Formally, this means that the probability density functions or probability mass functions in this class have the form

Limits and colimits in a category C are defined by means of diagrams in C. Formally, a diagram of type J in C is a functor from J to C:

Formally, this means symmetry under a sub-group of the Euclidean group of isometries in two or three dimensional Euclidean space.

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, this means classifying finitely generated groups with their word metric up to quasi-isometry.

Formally, however, the role also carries the title of " Klingon supreme commander " ( TNG's " Reunion "), which presumably means commander-in-chief of the military.

Formally we mean that is an ideal if it satisfies the following conditions:

Formally, we start with a category C with finite products ( i. e. C has a terminal object 1 and any two objects of C have a product ).

Formally it is precisely in allowing quantification over class variables α, β, etc., that we assume a range of values for these variables to refer to.

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, we have for the approximation to the full solution A, a series in the small parameter ( here called ), like the following:

Formally, we start with a metric space M and a subset X.

Formally, if we write F < sub > Δ </ sub >( x ) to mean the f-polynomial of Δ, then the h-polynomial of Δ is

Formally, for a countable set of events A < sub > 1 </ sub >, A < sub > 2 </ sub >, A < sub > 3 </ sub >, ..., we have

Formally, we define

Formally, we begin by considering some family of distributions for a random variable X, that is indexed by some θ.

Formally, let A be a real matrix of which we want to compute the eigenvalues, and let A < sub > 0 </ sub >:= A.

Formally we have:

Formally, we have

Formally, an antihomomorphism between X and Y is a homomorphism, where equals Y as a set, but has multiplication reversed: denoting the multiplication on Y as and the multiplication on as, we have.

Formally, the definition only requires some invertibility, so we can substitute for Q any matrix M whose eigenvalues do not include − 1.

Formally, given a finite set X, a collection C of subsets of X, all of size n, has Property B if we can partition X into two disjoint subsets Y and Z such that every set in C meets both Y and Z.

Formally, we define indices inductively using

Formally, we define a bad field as a structure of the form ( K, T ), where K is an algebraically closed field and T is an infinite, proper, distinguished subgroup of K, such that ( K, T ) is of finite Morley rank in its full language.

Formally, we want:.

Formally we can write the factor as,

Formally, if we denote the set of stable functions by S ( D ) and the stability radius by r ( f, D ), then:

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, 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, 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 )