Conjuntos excepcionais e alguns problemas de Mahler

Dissertação (mestrado)—Universidade de Brasília, Instituto de Ciências Exatas, Departamento de Matemática, 2017.Seja f uma função inteira e transcendente. Denotamos por Sf o conjunto de todos os α ∈ ´Q tais que f(α) ∈ ´Q (o conjunto excepcional de f). Nessa dissertação, mostraremos quais subconjuntos de ´Q podem ser o conjunto excepcional de alguma função inteira e transcendente. Além disso, trataremos de dois problemas de Mahler relacionados a propriedades de funções inteiras e transcendentes. Mostraremos que existem funções inteiras e transcendentes que levam um subconjunto dos números de Liouville nele mesmo e daremos uma resposta positiva ao Problema B de Mahler: Problema B: Existe uma função inteira e transcendente f(z) = Σn =0 ∞ a nz n com coeficientes racionais tal que f( ´Q ) ⊆ ´Q e f−1( ´Q ) ⊆ ´Q ? .Let f be an entire transcendental function. We denote by Sf the set of all α ∈ ´Q such that f(α) ∈ ´Q (exceptional set of f). Throughout this dissertation, we will show which subsets of ´Q can be the exceptional set of some entire transcendental function. Moreover, we will deal with two of Mahler’s problems related to properties of entire transcendental functions. We will show that there are entire transcendental functions that map a subset of Liouville numbers in itself and we will give a positive answer for Mahler’s Problem B: Problem B: Is there an entire transcendental function f(z) = Σn =0 ∞ a nz n with rational coefficients such that que f( ´Q ) ⊆ ´Q e f−1( ´Q ) ⊆ ´Q ?

Segmental paleotetraploidy revealed in sterlet (Acipenser ruthenus) genome by chromosome painting

BACKGROUND: Acipenseriformes take a basal position among Actinopteri and demonstrate a striking ploidy variation among species. The sterlet (Acipenser ruthenus, Linnaeus, 1758; ARUT) is a diploid 120-chromosomal sturgeon distributed in Eurasian rivers from Danube to Enisey. Despite a high commercial value and a rapid population decline in the wild, many genomic characteristics of sterlet (as well as many other sturgeon species) have not been studied. RESULTS: Cell lines from different tissues of 12 sterlet specimens from Siberian populations were established following an optimized protocol. Conventional cytogenetic studies supplemented with molecular cytogenetic investigations on obtained fibroblast cell lines allowed a detailed description of sterlet karyotype and a precise localization of 18S/28S and 5S ribosomal clusters. Localization of sturgeon specific HindIII repetitive elements revealed an increased concentration in the pericentromeric region of the acrocentric ARUT14, while the total sterlet repetitive DNA fraction (C(0)t30) produced bright signals on subtelomeric segments of small chromosomal elements. Chromosome and region specific probes ARUT1p, 5, 6, 7, 8 as well as 14 anonymous small sized chromosomes (probes A-N) generated by microdissection were applied in chromosome painting experiments. According to hybridization patterns all painting probes were classified into two major groups: the first group (ARUT5, 6, 8 as well as microchromosome specific probes C, E, F, G, H, and I) painted only a single region each on sterlet metaphases, while probes of the second group (ARUT1p, 7 as well as microchromosome derived probes A, B, D, J, K, M, and N) marked two genomic segments each on different chromosomes. Similar results were obtained on male and female metaphases. CONCLUSIONS: The sterlet genome represents a complex mosaic structure and consists of diploid and tetraploid chromosome segments. This may be regarded as a transition stage from paleotetraploid (functional diploid) to diploid genome condition. Molecular cytogenetic and genomic studies of other 120- and 240-chromosomal sturgeons are needed to reconstruct genome evolution of this vertebrate group