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Formally, given a G-bundle B and a map H → G ( which need not be an inclusion ),

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## Some Related Sentences

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 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__

**,**

__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__

**,**

__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__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__

**,**cl

**(**S ) denotes the smallest subset Y of M that contains S such that for each reaction

**(**A

**,**

__B__)

__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__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 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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