On a Theorem by Van Vleck Regarding Sturm Sequences

In 1900 E. B. Van Vleck proposed a very efficient method to compute the Sturm sequence of a polynomial p (x) ∈ Z[x] by triangularizing one of Sylvester’s matrices of p (x) and its derivative p′(x). That method works fine only for the case of complete sequences provided no pivots take place. In 1917, A. J. Pell and R. L. Gordon pointed out this “weakness” in Van Vleck’s theorem, rectified it but did not extend his method, so that it also works in the cases of: (a) complete Sturm sequences with pivot, and (b) incomplete Sturm sequences. Despite its importance, the Pell-Gordon Theorem for polynomials in Q[x] has been totally forgotten and, to our knowledge, it is referenced by us for the first time in the literature. In this paper we go over Van Vleck’s theorem and method, modify slightly the formula of the Pell-Gordon Theorem and present a general triangularization method, called the VanVleck-Pell-Gordon method, that correctly computes in Z[x] polynomial Sturm sequences, both complete and incomplete.

Sturm Sequences and Modified Subresultant Polynomial Remainder Sequences

ACM Computing Classification System (1998): F.2.1, G.1.5, I.1.2.

Three New Methods for Computing Subresultant Polynomial Remainder Sequences (PRS’S)

Given the polynomials f, g ∈ Z[x] of degrees n, m, respectively, with n > m, three new, and easy to understand methods — along with the more efficient variants of the last two of them — are presented for the computation of their subresultant polynomial remainder sequence (prs). All three methods evaluate a single determinant (subresultant) of an appropriate sub-matrix of sylvester1, Sylvester’s widely known and used matrix of 1840 of dimension (m + n) × (m + n), in order to compute the correct sign of each polynomial in the sequence and — except for the second method — to force its coefficients to become subresultants. Of interest is the fact that only the first method uses pseudo remainders. The second method uses regular remainders and performs operations in Q[x], whereas the third one triangularizes sylvester2, Sylvester’s little known and hardly ever used matrix of 1853 of dimension 2n × 2n. All methods mentioned in this paper (along with their supporting functions) have been implemented in Sympy and can be downloaded from the link http://inf-server.inf.uth.gr/~akritas/publications/subresultants.py

Zeros of Sequences of Partial Sums and Overconvergence

2000 Mathematics Subject Classification: 30B40, 30B10, 30C15, 31A15.

Number Sequences in an Integral Form with a Generalized Convolution Property and Somos-4 Hankel Determinants

MSC 2010: 11B83, 05A19, 33C45

Accurate estimation of isoelectric point of protein and peptide based on amino acid sequences

Motivation: In any macromolecular polyprotic system - for example protein, DNA or RNA - the isoelectric point - commonly referred to as the pI - can be defined as the point of singularity in a titration curve, corresponding to the solution pH value at which the net overall surface charge - and thus the electrophoretic mobility - of the ampholyte sums to zero. Different modern analytical biochemistry and proteomics methods depend on the isoelectric point as a principal feature for protein and peptide characterization. Protein separation by isoelectric point is a critical part of 2-D gel electrophoresis, a key precursor of proteomics, where discrete spots can be digested in-gel, and proteins subsequently identified by analytical mass spectrometry. Peptide fractionation according to their pI is also widely used in current proteomics sample preparation procedures previous to the LC-MS/MS analysis. Therefore accurate theoretical prediction of pI would expedite such analysis. While such pI calculation is widely used, it remains largely untested, motivating our efforts to benchmark pI prediction methods. Results: Using data from the database PIP-DB and one publically available dataset as our reference gold standard, we have undertaken the benchmarking of pI calculation methods. We find that methods vary in their accuracy and are highly sensitive to the choice of basis set. The machine-learning algorithms, especially the SVM-based algorithm, showed a superior performance when studying peptide mixtures. In general, learning-based pI prediction methods (such as Cofactor, SVM and Branca) require a large training dataset and their resulting performance will strongly depend of the quality of that data. In contrast with Iterative methods, machine-learning algorithms have the advantage of being able to add new features to improve the accuracy of prediction. Contact: yperez@ebi.ac.uk Availability and Implementation: The software and data are freely available at https://github.com/ypriverol/pIR. Supplementary information: Supplementary data are available at Bioinformatics online.

