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A cover system is a video game gameplay mechanic that allows a virtual avatar to hide from and avoid dangers, usually in a three-dimensional world. This method is a digital adaptation of the real-life military tactic of taking cover behind obstacles, for purposes of attaining protection from enemy ranged or area effect attacks, such as gunfire or explosions.
In mathematics, particularly topology, an atlas is a concept used to describe a manifold. An atlas consists of individual charts that, roughly speaking, describe individual regions of the manifold. In general, the notion of atlas underlies the formal definition of a manifold and related structures such as vector bundles and other fiber bundles.
Let M be a compact smooth manifold, f: M → M a diffeomorphism, and Df: TM → TM the differential of f.An f-invariant subset Λ of M is said to be hyperbolic, or to have a hyperbolic structure, if the restriction to Λ of the tangent bundle of M admits a splitting into a Whitney sum of two Df-invariant subbundles, called the stable bundle and the unstable bundle and denoted E s and E u.
Introduction to Smooth Manifolds. Graduate Texts in Mathematics. Vol. 218 (Second ed.). New York London: Springer-Verlag. ISBN 978-1-4419-9981-8. OCLC 808682771. Introduction to Smooth Manifolds, Springer-Verlag, Graduate Texts in Mathematics, 2002, 2nd edition 2012 [6] Fredholm Operators and Einstein Metrics on Conformally Compact Manifolds.
This atlas contains every chart that is compatible with the smooth structure. There is a natural one-to-one correspondence between smooth structures and maximal smooth atlases. Thus, we may regard a smooth structure as a maximal smooth atlas and vice versa. In general, computations with the maximal atlas of a manifold are rather unwieldy.
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Let M be a smooth manifold. A (smooth) singular k-simplex in M is defined as a smooth map from the standard simplex in R k to M. The group C k (M, Z) of singular k-chains on M is defined to be the free abelian group on the set of singular k-simplices in M. These groups, together with the boundary map, ∂, define a chain complex.
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