[permalink] [id link]

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

from
Wikipedia

## Some Related Sentences

Formally and if

__Formally__

**,**oxidation state

**is**

**the**hypothetical charge

**that**an atom would have

__if__all bonds

**to**atoms of different elements were 100 % ionic.

__Formally__

**,**

__if__d

**is**

**the**dimension of

**the**parameter

**,**

**and**n

**is**

**the**number of samples

**,**

__if__as

**and**as

**,**then

**the**model

**is**semi-parametric.

__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__

**,**

**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.

If R

**is****a**ring**,**let R denote**the**ring of polynomials in**the**indeterminate X over R. Hilbert proved**that**__if__R**is**" not too large ", in**the**sense**that**__if__R**is**Noetherian**,****the**same must**be**true for R.__Formally__**,**__Formally__

**,**

**a**hypothesis

**is**compared against its opposite or null hypothesis ("

__if__I release this ball

**,**it will not fall

**to**

**the**floor ").

__Formally__

**,**

**the**sequence of partial sums of some infinite summation converges

__if__for every fixed power of X

**the**coefficient stabilizes: there

**is**

**a**point beyond

**which**all further partial sums have

**the**same coefficient.

__Formally__

**,**

**a**

**function**ƒ

**is**real analytic

**on**an open

**set**D in

**the**real line

__if__for any x < sub > 0 </ sub > in D one can write

__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__

**,**two variables are inversely proportional ( or varying inversely

**,**or in inverse variation

**,**or in inverse proportion or in reciprocal proportion )

__if__one of

**the**variables

**is**directly proportional

**with**

**the**multiplicative inverse ( reciprocal ) of

**the**other

**,**or equivalently

__if__their product

**is**

**a**constant.

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__

**,**

__if__we write F < sub > Δ </ sub >( x )

**to**mean

**the**f-polynomial of Δ

**,**then

**the**h-polynomial of Δ

**is**

__Formally__

**,**

**a**cardinal κ

**is**

**defined**

**to**

**be**weakly compact

__if__it

**is**uncountable

**and**for every

**function**

**f**: < sup > 2 </ sup > →

Formally and M

__Formally__

**,**

**the**singular value decomposition of an m × n real or complex matrix

__M__

**is**

**a**factorization of

**the**form

__Formally__

**,**cl ( S ) denotes

**the**smallest subset Y of

__M__

**that**contains S such

**that**for each reaction ( A

**,**B )

__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__

**,**

**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__

**,**

**a**cardinal number κ

**is**λ-unfoldable

**if**

**and**only

**if**for every transitive model

__M__of cardinality κ of ZFC-minus-power

**set**such

**that**κ

**is**in

__M__

**and**

__M__contains all its sequences of length less than κ

**,**there

**is**

**a**non-trivial elementary embedding j of

__M__into

**a**transitive model

**with**

**the**critical point of j being κ

**and**j ( κ ) ≥ λ.

__Formally__

**,**

**the**definition only requires some invertibility

**,**so we can substitute for Q any matrix

__M__whose eigenvalues do not include − 1.

__Formally__

**,**

**the**normal bundle

**to**N in

__M__

**is**

**a**quotient bundle of

**the**tangent bundle

**on**

__M__: one has

**the**short exact sequence of vector bundles

**on**N:

Formally and is

__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__

**,**

**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__

**,**an inner product space

__is__

**a**vector space V over

**the**field together

**with**an inner product

**,**i. e.,

**with**

**a**map

*

__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__

**,**

**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.

__Formally__

**,**

**a**frame

__is__

**defined**

**to**

**be**

**a**lattice L in

**which**finite meets distribute over arbitrary joins

**,**i. e. every ( even infinite ) subset

0.088 seconds.