The GKZ system for the Hesse pencil of elliptic curves has more solutions than the period integrals. In this work we give different realizations and interpretations of the extra solution, in terms of oscillating integral, Eichler integral, chain integral on the elliptic curve, limit of a period of a certain compact Calabi–Yau threefold geometry, etc. We also highlight the role played by the orb...

We prove some new degeneracy results for integral points and entire curves on surfaces; in particular, we provide the first examples, to our knowledge, of a simply connected smooth variety whose sets of integral points are never Zariski-dense (and no entire curve has Zariski-dense image). Some of our results are connected with divisibility problems, i.e. the problem of describing the integral p...

The four-point integral of the minimal super Liouville gravity on the sphere is evaluated numerically. The integration procedure is based on the effective elliptic parameterization of the moduli space. The analysis is performed for a few different gravitational four-point amplitudes. The results agree with the analytic results recently obtained using the Higher super Liouville equations of motion.

A new calculus on fractal curves, such as the von Koch curve, is formulated. We define a Riemann-like integral along a fractal curve F , called F α-integral, where α is the dimension of F . A derivative along the fractal curve called Fα-derivative, is also defined. The mass function, a measurelike algorithmic quantity on the curves, plays a central role in the formulation. An appropriate algori...

For a simple planar curve in a nonautonomous flow, the fluxing index of a passively advected Lagrangian particle is the net number of its crossing the curve within a given time interval. Such indices naturally induce flux sets, equivalence classes of the particles at the initial time. Previously, we proposed donating regions as an explicit geometric construction of flux sets. In this work, we s...

Abstract. For an analysis of microlensing observational data in case of binary gravitational lenses as well as for an interpretation of observations of high magnification events in multiple images of a lensed quasar it is necessary to calculate for a given source the microlensing light curve by a fold caustic. This problem comes to the numerical calculation of a singular integral. We formulated...

Integral Photography images exhibit high intra-pixel as well as inter-elemental-image correlation. In this work, we present an efficient, omnidirectional Integral Photography compression scheme based on a Hilbert curve scan and a three dimensional transform technique. ©2007 Optical Society of America OCIS codes: (999.9999) Image compression; (100.6890) Three-dimensional image processing; (110.3...

We bound the j-invariant of integral points on a modular curve in terms of the congruence group defining the curve. We apply this to prove that the modular curve Xsplit(p ) has no non-trivial rational point if p is a sufficiently large prime number. Assuming the GRH, one can replace p by p. AMS 2000 Mathematics Subject Classification 11G18 (primary), 11G05, 11G16 (secondary).