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

The art of writing scientific reports

As the purpose of the report is to transmit as smoothly and as easily as possible, certain facts and ideas, to the average person likely to read it, it should be written in a full and simple enough manner to be comprehended by the least tutored, and still not be boring to the more learned readers

The effect of staggering a biplane

This investigation was carried out by request of the United States Air Service at the Massachusetts Institute of Technology wind tunnel in 1918. As the data collected may be of general interest, they are published here by the National Advisory Committee for Aeronautics. The lift, drag, and center of pressure travel are determined for a biplane with a stagger varying from +100% to -100%. It is found that the efficiency and the maximum lift increase with positive stagger. With large positive staggers the center of pressure is far forward and has a very slight travel with changes in lift coefficient

The Photographic Recording of Small Motions

Methods and equipment for recording small and sometimes rapid motions by photographic means are described, and the efficacy of photographic recording in such instances is evaluated. The optical system consisting of the light source, the mirror or prism for transmitting motion to the emergent beam, and a means of bringing the rays into focus on the film are discussed. Attention is given to the critical issue of mirror mounting. The film holder and the driving motor for the recording drum are described in detail. The authors conclude that the optical methods they describe are far more satisfactory than the recording pen, in compactness, in high natural period, and in elimination of friction. Costs are similar to mechanical methods. The development and reproduction of the record is an added complication, but the ease of duplicating the records is a decided advantage

The drawing of experimental curves

This report presents a discussion of how to determine the location of a line or surface from experimental data. What we desire to know practically is the number of ordinates required to obtain a certain probable precision in drawing a line or surface

N.A.C.A. Recording Air Speed Meter

A new type of air speed meter is described which was designed by the technical staff of the National Advisory Committee for Aeronautics. The instrument consists essentially of a tight metal diaphragm of high natural period which is acted upon by the pressure difference of a pitot-static head. The resulting deflection of this diaphragm is recorded optically on a moving film