Interlacing of zeros of orthogonal polynomials under modification of the measure

We investigate the mutual location of the zeros of two families of orthogonal polynomials. One of the families is orthogonal with respect to the measure dμ (x), supported on the interval (a, b) and the other with respect to the measure |x -c|τ|x -d|γdμ (x), where c and d are outside (a, b) We prove that the zeros of these polynomials, if they are of equal or consecutive degrees, interlace when either 0 < τ, γ ≤ 1 or γ = 0 and 0 < τ ≤ 2. This result is inspired by an open question of Richard Askey and it generalizes recent results on some families of orthogonal polynomials. Moreover, we obtain further statements on interlacing of zeros of specific orthogonal polynomials, such as the Askey-Wilson ones. © 2013 Elsevier Inc.

Monotonicity, interlacing and electrostatic interpretation of zeros of exceptional Jacobi polynomials

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

APPROXIMATE CALCULATION OF SUMS I: BOUNDS FOR THE ZEROS OF GRAM POLYNOMIALS

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Inequalities for zeros of Jacobi polynomials via Sturm's theorem: Gautschi's conjectures

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

On the zeros of a class of generalized hypergeometric polynomials

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Bounds for the zeros of symmetric kravchuk polynomials

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

Yang-Lee zeros of the two- and three-state Potts model defined on π3 Feynman diagrams

We present both analytical and numerical results on the position of partition function zeros on the complex magnetic field plane of the q=2 state (Ising) and the q=3 state Potts model defined on phi(3) Feynman diagrams (thin random graphs). Our analytic results are based on the ideas of destructive interference of coexisting phases and low temperature expansions. For the case of the Ising model, an argument based on a symmetry of the saddle point equations leads us to a nonperturbative proof that the Yang-Lee zeros are located on the unit circle, although no circle theorem is known in this case of random graphs. For the q=3 state Potts model, our perturbative results indicate that the Yang-Lee zeros lie outside the unit circle. Both analytic results are confirmed by finite lattice numerical calculations.

Comparative study between powers of sigmoid functions, MLP-backpropagation and polynomials in function approximation problems

Function approximation is a very important task in environments where the computation has to be based on extracting information from data samples in real world processes. So, the development of new mathematical model is a very important activity to guarantee the evolution of the function approximation area. In this sense, we will present the Polynomials Powers of Sigmoid (PPS) as a linear neural network. In this paper, we will introduce one series of practical results for the Polynomials Powers of Sigmoid, where we will show some advantages of the use of the powers of sigmiod functions in relationship the traditional MLP-Backpropagation and Polynomials in functions approximation problems.

Estimates of covariance functions for growth from birth to 630 days of age in Nelore cattle

Weight records of Brazilian Nelore cattle, from birth to 630 d of age, recorded every 3 mo, were analyzed using random regression models. Independent variables were Legendre polynomials of age at recording. The model of analysis included contemporary groups as fixed effects and age of dam as a linear and quadratic covariable. Mean trends were modeled through a cubic regression on orthogonal polynomials of age. Up to four sets of random regression coefficients were fitted for animals' direct and maternal, additive genetic, and permanent environmental effects. Changes in measurement error variances with age were modeled through a variance function. Orders of polynomial fit from three to six were considered, resulting in up to 77 parameters to be estimated. Models fitting random regressions modeled the pattern of variances in the data adequately, with estimates similar to those from corresponding univariate analysis. Direct heritability estimates decreased after birth and tended to be lowest at ages at which maternal effect estimates tended to be highest. Maternal heritability estimates increased after birth to a peak around 110 to 120 d of age and decreased thereafter. Additive genetic direct correlation estimates between weights at standard ages (birth, weaning, yearling, and final weight) were moderate to high and maternal genetic and environmental correlations were consistently high.

