Mathieu functions computational toolbox implemented in Matlab

The Mathieu functions are used to solve analytically some problems in elliptical cylinder coordinates. A computational toolbox was implemented in Matlab. Since the notation and normalization for Mathieu functions vary in the literature, we have included sufficient material to make this presentation self contained. Thus, all formulas required to get the Mathieu functions are given explicitly. Following the outlines in this presentation, the Mathieu functions could be readily implemented in other computer programs and used in different domains. Tables of numerical values are provided.Comment: 19pages,0figures,6table

Elliptic Cylindrical Invisibility Cloak, a Semianalytical Approach Using Mathieu Functions

An elliptic cylindrical wave expansion method by using Mathieu functions is developed to obtain the scattering field for a two-dimensional elliptic cylindrical invisibility cloak. The cloak material parameters are obtained from the spatial transformation approach. A near-ideal model of the invisibility cloak is set up to solve the boundary problem at the inner boundary in the cloak shell. The proposed design provides a more practical cloak geometry when compared to previous designs of elliptic cylindrical cloaks.Comment: 8 pages, 1 figure, articl

Elemental matrices for the finite element method in electromagnetics with quadratic triangular elements

The finite element method has become a preeminent simulation technique in electromagnetics. For problems involving anisotropic media and metamaterials, proper algorithms should be developed. It has been proved that discretizing in quadratic triangular elements may lead to an improved accuracy. Here we present a collection of elemental matrices evaluated analytically for quadratic triangular elements. They could be useful for the finite element method in advanced electromagnetics

Third-order triangular finite elements for waveguiding problems

Explicit relations of matrices for two-dimensional finite element method with third-order triangular elements are given. They are more simple than relations presented in other works and could be easily implemented in new algorithms for both isotropic and anisotropic materials. Numerical examples are given comparatively using second-order and third-order triangular elements for problems of wave propagation in rectangular waveguides which have analytic solutions.Comment: 12 pages, 2 figure

Distribution of Farey Fractions in Residue Classes and Lang--Trotter Conjectures on Average

We prove that the set of Farey fractions of order $T$, that is, the set \{\alpha/\beta \in \Q : \gcd(\alpha, \beta) = 1, 1 \le \alpha, \beta \le T\}, is uniformly distributed in residue classes modulo a prime $p$ provided T \ge p^{1/2 +\eps} for any fixed \eps>0. We apply this to obtain upper bounds for the Lang--Trotter conjectures on Frobenius traces and Frobenius fields ``on average'' over a one-parametric family of elliptic curves

Backward waves in a grounded bilayer slab containing double-negative (DNG) and double-positive (DPS) metamaterials

Simple dispersion relations for the guided modes in a grounded DNG/DPS bilayer slab are given in terms of normalized parameters. Relations corresponding to the grounded single-layer DNG slab are refound as specific cases. Numerical examples are given showing dispersion curves of the lower order modes and the respective total normalized power carried on the propagation direction. Snapshots obtained by the finite-difference time-domain method are provided showing the electromagnetic field inside the grounded DNG/DPS bilayer slabs. Since an important characteristic of the guided modes in the slab containing a DNG layer is the existence of a turning point (TP) at which the power carried by each mode of order m>0 equals zero and changes the sign, we present implicit relations at the TP for the normalized parameters of the guided modes in the grounded DNG/DPS and DNG slabs. We show that a thin DPS layer coating on the grounded DNG slab produces a shift of the TP on the dispersion curve.Comment: 14 pages, 8 figure

Mathieu functions approach to bidimensional scattering by dielectric elliptical cylinders

Two-dimensional scattering by homogeneous and layered dielectric elliptical cylinders is analyzed following an analytical approach using Mathieu functions. Closed-form relations for the expansion coefficients of the resulting electric field in the vicinity of the scatterer are provided. Numerical examples show the focalizing effect of dielectric elliptical cylinders illuminated normally to the axis. The influence of the confocal dielectric cover on the resulting scattered field is envisaged.Comment: 10 pages, 3 figure

The population of single and binary white dwarfs of the Galactic bulge

Recent Hubble Space Telescope observations have unveiled the white dwarf cooling sequence of the Galactic bulge. Although the degenerate sequence can be well fitted employing the most up-to-date theoretical cooling sequences, observations show a systematic excess of red objects that cannot be explained by the theoretical models of single carbon-oxygen white dwarfs of the appropriate masses. Here we present a population synthesis study of the white dwarf cooling sequence of the Galactic bulge that takes into account the populations of both single white dwarfs and binary systems containing at least one white dwarf. These calculations incorporate state-of-the-art cooling sequences for white dwarfs with hydrogen-rich and hydrogen-deficient atmospheres, for both white dwarfs with carbon-oxygen and helium cores, and also take into account detailed prescriptions of the evolutionary history of binary systems. Our Monte Carlo simulator also incorporates all the known observational biases. This allows us to model with a high degree of realism the white dwarf population of the Galactic bulge. We find that the observed excess of red stars can be partially attributed to white dwarf plus main sequence binaries, and to cataclysmic variables or dwarf novae. Our best fit is obtained with a higher binary fraction and an initial mass function slope steeper than standard values, as well as with the inclusion of differential reddening and blending. Our results also show that the possible contribution of double degenerate systems or young and thick-disk bulge stars is negligible.Comment: 10 pages, 9 figures, accepted for publication in MNRA

The population of white dwarf-main sequence binaries in the SDSS DR 12

We present a Monte Carlo population synthesis study of white dwarf-main sequence (WD+MS) binaries in the Galactic disk aimed at reproducing the ensemble properties of the entire population observed by the Sloan Digital Sky Survey (SDSS) Data Release 12. Our simulations take into account all known observational biases and use the most up-to-date stellar evolutionary models. This allows us to perform a sound comparison between the simulations and the observational data. We find that the properties of the simulated and observed parameter distributions agree best when assuming low values of the common envelope efficiency (0.2-0.3), a result that is in agreement with previous findings obtained by observational and population synthesis studies of close SDSS WD+MS binaries. We also show that all synthetic populations that result from adopting an initial mass ratio distribution with a positive slope are excluded by observations. Finally, we confirm that the properties of the simulated WD+MS binary populations are nearly independent of the age adopted for the thin disk, on the contribution of WD+MS binaries from the thick disk (0-17 per cent of the total population) and on the assumed fraction of the internal energy that is used to eject the envelope during the common envelope phase (0.1-0.5).Comment: accepted for publication by MNRA

Arithmetic properties of the Frobenius traces defined by a rational abelian variety (with two appendices by J-P. Serre)

Let $A$ be an abelian variety over $\mathbb{Q}$ of dimension $g$ such that the image of its associated absolute Galois representation $\rho_A$ is open in $\operatorname{GSp}_{2g}(\hat{\mathbb{Z}})$. We investigate the arithmetic of the traces $a_{1, p}$ of the Frobenius at $p$ in $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ under $\rho_A$, modulo varying primes $p$. In particular, we obtain upper bounds for the counting function $\#\{p \leq x: a_{1, p} = t\}$ and we prove an Erd\"os-Kac type theorem for the number of prime factors of $a_{1, p}$. We also formulate a conjecture about the asymptotic behaviour of $\#\{p \leq x: a_{1, p} = t\}$, which generalizes a well-known conjecture of S. Lang and H. Trotter from 1976 about elliptic curves.Comment: 37 pages including four figures, two appendices by J-P. Serre, and references; to appear in International Mathematics Research Notice