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Two metrics and on X are strongly or bilipschitz equivalent or uniformly equivalent if and only if there exist positive constants and such that, for every ,, (,) (,) (,).In contrast to the sufficient condition for topological equivalence listed above, strong equivalence requires that there is a single set of constants that holds for every pair of points in , rather than potentially different ...
An equivalence class of such metrics is known as a conformal metric or conformal class. Thus, a conformal metric may be regarded as a metric that is only defined "up to scale". Often conformal metrics are treated by selecting a metric in the conformal class, and applying only "conformally invariant" constructions to the chosen metric.
In mathematics, and specifically in measure theory, equivalence is a notion of two measures being qualitatively similar. Specifically, the two measures agree on which events have measure zero. Specifically, the two measures agree on which events have measure zero.
Measurement invariance or measurement equivalence is a statistical property of measurement that indicates that the same construct is being measured across some specified groups. [1] For example, measurement invariance can be used to study whether a given measure is interpreted in a conceptually similar manner by respondents representing ...
Evidently, conformality of metrics is an equivalence relation. Here are some formulas for conformal changes in tensors associated with the metric. (Quantities marked with a tilde will be associated with g ~ {\displaystyle {\tilde {g}}} , while those unmarked with such will be associated with g {\displaystyle g} .)
The equivalence between these two definitions can be seen as a particular case of the Monge–Kantorovich duality. From the two definitions above, it is clear that the total variation distance between probability measures is always between 0 and 2. To illustrate the meaning of the total variation distance, consider the following thought experiment.
Elementary equivalence; Equals sign; Equality (mathematics) Equality operator; Equipollence (geometry) Equivalence (measure theory) Equivalence class; Equivalence of categories; Equivalence of metrics; Equivalence relation; Equivalence test; Equivalent definitions of mathematical structures; Equivalent infinitesimal; Equivalent latitude ...
(This limit exists because the real numbers are complete.) This is only a pseudometric, not yet a metric, since two different Cauchy sequences may have the distance 0. But "having distance 0" is an equivalence relation on the set of all Cauchy sequences, and the set of equivalence classes is a metric space, the completion of M.