text: 7 This second order term contains all the elements oc- curring in the expansion, namely the bare propagators and hybridization lines. A graphical representation is shown in Fig. 2: in the upper left panel, atomic state propagators for a given state are represented by a pair of lines; the upper one representing spin up, the lower one spin down. Dotted lines represent empty states and solid lines represent occupied states, such that the full set of 2N states is represented by N lines. Additionally, thick (or bold) lines represent full propagators while thin lines stand for bare propagators. Full propagators are built out of all possible combinations of bare propagators and interaction vertices (shown to the upper right), with all interaction times integrated over. Every hybridization Figure 3. The matrix elements of the NCA self energy (two line connects two interactions, and each such line comes terms on the left with a single hybridization line) and of the with the minus sign of Eq. 35. The second order contri- OCA self energy (all four terms) in diagrammatic form. bution explicitly written above includes both the 2nd and 3rd term in the expansion of G shown in the bottom 00 part of Fig. 2, where an additional 8th order diagram is Each diagram in the bare expansion is contained in ex- also shown. actly one bold diagram, which has the same number of or The essence of the CTQMC method35,37,41,42 is that fewer vertices. Since the original expansion was conver- one can evaluate the sum of all diagrams by sampling gent, the sampling process of all diagrams gives the same them stochastically using the Metropolis algorithm. In exact answer—however, if the renormalized propagators particular, the ‘continuous time’ descriptor implies that, contain a large proportion of the most important con- unlike in previous algorithms,34systematic Trotter time- tributions, convergence occurs at lower diagram order, discretization errors are absent.61 In the hybridization is substantially faster, and the sign problem is greatly expansion continuous-time algorithm,36,37 the stochastic alleviated.50 sampling of diagrams is done by insertion of pairs of dot The NCA approximation to the propagator is de- operators and hybridization lines: Namely, in each Monte scribed pictorially in the ﬁrst two terms of Fig. 3 with Carlo step, a hybridization line and two dot operators are one hybridization line (see also Ref. 52, which uses a created, destroyed, or moved. This set of updates can be more compact notation). The OCA also includes the shown to respect ergodicity, and moves can be weighed two additional diagrams with two hybridization lines. in such a way that detailed balance is also maintained.37 The auxiliary state self energy Σ is inserted into the αα Since the leads are noninteracting both initially and un- causal Dyson equation der propagation by H , Wick’s theorem is respected for 0 lteoaedﬃ(tchieonutglyh snuomt floarrgtheesedtost)ofopdieargartaomrss, ahnadviintgisthpeosssaibmlee Gαα (t − t(cid:48)) = Gˆ (α0α) (t −ˆ t(cid:48)) (40) path—dot state as a function of time—in the form of t t1 × determinants, signiﬁcantly reducing the sign problem.36 + dt1 dt2 (cid:48) (cid:48) t t In bold-line CTQMC, rather than summing each di- − − − (cid:48) G(0) (t t ) Σ (t t ) G (t t ) agram explicitly, one ﬁrst obtains an approximation for αα 1 αα 1 2 αα 2 the atomic state propagator that is better than the bare atomic state propagator by dressing it using some ansatz and solved self-consistently. For Hamiltonians without for the self energy. If one chooses a self-consistent ap- an explicit time dependence, like the Anderson impurity proximation such as the NCA or OCA, this allows for the Hamiltonian considered here, the propagator is a func- relatively inexpensive computation of a renormalized, or tion of only time diﬀerences even outside of steady state bold line, propagator which contains an inﬁnite (though conditions. This can be seen directly from its deﬁnition still partial) set of diagrams. This is not the full prop- in Eq. 31. In terms of numerical eﬀort the cost of solving agator, but often represents a reasonable approximation the Dyson equation is therefore negligible even for very to it. It is then possible to write all additional diagrams long propagation times, within both the NCA and the in terms of this new bold propagator. To avoid double OCA. counting diagrams, one must take into account only dia- Bold diagrams no longer have the determinant struc- grams which do not contain parts already summed within ture of bare diagrams, since some permutations of lead the underlying approximation. In a Monte Carlo algo- operators are contained in the underlying partial sum- rithm this is feasible if one rejects any update that leads mation, while most are not. It is therefore necessary to an unwanted diagram, while ascertaining that still to explicitly treat the sum of individual hybridization all diagrams are generated (thus maintaining ergodicity). lines. In practice, cancellation eﬀects of diagrams with ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: This

text: second

text: order

text: term

text: contains

text: all

text: the

text: elements

text: oc-

text: curring

text: in

text: the

text: expansion,

text: namely

text: the

text: bare

text: propagators

text: and

text: hybridization

text: lines.

text: A

text: graphical

text: representation

text: is

text: shown

text: in

text: Fig.

text: in

text: the

text: upper

text: left

text: panel,

text: atomic

text: state

text: propagators

text: for

text: a

text: given

text: state

text: are

text: represented

text: by

text: a

text: pair

text: of

text: lines;

text: the

text: upper

text: one

text: representing

text: spin

text: up,

text: the

text: lower

text: one

text: spin

text: down.

text: Dotted

text: lines

text: represent

text: empty

text: states

text: and

text: solid

text: lines

text: represent

text: occupied

text: states,

text: such

text: that

text: the

text: full

text: set

text: of

text: states

text: is

text: represented

text: by

text: N

text: lines.

text: Additionally,

text: thick

text: (or

text: bold)

text: lines

text: represent

text: full

text: propagators

text: while

text: thin

text: lines

text: stand

text: for

text: bare

text: propagators.

text: Full

text: propagators

text: are

text: built

text: out

text: of

text: all

text: possible

text: combinations

text: of

text: bare

text: propagators

text: and

text: interaction

text: vertices

text: (shown

text: to

text: the

text: upper

text: right),

text: with

text: all

text: interaction

text: times

text: integrated

text: over.

text: Every

text: hybridization

text: Figure

text: The

text: matrix

text: elements

text: of

text: the

text: NCA

text: self

text: energy

text: (two

text: line

text: connects

text: two

text: interactions,

text: and

text: each

text: such

text: line

text: comes

text: terms

text: on

text: the

text: left

text: with

text: a

text: single

text: hybridization

text: line)

text: and

text: of

text: the

text: with

text: the

text: minus

text: sign

text: of

text: Eq.

text: The

text: second

text: order

text: contri-

text: OCA

text: self

text: energy

text: (all

text: four

text: terms)

text: in

text: diagrammatic

text: form.

text: bution

text: explicitly

text: written

text: above

text: includes

text: both

text: the

text: and

text: term

text: in

text: the

text: expansion

text: of

text: G

text: shown

text: in

text: the

text: bottom

text: part

text: of

text: Fig.

text: where

text: an

text: additional

text: order

text: diagram

text: is

text: Each

text: diagram

text: in

text: the

text: bare

text: expansion

text: is

text: contained

text: in

text: ex-

text: also

text: shown.

text: actly

text: one

text: bold

text: diagram,

text: which

text: has

text: the

text: same

text: number

text: of

text: or

text: The

text: essence

text: of

text: the

text: CTQMC

text: is

text: that

text: fewer

text: vertices.

text: Since

text: the

text: original

text: expansion

text: was

text: conver-

text: one

text: can

text: evaluate

text: the

text: sum

text: of

text: all

text: diagrams

text: by

text: sampling

text: gent,

text: the

text: sampling

text: process

text: of

text: all

text: diagrams

text: gives

text: the

text: same

text: them

text: stochastically

text: using

text: the

text: Metropolis

text: algorithm.

text: In

text: exact

text: answer—however,

text: if

text: the

text: renormalized

text: propagators

text: particular,

text: the

text: ‘continuous

text: time’

text: descriptor

text: implies

text: that,

text: contain

text: a

text: large

text: proportion

text: of

text: the

text: most

text: important

text: con-

text: unlike

text: in

text: previous

text: Trotter

text: time-

text: tributions,

text: convergence

text: occurs

text: at

text: lower

text: diagram

text: order,

text: discretization

text: errors

text: are

text: In

text: the

text: hybridization

text: is

text: substantially

text: faster,

text: and

text: the

text: sign

text: problem

text: is

text: greatly

text: expansion

text: continuous-time

text: the

text: stochastic

text: sampling

text: of

text: diagrams

text: is

text: done

text: by

text: insertion

text: of

text: pairs

text: of

text: dot

text: The

text: NCA

text: approximation

text: to

text: the

text: propagator

text: is

text: de-

text: operators

text: and

text: hybridization

text: lines:

text: Namely,

text: in

text: each

text: Monte

text: scribed

text: pictorially

text: in

text: the

text: ﬁrst

text: two

text: terms

text: of

text: Fig.

text: with

text: Carlo

text: step,

text: a

text: hybridization

text: line

text: and

text: two

text: dot

text: operators

text: are

text: one

text: hybridization

text: line

text: (see

text: also

text: Ref.

text: which

text: uses

text: a

text: created,

text: destroyed,

text: or

text: moved.

text: This

text: set

text: of

text: updates

text: can

text: be

text: more

text: compact

text: notation).

text: The

text: OCA

text: also

text: includes

text: the

text: shown

text: to

text: respect

text: ergodicity,

text: and

text: moves

text: can

text: be

text: weighed

text: two

text: additional

text: diagrams

text: with

text: two

text: hybridization

text: lines.

text: in

text: such

text: a

text: way

text: that

text: detailed

text: balance

text: is

text: also

text: The

text: auxiliary

text: state

text: self

text: energy

text: Σ

text: is

text: inserted

text: into

text: the

text: αα

text: Since

text: the

text: leads

text: are

text: noninteracting

text: both

text: initially

text: and

text: un-

text: causal

text: Dyson

text: equation

text: der

text: propagation

text: by

text: H

text: Wick’s

text: theorem

text: is

text: respected

text: for

text: lteoaedﬃ(tchieonutglyh

text: snuomt

text: floarrgtheesedtost)ofopdieargartaomrss,

text: ahnadviintgisthpeosssaibmlee

text: Gαα

text: (t

text: Gˆ

text: (t

text: −ˆ

text: path—dot

text: state

text: as

text: a

text: function

text: of

text: time—in

text: the

text: form

text: of

text: t

text: determinants,

text: signiﬁcantly

text: reducing

text: the

text: sign

text: t

text: t

text: In

text: bold-line

text: CTQMC,

text: rather

text: than

text: summing

text: each

text: di-

text: (t

text: t

text: Σ

text: (t

text: t

text: G

text: (t

text: t

text: agram

text: explicitly,

text: one

text: ﬁrst

text: obtains

text: an

text: approximation

text: for

text: αα

text: αα

text: αα

text: the

text: atomic

text: state

text: propagator

text: that

text: is

text: better

text: than

text: the

text: bare

text: atomic

text: state

text: propagator

text: by

text: dressing

text: it

text: using

text: some

text: ansatz

text: and

text: solved

text: self-consistently.

text: For

text: Hamiltonians

text: without

text: for

text: the

text: self

text: energy.

text: If

text: one

text: chooses

text: a

text: self-consistent

text: ap-

text: an

text: explicit

text: time

text: dependence,

text: like

text: the

text: Anderson

text: impurity

text: proximation

text: such

text: as

text: the

text: NCA

text: or

text: OCA,

text: this

text: allows

text: for

text: the

text: Hamiltonian

text: considered

text: here,

text: the

text: propagator

text: is

text: a

text: func-

text: relatively

text: inexpensive

text: computation

text: of

text: a

text: renormalized,

text: or

text: tion

text: of

text: only

text: time

text: diﬀerences

text: even

text: outside

text: of

text: steady

text: state

text: bold

text: line,

text: propagator

text: which

text: contains

text: an

text: inﬁnite

text: (though

text: conditions.

