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Embeddings of infinite graphs in surfaces without boundary are considered. Cellular embeddings are studied in details. Each rotation system of a locally finite graph G gives rise to a cellular embedding of G, and every cellular embedding with all 2-cells of finite size can be obtained in this way. The graphs which admit cellular embeddings with all cells finite are characterized. **Real Uggs Uk Sale** The ends of a graph G and the ends of the surfaces, in which G has cellular embeddings, are shown to be closely related. Finally, the genus of infinite graphs is considered.
Explicit error bounds are obtained for the well-known asymptotic expansion of **Ugg Cheap Uk** integrals of the form ∫abe−λp(x)q(x)dx, in which λ is a large positive parameter, p(x) and q(x) are real differentiable functions, and p′(x) has a simple zero in the finite or infinite range [a, b]. The bounds are expressed in terms of the supremum of a certain function, taken over [a, b], and are asymptotic to the absolute value of the first neglected term in the expansion, as λ → ∞. Several illustrative examples are given, including modified Bessel functions and the gamma function.