High-Order Iterative Scheme for a Viscoelastic Wave Equation and Numerical Results

In this paper, we consider a Robin problem for a viscoelastic wave equation. First, by the high-order iterative method coupled with the Galerkin method, the existence of a recurrent sequence via an N -order iterative scheme is established, and then the N -order convergent rate of the obtained sequence to the unique weak solution of the proposed model is also proved. Next, with N = 2 , a numerical algorithm given by the finite-difference method is constructed to approximate the solution via the 2-order iterative scheme. Moreover, the same algorithm for the single-iterative scheme generated by the 2-order iterative scheme is also considered. Finally, comparison with errors of the numerical solutions obtained by the single-iterative scheme and the 2-order iterative scheme shows that the convergent rate of the 2-order iterative scheme is faster than that of the single-iterative scheme.

A Survey on Old and New Approximations to the Function ϕ(x) Involved in LDPC Codes Density Evolution Analysis Using a Gaussian Approximation

Low Density Parity Check (LDPC) codes are currently being deeply analyzed through algorithms that require the capability of addressing their iterative decoding convergence performance. Since it has been observed that the probability distribution function of the decoder’s log-likelihood ratio messages is roughly Gaussian, a multiplicity of moderate entanglement strategies to this analysis has been suggested. The first of them was proposed in Chung et al.’s 2001 paper, where the recurrent sequence, characterizing the passage of messages between variable and check nodes, concerns the function ϕ(x), therein specified, and its inverse. In this paper, we review this old approximation to the function ϕ(x), one variant on it obtained in the same period (proposed in Ha et al.’s 2004 paper), and some new ones, recently published in two 2019 papers by Vatta et al. The objective of this review is to analyze the differences among them and their characteristics in terms of accuracy and computational complexity. In particular, the explicitly invertible, not piecewise defined approximation of the function ϕ(x), published in the second of the two abovementioned 2019 papers, is shown to have less relative error in any x than most of the other approximations. Moreover, its use conducts to an important complexity reduction, and allows better Gaussian approximated thresholds to be obtained.

Recurrent Sequence Evolution After Independent Gene Duplication

Background Convergent and parallel evolution provide unique insights into the mechanisms of natural selection. Some of the most striking convergent and parallel (collectively recurrent ) amino acid substitutions in proteins are adaptive, but there are also many that are selectively neutral. Accordingly, genome-wide assessment has shown that recurrent sequence evolution in orthologs is chiefly explained by nearly neutral evolution. For paralogs, more frequent functional change is expected because additional copies are generally not retained if they do not acquire their own niche. Yet, it is unknown to what extent recurrent sequence differentiation is discernible after independent gene duplications in different eukaryotic taxa. Results We develop a framework that detects patterns of recurrent sequence evolution in duplicated genes. This is used to analyze the genomes of 90 diverse eukaryotes. We find a remarkable number of families with a potentially predictable functional differentiation following gene duplication. In some protein families, more than ten independent duplications show a similar sequence-level differentiation between paralogs. Based on further analysis, the sequence divergence is found to be generally asymmetric. Moreover, about 6\% of the recurrent sequence evolution between paralog pairs can be attributed to recurrent differentiation of subcellular localization. Finally, we reveal the specific recurrent patterns for the gene families Hint1/Hint2, Sco1/Sco2 and vma11/vma3. Conclusions The presented methodology provides a means to study the biochemical underpinning of functional differentiation between paralogs. For instance, two abundantly repeated substitutions are identified between independently derived Sco1 and Sco2 paralogs. Such identified substitutions allow direct experimental testing of the biological role of these residues for the repeated functional differentiation. We also uncover a diverse set of families with recurrent sequence evolution and reveal trends in the functional and evolutionary trajectories of this hitherto understudied phenomenon.

