Rafael G. Campos

A fast algorithm for the linear canonical transform

In recent years there has been a renewed interest in finding fast algorithms to compute accurately the linear canonical transform (LCT) of a given function. This is driven by the large number of applications of the LCT in optics and signal processing. The well-known integral transforms: Fourier, fractional Fourier, bilateral Laplace and Fresnel transforms are special cases of the LCT. In this paper we obtain an O(NlogN) algorithm to compute the LCT by using a chirp-FFT-chirp transformation yielded by a convergent quadrature formula for the fractional Fourier transform. This formula gives a unitary discrete LCT in closed form. In the case of the fractional Fourier transform the algorithm computes this transform for arbitrary complex values inside the unitary circle and not only at the boundary. This chirp-FFT-chirp transform approximates the ordinary Fourier transform more precisely than just the FFT, since it comes from a convergent procedure for non-periodic functions.

Coincident frequencies and relative phases among brain activity and hormonal signals.

BackgroundFourier transform is a basic tool for analyzing biological signals and is computed for a finite sequence of data sample. The electroencephalographic (EEG) signals analyzed with this method provide only information based on the frequency range, for short periods. In some cases, for long periods it can be useful to know whether EEG signals coincide or have a relative phase between them or with other biological signals. Some studies have evidenced that sex hormones and EEG signals show oscillations in their frequencies across a period of 28 days; so it seems of relevance to seek after possible patterns relating EEG signals and endogenous sex hormones, assumed as long time-periodic functions to determine their typical periods, frequencies and relative phases.MethodsIn this work we propose a method that can be used to analyze brain signals and hormonal levels and obtain frequencies and relative phases among them. This method involves the application of a discrete Fourier Transform on previously reported datasets of absolute power of brain signals delta, theta, alpha1, alpha2, beta1 and beta2 and the endogenous estrogen and progesterone levels along 28 days.ResultsApplying the proposed method to exemplary datasets and comparing each brain signal with both sex hormones signals, we found a characteristic profile of coincident periods and typical relative phases. For the corresponding coincident periods the progesterone seems to be essentially in phase with theta, alpha1, alpha2 and beta1, while delta and beta2 go oppositely. For the relevant coincident periods, the estrogen goes in phase with delta and theta and goes oppositely with alpha2.ConclusionFindings suggest that the procedure applied here provides a method to analyze typical frequencies, or periods and phases between signals with the same period. It generates specific patterns for brain signals and hormones and relations among them.

Aliasing modes in the lattice Schwinger model

Abstract We study the Schwinger model on a lattice consisting of zeros of the Hermite polynomials that incorporates a lattice derivative and a discrete Fourier transform with many properties. Such a lattice produces a Klein–Gordon equation for the boson field and the exact value of the mass in the asymptotic limit if the boundaries are not taken into account. On the contrary, if the lattice is considered with boundaries new modes appear due to aliasing effects. In the continuum limit, however, this lattice yields also a Klein–Gordon equation with a reduced mass.

LATTICE CALCULATIONS ON THE SPECTRUM OF DIRAC AND DIRAC–KÄHLER OPERATORS

We use a lattice formulation to study the spectra of the Dirac and the Dirac–Kahler operators on the 2-sphere. This lattice formulation uses differentiation matrices which yield exact values for the derivatives of polynomials, preserving the Leibniz rule in subspaces of polynomials of low degree and therefore, this formulation can be used to study the fermion–boson symmetry on the lattice. In this context, we find that the free Dirac and Dirac–Kahler operators on the 2-sphere exhibit fermionic as well as bosonic-like eigensolutions for which the corresponding eigenvalues and the number of states are computed. In the Dirac case these solutions appear in doublets, except for the bosonic mode with zero eigenvalue, indicating the possible existence of a supersymmetry of the square of the Dirac operator.

Quadratures and integral transforms arising from generating functions

By using the explicit form of the eigenvectors of the finite Jacobi matrix associated to a family of orthogonal polynomials and some asymptotic expressions, we obtain quadrature formulas for the integral transforms arising from linear generating functions of the classical orthogonal polynomials. As a bypass product, we obtain simple and accurate Riemann–Steklov quadrature formulas and as an application of this quadrature formalism, we obtain the relationship between the fractional Fourier transform and the canonical coherent states.

Finite-size effects on a lattice calculation

We study in this Letter the finite-size effects of a non-periodic lattice on a lattice calculation. To this end we use a finite lattice equipped with a central difference derivative with homogeneous boundary conditions to calculate the bosonic mass associated to the Schwinger model. We found that the homogeneous boundary conditions produce absence of fermion doubling and chiral invariance, but we also found that in the continuum limit this lattice model does not yield the correct value of the boson mass as other models do. We discuss the reasons for this and, as a result, the matrix which cause the fermion doubling problem is identified.

A NONLOCAL DISCRETIZATION OF FIELDS

A nonlocal method to obtain discrete classical fields is presented. This technique relies on well-behaved matrix representations of the derivatives constructed on a nonequispaced lattice. The drawbacks of lattice theory like the fermion doubling or the breaking of chiral symmetry for the massless case are absent in this method.

A discrete scheme for the Dirac and Klein–Gordon equations

Abstract The 1+1 Klein–Gordon and Dirac equations are converted into finite-dimensional matrix equations by using a projection method. In the Dirac case, the discrete solution converges to the continuum propagator and it can also be rewritten in the form given by the checkerboard model of Feynman.

Derivatives and Locality in a Lattice

We study a nonlocal Dirac operator that preserves chiral symmetry an uniqueness. It is shown that this operator approaches to an ultralocal operator when the size of the lattice tends to zero.

AN IMPLEMENTATION OF THE COLLOCATION METHOD FOR INITIAL VALUE PROBLEMS

An implementation of the standard collocation method based on polynomial interpolation is presented in a matrix framework in this paper. The underlying differentiation matrix can be partitioned to yield a superconvergent implicit multistep-like method to solve the initial value problem numerically. The first- and second-order versions of this method are L-stable.