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

Formally, the convex hull may be defined as the intersection of all convex sets containing X or as the set of all convex combinations of points in X.

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, 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 styled as " Excelentísimo e Ilustrísimo Señor Profesor Doctor Don N, Rector Magnífico de la Universidad de X " ( Most Excellent and Illustrious Lord Professor Doctor Don N, Rector Magnificus of the University of X ), it is an office of high dignity within Spanish society, usually being highly respected.

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

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, assuming the axiom of choice, cardinality of a set X is the least ordinal α such that there is a bijection between X and α.

Formally, the mutual information of two discrete random variables X and Y can be defined as:

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

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, f < sub > X, Y </ sub >( x, y ) is the probability density function of ( X, Y ) with respect to the product measure on the respective supports of X and Y.

Formally, according to the Constitution, citizens of Turkmenistan have the right to set up political parties and other public associations, acting within the framework of the Constitution and laws, and public associations and groups of citizens have the right to nominate their candidates in accordance with the election law.

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, 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, it may even be taken back to Marx's earliest statements about a class becoming conscious of itself and developing a class identity.

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 speaking, a collation method typically defines a total order on a set of possible identifiers, called sort keys, which consequently produces a total preorder on the set of items of information ( items with the same identifier are not placed in any defined order ).

*, see On Formally Undecidable Propositions of Principia Mathematica and Related Systems for details on English translations.

Formally a random variable is considered to be a function on the possible outcomes.

Formally, a detailed balanced process is the one on which the probability to be in the state x and transit to x ' must be the same as to be in the state x ' and transit to x.

Formally, Gaussian curvature only depends on the Riemannian metric of the surface.

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 appointment remains a royal one made on the advice of the Lord Chancellor, but he / she no longer comments on individual applications, and merely supervises the process and reviews the panel's recommendations in general terms ( satisfying himself / herself that the process as operated is fair and efficient ).

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.

In late 1900s the day that of the City was founded is Jakin School is a Private School is Formally Part of Jakin School Board was Grades One through Twelve there is no Football field on this school. In 1966 When Jakin School was Closed were Sent to Blakely-Union Elementary, Junior High, and High School ( which later Became Early County High, Middle, and Elementary School ). while Carver school was built for African-American students grades One through Twelve. although late 1960s when Carver school was closed along with Kestler School was based in Damascus were sent to Washington High & Elementary at Blakely. In addition to small farm agriculture, Jakin's early economic growth resulted from turpentine.

Formally WPAZ, the station changed its call letters and name on October 28th, 2011.

Formally, the Sachs – Hornbostel is modeled on the Dewey Decimal classification.

Formally, a center of gravity is an application point of the resultant gravitational force on the body.

Formally named The King's College of Our Lady and Saint Nicholas in Cambridge, the college lies on the River Cam and faces out onto King's Parade in the centre of the city.

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

Formally educated African Americans, on the other hand, either disregarded black minstrelsy or openly disdained it.

Formally, the motion merely calls on parliament to thank the monarch or viceroy for the speech via an Address in Reply.

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 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, if the equilibrium strategy profile under the mechanism maps to the same goods allocation as a social choice function,

Formally, their designation is the letter Ž and the number.

Formally, the set of all context-free languages is identical to the set of languages accepted by pushdown automata ( PDA ).

Formally, the derivative of the function f at a is the limit

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, 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, 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, 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, Φ = kx − ωt is the phase.