On Sequences, Rational Functions and Decomposition

Our overall goal is to unify and extend some results in the literature related to the approximation of generating functions of finite and infinite sequences over a field by rational functions. In our approach, numerators play a significant role. We revisit a theorem of Niederreiter on (i) linear complexities and (ii) '$n^{th}$ minimal polynomials' of an infinite sequence, proved using partial quotients. We prove (i) and its converse from first principles and generalise (ii) to rational functions where the denominator need not have minimal degree. We prove (ii) in two parts: firstly for geometric sequences and then for sequences with a jump in linear complexity. The basic idea is to decompose the denominator as a sum of polynomial multiples of two polynomials of minimal degree; there is a similar decomposition for the numerators. The decomposition is unique when the denominator has degree at most the length of the sequence. The proof also applies to rational functions related to finite sequences, generalising a result of Massey. We give a number of applications to rational functions associated to sequences.Comment: Several more typos corrected. To appear in J. Applied Algebra in Engineering, Communication and Computing. The final publication version is available at Springer via http://dx.doi.org/10.1007/s00200-015-0256-

Minimal Polynomial Algorithms for Finite Sequences

We show that a straightforward rewrite of a known minimal polynomial algorithm yields a simpler version of a recent algorithm of A. Salagean.Comment: Section 2 added, remarks and references expanded. To appear in IEEE Transactions on Information Theory

Vacuum polarization energy of the Shifman-Voloshin soliton

We compute the vacuum polarization energy of soliton configurations in a model with two scalar fields in one space dimension using spectral methods. The second field represents an extension of the conventional $\phi^4$ kink soliton model. We find that the vacuum polarization energy destabilizes the soliton except when the fields have identical masses. In that case the model is equivalent to two independent $\phi^4$ models.Comment: nine pape

Digital overlaying of the universal transverse Mercator grid with LANDSAT data derived products

Picture elements of data from the LANDSAT multispectral scanner are correlated with the universal tranverse Mercator grid. In the procedure, a series of computer modules was used to make approximations of universal transverse Mercator grid locations for all picture elements from the grid locations of a limited number of known control points and to provide display and digital storage of the data. The software has been written in FORTRAN 4 language for a Varian 70-series computer

Spectral Methods for Coupled Channels with a Mass Gap

We develop a method to compute the vacuum polarization energy for coupled scalar fields with different masses scattering off a background potential in one space dimension. As an example we consider the vacuum polarization energy of a kink-like soliton built from two real scalar fields with different mass parameters.Comment: 14 pages, 5 figures, matches journal version, references added (surprisingly many

Simplified system displays complex curves corresponding to input data

Cathode ray oscilloscope displays curves or contours of complex shapes corresponding to sets of x,y coordinates. It requires few storage facilities and produces a rapid display of complex curves with a fewer number of commands than previous systems

Attractive Electromagnetic Casimir Stress on a Spherical Dielectric Shell

Based on calculations involving an idealized boundary condition, it has long been assumed that the stress on a spherical conducting shell is repulsive. We use the more realistic case of a Drude dielectric to show that the stress is attractive, matching the generic behavior of Casimir forces in electromagnetism. We trace the discrepancy between these two cases to interactions between the electromagnetic quantum fluctuations and the dielectric material.Comment: Five pages, one figure, pdflatex, matches version to be pusblished in Phys Lett