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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.
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Formally and assuming
Formally, assuming our intuition from the finite dimensional case carries over into the infinite dimensional setting, the measure should then be equal to
Formally and axiom
( Formally speaking, this then satisfies the premises of an axiom of well-founded induction, which asserts that these two conditions are sufficient for the proposition to hold for all x.
Formally and choice
Formally, if the equilibrium strategy profile under the mechanism maps to the same goods allocation as a social choice function,
Formally and cardinality
Formally, a cardinal number κ is λ-unfoldable if and only if for every transitive model M of cardinality κ of ZFC-minus-power set such that κ is in M and M contains all its sequences of length less than κ, there is a non-trivial elementary embedding j of M into a transitive model with the critical point of j being κ and j ( κ ) ≥ λ.
Formally and set
Formally, a binary operation on a set S is called associative if it satisfies the associative law:
Formally, the set of all context-free languages is identical to the set of languages accepted by pushdown automata ( PDA ).
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 ).
Formally, one has an implication, not an equivalence, so the solution set may get larger.
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, 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, the weighted mean of a non-empty set of data
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.
Formally, Gaussian integers are the set
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, 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, start with a set Ω and consider the sigma algebra Σ on Ω consisting of all subsets of Ω.
Formally, the movement is a rondo that acts as the theme for a set of eight variations, capped off by a dramatic coda.
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, starting with a linearly independent set of vectors
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 constraint satisfaction problem is defined as a triple, where is a set of variables, is a domain of values, and is a set of constraints.
Formally and X
Formally, a topological space X is called compact if each of its open covers has a finite subcover.
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, 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, sending X to and acting as the identity on maps is a functor ( indeed, an involution ).
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 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, their designation is the letter Ž and the number.
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, 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.
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