Formally, start with a set Ω and consider the sigma algebra Σ on Ω consisting of all subsets of Ω.

Formally, consider an economic model with different mathematical weightings placed on the utilities of each self.

Formally, if d is the dimension of the parameter, and n is the number of samples, if as and as, then the model is semi-parametric.

Formally, the singular value decomposition of an m × n real or complex matrix M is a factorization of the form

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, 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, Aff ( V ) is naturally isomorphic to a subgroup of, with V embedded as the affine plane, namely the stabilizer of this affine plane ; the above matrix formulation is the ( transpose of ) the realization of this, with the ( n × n and 1 × 1 ) blocks corresponding to the direct sum decomposition.

Formally, let f: < sup > n </ sup > → be the cost function which must be minimized.

Formally, given complex-valued functions f and g of a natural number variable n, one writes

Formally, a complex projective space is the space of complex lines through the origin of an ( n + 1 )- dimensional complex vector space.

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, an algebraic function in n variables over the field K is an element of the algebraic closure of the field of rational functions K ( x < sub > 1 </ sub >,..., x < sub > n </ sub >).

Formally, a composite number n = d · 2 < sup > s </ sup > + 1 with d being odd is called a strong pseudoprime to a relatively prime base a when one of the following conditions hold:

Formally, the use of a reduction is the function that sends each natural number n to the largest natural number m whose membership in the set B was queried by the reduction while determining the membership of n in A.

Formally, P is a symmetric polynomial, if for any permutation σ of the subscripts 1, 2, ..., n one has P ( X < sub > σ ( 1 )</ sub >, X < sub > σ ( 2 )</ sub >, …, X < sub > σ ( n )</ sub >) = P ( X < sub > 1 </ sub >, X < sub > 2 </ sub >, …, X < sub > n </ sub >).

Formally, a natural number n is called superabundant precisely when, for any m < n,

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, 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, 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, this is a solution of the matrix equation by Jacobi iteration.

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

Formally, the Mahalanobis distance of a multivariate vector from a group of values with mean and covariance matrix is defined as:

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

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.

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, 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, the ith row, jth column element of A < sup > T </ sup > is the jth row, ith column element of A:

Formally, the outcomes Y < sub > i </ sub > are described as being Bernoulli-distributed data, where each outcome is determined by an unobserved probability p < sub > i </ sub > that is specific to the outcome at hand, but related to the explanatory variables.

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, a ringed space ( X, O < sub > X </ sub >) is a topological space X together with a sheaf of rings O < sub > X </ sub > on 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, a cardinal κ is defined to be weakly compact if it is uncountable and for every function f: < sup > 2 </ sup > →

Formally, a Lie superalgebra is a ( nonassociative ) Z < sub > 2 </ sub >- graded algebra, or superalgebra, over a commutative ring ( typically R or C ) whose product, called the Lie superbracket or supercommutator, satisfies the two conditions ( analogs of the usual Lie algebra axioms, with grading ):

11 1 / 9 % interest .< ref > Formally, a discount of d % results in effective interest of

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

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

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 most of these approaches are similar to an artificial neural network, as inputs to a node are summed up and the result serves as input to a sigmoid function, e. g., but proteins do often control gene expression in a synergistic, i. e. non-linear, way.

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, 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, the person who directly draws the funds (" the payee ") instructs his or her bank to collect ( i. e., debit ) an amount directly from another's (" the payer's ") bank account designated by the payer and pay those funds into a bank account designated by the payee.