text: Case M binary black holes 7 ( (cid:16) (cid:17) We simulate massive binary populations over cosmic grid. time under the following assumptions. 4.1. Initial distribution = 0.2 + 2.7 × 10−4 m − 50 2 for m < 50M(cid:12), We sample massive binary systems by assuming that ωc 0.2 M(cid:12) for m ≥ 50M(cid:12). the primary mass follows a Kroupa IMF (Kroupa & m1 Weidner 2003). We adopt a ﬂat mass ratio distribution Yoon et al. (2006) produce models with step sizes of 0 1 ∈ . for q [0.1, 1], where q = m2/m1 (e.g. Sana et al. 2012; in ω , and our ﬁt lies close to the maximum ω for single Kobulnicky et al. 2014). For the distribution of orbital starsc to evolve on the standard evolutionary tcracks. On periods P we adopt dN/d log10 P ∝ (log10 P)−0.5 ap- the other hand, mixing in stars in a tidally locked binary propriate for O-type stars (Sana et al. 2012), where we is likely to be stronger than in single stars with the same ∈ extend the period range to log ( days) [0 075 3 5]. initial rotational frequency due to the stronger deviations 10 P/ . , . The lower limit is chosen by estimating the likely mini- from symmetry and particularly due to the extra reser- mum of the underlying distribution consistent with the voir of orbital angular momentum which can be fed back shortest-period system in the observed sample of 34 bi- into the stars as they evolve. The threshold initial rota- naries with constrained orbital periods (Sana et al. 2012) tion rate for chemically homogeneous evolution in single having a period of 1.41 days, or log ( days) = 0 15. stars is therefore probably a conservative lower limit for 10 P/ . The extension of the upper limit allows for wide binary the threshold in tidally locked binary stars. Future 3D stars (e.g., Sana et al. 2014) and single stars (see dis- hydrodynamical simulations will be needed to investigate cussion in de Mink & Belczynski 2015). In practice, we the threshold. We adopt the expression above as our cur- are only interested in short-period binaries, and the re- rent best guess for the threshold in binary systems. maining binaries are included in the simulation only for After drawing systems from the initial distributions, normalization. We assume that all binaries are circular, we compute the Keplerian rotation rate assuming that as expected for the short-period tidally locked systems the stellar spin is synchronized with the orbit, which of interest here. is appropriate for the short-period systems of interest We implicitly assume here that the distributions are (Zahn 1989). If the stellar rotation rate is larger than separable and that the distributions measured in resolved our threshold for chemically homogeneous evolution we nearby stellar populations are a fair approximation for follow the evolution further. the distribution of binary properties at higher redshift and lower metallicity. The latter assumption is consis- 4.4. Mass loss and formation of the black hole remnant tent with the observational data available so far, which We account for the mass loss driven by stellar winds show no statistically signiﬁcant trends with metallicity and by envelope ejection during the ﬁnal explosion, which or environment (e.g., Moe & Di Stefano 2013; Sana et al. aﬀects the orbital separation and the masses of the ﬁnal 2013). remnants. We adopt a simple parametrized approach, 4.2. Stellar radii which is suﬃcient given the approximate nature of this For the stellar radii at zero-age we use a ﬁt against calculation. Since the rates are dominated by systems zero-age main-sequence models computed with Eggle- formed at a metallicity of ≤ 0 004 we adopt typical Z . ton’s evolutionary code (Eggleton 1971) with updates by values for this metallicity. This is a conservative assump- Pols et al. (1995) and Glebbeek et al. (2008). To check if tion, since mass loss via radiatively driven stellar winds a star ﬁlls its Roche lobe, we compare its radius with the is reduced at lower metallicity (Vink et al. 2001). volume-equivalent eﬀective radius ﬁt of Eggleton (1983). We adopt = 0 1 for the fraction of mass that is lost fMS . during the main sequence evolution, and = 0 25 for fWR . the fraction of mass lost during post main sequence evo- lution as a Wolf-Rayet star. These are consistent with the results of Yoon et al. (2006), taking into account that 4.3. Threshold for homogeneous mixing their mass loss rates are over-predicted by their assump- We base our simulations on the grid of detailed models tion that the dredge-up of CNO is as eﬀective as Fe. by Yoon et al. (2006). These models are computed with a Both the mass fallback fraction (e.g., Fryer 1999; Bel- hydrodynamic stellar evolution code which includes the czynski et al. 2012; Fryer et al. 2012) and black-hole na- eﬀect of the centrifugal force on the stellar structure, tal kicks (e.g, Repetto et al. 2012; Janka 2013; Miller- chemical mixing and transport of angular momentum Jones 2014) accompanying BH-forming core collapse su- due to rotationally induced hydrodynamic instabilities pernovae are highly uncertain. We adopt = 0 1 for fSN . (Heger et al. 2000), and the transport of angular mo- the fraction of mass lost during the supernova explosion, mentum due to magnetic torque (Spruit 2002). consistent with the low mass loss predicted by Fryer & The threshold rotation rate for a single star to un- Heger (2011) and Fryer et al. (2012). The natal kicks dergo homogeneous evolution can be inferred from the are not expected to be important given the very com- grid of models by Yoon et al. (2006). They express the pact binaries considered here: even the high natal kicks p threshold as a function of the ratio of the equatorial ve- of ∼ 400 km/s claimed by Repetto & Nelemans (2015) locity to the Keplerian velocity, ω = v/v , where they for the most extreme BHs are lower than the typical or- c k ∼ deﬁne = with denoting the stellar mass bital velocity of 800 km/s. Moreover, evidence for such v Gm/r m and thke stellar radius, ignoring possible deformation. high kicks is disputed by Mandel (2016) and Belczynski r We have approximated the minimum for single stars et al. (2016), who show that the existing observations are ω to achieve rotationally-induced quasi-hocmogeneous evo- consistent with much lower natal kicks of (cid:46) 100 km/s. lution with the following analytic ﬁt for the = 0 004 Therefore, we ignore BH natal kicks in our analysis. Z . ##LTLine## ##LTLine##

text: Case

text: M

text: binary

text: black

text: holes

text: We

text: simulate

text: massive

text: binary

text: populations

text: over

text: cosmic

text: grid.

text: time

text: under

text: the

text: following

text: assumptions.

label: section

text: Initial

text: distribution

text: m

text: for

text: m

text: We

text: sample

text: massive

text: binary

text: systems

text: by

text: assuming

text: that

text: ωc

text: for

text: m

text: the

text: primary

text: mass

text: follows

text: a

text: Kroupa

text: IMF

text: (Kroupa

text: Weidner

text: We

text: adopt

text: a

text: ﬂat

text: mass

text: ratio

text: distribution

text: Yoon

text: et

text: al.

text: produce

text: models

text: with

text: step

text: sizes

text: of

text: for

text: q

text: where

text: q

text: (e.g.

