Hyperelliptic addition law

We construct an explicit form of the addition law for hyperelliptic Abelian vector functions $\wp$ and $\wp'$. The functions $\wp$ and $\wp'$ form a basis in the field of hyperelliptic Abelian functions, i.e., any function from the field can be expressed as a rational function of $\wp$ and $\wp'$.Comment: 18 pages, amslate

Solution of the Jacobi inversion problem on non-hyperelliptic curves

In this paper we propose a method of solving the Jacobi inversion problem in terms of multiply periodic $\wp$ functions, also called Kleinian $\wp$ functions. This result is based on the recently developed theory of multivariable sigma functions for $(n,s)$-curves. Considering $(n,s)$-curves as canonical representatives in the corresponding classes of bi-rationally equivalent plane algebraic curves, we claim that the Jacobi inversion problem on plane algebraic curves is solved completely. Explicit solutions on trigonal, tetragonal and pentagonal curves are given as an illustration.Comment: 29 page

The problem of differentiation of an Abelian function over its parameters

Theory of Abelian functions was a central topic of the 19th century mathematics. In mid-seventies of the last century a new wave arose of investigation in this field in response to the discovery that Abelian functions provide solutions of a number of challenging problems of modern Theoretical and Mathematical Physics. In a cycle of our joint papers published in 2000–05, we have developed a theory of multivariate sigma-function, an analogue of the classic Weierstrass sigma-function. A sigma-function is defined on a cover of U , where U is the space of a bundle p : U → B defined by a family of plane algebraic curves of fixed genus. The base B of the bundle is the space of the family parameters and a fiber J_b over b ∈ B is the Jacobi variety of the curve with the parameters b. A second logarithmic derivative of the sigma-function along the fiber is an Abelian function on J_b. Thus, one can generate a ring F of fiber-wise Abelian functions on U. The problem to find derivations of the ring F along the base B is a reformulation of the classic problem of differentiation of Abelian functions over parameters. Its solution is relevant to a number of topical applications. This work presents a solution of this problem recently found by the authors. Our method of solution essentially employs the results from Singularity Theory about vector fields tangent to the discriminant of a singularity y^n -x^s, gcd(n, s) = 1

Number of tweets per day made by Twitter accounts of managers and organizations.

<p>Number of tweets per day made by Twitter accounts of managers and organizations.</p

Extent of tweets, retweets and mentions during the Westgate mall attack.

<p>Extent of tweets, retweets and mentions during the Westgate mall attack.</p