127,771 research outputs found

    Vertex Splitting and Upper Embeddable Graphs

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    The weak minor G of a graph G is the graph obtained from G by a sequence of edge-contraction operations on G. A weak-minor-closed family of upper embeddable graphs is a set G of upper embeddable graphs that for each graph G in G, every weak minor of G is also in G. Up to now, there are few results providing the necessary and sufficient conditions for characterizing upper embeddability of graphs. In this paper, we studied the relation between the vertex splitting operation and the upper embeddability of graphs; provided not only a necessary and sufficient condition for characterizing upper embeddability of graphs, but also a way to construct weak-minor-closed family of upper embeddable graphs from the bouquet of circles; extended a result in J: Graph Theory obtained by L. Nebesk{\P}y. In addition, the algorithm complex of determining the upper embeddability of a graph can be reduced much by the results obtained in this paper

    Tensor models and embedded Riemann surfaces

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    Tensor models and, more generally, group field theories are candidates for higher-dimensional quantum gravity, just as matrix models are in the 2d setting. With the recent advent of a 1/N-expansion for coloured tensor models, more focus has been given to the study of the topological aspects of their Feynman graphs. Crucial to the aforementioned analysis were certain subgraphs known as bubbles and jackets. We demonstrate in the 3d case that these graphs are generated by matrix models embedded inside the tensor theory. Moreover, we show that the jacket graphs represent (Heegaard) splitting surfaces for the triangulation dual to the Feynman graph. With this in hand, we are able to re-express the Boulatov model as a quantum field theory on these Riemann surfaces.Comment: 9 pages, 7 fi

    Three-Loop Radiative-Recoil Corrections to Hyperfine Splitting in Muonium

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    We calculate three-loop radiative-recoil corrections to hyperfine splitting in muonium generated by the diagrams with the first order electron and muon polarization loop insertions in graphs with two exchanged photons. These corrections are enhanced by the large logarithm of the electron-muon mass ratio. The leading logarithm squared contribution was obtained a long time ago. Here we calculate the single-logarithmic and nonlogarithmic contributions. We previously calculated the three-loop radiative-recoil corrections generated by two-loop polarization insertions in the exchanged photons. The current paper therefore concludes calculation of all three-loop radiative-recoil corrections to hyperfine splitting in muonium generated by diagrams with closed fermion loop insertions in the exchanged photons. The new results obtained here improve the theory of hyperfine splitting, and affect the value of the electron-muon mass ratio extracted from experimental data on the muonium hyperfine splitting.Comment: 27 pages, 6 figures, 7 table
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