text: Sana

text: et

text: al.

text: in

text: ω

text: and

text: our

text: ﬁt

text: lies

text: close

text: to

text: the

text: maximum

text: ω

text: for

text: single

text: Kobulnicky

text: et

text: al.

text: For

text: the

text: distribution

text: of

text: orbital

text: starsc

text: to

text: evolve

text: on

text: the

text: standard

text: evolutionary

text: tcracks.

text: On

text: periods

text: P

text: we

text: adopt

text: dN/d

text: P

text: ap-

text: the

text: other

text: hand,

text: mixing

text: in

text: stars

text: in

text: a

text: tidally

text: locked

text: binary

text: propriate

text: for

text: O-type

text: stars

text: (Sana

text: et

text: al.

text: where

text: we

text: is

text: likely

text: to

text: be

text: stronger

text: than

text: in

text: single

text: stars

text: with

text: the

text: same

text: extend

text: the

text: period

text: range

text: to

text: log

text: days)

text: initial

text: rotational

text: frequency

text: due

text: to

text: the

text: stronger

text: deviations

text: P/

text: The

text: lower

text: limit

text: is

text: chosen

text: by

text: estimating

text: the

text: likely

text: mini-

text: from

text: symmetry

text: and

text: particularly

text: due

text: to

text: the

text: extra

text: reser-

text: mum

text: of

text: the

text: underlying

text: distribution

text: consistent

text: with

text: the

text: voir

text: of

text: orbital

text: angular

text: momentum

text: which

text: can

text: be

text: fed

text: back

text: shortest-period

text: system

text: in

text: the

text: observed

text: sample

text: of

text: bi-

text: into

text: the

text: stars

text: as

text: they

text: evolve.

text: The

text: threshold

text: initial

text: rota-

text: naries

text: with

text: constrained

text: orbital

text: periods

text: (Sana

text: et

text: al.

text: tion

text: rate

text: for

text: chemically

text: homogeneous

text: evolution

text: in

text: single

text: having

text: a

text: period

text: of

text: days,

text: or

text: log

text: days)

text: stars

text: is

text: therefore

text: probably

text: a

text: conservative

text: lower

text: limit

text: for

text: P/

text: The

text: extension

text: of

text: the

text: upper

text: limit

text: allows

text: for

text: wide

text: binary

text: the

text: threshold

text: in

text: tidally

text: locked

text: binary

text: stars.

text: Future

text: stars

text: (e.g.,

text: Sana

text: et

text: al.

text: and

text: single

text: stars

text: (see

text: dis-

text: hydrodynamical

text: simulations

text: will

text: be

text: needed

text: to

text: investigate

text: cussion

text: in

text: de

text: Mink

text: Belczynski

text: In

text: practice,

text: we

text: the

text: threshold.

text: We

text: adopt

text: the

text: expression

text: above

text: as

text: our

text: cur-

text: are

text: only

text: interested

text: in

text: short-period

text: binaries,

text: and

text: the

text: re-

text: rent

text: best

text: guess

text: for

text: the

text: threshold

text: in

text: binary

text: systems.

text: maining

text: binaries

text: are

text: included

text: in

text: the

text: simulation

text: only

text: for

text: After

text: drawing

text: systems

text: from

text: the

text: initial

text: distributions,

text: normalization.

text: We

text: assume

text: that

text: all

text: binaries

text: are

text: circular,

text: we

text: compute

text: the

text: Keplerian

text: rotation

text: rate

text: assuming

text: that

text: as

text: expected

text: for

text: the

text: short-period

text: tidally

text: locked

text: systems

text: the

text: stellar

text: spin

text: is

text: synchronized

text: with

text: the

text: orbit,

text: which

text: of

text: interest

text: here.

text: is

text: appropriate

text: for

text: the

text: short-period

text: systems

text: of

text: interest

text: We

text: implicitly

text: assume

text: here

text: that

text: the

text: distributions

text: are

text: (Zahn

text: If

text: the

text: stellar

text: rotation

text: rate

text: is

text: larger

text: than

text: separable

text: and

text: that

text: the

text: distributions

text: measured

text: in

text: resolved

text: our

text: threshold

text: for

text: chemically

text: homogeneous

text: evolution

text: we

text: nearby

text: stellar

text: populations

text: are

text: a

text: fair

text: approximation

text: for

text: follow

text: the

text: evolution

text: further.

text: the

text: distribution

text: of

text: binary

text: properties

text: at

text: higher

text: redshift

text: and

text: lower

text: metallicity.

text: The

text: latter

text: assumption

text: is

text: consis-

text: Mass

text: loss

text: and

text: formation

text: of

text: the

text: black

text: hole

text: remnant

text: tent

text: with

text: the

text: observational

text: data

text: available

text: so

text: far,

text: which

text: We

text: account

text: for

text: the

text: mass

text: loss

text: driven

text: by

text: stellar

text: winds

text: show

text: no

text: statistically

text: signiﬁcant

text: trends

text: with

text: metallicity

text: and

text: by

text: envelope

text: ejection

text: during

text: the

text: ﬁnal

text: explosion,

text: which

text: or

text: environment

text: (e.g.,

text: Moe

text: Di

text: Stefano

text: Sana

text: et

text: al.

text: aﬀects

text: the

text: orbital

text: separation

text: and

text: the

text: masses

text: of

text: the

text: ﬁnal

text: remnants.

text: We

text: adopt

text: a

text: simple

text: parametrized

text: approach,

text: Stellar

text: radii

text: which

text: is

text: suﬃcient

text: given

text: the

text: approximate

text: nature

text: of

text: this

text: For

text: the

text: stellar

text: radii

text: at

text: zero-age

text: we

text: use

text: a

text: ﬁt

text: against

text: calculation.

text: Since

text: the

text: rates

text: are

text: dominated

text: by

text: systems

text: zero-age

text: main-sequence

text: models

text: computed

text: with

text: Eggle-

text: formed

text: at

text: a

text: metallicity

text: of

text: we

text: adopt

text: typical

text: Z

text: ton’s

text: evolutionary

text: code

text: (Eggleton

text: with

text: updates

text: by

text: values

text: for

text: this

text: metallicity.

text: This

text: is

text: a

text: conservative

text: assump-

text: Pols

text: et

text: al.

text: and

text: Glebbeek

text: et

text: al.

text: To

text: check

text: if

text: tion,

text: since

text: mass

text: loss

text: via

text: radiatively

text: driven

text: stellar

text: winds

text: a

text: star

text: ﬁlls

text: its

text: Roche

text: lobe,

text: we

text: compare

text: its

text: radius

text: with

text: the

text: is

text: reduced

text: at

text: lower

text: metallicity

text: (Vink

text: et

text: al.

text: volume-equivalent

text: eﬀective

text: radius

text: ﬁt

text: of

text: Eggleton

text: We

text: adopt

text: for

text: the

text: fraction

text: of

text: mass

text: that

text: is

text: lost

text: fMS

text: during

text: the

text: main

text: sequence

text: evolution,

text: and

text: for

text: fWR

text: the

text: fraction

text: of

text: mass

text: lost

text: during

text: post

text: main

text: sequence

text: evo-

text: lution

text: as

text: a

text: Wolf-Rayet

text: star.

text: These

text: are

text: consistent

text: with

text: the

text: results

text: of

text: Yoon

text: et

text: al.

text: taking

text: into

text: account

text: that

text: Threshold

text: for

text: homogeneous

text: mixing

text: their

text: mass

text: loss

text: rates

text: are

text: over-predicted

text: by

text: their

text: assump-

text: We

text: base

text: our

text: simulations

text: on

text: the

text: grid

text: of

text: detailed

text: models

text: tion

text: that

text: the

text: dredge-up

text: of

text: CNO

text: is

text: as

text: eﬀective

text: as

text: Fe.

text: by

text: Yoon

text: et

text: al.

text: These

text: models

text: are

text: computed

text: with

text: a

text: Both

text: the

text: mass

text: fallback

text: fraction

text: (e.g.,

text: Fryer

text: Bel-

text: hydrodynamic

text: stellar

text: evolution

text: code

text: which

text: includes

text: the

text: czynski

text: et

text: al.

text: Fryer

text: et

text: al.

text: and

text: black-hole

text: na-

text: eﬀect

text: of

text: the

text: centrifugal

text: force

text: on

text: the

text: stellar

text: structure,

text: tal

text: kicks

text: (e.g,

text: Repetto

text: et

text: al.

text: Janka

text: Miller-

text: chemical

text: mixing

text: and

text: transport

text: of

text: angular

text: momentum

text: Jones

text: accompanying

text: BH-forming

text: core

text: collapse

text: su-

text: due

text: to

text: rotationally

text: induced

text: hydrodynamic

text: instabilities

text: pernovae

text: are

text: highly

text: uncertain.

text: We

text: adopt

text: for

text: fSN

text: (Heger

text: et

text: al.

text: and

text: the

text: transport

text: of

text: angular

text: mo-

text: the

text: fraction

text: of

text: mass

text: lost

text: during

text: the

text: supernova

text: explosion,

text: mentum

text: due

text: to

text: magnetic

text: torque

text: (Spruit

text: consistent

text: with

text: the

text: low

text: mass

text: loss

text: predicted

text: by

text: Fryer

text: The

text: threshold

text: rotation

text: rate

text: for

text: a

text: single

text: star

text: to

text: un-

text: Heger

text: and

text: Fryer

text: et

text: al.

text: The

text: natal

text: kicks

text: dergo

text: homogeneous

text: evolution

text: can

text: be

text: inferred

text: from

text: the

text: are

text: not

text: expected

text: to

text: be

text: important

text: given

text: the

text: very

text: com-

text: grid

text: of

text: models

text: by

text: Yoon

text: et

text: al.

text: They

text: express

text: the

text: pact

text: binaries

text: considered

text: here:

text: even

text: the

text: high

text: natal

text: kicks

text: p

text: threshold

text: as

text: a

text: function

text: of

text: the

text: ratio

text: of

text: the

text: equatorial

text: ve-

text: of

text: km/s

text: claimed

text: by

text: Repetto

text: Nelemans

text: locity

text: to

text: the

text: Keplerian

text: velocity,

text: ω

text: v/v

text: where

text: they

text: for

text: the

text: most

text: extreme

text: BHs

text: are

text: lower

text: than

text: the

text: typical

text: or-

text: c

text: k

text: deﬁne

text: with

text: denoting

text: the

text: stellar

text: mass

text: bital

text: velocity

text: of

text: km/s.

text: Moreover,

text: evidence

text: for

text: such

text: v

text: Gm/r

text: m

text: and

text: thke

text: stellar

text: radius,

text: ignoring

text: possible

text: deformation.

text: high

text: kicks

text: is

text: disputed

text: by

text: Mandel

text: and

text: Belczynski

text: r

text: We

text: have

text: approximated

text: the

text: minimum

text: for

text: single

text: stars

text: et

text: al.

text: who

text: show

text: that

text: the

text: existing

text: observations

text: are

text: ω

text: to

text: achieve

text: rotationally-induced

text: quasi-hocmogeneous

text: evo-

text: consistent

text: with

text: much

text: lower

text: natal

text: kicks

text: of

text: km/s.

text: lution

text: with

text: the

text: following

text: analytic

text: ﬁt

text: for

text: the

text: Therefore,

text: we

text: ignore

text: BH

text: natal

text: kicks

text: in

text: our

text: analysis.

text: Z

text: ##LTLine##

text: ##LTLine##


text: 4 Conclusion In conclusion, by using the previous results given in [8], and [33] we ob- tain the following commutative diagram among the two continuous optimal control problems (P ) and (P ), and two discrete optimal control problems → α → ∞ → ∞ (P ) and (P ) when h 0, α and (h, α) (0+, ), which can be h hα summarized by the following ﬁgure (Fig. 1): Problem  P  Problem  P   g , u , J (g )     g , u , J (g ) op gop  op op gop op (h,)  (0, ) h  0  h  0  g , u , J (g ) g , u , J (g ) oph hgoph h oph oph hgoph h oph     Problem P  Problem  P  h h Fig. 1: C omplete diagram for two continuous and two discrete optimal control and the associated state system solutions 1 References [1] D. R. Adams, S. M. Lenhart, and J. Yong, Optimal control of the obstacle for an elliptic variational inequality, Appl. Math. Optim., 38 (1998), pp. 121–140. [2] K. Ait Hadi, Optimal control of an obstacle problem: optimality condi- tions, IMA J. Math. Control Inform., 23 (2008), pp. 325–334. 18 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: Conclusion

text: In

label: paragraph

text: conclusion,

text: by

text: using

text: the

text: previous

text: results

text: given

text: in

text: and

text: we

text: ob-

text: tain

text: the

text: following

text: commutative

text: diagram

text: among

text: the

text: two

text: continuous

text: optimal

text: control

text: problems

text: (P

text: and

text: (P

text: and

text: two

text: discrete

text: optimal

text: control

text: problems

text: α

text: (P

text: and

text: (P

text: when

text: h

text: α

text: and

text: (h,

text: α)

text: which

text: can

text: be

text: h

text: hα

text: summarized

text: by

text: the

text: following

text: ﬁgure

text: (Fig.