Optimised phase disposition pulse-width modulation strategy for hybrid-clamped multilevel inverters using switching state sequences

This study describes an optimised modulation strategy based on switching state sequences for the hybrid-clamped multilevel converter. Two key control variables defined as 'phase shift angle' and 'switching state change' for a five-level hybrid-clamped inverter are proposed to improve all switches' operation, and by changing their values, different control methods can be obtained for modulation optimisation purposes. Two example methods can solve the voltage imbalance problem of the dc-link capacitors and furthermore avoid two switches' simultaneous switching transitions and improve the inverter's performance as compared with the traditional phase disposition pulse-width modulation strategy. A 6 kW prototype inverter is developed and a range of simulation and experiments are carried out for validation. It is found that simulation and experimental results are in a good agreement and the proposed modulation strategy is verified in terms of low-order harmonic reduction.

Ammonite biostratigraphy, lithofacies variations, and paleoceanographic implications for Barremian-Aptian sequences of northeastern Mexico

Two Barremian-Aptian sequences studied in Durango and Nuevo Leon States, northeastern Mexico include three lithic units which have been described as the Cupido Formation of Barremian-early Early Aptian age, its lateral equivalent, the Lower Tamaulipas Formation, and the La Peña Formation extending through the early Albian. ^ The present work improves the existing ammonite Aptian biozonation by considering constraints associated with a discontinuous spatial and temporal record of the different taxa within the La Peña Formation. ^ Four ammonite biozones are established: (1) The Dufrenoyia justinae Zone for the late Early Aptian, (2) The Burckhardtites nazasensis/Rhytidoplites robertsi Zone for the middle Aptian, (3) The Cheloniceras inconstans Zone for the early Late Aptian, and (4) The Hypacanthoplites cf. leanzae Zone for the late late Aptian. ^ Also, a detailed sedimentological analysis of the sections shed further light on the possible causes that controlled intermittent occurrences of the ammonites in relation to the prevailing paleoceanographic and paleoecologic conditions in northeastern Mexico during the late Barremian-Aptian. ^ Microfacies analyses show that the upper part of the Cupido facies are represented by biocalcirudite with rudists, biocalcarenites with oolites and algae, and rich benthonic foraminifera assemblages with ostracods. These facies are related to paleoceanographic conditions of sedimentation within a shallow-marine carbonate platform. Its lateral equivalent, deep-water facies extended to the southeast and it is represented by the Lower Tamaulipas Formation, which includes planktonic foraminifera, ostracods, and mollusk and echinoid fragments. The beginning of deposition of the La Peña Formation in the late Early Aptian is characterized by an increase in terrigenous materials and significant decrease in the abundance of benthic fauna. The La Peña Formation is recognized by an alternation of marls and shale limestones containing ammonites, planktonic foraminifera, ostracods, and radiolaria toward the top. Accumulation of the La Peña continued throughout the end of the Aptian and records changes in conditions of sedimentation and productivity in the water column, which abruptly terminated the carbonate deposition in the Cupido Platform. ^ Results of carbon/carbonate content analyses show that changes from the Cupido to the La Peña facies are also characterized by an increase of organic carbon, which indicate the onset of enhanced dysoxic/anoxic conditions in the lower water column. ^

Serial evolutionary networks of within-patient HIV-1 sequences reveal patterns of evolution of X4 strains

