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Property FW, differentiable structures and smoothability of singular actions
Related concepts (35)
Group action
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism group of the structure. It is said that the group acts on the space or structure. If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of Euclidean isometries acts on Euclidean space and also on the figures drawn in it.
Differential structure
In mathematics, an n-dimensional differential structure (or differentiable structure) on a set M makes M into an n-dimensional differential manifold, which is a topological manifold with some additional structure that allows for differential calculus on the manifold. If M is already a topological manifold, it is required that the new topology be identical to the existing one.
Countable set
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is countable if there exists an injective function from it into the natural numbers; this means that each element in the set may be associated to a unique natural number, or that the elements of the set can be counted one at a time, although the counting may never finish due to an infinite number of elements.
Local diffeomorphism
In mathematics, more specifically differential topology, a local diffeomorphism is intuitively a map between Smooth manifolds that preserves the local differentiable structure. The formal definition of a local diffeomorphism is given below. Let and be differentiable manifolds. A function is a local diffeomorphism, if for each point there exists an open set containing such that is open in and is a diffeomorphism.
Long line (topology)
In topology, the long line (or Alexandroff line) is a topological space somewhat similar to the real line, but in a certain way "longer". It behaves locally just like the real line, but has different large-scale properties (e.g., it is neither Lindelöf nor separable). Therefore, it serves as an important counterexample in topology. Intuitively, the usual real-number line consists of a countable number of line segments laid end-to-end, whereas the long line is constructed from an uncountable number of such segments.
Axiom of countable choice
The axiom of countable choice or axiom of denumerable choice, denoted ACω, is an axiom of set theory that states that every countable collection of non-empty sets must have a choice function. That is, given a function A with domain N (where N denotes the set of natural numbers) such that A(n) is a non-empty set for every n ∈ N, there exists a function f with domain N such that f(n) ∈ A(n) for every n ∈ N. The axiom of countable choice (ACω) is strictly weaker than the axiom of dependent choice (DC), which in turn is weaker than the axiom of choice (AC).
Property tax
A property tax (whose rate is expressed as a percentage or per mille, also called millage) is an ad valorem tax on the value of a property. The tax is levied by the governing authority of the jurisdiction in which the property is located. This can be a national government, a federated state, a county or other geographical region, or a municipality. Multiple jurisdictions may tax the same property. Often a property tax is levied on real estate. It may be imposed annually or at the time of a real estate transaction, such as in real estate transfer tax.
4-manifold
In mathematics, a 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different. There exist some topological 4-manifolds which admit no smooth structure, and even if there exists a smooth structure, it need not be unique (i.e. there are smooth 4-manifolds which are homeomorphic but not diffeomorphic).
Gamma ray
A gamma ray, also known as gamma radiation (symbol γ or ), is a penetrating form of electromagnetic radiation arising from the radioactive decay of atomic nuclei. It consists of the shortest wavelength electromagnetic waves, typically shorter than those of X-rays. With frequencies above 30 exahertz (3e19Hz), it imparts the highest photon energy. Paul Villard, a French chemist and physicist, discovered gamma radiation in 1900 while studying radiation emitted by radium.
Singular value
In mathematics, in particular functional analysis, the singular values, or s-numbers of a compact operator acting between Hilbert spaces and , are the square roots of the (necessarily non-negative) eigenvalues of the self-adjoint operator (where denotes the adjoint of ). The singular values are non-negative real numbers, usually listed in decreasing order (σ1(T), σ2(T), ...). The largest singular value σ1(T) is equal to the operator norm of T (see Min-max theorem).