Spectral and weak polynomial completeness for the porduct of nonsingular matrices

Let F be a field with at least four elements. In this paper, we identify all the pairs (A, B) of n x n nonsingular matrices over F , satisfying the following property: for every monic polynomial f(x) = xn + an-1xn-1 + … +a1x + aο over F, with a root in F and aο = (-1)n det(AB), there are nonsingular matrices X, Y ϵ Fnxn such that X A X-1 Y BY-1 has characteristic polynomial f (x). © 2014 © 2014 Taylor & Francis.

Spectral and weak polynomial completeness for the product of nonsingular matrices

Let F be a field with at least four elements. In this paper, we identify all the pairs (A, B) of n x n nonsingular matrices over F, satisfying the following property: for every monic polynomial f (x) = x(n) + a(n-1)x(n-1) +... + a(1)x + a(0) over F, with a root in F and a(0) = (-1)(n) det(AB), there are nonsingular matrices X, Y is an element of F-nxn such that XAX(-1)Y BY-1 has characteristic polynomial f (x).

Weakly spectrally complete pair of matrices

Let and be matrices over an algebraically closed field. Let be elements of such that and . We give necessary and sufficient condition for the existence of matrices and similar to and, respectively, such that has eigenvalues.

Efficient Legendre spectral tau algorithm for solving two-sided space-time Caputo fractional advection-dispersion equation

In this paper we present the operational matrices of the left Caputo fractional derivative, right Caputo fractional derivative and Riemann–Liouville fractional integral for shifted Legendre polynomials. We develop an accurate numerical algorithm to solve the two-sided space–time fractional advection–dispersion equation (FADE) based on a spectral shifted Legendre tau (SLT) method in combination with the derived shifted Legendre operational matrices. The fractional derivatives are described in the Caputo sense. We propose a spectral SLT method, both in temporal and spatial discretizations for the two-sided space–time FADE. This technique reduces the two-sided space–time FADE to a system of algebraic equations that simplifies the problem. Numerical results carried out to confirm the spectral accuracy and efficiency of the proposed algorithm. By selecting relatively few Legendre polynomial degrees, we are able to get very accurate approximations, demonstrating the utility of the new approach over other numerical methods.

A Review of Operational Matrices and Spectral Techniques for Fractional Calculus

Recently, operational matrices were adapted for solving several kinds of fractional differential equations (FDEs). The use of numerical techniques in conjunction with operational matrices of some orthogonal polynomials, for the solution of FDEs on finite and infinite intervals, produced highly accurate solutions for such equations. This article discusses spectral techniques based on operational matrices of fractional derivatives and integrals for solving several kinds of linear and nonlinear FDEs. More precisely, we present the operational matrices of fractional derivatives and integrals, for several polynomials on bounded domains, such as the Legendre, Chebyshev, Jacobi and Bernstein polynomials, and we use them with different spectral techniques for solving the aforementioned equations on bounded domains. The operational matrices of fractional derivatives and integrals are also presented for orthogonal Laguerre and modified generalized Laguerre polynomials, and their use with numerical techniques for solving FDEs on a semi-infinite interval is discussed. Several examples are presented to illustrate the numerical and theoretical properties of various spectral techniques for solving FDEs on finite and semi-infinite intervals.

An efficient numerical scheme for solving multi-dimensional fractional optimal control problems with a quadratic performance index

The shifted Legendre orthogonal polynomials are used for the numerical solution of a new formulation for the multi-dimensional fractional optimal control problem (M-DFOCP) with a quadratic performance index. The fractional derivatives are described in the Caputo sense. The Lagrange multiplier method for the constrained extremum and the operational matrix of fractional integrals are used together with the help of the properties of the shifted Legendre orthonormal polynomials. The method reduces the M-DFOCP to a simpler problem that consists of solving a system of algebraic equations. For confirming the efficiency and accuracy of the proposed scheme, some test problems are implemented with their approximate solutions.

Study of forbidden transitions in atomic systems

A Thesis submitted for the co-tutelle degree of Doctor in Physics at Universidade Nova de Lisboa and Université Pierre et Marie Curie

Comparison of different numerical methods for the solution of the time-fractional reaction-diffusion equation with variable diffusion coefficient

In this work we perform a comparison of two different numerical schemes for the solution of the time-fractional diffusion equation with variable diffusion coefficient and a nonlinear source term. The two methods are the implicit numerical scheme presented in [M.L. Morgado, M. Rebelo, Numerical approximation of distributed order reaction- diffusion equations, Journal of Computational and Applied Mathematics 275 (2015) 216-227] that is adapted to our type of equation, and a colocation method where Chebyshev polynomials are used to reduce the fractional differential equation to a system of ordinary differential equations

O regime de chuvas em Pesqueira

The authors studied the rainfall in Pesqueira (Pernambuco, Brasil) in a period of 48 years (1910 through 1957) by the method of orthogonal polynomials, degrees up to the fourth having been tried. None of them was significant, so that it seems that no trend is present. The mean observed was 679.00 mm., with standard error of the mean 205.5 mm., and a 30.3% coefficient of variation. The 95% level of probability would include annual rainfall from 263.9 up to 1094.1mm.

