Number of rational points of a singular curve

In this paper, we give a bound for the number of rational points of a complete, geometrically irreducible, algebraic curve defined over a finite field. We compare it with other known bounds and discuss its sharpness. We also show that the asymptotic Drinfeld-Vladut bound can be generalized to the case of singular curves.

     

Cross-sectional observations of the cholesteric texture in a Cano wedge cell

A novel procedure for an observation of a liquid crystal texture is proposed by using a UV curable liquid crystal. Optical observations from a cross-section of the Cano wedge cell are demonstrated and layered structures of cholesteric LC are visualized clearly. The structures near the Grandjean lines are then revealed in detail from images. From the observations, it is confirmed that the texture near the Grandjean line is similar to fingerprint texture     

Generation of the second harmonic in inhomogeneous magnetoactive plasma

The paper deals with the experimental study of conditions under which the second harmonic of a strong electromagnetic wave generating a spatially inhomogeneous plasma originates and exists. The second harmonic was detected by means of a h.f. probe inside the plasma and by a horn antenna outside the plasma. It was found that the beginning of generation of a strong second harmonic signal inside the plasma coincides with the overcoming of an evanescent layer and the free passage of the right-hand circularly polarized wave (RHCP wave) into an inhomogeneous plasma. Dependences of the second harmonic amplitude on the static magnetic field and the gas pressure are given.The authors wish to thank Dr. S. S. Moiseyev from FTI of Academy USSR (Kharkov) for valuable discussions concerning mechanisms of the second harmonic generation in an inhomogeneous plasma [4, 6]. To Ph.D.V. Kopeck); the authors would like to expresstheir thanks for many valuable discussions and his interest.     

Changes in light scattering from poly(ethylene terephthalate) and nylon 6 films upon stretching in relation to the deformation mechanism

Unoriented T-die flat films of nylon 6 and PET films annealed at 90°C were stretched in water at 80°C. Amorphous PET films were stretched in water at 65–75°C. Changes in the light scattering patterns from these samples upon stretching were investigated. One of the observed LS patterns from the stretched samples is the H_v eight-leaf pattern consisting of four lobes and streaks. In the nylon 6 and heat-treated PET showing this pattern, spherulitic patterns can be seen in polarization microscopy. The microscopic spherulitic superstructure may possibly be the factor responsible for producing the lobe-and-streak pattern. On the other hand, many microscopic eight-leaf patterns can be observed in amorphous unannealed PET showing the lobe-and-streak pattern. These microscopic patterns are due to retardation at stress concentrations around impurities and nuclei. The superstructure giving these microscopic patterns must be the origin of the lobe-and-streak pattern from unannealed PET. Another scattering pattern, the V_v cruciform pattern, was observed in both stretched nylon 6 and unannealed PET. This pattern is due to an orientation change across the slip lines observed under a polarizing microscope. It is noted (1) that the appearance of the slip lines in PET coincides with the occurrence of oriented crystallization on stretching, (2) that the lobe-and-streak pattern from PET in which orientation crystallization has taken place is fairly stable to heat treatment and does not disappear until just before melting, and (3) that the superstructures produced at low stretching seem to be deformed on further stretching, in accordance with affine deformation theory.