Asymptotics of zeros of polynomials arising from rational integrals

We prove that the zeros of the polynomials P.. (a) of degree m, defined by Boros and Moll via[GRAPHICS]approach the lemmiscate {zeta epsilon C: \zeta(2) - 1\ = Hzeta < 0}, as m --> infinity. (C) 2004 Elsevier B.V. All rights reserved.

MA160exam00

Exam questions and solutions in LaTex. Diagrams for the questions are all together in the support.zip file, as .eps files

MA160exam00pdf

Exam questions and solutions in PDF

Prime Rational Functions and Integral Polynomials

Let f(x) be a complex rational function. In this work, we study conditions under which f(x) cannot be written as the composition of two rational functions which are not units under the operation of function composition. In this case, we say that f(x) is prime. We give sufficient conditions for complex rational functions to be prime in terms of their degrees and their critical values, and we derive some conditions for the case of complex polynomials. We consider also the divisibility of integral polynomials, and we present a generalization of a theorem of Nieto. We show that if f(x) and g(x) are integral polynomials such that the content of g divides the content of f and g(n) divides f(n) for an integer n whose absolute value is larger than a certain bound, then g(x) divides f(x) in Z[x]. In addition, given an integral polynomial f(x), we provide a method to determine if f is irreducible over Z, and if not, find one of its divisors in Z[x].

Phase portraits of quadratic polynomial vector fields having a rational first integral of degree 3

In this paper, we classify all the global phase portraits of the quadratic polynomial vector fields having a rational first integral of degree 3. (C) 2008 Elsevier Ltd. All rights reserved.

Rational approximations of the Arrhenius integral using Jacobi fractions and gaussian quadrature

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Symmetric representation of an integral domain over a subdomain

Let F(θ) be a separable extension of degree n of a field F. Let Δ and D be integral domains with quotient fields F(θ) and F respectively. Assume that Δ ᴝ D. A mapping φ of Δ into the n x n D matrices is called a Δ/D rep if (i) it is a ring isomorphism and (ii) it maps d onto dI_n whenever d ϵ D. If the matrices are also symmetric, φ is a Δ/D symrep.

Every Δ/D rep can be extended uniquely to an F(θ)/F rep. This extension is completely determined by the image of θ. Two Δ/D reps are called equivalent if the images of θ differ by a D unimodular similarity. There is a one-to-one correspondence between classes of Δ/D reps and classes of Δ ideals having an n element basis over D.

The condition that a given Δ/D rep class contain a Δ/D symrep can be phrased in various ways. Using these formulations it is possible to (i) bound the number of symreps in a given class, (ii) count the number of symreps if F is finite, (iii) establish the existence of an F(θ)/F symrep when n is odd, F is an algebraic number field, and F(θ) is totally real if F is formally real (for n = 3 see Sapiro, “Characteristic polynomials of symmetric matrices” Sibirsk. Mat. Ž. 3 (1962) pp. 280-291), and (iv) study the case D = Z, the integers (see Taussky, “On matrix classes corresponding to an ideal and its inverse” Illinois J. Math. 1 (1957) pp. 108-113 and Faddeev, “On the characteristic equations of rational symmetric matrices” Dokl. Akad. Nauk SSSR 58 (1947) pp. 753-754).

The case D = Z and n = 2 is studied in detail. Let Δ’ be an integral domain also having quotient field F(θ) and such that Δ’ ᴝ Δ. Let φ be a Δ/Z symrep. A method is given for finding a Δ’/Z symrep ʘ such that the Δ’ ideal class corresponding to the class of ʘ is an extension to Δ’ of the Δ ideal class corresponding to the class of φ. The problem of finding all Δ/Z symreps equivalent to a given one is studied.

