[permalink] [id link]
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
Some Related Sentences
Formally and function
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, 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, 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, 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, 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 ƒ
Formally, in Euclidean space, the wave front set of ƒ is defined as the complement of the set of all pairs ( x < sub > 0 </ sub >, v ) such that there exists a test function with φ ( x < sub > 0 </ sub >) ≠ 0 and an open cone Γ containing v such that the estimate
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, 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, an inner product space is a vector space V over the field together with an inner product, i. e., with a map
* 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 and real
Formally, the singular value decomposition of an m × n real or complex matrix M is a factorization of the form
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, let A be a real matrix of which we want to compute the eigenvalues, and let A < sub > 0 </ sub >:= A.
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, the bounded Borel functional calculus of a self adjoint operator T on Hilbert space H is a mapping defined on the space of bounded complex-valued Borel functions f on the real line,