Recurrent Sequence Evolution After Independent Gene Duplication

Background Convergent and parallel evolution provide unique insights into the mechanisms of natural selection. Some of the most striking convergent and parallel (collectively recurrent ) amino acid substitutions in proteins are adaptive, but there are also many that are selectively neutral. Accordingly, genome-wide assessment has shown that recurrent sequence evolution in orthologs is chiefly explained by nearly neutral evolution. For paralogs, more frequent functional change is expected because additional copies are generally not retained if they do not acquire their own niche. Yet, it is unknown to what extent recurrent sequence differentiation is discernible after independent gene duplications in different eukaryotic taxa. Results We develop a framework that detects patterns of recurrent sequence evolution in duplicated genes. This is used to analyze the genomes of 90 diverse eukaryotes. We find a remarkable number of families with a potentially predictable functional differentiation following gene duplication. In some protein families, more than ten independent duplications show a similar sequence-level differentiation between paralogs. Based on further analysis, the sequence divergence is found to be generally asymmetric. Moreover, about 6\% of the recurrent sequence evolution between paralog pairs can be attributed to recurrent differentiation of subcellular localization. Finally, we reveal the specific recurrent patterns for the gene families Hint1/Hint2, Sco1/Sco2 and vma11/vma3. Conclusions The presented methodology provides a means to study the biochemical underpinning of functional differentiation between paralogs. For instance, two abundantly repeated substitutions are identified between independently derived Sco1 and Sco2 paralogs. Such identified substitutions allow direct experimental testing of the biological role of these residues for the repeated functional differentiation. We also uncover a diverse set of families with recurrent sequence evolution and reveal trends in the functional and evolutionary trajectories of this hitherto understudied phenomenon.

Recurrent Sequence Evolution After Independent Gene Duplication

Background Convergent and parallel evolution provide unique insights into the mechanisms of natural selection. Some of the most striking convergent and parallel (collectively recurrent ) amino acid substitutions in proteins are adaptive, but there are also many that are selectively neutral. Accordingly, genome-wide assessment has shown that recurrent sequence evolution in orthologs is chiefly explained by nearly neutral evolution. For paralogs, more frequent functional change is expected because additional copies are generally not retained if they do not acquire their own niche. Yet, it is unknown to what extent recurrent sequence differentiation is discernible after independent gene duplications in different eukaryotic taxa. Results We develop a framework that detects patterns of recurrent sequence evolution in duplicated genes. This is used to analyze the genomes of 90 diverse eukaryotes. We find a remarkable number of families with a potentially predictable functional differentiation following gene duplication. In some protein families, more than ten independent duplications show a similar sequence-level differentiation between paralogs. Based on further analysis, the sequence divergence is found to be generally asymmetric. Moreover, about 6\% of the recurrent sequence evolution between paralog pairs can be attributed to recurrent differentiation of subcellular localization. Finally, we reveal the specific recurrent patterns for the gene families Hint1/Hint2, Sco1/Sco2 and vma11/vma3. Conclusions The presented methodology provides a means to study the biochemical underpinning of functional differentiation between paralogs. For instance, two abundantly repeated substitutions are identified between independently derived Sco1 and Sco2 paralogs. Such identified substitutions allow direct experimental testing of the biological role of these residues for the repeated functional differentiation. We also uncover a diverse set of families with recurrent sequence evolution and reveal trends in the functional and evolutionary trajectories of this hitherto understudied phenomenon.

Parallel machine arithmetic for recurrent number systems in non-quadratic fields

The paper proposes a new method of synthesis of computer arithmetic systems for "error-free" parallel calculations. The difference between the proposed approach and calculations in traditional systems of Residue Number Systems for the direct sum of modular rings is the parallelization of calculations in non-quadratic extensions of simple finite fields whose elements are represented in number systems generated by sequences of powers of roots of the characteristic polynomial of the recurrent sequence.

Recurrent Sequence Evolution After Independent Gene Duplication

Background Convergent and parallel evolution provide unique insights into the mechanisms of natural selection. Some of the most striking convergent and parallel (collectively recurrent ) amino acid substitutions in proteins are adaptive, but there are also many that are selectively neutral. Genome-wide assessment of recurrent substitutions has only been performed for orthologs. These studies have revealed that the pervasiveness of recurrent substitutions is for a large part explained by purifying selection. At any position in a protein, only a subset of amino acids is allowed, increasing the chance of the same substitution happening in different lineages. ResultsWe developed a framework that detects patterns of recurrent differentiation in paralogs across 90 divergent eukaryotic genomes. A skew in recurrent substitutions serves as a proxy for a recurrent trend in function. We find remarkable examples of recurrent sequence evolution after independent duplication, in some cases involving more than ten different lineages where duplicates show a similar differentiation. We reveal the implicated functional patterns for the gene families Hint1/Hint2, Sco1/Sco2 and vma11/vma3. ConclusionsThe presented methodology provides a means to study the biochemical underpinning of functional differentiation between paralogs. For instance, two abundantly repeated substitutions are identified between independently derived Sco1 and Sco2 paralogs. Such identified substitutions allow direct experimental testing of the biological role of these residues for the repeated functional differentiation. The present study uncovers a diverse set of families with recurrent sequence evolution and reveals trends in the functional and evolutionary trajectories of this hitherto understudied phenomenon.