Variational studies and replica symmetry breaking in the generalization problem of the binary perceptron

We analyze the average performance of a general class of learning algorithms for the nondeterministic polynomial time complete problem of rule extraction by a binary perceptron. The examples are generated by a rule implemented by a teacher network of similar architecture. A variational approach is used in trying to identify the potential energy that leads to the largest generalization in the thermodynamic limit. We restrict our search to algorithms that always satisfy the binary constraints. A replica symmetric ansatz leads to a learning algorithm which presents a phase transition in violation of an information theoretical bound. Stability analysis shows that this is due to a failure of the replica symmetric ansatz and the first step of replica symmetry breaking (RSB) is studied. The variational method does not determine a unique potential but it allows construction of a class with a unique minimum within each first order valley. Members of this class improve on the performance of Gibbs algorithm but fail to reach the Bayesian limit in the low generalization phase. They even fail to reach the performance of the best binary, an optimal clipping of the barycenter of version space. We find a trade-off between a good low performance and early onset of perfect generalization. Although the RSB may be locally stable we discuss the possibility that it fails to be the correct saddle point globally. ©2000 The American Physical Society.

Lamé differential equations and electrostatics

The problem of existence and uniqueness of polynomial solutions of the Lamé differential equation A(x)y″ + 2B(x)y′ + C(x)y = 0, where A(x),B(x) and C(x) are polynomials of degree p + 1,p and p - 1, is under discussion. We concentrate on the case when A(x) has only real zeros aj and, in contrast to a classical result of Heine and Stieltjes which concerns the case of positive coefficients rj in the partial fraction decomposition B(x)/A(x) = ∑j p=0 rj/(x - aj), we allow the presence of both positive and negative coefficients rj. The corresponding electrostatic interpretation of the zeros of the solution y(x) as points of equilibrium in an electrostatic field generated by charges rj at aj is given. As an application we prove that the zeros of the Gegenbauer-Laurent polynomials are the points of unique equilibrium in a field generated by two positive and two negative charges. © 2000 American Mathematical Society.

Galilean covariance and non-relativistic Bhabha equations

We apply a five-dimensional formulation of Galilean covariance to construct non-relativistic Bhabha first-order wave equations which, depending on the representation, correspond either to the well known Dirac equation (for particles with spin 1/2) or the Duffin-Kemmer-Petiau equation (for spinless and spin 1 particles). Here the irreducible representations belong to the Lie algebra of the 'de Sitter group' in 4 + 1 dimensions, SO(5, 1). Using this approach, the non-relativistic limits of the corresponding equations are obtained directly, without taking any low-velocity approximation. As a simple illustration, we discuss the harmonic oscillator.

Energy and angular momentum of the gravitational field in the teleparallel geometry

The Hamiltonian formulation of the teleparallel equivalent of general relativity is considered. Definitions of energy, momentum and angular momentum of the gravitational field arise from the integral form of the constraint equations of the theory. In particular, the gravitational energy-momentum is given by the integral of scalar densities over a three-dimensional spacelike hypersurface. The definition for the gravitational energy is investigated in the context of the Kerr black hole. In the evaluation of the energy contained within the external event horizon of the Kerr black hole, we obtain a value strikingly close to the irreducible mass of the latter. The gravitational angular momentum is evaluated for the gravitational field of a thin, slowly rotating mass shell. © 2002 The American Physical Society.

Isotopic characterization of the metabasites and amphibolites of the Acungui and Setuva Groups of the Ribeira Belt

Pb, Rb, Sr, Sm and Nd isotope analyses were carried out on amphibolitic rocks from Campo Largo (PR), Rio Branco do Sul (PR) and Adrianópolis (PR) and for the metabasite from Adrianópolis and Apiai (SP), all belonging to the Açungui and Setuva Groups, southern portion of the Ribeira Belt. These occurrences were chosen because each exhibits geochemical data indicative of a mantle source, having different signatures. Geochronologic determinations are available only for the Apiai metagabbro with Neoproterozoic ages of 617 ± 4 Ma (U/Pb in zircon) and 839 ± 85 Ma (Rb/Sr, whole rock). Age determinations by Sm/Nd mineral isochron using whole rock, plagioclase and pyroxene yield values of 885 ± 53 Ma for the Apiai metagabbro and 820 ± 84 Ma for the Adrianópolis metabasite. These results agree within error with the Rb/Sr isochron of 839 ± 85 Ma may possibly be interpreted as the time of extraction of the magma from the mantle, as suggested by positive εNd(850) values (+2,76) from an Adrianópolis sample. An εNd values versus 87Sr/86Sr diagram shows that each studied occurrence plots in a different position, following approximately the mantle array trend, the most primitive being the metabasite of Adrian-acopolis, near the MORB field, and the most enriched being the amphibolite from the same region, near the enriched mantle and or crustal contamination field. The distinct difference in isotopic signatures between the amphibolite and metabasite suggests derivation from different mantle sources, as supported by available geochemical data. Nevertheless, some of the samples show isotopic evidence of the action of metamorphism, crustal contamination and weathering.