text: This

text: can

text: be

text: seen

text: directly

text: from

text: its

text: deﬁnition

text: still

text: partial)

text: set

text: of

text: diagrams.

text: This

text: is

text: not

text: the

text: full

text: prop-

text: in

text: Eq.

text: In

text: terms

text: of

text: numerical

text: eﬀort

text: the

text: cost

text: of

text: solving

text: agator,

text: but

text: often

text: represents

text: a

text: reasonable

text: approximation

text: the

text: Dyson

text: equation

text: is

text: therefore

text: negligible

text: even

text: for

text: very

text: to

text: it.

text: It

text: is

text: then

text: possible

text: to

text: write

text: all

text: additional

text: diagrams

text: long

text: propagation

text: times,

text: within

text: both

text: the

text: NCA

text: and

text: the

text: in

text: terms

text: of

text: this

text: new

text: bold

text: propagator.

text: To

text: avoid

text: double

text: OCA.

text: counting

text: diagrams,

text: one

text: must

text: take

text: into

text: account

text: only

text: dia-

text: Bold

text: diagrams

text: no

text: longer

text: have

text: the

text: determinant

text: struc-

text: grams

text: which

text: do

text: not

text: contain

text: parts

text: already

text: summed

text: within

text: ture

text: of

text: bare

text: diagrams,

text: since

text: some

text: permutations

text: of

text: lead

text: the

text: underlying

text: approximation.

text: In

text: a

text: Monte

text: Carlo

text: algo-

text: operators

text: are

text: contained

text: in

text: the

text: underlying

text: partial

text: sum-

text: rithm

text: this

text: is

text: feasible

text: if

text: one

text: rejects

text: any

text: update

text: that

text: leads

text: mation,

text: while

text: most

text: are

text: not.

text: It

text: is

text: therefore

text: necessary

text: to

text: an

text: unwanted

text: diagram,

text: while

text: ascertaining

text: that

text: still

text: to

text: explicitly

text: treat

text: the

text: sum

text: of

text: individual

text: hybridization

text: all

text: diagrams

text: are

text: generated

text: (thus

text: maintaining

text: ergodicity).

text: lines.

text: In

text: practice,

text: cancellation

text: eﬀects

text: of

text: diagrams

text: with

text: ##LTFigure##

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text: EQUIVARIANT ALEXANDROV GEOMETRY AND ORBIFOLD FINITENESS 13 (3) induces a convergence of X to X in the Hausdorff metric on the X ij closed subsets of ; and therefore G is, after factoring out any ineffective kernel of its action on X, isomorphic to Γ. → Proof. Fix Gromov–Hausdorff (cid:15) –aproximations f : X X which wit- i i i ness the Gromov–Hausdorff convergence of the underlying metric spaces. Using these approximations, it is possible to deﬁne a limiting G–action on X as follows. × → Consider the actions as continuous maps φ : G X X . Fixing × i i i a metric on G, the functions id f are Gromov–Hausdorff approxi- G i × × mations showing the convergence of G X to G X. By the Grove– i Petersen–Arzela`–Ascoli Theorem, one can extract from the equicontinuous subsequence φ a compact subsequence converging to a continuous map × → i φ: G X X [15, Appendix]. It is clear that this map is also an isomet- ric action. → ◦ Now pick approximations g : X X such that g f is close to the → i i i ◦i identity. Let h : X X be deﬁned by h = g f . Then h is i i i+1 i i+1 i i a Gromov–Hausdorff 5(cid:15) –approximation which is almost equivariant with i respect to the action of G, and so (h , id , id ) can be used as an equivariant i G G (cid:96) Gromov–Hausdorff r –aproximation. The quantity r depends both on (cid:15) i i i and on the rate of convergence of the φ to φ. i It is then possible to place a metric on the disjoint union X X i i+1 such that dist(x, h (x)) = r (see Burago, Burago and Ivanov [4, Corol- i i lary 7.3.28]). This metric can be rendered G-invariant by the usual av- eraging procedure, at a small cost—h is now a Gromov–Hausdorff 3r – i i approximation. The restriction of the metric to X and to X is un- i i+1 changed. Let d be the Hausdorff distance between X and X in this i i (cid:96) i+1 metric. (cid:96) Following Petersen [28, p297], pass to a subsequence so that d < 2i i for all i. Then, by gluing the metrics on each of the X X , a G– (cid:96) i i+1 invariant metric on X can be constructed, which restricts to the original (cid:96) i i X metric on each X . This space can be completed to in such a way that X i X = X X , and X converges to X in the Hausdorff sense in . i i i X Since X is dense in , the isometric G–action can be extended to an i i X isometric action on all of , and the extension to X is the limiting G-action constructed at the beginning of the argument. This action is, after factoring out any ineffective kernel, the limit of the (cid:3) G–actions in the equivariant Gromov–Hausdorff topology.

text: EQUIVARIANT

label: paragraph

text: ALEXANDROV

text: GEOMETRY

text: AND

text: ORBIFOLD

text: FINITENESS

label: list

text: induces

text: a

text: convergence

text: of

text: X

text: to

text: X

text: in

text: the

text: Hausdorff

text: metric

text: on

text: the

text: X

text: ij

text: closed

text: subsets

text: of

text: and

text: therefore

text: G

text: is,

text: after

text: factoring

text: out

text: any

text: ineffective

text: kernel

text: of

text: its

text: action

text: on

text: X,

text: isomorphic

text: to

text: Γ.

text: Proof.

text: Fix

text: Gromov–Hausdorff

text: –aproximations

text: f

text: X

text: X

text: which

text: wit-

text: i

text: i

text: i

text: ness

text: the

text: Gromov–Hausdorff

text: convergence

text: of

text: the

text: underlying

text: metric

text: spaces.

text: Using

text: these

text: approximations,

text: it

text: is

text: possible

text: to

text: deﬁne

text: a

text: limiting

text: G–action

text: on

text: X

text: as

text: follows.

text: Consider

text: the

text: actions

text: as

text: continuous

text: maps

text: φ

text: G

text: X

text: X

text: Fixing

text: i

text: i

text: i

text: a

text: metric

text: on

text: G,

text: the

text: functions

text: id

text: f

text: are

text: Gromov–Hausdorff

text: approxi-

text: G

text: i

text: mations

text: showing

text: the

text: convergence

text: of

text: G

text: X

text: to

text: G

text: X.

text: By

text: the

text: Grove–

text: i

text: Petersen–Arzela`–Ascoli

text: Theorem,

text: one

text: can

text: extract

text: from

text: the

text: equicontinuous

text: subsequence

text: φ

text: a

text: compact

text: subsequence

text: converging

text: to

text: a

text: continuous

text: map

text: i

text: φ:

text: G

text: X

text: X

text: Appendix].

text: It

text: is

text: clear

text: that

text: this

text: map

text: is

text: also

text: an

text: isomet-

text: ric

text: action.

text: Now

text: pick

text: approximations

text: g

text: X

text: X

text: such

text: that

text: g

text: f

text: is

text: close

text: to

text: the

text: i

text: i

text: i

text: ◦i

text: identity.

text: Let

text: h

text: X

text: X

text: be

text: deﬁned

text: by

text: h

text: g

text: f

text: Then

text: h

text: is

text: i

text: i

text: i

text: i

text: i

text: a

text: Gromov–Hausdorff

text: –approximation

text: which

text: is

text: almost

text: equivariant

text: with

text: i

text: respect

text: to

text: the

text: action

text: of

text: G,

text: and

text: so

text: (h

text: id

text: id

text: can

text: be

text: used

text: as

text: an

text: equivariant

text: i

text: G

text: G

text: Gromov–Hausdorff

text: r

text: –aproximation.

text: The

text: quantity

text: r

text: depends

text: both

text: on

text: i

text: i

text: i

text: and

text: on

text: the

text: rate

text: of

text: convergence

text: of

text: the

text: φ

text: to

text: φ.

text: i

text: It

text: is

text: then

text: possible

text: to

text: place

text: a

text: metric

text: on

text: the

text: disjoint

text: union

text: X

text: X

text: i

text: such

text: that

text: dist(x,

text: h

text: (x))

text: r

text: (see

text: Burago,

text: Burago

text: and

text: Ivanov

text: Corol-

text: i

text: i

text: lary

text: This

text: metric

text: can

text: be

text: rendered

text: G-invariant

text: by

text: the

text: usual

text: av-

text: eraging

text: procedure,

text: at

text: a

text: small

text: cost—h

text: is

text: now

text: a

text: Gromov–Hausdorff

text: i

text: i

text: approximation.

text: The

text: restriction

text: of

text: the

text: metric

text: to

text: X

text: and

text: to

text: X

text: is

text: un-

text: i

text: changed.

text: Let

text: d

text: be

text: the

text: Hausdorff

text: distance

text: between

text: X

text: and

text: X

text: in

text: this

text: i

text: i

text: metric.

text: Following

text: Petersen

text: pass

text: to

text: a

text: subsequence

text: so

text: that

text: d

text: i

text: for

text: all

text: i.

text: Then,

text: by

text: gluing

text: the

text: metrics

text: on

text: each

text: of

text: the

text: X

text: X

text: a

text: G–

text: i

text: invariant

text: metric

text: on

text: X

text: can

text: be

text: constructed,

text: which

text: restricts

text: to

text: the

text: original

text: i

text: i

text: X

text: metric

text: on

text: each

text: X

text: This

text: space

text: can

text: be

text: completed

text: to

text: in

text: such

text: a

text: way

text: that

text: X

text: i

text: X

text: X

text: X

text: and

text: X

text: converges

text: to

text: X

text: in

text: the

text: Hausdorff

text: sense

text: in

text: i

text: i

text: i

text: X

text: Since

text: X

text: is

text: dense

text: in

text: the

text: isometric

text: G–action

text: can

text: be

text: extended

text: to

text: an

text: i

text: i

text: X

text: isometric

text: action

text: on

text: all

text: of

text: and

text: the

text: extension

text: to

text: X

text: is

text: the

text: limiting

text: G-action

text: constructed

text: at

text: the

text: beginning

text: of

text: the

text: argument.

text: This

text: action

text: is,

text: after

text: factoring

text: out

text: any

text: ineffective

text: kernel,

text: the

text: limit

text: of

text: the

text: G–actions

text: in

text: the

text: equivariant

text: Gromov–Hausdorff

text: topology.


text: 14 JOHN HARVEY It is now possible to proceed to the proof of Theorem A. The proof relies on a result of the author from the general theory of transformation groups [18], given here as Theorem 5.4. This result, along with the justiﬁcation for its application here, is reviewed at the end of the paper, in Section . (cid:96) Proof of Theorem A. Envelop the convergence. By Lemma 3.4, one may assume by passing to a subsequence that there is X a G-invariant metric on = X X which restricts to the original metrics i i and actions on each of the X , with X converging to X in the Hausdorff i X i → metric on the closed subsets of . Fix approximations θ : X X. (cid:48) i i Let G be the ineffective kernel of the G–action on X (it will be shown (cid:48) later that this is trivial). Now (X , G), converges to (X, G/G ) in the equi- i → (cid:48) variant Gromov–Hausdorff topology. Let π : G G/G be the projection map. Then equivariant Gromov–Hausdorff approximations from (X , G) i to (X, Γ) which witness the convergence are given by the triple (θ , π, s) i where s is a (possibly discontinuous) section of π. The cohomogeneity is constant. X ∈ By applying the slice theorem to , as a sequence of points p X ∈ i i converges to p X the isotropy group G must be larger than G . In p pi particular, the principal orbits of X have dimension no greater than those of ≥ X . In other words, dim(X/G) dim(X /G). i i On the other hand, the orbit spaces X /G converge to X/G under a uni- i form lower curvature bound, so it follows that X /G and X/G have the i same dimension, and are therefore homeomorphic by Perelman’s Stability Theorem. The radius of the tubes is bounded. Here the term tube is used in the transformation groups sense, meaning the image of a slice under translation by the group. ∈ Let p X, and let p¯ be its image in X/G. As described by the author and Searle [19, section 3.4], a tube in an Alexandrov space around the orbit · G p can be constructed by choosing a strictly concave function h¯ on a neighborhood U of p¯ which achieves its maximum at p¯. This construction is due to Perelman [25]. → The gradient ﬂow of h¯ gives a retraction r¯: U p¯. The function h¯ lifts · to a function h on a neighborhood of G p. The gradient ﬂow of h then · gives a G–invariant retraction r onto G p, showing that neighborhood to be a tube around the orbit. By Perelman and Petrunin [26, Lemma 4.3], the construction of h¯ is such that strictly concave functions h¯ exist on neighborhoods in X /G converg- i i ing to h. Let p¯ be the maxima of the h¯ . Let r be such that, for large i, i i B (p¯ ) is contained in the domain of concavity of h¯ . r i i

text: JOHN

text: HARVEY

text: It

text: is

text: now

text: possible

text: to

text: proceed

text: to

text: the

text: proof

text: of

text: Theorem

text: A.

