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Formally and function
Formally, the derivative of the function f at a is the limit
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, 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, 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 and c
Formally, a ring is an Abelian group ( R, +), together with a second binary operation * such that for all a, b and c in R,

Formally and
* 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 coalgebra over a field K is a vector space C over K together with K-linear maps Δ: C C ⊗ C and ε: C K such that
Formally, a Hopf algebra is a ( associative and coassociative ) bialgebra H over a field K together with a K-linear map S: H H ( called the antipode ) such that the following diagram commutes:
Formally, let f: < sup > n </ sup > be the cost function which must be minimized.
Formally, given two categories C and D, an equivalence of categories consists of a functor F: C D, a functor G: D C, and two natural isomorphisms ε: FG I < sub > D </ sub > and η: I < sub > C </ sub >→ GF.
Formally, an unlabelled state transition system is a tuple ( S, →) where S is a set ( of states ) and ⊆ S × S is a binary relation over S ( of transitions ).
Formally, a frame on a homogeneous space G / H consists of a point in the tautological bundle G G / H.
Formally, complexification is a functor Vect < sub > R </ sup > Vect < sub > C </ sup >, from the category of real vector spaces to the category of complex vector spaces.
Formally, given two partially ordered sets ( S, ≤) and ( T, ≤), a function f: S T is an order-embedding if f is both order-preserving and order-reflecting, i. e. for all x and y in S, one has
Formally, given a G-bundle B and a map H G ( which need not be an inclusion ),
Formally it can be seen just as an ordinary function from X to the power set of Y, written as φ: X 2 < sup > Y </ sup >.

Formally and is
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 ).
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, 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, 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.

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