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A local trivialisation for is equivalently given by a local section : and the connection one-form and curvature can be pulled back along this smooth map. This gives the local connection one-form A α = s α ∗ ν ∈ Ω 1 ( U α , ad ( P ) ) {\displaystyle A_{\alpha }=s_{\alpha }^{*}\nu \in \Omega ^{1}(U_{\alpha },\operatorname {ad} (P ...
Local connectedness is, by definition, a local property of topological spaces, i.e., a topological property P such that a space X possesses property P if and only if each point x in X admits a neighborhood base of sets that have property P. Accordingly, all the "metaproperties" held by a local property hold for local connectedness. In particular:
which vanishes if and only if Γ i kj is symmetric on its lower indices. Given a metric connection with torsion, one can always find a single, unique connection that is torsion-free, this is the Levi-Civita connection. The difference between a Riemannian connection and its associated Levi-Civita connection is the contorsion tensor.
In mathematics, a local system (or a system of local coefficients) on a topological space X is a tool from algebraic topology which interpolates between cohomology with coefficients in a fixed abelian group A, and general sheaf cohomology in which coefficients vary from point to point. Local coefficient systems were introduced by Norman ...
An adequately defined interface will allow one team to test its implementation of the interface by simulating the opposing side with a simple communications simulator. Not knowing the business logic of the system on the far side of an interface makes it more likely that one will develop a system that does not break when the other system changes ...
Within object-oriented design, interfaces provide layers of abstraction that simplify code and create a barrier preventing coupling to dependencies.A system may become so coupled at multiple levels that it is no longer possible to make a change in one place without necessitating many additional changes. [1]
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Example of a high-level systems architecture for a computer. A system architecture is the conceptual model that defines the structure, behavior, and views of a system. [1] An architecture description is a formal description and representation of a system, organized in a way that supports reasoning about the structures and behaviors of the system.