Riemann-Roch machinery on a finite graph G. In this paper, we develop a general Riemann-Roch Theory for sub-lattices of the root lattice An by following the work of Baker and Norine, and establish connections between the Riemann-Roch theory and th...

We define and prove basic properties of Riemann surfaces, which we follow with a discussion of divisors and an elementary proof of the RiemannRoch theorem for compact Riemann surfaces. The Riemann-Roch theorem is used to prove the existence of a holomorphic embedding from any compact Riemann surface into n-dimensional complex projective space Pn. Using comparison principles such as Chow’s theor...

In order to study the properties of Lovelock gravity theories in low dimensions, we define the kth-order Riemann-Lovelock tensor as a certain quantity having a total 4k-indices, which is kth order in the Riemann curvature tensor and shares its basic algebraic and differential properties. We show that the kth order Riemann-Lovelock tensor is determined by its traces in dimensions 2k ≤ D < 4k. In...

In these notes we solve a class of Riemann-Hilbert (inverse monodromy) problems with an arbitrary quasi-permutation monodromy group. The solution is given in terms of Szegö kernel on the underlying Riemann surface. In particular, our construction provides a new class of solutions of the Schlesinger system. We present some results on explicit calculation of the corresponding tau-function, and de...

The theory of S2(δ) family of probability distributions is used to give a derivation of the functional equation of the Riemann xi function. The δ deformation of the xi function is formulated in terms of the S2(δ) distribution and shown to satisfy Riemann’s functional equation. Criteria for simplicity of roots of the xi function and for its simple roots to satisfy the Riemann hypothesis are form...