text: The

text: proof

text: relies

text: on

text: a

text: result

text: of

text: the

text: author

text: from

text: the

text: general

text: theory

text: of

text: transformation

text: groups

text: given

text: here

text: as

text: Theorem

text: This

text: result,

text: along

text: with

text: the

text: justiﬁcation

text: for

text: its

text: application

text: here,

text: is

text: reviewed

text: at

text: the

text: end

text: of

text: the

text: paper,

text: in

text: Section

text: Proof

text: of

text: Theorem

text: A.

text: Envelop

text: the

text: convergence.

text: By

text: Lemma

text: one

text: may

text: assume

text: by

text: passing

text: to

text: a

text: subsequence

text: that

text: there

text: is

text: X

text: a

text: G-invariant

text: metric

text: on

text: X

text: X

text: which

text: restricts

text: to

text: the

text: original

text: metrics

text: i

text: i

text: and

text: actions

text: on

text: each

text: of

text: the

text: X

text: with

text: X

text: converging

text: to

text: X

text: in

text: the

text: Hausdorff

text: i

text: X

text: i

text: metric

text: on

text: the

text: closed

text: subsets

text: of

text: Fix

text: approximations

text: θ

text: X

text: X.

text: i

text: i

text: Let

text: G

text: be

text: the

text: ineffective

text: kernel

text: of

text: the

text: G–action

text: on

text: X

text: (it

text: will

text: be

text: shown

text: later

text: that

text: this

text: is

text: trivial).

text: Now

text: (X

text: G),

text: converges

text: to

text: (X,

text: G/G

text: in

text: the

text: equi-

text: i

text: variant

text: Gromov–Hausdorff

text: topology.

text: Let

text: π

text: G

text: G/G

text: be

text: the

text: projection

text: map.

text: Then

text: equivariant

text: Gromov–Hausdorff

text: approximations

text: from

text: (X

text: G)

text: i

text: to

text: (X,

text: Γ)

text: which

text: witness

text: the

text: convergence

text: are

text: given

text: by

text: the

text: triple

text: (θ

text: π,

text: s)

text: i

text: where

text: s

text: is

text: a

text: (possibly

text: discontinuous)

text: section

text: of

text: π.

text: The

text: cohomogeneity

text: is

text: constant.

text: X

text: By

text: applying

text: the

text: slice

text: theorem

text: to

text: as

text: a

text: sequence

text: of

text: points

text: p

text: X

text: i

text: i

text: converges

text: to

text: p

text: X

text: the

text: isotropy

text: group

text: G

text: must

text: be

text: larger

text: than

text: G

text: In

text: p

text: pi

text: particular,

text: the

text: principal

text: orbits

text: of

text: X

text: have

text: dimension

text: no

text: greater

text: than

text: those

text: of

text: X

text: In

text: other

text: words,

text: dim(X/G)

text: dim(X

text: /G).

text: i

text: i

text: On

text: the

text: other

text: hand,

text: the

text: orbit

text: spaces

text: X

text: /G

text: converge

text: to

text: X/G

text: under

text: a

text: uni-

text: i

text: form

text: lower

text: curvature

text: bound,

text: so

text: it

text: follows

text: that

text: X

text: /G

text: and

text: X/G

text: have

text: the

text: i

text: same

text: dimension,

text: and

text: are

text: therefore

text: homeomorphic

text: by

text: Perelman’s

text: Stability

text: Theorem.

text: The

text: radius

text: of

text: the

text: tubes

text: is

text: bounded.

text: Here

text: the

text: term

text: tube

text: is

text: used

text: in

text: the

text: transformation

text: groups

text: sense,

text: meaning

text: the

text: image

text: of

text: a

text: slice

text: under

text: translation

text: by

text: the

text: group.

text: Let

text: p

text: X,

text: and

text: let

text: p¯

text: be

text: its

text: image

text: in

text: X/G.

text: As

text: described

text: by

text: the

text: author

text: and

text: Searle

text: section

text: a

text: tube

text: in

text: an

text: Alexandrov

text: space

text: around

text: the

text: orbit

text: G

text: p

text: can

text: be

text: constructed

text: by

text: choosing

text: a

text: strictly

text: concave

text: function

text: h¯

text: on

text: a

text: neighborhood

text: U

text: of

text: p¯

text: which

text: achieves

text: its

text: maximum

text: at

text: p¯.

text: This

text: construction

text: is

text: due

text: to

text: Perelman

text: The

text: gradient

text: ﬂow

text: of

text: h¯

text: gives

text: a

text: retraction

text: r¯:

text: U

text: p¯.

text: The

text: function

text: h¯

text: lifts

text: to

text: a

text: function

text: h

text: on

text: a

text: neighborhood

text: of

text: G

text: p.

text: The

text: gradient

text: ﬂow

text: of

text: h

text: then

text: gives

text: a

text: G–invariant

text: retraction

text: r

text: onto

text: G

text: p,

text: showing

text: that

text: neighborhood

text: to

text: be

text: a

text: tube

text: around

text: the

text: orbit.

text: By

text: Perelman

text: and

text: Petrunin

text: Lemma

text: the

text: construction

text: of

text: h¯

text: is

text: such

text: that

text: strictly

text: concave

text: functions

text: h¯

text: exist

text: on

text: neighborhoods

text: in

text: X

text: /G

text: converg-

text: i

text: i

text: ing

text: to

text: h.

text: Let

text: p¯

text: be

text: the

text: maxima

text: of

text: the

text: h¯

text: Let

text: r

text: be

text: such

text: that,

text: for

text: large

text: i,

text: i

text: i

text: B

text: (p¯

text: is

text: contained

text: in

text: the

text: domain

text: of

text: concavity

text: of

text: h¯

text: r

text: i

text: i


text: 18 JOHN HARVEY orbifold map are given there, of which this is the most na¨ıve notion, the reduced orbifold map. ∈ Let X be an orbifold, let p X, and let (U˜, Γ, π) be a chart over p with π(y) = p. The isotropy group of y will be called the local group at p, and will be written as Γ . It is uniquely deﬁned up to conjugacy in Γ, p and choosing a different chart does not change the isomorphism type of the group. In fact, one can always choose a linear chart over p such that the group of automorphisms is isomorphic to Γ . By this is meant a chart of the form R p ( n, Γ , π) where the action of Γ is via a faithful orthogonal representation p → p ρ : Γ (cid:44) O(n). Such a chart will be referred to as a linear chart around p p p. The representation is also uniquely determined up to isomorphism, and will be called the local action at p. The differential of the action of Γ at p the origin of the chart is also isomorphic to ρ . p A Riemannian metric on an orbifold can be given by ﬁxing a ﬁnite atlas and a partition of unity with respect to the corresponding cover, and choos- ing Riemannian metrics on the charts which are invariant with respect to the ﬁnite group action. An orbifold equipped with a Riemannian metric is called a Riemannian orbifold. Once the metric on the orbifold is given it can be lifted to the maximal atlas in a canonical manner. The various no- tions of curvature at points of an orbifold can then be deﬁned by reference to the curvature of the charts. ≥ It is straightforward to see that an orbifold with sectional curvature k is ≥ also an Alexandrov space with curvature k. The tangent cone at any point of an orbifold is then well-deﬁned, and coincides with the usual notion of tangent space for orbifolds. The notion of an extremal set now ﬁnds a very natural application in orbifolds. Proposition 4.3. Let X be an orbifold of dimension n, Γ a ﬁnite group, and → ρ: Γ (cid:44) O(n) a linear representation of Γ. Let Xρ be the closure of all points with local action ρ. Then Xρ is an extremal set of X. ∈ Proof. The result is clear where n = 1. Let p Xρ and consider the local action at p by Γ . The tangent cone at p is the cone on the quotient of the unit p sphere by Γ . Consider the image of those points in the unit sphere having p isotropy isomorphic to ρ. The closure of the cone on this set is T Xρ and p (cid:3) by induction it is extremal in T X. Since Xρ is closed, it is extremal. p The following lemma now shows that a linear chart around p can be ex- tended over any cone-like set around p. ∈ Lemma 4.4. Let X be an orbifold, and let p X. Let U be a cone-like set around p. Then there is a linear chart over U around p.

text: JOHN

text: HARVEY

text: orbifold

text: map

text: are

text: given

text: there,

text: of

text: which

text: this

text: is

text: the

text: most

text: na¨ıve

text: notion,

text: the

text: reduced

text: orbifold

text: map.

text: Let

text: X

text: be

text: an

text: orbifold,

text: let

text: p

text: X,

text: and

text: let

text: (U˜,

text: Γ,

text: π)

text: be

text: a

text: chart

text: over

text: p

text: with

text: π(y)

text: p.

text: The

text: isotropy

text: group

text: of

text: y

text: will

text: be

text: called

text: the

text: local

text: group

text: at

text: p,

text: and

text: will

text: be

text: written

text: as

text: Γ

text: It

text: is

text: uniquely

text: deﬁned

text: up

text: to

text: conjugacy

text: in

text: Γ,

text: p

text: and

text: choosing

text: a

text: different

text: chart

text: does

text: not

text: change

text: the

text: isomorphism

text: type

text: of

text: the

text: group.

text: In

text: fact,

text: one

text: can

text: always

text: choose

text: a

text: linear

text: chart

text: over

text: p

text: such

text: that

text: the

text: group

text: of

text: automorphisms

text: is

text: isomorphic

text: to

text: Γ

text: By

text: this

text: is

text: meant

text: a

text: chart

text: of

text: the

text: form

text: R

text: p

text: n,

text: Γ

text: π)

text: where

text: the

text: action

text: of

text: Γ

text: is

text: via

text: a

text: faithful

text: orthogonal

text: representation

text: p

text: p

text: ρ

text: Γ

text: O(n).

text: Such

text: a

text: chart

text: will

text: be

text: referred

text: to

text: as

text: a

text: linear

text: chart

text: around

text: p

text: p

text: p.

text: The

text: representation

text: is

text: also

text: uniquely

text: determined

text: up

text: to

text: isomorphism,

text: and

text: will

text: be

text: called

text: the

text: local

text: action

text: at

text: p.

text: The

text: differential

text: of

text: the

text: action

text: of

text: Γ

text: at

text: p

text: the

text: origin

text: of

text: the

text: chart

text: is

text: also

text: isomorphic

text: to

text: ρ

text: p

text: A

text: Riemannian

text: metric

text: on

text: an

text: orbifold

text: can

text: be

text: given

text: by

text: ﬁxing

text: a

text: ﬁnite

text: atlas

text: and

text: a

text: partition

text: of

text: unity

text: with

text: respect

text: to

text: the

text: corresponding

text: cover,

text: and

text: choos-

text: ing

text: Riemannian

text: metrics

text: on

text: the

text: charts

text: which

text: are

text: invariant

text: with

text: respect

text: to

text: the

text: ﬁnite

text: group

text: action.

