An important result on the topology of R < sup > n </ sup >, that is far from superficial, is Brouwer's invariance of domain.

An important consequence of the domain invariance theorem is that R < sup > n </ sup > cannot be homeomorphic to R < sup > m </ sup > if m ≠ n. Indeed, no non-empty open subset of R < sup > n </ sup > can be homeomorphic to any open subset of R < sup > m </ sup > in this case.

The initial result of this kind was proved by Emanuel Sperner, in relation with proofs of invariance of domain.

Detecting local maxima over scales of normalized derivative responses provides a general framework for obtaining scale invariance in this domain.

In linear algebra, similarity invariance is a property exhibited by a function whose value is unchanged under similarities of its domain.

Specifically, invariance ( or more appropriately covariance ) to local geometric transformations, such as rotations or local affine transformations, can be obtained by considering differential invariants under the appropriate class of transformations or alternatively by normalizing the Gaussian derivative operators to a locally determined coordinate frame determined from e. g. a preferred orientation in the image domain or by applying a preferred local affine transformation to a local image patch ( see the article on affine shape adaptation for further details ).

The dimension of a manifold is a topological property, meaning that any manifold homeomorphic to an n-manifold also has dimension n. It follows from invariance of domain that an n-manifold cannot be homeomorphic to an m-manifold for n ≠ m.

It follows from invariance of domain that Euclidean neighborhoods are always open sets.

Owing respectively to Green's theorem and the divergence theorem, such a field is necessarily conserved and free from sources or sinks, having net flux equal to zero through any open domain.

Then Goursat's theorem asserts that ƒ is analytic in an open complex domain Ω if and only if it satisfies the Cauchy – Riemann equation in the domain.

A sufficient condition for recovering s ( t ) ( and therefore S ( ƒ )) from just these samples is that the non-zero portion of s ( t ) be confined to a known interval of duration P, which is the frequency domain dual of the Nyquist – Shannon sampling theorem.

In mathematical analysis, the intermediate value theorem states that for each value between the least upper bound and greatest lower bound of the image of a continuous function there is at least one point in its domain that the function maps to that value.

A fundamental theorem states that every spline function of a given degree, smoothness, and domain partition, can be uniquely represented as a linear combination of B-splines of that same degree and smoothness, and over that same partition.

The key result is the structure theorem: If R is a principal ideal domain, and M is a finitely

This is similar ( but more categorical ) to concepts in group theory or module theory, where a given decomposition of an object into a direct sum is " not natural ", or rather " not unique ", as automorphisms exist that do not preserve the direct sum decomposition – see Structure theorem for finitely generated modules over a principal ideal domain # Uniqueness for example.

* Invariance of domain, a theorem in topology about homeomorphic subsets of Euclidean space

Note that there exists an instance of Bayes ' theorem for each point in the Domain of a function | domain.

In mathematics, a unique factorization domain ( UFD ) is a commutative ring in which every non-unit element, with special exceptions, can be uniquely written as a product of prime elements ( or irreducible elements ), analogous to the fundamental theorem of arithmetic for the integers.

There are social choice theory definitions of seemingly reasonable criteria that are a measure of the fairness of certain aspects of voting, including non-dictatorship, unrestricted domain, non-imposition, Pareto efficiency, and independence of irrelevant alternatives but Arrow's impossibility theorem states that no voting system can meet all these standards.

A remarkable theorem of L. Claborn ( Claborn 1966 ) asserts that for any abelian group G whatsoever, there exists a Dedekind domain R whose ideal class group is isomorphic to G. Later, C. R.

In view of the well known and exceedingly useful structure theorem for finitely generated modules over a principal ideal domain ( PID ), it is natural to ask for a corresponding theory for finitely generated modules over a Dedekind domain.

* By the Riesz representation theorem for linear functionals, if x is in the domain of A *, there is a unique vector z in H such that

This theorem can be extended to the time-dependent domain to develop time-dependent density functional theory ( TDDFT ), which can be used to describe excited states.

Invariance of domain is a theorem in topology about homeomorphic subsets of Euclidean space R < sup > n </ sup >.

Normally, to check that f is a homeomorphism, one would have to verify that both f and its inverse function f < sup >-1 </ sup > are continuous ; the theorem says that if the domain is an open subset of R < sup > n </ sup > and the image is also in R < sup > n </ sup >, then continuity of f < sup >-1 </ sup > is automatic.

Schaefer's dichotomy theorem handles the case when all the available relations are boolean operators, that is, for domain size 2.

In a deeper way, this theorem relates the topology of the domain of integration to the structure of the differential forms themselves ; the precise connection is known as De Rham's theorem.

In addition, the size distribution obtained from visual and radar observations of meteors may be extrapolated to the micrometeorite domain.

Beyond this, philosophy may urge the social sciences forward by asking the type of question that falls outside the present scope of social inquiry, but within its potential domain of relevance.

As audio signals may be electronically represented in either digital or analog format, signal processing may occur in either domain.

If the domain of F is a disjoint union of two or more intervals, then a different constant of integration may be chosen for each of the intervals.

It follows that the proof that follows may be adapted for any Euclidean domain.

Once the term of a copyright has expired, the formerly copyrighted work enters the public domain and may be freely used or exploited by anyone.

Most systems may be assumed to have a second order and single variable system response in the time domain ignoring multi-variable.

Students of control engineering may start with a linear control system course dealing with the time and complex-s domain, which requires a thorough background in elementary mathematics and Laplace transform ( called classical control theory ).

may consist of only one domain, or may consist of many domains and sub-domains, depending on the administrative authority delegated to the manager.

The full domain name may not exceed a total length of 253 characters in its external dotted-label specification.

In practice, some domain registries may have shorter limits.

Multiple domain names may be associated with an IP address.

For example, a scientist may find herself attempting to counter-argue philosophical skeptics, when Harris says she should be practically asking – as scientists would in any other domain – " why would we listen to a solipsist in the first place?

Expert witnesses may also deliver expert evidence about facts from the domain of their expertise.

As a ring, a field may be classified as a specific type of integral domain, and can be characterized by the following ( not exhaustive ) chain of class inclusions:

Depending on the problem domain some layouts may be better suited and easier to understand than others.

Of course, when a simple function is used to estimate data points from the original, interpolation errors are usually present ; however, depending on the problem domain and the interpolation method used, the gain in simplicity may be of greater value than the resultant loss in accuracy.

In international law interpretation is within the domain of the protagonists, but may also be conferred on judicial bodies such as the International Court of Justice, by the terms of the treaties or by consent of the parties.

The function ƒ ( x ) = x < sup > 2 </ sup > may or may not be invertible, depending on the domain.

Internet humor may also be regarded as humor that specifically relies on characteristics belonging to the Internet, such as the domain name slurls or the " geek " or " hacker " humor ( i. e., humor that would not exist if not for the Internet ), some of which can be considered ironic.

This domain may be involved in regulating the activity of JH1, and was likely a duplication of the JH1 domain which has undergone mutation post-duplication.