Formally, the theorem can be stated as follows:

Formally, the theorem is stated as follows: There exist unique integers q and r such that a = qd + r and 0 ≤ r < | d |, where | d | denotes the absolute value of d.

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

Formally, the theorem states:

Formally, one has an implication, not an equivalence, so the solution set may get larger.

Formally, the convex hull may be defined as the intersection of all convex sets containing X or as the set of all convex combinations of points in X.

Formally, powers with positive integer exponents may be defined by the initial condition

Formally, Minkowski space is a four-dimensional real vector space equipped with a nondegenerate, symmetric bilinear form with signature < tt >(−,+,+,+)</ tt > ( Some may also prefer the alternative signature < tt >(+,−,−,−)</ tt >; in general, mathematicians and general relativists prefer the former while particle physicists tend to use the latter.

Formally, one may connect the kernel and the cokernel by the exact sequence

Formally, it may even be taken back to Marx's earliest statements about a class becoming conscious of itself and developing a class identity.

Formally, define the set of lines in the plane P as L ( P ); then a rigid motion of the plane takes lines to lines – the group of rigid motions acts on the set of lines – and one may ask which lines are unchanged by an action.

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, a party in honor of the bride-to-be is never hosted by the bride-to-be, although she may participate in its planning.

However, shortly after this positive result, Kurt Gödel published On Formally Undecidable Propositions of Principia Mathematica and Related Systems ( 1931 ), showing that in any sufficiently strong axiomatic system there are true statements which cannot be proved in the system.

Formally the algorithm can be described as:

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, let S and T be finite sets and let F =

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

Formally the " head of state " can also personally be the " head of government " ( ex officio or by ad hoc cumulation such as an absolute monarch nominating himself ) but otherwise has formal precedence over the Head of Government and other ministers, whether he is their actual political superior ( absolute monarch, executive president ) or rather theoretical or ceremonial in character.

Formally, in the finite-dimensional case, if the linear map is represented as a multiplication by a matrix A and the translation as the addition of a vector, an affine map acting on a vector can be represented as

Formally, a decision problem is P-complete ( complete for the complexity class P ) if it is in P and that every problem in P can be reduced to it by using an appropriate reduction.

Formally, a detailed balanced process is the one on which the probability to be in the state x and transit to x ' must be the same as to be in the state x ' and transit to x.

If R is a ring, let R denote the ring of polynomials in the indeterminate X over R. Hilbert proved that if R is " not too large ", in the sense that if R is Noetherian, the same must be true for R. Formally,

Formally, a unique factorization domain is defined to be an integral domain R in which every non-zero and non-unit x of R can be written as a product ( including an empty product ) of irreducible elements p < sub > i </ sub > of R and a unit u:

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 self-inductance of a wire loop would be given by the above equation with i

Formally, a government would lay out its reasons for going to war, as well as its intentions in prosecuting it and the steps that might be taken to avert it.

Formally, let P be an expression in which the variable x is free.

Formally, if is any non-zero polynomial, it must be writable as.

Formally, a Menger sponge can be defined as follows:

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.

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

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, 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: