There is no triangulation of the torus with vertex degrees 5, 6, ..., 6, 7 and related results: Geometric proofs for combinatorial theorems

There is no 5,7-triangulation of the torus, that is, no triangulation with exactly two exceptional vertices, of degree 5 and 7. Similarly, there is no 3,5-quadrangulation. The vertices of a 2,4-hexangulation of the torus cannot be bicolored. Similar statements hold for 4,8-triangulations and 2,6-quadrangulations. We prove these results, of which the first two are known and the others seem to be new, as corollaries of a theorem on the holonomy group of a euclidean cone metric on the torus with just two cone points. We provide two proofs of this theorem: One argument is metric in nature, the other relies on the induced conformal structure and proceeds by invoking the residue theorem. Similar methods can be used to prove a theorem of Dress on infinite triangulations of the plane with exactly two irregular vertices. The non-existence results for torus decompositions provide infinite families of graphs which cannot be embedded in the torus.Comment: 14 pages, 11 figures, only minor changes from first version, to appear in Geometriae Dedicat

Luca Pacioli and his 1500 book de Viribus Quantitatis

Tese de mestrado, História e Filosofia das Ciências, Universidade de Lisboa, Faculdade de Ciências, 2015As the field grows, History of Science has become wider-ranging than a purely progress-oriented view of the history of Science. The History of Mathematics, even though more resilient, has shown to follow the same development. The present dissertation tries to contribute to the general study by shedding some light on a book which has been belittled, misinterpreted or ignored altogether, De Viribus Quantitatis, one of the major historical recreational mathematics books, and its author Luca Pacioli. This text aims to provide a modern updated survey of the content of this book for related studies, as well as a résumé of its contents.Com o crescimento do ramo de História das Ciências este tem vindo a desenvolver um olhar mais abrangente que a clássica visão dedicada ao progresso das ideias científicas. A História da Matemática, embora mais resiliente, também tem vindo a mostrar interesse em expandir os seus horizontes. No presente texto tentamos contribuir para o estudo geral destas disciplinas estudando um tratado que pouca atenção tem tido até ao momento, sendo até mesmo mal interpretado. Trata-se De Viribus Quantitatis, sendo este um dos maiores compêndio de matemática recreativa no seu contexto histórico. O seu autor, Luca Pacioli, sendo uma personalidade de grande interesse e mais conhecido por outras obras suas. Nestas páginas tentamos fornecer uma versão atualisada da documentação relativa ao tratado tal como um resumo dos seus conteúdos

First principles studies of Si-C alloys

This study involves the investigation of silicon-carbon systems using ab initio techniques. It was motivated by the search for off-50:50 alloys and a way to quantify the strengths of 2D silicon-carbon materials. The study also predicts some under-reported properties for three previously proposed hypothetical allotropes of carbon. Preferably stable off-50:50 structures are identified from a set of trial structures for silicon-rich and carbon-rich candidates and their conditions of stability and physical properties are identified. A two-dimensional equation of state is introduced and applied to analyze the relative strengths of various 2D silicon-carbon materials. Of the possible off-50:50 alloy combinations and candidate structures considered, only the pyrite-FeS2, glitter-SiC2 and t-BC2 structures for SiC2 are elastically and dynamically stable. Analysis of the instability of Si2C reveals that it seems likely that carbon rich alloys are more favorable to their silicon-rich counterparts due to the smaller size of the carbon atoms and the more compact carbon-carbon bonds which result in less distorted bonding that is less metallic. The stiffness of the silicon dicarbide structures rank, in increasing order with 3C-SiC included for comparison, as glitter --> pyrite --> 3C-SiC --> t-SiC2. The moduli values for t-SiC2 are very comparable to 3C-SiC since for both materials, all atoms are four-fold coordinated with t-SiC2 having similar but slightly distorted, strong covalent tetrahedral bonding. The pyrite and glitter structures exhibit metallic character whereas t-SiC2 is a semi-conductor. Not only has this work demonstrated that, in principle, off-50:50 alloys of carbon and silicon are plausible, it has also provided information on how the strength and elastic properties of these materials are effected by increased silicon content. This has filled in a significant lack of knowledge about these bulk systems. For 2D systems, an equation of state is proposed that equates in-plane pressure with a change in surface area. It extracts the layer modulus as one of its fit parameters, which measures a material's resilience to hydrostatic stretching and predicts the material's intrinsic strength. Graphene is the most resilient to stretching with the highest intrinsic strength of all structures considered followed by SiC. Buckled Si is the least resilient with the lowest strength. An off-50:50 planar alloy, called silagraphene, differs elastically from SiC but has a comparable strength due to the similarity of their layer modulus. The novel 2D equation of state presented here opens up new ways to study and compare the strength properties of mono or multi-layered 2D materials, especially how their resilience to isotropic stretching responds to in-plane pressure.Thesis (PhD)--University of Pretoria, 2013.Physicsunrestricte

An Eberhard-like theorem for pentagons and heptagons

Eberhard proved that for every sequence (p k ), 3≤k≤r, k≠6, of nonnegative integers satisfying Euler’s formula ∑ k≥3(6−k)p k =12, there are infinitely many values p 6 such that there exists a simple convex polyhedron having precisely p k faces of size k for every k≥3, where p k =0 if k>r. In this paper we prove a similar statement when nonnegative integers p k are given for 3≤k≤r, except for k=5 and k=7 (but including p 6). We prove that there are infinitely many values p 5,p 7 such that there exists a simple convex polyhedron having precisely p k faces of size k for every k≥3. We derive an extension to arbitrary closed surfaces, yielding maps of arbitrarily high face-width. Our proof suggests a general method for obtaining results of this kind

4.Uluslararası Öğrenciler Fen Bilimleri Kongresi Bildiriler Kitabı

Çevrimiçi ( XIII, 495 Sayfa ; 26 cm.)