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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

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## Some Related Sentences

Formally and signed

__Formally__

__signed__by Count Alexander Izvolsky

**,**Foreign Minister

**of**

**the**Russian Empire

**,**

**and**Sir Arthur Nicolson

**,**

**the**British Ambassador

**to**Russia

**,**

**the**British-Russian Convention

**of**1907 stipulated

**the**following:

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**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.

__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 Σ

__Formally__

**,**

**a**deterministic Büchi automaton

**is**

**a**tuple A

**=**

**(**Q

**,**

__Σ__

**,**δ

**,**q < sub > 0 </ sub >, F

**)**

**that**

**consists**

**of**

**the**following components:

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__).

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