Background The HIV virus is known for its ability to exploit numerous genetic and evolutionary mechanisms to ensure its proliferation, among them, high replication, mutation and recombination rates. Sliding MinPD, a recently introduced computational method [1], was used to investigate the patterns of evolution of serially-sampled HIV-1 sequence data from eight patients with a special focus on the emergence of X4 strains. Unlike other phylogenetic methods, Sliding MinPD combines distance-based inference with a nonparametric bootstrap procedure and automated recombination detection to reconstruct the evolutionary history of longitudinal sequence data. We present serial evolutionary networks as a longitudinal representation of the mutational pathways of a viral population in a within-host environment. The longitudinal representation of the evolutionary networks was complemented with charts of clinical markers to facilitate correlation analysis between pertinent clinical information and the evolutionary relationships. Results Analysis based on the predicted networks suggests the following:: significantly stronger recombination signals (p = 0.003) for the inferred ancestors of the X4 strains, recombination events between different lineages and recombination events between putative reservoir virus and those from a later population, an early star-like topology observed for four of the patients who died of AIDS. A significantly higher number of recombinants were predicted at sampling points that corresponded to peaks in the viral load levels (p = 0.0042). Conclusion Our results indicate that serial evolutionary networks of HIV sequences enable systematic statistical analysis of the implicit relations embedded in the topology of the structure and can greatly facilitate identification of patterns of evolution that can lead to specific hypotheses and new insights. The conclusions of applying our method to empirical HIV data support the conventional wisdom of the new generation HIV treatments, that in order to keep the virus in check, viral loads need to be suppressed to almost undetectable levels.

Towards the prediction of mutations in genomic sequences

Bio-systems are inherently complex information processing systems. Furthermore, physiological complexities of biological systems limit the formation of a hypothesis in terms of behavior and the ability to test hypothesis. More importantly the identification and classification of mutation in patients are centric topics in today's cancer research. Next generation sequencing (NGS) technologies can provide genome-wide coverage at a single nucleotide resolution and at reasonable speed and cost. The unprecedented molecular characterization provided by NGS offers the potential for an individualized approach to treatment. These advances in cancer genomics have enabled scientists to interrogate cancer-specific genomic variants and compare them with the normal variants in the same patient. Analysis of this data provides a catalog of somatic variants, present in tumor genome but not in the normal tissue DNA. In this dissertation, we present a new computational framework to the problem of predicting the number of mutations on a chromosome for a certain patient, which is a fundamental problem in clinical and research fields. We begin this dissertation with the development of a framework system that is capable of utilizing published data from a longitudinal study of patients with acute myeloid leukemia (AML), who's DNA from both normal as well as malignant tissues was subjected to NGS analysis at various points in time. By processing the sequencing data at the time of cancer diagnosis using the components of our framework, we tested it by predicting the genomic regions to be mutated at the time of relapse and, later, by comparing our results with the actual regions that showed mutations (discovered at relapse time). We demonstrate that this coupling of the algorithm pipeline can drastically improve the predictive abilities of searching a reliable molecular signature. Arguably, the most important result of our research is its superior performance to other methods like Radial Basis Function Network, Sequential Minimal Optimization, and Gaussian Process. In the final part of this dissertation, we present a detailed significance, stability and statistical analysis of our model. A performance comparison of the results are presented. This work clearly lays a good foundation for future research for other types of cancer.^

A heuristic algorithm to generate test program sequences for moving probe electronic test equipment

The electronics industry, is experiencing two trends one of which is the drive towards miniaturization of electronic products. The in-circuit testing predominantly used for continuity testing of printed circuit boards (PCB) can no longer meet the demands of smaller size circuits. This has lead to the development of moving probe testing equipment. Moving Probe Test opens up the opportunity to test PCBs where the test points are on a small pitch (distance between points). However, since the test uses probes that move sequentially to perform the test, the total test time is much greater than traditional in-circuit test. While significant effort has concentrated on the equipment design and development, little work has examined algorithms for efficient test sequencing. The test sequence has the greatest impact on total test time, which will determine the production cycle time of the product. Minimizing total test time is a NP-hard problem similar to the traveling salesman problem, except with two traveling salesmen that must coordinate their movements. The main goal of this thesis was to develop a heuristic algorithm to minimize the Flying Probe test time and evaluate the algorithm against a "Nearest Neighbor" algorithm. The algorithm was implemented with Visual Basic and MS Access database. The algorithm was evaluated with actual PCB test data taken from Industry. A statistical analysis with 95% C.C. was performed to test the hypothesis that the proposed algorithm finds a sequence which has a total test time less than the total test time found by the "Nearest Neighbor" approach. Findings demonstrated that the proposed heuristic algorithm reduces the total test time of the test and, therefore, production cycle time can be reduced through proper sequencing.

