Formally, the theorem can be stated as follows:

Gödel's first incompleteness theorem first appeared as " Theorem VI " in Gödel's 1931 paper On Formally Undecidable Propositions in Principia Mathematica and Related Systems I.

Formally, this follows from the convolution theorem in mathematics, which relates the Fourier transform of the power spectrum ( the intensity of each frequency ) to its autocorrelation.

Formally, the dimension theorem for vector spaces states that

Formally, Turán's theorem may be stated as follows.

Formally, the theorem states that, for every Sperner family S over an n-set,

Formally, the theorem states:

Formally organized vocational programs supported by federal funds allow high school students to gain experience in a field of work which is likely to lead to a full-time job on graduation.

Formally, a binary operation on a set S is called associative if it satisfies the associative law:

Formally, their designation is the letter Ž and the number.

Formally, a topological space X is called compact if each of its open covers has a finite subcover.

Formally, the set of all context-free languages is identical to the set of languages accepted by pushdown automata ( PDA ).

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

More rigorously, the divergence of a vector field F at a point p is defined as the limit of the net flow of F across the smooth boundary of a three dimensional region V divided by the volume of V as V shrinks to p. Formally,

Formally, the base is known as Naval Support Facility Diego Garcia ( the US activity ) or Permanent Joint Operating Base ( PJOB ) Diego Garcia ( the UK's term ).

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, oxidation state is the hypothetical charge that an atom would have if all bonds to atoms of different elements were 100 % ionic.

Formally, a bifunctor is a functor whose domain is a product category.

Formally, a set S is called finite if there exists a bijection

Formally, the system is said to have memory.

Formally, an inner product space is a vector space V over the field together with an inner product, i. e., with a map

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, 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, a profinite group is a Hausdorff, compact, and totally disconnected topological group: that is, a topological group that is also a Stone space.

Formally, this sharing of dynamics is referred to as universality, and systems with precisely the same critical exponents are said to belong to the same universality class.

Formally, a frame is defined to be a lattice L in which finite meets distribute over arbitrary joins, i. e. every ( even infinite ) subset

Formally, Φ = kx − ωt is the phase.

Formally stated, the FFT is a method for computing the discrete Fourier transform of a sampled signal.

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, the problem can be stated as follows: given a desired property, expressed as a temporal logic formula p, and a structure M with initial state s, decide if.

“ Formally stated, interactivity is an expression of the extent that in a given series of communication exchanges, any third ( or later ) transmission ( or message ) is related to the degree to which previous exchanges referred to even earlier transmissions ” ( Sheizaf Rafaeli, “ Interactivity, From New Media to Communication ,” pages 110-34 in Advanced Communicational Science: Merging Mass and Interpersonal Processes, ed.

Formally, a Menger sponge can be defined as follows:

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

Formally, the sets of free and bound names of a process in π – calculus are defined inductively as follows.

Formally, the set of LTL formulas over AP is inductively defined as follows:

a < sub > i </ sub >, a < sub > i + 1 </ sub >,..., which is a suffix of w. Formally, the satisfaction relation between a word and an LTL formula is defined as follows:

Formally, the construction is as follows.

Formally, the chain rule indeed follows from the above definition of conditional probability:

Formally, the expenditure function is defined as follows.

Formally, this can be expressed as follows.

Formally, this follows from the fact that the code is an injective map.

Formally, a cumulativity predicate CUM can be defined as follows, where capital X is a variable over sets, U is the universe of discourse, p is a mereological part structure on U, and is the mereological sum operation.

Formally, a quantization predicate QUA can be defined as follows, where is the universe of discourse, and is a variable over sets, and is a mereological part structure on with < math > < _p </ math > the mereological part-of relation.

Formally, the extension of circumscription that incorporate varying and fixed variables is as follows, where is the set of variables to minimize, the fixed variables, and the varying variables are those not in: