Given a faithful parameterization P(t) of a rational plane curve, an inversion formula t = f(x,y) gives the parameter value corresponding to a point (x,y) on the curve, where f is a rational function in x and y. We investigate the relationship between a point (x*, y*) not on the curve and the corresponding point P(t*) on the curve, where t* = f(x*, y*). It is shown that for a rational quadratic...

Let K = Q(ω), with ω3 = m a positive integer, be a pure cubic number field. We show that the elements α ∈ K× whose squares have the form a − ω for rational numbers a form a group isomorphic to the group of rational points on the elliptic curve Em : y2 = x3 − m. This result will allow us to construct unramified quadratic extensions of pure cubic number fields K.

We report the structural and transport properties of the Bi-substituted Co2MnO4 multiferroic materials. Samples synthesized using the solid state reaction route with the composition BixCo2−xMnO4 0x 0.3 exhibit a single phase behavior with a cubic spinel structure space group Fd3m . The lattice parameter was found to increase with the Bi substitution. The dc-conductivity studies reveal that all ...

In our contribution we study cubic and quintic Pythagorean Hodograph (PH) curves in the Euclidean and Minkowski planes. We analyze their control polygons and give necessary and sufficient conditions for cubic and quintic curves to be PH. In the case of Euclidean cubics the conditions are known and we provide a new proof. For the case of Minkowski cubics we formulate and prove a new simple geome...

Several recent papers have shown how to approximate the difference ∑i |ai−bi| or ∑ |ai−bi| between two functions, when the function values ai and bi are given in a data stream, and their order is chosen by an adversary. These algorithms use little space (much less than would be needed to store the entire stream) and little time to process each item in the stream. They approximate with small rel...

We give an explicit treatment of cubic function fields of characteristic at least five. This includes an efficient technique for converting such a field into standard form, formulae for the field discriminant and the genus, simple necessary and sufficient criteria for non-singularity of the defining curve, and a characterization of all triangular integral bases. Our main result is a description...

We present several Hermite-type interpolation methods for rational cubics. In case the input data come from a circular arc, the rational cubic will reproduce it. c © 2008 Elsevier Ltd. All rights reserved.

We consider diagonal cubic surfaces defined by an equation of the form ax + by + cz + dt = 0. Numerically, one can find all rational points of height 6 B for B in the range of up to 10 , thanks to a program due to D. J. Bernstein. On the other hand, there are precise conjectures concerning the constants in the asymptotics of rational points of bounded height due to Manin, Batyrev and the author...

Based on the local refinement algorithm addressed in [18], we analyze the linear independence of the bi-cubic T-spline blending functions corresponding to particular T-meshes.