Ammonite biostratigraphy, lithofacies variations, and paleoceanographic implications for barremian-aptian sequences of northeastern Mexico

Two Barremian-Aptian sequences studied in Durango and Nuevo Leon States, northeastern Mexico include three lithic units which have been described as the Cupido Formation of Barremian-early Early Aptian age, its lateral equivalent, the Lower Tamaulipas Formation, and the La Pena Formation extending through the early Albian. The present work improves the existing ammonite Aptian biozonation by considering constraints associated with a discontinuous spatial and temporal record of the different taxa within the La Pena Formation. Four ammonite biozones are established: 1) The Dufrenoyia justinae Zone for the late Early Aptian, 2) The Burckhardtites nazasensis/Rhytidoplites robertsi Zone for the middle Aptian, 3) The Cheloniceras inconstans Zone for the early Late Aptian, and 4) The Hypacanthoplites cf. leanzae Zone for the late late Aptian. Also, a detailed sedimentological analysis of the sections shed further light on the possible causes that controlled intermittent occurrences of the ammonites in relation to the prevailing paleoceanographic and paleoecologic conditions in northeastern Mexico during the late Barremian-Aptian. Microfacies analyses show that the upper part of the Cupido facies are represented by biocalcirudite with rudists, biocalcarenites with oolites and algae, and rich benthonic foraminifera assemblages with ostracods. These facies are related to paleoceanographic conditions of sedimentation within a shallow-marine carbonate platform. Its lateral equivalent, deep-water facies extended to the southeast and it is represented by the Lower Tamaulipas Formation, which includes planktonic foraminifera, ostracods, and mollusk and echinoid fragments. The beginning of deposition of the La Pena Formation in the late Early Aptian is characterized by an increase in terrigenous materials and significant decrease in the abundance of benthic fauna. The La Pena Formation is recognized by an alternation of marls and shale limestones containing ammonites, planktonic foraminifera, ostracods, and radiolaria toward the top. Accumulation of the La Pena continued throughout the end of the Aptian and records changes in conditions of sedimentation and productivity in the water column, which abruptly terminated the carbonate deposition in the Cupido Platform. Results of carbon/carbonate content analyses show that changes from the Cupido to the La Pena facies are also characterized by an increase of organic carbon, which indicate the onset of enhanced dysoxic/anoxic conditions in the lower water column.

(Table 1) Individual codes, haplotype codes, GenBank accession numbers for DNA sequences and sample locality of the analysed Betamorpha fusiformis specimens from the ANDEEP III expedition

Based on our current knowledge about population genetics, phylogeography and speciation, we begin to understand that the deep sea harbours more species than suggested in the past. Deep-sea soft-sediment environment in particular hosts a diverse and highly endemic invertebrate fauna. Very little is known about evolutionary processes that generate this remarkable species richness, the genetic variability and spatial distribution of deep-sea animals. In this study, phylogeographic patterns and the genetic variability among eight populations of the abundant and widespread deep-sea isopod morphospecies Betamorpha fusiformis [Barnard, K.H., 1920. Contributions to the crustacean fauna of South Africa. 6. Further additions to the list of marine isopods. Annals of the South African Museum 17, 319-438] were examined. A fragment of the mitochondrial 16S rRNA gene of 50 specimens and the complete nuclear 18S rRNA gene of 7 specimens were sequenced. The molecular data reveal high levels of genetic variability of both genes between populations, giving evidence for distinct monophyletic groups of haplotypes with average p-distances ranging from 0.0470 to 0.1440 (d-distances: 0.0592-0.2850) of the 16S rDNA, and 18S rDNA p-distances ranging between 0.0032 and 0.0174 (d-distances: 0.0033-0.0195). Intermediate values are absent. Our results show that widely distributed benthic deep-sea organisms of a homogeneous phenotype can be differentiated into genetically highly divergent populations. Sympatry of some genotypes indicates the existence of cryptic speciation. Flocks of closely related but genetically distinct species probably exist in other widespread benthic deep-sea asellotes and other Peracarida. Based on existing data we hypothesize that many widespread morphospecies are complexes of cryptic biological species (patchwork hypothesis).