Landau and Kolmogoroff type polynomial inequalities II

Let 0 < j < m ≤ n. Kolmogoroff type inequalities of the form ∥f(j)∥2 ≤ A∥f(m)∥ 2 + B∥f∥2 which hold for algebraic polynomials of degree n are established. Here the norm is defined by ∫ f2(x)dμ(x), where dμ(x) is any distribution associated with the Jacobi, Laguerre or Bessel orthogonal polynomials. In particular we characterize completely the positive constants A and B, for which the Landau weighted polynomial inequalities ∥f′∥ 2 ≤ A∥f″∥2 + B∥f∥ 2 hold. © Dynamic Publishers, Inc.

Bovine serum albumin displays luciferase-like activity in presence of luciferyl adenylate: Insights on the origin of protoluciferase activity and bioluminescence colours

Luciferyl adenylate, the key intermediate in beetle bioluminescence, is produced through adenylation of D-luciferin by beetle luciferases and also by mealworm luciferase-like enzymes which produce a weak red chemiluminescence. However, luciferyl adenylate is only weakly chemiluminescent in water at physiological pH and it is unclear how efficient bioluminescence evolved from its weak chemiluminescent properties. We found that bovine serum albumin (BSA) and neutral detergents enhance luciferyl adenylate chemiluminescence by three orders of magnitude, simulating the mealworm luciferase-like enzyme chemiluminescence properties. These results suggest that the beetle protoluciferase activity arose as an enhanced luciferyl adenylate chemiluminescence in the protein environment of the ancestral AMP-ligase. The predominance of luciferyl adenylate chemiluminescence in the red region under most conditions suggests that red luminescence is a more primitive condition that characterized the original stages of protobioluminescence, whereas yellow-green bioluminescence may have evolved later through the development of a more structured and hydrophobic active site. Copyright © 2006 John Wiley & Sons, Ltd.

Multi-trait and random regression mature weight heritability and breeding value estimates in Nelore cattle

Mature weight breeding values were estimated using a multi-trait animal model (MM) and a random regression animal model (RRM). Data consisted of 82 064 weight records from 8 145 animals, recorded from birth to eight years of age. Weights at standard ages were considered in the MM. All models included contemporary groups as fixed effects, and age of dam (linear and quadratic effects) and animal age as covariates. In the RRM, mean trends were modelled through a cubic regression on orthogonal polynomials of animal age and genetic maternal and direct and maternal permanent environmental effects were also included as random. Legendre polynomials of orders 4, 3, 6 and 3 were used for animal and maternal genetic and permanent environmental effects, respectively, considering five classes of residual variances. Mature weight (five years) direct heritability estimates were 0.35 (MM) and 0.38 (RRM). Rank correlation between sires' breeding values estimated by MM and RRM was 0.82. However, selecting the top 2% (12) or 10% (62) of the young sires based on the MM predicted breeding values, respectively 71% and 80% of the same sires would be selected if RRM estimates were used instead. The RRM modelled the changes in the (co)variances with age adequately and larger breeding value accuracies can be expected using this model. © South African Society for Animal Science.

Zeros of Jacobi-Sobolev orthogonal polynomials following non-coherent pair of measures

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Sharp Turán inequalities via very hyperbolic polynomials

We present new sharp inequalities for the Maclaurin coefficients of an entire function from the Laguerre-Pólya class. They are obtained by a new technique involving the so-called very hyperbolic polynomials. The results may be considered as extensions of the classical Turán inequalities. © 2010 Elsevier Inc.

