Summability of subsequences of a divergent sequence by regular matrices II

C. Stuart proved in [27, Proposition 7] that the Ces?ro matrix C1 cannot sum almost every subsequence of a bounded divergent sequence. At the end of the paper he remarked ?It seems likely that this proposition could be generalized for any regular matrix, but we do not have a proof of this?. In [4, Theorem 3.1] Stuart?s conjecture is confirmed, and it is even extended to the more general case of divergent sequences. In this note we show that [4, Theorem 3.1] is a special case of Theorem 3.5.5 in [24] by proving that the set of all index sequences with positive density is of the second category. For the proof of that a decisive hint was given to the author by Harry I. Miller a few months before he passed away on 17 December 2018.

Ideal convergent subsequences and rearrangements for divergent sequences of functions

Let 𝓙 be an ideal on ℕ which is analytic or coanalytic. Assume that (

An alignment method for nucleic acid sequences against annotated genomes

MotivationBiological sequence alignment is fundamental to their further interpretation. Current alignment algorithms typically align either nucleic acid or amino acid sequences. Using only nucleic acid sequence similarity, divergent sequences cannot be aligned reliably because of the limited alphabet and genetic saturation. To align divergent coding nucleic acid sequences, one can align using the translated amino acid sequences. This requires the detection of the correct open reading frame, is prone to eventual frame shift errors, and typically requires the treatment of genes separately. It was our motivation to design a nucleic acid sequence alignment algorithm to align a nucleic acid sequence against a (reference) genome sequence, that works equally well for similar and divergent sequences, and produces an optimal alignment considering simultaneously the alignment of all annotated coding sequences.ResultsWe define a genome alignment score for evaluating the quality of an alignment of a nucleic acid query sequence against a reference genome sequence, for which coding sequence features have been annotated (for example in a GenBank record). The genome alignment score combines the a ne gap score for the nucleic acid sequence with an a ne gap score for all amino acid alignments resulting from coding sequences in open reading frames contained within the query sequence. We present a Dynamic Programming algorithm to compute the optimal global or local alignment using this genomic alignment score and provide a formal proof of correctness. This algorithm allows the alignment of nucleic acid sequences from closely related and highly divergent sequences within the same software and using the same parameters, automatically correcting any eventual frame shift errors and produces at the same time the aligned translated amino acid sequences of all relevant coding sequence features.AvailabilityThe software is available as a web application at http://www.genomedetective.com/app/aga and as command-line application at https://github.com/emweb/aga

A Note on The Abel Matrix Transformations

The purpose of this research is to investigate the effect of applying At to convergent sequences, bounded sequences, divergent sequences, and absolutely convergent sequences. We considering and answer the following interesting main research questions.