text: Problem

font: default

text: P

text: Problem

text: P

text: g

text: u

text: J

text: (g

text: g

text: u

text: J

text: (g

text: op

text: gop

text: op

text: op

text: gop

text: op

text: (h,)

text: h

text: 0

text: h

text: 0

text: g

text: u

text: J

text: (g

text: g

text: u

text: J

text: (g

text: oph

text: hgoph

text: h

text: oph

text: oph

text: hgoph

text: h

text: oph

text: Problem

text: P

text: Problem

text: P

text: h

text: h

text: Fig.

text: C

text: omplete

text: diagram

text: for

text: two

text: continuous

text: and

text: two

text: discrete

text: optimal

text: control

text: and

text: the

text: associated

text: state

text: system

text: solutions

text: References

text: D.

text: R.

text: Adams,

text: S.

text: M.

text: Lenhart,

text: and

text: J.

text: Yong,

text: Optimal

text: control

text: of

text: the

text: obstacle

text: for

text: an

text: elliptic

text: variational

text: inequality,

text: Appl.

text: Math.

text: Optim.,

text: pp.

text: K.

text: Ait

text: Hadi,

text: Optimal

text: control

text: of

text: an

text: obstacle

text: problem:

text: optimality

text: condi-

text: tions,

text: IMA

text: J.

text: Math.

text: Control

text: Inform.,

text: pp.

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##


text: J (g ) = min J (g) (1.6) α opα ∈ α g H → R where the quadratic cost functional J : H + was deﬁned by: α 0 1(cid:107) (cid:107) M (cid:107) (cid:107) J (g) = u 2 + g 2 (1.7) α 2 αg H 2 H with M > 0 a given constant and u is the corresponding solution of the αg ∈ elliptic variational inequality (1.3) associated to the control g H. Several optimal control problems are governed by elliptic variational inequalities ([1],[2],[3],[5],[6], [13],[14],[27],[31],[32],[39]) and there exists an abundant literature about continuous and numerical analysis of optimal control problems governed by elliptic variational equalities or inequalities ([4],[10],[11],[15],[16], [17],[18],[19],[20], [21],[22], [23],[24],[26], [29],[30],[35], [36],[40]) and by parabolic variational equalities or inequalities ([7], [28]). The objective of this work is to make the numerical analysis of the con- tinuous optimal control problem (P ) which is governed by the elliptic vari- α ational inequality (1.3) by proving the convergence of a discrete solution to the solution of the continuous optimal control problem. In Section 2, we establish the discrete elliptic variational inequality (2.3) which is the discrete formulation of the continuous elliptic variational in- equality (1.3), and we obtain that these discrete problem has unique so- lutions for all positive h. Moreover, we deﬁne a family (P ) of discrete hα optimal control problems (2.8) and, we obtain several properties for the state system (2.3) and for the discrete cost functional J deﬁned in (2.7). hα In Section 3, on adequate functional spaces, we obtain a result of global → strong convergence when the parameter h 0 (for each α > 0) and when → ∞ α (for each h > 0) for the discrete state sytems and for the discrete optimal optimal problem corresponding to (P ). We end this work prov- α ing the double convergence of the discrete optimal solutions of (P ) when → ∞ hα (h, α) (0, ) obtaining a complete commutative diagram among two dis- crete and two continuous optimal control problems given en Fig. 1. We generalize recent results obtained for optimal control problems governed by elliptic variational equalities given in [37, 38]. 2 Properties of the discretization of the problem (P ) α 3 ##LTLine## ##LTLine##

text: J

label: equation

text: (g

text: min

text: J

text: (g)

text: α

text: opα

text: α

text: g

text: H

text: R

text: where

text: the

text: quadratic

text: cost

text: functional

text: J

text: H

text: was

text: deﬁned

text: by:

text: α

text: M

text: J

text: (g)

text: u

text: g

text: α

text: αg

text: H

text: H

text: with

text: M

text: a

text: given

text: constant

text: and

text: u

text: is

text: the

text: corresponding

text: solution

text: of

text: the

text: αg

text: elliptic

text: variational

text: inequality

text: associated

text: to

text: the

text: control

text: g

text: H.

text: Several

text: optimal

text: control

text: problems

text: are

text: governed

text: by

text: elliptic

text: variational

text: inequalities

text: and

text: there

text: exists

text: an

text: abundant

text: literature

text: about

text: continuous

text: and

text: numerical

text: analysis

text: of

text: optimal

text: control

text: problems

text: governed

text: by

text: elliptic

text: variational

text: equalities

text: or

text: inequalities

text: and

text: by

text: parabolic

text: variational

text: equalities

text: or

text: inequalities

text: The

text: objective

text: of

text: this

text: work

text: is

text: to

text: make

text: the

text: numerical

text: analysis

text: of

text: the

text: con-

text: tinuous

text: optimal

text: control

text: problem

text: (P

text: which

text: is

text: governed

text: by

text: the

text: elliptic

text: vari-

text: α

text: ational

text: inequality

text: by

text: proving

text: the

text: convergence

text: of

text: a

text: discrete

text: solution

text: to

text: the

text: solution

text: of

text: the

text: continuous

text: optimal

text: control

text: problem.

text: In

text: Section

text: we

text: establish

text: the

text: discrete

text: elliptic

text: variational

text: inequality

text: which

text: is

text: the

text: discrete

text: formulation

text: of

text: the

text: continuous

text: elliptic

text: variational

text: in-

text: equality

text: and

text: we

text: obtain

text: that

text: these

text: discrete

text: problem

text: has

text: unique

text: so-

text: lutions

text: for

text: all

text: positive

text: h.

text: Moreover,

text: we

text: deﬁne

text: a

text: family

text: (P

text: of

text: discrete

text: hα

text: optimal

text: control

text: problems

text: and,

text: we

text: obtain

text: several

text: properties

text: for

text: the

text: state

text: system

text: and

text: for

text: the

text: discrete

text: cost

text: functional

text: J

text: deﬁned

text: in

text: hα

text: In

text: Section

text: on

text: adequate

text: functional

text: spaces,

text: we

text: obtain

text: a

text: result

text: of

text: global

text: strong

text: convergence

text: when

text: the

text: parameter

text: h

text: (for

text: each

text: α

text: and

text: when

text: α

text: (for

text: each

text: h

text: for

text: the

text: discrete

text: state

text: sytems

text: and

text: for

text: the

text: discrete

text: optimal

text: optimal

text: problem

text: corresponding

text: to

text: (P

text: We

text: end

text: this

text: work

text: prov-

text: α

text: ing

text: the

text: double

text: convergence

text: of

text: the

text: discrete

text: optimal

text: solutions

text: of

text: (P

text: when

text: hα

text: (h,

text: α)

text: obtaining

text: a

text: complete

text: commutative

text: diagram

text: among

text: two

text: dis-

text: crete

text: and

text: two

text: continuous

text: optimal

text: control

text: problems

text: given

text: en

text: Fig.

