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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 complex-valued functions f and g of a natural number variable n, one writes
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 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 B
Formally, if there exists some B ≥ 0 such that
Formally, cl ( S ) denotes the smallest subset Y of M that contains S such that for each reaction ( A, B )
Formally said, the intersection of each two of them is empty ( the null event ): A and B
Formally, a bicategory B consists of:
Formally, the issue is that interfertile " able to interbreed " is not a transitive relation – if A can breed with B, and B can breed with C, it does not follow that A can breed with C – and thus does not define an equivalence relation.
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, an encoding of a language A into language B is a mapping of all terms of A into B.
Formally trained in music composition, Dargel studied with Oliveros, John Luther Adams, and Lewis Nielson, and received a B. A.
Formally, a biased graph Ω is a pair ( G, B ) where B is a linear class of circles ; this by definition is a class of circles that satisfies the theta-graph property mentioned above.

Formally and map
Formally, an inner product space is a vector space V over the field together with an inner product, i. e., with a map
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, scalar multiplication is a linear map, inducing a map ( send a scalar λ to the corresponding scalar transformation, multiplication by λ ) exhibiting End ( M ) as a R-algebra.
Formally, a Hopf algebra is a ( associative and coassociative ) bialgebra H over a field K together with a K-linear map S: H H ( called the antipode ) such that the following diagram commutes:
Formally, this follows from the fact that the code is an injective map.

Formally and H
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, the index of H in G is defined as the number of cosets of H in G. ( The number of left cosets of H in G is always equal to the number of right cosets.
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, a frame on a homogeneous space G / H consists of a point in the tautological bundle G G / H.
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,

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