Integral right triangle and rhombus pairs with a common area and a common perimeter

We prove that there are infinitely many integral right triangle and $\theta$-integral rhombus pairs with a common area and a common perimeter by the theory of elliptic curves.Comment: Final versio

Generic spectral properties of right triangle billiards

This article presents a new method to calculate eigenvalues of right triangle billiards. Its efficiency is comparable to the boundary integral method and more recently developed variants. Its simplicity and explicitness however allow new insight into the statistical properties of the spectra. We analyse numerically the correlations in level sequences at high level numbers (>10^5) for several examples of right triangle billiards. We find that the strength of the correlations is closely related to the genus of the invariant surface of the classical billiard flow. Surprisingly, the genus plays and important role on the quantum level also. Based on this observation a mechanism is discussed, which may explain the particular quantum-classical correspondence in right triangle billiards. Though this class of systems is rather small, it contains examples for integrable, pseudo integrable, and non integrable (ergodic, mixing) dynamics, so that the results might be relevant in a more general context.Comment: 18 pages, 8 eps-figures, revised: stylistic changes, improved presentatio

Integral points in rational polygons: a numerical semigroup approach

In this paper we use an elementary approach by using numerical semigroups (specifically, those with two generators) to give a formula for the number of integral points inside a right-angled triangle with rational vertices. This is the basic case for computing the number of integral points inside a rational (not necessarily convex) polygon.Fondo Europeo de Desarrollo RegionalFondo Social EuropeoAgence nationale de la recherch

Interferometer predictions with triangulated images: solving the multi-scale problem

Interferometers play an increasingly important role for spatially resolved observations. If employed at full potential, interferometry can probe an enormous dynamic range in spatial scale. Interpretation of the observed visibilities requires the numerical compu- tation of Fourier integrals over the synthetic model images. To get the correct values of these integrals, the model images must have the right size and resolution. Insufficient care in these choices can lead to wrong results. We present a new general-purpose scheme for the computation of visibilities of radiative transfer images. Our method requires a model image that is a list of intensities at arbitrarily placed positions on the image-plane. It creates a triangulated grid from these vertices, and assumes that the intensity inside each triangle of the grid is a linear function. The Fourier integral over each triangle is then evaluated with an analytic expression and the complex visibility of the entire image is then the sum of all triangles. The result is a robust Fourier trans- form that does not suffer from aliasing effects due to grid regularities. The method automatically ensures that all structure contained in the model gets reflected in the Fourier transform.Comment: 9 pages, 7 figures, accepted for publication in MNRA

Scalar Quantum Field Theory on Fractals

We construct a family of measures for random fields based on the iterated subdivision of simple geometric shapes (triangles, squares, tetrahedrons) into a finite number of similar shapes. The intent is to construct continuum limits of scale invariant scalar field theories, by imitating Wiener's construction of the measure on the space of functions of one variable. These are Gaussian measures, except for one example of a non-Gaussian fixed point for the Ising model on a fractal. In the continuum limits what we construct have correlation functions that vary as a power of distance. In most cases this is a positive power (as for the Wiener measure) but we also find a few examples with negative exponent. In all cases the exponent is an irrational number, which depends on the particular subdivision scheme used. This suggests that the continuum limits corresponds to quantum field theories (random fields) on spaces of fractional dimension

Primordial non-Gaussianity and the CMB bispectrum

We present a new formalism, together with efficient numerical methods, to directly calculate the CMB bispectrum today from a given primordial bispectrum using the full linear radiation transfer functions. Unlike previous analyses which have assumed simple separable ansatze for the bispectrum, this work applies to a primordial bispectrum of almost arbitrary functional form, for which there may have been both horizon-crossing and superhorizon contributions. We employ adaptive methods on a hierarchical triangular grid and we establish their accuracy by direct comparison with an exact analytic solution, valid on large angular scales. We demonstrate that we can calculate the full CMB bispectrum to greater than 1% precision out to multipoles l<1800 on reasonable computational timescales. We plot the bispectrum for both the superhorizon ('local') and horizon-crossing ('equilateral') asymptotic limits, illustrating its oscillatory nature which is analogous to the CMB power spectrum

Polynomial integration on regions defined by a triangle and a conic

We present an efficient solution to the following problem, of relevance in a numerical optimization scheme: calculation of integrals of the type $$\iint_{T \cap \{f\ge0\}} \phi_1\phi_2 \, dx\,dy$$ for quadratic polynomials $f,\phi_1,\phi_2$ on a plane triangle $T$. The naive approach would involve consideration of the many possible shapes of $T\cap\{f\geq0\}$ (possibly after a convenient transformation) and parameterizing its border, in order to integrate the variables separately. Our solution involves partitioning the triangle into smaller triangles on which integration is much simpler.Comment: 8 pages, accepted by ISSAC 201