Assignment of Putative Functions to Membrane "Hypothetical Proteins" from the Trypanosoma cruzi Genome

Protozoan parasites cause thousands of deaths each year in developing countries. The genome projects of these parasites opened a new era in the identification of therapeutic targets. However, the putative function could be predicted for fewer than half of the protein-coding genes. In this work, all Trypanosoma cruzi proteins containing predicted transmembrane spans were processed through an automated computational routine and further analyzed in order to assign the most probable function. The analysis consisted of dissecting the whole predicted protein in different regions. More than 5,000 sequences were processed, and the predicted biological functions were grouped into 19 categories according to the hits obtained after analysis. One focus of interest, due to the scarce information available on trypanosomatids, is the proteins involved in signal-transduction processes. In the present work, we identified 54 proteins belonging to this group, which were individually analyzed. The results show that by means of a simple pipeline it was possible to attribute probable functions to sequences annotated as coding for "hypothetical proteins.'' Also, we successfully identified the majority of candidates participating in the signal-transduction pathways in T. cruzi.

Continuity of Lyapunov functions and of energy level for generalized gradient semigroup

The global attractor of a gradient-like semigroup has a Morse decomposition. Associated to this Morse decomposition there is a Lyapunov function (differentiable along solutions)-defined on the whole phase space- which proves relevant information on the structure of the attractor. In this paper we prove the continuity of these Lyapunov functions under perturbation. On the other hand, the attractor of a gradient-like semigroup also has an energy level decomposition which is again a Morse decomposition but with a total order between any two components. We claim that, from a dynamical point of view, this is the optimal decomposition of a global attractor; that is, if we start from the finest Morse decomposition, the energy level decomposition is the coarsest Morse decomposition that still produces a Lyapunov function which gives the same information about the structure of the attractor. We also establish sufficient conditions which ensure the stability of this kind of decomposition under perturbation. In particular, if connections between different isolated invariant sets inside the attractor remain under perturbation, we show the continuity of the energy level Morse decomposition. The class of Morse-Smale systems illustrates our results.

Differential calculus and integration of generalized functions over membranes

In this paper we continue the development of the differential calculus started in Aragona et al. (Monatsh. Math. 144: 13-29, 2005). Guided by the so-called sharp topology and the interpretation of Colombeau generalized functions as point functions on generalized point sets, we introduce the notion of membranes and extend the definition of integrals, given in Aragona et al. (Monatsh. Math. 144: 13-29, 2005), to integrals defined on membranes. We use this to prove a generalized version of the Cauchy formula and to obtain the Goursat Theorem for generalized holomorphic functions. A number of results from classical differential and integral calculus, like the inverse and implicit function theorems and Green's theorem, are transferred to the generalized setting. Further, we indicate that solution formulas for transport and wave equations with generalized initial data can be obtained as well.

On the Hilbert transform in the framework of generalized functions

In this paper, a definition of the Hilbert transform operating on Colombeau's temperated generalized functions is given. Similar results to some theorems that hold in the classical theory, or in certain subspaces of Schwartz distributions, have been obtained in this framework.

On the zeros of polynomials: an extension of the Enestrom-Kakeya theorem

This paper presents an extension of the Enestrom-Kakeya theorem concerning the roots of a polynomial that arises from the analysis of the stability of Brown (K, L) methods. The generalization relates to relaxing one of the inequalities on the coefficients of the polynomial. Two results concerning the zeros of polynomials will be proved, one of them providing a partial answer to a conjecture by Meneguette (1994)[6]. (C) 2011 Elsevier Inc. All rights reserved.

Reconstruction of spatiotemporal capture data by means of orthogonal functions: the case of skipjack tuna (Katsuwonus pelamis) in the central-east Atlantic

[EN] The information provided by the International Commission for the Conservation of Atlantic Tunas (ICCAT) on captures of skipjack tuna (Katsuwonus pelamis) in the central-east Atlantic has a number of limitations, such as gaps in the statistics for certain fleets and the level of spatiotemporal detail at which catches are reported. As a result, the quality of these data and their effectiveness for providing management advice is limited. In order to reconstruct missing spatiotemporal data of catches, the present study uses Data INterpolating Empirical Orthogonal Functions (DINEOF), a technique for missing data reconstruction, applied here for the first time to fisheries data. DINEOF is based on an Empirical Orthogonal Functions decomposition performed with a Lanczos method. DINEOF was tested with different amounts of missing data, intentionally removing values from 3.4% to 95.2% of data loss, and then compared with the same data set with no missing data. These validation analyses show that DINEOF is a reliable methodological approach of data reconstruction for the purposes of fishery management advice, even when the amount of missing data is very high.