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Formally and given
Formally given to bishop Philip of Senj, the permission to use the Glagolitic liturgy ( the Roman Rite conducted in Slavic language instead of Latin, not the Byzantine rite ), actually extended to all Croatian lands, mostly along the Adriatic coast.
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, the two parts are given by the following expression, where is the number being encoded:
Formally the self-inductance of a wire loop would be given by the above equation with i
Formally, the statement that " value decreases over time " is given by defining the linear differential operator as:
Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces.
Formally, the problem can be stated as follows: given a desired property, expressed as a temporal logic formula p, and a structure M with initial state s, decide if.
Formally, the Secretary-General is given the same rank as a High Commissioner or ambassador.
Formally, given such that, the corresponding characteristic number is:
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, given a vector field v, a vector potential is a vector field A such that
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, given a partially ordered set ( P, ≤), then an element g of a subset S of P is the greatest element of S if
Formally, a deterministic algorithm computes a mathematical function ; a function has a unique value for any given input, and the algorithm is a process that produces this particular value as output.
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, 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, given a G-bundle B and a map H → G ( which need not be an inclusion ),
Formally, given a graph G, a vertex labeling is a function mapping vertices of G to a set of labels.
Formally, if and only if, for any given, there exists an such that
Formally, given two mappings
Formally this mission was declared to be only in pursuit of the Russian vessels and ammunition taken to Anzali by the White Russian counter-revolutionary General Denikin, who had been given asylum by British forces in Anzali.
Formally, a married or widowed woman can be called by the given name of her husband ( Madame ( given name of husband ) family name or Madame veuve ( given name of husband ) family name ); this is now slightly out of fashion.

Formally and complex-valued
Formally, the bounded Borel functional calculus of a self adjoint operator T on Hilbert space H is a mapping defined on the space of bounded complex-valued Borel functions f on the real line,

Formally and functions
Formally, the Congress serves two functions: to approve changes to the Party constitution regarding policy and to elect a Central Committee, about 300 strong.
Formally, this means that the probability density functions or probability mass functions in this class have the form
Formally, a homotopy between two continuous functions f and g from a
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, if we denote the set of stable functions by S ( D ) and the stability radius by r ( f, D ), then:

Formally and f
Formally, the derivative of the function f at a is the limit
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, suppose U is an open subset of the complex plane C, a is an element of U and f: U
Formally, a cardinal κ is defined to be weakly compact if it is uncountable and for every function f: < sup > 2 </ sup > →
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, let f: < sup > n </ sup > → be the cost function which must be minimized.
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, if f is a harmonic function, then f cannot exhibit a true local maximum within the domain of definition of f. In other words, either f is a constant function, or, for any point inside the domain of f, there exist other points arbitrarily close to at which f takes larger values.

Formally and g
Formally, if T is a complete sufficient statistic for θ and E ( g ( T )) = τ ( θ ) then g ( T ) is the minimum-variance unbiased estimator ( MVUE ) of τ ( θ ).
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.

Formally and natural
Formally, a rigged Hilbert space consists of a Hilbert space H, together with a subspace Φ which carries a finer topology, that is one for which the natural inclusion
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, a natural number n is called superabundant precisely when, for any m < n,
Formally, a natural number n is called highly abundant if and only if for all natural numbers m < n,
Formally, citizen science has been defined as " the systematic collection and analysis of data ; development of technology ; testing of natural phenomena ; and the dissemination of these activities by researchers on a primarily avocational basis ".
Formally, the right Kan extension of along consists of a functor and a natural transformation which is couniversal with respect to the specification, in the sense that for any functor and natural transformation, a unique natural transformation is defined and fits into a commutative diagram

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