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 ?

Applying the Multiple Scattering (MS) Method to Evaluate the Current Response on a Cable Harness Due to an Incident Plane Wave

In this paper, the MS method is applied to solve an immunity problem of harness-body co-simulation. In this immunity problem, the structure under study consists of two parts: the cable harness part (harness) and the metal surface part (body). The harness is solved using the generalized multiconductor transmission-line (GMTL) solver and the body is solved using the mixed-potential integral equation (MPIE) solver. The interactions between the harness and the body are achieved through a series of iterations called MS. The calculation results show good correlation with the reference results, which validates the accuracy of the MS method

A Generalized Multiple-Scattering Method for Modeling a Cable Harness with Ground Connections to a Nearby Metal Surface

This paper proposes a generalized multiple-scattering (GMS) method to evaluate the current distribution on a cable harness with ground connections to a nearby metal surface. The GMS method is a hybrid method combining the transmission line theory and the method of moments. The GMS method uses the generalized multiconductor transmission line (GMTL) solver for the cable harness part and the mixed-potential integral equation (MPIE) solver for the rest of the structure including the metal surface and the grounding wires. Neither the GMTL nor the MPIE solver alone takes into account the mutual interactions between the cable harness and the rest of the structure. Therefore, an iterative scheme is arranged in the GMS method to compensate the above-mentioned interactions. These interactions occur via not only field couplings, but also current conducting through the grounding points on the cable harness. A numerical test case is provided to benchmark the proposed GMS method

Evaluating Field Interactions Between Multiple Wires and the Nearby Surface Enabled by a Generalized MTL Approach

Interactions between a cable harness containing multiple wires and the nearby metal surface can be evaluated by full-wave methods. Though these methods can calculate the interactions with great accuracy, they have long simulation times and large memory requirements when dealing with complex wire structures. The multiple scattering (MS) approach by treating the cable harness and the surface separately using different algorithms has been proven to be superior to the full-wave methods when evaluating these interactions. However, the cable harness solver in the previous MS approach is restricted to two-wire structures since the per-unit-length (pul) inductance (L) and capacitance (C) are derived based on the antenna and differential modes assumption between two wires. In this paper, a generalized multiconductor transmission-line (GMTL) approach is proposed to overcome the two-wire limitation. In the GMTL approach, all wires take the infinity as the reference. The extraction of the pul L and C for the cable harness is not limited by the number of wires. Thus, the GMTL approach can conveniently model multiple wire structures. The application of the GMTL approach to the multiple wires enables the MS approach to accurately evaluate the interactions between the cable harness and the metal surface