Hierarchical designs can be recursively " exploded " (" flattened ") by creating a new copy ( with a new name ) of each definition each time it is used.

This definition can be applied recursively to the-tuple:

This definition can be applied recursively:

These models enable some nonrecursive sets of numbers or languages ( including all recursively enumerable sets of languages ) to be " learned in the limit "; whereas, by definition, only recursive sets of numbers or languages could be identified by a Turing machine.

( The definition of the coproduct is extended recursively by the rule ).

According to the Church-Turing thesis, any effectively calculable function is calculable by a Turing machine, and thus a set S is recursively enumerable if and only if there is some algorithm which yields an enumeration of S. This cannot be taken as a formal definition, however, because the Church-Turing thesis is an informal conjecture rather than a formal axiom.

Other texts use the definition in terms of enumerations, which is equivalent for recursively enumerable sets.

While there is no generally accepted formal definition of " algorithm ," an informal definition could be " a set of rules that precisely defines a sequence of operations.

Its domain is the powerset of A ( with the empty set removed ), and so makes sense for any set A, whereas with the definition used elsewhere in this article, the domain of a choice function on a collection of sets is that collection, and so only makes sense for sets of sets.

Simply storing the 24-bit color of each pixel in this image would require 1. 62 million bits, but a small computer program can reproduce these 1. 62 million bits using the definition of the Mandelbrot set and the coordinates of the corners of the image.

Also running on a JOHANNIAC, the Logic Theory Machine constructed proofs from a small set of propositional axioms and three deduction rules: modus ponens, ( propositional ) variable substitution, and the replacement of formulas by their definition.

For a rigorous definition of basis with a continuous set of indices and consequently for a rigorous definition of position and momentum basis see.

Unfortunately in the literature the definition is given in two variants: Despite the fact that Bernoulli defined B < sub > 1 </ sub > = 1 / 2 ( now known as " second Bernoulli numbers "), some authors set B < sub > 1 </ sub > = − 1 / 2 (" first Bernoulli numbers ").

The basic idea of his proof is that a proposition that holds of x if x = n for some natural number n can be called a definition for n, and that the set

These categories surely have some objects that are " special " in a certain way, such as the empty set or the product of two topologies, yet in the definition of a category, objects are considered to be atomic, i. e., we do not know whether an object A is a set, a topology, or any other abstract concept – hence, the challenge is to define special objects without referring to the internal structure of those objects.

By definition a set S is countable if there exists an injective function

In practice, the above definition is rarely used because in virtually all cases, the curl operator can be applied using some set of curvilinear coordinates, for which simpler representations have been derived.

This definition of cofinality relies on the axiom of choice, as it uses the fact that every non-empty set of cardinal numbers has a least member.

A definition is a passage that explains the meaning of a term ( a word, phrase, or other set of symbols ), or a type of thing.

A precising definition extends the descriptive dictionary definition ( lexical definition ) of a term for a specific purpose by including additional criteria that narrow down the set of things meeting the definition.

An intensional definition, also called a coactive definition, specifies the necessary and sufficient conditions for a thing being a member of a specific set.

Any definition that attempts to set out the essence of something, such as that by genus and differentia, is an intensional definition.

# A definition must set out the essential attributes of the thing defined.

Some authors also require the domain of the Euclidean function be the entire ring R ; this can always be accommodated by adding 1 to the values at all nonzero elements, and defining the function to be 0 at the zero element of R, but the result is somewhat awkward in the case of K. The definition is sometimes generalized by allowing the Euclidean function to take its values in any well-ordered set ; this weakening does not affect the most important implications of the Euclidean property.

If R is an integral domain then any two gcd's of a and b must be associate elements, since by definition either one must divide the other ; indeed if a gcd exists, any one of its associates is a gcd as well.

Most of these languages only define an upper ontology with generic concepts, whereas the domain concepts are not part of the language definition.

The William and Flora Hewlett Foundation definition describes open educational resources as either " resid in the public domain or [...] released under an intellectual property license that permits their free use or re-purposing by others.

The combination of the language definition, a program, and the program's inputs must fully specify the external behavior that occurs when the program is executed, within the domain of control of that program.

Coherence is the quality of a wave to display well defined phase relationship in different regions of its domain of definition.

The previous three statements give the definition of a Dedekind domain, and hence every principal ideal domain is a Dedekind domain.

The Riemann integral generalises to the improper Riemann integral to measure functions whose domain of definition is not a closed interval.

An ontology renders shared vocabulary and taxonomy which models a domain with the definition of objects and / or concepts and their properties and relations.

The creation of domain ontologies is also fundamental to the definition and use of an enterprise architecture framework.

In mathematics, the domain of definition or simply the domain of a function is the set of " input " or argument values for which the function is defined.

The definition of Julia and Fatou sets easily carries over to the case of certain maps whose image contains their domain ; most notably transcendental meromorphic functions and Adam Epstein's finite-type maps.

By definition, switches may not bridge IP traffic between VLANs as it would violate the integrity of the VLAN broadcast domain.

However, the provinces may use the " notwithstanding clause " only on legislation that they otherwise have the authority to enact, and the Supreme Court ruled in Reference re Same-Sex Marriage that the definition of marriage is within the exclusive domain of the Canadian Parliament.

An immediate consequence of the definition is that every principal ideal domain ( PID ) is a Dedekind domain.

In fact, this is the definition of a Dedekind domain used in Bourbaki's " Commutative algebra ".

An ontology renders shared vocabulary and taxonomy which models a domain with the definition of objects and / or concepts and their properties and relations.

The creation of domain ontologies is also fundamental to the definition and use of an enterprise architecture framework.

The above definition is common in many applications today, and prominent in lattice and domain theory.