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Formally, the subhypergraph induced by a subset of is defined as

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## Some Related Sentences

Formally and by

__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__

**,**

**the**set

**of**all context-free languages

**is**identical to

**the**set

**of**languages accepted

__by__pushdown automata ( PDA ).

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__appoints**the**Prime Minister**of**Finland selected__by__**the**Parliament**,**and formally appoints**the**rest**of****the**cabinet ( Council**of**State )**as**proposed__by__**the**Prime Minister__Formally__

**the**Prince-Electors elected

**a**King

**of**

**the**Romans

**,**who was elected in Germany but became Holy Roman Emperor only when crowned

__by__

**the**Pope.

*

__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__

**,**these failed when they were rejected

__by__

**the**Church

**of**England's General Synod in 1972 ; conversations and co-operation continued

**,**however

**,**leading in 2003 to

**the**signing

**of**

**a**covenant between

**the**two churches.

__Formally__

**the**" head

**of**state " can also personally be

**the**" head

**of**government " ( ex officio or

__by__ad hoc cumulation such

**as**an absolute monarch nominating himself ) but otherwise has formal precedence over

**the**Head

**of**Government and other ministers

**,**whether he

**is**their actual political superior ( absolute monarch

**,**executive president ) or rather theoretical or ceremonial in character.

__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**

Limits and colimits in

**a**category C are**defined**__by__means**of**diagrams in C.__Formally__**,****a**diagram**of**type J in C**is****a**functor from J to C:__Formally__named " United Press Associations " for incorporation and legal purposes

**,**but publicly known and identified

**as**United Press or UP

**,**it was created

__by__

**the**1907 uniting

**of**three smaller news syndicates

__by__

**the**Midwest newspaper publisher E. W. Scripps.

__Formally__

**,**most

**of**these people were convicted

__by__NKVD troikas (" triplets ")– special courts martial.

__Formally__

**,**collective noun forms such

**as**“

**a**group

**of**people ” are represented

__by__second-order variables

**,**or

__by__first-order variables standing for sets ( which are well-defined objects in mathematics and logic ).

__Formally__

**,**

**a**decision problem

**is**P-complete ( complete for

**the**complexity class P ) if it

**is**in P and that every problem in P can be reduced to it

__by__using an appropriate reduction.

__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**transductive support vector machine

**is**

**defined**

__by__

**the**following primal optimization problem:

__Formally__

**,**DFS

**is**an uninformed search that progresses

__by__expanding

**the**first child node

**of**

**the**search tree that appears and thus going deeper and deeper until

**a**goal node

**is**found

**,**or until it hits

**a**node that has no children.

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.

Formally and subset

__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__

**,**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__

**,**cl ( S ) denotes

**the**smallest

__subset__Y

**of**M that contains S such that for each reaction ( A

**,**B )

__Formally__

**,**

**the**case where only

**a**

__subset__

**of**parameters

**is**

**defined**

**is**still

**a**composite hypothesis ; nonetheless

**,**

**the**term point hypothesis

**is**often applied in such cases

**,**particularly where

**the**hypothesis test can be structured in such

**a**way that

**the**distribution

**of**

**the**test statistic (

**the**distribution under

**the**null hypothesis ) does not depend on

**the**parameters whose values have not been specified under

**the**point null hypothesis.

__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**partially ordered set ( P

**,**≤)

**is**bounded complete if

**the**following holds for any

__subset__S

**of**P:

__Formally__

**,**let P be

**a**poset ( partially ordered set ), and let F be

**a**filter on P ; that

**is**

**,**F

**is**

**a**

__subset__

**of**P such that:

Formally and is

__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__

**,**an inner product space

__is__

**a**vector space V over

**the**field together with an inner product

**,**i. e., with

**a**map

__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__

**,**

**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.

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