Instanton bundles on Fano threefolds

We introduce the notion of an instanton bundle on a Fano threefold of index 2. For such bundles we give an analogue of a monadic description and discuss the curve of jumping lines. The cases of threefolds of degree 5 and 4 are considered in a greater detail.Comment: 31 page, to appear in CEJ

Accuracy and Stability of Computing High-Order Derivatives of Analytic Functions by Cauchy Integrals

High-order derivatives of analytic functions are expressible as Cauchy integrals over circular contours, which can very effectively be approximated, e.g., by trapezoidal sums. Whereas analytically each radius r up to the radius of convergence is equal, numerical stability strongly depends on r. We give a comprehensive study of this effect; in particular we show that there is a unique radius that minimizes the loss of accuracy caused by round-off errors. For large classes of functions, though not for all, this radius actually gives about full accuracy; a remarkable fact that we explain by the theory of Hardy spaces, by the Wiman-Valiron and Levin-Pfluger theory of entire functions, and by the saddle-point method of asymptotic analysis. Many examples and non-trivial applications are discussed in detail.Comment: Version 4 has some references and a discussion of other quadrature rules added; 57 pages, 7 figures, 6 tables; to appear in Found. Comput. Mat

Areas y logaritmos (Lecciones populares de matemáticas), 2a. ed., 1981

Livro disponibilizado por aluno do curso de graduação de Matemática. Encontra-se parte de acerco pessoal. Florianópolis/SC.Exposição da teoria geométrica dos logaritmos quando os mesmos podem ser reconhecidos como certas áreas. El lector necesita tener conocimientos primarios sobre las funciones más simples y su representación gráfica, la progresión geométrica y la noción del límite. Los alumnos de los grados superiores de la secundaria ya conocen todo esto. Todas las propiedades de los logaritmos se deducen del análisis de las propiedades respectivas de las áreas. Junto con esto la conferencia proporciona las más simples nociones y propiedades del cálculo integral

Escaping points of entire functions of small growth

Let f be a transcendental entire function and let I(f) denote the set of points that escape to infinity under iteration. We give conditions which ensure that, for certain functions, I(f) is connected. In particular, we show that I(f) is connected if f has order zero and sufficiently small growth or has order less than 1/2 and regular growth. This shows that, for these functions, Eremenko’s conjecture that I(f) has no bounded components is true. We also give a new criterion related to I(f) which is sufficient to ensure that f has no unbounded Fatou components

Stability in an overdetermined problem for the Green's function

In the plane, we consider the problem of reconstructing a domain from the normal derivative of its Green's function (with fixed pole) relative to the Dirichlet problem for the Laplace operator. By means of the theory of conformal mappings, we derive stability estimates of Holder type