Sequences of compatible periodic hybrid orbits of prefractal Koch snowflake billiards

The Koch snowflake KS is a nowhere differentiable curve. The billiard table Omega(KS) with boundary KS is, a priori, not well defined. That is, one cannot a priori determine the minimal path traversed by a billiard ball subject to a collision in the boundary of the table. It is this problem which makes Omega(KS) such an interesting, yet difficult, table to analyze. In this paper, we approach this problem by approximating (from the inside) Omega(KS) by well-defined (prefractal) rational polygonal billiard tables Omega(KS_n). We first show that the flat surface S(KS_n) determined from the rational billiard Omega(KS_n) is a branched cover of the singly punctured hexagonal torus. Such a result, when combined with the results of [Gut2], allows us to define a sequence of compatible orbits of prefractal billiards. We define a hybrid orbit of a prefractal billiard Omega(KS_n) and show that every dense orbit of a prefractal billiard is a dense hybrid orbit of Omega(KS_n). This result is key in obtaining a topological dichotomy for a sequence of compatible orbits. Furthermore, we determine a sufficient condition for a sequence of compatible orbits to be a sequence of compatible periodic hybrid orbits. We then examine the limiting behavior of a sequence of compatible periodic hybrid orbits. We show that the trivial limit of particular (eventually) constant sequences of compatible hybrid orbits constitutes an orbit of Omega(KS). In addition, we show that the union of two suitably chosen nontrivial polygonal paths connects two elusive limit points of the Koch snowflake. Finally, we discuss how it may be possible for our results to be generalized to other fractal billiard tables and how understanding the structures of the Veech groups of the prefractal billiards may help in determining `fractal flat surfaces' naturally associated with the billiard flows.Comment: 21 pages, 15 figure

On the Gerasimov-Drell-Hearn sum rule for the deuteron

The Gerasimov-Drell-Hearn sum rule is evaluated for the deuteron by explicit integration up to 550 MeV including contributions from the photodisintegration channel and from coherent and incoherent single pion production as well. The photodisintegration channel converges fast enough in this energy range and gives a large negative contribution, essentially from the $^1S_0$ resonant state near threshold. Its absolute value is about the same size as the sum of proton and neutron GDH values. It is only partially cancelled by the single pion production contribution. But the incoherent channel has not reached convergence at 550 MeV.Comment: 6 pages latex including 3 postscript figures, talk at the 15th Int. Conf. on Few-Body Problems in Physics, Groningen, Netherlands, 22-26 July 1997. To be published in Nucl. Phys.

Avaliação da severidade da murcha de fusário em tomateiro em diferentes níveis de água no solo por meio do teor de clorollila das folhas.

O objetivo deste experimento foi avaliar a severidade da murcha de fusário em 4 diferentes niveis de água no solo por meio do teor de clorofila das folhas.Resumo 820-1

Avaliação da severidade da murcha de fusário em tomateiro em diferentes níveis de água no solo por meio da temperatura na superficie foliar.

O objetivo do trabalho foi avaliar o efeito do manejo de água de irrigação por meio de diferentes níveis de água no solo sobre o desenvolvimento temporal da murcha de fusário em tomateiro.Resumo 849-1

Laplace Operators on Fractals and Related Functional Equations

We give an overview over the application of functional equations, namely the classical Poincar\'e and renewal equations, to the study of the spectrum of Laplace operators on self-similar fractals. We compare the techniques used to those used in the euclidean situation. Furthermore, we use the obtained information on the spectral zeta function to define the Casimir energy of fractals. We give numerical values for this energy for the Sierpi\'nski gasket

Progresso temporal de Septoria lycopersici em tomateiro orgânico em sistemas de irrigação distintos.

O objetivo deste trabalho foi analisar o progresso temporal da septoriose causada por Septoria lycopersici, em distintos sistema de irrigação