Kernel polynomials from L-orthogonal polynomials

A positive measure ψ defined on [a,b] such that its moments μn=∫a btndψ(t) exist for n=0,±1,±2,⋯, is called a strong positive measure on [a,b]. If 0≤apolynomials {Qn}, defined by ∫a bt-n+sQn(t)dψ(t)=0, s=0,1,⋯,n-1, is known to exist. We refer to these polynomials as the L-orthogonal polynomials with respect to the strong positive measure ψ. The purpose of this manuscript is to consider some properties of the kernel polynomials associated with these L-orthogonal polynomials. As applications, we consider the quadrature rules associated with these kernel polynomials. Associated eigenvalue problems and numerical evaluation of the nodes and weights of such quadrature rules are also considered. © 2010 IMACS. Published by Elsevier B.V. All rights reserved.

Multivariate sobolev-type orthogonal polynomials

Multivariate orthogonal polynomials associated with a Sobolev-type inner product, that is, an inner product defined by adding to a measure the evaluation of the gradients in a fixed point, are studied. Orthogonal polynomials and kernel functions associated with this new inner product can be explicitly expressed in terms of those corresponding with the original measure. We apply our results to the particular case of the classical orthogonal polynomials on the unit ball, and we obtain the asymptotics of the kernel functions. © 2011 Universidad de Jaén.

Random regression models using Legendre polynomials or linear splines for test-day milk yield of dairy Gyr (Bos indicus) cattle

Studies investigating the use of random regression models for genetic evaluation of milk production in Zebu cattle are scarce. In this study, 59,744 test-day milk yield records from 7,810 first lactations of purebred dairy Gyr (Bos indicus) and crossbred (dairy Gyr × Holstein) cows were used to compare random regression models in which additive genetic and permanent environmental effects were modeled using orthogonal Legendre polynomials or linear spline functions. Residual variances were modeled considering 1, 5, or 10 classes of days in milk. Five classes fitted the changes in residual variances over the lactation adequately and were used for model comparison. The model that fitted linear spline functions with 6 knots provided the lowest sum of residual variances across lactation. On the other hand, according to the deviance information criterion (DIC) and Bayesian information criterion (BIC), a model using third-order and fourth-order Legendre polynomials for additive genetic and permanent environmental effects, respectively, provided the best fit. However, the high rank correlation (0.998) between this model and that applying third-order Legendre polynomials for additive genetic and permanent environmental effects, indicates that, in practice, the same bulls would be selected by both models. The last model, which is less parameterized, is a parsimonious option for fitting dairy Gyr breed test-day milk yield records. © 2013 American Dairy Science Association.

The monetary crisis, dollarization and the exchange rate

Includes bibliography

Are NORs Always Located on Homeologous Chromosomes? A FISH Investigation with rDNA and Whole Chromosome Probes in Gymnotus Fishes (Gymnotiformes)

Gymnotus (Gymnotiformes, Gymnotidae) is the most diverse known Neotropical electric knife fish genus. Cytogenetic studies in Gymnotus demonstrate a huge karyotypic diversity for this genus, with diploid numbers ranging from 34 to 54. The NOR are also variable in this genus, with both single and multiple NORs described. A common interpretation is that the single NOR pair is a primitive trait while multiple NORs are derivative. However this hypothesis has never been fully tested. In this report we checked if the NOR-bearing chromosome and the rDNA site are homeologous in different species of the genus Gymnotus: G. carapo (2n = 40, 42, 54), G. mamiraua (2n = 54), G. arapaima (2n = 44), G. sylvius (2n = 40), G. inaequilabiatus (2n = 54) and G. capanema (2n = 34), from the monophyletic group G. carapo (Gymnotidae-Gymnotiformes), as well as G. jonasi (2n = 52), belonging to the G1 group. They were analyzed with Fluorescence in situ hybridization (FISH) using 18S rDNA and whole chromosome probes of the NOR-bearing chromosome 20 (GCA20) of G. carapo (cytotype 2n = 42), obtained by Fluorescence Activated Cell Sorting. All species of the monophyletic G. carapo group show the NOR in the same single pair, confirmed by hybridization with CGA20 whole chromosome probe. In G. jonasi the NORs are multiple, and located on pairs 9, 10 and 11. In G. jonasi the GCA20 chromosome probe paints the distal half of the long arm of pair 7, which is not a NOR-bearing chromosome. Thus these rDNA sequences are not always in the homeologous chromosomes in different species thus giving no support to the hypothesis that single NOR pairs are primitive traits while multiple NORs are derived. The separation of groups of species in the genus Gymnotus proposed by phylogenies with morphologic and molecular data is supported by our cytogenetic data. © 2013 Milhomem et al.