text: We

text: generalize

text: recent

text: results

text: obtained

text: for

text: optimal

text: control

text: problems

text: governed

text: by

text: elliptic

text: variational

text: equalities

text: given

text: in

text: Properties

text: of

text: the

text: discretization

text: of

text: the

text: problem

text: (P

text: α

text: ##LTLine##

text: ##LTLine##


text: where u is the unique solution of the elliptic variational inequality (2.3) hαg ∈ for a given control g H and a given parameter α > 0. Then, we estab- lish the following discrete distributed optimal control problem (P ): Find ∈ hα g H such that op hα J (g ) = min J (g). (2.8) hα ophα g ∈H hα We remark that the discrete (in the space) distributed optimal control prob- lem (P ) is still an inﬁnite dimensional optimal control problem since the hα control space H is not discretized. ∈ Theorem 2.2. For the control g H, the parameters α > 0 and h > 0, we have: a) ∞ lim J (g) = . (cid:107) (cid:107) →∞ hα g H ≥ (cid:107) (cid:107) − (cid:107) (cid:107) b) J (g) M g 2 C g for some constant C independent of hα 2 H H h > 0. c) The functional J is a lower weakly semi-continuous application in hα H. d) For each h > 0 and α > 0, there exists a solution of the discrete distributed optimal control problem (2.8). Proof. From the deﬁnition of J (g) we obtain a) and b). hα (cid:107) (cid:107) (cid:107) − (cid:107) − c) Let g (cid:42) g in H weak, then by using the equality g 2 = g g 2 (cid:107) (cid:107) n (cid:107) (cid:107) ≤ (cid:107) n(cid:107) H n H g 2 + 2(g , g) we obtain that g lim inf →∞ g . Therefore, we H n H H n n H have ≥ 1(cid:107) (cid:107) M (cid:107) (cid:107) lim inf J (g ) u 2 + g 2 = J (g). n→∞ hα n 2 hαg H 2 H hα d) It follows from [27]. ∈ Lemma 2.3. If the continuous state system has the regularity u Hr(Ω) ≤ ∈ αg (1 < r 2) for g H and α > 0, then we have the following estimation ∀ ∈ g H: | − | ≤ − r 1 Jhα(g) Jα(g) C(α)h 2 (2.9) 6 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

text: u

text: is

text: the

text: unique

text: solution

text: of

text: the

text: elliptic

text: variational

text: inequality

text: hαg

text: for

text: a

text: given

text: control

text: g

text: H

text: and

text: a

text: given

text: parameter

text: α

text: Then,

text: we

text: estab-

text: lish

text: the

text: following

text: discrete

text: distributed

text: optimal

text: control

text: problem

text: (P

text: Find

text: hα

text: g

text: H

text: such

text: that

text: op

text: hα

text: J

text: (g

text: min

text: J

text: (g).

text: hα

text: ophα

text: g

text: ∈H

text: hα

text: We

text: remark

text: that

text: the

text: discrete

text: (in

text: the

text: space)

text: distributed

text: optimal

text: control

text: prob-

text: lem

text: (P

text: is

text: still

text: an

text: inﬁnite

text: dimensional

text: optimal

text: control

text: problem

text: since

text: the

text: hα

text: control

text: space

text: H

text: is

text: not

text: discretized.

text: Theorem

text: For

text: the

text: control

text: g

text: H,

text: the

text: parameters

text: α

text: and

text: h

text: we

text: have:

text: a)

text: lim

text: J

text: (g)

text: hα

text: g

text: H

text: b)

text: J

text: (g)

text: M

text: g

text: C

text: g

text: for

text: some

text: constant

text: C

text: independent

text: of

text: hα

text: H

text: H

text: h

text: c)

text: The

text: functional

text: J

text: is

text: a

text: lower

text: weakly

text: semi-continuous

text: application

text: in

text: hα

text: H.

text: d)

text: For

text: each

text: h

text: and

text: α

text: there

text: exists

text: a

text: solution

text: of

text: the

text: discrete

text: distributed

text: optimal

text: control

text: problem

text: Proof.

text: From

text: the

text: deﬁnition

text: of

text: J

text: (g)

text: we

text: obtain

text: a)

text: and

text: b).

text: hα

text: c)

text: Let

text: g

text: g

text: in

text: H

text: weak,

text: then

text: by

text: using

text: the

text: equality

text: g

text: g

text: g

text: n

text: H

text: n

text: H

text: g

text: g)

text: we

text: obtain

text: that

text: g

text: lim

text: inf

text: g

text: Therefore,

text: we

text: H

text: n

text: H

text: H

text: n

text: n

text: H

text: have

text: M

text: lim

text: inf

text: J

text: (g

text: u

text: g

text: J

text: (g).

text: n→∞

text: hα

text: n

text: hαg

text: H

text: H

text: hα

text: d)

text: It

text: follows

text: from

text: Lemma

text: If

text: the

text: continuous

text: state

text: system

text: has

text: the

text: regularity

text: u

text: Hr(Ω)

text: αg

text: r

text: for

text: g

text: H

text: and

text: α

text: then

text: we

text: have

text: the

text: following

text: estimation

text: g

text: H:

text: r

text: Jhα(g)

text: Jα(g)

text: C(α)h

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: where C is a positive constant independent of h > 0. Proof. By deﬁnition of the discrete cost functional J , we have: hα − 1 (cid:107) (cid:107) −(cid:107) (cid:107) 1(cid:107) − (cid:107) − J (g) J (g) = ( u 2 u 2 ) = u u 2 + (u , u u ) hα α 2 hαg H αg H 2 hαg αg H αg hαg αg H and therefore, if we apply (2.6), it results: | − | ≤ 1(cid:107) − (cid:107) (cid:107) (cid:107) (cid:107) − (cid:107) ≤ − r 1 Jhα(g) Jα(g) ( uhαg uαg H + uαg H) uhαg uαg H C(α) h 2 , 2 and (2.9) holds. 3 Results of Convergence (cid:84) → 3.1 Convergence when h 0 ∈ ≤ ∈ Theorem 3.1. Let u K Hr(Ω), (1 < r 2) and u K be the αg + hαg +h solutions of the elliptic variational inequalities (1.3) and (2.3) respectively ∈ → → for the control g H, then u u in V when h 0+. hαg αg Proof. Similarly to the part a) of the Lemma 2.1, we can show that there (cid:107) (cid:107) ≤ ∀ exist a constant C > 0 such that u C, h > 0. Therefore, we ∈ hαg V conclude that there exists η V so that u (cid:42) η in V (in H strong) → ∈ α hαg α ∈ as h 0+ and η K . On the other hand, given v K let be v = ∈ + → + h Π(v) K for each h such that v v in V when h goes to zero. Now, +h ∈ h by considering v K in the discrete elliptic variational inequality (2.3) h +h we get: − ≥ − − − − a (u , v u ) (g, v u ) (q, v u ) +α(b, v u ) (3.1) α hαg h hαg h hαg H h hαg Q h hαg R → and when we pass to the limit as h 0+ in (3.1) by using that the bilinear form a is lower weak semi-continuous in V , we obtain: − ≥ − − − − ∀ ∈ a (η , v η ) (g, v η ) (q, v η ) + α(b, v η ) , v K α α α α H α Q α R + 7 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

text: C

text: is

text: a

text: positive

text: constant

text: independent

text: of

text: h

text: Proof.

