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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.

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## Some Related Sentences

Informally and G

In

**the**mathematical field**of**graph theory**,****a**spanning tree T**of****a**connected**,**undirected graph__G__**is****a**tree composed**of**all**the**vertices and some ( or perhaps all )**of****the**edges**of**__G__**.**__Informally__**,****a**spanning tree**of**__G__**is****a**selection**of**edges**of**__G__that form**a**tree spanning every vertex**.**__Informally__

**,**this reads as follows: something exists with

**the**property F

**,**there

**is**

**only**one such thing

**,**and this unique thing also

**has**

**the**property

__G__

**.**

__Informally__

**,**this means that

**if**

__G__acts unitarily

**on**

**a**Hilbert space and

**has**

**"**almost invariant vectors ", then

**it**

**has**

**a**nonzero invariant vector

**.**

__Informally__

**,**

**if**F

**is**'= y ',

__G__

**is**'

**is**Pegasus ', and we substitute ' Pegasus ' for y

**,**then ( 4 ) appears

**to**allow us

**to**infer from ' everything identical with Pegasus

**is**Pegasus ' that something

**is**identical with Pegasus

**.**

Informally and has

__Informally__

**,**

**the**expression

**"**infinitesimal calculus

**"**became commonly used

**to**refer

**to**Weierstrass ' approach but

__has__become something

**of**

**a**dead metaphor

**.**

__Informally__

**,**

**a**measure

__has__

**the**property

**of**being monotone function | monotone in

**the**sense that

**if**A

**is**

**a**subset

**of**B

**,**

**the**measure

**of**A

**is**less than or equal

**to**

**the**measure

**of**B

**.**

__Informally__this

__has__been called

**the**

**"**Step Boundary ", although

**the**step-like shape

**of**

**the**boundary was not apparent for several decades —

**the**source

**of**

**the**Arkansas

**,**believed

**to**be near

**the**42nd parallel

**,**was not known until John C

**.**Frémont located

**it**in

**the**1840s

**,**hundreds

**of**miles south

**of**

**the**42nd parallel

**.**

__Informally__

**,**

**the**market spilled into

**the**surrounding neighborhood

**,**both in terms

**of**street vendors and in terms

**of**

**the**nearby Magazin Universal (" Universal Store ") named Bucur Obor

**,**

**a**large commercial building that

__has__been parcelled up into hundreds

**of**small

**,**independent retail stores

**.**

__Informally__

**,**one might say that

**the**Dirac string carries away

**the**

**"**excess curvature

**"**that would otherwise prevent F from being

**a**closed form

**,**as one

__has__that everywhere except at

**the**location

**of**

**the**monopole

**.**

Informally and above

__Informally__

**,**flak jacket may refer

**to**

**a**ballistic vest

**,**particularly Type III and

__above__

**,**which

**is**reinforced with either steel

**,**titanium

**,**ceramic

**,**or polyethylene plates

**.**

Informally and if

Such

**a**definition can be formulated in terms**of**equivalence classes**of**smooth functions**on**M**.**__Informally__**,**we will say that two smooth functions f and g are equivalent at**a**point x__if__they**have****the**same first-order behavior near x**.**__Informally__

**,**an object

**is**reachable

__if__

**it**

**is**referenced

**by**at least one variable in

**the**program

**,**either directly or through references from other reachable objects

**.**

__Informally__

**the**Church – Turing thesis states that

__if__some method ( algorithm ) exists

**to**carry out

**a**calculation

**,**then

**the**same calculation can also be carried out

**by**

**a**Turing machine ( as well as

**by**

**a**recursively definable function

**,**and

**by**

**a**λ-function ).

__Informally__

**,**

**a**relational database table

**is**often described as

**"**normalized

**"**

__if__

**it**

**is**in

**the**Third Normal Form

**.**

__Informally__speaking

**,**

**the**prime number theorem states that

__if__

**a**random integer

**is**selected in

**the**range

**of**zero

**to**some large integer N

**,**

**the**probability that

**the**selected integer

**is**prime

**is**about 1 / ln ( N ), where ln ( N )

**is**

**the**natural logarithm

**of**N

**.**For example

**,**among

**the**positive integers up

**to**and including N = 10 < sup > 3 </ sup > about one in seven numbers

**is**prime

**,**whereas up

**to**and including N = 10 < sup > 10 </ sup > about one in 23 numbers

**is**prime ( where ln ( 10 < sup > 3 </ sup >)= 6

**.**90775528

**.**and ln ( 10 < sup > 10 </ sup >)= 23

**.**0258509 ).

__Informally__

**,**

**a**set

**of**strategies

**is**

**a**Nash equilibrium

__if__no player can do better

**by**unilaterally changing his or her strategy

**.**

In other words

**,**L can be solved in polynomial time**by**an oracle machine with an oracle for H**.**__Informally__**,**we can think**of**an algorithm that can call such an oracle machine as**a**subroutine for solving H**,**and solves L in polynomial time**,**__if__**the**subroutine call takes**only**one step**to**compute**.**__Informally__

**,**two sequences (

**a**< sub > n </ sub >) and ( b < sub > n </ sub >) become closer and closer

__if__and

**only**

__if__more and more

**of**their terms agree exactly

**.**

__Informally__

**,**this means that arguments

**of**this form do not give good reason

**to**establish their conclusions

**,**even

__if__their premises are true

**.**

__Informally__

**,**

**a**set

**of**natural numbers A

**is**Turing reducible

**to**

**a**set B

__if__there

**is**an oracle machine that correctly tells whether numbers are in A when run with B as

**the**oracle set ( in this case

**,**

**the**set A

**is**also

**said**

**to**be ( relatively ) computable from B and recursive in B ).

__Informally__

**,**

__if__P

**is**any polyhedron or polytope

**,**and tP

**is**

**the**polytope formed

**by**expanding P

**by**

**a**factor

**of**t in each dimension

**,**then L ( int P

**,**t )

**is**

**the**number

**of**integer lattice points in tP

**.**

__Informally__

**,**subvarieties are in general position

__if__they cannot be described more simply than others

**.**

__Informally__

**,**two sets are close in

**the**Hausdorff distance

__if__every point

**of**either set

**is**close

**to**some point

**of**

**the**other set

**.**

__Informally__

**,**two crystals tend

**to**be in

**the**same crystal system

__if__they

**have**similar symmetries

**,**though there are many exceptions

**to**this

**.**

__Informally__

**,**given

**the**prime factorization

**of**m

**,**take b

**to**be

**the**product

**of**

**the**prime factors

**of**m that

**have**an odd exponent (

__if__there are none

**,**then take b

**to**be 1 ).

__Informally__

**,**

**a**thick object in our space

**is**simply connected

__if__

**it**consists

**of**one piece and does not

**have**any

**"**holes

**"**that pass all

**the**way through

**it**

**.**

__Informally__

**,**

**a**language L

**is**in MA

__if__for all strings in

**the**language

**,**there

**is**

**a**polynomial sized proof that Merlin can send Arthur

**to**convince him

**of**this fact with high probability

**,**and for all strings not in

**the**language there

**is**no proof that convinces Arthur with high probability

**.**

__Informally__put

**,**

**the**idea behind

**the**proof

**of**

**the**no-trade theorem

**is**that

__if__there

**is**common knowledge about

**the**structure

**of**

**a**market

**,**then any bid or offer ( i

**.**e

**.**attempt

**to**initiate

**a**trade ) will reveal

**the**bidder's private knowledge and will be incorporated into market prices even before anyone accepts

**the**bid or offer

**,**so no profit will result

**.**

Informally and is

__Informally__put

**,**

**the**axiom

**of**choice says that given any collection

**of**bins

**,**each containing at least one object

**,**

**it**

__is__possible

**to**make

**a**selection

**of**exactly one object from each bin

**.**

__Informally__

**,**

**it**

__is__

**the**similarity between observations as

**a**function

**of**

**the**time separation between them

**.**

__Informally__we can think

**of**elements

**of**

**the**Lie algebra as elements

**of**

**the**

**group**that are

**"**infinitesimally close

**"**

**to**

**the**identity

**,**and

**the**Lie bracket

__is__something

**to**do with

**the**commutator

**of**two such infinitesimal elements

**.**

__Informally__

**,**word formation rules form

**"**new words

**"**( that

__is__

**,**new lexemes ), while inflection rules yield variant forms

**of**

**the**

**"**same

**"**word ( lexeme ).

__Informally__

**,**he may

**have**been known as

**"**Dickon ", according

**to**

**a**sixteenth-century legend

**of**

**a**note

**,**warning

**of**treachery

**,**that was sent

**to**

**the**Duke

**of**Norfolk

**on**

**the**eve

**of**Bosworth:

**"**Jack

**of**Norffolke be not

**to**bolde ,/ For Dyckon thy maister

__is__bought and solde ".

__Informally__

**,**

**a**permutation

**of**

**a**set

**of**objects

__is__an arrangement

**of**those objects into

**a**particular order

**.**

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