Rational parametrization of conchoids to algebraic curves

We study the rationality of each of the components of the conchoid to an irreducible algebraic affine plane curve, excluding the trivial cases of the lines through the focus and the circle centered at the focus and radius the distance involved in the conchoid. We prove that conchoids having all their components rational can only be generated by rational curves. Moreover, we show that reducible conchoids to rational curves have always their two components rational. In addition, we prove that the rationality of the conchoid component, to a rational curve, does depend on the base curve and on the focus but not on the distance. As a consequence, we provide an algorithm that analyzes the rationality of all the components of the conchoid and, in the affirmative case, parametrizes them. The algorithm only uses a proper parametrization of the base curve and the focus and, hence, does not require the previous computation of the conchoid. As a corollary, we show that the conchoid to the irreducible conics, with conchoid-focus on the conic, are rational and we give parametrizations. In particular we parametrize the Limaçons of Pascal. We also parametrize the conchoids of Nicomedes. Finally, we show how to find the foci from where the conchoid is rational or with two rational components

Total Degree Formula for the Generic Offset to a Parametric Surface

We provide a resultant-based formula for the total degree w.r.t. the spatial variables of the generic offset to a parametric surface. The parametrization of the surface is not assumed to be proper.Comment: Preprint of an article to be published at the International Journal of Algebra and Computation, World Scientific Publishing, DOI:10.1142/S021819671100680

An Algebraic Analysis of Conchoids to Algebraic Curves

We study the conchoid to an algebraic affine plane curve C from the perspective of algebraic geometry, analyzing their main algebraic properties. Beside C, the notion of conchoid involves a point A in the affine plane (the focus) and a nonzero field element d (the distance).We introduce the formal definition of conchoid by means of incidence diagrams.We prove that the conchoid is a 1-dimensional algebraic set having atmost two irreducible components. Moreover, with the exception of circles centered at the focus A and taking d as its radius, all components of the corresponding conchoid have dimension 1. In addition, we introduce the notions of special and simple components of a conchoid. Furthermore we state that, with the exception of lines passing through A, the conchoid always has at least one simple component and that, for almost every distance, all the components of the conchoid are simple. We state that, in the reducible case, simple conchoid components are birationally equivalent to C, and we show how special components can be used to decide whether a given algebraic curve is the conchoid of another curve

Computing the Singularities of Rational Surfaces

Given a rational projective parametrization \cP(\ttt,\sss,\vvv) of a rational projective surface \cS we present an algorithm such that, with the exception of a finite set (maybe empty) \cB of projective base points of \cP, decomposes the projective parameter plane as \projdos\setminus \cB=\cup_{k=1}^{\ell} \cSm_k such that if (\ttt_0:\sss_0:\vvv_0)\in \cSm_k then \cP(\ttt_0,\sss_0,\vvv_0) is a point of \cS of multiplicity $k$.Comment: In this new version, we only have changed the thanks. In particular, we have written: This work was developed, and partially supported, under the research project MTM2008-04699-C03-01 "Variedades param\'etricas: algoritmos y aplicaciones", Ministerio de Ciencia e Innovaci\'on, Spain and by "Fondos Europeos de Desarrollo Regional" of the European Unio

r-norm bounds and metric properties for zero loci of real analytic functions

We consider the problem of deciding whether or not a zero locus, X, of multivariate real analytic functions crosses a given r-norm ball in the real n-dimensional affine space. We perform a local study of the problem, and we provide both necessary and sufficient conditions to answer the question. Our conditions derive from the analysis of differential geometric properties of X at the center of the ball. An algorithm to evaluate r-norms distances is proposed

SVMs for Automatic Speech Recognition: a Survey

Hidden Markov Models (HMMs) are, undoubtedly, the most employed core technique for Automatic Speech Recognition (ASR). Nevertheless, we are still far from achieving high-performance ASR systems. Some alternative approaches, most of them based on Artificial Neural Networks (ANNs), were proposed during the late eighties and early nineties. Some of them tackled the ASR problem using predictive ANNs, while others proposed hybrid HMM/ANN systems. However, despite some achievements, nowadays, the preponderance of Markov Models is a fact. During the last decade, however, a new tool appeared in the field of machine learning that has proved to be able to cope with hard classification problems in several fields of application: the Support Vector Machines (SVMs). The SVMs are effective discriminative classifiers with several outstanding characteristics, namely: their solution is that with maximum margin; they are capable to deal with samples of a very higher dimensionality; and their convergence to the minimum of the associated cost function is guaranteed. These characteristics have made SVMs very popular and successful. In this chapter we discuss their strengths and weakness in the ASR context and make a review of the current state-of-the-art techniques. We organize the contributions in two parts: isolated-word recognition and continuous speech recognition. Within the first part we review several techniques to produce the fixed-dimension vectors needed for original SVMs. Afterwards we explore more sophisticated techniques based on the use of kernels capable to deal with sequences of different length. Among them is the DTAK kernel, simple and effective, which rescues an old technique of speech recognition: Dynamic Time Warping (DTW). Within the second part, we describe some recent approaches to tackle more complex tasks like connected digit recognition or continuous speech recognition using SVMs. Finally we draw some conclusions and outline several ongoing lines of research

Espacios acoplados en la Mezquita-Catedral de Córdoba: el sonido de los límites

The Cathedral-Mosque of Córdoba represented, in its day, a remarkable form of intervention in a pre-existing building, being an important place of worship for islamic culture, which the conquering christian civilization wished to occupy. Intervention of this type usually consisted of a new construction which would guarantee ideological domination over the previous culture and wipe out all traces of it. However, in this case the moslem building underwent a metamorphosis to adapt it to the new requirements of the christian faith, but without losing its individual character. The horizontal, abstract space of the mosque was transformed by the insertion of two spaces which were clearly western and vertical: two cathedrals, one gothic and the other renaissance.These two cathedral spaces present welldefined typologies, the acoustic features of which have in some cases been studied and are generally known. However, in the Cathedral-Mosque of Córdoba, this behaviour differs from what would normally be expected of these typologies, with some special acoustic features being generated which are the object of the present study.It is the boundaries delimiting the different areas - virtual and non-material- which are responsible for the improved acoustics in each of the spaces. These limits behave, rather than as boundaries between two spaces, as energizing elements which are fed by the tension provoked by the differences between them.La Mezquita-Catedral de Córdoba supuso, en su momento, una forma de intervención singular sobre un edificio preexistente, siendo un lugar de culto destacable para la cultura islámica, que la civilización cristiana conquistadora quería ocupar. Usualmente esa intervención consistía en una nueva construcción que garantizase el dominio ideológico sobre la cultura precedente, borrando sus huellas. Sin embargo, en este caso se produce una metamorfosis del edificio musulmán para adecuarlo a las nuevas exigencias de la religión cristiana, pero sin perder su propio carácter. El espacio horizontal y abstracto de la mezquita se transforma mediante la inserción de dos espacios claramente occidentales y verticales: dos catedrales, una gótica y otra renacentista.Esos dos espacios catedralicios presentan tipologías bien definidas, cuyas características acústicas en algunos casos han sido estudiadas y son conocidas de modo general. Sin embargo, en la Mezquita-Catedral de Córdoba, este comportamiento difiere del que correspondería a esas tipologías, generándose unas conductas acústicas singulares que son objeto de estudio en este trabajo.Los límites que conforman los diferentes recintos —virtuales y no materiales— son los responsables de la mejora en el comportamiento acústico de cada uno de los espacios. Estos límites, más que fronteras entre dos espacios, se comportan como elementos energéticos que se nutren de la tensión provocada por las diferencias entre ellos