In 1990s several papers studied a strong tridiagonal M-basis that did not possess rank one density property. We offer a new method for the study of more generic finite-band M-bases, employing a graph theory framework. We determine the necessary and sufficient conditions for rank one density property in this class of M-bases. Also we give some sufficient conditions concerning k point density property.

In the early 1990s the works of Larson, Wogen and Argyros, Lambrou, Longstaff disclosed an example of a strong tridiagonal $M$-basis that was not rank one dense. Later Katavolos, Lambrou and Papadakis studied $k$ point density property of this example. In this paper we present new methods for the analysis of $k$ point density and rank one density properties for band-diagonal $M$-bases.

A special class of complete minimal systems with complete biorthogonal system in a Hilbert space is considered. This class was introduced by Azoff and Shehada. The paper studies conditions under which there exists a linear summation method for Fourier series with respect to the Azoff–Shehada system. A construction of a linear summation method of the Fourier series for a given vector is presented, as well as a construction of a universal linear summation method.

Recent advances in single-cell genomics provide an alternative to gene-centric metagenomics studies, enabling whole genome sequencing of uncultivated bacteria. However, single-cell assembly projects are challenging due to (i) the highly non-uniform read coverage, and (ii) a greatly elevated number of chimeric reads and read pairs. While recently developed single-cell assemblers have addressed the former challenge, methods for assembling highly chimeric reads remain poorly explored. We present algorithms for identifying chimeric edges and resolving complex bulges in de Bruijn graphs, which significantly improve single-cell assemblies. We further describe applications of the single-cell assembler SPAdes to a new approach for capturing and sequencing “dark matter of life” that forms small pools of randomly selected single cells (called a mini-metagenome) and further sequences all genomes from the mini-metagenome at once. We demonstrate that SPAdes enables sequencing mini-metagenomes and benchmark it against various assemblers. On single-cell bacterial datasets, SPAdes improves on the recently developed E+V-SC and IDBA-UD assemblers specifically designed for single-cell sequencing. For standard (multicell) datasets, SPAdes also improves on A5, ABySS, CLC, EULER-SR, Ray, SOAPdenovo, and Velvet.

Recent advances in single-cell genomics provide an alternative to largely gene-centric metagenomics studies, enabling whole-genome sequencing of uncultivated bacteria. However, single-cell assembly projects are challenging due to (i) the highly nonuniform read coverage and (ii) a greatly elevated number of chimeric reads and read pairs. While recently developed single-cell assemblers have addressed the former challenge, methods for assembling highly chimeric reads remain poorly explored. We present algorithms for identifying chimeric edges and resolving complex bulges in de Bruijn graphs, which significantly improve single-cell assemblies. We further describe applications of the single-cell assembler SPAdes to a new approach for capturing and sequencing "microbial dark matter" that forms small pools of randomly selected single cells (called a mini-metagenome) and further sequences all genomes from the mini-metagenome at once. On single-cell bacterial datasets, SPAdes improves on the recently developed E+V-SC and IDBA-UD assemblers specifically designed for single-cell sequencing. For standard (cultivated monostrain) datasets, SPAdes also improves on A5, ABySS, CLC, EULER-SR, Ray, SOAPdenovo, and Velvet. Thus, recently developed single-cell assemblers not only enable single-cell sequencing, but also improve on conventional assemblers on their own turf. SPAdes is available for free online download under a GPLv2 license.

The lion's share of bacteria in various environments cannot be cloned in the laboratory and thus cannot be sequenced using existing technologies. A major goal of single-cell genomics is to complement gene-centric metagenomic data with whole-genome assemblies of uncultivated organisms. Assembly of single-cell data is challenging because of highly non-uniform read coverage as well as elevated levels of sequencing errors and chimeric reads. We describe SPAdes, a new assembler for both single-cell and standard (multicell) assembly, and demonstrate that it improves on the recently released E+V-SC assembler (specialized for single-cell data) and on popular assemblers Velvet and SoapDeNovo (for multicell data). SPAdes generates single-cell assemblies, providing information about genomes of uncultivatable bacteria that vastly exceeds what may be obtained via traditional metagenomics studies. SPAdes is available online ( http://bioinf.spbau.ru/spades ). It is distributed as open source software.