text: An

text: orbifold

text: equipped

text: with

text: a

text: Riemannian

text: metric

text: is

text: called

text: a

text: Riemannian

text: orbifold.

text: Once

text: the

text: metric

text: on

text: the

text: orbifold

text: is

text: given

text: it

text: can

text: be

text: lifted

text: to

text: the

text: maximal

text: atlas

text: in

text: a

text: canonical

text: manner.

text: The

text: various

text: no-

text: tions

text: of

text: curvature

text: at

text: points

text: of

text: an

text: orbifold

text: can

text: then

text: be

text: deﬁned

text: by

text: reference

text: to

text: the

text: curvature

text: of

text: the

text: charts.

text: It

text: is

text: straightforward

text: to

text: see

text: that

text: an

text: orbifold

text: with

text: sectional

text: curvature

text: k

text: is

text: also

text: an

text: Alexandrov

text: space

text: with

text: curvature

text: k.

text: The

text: tangent

text: cone

text: at

text: any

text: point

text: of

text: an

text: orbifold

text: is

text: then

text: well-deﬁned,

text: and

text: coincides

text: with

text: the

text: usual

text: notion

text: of

text: tangent

text: space

text: for

text: orbifolds.

text: The

text: notion

text: of

text: an

text: extremal

text: set

text: now

text: ﬁnds

text: a

text: very

text: natural

text: application

text: in

text: orbifolds.

text: Proposition

text: Let

text: X

text: be

text: an

text: orbifold

text: of

text: dimension

text: n,

text: Γ

text: a

text: ﬁnite

text: group,

text: and

text: ρ:

text: Γ

text: O(n)

text: a

text: linear

text: representation

text: of

text: Γ.

text: Let

text: Xρ

text: be

text: the

text: closure

text: of

text: all

text: points

text: with

text: local

text: action

text: ρ.

text: Then

text: Xρ

text: is

text: an

text: extremal

text: set

text: of

text: X.

text: Proof.

text: The

text: result

text: is

text: clear

text: where

text: n

text: Let

text: p

text: Xρ

text: and

text: consider

text: the

text: local

text: action

text: at

text: p

text: by

text: Γ

text: The

text: tangent

text: cone

text: at

text: p

text: is

text: the

text: cone

text: on

text: the

text: quotient

text: of

text: the

text: unit

text: p

text: sphere

text: by

text: Γ

text: Consider

text: the

text: image

text: of

text: those

text: points

text: in

text: the

text: unit

text: sphere

text: having

text: p

text: isotropy

text: isomorphic

text: to

text: ρ.

text: The

text: closure

text: of

text: the

text: cone

text: on

text: this

text: set

text: is

text: T

text: Xρ

text: and

text: p

text: by

text: induction

text: it

text: is

text: extremal

text: in

text: T

text: X.

text: Since

text: Xρ

text: is

text: closed,

text: it

text: is

text: extremal.

text: p

text: The

text: following

text: lemma

text: now

text: shows

text: that

text: a

text: linear

text: chart

text: around

text: p

text: can

text: be

text: ex-

text: tended

text: over

text: any

text: cone-like

text: set

text: around

text: p.

text: Lemma

text: Let

text: X

text: be

text: an

text: orbifold,

text: and

text: let

text: p

text: X.

text: Let

text: U

text: be

text: a

text: cone-like

text: set

text: around

text: p.

text: Then

text: there

text: is

text: a

text: linear

text: chart

text: over

text: U

text: around

text: p.


text: EQUIVARIANT ALEXANDROV GEOMETRY AND ORBIFOLD FINITENESS 3 The ﬁrst such result is that of Weinstein, who showed that, for δ > 0, M · · 1,·,·(2n), the class of unformly pinched positively curved manifolds of δ, , even dimension, has only ﬁnitely many members up to homotopy [34]. (cid:54) M · Shortly after this, Cheeger showed that, for n = 4, K·,D, (n) has ﬁnitely k, ,v many simply connected members up to diffeomorphism [6]. In [27], Pe- ters shows that this result still holds for n = 4, as well as removing the hypothesis of simple connectivity. Grove and Petersen removed the upper bound on sectional curvature, and M· · obtained ﬁniteness of ,D· , (n) up to homotopy [14]. Shortly afterwards, k, ,v in collaboration with Wu, this result was improved to show ﬁniteness up ≥ to homeomorphism for n 4 [16]. As long as the dimension is not four, the work of Kirby and Siebenmann [21] implies ﬁniteness up to diffeomor- phism. · · Perelman’s Stability Theorem [24] showed that Alex ,D· , (n), the corre- k, ,v sponding class of Alexandrov spaces, is ﬁnite up to homeomorphism. A fortiori, this generalizes the Grove–Petersen–Wu ﬁniteness result to all di- mensions. Theorem B uses the equivariant version of the Stability Theorem to gen- eralize the homeomorphism ﬁniteness result of Grove, Petersen and Wu to the area of Riemannian orbifolds. An orbifold is a mild generalization of a manifold, and, to give just a few examples, the concept has found applica- tions in Thurston’s work on the Geometrization Conjecture [33], the con- struction of a new positively curved manifold by Dearricott [7] and Grove– Verdiani–Ziller [17], and string theory, such as Dixon, Harvey, Vafa and Witten’s conformal ﬁeld theory built on a quotient of a torus [8]. The same M O convenient notation can be used for orbifolds, here replacing with . The ﬁrst ﬁniteness result for orbifolds is that of Fukaya [10], who gener- O · alized the result of Cheeger, showing that a subclass of K·,D, (n) is ﬁnite k, ,v up to orbifold diffeomorphism. Fukaya used a much more restrictive def- inition of orbifold, considering only the orbit spaces of global actions by ﬁnite groups on Riemannian manifolds. This corresponds to what Thurston called a “good” orbifold [33]. O· · Working in dimension two, Proctor and Stanhope showed that ,D· , (2) is k, ,v ﬁnite up to orbifold diffeomorphism [30], providing a ﬁrst generalization of the result of Grove, Petersen and Wu. The homeomorphism ﬁniteness result was then shown in all dimensions by Proctor, provided the orbifold has only isolated singularities [29]. Here that assumption is removed, completing the generalization of Grove–Petersen–Wu’s homeomorphism ﬁniteness. O· · Theorem B. For any k, D, v, n, the class ,D· , (n) has only ﬁnitely many k, ,v members up to orbifold homeomorphism.

text: EQUIVARIANT

text: ALEXANDROV

text: GEOMETRY

text: AND

text: ORBIFOLD

text: FINITENESS

text: The

text: ﬁrst

text: such

text: result

text: is

text: that

text: of

text: Weinstein,

text: who

text: showed

text: that,

text: for

text: δ

text: M

text: the

text: class

text: of

text: unformly

text: pinched

text: positively

text: curved

text: manifolds

text: of

text: δ,

text: even

text: dimension,

text: has

text: only

text: ﬁnitely

text: many

text: members

text: up

text: to

text: homotopy

font: default

text: M

text: Shortly

text: after

text: this,

text: Cheeger

text: showed

text: that,

text: for

text: n

text: K·,D,

text: (n)

text: has

text: ﬁnitely

text: k,

text: ,v

text: many

text: simply

text: connected

text: members

text: up

text: to

text: diffeomorphism

text: In

text: Pe-

text: ters

text: shows

text: that

text: this

text: result

text: still

text: holds

text: for

text: n

text: as

text: well

text: as

text: removing

text: the

text: hypothesis

text: of

text: simple

text: connectivity.

text: Grove

text: and

text: Petersen

text: removed

text: the

text: upper

text: bound

text: on

text: sectional

text: curvature,

text: and

text: M·

text: obtained

text: ﬁniteness

text: of

text: ,D·

text: (n)

text: up

text: to

text: homotopy

text: Shortly

text: afterwards,

text: k,

text: ,v

text: in

text: collaboration

text: with

text: Wu,

text: this

text: result

text: was

text: improved

text: to

text: show

text: ﬁniteness

text: up

text: to

text: homeomorphism

text: for

text: n

text: As

text: long

text: as

text: the

text: dimension

text: is

text: not

text: four,

text: the

text: work

text: of

text: Kirby

text: and

text: Siebenmann

text: implies

text: ﬁniteness

text: up

text: to

text: diffeomor-

text: phism.

text: Perelman’s

text: Stability

text: Theorem

text: showed

text: that

text: Alex

text: ,D·

text: (n),

text: the

text: corre-

text: k,

text: ,v

text: sponding

text: class

text: of

text: Alexandrov

text: spaces,

text: is

text: ﬁnite

text: up

text: to

text: homeomorphism.

text: A

text: fortiori,

text: this

text: generalizes

text: the

text: Grove–Petersen–Wu

text: ﬁniteness

text: result

text: to

text: all

text: di-

text: mensions.

text: Theorem

text: B

text: uses

text: the

text: equivariant

text: version

text: of

text: the

text: Stability

text: Theorem

text: to

text: gen-

text: eralize

text: the

text: homeomorphism

text: ﬁniteness

text: result

text: of

text: Grove,

text: Petersen

text: and

text: Wu

text: to

text: the

text: area

text: of

text: Riemannian

text: orbifolds.

text: An

text: orbifold

text: is

text: a

text: mild

text: generalization

text: of

text: a

text: manifold,

text: and,

text: to

text: give

text: just

text: a

text: few

text: examples,

text: the

text: concept

text: has

text: found

text: applica-

text: tions

text: in

text: Thurston’s

text: work

text: on

text: the

text: Geometrization

text: Conjecture

text: the

text: con-

text: struction

text: of

text: a

text: new

text: positively

text: curved

text: manifold

text: by

text: Dearricott

text: and

text: Grove–

text: Verdiani–Ziller

text: and

text: string

text: theory,

text: such

text: as

text: Dixon,

text: Harvey,

text: Vafa

text: and

text: Witten’s

text: conformal

text: ﬁeld

text: theory

text: built

text: on

text: a

text: quotient

text: of

text: a

text: torus

text: The

text: same

text: M

text: O

text: convenient

text: notation

text: can

text: be

text: used

text: for

text: orbifolds,

text: here

text: replacing

text: with

text: The

text: ﬁrst

text: ﬁniteness

text: result

text: for

text: orbifolds

text: is

text: that

text: of

text: Fukaya

text: who

text: gener-

text: O

text: alized

text: the

text: result

text: of

text: Cheeger,

text: showing

text: that

text: a

text: subclass

text: of

text: K·,D,

text: (n)

text: is

text: ﬁnite

text: k,

text: ,v

text: up

text: to

text: orbifold

text: diffeomorphism.

text: Fukaya

text: used

text: a

text: much

text: more

text: restrictive

text: def-

text: inition

text: of

text: orbifold,

text: considering

text: only

text: the

text: orbit

text: spaces

text: of

text: global

text: actions

text: by

text: ﬁnite

text: groups

text: on

text: Riemannian

text: manifolds.

text: This

text: corresponds

text: to

text: what

text: Thurston

text: called

text: a

text: “good”

text: orbifold

text: O·

text: Working

text: in

text: dimension

text: two,

text: Proctor

text: and

text: Stanhope

text: showed

text: that

text: ,D·

text: is

text: k,

text: ,v

text: ﬁnite

text: up

text: to

text: orbifold

text: diffeomorphism

text: providing

text: a

text: ﬁrst

text: generalization

text: of

text: the

text: result

text: of

text: Grove,

text: Petersen

text: and

text: Wu.