text: By

text: deﬁnition

text: of

text: the

text: discrete

text: cost

text: functional

text: J

text: we

text: have:

text: hα

text: J

text: (g)

text: J

text: (g)

text: u

text: u

text: u

text: u

text: (u

text: u

text: u

text: hα

text: α

text: hαg

text: H

text: αg

text: H

text: hαg

text: αg

text: H

text: αg

text: hαg

text: αg

text: H

text: and

text: therefore,

text: if

text: we

text: apply

text: it

text: results:

text: r

text: Jhα(g)

text: Jα(g)

text: uhαg

text: uαg

text: H

text: uαg

text: H)

text: uhαg

text: uαg

text: H

text: C(α)

text: h

text: and

text: holds.

text: Results

text: of

text: Convergence

text: Convergence

text: when

text: h

text: Theorem

text: Let

text: u

text: K

text: Hr(Ω),

text: r

text: and

text: u

text: K

text: be

text: the

text: αg

text: hαg

text: +h

text: solutions

text: of

text: the

text: elliptic

text: variational

text: inequalities

text: and

text: respectively

text: for

text: the

text: control

text: g

text: H,

text: then

text: u

text: u

text: in

text: V

text: when

text: h

text: hαg

text: αg

text: Proof.

text: Similarly

text: to

text: the

text: part

text: a)

text: of

text: the

text: Lemma

text: we

text: can

text: show

text: that

text: there

text: exist

text: a

text: constant

text: C

text: such

text: that

text: u

text: C,

text: h

text: Therefore,

text: we

text: hαg

text: V

text: conclude

text: that

text: there

text: exists

text: η

text: V

text: so

text: that

text: u

text: η

text: in

text: V

text: (in

text: H

text: strong)

text: α

text: hαg

text: α

text: as

text: h

text: and

text: η

text: K

text: On

text: the

text: other

text: hand,

text: given

text: v

text: K

text: let

text: be

text: v

text: h

text: Π(v)

text: K

text: for

text: each

text: h

text: such

text: that

text: v

text: v

text: in

text: V

text: when

text: h

text: goes

text: to

text: zero.

text: Now,

text: +h

text: h

text: by

text: considering

text: v

text: K

text: in

text: the

text: discrete

text: elliptic

text: variational

text: inequality

text: h

text: +h

text: we

text: get:

text: a

text: (u

text: v

text: u

text: (g,

text: v

text: u

text: (q,

text: v

text: u

text: +α(b,

text: v

text: u

text: α

text: hαg

text: h

text: hαg

text: h

text: hαg

text: H

text: h

text: hαg

text: Q

text: h

text: hαg

text: R

text: and

text: when

text: we

text: pass

text: to

text: the

text: limit

text: as

text: h

text: in

text: by

text: using

text: that

text: the

text: bilinear

text: form

text: a

text: is

text: lower

text: weak

text: semi-continuous

text: in

text: V

text: we

text: obtain:

text: a

text: (η

text: v

text: η

text: (g,

text: v

text: η

text: (q,

text: v

text: η

text: α(b,

text: v

text: η

text: v

text: K

text: α

text: α

text: α

text: α

text: H

text: α

text: Q

text: α

text: R

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 6 Mordecai-Mark Mac Low It is worth noting that in a periodic box simulation of supernova-driven turbulence, neglecting stratiﬁcation, but run at higher resolution, Padoan et al. (2015) ﬁnd velocity dispersions after 6 Myr that agree with our results. However Padoan (priv. comm., 2015) ﬁnds similar velocity dispersions in their model even prior to the onset of self-gravity, in contradiction to our results. Further investigation will be needed to establish if the dif- ference is due to their higher resolution or their neglect of stratiﬁcation and conﬁnement within the periodic box. Figure 5. Vertical projection through a sub-region of a supernova-driven, stratiﬁed, magnetized model of the interstellar medium at times of 0 and 3 Myr after self-gravity has been turned on. × × (The full domain is a 1 1 20 kpc vertical cut through the galactic disk.) Prompt collapse occurs, forming ﬁlaments strongly resembling observed IRDCs, with star-forming cores strung along them. Adapted from Iba´n˜ez-Mej´ıa et al. (2015). Figure 6. Radius-velocity dispersion relation for clouds drawn from the models prior to the action of self-gravity (yellow dots in all three panels), and for clouds at times of 2 Myr (left), 4 Myr (center) and 6 Myr (right) after self-gravity begins to act. For comparison, the observed relation derived from the data presented by Heyer et al. (2009) is shown by the dashed line. The shaded region shows clouds under 10 zones whose velocity dispersion is likely to be strongly suppressed by numerical viscosity. Adapted from Iba´n˜ez-Mej´ıa et al. (2015). ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: Mordecai-Mark

text: Mac

text: Low

text: It

text: is

text: worth

text: noting

text: that

text: in

text: a

text: periodic

text: box

text: simulation

text: of

text: supernova-driven

text: turbulence,

text: neglecting

text: stratiﬁcation,

text: but

text: run

text: at

text: higher

text: resolution,

text: Padoan

text: et

text: al.

text: ﬁnd

text: velocity

text: dispersions

text: after

text: Myr

text: that

text: agree

text: with

text: our

text: results.

text: However

text: Padoan

text: (priv.

text: comm.,

text: ﬁnds

text: similar

text: velocity

text: dispersions

text: in

text: their

text: model

text: even

text: prior

text: to

text: the

text: onset

text: of

text: self-gravity,

text: in

text: contradiction

text: to

text: our

text: results.

text: Further

text: investigation

text: will

text: be

text: needed

text: to

text: establish

text: if

text: the

text: dif-

text: ference

text: is

text: due

text: to

text: their

text: higher

text: resolution

text: or

text: their

text: neglect

text: of

text: stratiﬁcation

text: and

text: conﬁnement

text: within

text: the

text: periodic

text: box.

text: Figure

text: Vertical

label: caption

text: projection

text: through

text: a

text: sub-region

text: of

text: a

text: supernova-driven,

text: stratiﬁed,

text: magnetized

text: model

text: of

text: the

text: interstellar

text: medium

text: at

text: times

text: of

text: and

text: Myr

text: after

text: self-gravity

text: has

text: been

text: turned

text: on.

