Formally, a binary operation on a set S is called associative if it satisfies the associative law:

Formally, a topological space X is called compact if each of its open covers has a finite subcover.

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 set S is called finite if there exists a bijection

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

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, 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.

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 hypothesis is compared against its opposite or null hypothesis (" if I release this ball, it will not fall to the floor ").

Formally, the sequence of partial sums of some infinite summation converges if for every fixed power of X the coefficient stabilizes: there is a point beyond which all further partial sums have the same coefficient.

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, 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, two variables are inversely proportional ( or varying inversely, or in inverse variation, or in inverse proportion or in reciprocal proportion ) if one of the variables is directly proportional with the multiplicative inverse ( reciprocal ) of the other, or equivalently if their product is a constant.

Informally, G has the above presentation if it is the " freest group " generated by S subject only to the relations R. Formally, the group G is said to have the above presentation if it is isomorphic to the quotient of a free group on S by the normal subgroup generated by the relations R.

Formally, if there exists some B ≥ 0 such that

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

Formally, a cardinal κ is defined to be weakly compact if it is uncountable and for every function f: < sup > 2 </ sup > →

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, this means that we want a function to be monotonic.

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:

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

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, a deterministic Büchi automaton is a tuple A = ( Q, Σ, δ, q < sub > 0 </ sub >, F ) that consists of the following components:

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 absolute coequalizer of a pair in a category C is a coequalizer as defined above but with the added property that given any functor F ( Q ) together with F ( q ) is the coequalizer of F ( f ) and F ( g ) in the category D. Split coequalizers are examples of absolute coequalizers.

Formally, a product term P in a sum of products is an implicant of the Boolean function F if P implies F. More precisely:

Formally described, it alternates slow sections in a modal F with faster sections, " Neue Kraft fühlend " ( with renewed strength ), in D. The slow sections each have two elements, ( 1 ) a passage reminiscent of the opening of the first movement in which the instruments overlap each other with a brief motive ; ( 2 ) a chorale, the actual song.

Formally called The Commissioner's Cup, it was renamed The Sasser Cup after former Commissioner George F. " Buddy " Sasser.

Formally, let P be a poset ( partially ordered set ), and let F be a filter on P ; that is, F is a subset of P such that: