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A continuous map f : X → Y remains continuous if the topology on Y becomes coarser or the topology on X finer. An open (resp. closed) map f : X → Y remains open (resp. closed) if the topology on Y becomes finer or the topology on X coarser. One can also compare topologies using neighborhood bases.
A coarse structure on a set is a collection of subsets of (therefore falling under the more general categorization of binary relations on ) called controlled set s, and so that possesses the identity relation, is closed under taking subsets, inverses, and finite unions, and is closed under composition of relations. Explicitly:
This article deals with a component of numerical methods. For coarse space in topology, see coarse structure. In numerical analysis, coarse problem is an auxiliary system of equations used in an iterative method for the solution of a given larger system of equations. A coarse problem is basically a version of the same problem at a lower ...
In mathematics, coarse topology is a term in comparison of topologies which specifies the partial order relation of a topological structure to other one(s). Specifically, the coarsest topology may refer to: Initial topology, the most coarse topology in a certain category of topologies; Trivial topology, the most coarse topology possible on a ...
Gross anatomy (also called topographical anatomy, regional anatomy, or anthropotomy) is the study of anatomical structures that can be seen by the naked eye. [1] Microscopic anatomy is the study of minute anatomical structures assisted with microscopes , which includes histology (the study of the organization of tissues), [ 1 ] and cytology ...
The formula plays a decisive role in the modern study of isoperimetric problems. For smooth functions the formula is a result in multivariate calculus which follows from a change of variables. More general forms of the formula for Lipschitz functions were first established by Herbert Federer (Federer 1959), and for BV functions by Fleming ...
(More generally, coarse grid unknowns can be particular linear combinations of fine grid unknowns.) Thus, AMG methods become black-box solvers for certain classes of sparse matrices . AMG is regarded as advantageous mainly where geometric multigrid is too difficult to apply, [ 20 ] but is often used simply because it avoids the coding necessary ...
Granularity (also called graininess) is the degree to which a material or system is composed of distinguishable pieces, "granules" or "grains" (metaphorically). It can either refer to the extent to which a larger entity is subdivided, or the extent to which groups of smaller indistinguishable entities have joined together to become larger distinguishable entities.