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Conversely, given a Hopf algebra whose coalgebra structure is given above, we can recover a formal group law F from it.

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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 algebra

__Conversely__

**,**many theorems that hold in universal

__algebra__do not generalise all the way to category theory

**.**

__Conversely__

**,**once

**a**basis for A has been chosen

**,**the products of basis elements

**can**be set arbitrarily

**,**and then extended in

**a**unique way to

**a**bilinear operator on A

**,**i

**.**e., so the resulting multiplication satisfies the

__algebra__laws

**.**

__Conversely__

**,**

**given**any topological space X

**,**the collection of subsets of X that are clopen ( both closed and open )

**is**

**a**Boolean

__algebra__

**.**

__Conversely__

**,**symmetric tensors are defined as invariants:

**given**the natural action of the symmetric

**group**on the tensor

__algebra__

**,**the symmetric tensors are the subspace on which the symmetric

**group**acts trivially

**.**

__Conversely__

**,**the normed division algebras immediately give rise to trialities by taking each V < sub > i </ sub > equal to the division

__algebra__

**,**and using the inner product on the

__algebra__to dualize the multiplication into

**a**trilinear form

**.**

__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

**.**

Conversely and whose

__Conversely__

**,**the existence of Bletchley Park

**,**and of the decrypting achievements there

**,**was never officially shared with the Soviet Union

**,**

__whose__war effort would have greatly benefited

**from**regular decrypting of German messages relating to the Eastern Front

**.**

__Conversely__

**,**since Imhotep was known as the " Son of Ptah ," his mother was sometimes claimed to be Sekhmet

**,**the patron of Upper Egypt

__whose__consort was Ptah

**.**

__Conversely__

**,**citizens may find themselves ineligible because they do not currently reside within the United States ; exceptions apply for children of military parent ( s ) who were born overseas

**,**were disabled or became blind overseas

**,**or first applied for benefits overseas and for students studying abroad who were eligible for SSI in the month prior to leaving the US

**,**

__whose__absence will be for less than 1 year

**,**and who are studying to enhance their ability to perform substantial gainful activity

**,**sponsored by an educational institution in the US

**,**and would not be available to the individual in the US

**.**

__Conversely__

**,**nominating the party leader

__whose__party maintains

**a**plurality and who are already familiar with their party manifesto facilitates

**a**smoother nomination process

**.**

Equipped with this notion of length

**,****a**Riemannian manifold**is****a**metric space**,**meaning that**it**has**a**distance function d ( p**,**q )__whose__value at**a**pair of points p and q**is**the distance**from**p to q**.**__Conversely__**,**the metric tensor itself**is**the derivative of the distance function ( taken in**a**suitable manner ).__Conversely__

**,**nominating the party leader

__whose__party maintains

**a**plurality and who are already familiar with their party manifesto facilitates

**a**smoother nomination process

**.**

__Conversely__

**,**the valuation on

**a**discrete valuation ring

**can**be extended to

**a**valuation on the quotient field giving

**a**discrete valued field

**,**

__whose__associated discrete valuation ring

**is**just

**.**

__Conversely__this book

**,**

__whose__discovery

**is**touted in 2 Kings

**,**was therefore believed to have been of an early enough authorship to validate

**,**not only its inclusion

**,**but the book's ultimate placement as the 5th of the " Five Books of Moses ".

__Conversely__

**,**Clover points out that the villain of slasher films

**is**often

**a**male

__whose__masculinity

**,**and sexuality more generally

**,**are in crisis

**.**

To

**a**graph G there corresponds**a**signed complete graph Σ on the same vertex set**,**__whose__edges are signed negative if in G and positive if not in G**.**__Conversely__**,**G**is**the subgraph of Σ that consists of all vertices and all negative edges**.**__Conversely__

**,**in the United States Navy

**,**the term " shipmate "

**can**be the derogatory term used by officers and senior enlisted to refer to

**a**sailor

__whose__name they have not bothered to remember or read off of their uniform ( Example: Hey!

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