We give a simple condition for a linear recurrence (mod 2W) of degree r to have the maximal possible period 2M_1(2'' 1). It follows that the period is maximal in the cases of interest for pseudorandom number generation, i.e., for three-term linear recurrences defined by trinomials which are primitive (mod 2) and of degree r > 2 . We consider the enumeration of certain exceptional polynomials wh...

Once on the forefront of mathematical research in America, the asymptotics of the solutions of linear recurrence equations is now almost forgotten, especially by the people who need it most, namely combinatorists and computer scientists. Here we present this theory in a concise form and give a number of examples that should enable the practicing combinatorist and computer scientist to include t...

A binomial identity ((1) below), which relates the famous Apéry numbers and the sums of cubes of binomial coefficients (for which Franel has established a recurrence relation almost 100 years ago), can be seen as a particular instance of a Legendre transform between sequences. A proof of this identity can be based on the more general fact that the Apéry and Franel recurrence relations themselve...

ergy To demonstrate the existence of kinks in ∆G(N ; d) more clearly, we plot smoothed derivatives of ∆G(N ; d) in Figure 2b of the main text. Directly computing derivatives of ∆G(N) using finite difference, for example, using ∂∆G/∂N = ∆G(N + 1) − ∆G(N), results in noisy estimates due to the numerical error in ∆G(N) (Figure S1). To minimize such effects, we first smooth ∆G(N) using a rectangula...

written for a particular gain. Then KGH = −1 so that a) when the amplitude of KGH is 1 or 0 dB and b) phase is 180o then we have the system just become unstable. Both the conditions have to be fulfilled. Hence, if we ask the question: How far is the system from instability ? We get two answers. The gain increment required at the frequency where phase is 180o and the phase increment required at ...

The paper deals with testing of selected heuristic optimization methods and their evaluation. We have proposed different techniques which express the success of the optimization method in different ways (the method success, the difference between optimum and local extreme, the distances of quartiles, the number of simulation experiments until the optimum was found). These evaluation techniques ...

SUMMARY High-throughput RNA sequencing (RNA-seq) is rapidly emerging as a major quantitative transcriptome profiling platform. Here, we present DEGseq, an R package to identify differentially expressed genes or isoforms for RNA-seq data from different samples. In this package, we integrated three existing methods, and introduced two novel methods based on MA-plot to detect and visualize gene ex...

The capability of the dimensions measurement by ten appraisers of ten samples for Charpy Vnotch pendulum impact test was evaluated by Youden plot, analysis of measurement uncertainties, Measurement Systems Analysis (MSA), %P/T, Z-score, Mandel’s statistics and ttest. The group of “better” appraisers given by Youden plot was validated by majority of other methods.