Related papers: Solution of Mumford's second problem
The paper develops the result of second Thomae theorem in hyperelliptic case. The main formula, called general Thomae formula, provides expressions for values at zero of the lowest non-vanishing derivatives of theta functions with singular…
In this paper we prove a Thomae derivative formula for trigonal curves admitting a non-singular affine model. This formula relates the derivatives of theta functions with rational characteristics on the curve to explicit expressions in the…
We give an explicit formula of the normalized Mumford form which expresses the second tautological line bundle by the Hodge line bundle defined on the moduli space of algebraic curves of any genus. This formula is represented by an infinite…
We evaluate 35 different theta derivatives with rational characteristics using the method of Matsuda, in which the logarithmic derivatives at $z=0$ of the triple product expansions of the associated theta functions become linear…
In any positive genus case, we show an explicit formula of the Mumford forms expressed by infinite products like Selberg type zeta values for Schottky groups. This result is considered as an extension of the formula in terms of Ramanujan's…
In this paper the Mittag-Leffler function is given through the exponential functions for any rational derivatives of m/n order, where m<n, n>1 are natural irreducible numbers (if n=1 then m is also equal to unity). Unlike the previous…
This note describes continued fraction representations for the rational approximations to the zeta function recently found by the author. It is tempting to think that these continued fractions might be analysed using a souped up version of…
We construct the theta function solution to the initial value problem for the generalized periodic discrete Toda equation.
The Riemann theta function is a complex-valued function of g complex variables. It appears in the construction of many (quasi-) periodic solutions of various equations of mathematical physics. In this paper, algorithms for its computation…
This paper is an annotated list of transformation properties and identities satisfied by the four theta functions $\theta _1$, $\theta _2$, $\theta _3$, $\theta _4$ of one complex variable, presented in a ready-to-use form. An attempt is…
We propose a conjecture for the exact expression of the dynamical zeta function for a family of birational transformations of two variables, depending on two parameters. This conjectured function is a simple rational expression with integer…
This article describes a sequence of rational functions which converges locally uniformly to the zeta function. The numerators (and denominators) of these rational functions can be expressed as characteristic polynomials of matrices that…
In this paper, we obtain analogues of Jacobi's derivative formula in terms of the theta constants with rational characteristics. For this purpose, we use the arithmetic formulas of the number of representations of a natural number…
In this paper, we first deduce the explicit formulas for the projector of the $n$th level fractional derivative and for its Laplace transform. Then the fractional relaxation equation with the $n$th level fractional derivative is discussed.…
We find identities between theta constants with rational characteristics evaluated at period matrix of $R,$ a cyclic 3 sheeted cover of the sphere with $3k$ branch points $\lambda_1...\lambda_{3k}.$ These identities follow from Thomae…
We give an application of Mumford's theory of canonical theta characteristics to a Diophantine problem in characteristic two. We prove that a smooth plane curve over a global field of characteristic two is defined by the determinant of a…
Following Douady-Hubbard and Bartholdi-Nekrashevych, we give an algebraic formulation of Thurston's characterization of rational functions. The techniques developed are applied to the analysis of the dynamics on the set of free homotopy…
Explicit solutions of differential equations of complex fractional orders with respect to functions and with continuous variable coefficients are established. The representations of solutions are given in terms of some convergent infinite…
While looking for abductive explanations of a given set of manifestations, an ordering between possible solutions is often assumed. The complexity of finding/verifying optimal solutions is already known. In this paper we consider the…
The asymptotic behavior (such as convergence to an equilibrium, convergence to a 2-cycle, and divergence to infinity) of solutions of the following multi-parameter, rational, second order difference equation x_{n+1} =(ax_{n}^3+…