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The difference between uniform continuity and (ordinary) continuity is that, in uniform continuity there is a globally applicable (the size of a function domain interval over which function value differences are less than ) that depends on only , while in (ordinary) continuity there is a locally applicable that depends on both and . So uniform ...
Hutton's Unconformity at Jedburgh. Above: John Clerk of Eldin's 1787 illustration. Below: 2003 photograph. Uniformitarianism, also known as the Doctrine of Uniformity or the Uniformitarian Principle, [1] is the assumption that the same natural laws and processes that operate in our present-day scientific observations have always operated in the universe in the past and apply everywhere in the ...
A nonempty collection of subsets of is a uniform structure (or a uniformity) if it satisfies the following axioms: If U ∈ Φ {\displaystyle U\in \Phi } then Δ ⊆ U , {\displaystyle \Delta \subseteq U,} where Δ = { ( x , x ) : x ∈ X } {\displaystyle \Delta =\{(x,x):x\in X\}} is the diagonal on X × X . {\displaystyle X\times X.}
In the general theory of uniform spaces, a uniform space is called a complete uniform space if each Cauchy filter on converges to some point of in the topology induced by the uniformity. When X {\displaystyle X} is a TVS, the topology induced by the canonical uniformity is equal to X {\displaystyle X} 's given topology (so convergence in this ...
For example, in order theory, an order-preserving function : between particular types of partially ordered sets and is continuous if for each directed subset of , we have () = (). Here sup {\displaystyle \,\sup \,} is the supremum with respect to the orderings in X {\displaystyle X} and Y , {\displaystyle Y,} respectively.
There is another notion of uniformity, slightly different than uniform integrability, which also has many applications in probability and measure theory, and which does not require random variables to have a finite integral [8] Definition: Suppose (,,) is a probability space.
In the theory of differential equations, Lipschitz continuity is the central condition of the Picard–Lindelöf theorem which guarantees the existence and uniqueness of the solution to an initial value problem. A special type of Lipschitz continuity, called contraction, is used in the Banach fixed-point theorem. [2]
Topological vector spaces are prominent examples of topological groups and every topological group has an associated canonical uniformity. Definition: [7] A family H of maps from T into Y is said to be equicontinuous at t ∈ T if for every neighborhood V of 0 in Y, there exists some neighborhood U of t in T such that h(U) ⊆ h(t) + V for ...