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

Sturm Sequences and Modified Subresultant Polynomial Remainder Sequences

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

Bone speed of sound, biochemical markers of bone turnover and IGF-1 in competetive synchronized swimmers

The purpose of this study was to compare bone speed of sound (SOS) measured by quantitative ultrasound, circulating levels of IGF- 1 and biochemical markers of bone turnover in pre- (Pr) and post-menarcheal (Po) synchronized swimmers (SS) and controls (NS). Seventy participants were recruited: 8 PrSS, 22 PoSS, 20 PrNS, and 20 PoNS. Anthropometric measures of height, weight, skeletal maturity and percent body fat were taken, and dietary intake evaluated using 24-hour recall. Bone SOS was measured at the distal radius and mid-tibia and blood samples analyzed for IGF-1, osteocalcin, NTx, and 25-OH vitamin D. Results demonstrated maturational effects on bone SOS, IGF-1 and bone turnover (p<0.05), with no differences observed between SS and NS. Main effects were observed for a reduced caloric intake in SS compared to NS (p<0.05). Therefore, SS does not offer additive affects on bone strength but imparts no adverse affects to skeletal health in these athletes.

Análise do risco da ocorrência da síndrome reprodutiva e respiratória dos suínos(PRRS) em granjas de suínos tecnificadas do Estado de São Paulo: estudo do perfil descritivo da situação atual

Pós-graduação em Medicina Veterinária - FMVZ

On the Remainders Obtained in Finding the Greatest Common Divisor of Two Polynomials

In 1917 Pell (1) and Gordon used sylvester2, Sylvester’s little known and hardly ever used matrix of 1853, to compute(2) the coefficients of a Sturmian remainder — obtained in applying in Q[x], Sturm’s algorithm on two polynomials f, g ∈ Z[x] of degree n — in terms of the determinants (3) of the corresponding submatrices of sylvester2. Thus, they solved a problem that had eluded both J. J. Sylvester, in 1853, and E. B. Van Vleck, in 1900. (4) In this paper we extend the work by Pell and Gordon and show how to compute (2) the coefficients of an Euclidean remainder — obtained in finding in Q[x], the greatest common divisor of f, g ∈ Z[x] of degree n — in terms of the determinants (5) of the corresponding submatrices of sylvester1, Sylvester’s widely known and used matrix of 1840. (1) See the link http://en.wikipedia.org/wiki/Anna_Johnson_Pell_Wheeler for her biography (2) Both for complete and incomplete sequences, as defined in the sequel. (3) Also known as modified subresultants. (4) Using determinants Sylvester and Van Vleck were able to compute the coefficients of Sturmian remainders only for the case of complete sequences. (5) Also known as (proper) subresultants.

Sobre a integração indefinida de funções racionais complexas: teoria e implementação de algoritmos racionais

We present indefinite integration algorithms for rational functions over subfields of the complex numbers, through an algebraic approach. We study the local algorithm of Bernoulli and rational algorithms for the class of functions in concern, namely, the algorithms of Hermite; Horowitz-Ostrogradsky; Rothstein-Trager and Lazard-Rioboo-Trager. We also study the algorithm of Rioboo for conversion of logarithms involving complex extensions into real arctangent functions, when these logarithms arise from the integration of rational functions with real coefficients. We conclude presenting pseudocodes and codes for implementation in the software Maxima concerning the algorithms studied in this work, as well as to algorithms for polynomial gcd computation; partial fraction decomposition; squarefree factorization; subresultant computation, among other side algorithms for the work. We also present the algorithm of Zeilberger-Almkvist for integration of hyperexpontential functions, as well as its pseudocode and code for Maxima. As an alternative for the algorithms of Rothstein-Trager and Lazard-Rioboo-Trager, we yet present a code for Benoulli’s algorithm for square-free denominators; and another for Czichowski’s algorithm, although this one is not studied in detail in the present work, due to the theoretical basis necessary to understand it, which is beyond this work’s scope. Several examples are provided in order to illustrate the working of the integration algorithms in this text

Investigating the Function of a Small Secreted Protein Family in Physcomitrella Patens

The correct development of multicellular organisms depends upon the perception of signals secreted by cells in order to co-ordinate cell differentiation. The Physcomitrella patens genome encodes many components of potential signaling systems, including putative receptor proteins and putative secreted protein ligands, yet at present little characterization of these proteins has been carried out. We are currently attempting to characterize the expression pattern and function of a family of 6 secreted proteins exhibiting homology to PrsS, the ligand that controls self-incompatibility (SI) in Papaver rhoeas (field poppy). In poppy, PrsS interacts a receptor on the surface of pollen tubes, PrpS causing SI by programmed cell death. Homologues of this protein (SPH – S-Protein Homologues) exist in dicotyledonous plants and bryophytes but not in other plant taxa. We aim to determine spatiotemporal expression differences between these proteins via reporter gene analysis and qPCR of cDNA. In addition we are in the process of creating targeted gene knockouts for all 6 of the genes in P. patens. We are also searching for receptors of PrpS in Physcomitrella using a bioinformatic strategy alongside phage display. In accomplishing this we hope to determine the function of a small novel secreted protein family in Physcomitrella but in addition we also hope to elucidate the function of SPH proteins in Arabidopsis.