Formally, a deterministic Büchi automaton is a tuple A = ( Q, Σ, δ, q < sub > 0 </ sub >, F ) that consists of the following components:

Formally, a deterministic finite automaton may be defined by the tuple where is the set of states of the automaton, is the set of input symbols, is the transition function that takes a state and an input symbol to a new state, is the initial state of the automaton, and is the set of accepting or final states of the automaton.

Formally the algorithm can be described as:

Formally, there is a clear distinction: " DFT " refers to a mathematical transformation or function, regardless of how it is computed, whereas " FFT " refers to a specific family of algorithms for computing DFTs.

Formally, dependence refers to any situation in which random variables do not satisfy a mathematical condition of probabilistic independence.

Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I (" On Formally Undecidable Propositions of Principia Mathematica and Related Systems I ") is a paper in mathematical logic by Kurt Gödel.

Formally, consider an economic model with different mathematical weightings placed on the utilities of each self.

Formally, the derivative of the function f at a is the limit

Formally, if M is a set, the identity function f on M is defined to be that function with domain and codomain M which satisfies

Formally a random variable is considered to be a function on the possible outcomes.

Formally, the discrete cosine transform is a linear, invertible function ( where denotes the set of real numbers ), or equivalently an invertible N × N square matrix.

Formally, we are given a set of hypotheses and a set of manifestations ; they are related by the domain knowledge, represented by a function that takes as an argument a set of hypotheses and gives as a result the corresponding set of manifestations.

Formally, a function ƒ is real analytic on an open set D in the real line if for any x < sub > 0 </ sub > in D one can write

Formally, an elliptic function is a function meromorphic on for which there exist two non-zero complex numbers and with ( in other words, not parallel ), such that and for all.

Formally, if is an open subset of the complex plane, a point of, and is a holomorphic function, then is called a removable singularity for if there exists a holomorphic function which coincides with on.

Formally, the problem of supervised pattern recognition can be stated as follows: Given an unknown function ( the ground truth ) that maps input instances to output labels, along with training data assumed to represent accurate examples of the mapping, produce a function that approximates as closely as possible the correct mapping.

Formally, a statistic s is a measurable function of X ; thus, a statistic s is evaluated on a random variable X, taking the value s ( X ), which is itself a random variable.

Formally, the discrete sine transform is a linear, invertible function F: R < sup > N </ sup > < tt >-></ tt > R < sup > N </ sup > ( where R denotes the set of real numbers ), or equivalently an N × N square matrix.

Formally, the discrete Hartley transform is a linear, invertible function H: R < sup > n </ sup > < tt >-></ tt > R < sup > n </ sup > ( where R denotes the set of real numbers ).

Formally, the integral is the inner product of the luminosity function with the light spectrum.

Formally, a cardinal κ is defined to be weakly compact if it is uncountable and for every function f: < sup > 2 </ sup > →

Formally, an analytic function ƒ ( z ) of the real or complex variables z < sub > 1 </ sub >,…, z < sub > n </ sub > is transcendental if z < sub > 1 </ sub >, …, z < sub > n </ sub >, ƒ ( z ) are algebraically independent, i. e., if ƒ is transcendental over the field C ( z < sub > 1 </ sub >, …, z < sub > n </ sub >).

Formally, the Cantor function c: → is defined as follows:

Formally, this means that we want a function to be monotonic.

Formally, this means that, for some function f, the image f ( D ) of a directed set D ( i. e. the set of the images of each element of D ) is again directed and has as a least upper bound the image of the least upper bound of D. One could also say that f preserves directed suprema.

Formally, let be a stochastic process and let represent the cumulative distribution function of the joint distribution of at times.

Formally, an ultrametric space is a set of points with an associated distance function ( also called a metric )

* Formally, when working over the reals, as here, this is accomplished by considering the limit as ε → 0 ; but the " infinitesimal " language generalizes directly to Lie groups over general rings.

Formally, these failed when they were rejected by the Church of England's General Synod in 1972 ; conversations and co-operation continued, however, leading in 2003 to the signing of a covenant between the two churches.

Formally, the case where only a subset of parameters is defined is still a composite hypothesis ; nonetheless, the term point hypothesis is often applied in such cases, particularly where the hypothesis test can be structured in such a way that the distribution of the test statistic ( the distribution under the null hypothesis ) does not depend on the parameters whose values have not been specified under the point null hypothesis.

; Electronics: Formally a fixture in radio shacks, now used for assembly and repair of all sorts of electronic equipment including communication, computer, and home entertainment items.

; Formally real field

Formally, Aff ( V ) is naturally isomorphic to a subgroup of, with V embedded as the affine plane, namely the stabilizer of this affine plane ; the above matrix formulation is the ( transpose of ) the realization of this, with the ( n × n and 1 × 1 ) blocks corresponding to the direct sum decomposition.

Formally, HPGCC is a cross-compiler ; it compiles code for the ARM-based HP calculators, but runs on a PC rather than the target system.

Formally, Korea's governmental FTC ( Fair Trade Commission: 공정거래위원회 ; 公正去來委員會 ) announces a limited Chaebol list every year as size of industrial assets ( not including financial companies ).

Formally the euro is the only legal tender in Büsingen ; however, the Swiss franc is far more popular in everyday use as shopping is generally done in nearby Schaffhausen instead of across the border ( s ) in Germany.

Formally, the algorithm's performance will be a random variable determined by the random bits ; thus either the running time, or the output ( or both ) are random variables.

Formally, Chrétien and Martin have remained publicly respectful of each other ; while much of the verbal sparring was between their supporters.

Formally, a squeeze preserves the hyperbolic metric expressed in the form ; in a different coordinate system.

Bābak Khorram-Din ( Formally known as " Pāpak " meaning " Young Father ") (, alternative spelling: Bābak Xorramdin ; 795, according to some other sources 798 — January 838 ) was one of the main PersianArmenian historian Vardan Areweltsʻi, ca.

Formally described, it alternates slow sections in a modal F with faster sections, " Neue Kraft fühlend " ( with renewed strength ), in D. The slow sections each have two elements, ( 1 ) a passage reminiscent of the opening of the first movement in which the instruments overlap each other with a brief motive ; ( 2 ) a chorale, the actual song.

Formally, a Markov partition is used to provide a finite cover for the smooth system ; each set of the cover is associated with a single symbol, and the sequences of symbols result as a trajectory of the system moves from one of the covering sets to another.

Formally, citizen science has been defined as " the systematic collection and analysis of data ; development of technology ; testing of natural phenomena ; and the dissemination of these activities by researchers on a primarily avocational basis ".

Formally, at least, the European Commission has primarily a monitoring role ; in practice, however, there is considerable scope for it to help set the policy agenda and persuade reluctant Member States to implement agreed policies.

Formally, let P be a poset ( partially ordered set ), and let F be a filter on P ; that is, F is a subset of P such that:

Formally ;

Formally, a biased graph Ω is a pair ( G, B ) where B is a linear class of circles ; this by definition is a class of circles that satisfies the theta-graph property mentioned above.