Bernstein polynomials in element-free Galerkin method

In the recent decades, meshless methods (MMs), like the element-free Galerkin method (EFGM), have been widely studied and interesting results have been reached when solving partial differential equations. However, such solutions show a problem around boundary conditions, where the accuracy is not adequately achieved. This is caused by the use of moving least squares or residual kernel particle method methods to obtain the shape functions needed in MM, since such methods are good enough in the inner of the integration domains, but not so accurate in boundaries. This way, Bernstein curves, which are a partition of unity themselves,can solve this problem with the same accuracy in the inner area of the domain and at their boundaries.

Bandlimited Airy Pulses for Invariant Propagation in Single-Mode Fibers

By spectral analysis, and using joint time-frequency representations, we present the theoretical basis to design invariant bandlimited Airy pulses with an arbitrary degree of robustness and an arbitrary range of single-mode fiber chromatic dispersion. The numerically simulated examples confirm the theoretically predicted pulse partial invariance in the propagation of the pulse in the fiber.

Differential resultant formulas for linear od-polynomials.

Ponencia

On some sampling-related frames in U-invariant spaces

This paper concerns the characterization as frames of some sequences in U-invariant spaces of a separable Hilbert space H where U denotes an unitary operator defined on H ; besides, the dual frames having the same form are also found. This general setting includes, in particular, shift-invariant or modulation-invariant subspaces in L2 (R), where these frames are intimately related to the generalized sampling problem. We also deal with some related perturbation problems. In so doing, we need that the unitary operator U belongs to a continuous group of unitary operators.

Generalized sampling in U-invariant subspaces

In this work we carry out some results in sampling theory for U-invariant subspaces of a separable Hilbert space H, also called atomic subspaces. These spaces are a generalization of the well-known shift- invariant subspaces in L2 (R); here the space L2 (R) is replaced by H, and the shift operator by U. Having as data the samples of some related operators, we derive frame expansions allowing the recovery of the elements in Aa. Moreover, we include a frame perturbation-type result whenever the samples are affected with a jitter error.

Differential resultants of super essential systems of linear OD-polynomials

The sparse differential resultant dres(P) of an overdetermined system P of generic nonhomogeneous ordinary differential polynomials, was formally defined recently by Li, Gao and Yuan (2011). In this note, a differential resultant formula dfres(P) is defined and proved to be nonzero for linear "super essential" systems. In the linear case, dres(P) is proved to be equal, up to a nonzero constant, to dfres(P*) for the supper essential subsystem P* of P.

Learning Bayesian networks from data by the incremental compilation of new network polynomials

Probabilistic graphical models are a huge research field in artificial intelligence nowadays. The scope of this work is the study of directed graphical models for the representation of discrete distributions. Two of the main research topics related to this area focus on performing inference over graphical models and on learning graphical models from data. Traditionally, the inference process and the learning process have been treated separately, but given that the learned models structure marks the inference complexity, this kind of strategies will sometimes produce very inefficient models. With the purpose of learning thinner models, in this master thesis we propose a new model for the representation of network polynomials, which we call polynomial trees. Polynomial trees are a complementary representation for Bayesian networks that allows an efficient evaluation of the inference complexity and provides a framework for exact inference. We also propose a set of methods for the incremental compilation of polynomial trees and an algorithm for learning polynomial trees from data using a greedy score+search method that includes the inference complexity as a penalization in the scoring function.

Conditional density approximations with mixtures of polynomials

Mixtures of polynomials (MoPs) are a non-parametric density estimation technique especially designed for hybrid Bayesian networks with continuous and discrete variables. Algorithms to learn one- and multi-dimensional (marginal) MoPs from data have recently been proposed. In this paper we introduce two methods for learning MoP approximations of conditional densities from data. Both approaches are based on learning MoP approximations of the joint density and the marginal density of the conditioning variables, but they differ as to how the MoP approximation of the quotient of the two densities is found. We illustrate and study the methods using data sampled from known parametric distributions, and we demonstrate their applicability by learning models based on real neuroscience data. Finally, we compare the performance of the proposed methods with an approach for learning mixtures of truncated basis functions (MoTBFs). The empirical results show that the proposed methods generally yield models that are comparable to or significantly better than those found using the MoTBF-based method.

A finite class of orthogonal functions generated by Routh-Romanovski polynomials

It is known that some orthogonal systems are mapped onto other orthogonal systems by the Fourier transform. In this article we introduce a finite class of orthogonal functions, which is the Fourier transform of Routh-Romanovski orthogonal polynomials, and obtain its orthogonality relation using Parseval identity.