Limit cycles of cubic polynomial differential systems with rational first integrals of degree 2

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Levy stable distributions via associated integral transform

We present a method of generation of exact and explicit forms of one-sided, heavy-tailed Levy stable probability distributions g(alpha)(x), 0 <= x < infinity, 0 < alpha < 1. We demonstrate that the knowledge of one such a distribution g a ( x) suffices to obtain exactly g(alpha)p ( x), p = 2, 3, .... Similarly, from known g(alpha)(x) and g(beta)(x), 0 < alpha, beta < 1, we obtain g(alpha beta)( x). The method is based on the construction of the integral operator, called Levy transform, which implements the above operations. For a rational, alpha = l/k with l < k, we reproduce in this manner many of the recently obtained exact results for g(l/k)(x). This approach can be also recast as an application of the Efros theorem for generalized Laplace convolutions. It relies solely on efficient definite integration. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4709443]

Implementation of pseudo J-integral based Paris’ law for fatigue cracking in asphalt mixtures and pavements

Pavement analysis and design for fatigue cracking involves a number of practical problems like material assessment/screening and performance prediction. A mechanics-aided method can answer these questions with satisfactory accuracy in a convenient way when it is appropriately implemented. This paper presents two techniques to implement the pseudo J-integral based Paris’ law to evaluate and predict fatigue cracking in asphalt mixtures and pavements. The first technique, quasi-elastic simulation, provides a rational and appropriate reference modulus for the pseudo analysis (i.e., viscoelastic to elastic conversion) by making use of the widely used material property: dynamic modulus. The physical significance of the quasi-elastic simulation is clarified. Introduction of this technique facilitates the implementation of the fracture mechanics models as well as continuum damage mechanics models to characterize fatigue cracking in asphalt pavements. The second technique about modeling fracture coefficients of the pseudo J-integral based Paris’ law simplifies the prediction of fatigue cracking without performing fatigue tests. The developed prediction models for the fracture coefficients rely on readily available mixture design properties that directly affect the fatigue performance, including the relaxation modulus, air void content, asphalt binder content, and aggregate gradation. Sufficient data are collected to develop such prediction models and the R2 values are around 0.9. The presented case studies serve as examples to illustrate how the pseudo J-integral based Paris’ law predicts fatigue resistance of asphalt mixtures and assesses fatigue performance of asphalt pavements. Future applications include the estimation of fatigue life of asphalt mixtures/pavements through a distinct criterion that defines fatigue failure by its physical significance.

Cognitive behaviour therapy vs rational-emotive education : impact on children's self-talk, self-esteem and irrational beliefs

This study investigated the impact of Cognitive-Behavioural Therapy (CBT) and Rational~Emotive Education (REE) self-enhancement programs on children's self-talk, self-esteem and irrational beliefs. A total of 116 children (50.9% girls) with a mean age of 9.8 years attending Years 4 and 6 at two primary schools participated in the study. eBT resulted in a reduction in negative self-talk while REE seemed to enhance independence beliefs. Both programs were associated with increased positive self-talk and with having increased rationality in Conformity and Discomfort Intolerance beliefs.

Modeling corneal surfaces with rational functions for high-speed videokeratoscopy data compression

High-speed videokeratoscopy is an emerging technique that enables study of the corneal surface and tear-film dynamics. Unlike its static predecessor, this new technique results in a very large amount of digital data for which storage needs become significant. We aimed to design a compression technique that would use mathematical functions to parsimoniously fit corneal surface data with a minimum number of coefficients. Since the Zernike polynomial functions that have been traditionally used for modeling corneal surfaces may not necessarily correctly represent given corneal surface data in terms of its optical performance, we introduced the concept of Zernike polynomial-based rational functions. Modeling optimality criteria were employed in terms of both the rms surface error as well as the point spread function cross-correlation. The parameters of approximations were estimated using a nonlinear least-squares procedure based on the Levenberg-Marquardt algorithm. A large number of retrospective videokeratoscopic measurements were used to evaluate the performance of the proposed rational-function-based modeling approach. The results indicate that the rational functions almost always outperform the traditional Zernike polynomial approximations with the same number of coefficients.

Bit-pattern based integral attack

Integral attacks are well-known to be effective against byte-based block ciphers. In this document, we outline how to launch integral attacks against bit-based block ciphers. This new type of integral attack traces the propagation of the plaintext structure at bit-level by incorporating bit-pattern based notations. The new notation gives the attacker more details about the properties of a structure of cipher blocks. The main difference from ordinary integral attacks is that we look at the pattern the bits in a specific position in the cipher block has through the structure. The bit-pattern based integral attack is applied to Noekeon, Serpent and present reduced up to 5, 6 and 7 rounds, respectively. This includes the first attacks on Noekeon and present using integral cryptanalysis. All attacks manage to recover the full subkey of the final round.