To obtain efficient cryptosystems based on hyperelliptic curves, we studied genus 2 isomorphism classes of hyperelliptic curves in characteristic 2. We found general and optimal form for these curves, just as the short Weierstrass form for elliptic curves. We studied the security and the arithmetic on their jacobian. We also rewrote and optimized the formulas of Lange in characteristic 2, and w...

A new problem is studied, that is to find nonlinear differential equations with special solutions expressed via the Weierstrass function. A method is discussed to construct nonlinear ordinary differential equations with exact solutions. The main step of our method is the assumption that nonlinear differential equations have exact solutions which are general solution of the simplest integrable e...

In this paper a new Weierstrass semi-rational expansion method is developed via the Weierstrass elliptic function ℘(ξ; g2, g3). With the aid of Maple, we choose the coupled water wave equation and the generalized Hirota-Satsuma coupled KdV equation to illustrate the method. As a consequence, it is shown that the method is powerful to obtain many types of new doubly periodic solutions in terms o...

LetX denote the rational curve with n+1 nodes obtained from the Riemann sphere by identifying 0 with∞ and ζj with −ζj for j = 0, 1, . . . , n−1, where ζ is a primitive (2n)th root of unity. We show that if n is even, then X has no smooth Weierstrass points, while if n is odd, then X has 2n smooth Weierstrass points. C. Widland [14] showed that the rational curve with three nodes obtained from P...

This work was initially motivated by Miranda’s work on elliptic Weierstrass threefolds. An elliptic variety is a complex (irreducible and reduced) projective variety together with a morphism π : X → S whose fiber over a general point is a smooth elliptic curve. For example, if the elliptic fibration has a section, X can be locally expressed as a hypersurface [De]. This is called the Weierstrass...

A new method of constructing elliptic finite-gap solutions of the stationary Korteweg-de Vries (KdV) hierarchy, based on a theorem due to Picard, is illustrated in the concrete case of the Lamé-Ince potentials −s(s+1)P(z), s ∈ N (P(.) the elliptic Weierstrass function). Analogous results are derived in the context of the stationary modified Korteweg-de Vries (mKdV) hierarchy for the first time.

The Weierstrass nowhere diierentiable function, and functions constructed from similar innnite series, have been studied often as examples of functions whose graph is a fractal. Though there is a simple formula for the Hausdorr dimension of the graph which is widely accepted, it has not been rigorously proved to hold. We prove that if arbitrary phases are included in each term of the summation ...

Here ~9(x):p(x; wl,w3) denotes the elliptic Weierstrass function with fundamental periods 2wl and 2w3 (Im(w3/Wl)#0). In the special case where Wl is real and w3 is purely imaginary, the potential q(x) in (1.1) is real-valued and Ince's striking result [51], in modern spectral-theoretic terminology, yields that the spectrum of the unique self-adjoint operator associated with the differential exp...

here, the concept of quantum potential that has been illustrated in extended phase space (eps) in previous works is explored for its marginal behavior. unlike in configuration space, different representations of the quantum mechanics can be found in eps when exploiting appropriate canonical transformations. these canonical transformations revealed that there exist several representations in whi...

QCD sum rules are based on the Operator Product Expansion of current correlators, and on QCD-hadron duality. An extension of this program to finite temperature is discussed. This allows for a study of deconfinement and chiral-symmetry restoration. In addition, it is possible to relate certain hadronic matrix elements to expectation values of quark and gluon field operators by using thermal Fini...