text: The

text: homeomorphism

text: ﬁniteness

text: result

text: was

text: then

text: shown

text: in

text: all

text: dimensions

text: by

text: Proctor,

text: provided

text: the

text: orbifold

text: has

text: only

text: isolated

text: singularities

text: Here

text: that

text: assumption

text: is

text: removed,

text: completing

text: the

text: generalization

text: of

text: Grove–Petersen–Wu’s

text: homeomorphism

text: ﬁniteness.

text: O·

text: Theorem

text: B.

text: For

text: any

text: k,

text: D,

text: v,

text: n,

text: the

text: class

text: ,D·

text: (n)

text: has

text: only

text: ﬁnitely

text: many

text: k,

text: ,v

text: members

text: up

text: to

text: orbifold

text: homeomorphism.


text: 6 G. Liu et al. Figure 4. Intensity maps of the [O iii] line in the 12 unobscured quasars in our sample, shown on a logarithmic scale. The tick marks of the color bars are in − − − − − units of 10 14 erg 1 s 1 cm 2 arcsec 2. In each spaxel, the [O iii] ﬂux is calculated from a multi-Gaussian ﬁt to the spectral proﬁle of the line. The seeing at the observing site is depicted by the open circle whose diameter is the FWHM of the PSF. SDSS J0924+0642 is the only object that is likely unresolved. Figure 5. Radial proﬁles of the surface brightness of [O iii] emission (solid lines), of quasar continuum emission (dashed lines), and of the PSF (dotted lines). − − − − [O iii] and quasar continuum proﬁles are clipped at 5σ. The peak [O iii] surface brightness (I0) marked on each panel is in units of 10 14 erg s 1 cm 2 arcsec 2. Also given in∝eac−h panel is the absolute value of the best-ﬁtting exponent of power-law ﬁts (η) to the ou−ter regions of the−nebul−ae along −the semi-major axes (i.e. IR,[OIII] R η). Vertical dash-dot lines mark the isophotal radii at a surface brightness level of 10 15/(1 + z)4 erg s 1 cm 2 arcsec 1 (i.e. Rint listed in Table 2). − ple. With marginally or unresolved objects excluded from both type 83 and 576 km s 1 among the twelve unobscured quasars. The same 1 and type 2 samples, the median ellipticity measured at a sur- measurement performed in the obscured sample yielded values of − − − − (cid:104) (cid:105) − face b±rightness of 10 15/(1 + z)4 erg s 1 cm (cid:104)2 ar(cid:105)csec 2 is ±(cid:15)iTn1t = 89–522 km s 1 (Liu et al. 2013b). 0.14 0.11 for the unobscured objects and (cid:15)iTn2t = 0.14 0.11. The [O iii] velocity widths are just as high as t(cid:104)hose(cid:105)seen in type The logrank test gives a probability of 1.00 that both samples fol- 2 quasars. Both the luminosity-weigthed overall W (measured 80 low the same distribution. Therefore, we do not observe signiﬁcant from the SDSS ﬁber spectra which capture the integrated emission morphological diﬀerence between the two samples. from the nebulae) and the maxima of W in the spatially resolved 80 Radial proﬁles. The radial distribution of [O iii] emission from maps (excluding the 5% highest values) are listed in Table 2. The the unobscured quasars also follow loci very similar to those of the maximal values are ∼1000 km s−1 or greater in most of the sample, obscured sample. When the outer (R (cid:38) 1(cid:48)(cid:48)) part of the [O iii] sur- reaching 2000 km s−1. The object with the smallest line-of-sight ∝ − f4acmeabrgriignhatlnlyesosrpurnorﬁelseolivseﬁdt qbuyaasaprsowareer elaxwcluIdR,e[Od,IIIw] e ﬁRndη tahnadt tthhee vneelaorclyitycodnisstpaenrtsWion a∼pp3e0a0rskmto sb−e1 SacDrSosSsJt0h7e5n3e+bu3l1a5.3 which has a 80 best-ﬁt e(cid:104)xpo(cid:105)nent η ra±nges from 2.58 to 4.68 (Table 1) and has a In Figure 8, we show the distributions of the overall line median ηT1 = 3.88 0.70, well consistent with the results from widths (cid:104)W (cid:105) and the maximum velocity diﬀerence ∆v for both the obscured q(cid:104)uas(cid:105)ar sample±which has η = 3.00–5.70 and a me- type 1 and80type 2 quasars. Excluding the only likelymuaxnresolved dian value of ηT2 = 3.53 0.87. The logrank test shows that the target, SDSS J0924+0642, we do not ﬁnd a signiﬁcant diﬀer- probabilities that the two samples are drawn from the same parent (cid:104) (cid:105) ence between W values in type 1 and type 2 objects, with the distribution is 0.28. 80 Kolmogorov-Smirnov (K-S) test giving a p = 0.37 probability that Hence, we conclude that no signiﬁcant diﬀerence in sizes, they are drawn from the same distribution. The median ∆v is morphology or radial distribution is seen between the gas nebulae (cid:104) (cid:105) ± max slightly larger in the unobscured sample ( ∆vT1 = 238 138 km of unobscured and obscured quasar samples. − (cid:104) (cid:105) max ± − s 1) than in the obscured quasars ( ∆vT2 = 161 146 km s 1), 4.2 Kinematic properties max although the diﬀerence is not statistically signiﬁcant (the K-S test To characterize the line-of-sight velocity and velocity disper- yields probability p = 0.15 that the two samples follow the same distribution). sion of the ionized gas, in every spaxel we use the multi-Gaussian ﬁts to the [O iii] line to measure the median velocity (v ) and the We ﬁnd no correlation between the two quantities for the ob- med velocity interval that contains 80% of the total emission centered scured sample (Liu et al. 2013b). A weak correlation might exist at the median velocity (W ). These parameters, ﬁrst introduced by among the unobscured quasars (the Kendall rank correlation coef- 80 Whittle (1985), were also used to characterize the kinematics of the ﬁcient τ = 0.60 with probability p = 0.01 that no correlation is [O iii] emission in the obscured quasar sample (Liu et al. 2013b). present). The spatial distributions of v and W are shown in Figures 6 and The radial proﬁles of W are almost ﬂat at projected distances med 80 (cid:46) 80 7. R 5 kpc. At larger radii, the scatter increases with R, while W ∼ 80 Across every nebula, the median velocity of [O iii] emission appears to decrease in 10/12 of the sample (by 5 % per kilopar- − varies by up to a few hundred km s 1. Although some noisy spax- sec from the brightness center) and remains roughly constant in els are present, most of the velocity variation proceeds in a smooth the two remaining objects (J0753+3153 and J0924+0642). This fashion from one spaxel to the next, which implies that the velocity is very similar to the proﬁles seen in type 2 quasars. As we dis- proﬁles of [O iii] vary among the spaxels. The well-resolved veloc- cussed previously (Liu et al. 2013b), at least some of this decline ity structure conﬁrms that the [O iii] emission is spatially resolved is due to the decrease in the signal-to-noise ratio which prevents in our targets. Several of the objects show well-organized velocity us from ﬁtting multiple Gaussian components to the [O iii] in the structures, with blue-shifted emission predominantly on one side outer parts of the nebulae and from reliably measuring weak broad of the nebula and redshifted emission predominantly on the other bases to the [O iii] emission. However, we concluded that this ef- and with the line of nodes centered at the brightness peak (e.g., fect was not suﬃcient to explain the entirety of the W decline and − 80 SDSS J0304+0022, SDSS J0311 0707, and SDSS J0935+5348). that it was likely that the line-of-sight velocity width was in fact In Table 2 we report the maximum diﬀerence in v between slightly decreasing with projected distance. Given the similarity of med the redshifted and the blueshifted regions ∆v after excluding the W proﬁles between the two samples, the same arguments apply max 80 5% highest and 5% lowest v values to minimize the eﬀect of for the unobscured quasars as well. A declining W radial proﬁle med 80 noise. The maximum projected velocity diﬀerence ranges between becomes more plausible when the PSF smearing eﬀect is taken into (cid:13) c 2013 RAS, MNRAS 000, 1–11

text: G.

text: Liu

text: et

text: al.

text: Figure

text: Intensity

label: caption

text: maps

text: of

text: the

text: [O

text: iii]

font: ITZUGB+rtxsc

text: line

text: in

text: the

text: unobscured

text: quasars

text: in

text: our

text: sample,

text: shown

text: on

text: a

text: logarithmic

text: scale.

text: The

text: tick

text: marks

text: of

text: the

text: color

text: bars

text: are

text: in

font: QWSYZL+txsy

text: units

text: of

text: erg

text: s

text: cm

text: arcsec

text: In

text: each

text: spaxel,

text: the

text: [O

text: iii]

text: ﬂux

text: is

text: calculated

text: from

text: a

text: multi-Gaussian

text: ﬁt

text: to

text: the

text: spectral

text: proﬁle

text: of

text: the

text: line.

text: The

text: seeing

text: at

text: the

text: observing

text: site

text: is

text: depicted

text: by

text: the

text: open

text: circle

text: whose

text: diameter

text: is

text: the

text: FWHM

text: of

text: the

text: PSF.

text: SDSS

text: is

text: the

text: only

text: object

text: that

text: is

text: likely

text: unresolved.

text: Figure

text: Radial

text: proﬁles

text: of

text: the

text: surface

text: brightness

text: of

text: [O

text: iii]

text: emission

text: (solid

text: lines),

text: of

text: quasar

text: continuum

text: emission

text: (dashed

text: lines),

text: and

text: of

text: the

text: PSF

text: (dotted

text: lines).

text: [O

text: iii]

text: and

text: quasar

text: continuum

text: proﬁles

text: are

text: clipped

text: at

text: The

text: peak

text: [O

text: iii]

text: surface

text: brightness

text: marked

text: on

text: each

text: panel

text: is

text: in

text: units

text: of

text: erg

text: s

text: cm

text: arcsec

text: Also

text: given

text: in∝eac−h

text: panel

text: is

text: the

text: absolute

text: value

text: of

text: the

text: best-ﬁtting

text: exponent

text: of

text: power-law

text: ﬁts

text: (η)

text: to

text: the

text: ou−ter

text: regions

text: of

text: the−nebul−ae

text: along

text: −the

text: semi-major

text: axes

text: (i.e.

text: IR,[OIII]

text: R

text: η).

text: Vertical

text: dash-dot

text: lines

text: mark

text: the

text: isophotal

text: radii

text: at

text: a

text: surface

text: brightness

text: level

text: of

text: erg

text: s

text: cm

text: arcsec

text: (i.e.

text: Rint

text: listed

text: in

text: Table

text: ple.

text: With

text: marginally

text: or

text: unresolved

text: objects

text: excluded

text: from

text: both

text: type

text: and

text: km

text: s

text: among

text: the

text: twelve

text: unobscured

text: quasars.

text: The

text: same

text: and

text: type

text: samples,

text: the

text: median

text: ellipticity

text: measured

text: at

text: a

text: sur-

text: measurement

text: performed

text: in

text: the

text: obscured

text: sample

text: yielded

text: values

text: of

text: face

text: b±rightness

text: of

text: erg

text: s

text: cm

text: is

text: km

text: s

text: (Liu

text: et

text: al.

text: for

text: the

text: unobscured

text: objects

text: and

text: The

text: [O

text: iii]

text: velocity

text: widths

text: are

text: just

text: as

text: high

text: as

text: in

text: type

text: The

text: logrank

text: test

text: gives

text: a

text: probability

text: of

text: that

text: both

text: samples

text: fol-

text: quasars.

text: Both

text: the

text: luminosity-weigthed

text: overall

text: W

text: (measured

text: low

text: the

text: same

text: distribution.