Estudo sôbre as temperaturas médias em Campinas

This paper deals with the study by orthogonal polynomials of trends in the mean annual and mean monthly temperatures (in degrees Centigrade) in Campinas (State of São Paulo, Brasil), from 1890 up to 1956. Only 4 months were studied (January, April, July and October) taken as typical of their respective season. For the annual averages both linear and quadratic components were significant, the regression equation being y = 19.95 - 0.0219 x + 0.00057 x², where y is the temperature (in degrees Centigrade) and x is the number of years after 1889. Thus 1890 corresponds to x = 1, 1891, to x = 2, etc. The equation shows a minimum for the year 1908, with a calculated mean y = 19.74. The expected means by the regression equation are given below. Anual temperature means for Campinas (SP, Brasil) calculated by the regression equation Year Annual mean (Degrees Centigrade) 1890 19.93 1900 10.78 1908 19.74 (minimum) 1010 19.75 1920 19.82 1930 20.01 1940 20.32 1950 20.74 1956 21.05 The mean for 67 years was 20.08°C with standard error of the mean 0.08°G. For January the regression equation was y = 23.08 - 0.0661 x + 0.00122 x², with a minimum of 22.19°C for 1916. The average for 67 years was 22.70°C, with standard error 0.12°C. For April no component of regression was significant. The average was 20.42°C, with standard error 0.13°C. For July the regression equation was of first degree, y = 16.01 + 0.0140X. The average for 67 years was 16.49°C, with standard error of the mean 0.14°C. Finally, for October the regression equation was y = 20.55 - 0.0362x + 0.00078x², with a minimum of 20.13°C for 1912. The average was 20.52°C, with standard error of the mean equal to 0.14°C.

Sex difference and the role of leptin in the association between high-sensitivity C-reactive protein and adiposity in two different populations.

Elevated high-sensitivity C-reactive protein (hs-CRP) concentration is associated with an increased risk of cardiovascular disease but this association seems to be largely mediated via conventional cardiovascular risk factors. In particular, the association between hs-CRP and obesity has been extensively demonstrated and correlations are stronger in women than men. We used fractional polynomials-a method that allows flexible modeling of non linear relations-to investigate the dose/response mathematical relationship between hs-CRP and several indicators of adiposity in Caucasians (Switzerland) and Africans (Seychelles) surveyed in two population-based studies. This relationship was non-linear exhibiting a steeper slope for low levels of hs-CRP and a higher level in women. The observed sex difference in the relationship between hs-CRP and adiposity almost disappeared upon adjustment for leptin, suggesting that these sex differences might be partially mediated, by leptin. All these relationship were similar in Caucasians and Africans. This is the first report on a non-linear relation, stratified by gender, between hs-CRP and adiposity.

Gains from switching and evolutionary stability in multi-player matrix games.

In this paper we unify, simplify, and extend previous work on the evolutionary dynamics of symmetric N-player matrix games with two pure strategies. In such games, gains from switching strategies depend, in general, on how many other individuals in the group play a given strategy. As a consequence, the gain function determining the gradient of selection can be a polynomial of degree N-1. In order to deal with the intricacy of the resulting evolutionary dynamics, we make use of the theory of polynomials in Bernstein form. This theory implies a tight link between the sign pattern of the gains from switching on the one hand and the number and stability of the rest points of the replicator dynamics on the other hand. While this relationship is a general one, it is most informative if gains from switching have at most two sign changes, as is the case for most multi-player matrix games considered in the literature. We demonstrate that previous results for public goods games are easily recovered and extended using this observation. Further examples illustrate how focusing on the sign pattern of the gains from switching obviates the need for a more involved analysis.

Statistical treatment of grain-size curves and empirical distributions: densities as compositions?

The preceding two editions of CoDaWork included talks on the possible considerationof densities as infinite compositions: Egozcue and D´ıaz-Barrero (2003) extended theEuclidean structure of the simplex to a Hilbert space structure of the set of densitieswithin a bounded interval, and van den Boogaart (2005) generalized this to the setof densities bounded by an arbitrary reference density. From the many variations ofthe Hilbert structures available, we work with three cases. For bounded variables, abasis derived from Legendre polynomials is used. For variables with a lower bound, westandardize them with respect to an exponential distribution and express their densitiesas coordinates in a basis derived from Laguerre polynomials. Finally, for unboundedvariables, a normal distribution is used as reference, and coordinates are obtained withrespect to a Hermite-polynomials-based basis.To get the coordinates, several approaches can be considered. A numerical accuracyproblem occurs if one estimates the coordinates directly by using discretized scalarproducts. Thus we propose to use a weighted linear regression approach, where all k-order polynomials are used as predictand variables and weights are proportional to thereference density. Finally, for the case of 2-order Hermite polinomials (normal reference)and 1-order Laguerre polinomials (exponential), one can also derive the coordinatesfrom their relationships to the classical mean and variance.Apart of these theoretical issues, this contribution focuses on the application of thistheory to two main problems in sedimentary geology: the comparison of several grainsize distributions, and the comparison among different rocks of the empirical distribution of a property measured on a batch of individual grains from the same rock orsediment, like their composition

Heat capacity measurements by differential scanning calorimetry in the Pd-Pb, Pd-Sn and Pd-ln systems

Molar heat capacities at constant pressure of six solid solutions and 11 intermediate phases in the Pd-Pb, Pd-Sn and Pd-In systems were determined each 10 K by differential scanning calorimetry from 310 to 1000 K, The experimental values have been fitted by polynomials C-p = a + bT + cT(2) + d/T-2. Results are given, discussed and compared with available literature data. (C) 2001 Elsevier Science B.V, AII rights reserved.

Bernstein's problem on weighted polynomial approximation

We formulate a necessary and sufficient condition for polynomials to be dense in a space of continuous functions on the real line, with respect to Bernstein's weighted uniform norm. Equivalently, for a positive finite measure [lletra "mu" minúscula de l'alfabet grec] on the real line we give a criterion for density of polynomials in Lp[lletra "mu" minúscula de l'alfabet grec entre parèntesis].