text: (The

text: full

text: domain

text: is

text: a

text: kpc

text: vertical

text: cut

text: through

text: the

text: galactic

text: disk.)

text: Prompt

text: collapse

text: occurs,

text: forming

text: ﬁlaments

text: strongly

text: resembling

text: observed

text: IRDCs,

text: with

text: star-forming

text: cores

text: strung

text: along

text: them.

text: Adapted

text: from

text: Iba´n˜ez-Mej´ıa

text: et

text: al.

text: Figure

text: Radius-velocity

text: dispersion

text: relation

text: for

text: clouds

text: drawn

text: from

text: the

text: models

text: prior

text: to

text: the

text: action

text: of

text: self-gravity

text: (yellow

text: dots

text: in

text: all

text: three

text: panels),

text: and

text: for

text: clouds

text: at

text: times

text: of

text: Myr

text: (left),

text: Myr

text: (center)

text: and

text: Myr

text: (right)

text: after

text: self-gravity

text: begins

text: to

text: act.

text: For

text: comparison,

text: the

text: observed

text: relation

text: derived

text: from

text: the

text: data

text: presented

text: by

text: Heyer

text: et

text: al.

text: is

text: shown

text: by

text: the

text: dashed

text: line.

text: The

text: shaded

text: region

text: shows

text: clouds

text: under

text: zones

text: whose

text: velocity

text: dispersion

text: is

text: likely

text: to

text: be

text: strongly

text: suppressed

text: by

text: numerical

text: viscosity.

text: Adapted

text: from

text: Iba´n˜ez-Mej´ıa

text: et

text: al.

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##


text: Atomic and Molecular Phases of the ISM 7 6. Summary In this review, I have addressed a few current topics in the theory of the atomic and molecular interstellar medium. I have focused on the following questions, with my current best guesses as to the answers. • Do cold streams contribute to gas accretion onto galaxies only at early times and for small galaxies, or far more generally as was originally claimed? The former appears to be the result of the numerical simulations most likely to be reliable. • How small is the actual ﬁlling factor of dense, molecular gas in an interstellar medium when it is observationally determined to be completely molecular? Quite small, with the rest of the volume occupied by low-density, hot gas with temperatures exceeding 106 K. • Does molecular hydrogen drive star formation, or rather does star formation produce molecular hydrogen and other high density tracers as a by-product? Molecular hydrogen and other molecules appear to function far more as a tracer than a cause of star formation, particularly since atomic line cooling is nearly as eﬀective as molecular line cooling. • Are the observed velocity dispersions in molecular clouds caused by supernova-driven turbulence, or by gravitational collapse? Gravitational collapse seems more likely, but this question does not yet appear to be settled. References Ballesteros-Paredes, J., Hartmann, L. W., Va´zquez-Semadeni, E., Heitsch, F., & Zamora-Avil´es, M. A. 2011, MNRAS, 411, 65 Bigiel, F., Leroy, A., Walter, F., et al. 2008, AJ, 136, 2846 Birnboim, Y., & Dekel, A. 2003, MNRAS, 345, 349 Bouch´e, N., Dekel, A., Genzel, R., et al. 2010, ApJ, 718, 1001 Clark, P. C., & Glover, S. C. O. 2014, MNRAS, 444, 2396 Dekel, A., & Birnboim, Y. 2006, MNRAS, 368, 2 Dekel, A., Birnboim, Y., Engel, G., et al. 2009, Nature, 457, 451 Dekel, A., & Mandelker, N. 2014, MNRAS, 444, 2071 Dekel, A., Zolotov, A., Tweed, D., et al. 2013, MNRAS, 435, 999 Fardal, M. A., Katz, N., Gardner, J. P., et al. 2001, ApJ, 562, 605 Field, G. B., Goldsmith, D. W., & Habing, H. J. 1969, ApJ (Letters), 155, L149 Gao, Y., & Solomon, P. M. 2004, ApJ, 606, 271 Gatto, A., Walch, S., Mac Low, M.-M. et al. 2015, MNRAS, 449, 1057 Goerdt, T., Ceverino, D., Dekel, A., & Teyssier, R. 2015, MNRAS, 454, 637 Hartmann, L., Ballesteros-Paredes, J., & Bergin, E. A. 2001, ApJ, 562, 852 Heyer, M., Krawczyk, C., Duval, J., & Jackson, J. M. 2009, ApJ, 699, 1092 Hill, A. S., Joung, M. R., Mac Low, M.-M., et al. 2012, ApJ, 761, 189 Iba´n˜ez-Mej´ıa, J. C., Mac Low, M.-M., Klessen, R. S., & Baczynski, C. 2015, ApJ, submitted, arXiv:1511.05602 Jackson, J. M., Rathborne, J. M., Shah, R. Y., et al. 2006, ApJ (Suppl.), 163, 145 Joung, M. K. R., & Mac Low, M.-M. 2006, ApJ, 653, 1266 Joung, M. R., Mac Low, M.-M., & Bryan, G. L. 2009, ApJ, 704, 137 Kereˇs, D., Katz, N., Weinberg, D. H., & Dav´e, R. 2005, MNRAS, 363, 2 Kereˇs, D., Katz, N., Fardal, M., Dav´e, R., & Weinberg, D. H. 2009, MNRAS, 395, 160 Keto, E. 2002, ApJ, 580, 980 Krumholz, M. R., Leroy, A. K., & McKee, C. F. 2011, ApJ, 731, 25 Lada, C. J., Lombardi, M., & Alves, J. F. 2010, ApJ, 724, 687 Leroy, A. K., Walter, F., Brinks, E., et al. 2008, AJ, 136, 2782 Li, M., Ostriker, J. P., Cen, R., Bryan, G. L., & Naab, T. 2015, ApJ, 814, 4 Nelson, D., Vogelsberger, M., Genel, S., et al. 2013, MNRAS, 429, 3353 Ocvirk, P., Pichon, C., & Teyssier, R. 2008, MNRAS, 390, 1326

text: Atomic

text: and

text: Molecular

text: Phases

text: of

text: the

text: ISM

text: Summary

text: In

text: this

text: review,

text: I

text: have

text: addressed

text: a

text: few

text: current

text: topics

text: in

text: the

text: theory

text: of

text: the

text: atomic

text: and

text: molecular

text: interstellar

text: medium.

text: I

text: have

text: focused

text: on

text: the

text: following

text: questions,

text: with

text: my

text: current

text: best

text: guesses

text: as

text: to

text: the

text: answers.

label: list

text: Do

text: cold

text: streams

text: contribute

text: to

text: gas

text: accretion

text: onto

text: galaxies

text: only

text: at

text: early

text: times

text: and

text: for

text: small

text: galaxies,

text: or

text: far

text: more

text: generally

text: as

text: was

text: originally

text: claimed?