text: Therefore,

text: we

text: do

text: not

text: observe

text: signiﬁcant

text: from

text: the

text: SDSS

text: ﬁber

text: spectra

text: which

text: capture

text: the

text: integrated

text: emission

text: morphological

text: diﬀerence

text: between

text: the

text: two

text: samples.

text: from

text: the

text: nebulae)

text: and

text: the

text: maxima

text: of

text: W

text: in

text: the

text: spatially

text: resolved

text: Radial

text: proﬁles.

text: The

text: radial

text: distribution

text: of

text: [O

text: iii]

text: emission

text: from

text: maps

text: (excluding

text: the

text: highest

text: values)

text: are

text: listed

text: in

text: Table

text: The

text: the

text: unobscured

text: quasars

text: also

text: follow

text: loci

text: very

text: similar

text: to

text: those

text: of

text: the

text: maximal

text: values

text: are

text: km

text: or

text: greater

text: in

text: most

text: of

text: the

text: sample,

text: obscured

text: sample.

text: When

text: the

text: outer

text: (R

text: part

text: of

text: the

text: [O

text: iii]

text: sur-

text: reaching

text: km

text: The

text: object

text: with

text: the

text: smallest

text: line-of-sight

text: qbuyaasaprsowareer

text: elaxwcluIdR,e[Od,IIIw]

text: e

text: ﬁRndη

text: tahnadt

text: tthhee

text: vneelaorclyitycodnisstpaenrtsWion

text: which

text: has

text: a

text: best-ﬁt

text: η

text: ra±nges

text: from

text: to

text: (Table

text: and

text: has

text: a

text: In

text: Figure

text: we

text: show

text: the

text: distributions

text: of

text: the

text: overall

text: line

text: median

text: well

text: consistent

text: with

text: the

text: results

text: from

text: widths

text: and

text: the

text: maximum

text: velocity

text: diﬀerence

text: ∆v

text: for

text: both

text: the

text: obscured

text: sample±which

text: has

text: η

text: and

text: a

text: me-

text: type

text: quasars.

text: Excluding

text: the

text: only

text: likelymuaxnresolved

text: dian

text: value

text: of

text: The

text: logrank

text: test

text: shows

text: that

text: the

text: target,

text: SDSS

text: we

text: do

text: not

text: ﬁnd

text: a

text: signiﬁcant

text: diﬀer-

text: probabilities

text: that

text: the

text: two

text: samples

text: are

text: drawn

text: from

text: the

text: same

text: parent

text: ence

text: between

text: W

text: values

text: in

text: type

text: and

text: type

text: objects,

text: with

text: the

text: distribution

text: is

text: Kolmogorov-Smirnov

text: (K-S)

text: test

text: giving

text: a

text: p

text: probability

text: that

text: Hence,

text: we

text: conclude

text: that

text: no

text: signiﬁcant

text: diﬀerence

text: in

text: sizes,

text: they

text: are

text: drawn

text: from

text: the

text: same

text: distribution.

text: The

text: median

text: ∆v

text: is

text: morphology

text: or

text: radial

text: distribution

text: is

text: seen

text: between

text: the

text: gas

text: nebulae

text: max

text: slightly

text: larger

text: in

text: the

text: unobscured

text: sample

text: km

text: of

text: unobscured

text: and

text: obscured

text: quasar

text: samples.

text: max

text: s

text: than

text: in

text: the

text: obscured

text: quasars

text: km

text: s

text: Kinematic

text: properties

text: max

text: although

text: the

text: diﬀerence

text: is

text: not

text: statistically

text: signiﬁcant

text: (the

text: K-S

text: test

text: To

text: characterize

text: the

text: line-of-sight

text: velocity

text: and

text: velocity

text: disper-

text: yields

text: probability

text: p

text: that

text: the

text: two

text: samples

text: follow

text: the

text: same

text: distribution).

text: sion

text: of

text: the

text: ionized

text: gas,

text: in

text: every

text: spaxel

text: we

text: use

text: the

text: multi-Gaussian

text: ﬁts

text: to

text: the

text: [O

text: iii]

text: line

text: to

text: measure

text: the

text: median

text: velocity

text: (v

text: and

text: the

text: We

text: ﬁnd

text: no

text: correlation

text: between

text: the

text: two

text: quantities

text: for

text: the

text: ob-

text: med

text: velocity

text: interval

text: that

text: contains

text: of

text: the

text: total

text: emission

text: centered

text: scured

text: sample

text: (Liu

text: et

text: al.

text: A

text: weak

text: correlation

text: might

text: exist

text: at

text: the

text: median

text: velocity

text: (W

text: These

text: parameters,

text: ﬁrst

text: introduced

text: by

text: among

text: the

text: unobscured

text: quasars

text: (the

text: Kendall

text: rank

text: correlation

text: coef-

text: Whittle

text: were

text: also

text: used

text: to

text: characterize

text: the

text: kinematics

text: of

text: the

text: ﬁcient

text: τ

text: with

text: probability

text: p

text: that

text: no

text: correlation

text: is

text: [O

text: iii]

text: emission

text: in

text: the

text: obscured

text: quasar

text: sample

text: (Liu

text: et

text: al.

text: present).

text: The

text: spatial

text: distributions

text: of

text: v

text: and

text: W

text: are

text: shown

text: in

text: Figures

text: and

text: The

text: radial

text: proﬁles

text: of

text: W

text: are

text: almost

text: ﬂat

text: at

text: projected

text: distances

text: med

text: R

text: kpc.

text: At

text: larger

text: radii,

text: the

text: scatter

text: increases

text: with

text: R,

text: while

text: W

text: Across

text: every

text: nebula,

text: the

text: median

text: velocity

text: of

text: [O

text: iii]

text: emission

text: appears

text: to

text: decrease

text: in

text: of

text: the

text: sample

text: (by

text: per

text: kilopar-

text: varies

text: by

text: up

text: to

text: a

text: few

text: hundred

text: km

text: s

text: Although

text: some

text: noisy

text: spax-

text: sec

text: from

text: the

text: brightness

text: center)

text: and

text: remains

text: roughly

text: constant

text: in

text: els

text: are

text: present,

text: most

text: of

text: the

text: velocity

text: variation

text: proceeds

text: in

text: a

text: smooth

text: the

text: two

text: remaining

text: objects

text: and

text: This

text: fashion

text: from

text: one

text: spaxel

text: to

text: the

text: next,

text: which

text: implies

text: that

text: the

text: velocity

text: is

text: very

text: similar

text: to

text: the

text: proﬁles

text: seen

text: in

text: type

text: quasars.

text: As

text: we

text: dis-

text: proﬁles

text: of

text: [O

text: iii]

text: vary

text: among

text: the

text: spaxels.

text: The

text: well-resolved

text: veloc-

text: cussed

text: previously

text: (Liu

text: et

text: al.

text: at

text: least

text: some

text: of

text: this

text: decline

text: ity

text: structure

text: conﬁrms

text: that

text: the

text: [O

text: iii]

text: emission

text: is

text: spatially

text: resolved

text: is

text: due

text: to

text: the

text: decrease

text: in

text: the

text: signal-to-noise

text: ratio

text: which

text: prevents

text: in

text: our

text: targets.

text: Several

text: of

text: the

text: objects

text: show

text: well-organized

text: velocity

text: us

text: from

text: ﬁtting

text: multiple

text: Gaussian

text: components

text: to

text: the

text: [O

text: iii]

text: in

text: the

text: structures,

text: with

text: blue-shifted

text: emission

text: predominantly

text: on

text: one

text: side

text: outer

text: parts

text: of

text: the

text: nebulae

text: and

text: from

text: reliably

text: measuring

text: weak

text: broad

text: of

text: the

text: nebula

text: and

text: redshifted

text: emission

text: predominantly

text: on

text: the

text: other

text: bases

text: to

text: the

text: [O

text: iii]

text: emission.

text: However,

text: we

text: concluded

text: that

text: this

text: ef-

text: and

text: with

text: the

text: line

text: of

text: nodes

text: centered

text: at

text: the

text: brightness

text: peak

text: (e.g.,

text: fect

text: was

text: not

text: suﬃcient

text: to

text: explain

text: the

text: entirety

text: of

text: the

text: W

text: decline

text: and

text: SDSS

text: SDSS

text: and

text: SDSS

text: that

text: it

text: was

text: likely

text: that

text: the

text: line-of-sight

text: velocity

text: width

text: was

text: in

text: fact

text: In

text: Table

text: we

text: report

text: the

text: maximum

text: diﬀerence

text: in

text: v

text: between

text: slightly

text: decreasing

text: with

text: projected

text: distance.

text: Given

text: the

text: similarity

text: of

text: med

text: the

text: redshifted

text: and

text: the

text: blueshifted

text: regions

text: ∆v

text: after

text: excluding

text: the

text: W

text: proﬁles

text: between

text: the

text: two

text: samples,

text: the

text: same

text: arguments

text: apply

text: max

text: highest

text: and

text: lowest

text: v

text: values

text: to

text: minimize

text: the

text: eﬀect

text: of

text: for

text: the

text: unobscured

text: quasars

text: as

text: well.

text: A

text: declining

text: W

text: radial

text: proﬁle

text: med

text: noise.

