On Generating Discrete Orthogonal Bivariate Polynomials

In this paper we present an algorithm for recursively generating orthogonal bivariate polynomials on a discrete set S ². For this purpose we employ commuting pairs of real symmetric matrices H, K ^n×n to obtain, in a certain sense, a two dimensional Hermitian Lanczos method. The resulting algorithm relies on a recurrence having a slowly growing length. Practical implementation issues an applications are considered. The method can be generalized to compute orthogonal polynomials depending on an arbitrary number of variables.     

On G 2 Continuous Spline Interpolation of Curves in ℝ d

In this paper the problem of G ² continuous interpolation of curves in ^d by polynomial splines of degree n is studied. The interpolation of the data points and two tangent directions at the boundary is considered. The case n = r + 2 = d, where r is the number of interior points interpolated by each segment of the spline curve, is studied in detail. It is shown that the problem is uniquely solvable asymptotically, e., when the data points are sampled regularly and sufficiently dense, and lie on a regular, convex parametric curve in ^d . In this case the optimal approximation order is also determined.     

Nef Divisors in Codimension One on the Moduli Space of Stable Curves

Let M _g be the moduli space of smooth curves of genus g 3, and M¯ _g the Deligne-Mumford compactification in terms of stable curves. Let M¯ _g ^[1] be an open set of M¯ _g consisting of stable curves of genus g with one node at most. In this paper, we determine the necessary and sufficient condition to guarantee that a -divisor D on M¯ _g is nef over M¯ _g ^[1], that is, (D · C) 0 for all irreducible curves C on M¯ _g with C M¯ _g ^[1] .     

Establishment of The Power—Time Curve Equation of Bacterial Growth in the Log Phase

The power-time curves of two species of bacteria, Vibro metschnikovii, Vibro bollisae were determined calorimetrically by using a 2277 bioactivity monitor. The power-time curve equation of bacterial growth in the log phase can be expressed as

Homogeneity and the disjoint arcs property

Some previous results of the author towards a classification of homogeneous metric continua are improved. The disjoint arcs property is fully revealed in this context. In particular, closed -manifolds, , are characterized as those homogeneous continua which do not have the disjoint arcs property.

     

Cyclic torsion of elliptic curves

Let be an elliptic curve over a number field such that
and let denote the number of roots of unity in . Ross proposed a question: Is isogenous over to an elliptic curve such that is cyclic of order dividing ? A counter-example of this question is given. We show that is isogenous to such that . In case has complex multiplication and , we obtain certain criteria whether or not is isogenous to such that .

     

Weight distributions of geometric Goppa codes

The in general hard problem of computing weight distributions of linear codes is considered for the special class of algebraic-geometric codes, defined by Goppa in the early eighties. Known results restrict to codes from elliptic curves. We obtain results for curves of higher genus by expressing the weight distributions in terms of -series. The results include general properties of weight distributions, a method to describe and compute weight distributions, and worked out examples for curves of genus two and three.

     

Hysteresis phenomena in the light self-diffraction in a dense resonant medium

On the basis of the modified Bloch equations a theory is developed of the light self-diffraction in a thin film of a dense resonant medium with allowance for the dipole-dipole interactions of atoms. It is shown that for certain values of intensity of an external field, there occurs in the film the light-induced first-kind phase transition which causes the emergence of regions (domains) with the high excitation of atoms. The switching effects of the system dense resonant medium + periodically-modulated field under consideration result in a hysteresis behaviour in the dependence of the diffraction efficiency on the adiabatically changing intensity of the external field. The predicted hysteresis offers additional potentialities in the investigation of the dynamics of the first-kind phase transition in dense media and outlook for usage of the film as a switching logical element.     

On the Diameter of Plane Curves

Recently, differential geometric properties of embedded projective varieties have gained increasing interest. In this note, we consider plane algebraic curves equipped with the Fubini--Study metric from₂ () and give an estimate for the diameter in terms of the degree, initiated in a paper by F. A. Bogomolov.     

MSE's of prediction in growth curve model with covariance structures

We consider the growth curve model with covariance structures: positive-definite, uniform covariance structure and serial covariance structure. Two types of prediction problems are studied in this paper. One is called the conditional prediction problem and the other is called the extended prediction problem. For both types of prediction problems, the mean squared error for a serial covariance structure is obtained for the estimates based on the conditional expectation: the mean squared error for an unrestricted covariance structure is compared with the mean squared error for a uniform covariance structure or a serial covariance structure. These results are exemplified by two sets of real data.This research was supported in part by Grant-in-Aid for general Scientific Research, The Ministry of Education, Science and Culture under Contract Number 03640239.