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Even though flt is a number theoretic result we prove that the result depends on the topological as well as the field structure of the underlying space.
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What values of the Standard Model hypercharges result in a mathematically consistent quantum field theory? We show that the constraints imposed by the lack of gauge anomalies can be recast as the equation x^3 + y^3 = z^3. If hypercharge is quantised, then x, y and z must be integers. The trivial (and only) solutions, with x=0 or y=0, reproduce the hypercharge assignments seen in Nature. This argument does not rely on the mixed gauge-gravitational anomaly, which is automatically vanishing if hypercharge is quantised and the gauge anomalies vanish.
Considering $mathbb{Z}_n$ the ring of integers modulo $n$, the classical Fermat-Euler theorem establishes the existence of a specific natural number $varphi(n)$ satisfying the following property: $ x^{varphi(n)}=1%hspace{1.0cm}text{for all}hspace{0.2cm}xin mathbb{Z}_n^*, $ for all $x$ belonging to the group of units of $mathbb{Z}_n$. In this manuscript, this result is extended to a class of rings that satisfies some mild conditions.
For the four-color theorem that has been developed over one and half centuries, all people believe it right but without complete proof convincing all1-3. Former proofs are to find the basic four-colorable patterns on a planar graph to reduce a map coloring4-6, but the unavoidable set is almost limitless and required recoloring hardly implements by hand7-14. Another idea belongs to formal proof limited to logical operation15. However, recoloring or formal proof way may block people from discovering the inherent essence of a coloring graph. Defining creation and annihilation operations, we show that four colors are sufficient to color a map and how to color it. We find what trapped vertices and boundary-vertices are, and how they decide how many colors to be required in coloring arbitrary maps. We reveal that there is the fourth color for new adding vertex differing from any three coloring vertices in creation operation. To implement a coloring map, we also demonstrate how to color an arbitrary map by iteratively using creation and annihilation operations. We hope our hand proof is beneficial to understand the mechanisms of the four-color theorem.
The purpose of these notes is to present a fairly complete proof of the classification Theorem for compact surfaces. Other presentations are often quite informal (see the references in Chapter V) and we have tried to be more rigorous. Our main source of inspiration is the beautiful book on Riemann Surfaces by Ahlfors and Sario. However, Ahlfors and Sarios presentation is very formal and quite compact. As a result, uninitiated readers will probably have a hard time reading this book. Our goal is to help the reader reach the top of the mountain and help him not to get lost or discouraged too early. This is not an easy task! We provide quite a bit of topological background material and the basic facts of algebraic topology needed for understanding how the proof goes, with more than an impressionistic feeling. We hope that these notes will be helpful to readers interested in geometry, and who still believe in the rewards of serious hiking!
For a given graph $G(V,E)$ and one of its dominating set $S$, the subgraph $Gleft[Sright]$ induced by $S$ is a called a dominating tree if $Gleft[Sright]$ is a tree. Not all graphs has a dominating tree, we will show that a graph without cut vertices has at least one dominating tree. Analogously, if $Gleft[Sright]$ is a forest, then it is called a dominating forest. As special structures of graphs, dominating tree and dominating forest have many interesting application, and we will focus on its application on the problem of planar graph coloring.
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