Thesis (doctoral)--Freie Universität Berlin, 1979. / Bibliography: p. 205-216.

Inaug.-Diss.--Freiburg i.d. Schweiz. / "Separatabzug aus Band 65 des Geschichtsfreundes; Mitteilungen des historischen Vereins der fünf Orte Luzern, Uri, Schwyz, Unterwalden und Zug." Includes bibliographical references.

No description available.

A Steiner system T with parameters (5,6,12) is a collection of 6-element sets, called hexads, of a 12-element set [omega], such that any 5 of the 12 elements belong to exactly one hexad. In this project we construct a graph whose vertices are the orbits of S₁₂ on T x T, where T is the set of all Steiner systems S(5,6,12). Two vertices are joined if an orbit is taken into another under the action of a transposition. The number of hexads common to two Steiner systems are also given. We also prove that any two Steiner systems with parameters (5,6,12) can intersect only in 0, 12, 24, 36, or 60 hexads.

In 1992 Du and Hwang published a paper confirming the correctness of a well known 1968 conjecture of Gilbert and Pollak suggesting that the Euclidean Steiner ratio for the plane is 2/3. The original objective of this thesis was to adapt the technique used in this proof to obtain results for other Minkowski spaces. In an attempt to create a rigorous and complete version of the proof, some known results were given new proofs (results for hexagonal trees and for the rectilinear Steiner ratio) and some new results were obtained (on approximation of Steiner ratios and on transforming Steiner trees). The most surprising result, however, was the discovery of a fundamental gap in the proof of Du and Hwang. We give counter examples demonstrating that a statement made about inner spanning trees, which plays an important role in the proof, is not correct. There seems to be no simple way out of this dilemma, and whether the Gilbert-Pollak conjecture is true or not for any number of points seems once again to be an open question. Finally we consider the question of whether Du and Hwang's strategy can be used for cases where the number of points is restricted. After introducing some extra lemmas, we are able to show that the Gilbert-Pollak conjecture is true for 7 or fewer points. This is an improvement on the 1991 proof for 6 points of Rubinstein and Thomas. / Mathematical Sciences / Ph. D. (Mathematics)

Made available in DSpace on 2015-04-22T22:16:13Z (GMT). No. of bitstreams: 1 Alcides de Castro Amorim Neto.pdf: 1160215 bytes, checksum: 9d45c01bd11518c19b4c0577a41c7455 (MD5) Previous issue date: 2007-05-26 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / An old problem and of great application in the Applied Mathematics is known as problem of Steiner, with consists of the determination of a point that minimizes certain distances, problem this that was studied by famous mathematical as Fermat and Torricelli. A fundamental result in biochemistry and molecular modelling is the determination of the configurations of minimum energy (MECs) for structures such macromoleculares like proteins and DNA. The Steiner minimal trees (SMT) are seen as a useful algorithm paradigm to model these structures. In this work, we will examine how SMTs and the value of the ratio Steiner (½) compared with the MSTs are correlated with the energies MECs in a way physically significant. We verified that carbon and nitrogen atoms are Steiner points in the proteins minimal Steiner trees. / Um problema antigo e de grande aplicação na Matemática Aplicada é conhecido como problema de Steiner, que consiste na determinação de um ponto que minimize certas distâncias, problema este que foi estudado por outros matemáticos renomados como Fermat e Torricelli. Um dos resultados fundamentais em bioquímica e modelagem molecular é a determinação das Configurações de Energia Mínima (MECs) para estruturas macromoleculares tais como proteínas e DNA. As árvores mínimas de Steiner (SMTs) servem de base para elaboração de algoritmos úteis para modelar estas estruturas. Nesta dissertação, faremos uma revisão bibliográfica sobre o problema de Steiner e verificaremos, através de resultados da literatura, como as SMTs e o valor da razão de Steiner (½) comparado com as árvores geradoras mínimas MSTs estão correlacionadas com as MECs de uma maneira fisicamente significativa. Uma das observações relevantes é que os átomos de carbono e nitrogênio atuam como pontos de Steiner nas árvores mínimas de Steiner das proteínas.

In 1992 Du and Hwang published a paper confirming the correctness of a well known 1968 conjecture of Gilbert and Pollak suggesting that the Euclidean Steiner ratio for the plane is 2/3. The original objective of this thesis was to adapt the technique used in this proof to obtain results for other Minkowski spaces. In an attempt to create a rigorous and complete version of the proof, some known results were given new proofs (results for hexagonal trees and for the rectilinear Steiner ratio) and some new results were obtained (on approximation of Steiner ratios and on transforming Steiner trees). The most surprising result, however, was the discovery of a fundamental gap in the proof of Du and Hwang. We give counter examples demonstrating that a statement made about inner spanning trees, which plays an important role in the proof, is not correct. There seems to be no simple way out of this dilemma, and whether the Gilbert-Pollak conjecture is true or not for any number of points seems once again to be an open question. Finally we consider the question of whether Du and Hwang's strategy can be used for cases where the number of points is restricted. After introducing some extra lemmas, we are able to show that the Gilbert-Pollak conjecture is true for 7 or fewer points. This is an improvement on the 1991 proof for 6 points of Rubinstein and Thomas. / Mathematical Sciences / Ph. D. (Mathematics)

In this paper, we describe the facial structure of the steiner problem in a directed graph by formulating it as a set covering problem. We first characterize trivial facets and derive a necessary condition for nontrivial facets. We also introduce a class of valid inequalities with 0-1 coefficients and show when such inequalities define facets.

Diss.--Philos.--Oxford--University of Oxford. / Bibliogr. p. 205-226. Index.

Thesis (B.A.)--Haverford College, Dept. of Philosophy, 2004. / Includes bibliographical references.