The Asymptotics of Wilkinson's Iteration: Loss of Cubic Convergence

One of the most widely used methods for eigenvalue computation is the $QR$ iteration with Wilkinson's shift: here the shift $s$ is the eigenvalue of the bottom $2\times 2$ principal minor closest to the corner entry. It has been a long-standing conjecture that the rate of convergence of the algorithm is cubic. In contrast, we show that there exist matrices for which the rate of convergence is strictly quadratic. More precisely, let $T_X$ be the $3 \times 3$ matrix having only two nonzero entries $(T_X)_{12} = (T_X)_{21} = 1$ and let $T_L$ be the set of real, symmetric tridiagonal matrices with the same spectrum as $T_X$. There exists a neighborhood $U \subset T_L$ of $T_X$ which is invariant under Wilkinson's shift strategy with the following properties. For $T_0 \in U$, the sequence of iterates $(T_k)$ exhibits either strictly quadratic or strictly cubic convergence to zero of the entry $(T_k)_{23}$. In fact, quadratic convergence occurs exactly when $\lim T_k = T_X$. Let $X$ be the union of such quadratically convergent sequences $(T_k)$: the set $X$ has Hausdorff dimension 1 and is a union of disjoint arcs $X^\sigma$ meeting at $T_X$, where $\sigma$ ranges over a Cantor set.Comment: 20 pages, 8 figures. Some passages rewritten for clarit

Simple sequence repeat variation in the Daphnia pulex genome

Background: Simple sequence repeats (SSRs) are highly variable features of all genomes. Their rapid evolution makes them useful for tracing the evolutionary history of populations and investigating patterns of selection and mutation across gnomes. The recently sequenced Daphnia pulex genome provides us with a valuable data set to study the mode and tempo of SSR evolution, without the inherent biases that accompany marker selection. Results: Here we catalogue SSR loci in the Daphnia pulex genome with repeated motif sizes of 1-100 nucleotides with a minimum of 3 perfect repeats. We then used whole genome shotgun reads to determine the average heterozygosity of each SSR type and the relationship that it has to repeat number, motif size, motif sequence, and distribution of SSR loci. We find that SSR heterozygosity is motif specific, and positively correlated with repeat number as well as motif size. For non-repeat unit polymorphisms, we identify a motif-dependent end-nucleotide polymorphism bias that may contribute to the patterns of abundance for specific homopolymers, dimers, and trimers. Our observations confirm the high frequency of multiple unit variation (multistep) at large microsatellite loci, and further show that the occurrence of multiple unit variation is dependent on both repeat number and motif size. Using the Daphnia pulex genetic map, we show a positive correlation between dimer and trimer frequency and recombination. Conclusions: This genome-wide analysis of SSR variation in Daphnia pulex indicates that several aspects of SSR variation are motif dependent and suggests that a combination of unit length variation and end repeat biased base substitution contribute to the unique spectrum of SSR repeat loci

Escherichia coli Frameshift Mutation Rate Depends on the Chromosomal Context but Not on the GATC Content Near the Mutation Site

Different studies have suggested that mutation rate varies at different positions in the genome. In this work we analyzed if the chromosomal context and/or the presence of GATC sites can affect the frameshift mutation rate in the Escherichia coli genome. We show that in a mismatch repair deficient background, a condition where the mutation rate reflects the fidelity of the DNA polymerization process, the frameshift mutation rate could vary up to four times among different chromosomal contexts. Furthermore, the mismatch repair efficiency could vary up to eight times when compared at different chromosomal locations, indicating that detection and/or repair of frameshift events also depends on the chromosomal context. Also, GATC sequences have been proved to be essential for the correct functioning of the E. coli mismatch repair system. Using bacteriophage heteroduplexes molecules it has been shown that GATC influence the mismatch repair efficiency in a distance- and number-dependent manner, being almost nonfunctional when GATC sequences are located at 1 kb or more from the mutation site. Interestingly, we found that in E. coli genomic DNA the mismatch repair system can efficiently function even if the nearest GATC sequence is located more than 2 kb away from the mutation site. The results presented in this work show that even though frameshift mutations can be efficiently generated and/or repaired anywhere in the genome, these processes can be modulated by the chromosomal context that surrounds the mutation site

Direct and Inverse Computation of Jacobi Matrices of Infinite Homogeneous Affine I.F.S

We introduce a new set of algorithms to compute Jacobi matrices associated with measures generated by infinite systems of iterated functions. We demonstrate their relevance in the study of theoretical problems, such as the continuity of these measures and the logarithmic capacity of their support. Since our approach is based on a reversible transformation between pairs of Jacobi matrices, we also discuss its application to an inverse / approximation problem. Numerical experiments show that the proposed algorithms are stable and can reliably compute Jacobi matrices of large order.Comment: 20 pages 6 figure

The Baker's Yeast Diploid Genome Is Remarkably Stable in Vegetative Growth and Meiosis

Accurate estimates of mutation rates provide critical information to analyze genome evolution and organism fitness. We used whole-genome DNA sequencing, pulse-field gel electrophoresis, and comparative genome hybridization to determine mutation rates in diploid vegetative and meiotic mutation accumulation lines of Saccharomyces cerevisiae. The vegetative lines underwent only mitotic divisions while the meiotic lines underwent a meiotic cycle every ∼20 vegetative divisions. Similar base substitution rates were estimated for both lines. Given our experimental design, these measures indicated that the meiotic mutation rate is within the range of being equal to zero to being 55-fold higher than the vegetative rate. Mutations detected in vegetative lines were all heterozygous while those in meiotic lines were homozygous. A quantitative analysis of intra-tetrad mating events in the meiotic lines showed that inter-spore mating is primarily responsible for rapidly fixing mutations to homozygosity as well as for removing mutations. We did not observe 1–2 nt insertion/deletion (in-del) mutations in any of the sequenced lines and only one structural variant in a non-telomeric location was found. However, a large number of structural variations in subtelomeric sequences were seen in both vegetative and meiotic lines that did not affect viability. Our results indicate that the diploid yeast nuclear genome is remarkably stable during the vegetative and meiotic cell cycles and support the hypothesis that peripheral regions of chromosomes are more dynamic than gene-rich central sections where structural rearrangements could be deleterious. This work also provides an improved estimate for the mutational load carried by diploid organisms

African Linguistics in Central and Eastern Europe, and in the Nordic Countries

Non peer reviewe