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Conversely, given any Lie algebra with a direct sum decomposition satisfying these three conditions, the linear map σ, equal to the identity on and minus the identity on, is an involutive automorphism.

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## Some Related Sentences

Conversely and given

__Conversely__

**,**if

**a**Boolean ring A

**is**

__given__

**,**we can turn it into

**a**Boolean

**algebra**by defining x ∨ y := x + y + ( x · y )

**and**x ∧ y := x · y

**.**

__Conversely__

**,**there are other stars that never rise above

**the**horizon

**,**as seen from

**any**

__given__point

**on**

**the**Earth's surface ( except exactly

**on**

**the**equator ).

__Conversely__

**,**

__given__

**a**groupoid G in

**the**algebraic sense

**,**let G < sub > 0 </ sub > be

**the**set of all elements of

**the**form x * x < sup >− 1 </ sup >

**with**x varying through G

**and**define G ( x * x < sup >-1 </ sup >, y * y < sup >-1 </ sup >) as

**the**set of all elements f such that y * y < sup >-1 </ sup > * f * x * x < sup >-1 </ sup > exists

**.**

__Conversely__

**,**

__given__central idempotents

**a**< sub > 1 </ sub >,...,

**a**< sub > n </ sub > in R that are pairwise orthogonal

**and**have

**sum**1

**,**then R

**is**

**the**

**direct**

**sum**of

**the**rings Ra < sub > 1 </ sub >,…, Ra < sub > n </ sub >.

Given

**a**field ordering ≤ as in Def 1**,****the**elements such that x ≥ 0 forms**a**positive cone of F**.**__Conversely__**,**__given__**a**positive cone P of F as in Def 2**,**one can associate**a**total ordering ≤< sub > P </ sub > by setting x ≤ y**to**mean y − x ∈ P**.**This total ordering ≤< sub > P </ sub > satisfies**the**properties of Def 1**.**__Conversely__

**,**only

**the**wiki users interested in

**a**

__given__project need look at its associated wiki pages

**,**in contrast

**to**high-traffic mailing lists which may burden subscribers

**with**many messages

**,**regardless of their relevance

**.**

__Conversely__

**,**

__given__

**a**harmonic function

**,**it

**is**

**the**real part of

**an**analytic function

**,**( at least locally ).

Each convex set containing X must ( by

**the**assumption that it**is**convex ) contain all convex combinations of points in X**,**so**the**set of all convex combinations**is**contained in**the**intersection of all convex sets containing X**.**__Conversely__**,****the**set of all convex combinations**is**itself**a**convex set containing X**,**so it also contains**the**intersection of all convex sets containing X**,****and**therefore**the**sets__given__by**these**two definitions must be**equal****.**__Conversely__

