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Formally, a biased graph Ω is a pair ( G, B ) where B is a linear class of circles ; this by definition is a class of circles that satisfies the theta-graph property mentioned above.

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## Some Related Sentences

Formally and graph

__Formally__

**,**

**a**vertex cover

**of**

**a**

__graph__

**G**

**is**

**a**set C

**of**vertices such

**that**each edge

**of**

**G**

**is**incident to at least one vertex in C

**.**The set C

**is**said to cover

**the**edges

**of**

**G**

**.**The following figure shows examples

**of**vertex covers in two graphs

**(**and

**the**set C

**is**marked with red ).

__Formally__

**,**

**a**

__graph__rewriting system usually consists

**of**

**a**set

**of**

__graph__rewrite rules

**of**

**the**form

**,**with being called pattern

__graph__

**(**or left-hand side

**)**and being called replacement

__graph__

**(**or right-hand side

**of**

**the**rule ).

__Formally__

**,**

**a**

__graph__

**is**symmetric with respect to

**the**origin if it

**is**unchanged when reflected across both

**the**x-axis and y-axis

**.**

__Formally__

**,**given

**a**

__graph__

**G**

**,**

**a**vertex labeling

**is**

**a**function mapping vertices

**of**

**G**to

**a**set

**of**labels

**.**

__Formally__

**,**

**a**signed

__graph__Σ

**is**

**a**

**pair**

**(**

**G**

**,**σ

**)**

**that**consists

**of**

**a**

__graph__

**G**=

**(**V

**,**E

**)**and

**a**sign mapping or signature σ from E to

**the**sign group

__Formally__

**,**

**the**upper density

**of**

**a**

__graph__

**G**

**is**

**the**infimum

**of**

**the**values α such

**that**

**the**finite subgraphs

**of**

**G**with density α have

**a**bounded number

**of**vertices

**.**

Formally and Ω

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__

**,**oxidation state

__is__

**the**hypothetical charge

**that**an atom would have if all bonds to atoms

**of**different elements were 100 % ionic

**.**

__Formally__

**,**an inner product space

__is__

**a**vector space V over

**the**field together with an inner product

**,**i

**.**e., with

**a**map

__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**

*

__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

Formally and pair

__Formally__

**,**

**a**hypergraph

**is**

**a**

__pair__

**where**

**is**

**a**set

**of**elements called nodes or vertices

**,**and

**is**

**a**set

**of**non-empty subsets

**of**called hyperedges or edges

**.**

__Formally__

**,**an absolute coequalizer

**of**

**a**

__pair__in

**a**category C

**is**

**a**coequalizer as defined

**above**but with

**the**added

**property**

**that**given any functor F

**(**Q

**)**together with F

**(**q

**)**

**is**

**the**coequalizer

**of**F

**(**f

**)**and F

**(**g

**)**in

**the**category D

**.**Split coequalizers are examples

**of**absolute coequalizers

**.**

__Formally__

**,**each

**of**

**the**following definitions defines

**a**concrete category

**,**and every

__pair__

**of**these categories can be shown to be concretely isomorphic

**.**

Formally and G

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__

**,**

**the**index

**of**H in

__G__

**is**defined as

**the**number

**of**cosets

**of**H in

__G__

**.**

**(**The number

**of**left cosets

**of**H in

__G__

**is**always equal to

**the**number

**of**right cosets

**.**

__Formally__

**,**given two categories C and D

**,**an equivalence

**of**categories consists

**of**

**a**functor F: C → D

**,**

**a**functor

__G__: D → C

**,**and two natural isomorphisms ε: FG → I < sub > D </ sub > and η: I < sub > C </ sub >→ GF

**.**

__Formally__

**,**

**a**frame on

**a**homogeneous space

__G__/ H consists

**of**

**a**point in

**the**tautological bundle

__G__→

__G__/ H

**.**

__Formally__

**,**let

__G__be

**a**Coxeter group with reduced root system R and k < sub > v </ sub >

**a**multiplicity function on R

**(**so k < sub > u </ sub > = k < sub > v </ sub > whenever

**the**reflections σ < sub > u </ sub > and σ < sub > v </ sub > corresponding to

**the**roots u and v are conjugate in

__G__).

Formally and B

__Formally__

**,**cl

**(**S

**)**denotes

**the**smallest subset Y

**of**M

**that**contains S such

**that**for each reaction

**(**A

**,**

__B__

**)**

__Formally__

**,**given

**a**finite set X

**,**

**a**collection C

**of**subsets

**of**X

**,**all

**of**size n

**,**has Property

__B__if we can partition X into two disjoint subsets Y and Z such

**that**every set in C meets both Y and Z

**.**

__Formally__

**,**

**the**issue

**is**

**that**interfertile " able to interbreed "

**is**not

**a**transitive relation – if A can breed with

__B__

**,**and

__B__can breed with C

**,**it does not follow

**that**A can breed with C – and thus does not define an equivalence relation

**.**

__Formally__

**,**

**the**use

**of**

**a**reduction

**is**

**the**function

**that**sends each natural number n to

**the**largest natural number m whose membership in

**the**set

__B__was queried

**by**

**the**reduction while determining

**the**membership

**of**n in A

**.**

__Formally__trained in music composition

**,**Dargel studied with Oliveros

**,**John Luther Adams

**,**and Lewis Nielson

**,**and received

**a**

__B__

**.**A

**.**

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