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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 >.

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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.

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 central

__Conversely__

**,**some regions such as Nuristan

**,**

**in**

**the**northeast

**,**

**and**Hazarajat

**,**

**in**

**the**

__central__mountains

**of**Afghanistan

**,**were virtually untouched by

**the**fighting

**,**

**and**lived

**in**almost complete independence.

__Conversely__

**,**often

**in**film noir

**and**crime dramas

**,**

**the**alternative narrative

**is**

**a**fiction created by

**a**

__central__character

**,**intentionally — as

**in**The Usual Suspects — or unintentionally — as

**in**Angel Heart.

__Conversely__

**,**if

**the**

__central__star's energy output significantly increases

**,**

**the**oceans may evaporate

**,**resulting

**in**

**a**greenhouse effect ; this may preclude

**the**oceans from reforming.

__Conversely__

**,**she

**is**not as

__central__

**in**most forms

**of**established Christianity as she

**is**

**in**Gnosticism

**,**but to some Christian mystics

**the**Sophia

**is**

**a**very important concept.

Conversely and <

__Conversely__

**,**some elements do not maintain distinct allotropes

**in**different phases – for example phosphorus has numerous solid allotropes

**,**which all revert to

**the**same P

__<__

**sub**

**>**4

**</**

**sub**

**>**form when melted to

**the**liquid state.

__Conversely__

**,**if ƒ: C → C

**is**

**a**function which

**is**differentiable when regarded as

**a**function on

**R**

__<__sup

**>**2

**</**sup >,

**then**ƒ

**is**complex differentiable if

**and**only if

**the**Cauchy – Riemann equations hold.

__Conversely__

**,**if

**the**measured potential

**is**negative

**,**

**the**electrode

**is**more oxidizing than

**the**SHE ( such as Zn

**in**ZnSO

__<__

**sub**

**>**4

**</**

**sub**

**>**where

**the**standard electrode potential

**is**− 0. 76 V ).

__Conversely__

**,**if

**the**data

**are**sparse —

**that**

**is**

**,**if only K out

**of**N Fourier coefficients

**are**nonzero —

**then**

**the**complexity can be reduced to O ( K log N log ( N / K )),

**and**this has been demonstrated to lead to practical speedups compared to an ordinary FFT for N / K

**>**32

**in**

**a**large-N example ( N = 2

__<__sup

**>**22

**</**sup >) using

**a**probabilistic approximate algorithm ( which estimates

**the**largest K coefficients to several decimal places ).

__Conversely__

**,**when one wants to compute an arbitrary number ( N )

**of**discrete samples

**of**one cycle

**of**

**a**continuous DTFT

**,**it can be done by computing

**the**relatively simple DFT

**of**s

__<__

**sub**

**>**N

**</**

**sub**>, as defined above.

__Conversely__

**,**

**a**high pH

**,**low CO

__<__

**sub**

**>**2

**</**

**sub**>, or low 2

**,**3 BPG favors

**the**relaxed form which can better bind oxygen.

__Conversely__we

**have**

**a**b if

**and**only if

**a**

__<__b or

**a**~ b. This

**is**

**the**reason for using

**the**notation ""; "≤" can be confusing for

**a**preorder

**that**

**is**not anti-symmetric

**,**it may suggest

**that**

**a**≤ b implies

**that**

**a**

__<__b or

**a**

__Conversely__

**,**if dφ

__<__

**sub**

**>**x

**</**

**sub**

**>**

**is**an isomorphism

**then**there

**is**an open neighborhood U

**of**x such

**that**φ maps U diffeomorphically onto its image.

__Conversely__every positive functional φ gives

**a**corresponding inner product

__<__ƒ

**,**g ><

**sub**

**>**φ

**</**

**sub**

**>**= φ ( g * ƒ ).

__Conversely__

**,**if M

**is**positive definite

**,**it has an eigendecomposition P

__<__sup >−

**1**

**</**sup

**>**DP where P

**is**unitary

**,**D diagonal

**,**

**and**all diagonal elements

**of**D

**are**real

**and**positive.

__Conversely__

**,**any Hermitian positive semidefinite matrix M can be written as M = A

__<__sup >*</ sup

**>**A ; this

**is**

**the**Cholesky decomposition.

__Conversely__

**,**any choice

**of**μ

**,**full rank matrix U

**,**

**and**positive diagonal entries Λ

__<__

**sub**

**>**i

**</**

**sub**

**>**yields

**a**non-singular multivariate normal distribution.

__Conversely__

**,**for

**the**previous range from 2

__<__sup

**>**51

**</**

**sub**

**>**to 2

__<__sup

**>**52

**</**

**sub**>,

**the**spacing

**is**0. 5

**,**etc.

Conversely and >

__Conversely__

**,**

**the**interior

**of**

**the**box has

**a**constant

**,**zero potential energy .< ref

__>__Actually

**,**any constant

**,**finite potential can be specified within

**the**box.

0.147 seconds.