We give an explicit construction of a closed curve with constant torsion and everywhere positive curvature. We also discuss the restrictions on closed curves of constant torsion when they are constrained to lie on convex surfaces.

The k-distance match count between the probe and the target is defined as the number of ungapped alignments between the two sequences that have exactly k mismatches, and the k-neighbor match count is defined as the sum of the j-distance match counts for j between 0 and k. We derive a novel formula for the probability of a k-distance match. This formula is based on the assumption that the target is strand-symmetric Bernoulli text (i.e. nucleotides are independently, identically distributed in the target and satisfy Chargaff's second parity rule). Our model predicts that the GC-content in both the probe and the target significantly affects the match count expectation. The ratio of k-neighbor match counts in two distinct genomes for a given probe is a measure of its specificity. We calculated such ratios for pairs of bacterial genomes with different combinations of length, GC-content and phylogenetic distance. Examination of the extreme values of these ratios indicates that probes with a high discriminative power exist for each tested pair.

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Noise-compensated autoregressive (AR) analysis is a problem insufficiently explored with regard to the accuracy of the estimate. This paper studies comprehensively the lower limit of the estimation variance, presenting the asymptotic Cramér-Rao bound (CRB) for Gaussian processes and additive Gaussian noise. This novel result is obtained by using a frequency-domain perspective of the problem as well as an unusual parametrization of an AR model. The Wiener filter rule appears as the distinctive building element in the Fisher information matrix. The theoretical analysis is validated numerically, showing that the proposed CRB is attained by competitive ad hoc estimation methods under a variety of Gaussian color noise and realistic scenarios.Index Terms-Additive Gaussian color noise, autoregressive analysis, Cramér-Rao bound, noise compensation.