Searcharxiv⌕ Search

arXiv subjects

P. Freitas

Publications and source records attributed to P. Freitas.

12 recordsLinked to original sources
arXiv2024

Recurrence formulae for spectral determinants

We develop a unified method to study spectral determinants for several different manifolds, including spheres and hemispheres, and projective spaces. This is a direct consequence of an approach based on deriving recursion relations for the corresponding zeta functions, which we are then able to solve explicitly. Apart from new applications such as hemispheres, we also believe that the resulting formulae in the cases for which expressions for the determinant were already known are simpler and easier to compute in general, when compared to those resulting from other approaches.

math.SP↗
arXiv2019

Committee machines -- a universal method to deal with non-idealities in memristor-based neural networks

Artificial neural networks are notoriously power- and time-consuming when implemented on conventional von Neumann computing systems. Consequently, recent years have seen an emergence of research in machine learning hardware that strives to bring memory and computing closer together. A popular approach is to realise artificial neural networks in hardware by implementing their synaptic weights using memristive devices. However, various device- and system-level non-idealities usually prevent these physical implementations from achieving high inference accuracy. We suggest applying a well-known concept in computer science -- committee machines -- in the context of memristor-based neural networks. Using simulations and experimental data from three different types of memristive devices, we show that committee machines employing ensemble averaging can successfully increase inference accuracy in physically implemented neural networks that suffer from faulty devices, device-to-device variability, random telegraph noise and line resistance. Importantly, we demonstrate that the accuracy can be improved even without increasing the total number of memristors.

cs.ET↗
arXiv2019

Extremal eigenvalues of the Dirichlet biharmonic operator on rectangles

We study the behaviour of extremal eigenvalues of the Dirichlet biharmonic operator over rectangles with a given fixed area. We begin by proving that the principal eigenvalue is minimal for a rectangle for which the ratio between the longest and the shortest side lengths does not exceed $1.066459$. We then consider the sequence formed by the minimal $k^{\rm th}$ eigenvalue and show that the corresponding sequence of minimising rectangles converges to the square as $k$ goes to infinity.

math.SP↗
arXiv2019

On the behaviour of clamped plates under large compression

We determine the asymptotic behaviour of eigenvalues of clamped plates under large compression, by relating this problem to eigenvalues of the Laplacian with Robin boundary conditions. Using the method of fundamental solutions, we then carry out a numerical study of the extremal domains for the first eigenvalue, from which we see that these depend on the value of the compression, and start developing a boundary structure as this parameter is increased. The corresponding number of nodal domains of the first eigenfunction of the extremal domain also increases with the compression.

math.SP↗
arXiv2008

Characterization and parameterization of the singular manifold of a simple 6-6 Stewart platform

This paper presents a study of the characterization of the singular manifold of the six-degree-of-freedom parallel manipulator commonly known as the Stewart platform. We consider a platform with base vertices in a circle and for which the bottom and top plates are related by a rotation and a contraction. It is shown that in this case the platform is always in a singular configuration and that the singular manifold can be parameterized by a scalar parameter.

math-ph↗
arXiv2006

Short-range effects and magnetization reversal in Co$_{80}$Fe$_{20}$ thin films: a MOKE magnetometry/domain imaging and AMR study

A MOKE magnetometry unit simultaneously sensitive to both in-plane magnetization components, based on an intensity differential detection method, allows us to observe the uniaxial anisotropy impressed during CoFe-deposition and to discriminate the magnetization processes under a magnetic field parallel and perpendicular to such axes. Our MOKE imaging unit, using a CCD camera for Kerr effect domain visualization provides direct evidence on the dominant M-processes, namely domain wall motion and moment rotation. Further magnetic information was obtained by AMR measurements due to the dependence of the electrical resistivity on the short-range spin disorder and also on the angle between the electrical current direction (I) and the spontaneous magnetization ($\emph{\textbf{M}}_{S}$).

cond-mat.mtrl-sci↗
arXiv2004

Geometrically induced discrete spectrum in curved tubes

The Dirichlet Laplacian in curved tubes of arbitrary cross-section rotating with respect to the Tang frame along infinite curves in Euclidean spaces of arbitrary dimension is investigated. If the reference curve is not straight and its curvatures vanish at infinity, we prove that the essential spectrum as a set coincides with the spectrum of the straight tube of the same cross-section and that the discrete spectrum is not empty.

math.SP↗
arXiv2004

Integrals of polylogarithmic functions, recurrence relations, and associated Euler sums

We show that integrals of the form \[ \dint_{0}^{1} x^{m}{\rm Li}_{p}(x){\rm Li}_{q}(x)dx, (m\geq -2, p,q\geq 1) \] and \[ \dint_{0}^{1} \frac{\ds \log^{r}(x){\rm Li}_{p}(x){\rm Li}_{q}(x)}{\ds x}dx, (p,q,r\geq 1) \] satisfy certain recurrence relations which allow us to write them in terms of Euler sums. From this we prove that, in the first case for all $m,p,q$, and in the second when $p+q+r$ is even, these integrals are reducible to zeta values. In the case of odd $p+q+r$, we combine the known results for Euler sums with the information obtained from the problem in this form, to give an estimate on the number of {\it new} constants which are needed to express the above integrals for a given weight $p+q+r$. The proofs are constructive, giving a method for the evaluation of these and other similar integrals, and we present a selection of explicit evaluations in the last section.

math.CA↗
arXiv2004

A lower bound to the spectral threshold in curved tubes

We consider the Laplacian in curved tubes of arbitrary cross-section rotating together with the Frenet frame along curves in Euclidean spaces of arbitrary dimension, subject to Dirichlet boundary conditions on the cylindrical surface and Neumann conditions at the ends of the tube. We prove that the spectral threshold of the Laplacian is estimated from below by the lowest eigenvalue of the Dirichlet Laplacian in a torus determined by the geometry of the tube.

math-ph↗
arXiv2000

Closed nodal lines and interior hot spots of the second eigenfunction of the Laplacian on surfaces

We build a one-parameter family of S^{1}-invariant metrics on the unit disc with fixed total area for which the second eigenvalue of the Laplace operator in the case of both Neumann and Dirichlet boundary conditions is simple and has an eigenfunction with a closed nodal line. In the case of Neumann boundary conditions, we also prove that this eigenfunction attains its maximum at an interior point, and thus provide a counterexample to the hot spots conjecture on a simply connected surface. This is a consequence of the stronger result that within this family of metrics any given (finite) number of S^{1}-invariant eigenvalues can be made to be arbitrarily small, while the non-invariant spectrum becomes arbitrarily large.

math.SP↗