text: The

text: maximum

text: projected

text: velocity

text: diﬀerence

text: ranges

text: between

text: becomes

text: more

text: plausible

text: when

text: the

text: PSF

text: smearing

text: eﬀect

text: is

text: taken

text: into

text: c

text: RAS,

text: MNRAS


text: 8 G. Liu et al. Figure 9. Size-luminosity relation of the radio-quiet/weak unobscured quasars from our sample (ﬁlled circles) in terms of [O iii] (top) and mid-IR (bottom) luminosities. Also included are previous observations of type 2 quasars (Liu et al. 2013a; Hainline et al. 2013; Greene et al. 2011; Humphrey et al. 2010), type 1 quasars (Husemann et al. 2013) and Seyfert 2 galaxies (Bennert et al. 2006; Fraquelli et al. 2003). R is the isophotal radius at the surface brightness of − − − − int 10 15/(1 + z)4 erg s 1 cm 2 arcsec 2 (Liu et al. 2013a), and νL±ν,8 µm is the rest-frame 8 µm luminosity obtained by interpolati±ng between WISE bands. Our linear ﬁt to the Rint vs. L[O III] relation results in a slope of 0.23 0.02 (solid black line),±consistent with Liu et al. (2013a, 0.25 ±0.02, grey line). Our best-ﬁt piecewise linear ﬁt to±the Rint vs. νLν, 8 µm relation (solid black line) has a slope of 0.44 0.02, a break at logνLνb,r8 µm = 44.95 0.06, a−maximum radius of logRmax/(pc)±= 4.06 0.01, well consistent with the result of Hainline et al. (2013, gray line). For the objects with νLν, 8 µm < 1045.5 erg s 1, we ﬁnd a best-ﬁt slope of 0.37 0.03. This result is consistent with that of Hainline et al. (2013, with typical limits on the amount of obscuration in type 2 objects ± ± slope = 0.41 0.02, intercept = 3.65 0.02) and is depicted by a dot- (Zakamska et al. 2004; also see Cleary et al. 2007, Haas et al. 2008 ted line in Figure 9. If the ﬂattening is characterized by a plateau, and Lacy et al. 2013 for extinction studies in mid-IR). we ﬁnd the following best-ﬁtting piecewise linear ﬁt for all data Thus there are two possible interpretations of Figure 9. The  (cid:16) (cid:17) ploogin(cid:32)tRspicbntu(cid:33)t =theu(p0p.e4r4±l±im0i.t0s:2)lofgor 1ν0νL4L4ννe,,8r8gµµmsm−1<+10(344..6954±±0.060.e0r2g)s−1; (3) ﬁptqtyhrurpeosatebso;eaibsrtsshoctefiuhnsrateh0twde.t3ehobedmnoeel8uxoss.mtmµIencmotorrtniehmcciislonluucntdrmaoiesncieshtn,hirocasosaitminltloycyauetirolcRufnm[lOequwiIunmIIa]osisaunaansomrdsstpihtLrlayee[nOgioaIstIfIhr]deaualnenrsesoaasmbscsoocpcuflsuerirtamheotdee-f 4.06 0.01 ilar between the two, this implies that there exists an upper limit for νLν,8µm (cid:62) 1044.95±0.06 erg s−1. on both the size (10 kpc) and the luminosity of the narrow-line re- gion and neither of these values further increases with bolometric This piecewise linear equation is shown by a solid black line in luminosity. the same ﬁgure, and is also in good agreement with Hainline et al. The second interpretation is that L is a reasonably ± ± [OIII] (2013, slope = 0.47 0.02, intercept = 3.67 0.02, break luminosity isotropic measure of quasar luminosity, unbiased with respect to ± − ± = 1044.86 0.05 erg s 1, and the limiting radius = 104.07 0.03 pc). quasar type. Then the diﬀerence in the 8 µm luminosity is due to We must be cautious in interpreting the ﬂattening seen in Fig- circumnuclear obscuration. As for the sizes, R are statistically [OIII] ure 9. In particular, our type 1 quasars have been intentionally se- indistinguishable in type 1 and type 2 samples, and we are unable lected to be matched in [O iii] luminosity to the type 2 quasars to comment on the existence of the upper limit to the size, since our from Liu et al. (2013a,b), and yet the former appear to be more new data do not in fact lead to an increase in the range of intrinsic ∼ luminous by a factor of 3 than the latter in mid-infrared (mid- luminosities probed. We are inclined to accept this latter interpre- IR). The signiﬁcant IR-to-[O iii] ratio diﬀerence points to the in- tation because the mid-IR spectral energy distributions of the two trinsic diﬀerence in the spectral energy distribution (SED) of the samples are undeniably diﬀerent. In order to reliably probe the ﬂat- two populations. The mid-IR continuum of type 2 quasars is much tening of the size-luminosity relationship (Netzer et al. 2004), ob- redder than that of type 1s (Liu et al. 2013b), which likely means jects with higher [O iii] and IR luminosities must be observed and that even the mid-IR emission is not an entirely isotropic lumi- quasars of diﬀerent types need to be considered separately. nosity indicator. Speciﬁcally, the slopes of the IR SEDs (deﬁned The derived B-band absolute magnitude of our sample is ∝ − ± as ν−Lν ±να) measured between rest-frame 5 and 12 µm are 25.6 0.7 mag on average (Table 1), higher than that of the 19 α = 0.33 0.2−7 for ±the type 1 sample (median and standard devi- radio-quiet type 1 quasars studied in Husemann et a−l. (200±8) and ation) and α = 1.13 0.55 for the type 2 sample. In type 2 objects Husemann et al. (2013) located at z = 0.06–0.24 ( 23.5 0.08 the thermal emission from hot dust seen at mid-IR wavelengths mag). The detection rate of extended nebulae around our radio- can be aﬀected by obscuration when propagating through the sur- quiet/weak quasars (11/12) is signiﬁcantly higher than that of theirs rounding much colder material in order to reach the observer, which (6/19), but the physical extents are comparable (Figure 9). Among would result in both its reddening and overall suppression. This hy- their 6 detected radio-quiet nebulae, 3 show one-sided or biconical pothesis is further strengthened by the observation that many type morphology and the rest are round, in contrast to our quasar neb- 2 quasars show silicate absorption which is a bi-product of this pro- ulae that are morphologically smoother and rounder in every case ∼ cess (Knacke & Thomson 1973; Zakamska et al. 2008; Hao et al. (ellipticity 0.1). 2005). If the [O iii] line is a more or less isotropic indicator of the bolometric luminosity of the quasar, then type 1’s would appear 5.2 [O iii]-to-Hβ ratio more luminous in the mid-IR than their [O iii]-matched type 2 coun- terparts, as we see in Figure 9. Ryan C. Hickox et al. (private com- The [O iii]-to-Hβ ratio is a powerful diagnostic of ionization muni−cation) ﬁnd a similar (0.3–0.4 dex) diﬀerence at L[OIII] > 1043 conditions in the nebula. In particular, in type 2 quasars this ratio erg s 1 between the type 1 and 2 spectral energy distributions when remains nearly constant over most of the nebula and then rapidly averaged over large SDSS samples. Our knowledge of the details of declines with the distance from the central source, which we inter- the mid-IR opacity curve of dust suﬀers from large systematic un- pret as the signature that the line-emitting clouds become fully ion- certainties (Smith et al. 2007). To crudely estimate the amount of ized and matter-bounded in the outer parts of the nebulae (Liu et al. extinction necessary to produce such diﬀerence in color between 2013a). In type 1 quasars, this is a diﬃcult measurement to replicate ∝ type 1 and type 2 quasars, we assume that dust opacity is 1/λ because of the contribution to the Hβ proﬁle from the broad-line re- over optical-to-IR range and compute reddening in the simplistic gion of the quasar. “screen of cold dust” approximation. To produce a > 0.3 dex red- To perform this measurement, in every spaxel we subtract the dening of the unobscured quasar continuum in the 5–12 µm range, overall quasar continuum and Fe ii, and then we ﬁt the Hβ line us- AV > 13 mag worth of dust obscuration would be required, in line ing a combination of Gaussians to isolate the narrow component. (cid:13) c 2013 RAS, MNRAS 000, 1–11 ##LTLine## ##LTLine##

text: G.

text: Liu

text: et

text: al.

text: Figure

text: Size-luminosity

text: relation

text: of

text: the

text: radio-quiet/weak

text: unobscured

text: quasars

text: from

text: our

text: sample

text: (ﬁlled

text: circles)

text: in

text: terms

text: of

text: [O

text: iii]

text: (top)

text: and

text: mid-IR

text: (bottom)

text: luminosities.

text: Also

text: included

text: are

text: previous

text: observations

text: of

text: type

text: quasars

text: (Liu

text: et

text: al.

text: Hainline

text: et

text: al.

text: Greene

text: et

text: al.

text: Humphrey

text: et

text: al.

text: type

text: quasars

text: (Husemann

text: et

text: al.

text: and

text: Seyfert

text: galaxies

text: (Bennert

text: et

text: al.

text: Fraquelli

text: et

text: al.

text: R

text: is

text: the

text: isophotal

text: radius

text: at

text: the

text: surface

text: brightness

text: of

text: int

font: PXCPBB+rtxr

text: erg

text: s

text: cm

text: arcsec

text: (Liu

text: et

text: al.

text: and

text: νL±ν,8

text: µm

text: is

text: the

text: rest-frame

text: µm

text: luminosity

text: obtained

text: by

text: interpolati±ng

text: between

text: WISE

text: bands.

text: Our

text: linear

text: ﬁt

text: to

text: the

text: Rint

text: vs.

text: L[O

text: III]

text: relation

text: results

text: in

text: a

text: slope

text: of

text: (solid

text: black

text: line),±consistent

text: with

text: Liu

text: et

text: al.

text: grey

text: line).

text: Our

text: best-ﬁt

text: piecewise

text: linear

text: ﬁt

text: to±the

text: Rint

text: vs.

text: νLν,

text: µm

text: relation

text: (solid

text: black

text: line)

text: has

text: a

text: slope

text: of

text: a

text: break

text: at

text: µm

text: a−maximum

text: radius

text: of

text: logRmax/(pc)±=

text: well

text: consistent

text: with

text: the

text: result

text: of

text: Hainline

text: et

text: al.

text: gray

text: line).

text: For

text: the

text: objects

text: with

text: νLν,

text: µm

text: erg

text: s

text: we

text: ﬁnd

text: a

text: best-ﬁt

text: slope

text: of

text: This

text: result

text: is

text: consistent

text: with

text: that

text: of

text: Hainline

text: et

text: al.

text: with

text: typical

text: limits

text: on

text: the

text: amount

text: of

text: obscuration

text: in

text: type

text: objects

text: slope

text: intercept

text: and

text: is

text: depicted

text: by

text: a

text: dot-

text: (Zakamska

text: et

text: al.

text: also

text: see

text: Cleary

text: et

text: al.

text: Haas

text: et

text: al.

text: ted

text: line

text: in

text: Figure

text: If

text: the

text: ﬂattening

text: is

text: characterized

text: by

text: a

text: plateau,

text: and

text: Lacy

text: et

text: al.

text: for

text: extinction

text: studies

text: in

text: mid-IR).

text: we

text: ﬁnd

text: the

text: following

text: best-ﬁtting

text: piecewise

text: linear

text: ﬁt

text: for

text: all

text: data

text: Thus

text: there

text: are

text: two

text: possible

text: interpretations

text: of

text: Figure

text: The

text: ilar

text: between

text: the

text: two,

text: this

text: implies

text: that

text: there

text: exists

text: an

text: upper

text: limit

text: for

text: erg

text: on

text: both

text: the

text: size

text: kpc)

text: and

text: the

text: luminosity

text: of

text: the

text: narrow-line

text: re-

text: gion

text: and

text: neither

text: of

text: these

text: values

text: further

text: increases

text: with

text: bolometric

text: This

text: piecewise

text: linear

text: equation

text: is

text: shown

text: by

text: a

text: solid

text: black

text: line

text: in

text: luminosity.

text: the

text: same

text: ﬁgure,

text: and

text: is

text: also

text: in

text: good

text: agreement

text: with

text: Hainline

text: et

text: al.

text: The

text: second

text: interpretation

text: is

text: that

text: L

text: is

text: a

text: reasonably

text: [OIII]

text: slope

text: intercept

text: break

text: luminosity

text: isotropic

text: measure

text: of

text: quasar

text: luminosity,

text: unbiased

text: with

text: respect

text: to

text: erg

text: s

text: and

text: the

text: limiting

text: radius

text: pc).

text: quasar

text: type.

text: Then

text: the

text: diﬀerence

text: in

text: the

text: µm

text: luminosity

text: is

text: due

text: to

text: We

text: must

text: be

text: cautious

text: in

text: interpreting

text: the

text: ﬂattening

text: seen

text: in

text: Fig-

text: circumnuclear

text: obscuration.

text: As

text: for

text: the

text: sizes,

text: R

text: are

text: statistically

text: [OIII]

text: ure

text: In

text: particular,

text: our

text: type

text: quasars

text: have

text: been

text: intentionally

text: se-

text: indistinguishable

text: in

text: type

text: and

text: type

text: samples,

text: and

text: we

text: are

text: unable

text: lected

text: to

text: be

text: matched

text: in

text: [O

text: iii]

text: luminosity

text: to

text: the

text: type

text: quasars

text: to

text: comment

text: on

text: the

text: existence

text: of

text: the

text: upper

text: limit

text: to

text: the

text: size,

text: since

text: our

text: from

text: Liu

text: et

text: al.

text: and

text: yet

text: the

text: former

text: appear

text: to

text: be

text: more

text: new

text: data

text: do

text: not

text: in

text: fact

text: lead

text: to

text: an

text: increase

text: in

text: the

text: range

text: of

text: intrinsic

text: luminous

text: by

text: a

text: factor

text: of

text: than

text: the

text: latter

text: in

text: mid-infrared

text: (mid-

text: luminosities

text: probed.

text: We

text: are

text: inclined

text: to

text: accept

text: this

text: latter

text: interpre-

text: IR).

text: The

text: signiﬁcant

text: IR-to-[O

text: iii]

text: ratio

text: diﬀerence

text: points

text: to

text: the

text: in-

text: tation

text: because

text: the

text: mid-IR

text: spectral

text: energy

text: distributions

text: of

text: the

text: two

text: trinsic

text: diﬀerence

text: in

text: the

text: spectral

text: energy

text: distribution

text: (SED)

text: of

text: the

text: samples

text: are

text: undeniably

text: diﬀerent.

