Kolmogorov used this theorem to define several functions of strings, including complexity, randomness, and information.

Under suitable topological restrictions, a suitably " consistent " collection of finite-dimensional distributions can be used to define a stochastic process ( see Kolmogorov extension in the next section ).

In this case, the probability P ( Y = y ) = 0, and the Borel – Kolmogorov paradox demonstrates the ambiguity of attempting to define conditional probability along these lines.

As CISC became a catch-all term meaning anything that's not a load-store ( RISC ) architecture, it's not the number of instructions, nor the complexity of the implementation or of the instructions themselves, that define CISC, but the fact that arithmetic instructions also perform memory accesses.

This diversity of uses and meanings, combined with the complexity of the feelings involved, makes love unusually difficult to consistently define, compared to other emotional states.

In particular, files of random data cannot be consistently compressed by any conceivable lossless data compression algorithm: indeed, this result is used to define the concept of randomness in algorithmic complexity theory.

In negative theology, it is accepted that the Divine is ineffable, an abstract experience that can only be recognized or remembered — that is, human beings cannot describe in words the essence of the perfect good that is unique to the individual, nor can they define the Divine, in its immense complexity, related to the entire field of reality, and therefore all descriptions if attempted will be ultimately false and conceptualization should be avoided ; in effect, it eludes definition by definition:

In mathematical logic, the arithmetical hierarchy, arithmetic hierarchy or Kleene-Mostowski hierarchy classifies certain sets based on the complexity of formulas that define them.

His method of dealing with ecological complexity was to define an ideal form of vegetation -- the climax community -- and describe other forms of vegetation as deviations from that ideal.

The measure DSPACE is used to define complexity classes, sets of all of the decision problems which can be solved using a certain amount of memory space.

The resource DTIME is used to define complexity classes, sets of all of the decision problems which can be solved using a certain amount of computation time.

Any proper complexity function can be used to define a complexity class, but only certain classes are useful to study.

The following complexity classes are useful to define for ATMs:

Thus the number of quantifier alternations required to define a set gives a measure of the complexity of the set.

Depending on the transformation used this can be used to define complexity classes on a set of problems.

Because it is able to define languages of such complexity, it is also Turing-complete itself.

The Blum axioms can be used to define an abstract computational complexity theory on the set of computable functions.

This connection between complexity and the logic of finite structures allows results to be transferred easily from one area to the other, facilitating new proof methods and providing additional evidence that the main complexity classes are somehow " natural " and not tied to the specific abstract machines used to define them.

* Cohen and Manion ( 2000 ) define triangulation as an " attempt to map out, or explain more fully, the richness and complexity of human behavior by studying it from more than one standpoint.

His other research interests include algorithmic information theory, in which the concepts of information and randomness are developed in terms of the input / output relation of universal computers, and the analogous use of universal computers to define the intrinsic complexity or " logical depth " of a physical state as the time required by a universal computer to simulate the evolution of the state from a random initial state.

The term " social complexity " is thus not without problems, and qualifiers are typically applied by anthropologists and archaeologists when using this term to define more precisely the phenomenon that is being described as complex.

A BRMS or Business Rule Management System is a software system used to define, deploy, execute, monitor and maintain the variety and complexity of decision logic that is used by operational systems within an organization or enterprise.

How do we define it ''??

But first, we must define two terms so that their meaning will be clearly understood: form -- any unique sequence of alphabetic characters that can appear in a language preceded and followed by a space ; ;

For a single stage we may define Af where the maximization is by choice of Af.

In summary, while we stress that constructive engagement between anthropology and the military is possible, CEAUSSIC suggests that the AAA emphasize the incompatibility of HTS with disciplinary ethics and practice for job seekers and that it further recognize the problem of allowing HTS to define the meaning of “ anthropology ” within DoD.

To define angles in an abstract real inner product space, we replace the Euclidean dot product ( · ) by the inner product, i. e.

How best to define the term “ art ” is a subject of constant contention ; many books and journal articles have been published arguing over even the basics of what we mean by the term “ art ”.

Time efficiency estimates depend on what we define to be a step.

If we define r < sub > i </ sub > as the displacement of particle i from the center of mass, and v < sub > i </ sub > as the velocity of particle i with respect to the center of mass, then we have

If we define the function f ( n ) = A ( n, n ), which increases both m and n at the same time, we have a function of one variable that dwarfs every primitive recursive function, including very fast-growing functions such as the exponential function, the factorial function, multi-and superfactorial functions, and even functions defined using Knuth's up-arrow notation ( except when the indexed up-arrow is used ).

Some philosophers deny that the concept of " being " has any meaning at all, since we only define an object's existence by its relation to other objects, and actions it undertakes.

If X is a Banach space and K is the underlying field ( either the real or the complex numbers ), then K is itself a Banach space ( using the absolute value as norm ) and we can define the continuous dual space as X ′ = B ( X, K ), the space of continuous linear maps into K.

The real line R with its usual topology is a locally compact Hausdorff space, hence we can define a Borel measure on it.

For the case of a non-commutative base ring R and a right module M < sub > R </ sub > and a left module < sub > R </ sub > N, we can define a bilinear map, where T is an abelian group, such that for any n in N, is a group homomorphism, and for any m in M, is a group homomorphism too, and which also satisfies

Starting from any ket in this Hilbert space, we can define a complex scalar function of r, known as a wavefunction:

Christopher Hitchens was offended by the notion of Clinton as the first black president noting " we can still define blackness by the following symptoms: alcoholic mothers, under-the-bridge habits ... the tendency to sexual predation and shameless perjury about the same ".

" In a general way, we can define computing to mean any goal-oriented activity requiring, benefiting from, or creating computers.

In fact, what we have done is define a category of categories and functors – the objects are categories, and the morphisms ( between categories ) are functors.

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.

But how can we define the empty set without referring to elements, or the product topology without referring to open sets?

If we define tangent covectors in terms of equivalence classes of smooth maps vanishing at a point then the definition of the pullback is even more straightforward.

First we must also define where our view point is, that is, from what vantage point will the scene be drawn.

If S is an arbitrary set, then the set S < sup > N </ sup > of all sequences in S becomes a complete metric space if we define the distance between the sequences ( x < sub > n </ sub >) and ( y < sub > n </ sub >) to be, where N is the smallest index for which x < sub > N </ sub > is distinct from y < sub > N </ sub >, or 0 if there is no such index.