**,**surveys conducted among living donors postoperatively

**and**in

**a**period of five years following

**the**procedure have shown extreme regret in

**a**majority of

**the**donors

**,**who said that

__given__

**the**chance

**to**repeat

**the**procedure

**,**they would not

**.**

__Conversely__

**,**

__given__

**an**ordered tree

**,**

**and**conventionally draw

**the**root at

**the**top

**,**then

**the**child nodes in

**an**ordered tree can be drawn left-to-right

**,**yielding

**an**essentially unique planar embedding ( up

**to**embedded homotopy

**,**i

**.**e., moving

**the**edges

**and**nodes without crossing ).

__Conversely__

**,**

**the**influence of

**the**data at

**any**

__given__point

**on**

**the**initial line propagates

**with**

**the**finite velocity c: there

**is**no effect outside

**a**triangle through that point whose sides are characteristic curves

**.**

__Conversely__

**,**

__given__

**any**harmonic function in two dimensions

**,**it

**is**

**the**real part of

**an**analytic function

**,**at least locally

**.**

__Conversely__

**,**if ( A

**,**m

**,**e

**,**inv )

**is**

**a**group object in one of those categories

**,**then m necessarily coincides

**with**

**the**

__given__operation

**on**A

**,**e

**is**

**the**inclusion of

**the**

__given__

**identity**element

**on**A

**,**inv

**is**

**the**inversion operation

**and**A

**with**

**the**

__given__operation

**is**

**an**abelian group

**.**

__Conversely__

**,**if

**the**Turing Machine

**is**expected polynomial-time ( for

**any**

__given__x ), then

**a**considerable fraction of

**the**runs must be polynomial-time bounded

**,**

**and**

**the**coin sequence used in such

**a**run will be

**a**witness

**.**

__Conversely__

**,**

__given__

**any**ring

**,**we can form

**a**category by taking objects A < sub > n </ sub > indexed by

**the**set of natural numbers ( including zero )

**and**letting

**the**hom-set of morphisms from

**to**be

**the**set of-by-matrices over

**,**

**and**where composition

**is**

__given__by matrix multiplication

**.**

__Conversely__

**,**it

**is**hard

**to**believe

**,**

__given__

**the**length

**and**intensity of

**the**struggle between Máel Sechnaill

**and**Brian

**,**that

**the**High King would surrender his title without

**a**fight

**.**

__Conversely__

**,**

**an**algorithm

**to**test for solvability in arbitrary integers could be used

**to**test

**a**

__given__equation for solvability in natural numbers by applying that supposed algorithm

**to**

**the**equation obtained from

**the**

__given__equation by replacing each unknown by

**the**

**sum**of

**the**squares of four new unknowns

**.**

__Conversely__

**,**

**a**disease that

**is**easily transmitted but has

**a**short duration might spread widely during 2002 but

**is**likely

**to**have

**a**low prevalence at

**any**

__given__point in 2003 ( due

**to**its short duration ) but

**a**high incidence during 2002 ( as many people develop

**the**disease ).

Conversely and any

__Conversely__

**,**

**a**subset R defines

**a**binary function if

**and**only if

**,**for

__any__x in X

**and**y in Y

**,**there exists

**a**unique z in Z such that ( x

**,**y

**,**z ) belongs

**to**R

**.**

__Conversely__

**the**equilibrium position

**is**said

**to**be " far

**to**

**the**left " if hardly

__any__product

**is**formed from

**the**reactants

**.**

* The spectrum of

__any__bounded**linear**operator**on****a**Banach space**is****a**nonempty compact subset of**the**complex numbers C**.**__Conversely__**,**__any__compact subset of C arises in this manner**,**as**the**spectrum of some bounded**linear**operator**.**__Conversely__

**,**

__any__surjection between sets determines

**a**partition

**on**its domain

**,**

**the**set of preimages of singletons in

**the**codomain

**.**

__Conversely__

**,**Derek Bickerton

**and**others argue that

**the**advent of abstract words provided

**a**mental basis for analyzing higher-order relations

**,**

**and**that

__any__communication system that remotely resembles human language utterly relies

**on**cognitive architecture that co-evolved alongside language

**.**

__Conversely__

**,**

**a**person may feel like he belongs

**to**one state without having

__any__legal relationship

**to**it

**.**

__Conversely__

**,**

**the**interior of

**the**box has

**a**constant

**,**zero potential energy .< ref > Actually

**,**

__any__constant

**,**finite potential can be specified within

**the**box

**.**

__Conversely__

**,**extremely thin jewelry can cause

**the**same tearing in what

**is**commonly referred

**to**as

**the**" cheese cutter effect ", either during sudden torsion or over

**a**long period of wearing

**,**especially if

**the**thin jewelry bears

__any__weight

**.**

__Conversely__

**,**

__any__Hermitian positive semidefinite matrix M can be written as M = A < sup >*</ sup > A ; this

**is**

**the**Cholesky

**decomposition**

**.**

__Conversely__

**,**if

**a**double sharp were added

**to**

__any__other note not containing

**a**sharp or flat as indicated by

**the**key signature then

**the**note will be raised two semitones or

**a**whole step

**with**respect

**to**

**the**chromatic scale

**.**

__Conversely__

**,**supplementation

**with**L-phenylalanine

**,**L-tyrosine

**,**L-DOPA

**,**or

__any__of

**the**cofactors will increase their respective concentrations

**.**

__Conversely__

**,**

__any__choice of μ

**,**full rank matrix U

**,**

**and**positive diagonal entries Λ < sub > i </ sub > yields

**a**non-singular multivariate normal distribution

**.**

It

**is****a**fact that**the**ring**is****a**principal ideal ring ; that**is****,**for__any__ideal I in**,**there exists**an**integer n in I such that every element of I**is****a**multiple of n**.**__Conversely__**,****the**set of all multiples of**an**arbitrary integer n**is**necessarily**an**ideal**,****and****is**usually denoted by ( n ).0.080 seconds.