text: In

text: order

text: to

text: reliably

text: probe

text: the

text: ﬂat-

text: two

text: populations.

text: The

text: mid-IR

text: continuum

text: of

text: type

text: quasars

text: is

text: much

text: tening

text: of

text: the

text: size-luminosity

text: relationship

text: (Netzer

text: et

text: al.

text: ob-

text: redder

text: than

text: that

text: of

text: type

text: (Liu

text: et

text: al.

text: which

text: likely

text: means

text: jects

text: with

text: higher

text: [O

text: iii]

text: and

text: IR

text: luminosities

text: must

text: be

text: observed

text: and

text: that

text: even

text: the

text: mid-IR

text: emission

text: is

text: not

text: an

text: entirely

text: isotropic

text: lumi-

text: quasars

text: of

text: diﬀerent

text: types

text: need

text: to

text: be

text: considered

text: separately.

text: nosity

text: indicator.

text: Speciﬁcally,

text: the

text: slopes

text: of

text: the

text: IR

text: SEDs

text: (deﬁned

text: The

text: derived

text: B-band

text: absolute

text: magnitude

text: of

text: our

text: sample

text: is

text: as

text: ν−Lν

text: ±να)

text: measured

text: between

text: rest-frame

text: and

text: µm

text: are

text: mag

text: on

text: average

text: (Table

text: higher

text: than

text: that

text: of

text: the

text: α

text: for

text: ±the

text: type

text: sample

text: (median

text: and

text: standard

text: devi-

text: radio-quiet

text: type

text: quasars

text: studied

text: in

text: Husemann

text: et

text: a−l.

text: and

text: ation)

text: and

text: α

text: for

text: the

text: type

text: sample.

text: In

text: type

text: objects

text: Husemann

text: et

text: al.

text: located

text: at

text: z

text: the

text: thermal

text: emission

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text: hot

text: dust

text: seen

text: at

text: mid-IR

text: wavelengths

text: mag).

text: The

text: detection

text: rate

text: of

text: extended

text: nebulae

text: around

text: our

text: radio-

text: can

text: be

text: aﬀected

text: by

text: obscuration

text: when

text: propagating

text: through

text: the

text: sur-

text: quiet/weak

text: quasars

text: is

text: signiﬁcantly

text: higher

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text: that

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text: theirs

text: rounding

text: much

text: colder

text: material

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text: the

text: observer,

text: which

text: but

text: the

text: physical

text: extents

text: are

text: comparable

text: (Figure

text: Among

text: would

text: result

text: in

text: both

text: its

text: reddening

text: and

text: overall

text: suppression.

text: This

text: hy-

text: their

text: detected

text: radio-quiet

text: nebulae,

text: show

text: one-sided

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text: biconical

text: pothesis

text: is

text: further

text: strengthened

text: by

text: the

text: observation

text: that

text: many

text: type

text: morphology

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text: rest

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text: contrast

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text: our

text: quasar

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text: quasars

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text: Zakamska

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text: Hao

text: et

text: al.

text: (ellipticity

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text: the

text: [O

text: iii]

text: line

text: is

text: a

text: more

text: or

text: less

text: isotropic

text: indicator

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text: the

text: bolometric

text: luminosity

text: of

text: the

text: quasar,

text: then

text: type

text: would

text: appear

text: [O

text: iii]-to-Hβ

text: ratio

text: more

text: luminous

text: in

text: the

text: mid-IR

text: than

text: their

text: [O

text: iii]-matched

text: type

text: coun-

text: terparts,

text: as

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text: in

text: Figure

text: Ryan

text: C.

text: Hickox

text: et

text: al.

text: (private

text: com-

text: The

text: [O

text: iii]-to-Hβ

text: ratio

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text: a

text: powerful

text: diagnostic

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text: ionization

text: muni−cation)

text: ﬁnd

text: a

text: similar

text: dex)

text: diﬀerence

text: at

text: L[OIII]

text: conditions

text: in

text: the

text: nebula.

text: In

text: particular,

text: in

text: type

text: quasars

text: this

text: ratio

text: erg

text: s

text: between

text: the

text: type

text: and

text: spectral

text: energy

text: distributions

text: when

text: remains

text: nearly

text: constant

text: over

text: most

text: of

text: the

text: nebula

text: and

text: then

text: rapidly

text: averaged

text: over

text: large

text: SDSS

text: samples.

text: Our

text: knowledge

text: of

text: the

text: details

text: of

text: declines

text: with

text: the

text: distance

text: from

text: the

text: central

text: source,

text: which

text: we

text: inter-

text: the

text: mid-IR

text: opacity

text: curve

text: of

text: dust

text: suﬀers

text: from

text: large

text: systematic

text: un-

text: pret

text: as

text: the

text: signature

text: that

text: the

text: line-emitting

text: clouds

text: become

text: fully

text: ion-

text: certainties

text: (Smith

text: et

text: al.

text: To

text: crudely

text: estimate

text: the

text: amount

text: of

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text: matter-bounded

text: in

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text: the

text: nebulae

text: (Liu

text: et

text: al.

text: extinction

text: necessary

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text: produce

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text: in

text: color

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text: type

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text: this

text: is

text: a

text: diﬃcult

text: measurement

text: to

text: replicate

text: type

text: and

text: type

text: quasars,

text: we

text: assume

text: that

text: dust

text: opacity

text: is

text: because

text: of

text: the

text: contribution

text: to

text: the

text: Hβ

text: proﬁle

text: from

text: the

text: broad-line

text: re-

text: over

text: optical-to-IR

text: range

text: and

text: compute

text: reddening

text: in

text: the

text: simplistic

text: gion

text: of

text: the

text: quasar.

text: “screen

text: of

text: cold

text: dust”

text: approximation.

text: To

text: produce

text: a

text: dex

text: red-

text: To

text: perform

text: this

text: measurement,

text: in

text: every

text: spaxel

text: we

text: subtract

text: the

text: dening

text: of

text: the

text: unobscured

text: quasar

text: continuum

text: in

text: the

text: µm

text: range,

text: overall

text: quasar

text: continuum

text: and

text: Fe

text: ii,

text: and

text: then

text: we

text: ﬁt

text: the

text: Hβ

text: line

text: us-

text: AV

text: mag

text: worth

text: of

text: dust

text: obscuration

text: would

text: be

text: required,

text: in

text: line

text: ing

text: a

text: combination

text: of

text: Gaussians

text: to

text: isolate

text: the

text: narrow

text: component.

text: c

text: RAS,

text: MNRAS

text: ##LTLine##

text: ##LTLine##


text: (cid:90) (cid:0) (cid:1) ∂fˆ 1 − · − − · (ω) = lim f(x) e i(ω x+(cid:15)xj) e iω x dx. (14) → ∂ωj (cid:15)∈C0 Rd (cid:15) (cid:15) ∈ R Fix λ satisfying Im ω < λ < γ. For suﬃciently small (cid:15), the integrand ∈ R in (14) is absolutely bounded over all x d by | | | | xf(x) eλ x , ∈ P(cid:48) ∩ R which is integrable since f L1( d). By dominated convergence, the (cid:90)γ limit and integral in (14) can be interchanged, yielding ∂fˆ − − · (ω) = ( ix )f(x)e iω x dx. j ∂ω R j d ˆ ˆ This shows that all complex ﬁrst partials of f(ω) exist; thus f is analytic at ω. ˆ Using dominated convergence to iteratively evaluate derivatives of f(ω) (cid:88) as in (14), we obtain the Taylor expansion − ( iω)α ˆ f(ω) = µ . α α! | |≥ | α 0 { } R By analytic continuation, the moments µ | ≥ uniquely determine fˆ on d. α α 0R Since the Fourier transform is one-to-one on L1( d), f is uniquely determined by its moments. This completes the proof. This theorem can be described as the strongest of its kind in two senses. First (Section ), the theorem is false if the exponential decay required by the ∈ P condition f is relaxed to algebraic decay. γ Second, if the above genericity condition ?? is relaxed even mildly, the theorem becomes false. Indeed, if we relax the condition to require that the regular (instead of all) zeroes of any ﬁxed order derivative ψ(n) never contain the regular zeroes of any other ﬁxed order derivative ψ(n), the theorem is false. R The simplest example of this involves on 1 a wavelet based on the Gaus- sian, for which a one term and a two term Dirac delta initial condition have the same zeroes. 16 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: equation

text: ∂fˆ

text: (ω)

text: lim

text: f(x)

text: e

text: i(ω

text: e

text: iω

text: x

text: dx.

text: ∂ωj

text: Rd

text: R

text: Fix

text: λ

text: satisfying

text: Im

text: ω

text: λ

text: γ.

text: For

text: suﬃciently

text: small

text: the

text: integrand

text: R

text: in

text: is

text: absolutely

text: bounded

text: over

text: all

text: x

text: d

text: by

text: xf(x)

text: eλ

text: x

text: R

text: which

text: is

text: integrable

text: since

text: f

text: d).

text: By

text: dominated

text: convergence,

text: the

text: limit

text: and

text: integral

text: in

text: can

text: be

text: interchanged,

text: yielding

text: ∂fˆ

text: (ω)

text: ix

text: )f(x)e

text: iω

text: x

text: dx.

text: j

text: ∂ω

text: R

text: j

text: d

text: ˆ

text: ˆ

text: This

text: shows

text: that

text: all

text: complex

text: ﬁrst

text: partials

text: of

text: f(ω)

text: exist;

text: thus

text: f

text: is

text: analytic

text: at

text: ω.

text: ˆ

text: Using

text: dominated

text: convergence

text: to

text: iteratively

text: evaluate

text: derivatives

text: of

text: f(ω)

text: as

text: in

text: we

text: obtain

text: the

text: Taylor

text: expansion

text: iω)α

text: ˆ

text: f(ω)

text: µ

text: α

text: α!

text: α

text: R

text: By

text: analytic

text: continuation,

text: the

text: moments

text: µ

text: uniquely

text: determine

text: fˆ

text: on

text: d.

text: α

text: α

text: Since

text: the

text: Fourier

text: transform

text: is

text: one-to-one

text: on

text: d),

text: f

text: is

text: uniquely

text: determined

text: by

text: its

text: moments.

text: This

text: completes

text: the

text: proof.

text: This

text: theorem

text: can

text: be

text: described

text: as

text: the

text: strongest

text: of

text: its

text: kind

text: in

text: two

text: senses.

text: First

text: (Section

text: the

text: theorem

text: is

text: false

text: if

text: the

text: exponential

text: decay

text: required

text: by

text: the

text: P

text: condition

text: f

text: is

text: relaxed

text: to

text: algebraic

text: decay.

text: γ

text: Second,

text: if

text: the

text: above

text: genericity

text: condition

text: is

text: relaxed

text: even

text: mildly,

text: the

text: theorem

text: becomes

text: false.

text: Indeed,

text: if

text: we

text: relax

text: the

text: condition

text: to

text: require

text: that

text: the

text: regular

text: (instead

text: of

text: all)

text: zeroes

text: of

text: any

text: ﬁxed

text: order

text: derivative

text: ψ(n)

text: never

text: contain

text: the

text: regular

text: zeroes

text: of

text: any

text: other

text: ﬁxed

text: order

text: derivative

text: ψ(n),

text: the

text: theorem

text: is

text: false.

text: R

text: The

text: simplest

text: example

text: of

text: this

text: involves

text: on

text: a

text: wavelet

text: based

text: on

text: the

text: Gaus-

text: sian,

text: for

text: which

text: a

text: one

text: term

text: and

text: a

text: two

text: term

text: Dirac

text: delta

text: initial

text: condition

text: have

text: the

text: same

text: zeroes.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##