Bivariate Krawtchouk polynomials: Inversion and connection problems with the NAVIMA algorithm

In this paper we present a recurrent procedure to solve an inversion problem for monic bivariate Krawtchouk polynomials written in vector column form, giving its solution explicitly. As a by-product, a general connection problem between two vector column of monic bivariate Krawtchouk families is also explicitly solved. Moreover, in the non monic case and also for Krawtchouk families, several expansion formulas are given, but for polynomials written in scalar form.

Solution of the pulse width modulation problem using orthogonal polynomials and Korteweg-de Vries equations

The mathematical underpinning of the pulse width modulation (PWM) technique lies in the attempt to represent “accurately” harmonic waveforms using only square forms of a fixed height. The accuracy can be measured using many norms, but the quality of the approximation of the analog signal (a harmonic form) by a digital one (simple pulses of a fixed high voltage level) requires the elimination of high order harmonics in the error term. The most important practical problem is in “accurate” reproduction of sine-wave using the same number of pulses as the number of high harmonics eliminated. We describe in this paper a complete solution of the PWM problem using Padé approximations, orthogonal polynomials, and solitons. The main result of the paper is the characterization of discrete pulses answering the general PWM problem in terms of the manifold of all rational solutions to Korteweg-de Vries equations.

Functionality of major histocompatibility complex class II molecules in mice doubly deficient for invariant chain and H-2M complexes

By combining two previously generated null mutations, Ii° and M°, we produced mice lacking the invariant chain and H-2M complexes, both required for normal cell-surface expression of major histocompatibility complex class II molecules loaded with the usual diverse array of peptides. As expected, the maturation and transport of class II molecules, their expression at the cell surface, and their capacity to present antigens were quite similar for cells from Ii°M° double-mutant mice and from animals carrying just the Ii° mutation. More surprising were certain features of the CD4+ T cell repertoire selected in Ii°M° mice: many fewer cells were selected than in Ii+M° animals, and these had been purged of self-reactive specificities, unlike their counterparts in Ii+M° animals. These findings suggest (i) that the peptides carried by class II molecules on stromal cells lacking H-2M complexes may almost all derive from invariant chain and (ii) that H-2M complexes edit the peptide array displayed on thymic stromal cells in the absence of invariant chain, showing that it can edit, in vivo, peptides other than CLIP.

Invariant size–frequency distributions along a latitudinal gradient in marine bivalves

In the most extensive analysis of body size in marine invertebrates to date, we show that the size–frequency distributions of northeastern Pacific bivalves at the provincial level are surprisingly invariant in modal and median size as well as size range, despite a 4-fold change in species richness from the tropics to the Arctic. The modal sizes and shapes of these size–frequency distributions are consistent with the predictions of an energetic model previously applied to terrestrial mammals and birds. However, analyses of the Miocene–Recent history of body sizes within 82 molluscan genera show little support for the expectation that the modal size is an evolutionary attractor over geological time.

In vitro roles of invariant helix–turn–helix motif residue R383 in σ54 (σN)

In vitro DNA-binding and transcription properties of σ54 proteins with the invariant Arg383 in the putative helix–turn–helix motif of the DNA-binding domain substituted by lysine or alanine are described. We show that R383 contributes to maintaining stable holoenzyme–promoter complexes in which limited DNA opening downstream of the –12 GC element has occurred. Unlike wild-type σ54, holoenzymes assembled with the R383A or R383K mutants could not form activator-independent, heparin-stable complexes on heteroduplex Sinorhizobium meliloti nifH DNA mismatched next to the GC. Using longer sequences of heteroduplex DNA, heparin-stable complexes formed with the R383K and, to a lesser extent, R383A mutant holoenzymes, but only when the activator and a hydrolysable nucleotide was added and the DNA was opened to include the –1 site. Although R383 appears inessential for polymerase isomerisation, it makes a significant contribution to maintaining the holoenzyme in a stable complex when melting is initiating next to the GC element. Strikingly, Cys383-tethered FeBABE footprinting of promoter DNA strongly suggests that R383 is not proximal to promoter DNA in the closed complex. This indicates that R383 is not part of the regulatory centre in the σ54 holoenzyme, which includes the –12 promoter region elements. R383 contributes to several properties, including core RNA polymerase binding and to the in vivo stability of σ54.