منابع مشابه
Proof of Poincare ' S Geometric Theorem
In a paper recently published in the Rendiconti del Circolo Matemático di Palermo (vol. 33, 1912, pp. 375-407) and entitled Sur un théorème de Géométrie, Poincaré enunciated a theorem of great importance, in particular for the restricted problem of three bodies; but, having only succeeded in treating a variety of special cases after long-continued efforts, he gave out the theorem for the consid...
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Mapping degree, intersection number, and the index of a zero of a vector field are defined. The Poincare-Hopf theorem, which states that under reasonable conditions the sum of the indices of a vector field equals the Euler characteristic of the manifold, is proven. Some consequences are discussed.
متن کاملA graph theoretical Poincare-Hopf Theorem
We introduce the index i(v) = 1− χ(S−(v)) for critical points of a locally injective function f on the vertex set V of a simple graph G = (V,E). Here S−(v) = {w ∈ S(v) | f(w) − f(v) < 0 } is the subgraph of the unit sphere S(v) generated by the vertices {w ∈ V | (v, w) ∈ E }. We prove that the sum ∑ v∈V i(v) of the indices always is equal to the Euler characteristic χ(G) of the graph G. This is...
متن کاملFermat’s Last Theorem
Fermat learned his number theory from the books of Diophantus; it was in the margins of his copy that he wrote down that he had discovered a truly marvelous proof of the fact that X + Y n = Z has no solutions in natural numbers for n > 2, and that the margin was too small to contain it. The books of Diophantus, including these remarks, were published posthumously by his son. By the early 1800s,...
متن کاملFermat's Last Theorem
The authors would like to give special thanks to N. Boston, K. Buzzard, and B. Conrad for providing so much valuable feedback on earlier versions of this paper. They are also grateful to A. Agboola, M. Bertolini, B. Edixhoven, J. Fearnley, R. Gross, L. Guo, F. Jarvis, H. Kisilevsky, E. Liverance, J. Manoharmayum, K. Ribet, D. Rohrlich, M. Rosen, R. Schoof, J.-P. Serre, C. Skinner, D. Thakur, J....
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ژورنال
عنوان ژورنال: The Astronomical Journal
سال: 1964
ISSN: 0004-6256
DOI: 10.1086/109426