On Arrangements of Real Roots of a Real Polynomial and Its Derivatives

2000 Mathematics Subject Classification: 12D10.

On a Stratification Defined by Real Roots of Polynomials

2000 Mathematics Subject Classification: 12D10

FLQ, the Fastest Quadratic Complexity Bound on the Values of Positive Roots of Polynomials

In this paper we present F LQ, a quadratic complexity bound on the values of the positive roots of polynomials. This bound is an extension of FirstLambda, the corresponding linear complexity bound and, consequently, it is derived from Theorem 3 below. We have implemented FLQ in the Vincent-Akritas-Strzeboński Continued Fractions method (VAS-CF) for the isolation of real roots of polynomials and compared its behavior with that of the theoretically proven best bound, LM Q. Experimental results indicate that whereas F LQ runs on average faster (or quite faster) than LM Q, nonetheless the quality of the bounds computed by both is about the same; moreover, it was revealed that when VAS-CF is run on our benchmark polynomials using F LQ, LM Q and min(F LQ, LM Q) all three versions run equally well and, hence, it is inconclusive which one should be used in the VAS-CF method.

GeoGebra e o método de Briot & Bouquet para a resolução gráfica de equações cúbicas

In the mid-nineteenth century, french mathematicians Briot and Bouquet have proposed an intriguing graphical method for solving cubic equations "depressed" - the third degree equations that do not have the quadratic term. The proposal is simple geometric construction, though based on an ingenious algebra. We propose here the verification and testing graphical method through an instructional sequence using the software GeoGebra also present the ingenious algebraic development that resulted in this graphic method for determination of real roots of a cubic equation of the type x³ + px + q = 0 where p and q are real numbers. The method states that these solutions are summarized in the abscissas of the points of intersection of the circumference containing the origin and the center C (-q/2, 1-p/2) with the parable y = x².

Equações diferenciais implícitas com descontinuidade

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Paraconsistent Multivalued Logic and Coincidentia Oppositorum: Evaluation with Complex Numbers

Paraconsistent logic admits that the contradiction can be true. Let p be the truth values and P be a proposition. In paraconsistent logic the truth values of contradiction is . This equation has no real roots but admits complex roots . This is the result which leads to develop a multivalued logic to complex truth values. The sum of truth values being isomorphic to the vector of the plane, it is natural to relate the function V to the metric of the vector space R2. We will adopt as valuations the norms of vectors. The main objective of this paper is to establish a theory of truth-value evaluation for paraconsistent logics with the goal of using in analyzing ideological, mythical, religious and mystic belief systems.

On the Various Bisection Methods Derived from Vincent’s Theorem

In 2000 A. Alesina and M. Galuzzi presented Vincent’s theorem “from a modern point of view” along with two new bisection methods derived from it, B and C. Their profound understanding of Vincent’s theorem is responsible for simplicity — the characteristic property of these two methods. In this paper we compare the performance of these two new bisection methods — i.e. the time they take, as well as the number of intervals they examine in order to isolate the real roots of polynomials — against that of the well-known Vincent-Collins-Akritas method, which is the first bisection method derived from Vincent’s theorem back in 1976. Experimental results indicate that REL, the fastest implementation of the Vincent-Collins-Akritas method, is still the fastest of the three bisection methods, but the number of intervals it examines is almost the same as that of B. Therefore, further research on speeding up B while preserving its simplicity looks promising.

Discriminant Sets of Families of Hyperbolic Polynomials of Degree 4 and 5

∗ Research partially supported by INTAS grant 97-1644

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.

Process Evaluation of the Roots of Empathy Program

This paper presents the findings of a qualitative process evaluation of the Roots of Empathy (ROE) programme. ROE is a universal, classroom-based intervention, which aims to enhance social and emotional learning of primary (elementary) school children. Effective delivery of such complex social interventions in real-world settings requires in-depth knowledge and understanding of factors that interact to influence implementation and fidelity. A case study methodology was employed with six schools, to explore the views of key actors and stakeholders involved in the delivery and receipt of the programme. Overall, ROE was delivered with high fidelity and the programme was viewed positively across the schools. However, one issue was the varied level of interest and awareness of the programme from parents.

Real convergence in per capita output and growth in Eurozone: A time-series approach

The real convergence hypothesis has spurred a myriad of empirical tests and approaches in the economic literature. This Work Project intends to test for real output and growth convergence in all N(N-1)/2 possible pairs of output and output growth gaps of 14 Eurozone countries. This paper follows a time-series approach, as it tests for the presence of unit roots and persistence changes in the above mentioned pairs of output gaps, as well as for the existence of growth convergence with autoregressive models. Overall, significantly greater evidence has been found to support growth convergence rather than output convergence in our sample.

Identification of reference genes for expression analysis by real-time quantitative PCR in drought-stressed soybean

O objetivo deste trabalho foi validar, pela técnica de PCR quantitativo em tempo real (RT-qPCR) genes para serem utilizados como referência em estudos de expressão gênica em soja, em ensaios de estresse hídrico. Foram avaliados quatro genes comumente utilizados em soja: Gmβ-actin, GmGAPDH, GmLectin e GmRNAr18S. O RNA total foi extraído de seis amostras: três amostras de raízes em sistema de hidroponia com diferentes intensidades de déficit hídrico (0, 25, 50, 75 e 100 minutos de estresse hídrico), e três amostras de folhas de plantas cultivadas em areia com diferentes umidades do solo (15, 5 e 2,5% de umidade gravimétrica). Os dados brutos do intervalo cycle threshold (Ct) foram analisados, e a eficiência de cada iniciador foi calculada para uma analise da Ct entre as diferentes amostras. A aplicação do programa GeNorm foi utilizada para a avaliação dos melhores genes de referência, de acordo com a estabilidade. O GmGAPDH foi o gene menos estável, com o maior valor médio de estabilidade de expressão (M), e os genes mais estáveis, com menor valor de M, foram o Gmβ-actin e GmRNAr18S, tanto nas amostras de raízes como nas de folhas. Estes genes podem ser usados em RT-qPCR como gens de referência para análises de expressão gênica.