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.

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.

How to Apply for Scholarships and Grants (2016)

A short guide highlighting federal and state scholarships and grants, including Pell Grant, career-related training grants, opportunities for foster youth and state-funded scholarships.

Incidência de fungos em pós-colheita de banana 'Prata anã'(Musa AAB)

Este trabalho teve por objetivo determinar a ocorrência e a freqüência de fungos em banana 'Prata anã' e elucidar o agente causal das podridões em pós-colheita de frutos provenientes do norte de Minas Gerais. Dois métodos de isolamento foram adotados: diluição em placas, a partir da lavagem de frutos verdes, e direto de frutos maduros. Os fungos Colletotrichum musae, Trichoderma harzianum, Fusarium equisetii, Penicillium sp. Aspergillus parasiticus, Trichothecium roseum, Colletotrichum acutatum, Alternaria sp., Cladosporium musae e Curvularia lunata foram os mais freqüentemente associados aos frutos. A patogenicidade desses fungos foi testada pela substituição de discos da casca de frutos verdes por discos de micélio. Colletotrichum musae apresentou área média lesionada em torno do ponto de inoculação igual a 5,8 cm², enquanto para os demais fungos testados não passou de 1,50 cm². Os resultados mostraram que C. musae é o agente primário das podridões dos frutos examinados com 100 % de incidência e os demais fungos limitaram-se a necrosar os ferimentos em torno do ponto de inoculação. O modo de infecção latente, causada por C. musae, parece favorecer, primeiramente, a colonização interna dos tecidos e, posteriormente, a ação dos fungos oportunistas, que aceleram as podridões nos frutos e na coroa.