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

Rational conchoid and offset constructions: algorithms and implementation

This paper is framed within the problem of analyzing the rationality of the components of two classical geometric constructions, namely the offset and the conchoid to an algebraic plane curve and, in the affirmative case, the actual computation of parametrizations. We recall some of the basic definitions and main properties on offsets (see [13]), and conchoids (see [15]) as well as the algorithms for parametrizing their rational components (see [1] and [16], respectively). Moreover, we implement the basic ideas creating two packages in the computer algebra system Maple to analyze the rationality of conchoids and offset curves, as well as the corresponding help pages. In addition, we present a brief atlas where the offset and conchoids of several algebraic plane curves are obtained, their rationality analyzed, and parametrizations are provided using the created packages

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

Dos nuevas especies y una subespecie de campodeidos cavernícolas de la cornisa cantábrica (Diplura, Campodeidae)

A total of 139 specimens of campodeid diplurans, collected from 27 caves of Cantabrian Cornice (Spain) are studied and two new species and one subspecies are described: Podocampa asturiana n. sp., Podocampa asturiana riberiensis n. ssp. and Litocampa zaldivarae n. sp. P. asturiana n. sp. differs from its most closely related species, P. group fragiloides (an endogean species) by troglomorphic characters; P. asturiana riberiensis n. ssp. is distinct from the type species by the number of trochanteral bacilliform sensilla; and L. zaldivarae n. sp. is easely recognized from Litocampa espanoli by the number of macrochaetae posterior lateral in IV urotergite. These new discoveries show the value and diversity of this family of apterygote insects in the Cantabrian subterranean environment.Se han estudiado un total de 139 ejemplares de zipluros campodeidos, recolectados en 27 grutas de la cornisa cantábrica y se han descrito dos nuevas especies y una subespecie: Podocampa asturiana sp. n., Podocampa asturiana riberiensis ssp. n. y Litocampa zaldivarae sp. n. P. asturiana sp. n. difiere de la especie más próxima, Podocampa grupo fragiloides (una forma de hábitat endógeo), por características relacionadas con su facies cavernícola; P. asturiana riberiensis ssp. n. difiere de la especie tipo por el número de sensilos baciliformes trocanterales; y L. zaldivarae sp. n. es fácilmente distinguible de Litocampa espanoli por el número de macroquetas laterales posteriores del IV uroterguito. Estos nuevos hallazgos indican la riqueza y diversidad de esta familia de insectos apterigotos en el medio subterráneo cantábrico

Upper Hessenberg and Toeplitz Bohemians

We also acknowledge the support of the Ontario Graduate Institution, The National Science & Engineering Research Council of Canada, the University of Alcala, the Rotman Institute of Philosophy, the Ontario Research Centre of Computer Algebra, and Western University. Part of this work was developed while R. M. Corless was visiting the University of Alcala, in the frame of the project Giner de los Rios. J.R. Sendra is member of the Research Group ASYNACS (Ref. CT-CE2019/683).A set of matrices with entries from a fixed finite population P is called “Bohemian”. The mnemonic comes from BOunded HEight Matrix of Integers, BOHEMI, and although the population P need not be solely made up of integers, it frequently is. In this paper we look at Bohemians, specifically those with population {−1,0,+1} and sometimes other populations, for instance {0,1,i,−1,−i}. More, we specialize the matrices to be upper Hessenberg Bohemian. We then study the characteristic polynomials of these matrices, and their height, that is the infinity norm of the vector of monomial basis coefficients. Focusing on only those matrices whose characteristic polynomials have maximal height allows us to explicitly identify these polynomials and give useful bounds on their height, and conjecture an accurate asymptotic formula. The lower bound for the maximal characteristic height is exponential in the order of the matrix; in contrast, the height of the matrices remains constant. We give theorems about the number of normal matrices and the number of stable matrices in these families.Agencia Estatal de Investigació

Modelo de supervivencia para los estadíos poslarvales del pejerrey Odontesthes bonariensis

La supervivencia de los peces adultos es representada por el clásico modelo exponencial. Sin embargo, el mismo no estima en forma apropiada la densidad de las etapas de larva y juvenil. Se propone un modelo que incorpora algunos parámetros y que ajusta mejor la supervivencia de la totalidad de estadíos por la que atraviesan los individuos hasta la etapa de adulto. Se asume que los individuos nacen con una “susceptibilidad”, que incrementa su mortalidad y que disminuye con la edad. La validez de este modelo fue contrastada mediante su ajuste a sucesivas cohortes de pejerrey obtenidas en la Laguna Chascomús, entre 1974 – 1977 y 1981 – 1982. Durante este período se reconoce un progresivo deterioro del sistema. Se asignó la edad mediante el análisis de la progresión modal de las distribuciones de talla. Fue posible discriminar 41 cohortes para el periodo de estudio. A los datos ordenados por clase de edad se les ajustó el modelo generalizado de crecimiento de von Bertalaffy, el modelo exponencial de supervivencia y el modelo propuesto a los rendimientos. El modelo solucionaría la discrepancia entre la fecundidad observada por recuento de gametas y la natalidad estimada (No) por modelo clásico

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

Encoding points on hyperelliptic curves over finite fields in deterministic polynomial time

We present families of (hyper)elliptic curve which admit an efficient deterministic encoding function