text: The

text: former

text: appears

text: to

text: be

text: the

text: result

text: of

text: the

text: numerical

text: simulations

text: most

text: likely

text: to

text: be

text: reliable.

text: How

text: small

text: is

text: the

text: actual

text: ﬁlling

text: factor

text: of

text: dense,

text: molecular

text: gas

text: in

text: an

text: interstellar

text: medium

text: when

text: it

text: is

text: observationally

text: determined

text: to

text: be

text: completely

text: molecular?

text: Quite

text: small,

text: with

text: the

text: rest

text: of

text: the

text: volume

text: occupied

text: by

text: low-density,

text: hot

text: gas

text: with

text: temperatures

text: exceeding

text: K.

text: Does

text: molecular

text: hydrogen

text: drive

text: star

text: formation,

text: or

text: rather

text: does

text: star

text: formation

text: produce

text: molecular

text: hydrogen

text: and

text: other

text: high

text: density

text: tracers

text: as

text: a

text: by-product?

text: Molecular

text: hydrogen

text: and

text: other

text: molecules

text: appear

text: to

text: function

text: far

text: more

text: as

text: a

text: tracer

text: than

text: a

text: cause

text: of

text: star

text: formation,

text: particularly

text: since

text: atomic

text: line

text: cooling

text: is

text: nearly

text: as

text: eﬀective

text: as

text: molecular

text: line

text: cooling.

text: Are

text: the

text: observed

text: velocity

text: dispersions

text: in

text: molecular

text: clouds

text: caused

text: by

text: supernova-driven

text: turbulence,

text: or

text: by

text: gravitational

text: collapse?

text: Gravitational

text: collapse

text: seems

text: more

text: likely,

text: but

text: this

text: question

text: does

text: not

text: yet

text: appear

text: to

text: be

text: settled.

text: References

text: Ballesteros-Paredes,

text: J.,

text: Hartmann,

text: L.

text: W.,

text: Va´zquez-Semadeni,

text: E.,

text: Heitsch,

text: F.,

text: Zamora-Avil´es,

text: M.

text: A.

text: MNRAS,

text: Bigiel,

text: F.,

text: Leroy,

text: A.,

text: Walter,

text: F.,

text: et

text: al.

text: AJ,

text: Birnboim,

text: Y.,

text: Dekel,

text: A.

text: MNRAS,

text: Bouch´e,

text: N.,

text: Dekel,

text: A.,

text: Genzel,

text: R.,

text: et

text: al.

text: ApJ,

text: Clark,

text: P.

text: C.,

text: Glover,

text: S.

text: C.

text: O.

text: MNRAS,

text: Dekel,

text: A.,

text: Birnboim,

text: Y.

text: MNRAS,

text: Dekel,

text: A.,

text: Birnboim,

text: Y.,

text: Engel,

text: G.,

text: et

text: al.

text: Nature,

text: Dekel,

text: A.,

text: Mandelker,

text: N.

text: MNRAS,

text: Dekel,

text: A.,

text: Zolotov,

text: A.,

text: Tweed,

text: D.,

text: et

text: al.

text: MNRAS,

text: Fardal,

text: M.

text: A.,

text: Katz,

text: N.,

text: Gardner,

text: J.

text: P.,

text: et

text: al.

text: ApJ,

text: Field,

text: G.

text: B.,

text: Goldsmith,

text: D.

text: W.,

text: Habing,

text: H.

text: J.

text: ApJ

text: (Letters),

text: Gao,

text: Y.,

text: Solomon,

text: P.

text: M.

text: ApJ,

text: Gatto,

text: A.,

text: Walch,

text: S.,

text: Mac

text: Low,

text: M.-M.

text: et

text: al.

text: MNRAS,

text: Goerdt,

text: T.,

text: Ceverino,

text: D.,

text: Dekel,

text: A.,

text: Teyssier,

text: R.

text: MNRAS,

text: Hartmann,

text: L.,

text: Ballesteros-Paredes,

text: J.,

text: Bergin,

text: E.

text: A.

text: ApJ,

text: Heyer,

text: M.,

text: Krawczyk,

text: C.,

text: Duval,

text: J.,

text: Jackson,

text: J.

text: M.

text: ApJ,

text: Hill,

text: A.

text: S.,

text: Joung,

text: M.

text: R.,

text: Mac

text: Low,

text: M.-M.,

text: et

text: al.

text: ApJ,

text: Iba´n˜ez-Mej´ıa,

text: J.

text: C.,

text: Mac

text: Low,

text: M.-M.,

text: Klessen,

text: R.

text: S.,

text: Baczynski,

text: C.

text: ApJ,

text: submitted,

text: Jackson,

text: J.

text: M.,

text: Rathborne,

text: J.

text: M.,

text: Shah,

text: R.

text: Y.,

text: et

text: al.

text: ApJ

text: (Suppl.),

text: Joung,

text: M.

text: K.

text: R.,

text: Mac

text: Low,

text: M.-M.

text: ApJ,

text: Joung,

text: M.

text: R.,

text: Mac

text: Low,

text: M.-M.,

text: Bryan,

text: G.

text: L.

text: ApJ,

text: Kereˇs,

text: D.,

text: Katz,

text: N.,

text: Weinberg,

text: D.

text: H.,

text: Dav´e,

text: R.

text: MNRAS,

text: Kereˇs,

text: D.,

text: Katz,

text: N.,

text: Fardal,

text: M.,

text: Dav´e,

text: R.,

text: Weinberg,

text: D.

text: H.

text: MNRAS,

text: Keto,

text: E.

text: ApJ,

text: Krumholz,

text: M.

text: R.,

text: Leroy,

text: A.

text: K.,

text: McKee,

text: C.

text: F.

text: ApJ,

text: Lada,

text: C.

text: J.,

text: Lombardi,

text: M.,

text: Alves,

text: J.

text: F.

text: ApJ,

text: Leroy,

text: A.

text: K.,

text: Walter,

text: F.,

text: Brinks,

text: E.,

text: et

text: al.

text: AJ,

text: Li,

text: M.,

text: Ostriker,

text: J.

text: P.,

text: Cen,

text: R.,

text: Bryan,

text: G.

text: L.,

text: Naab,

text: T.

text: ApJ,

text: Nelson,

text: D.,

text: Vogelsberger,

text: M.,

text: Genel,

text: S.,

text: et

text: al.

text: MNRAS,

text: Ocvirk,

text: P.,

text: Pichon,

text: C.,

text: Teyssier,

text: R.

text: MNRAS,


text: 11 γ = = γ = = γ = = RadRx, p 1.0 RadRx, p 2.0 RadRx, p 3.0 min min min 1042 3M 1042 3M 1042 3M 6M 6M 6M 1040 10M 1040 10M 1040 10M 1038 1038 1038 1036 1036 1036 1034 1034 1034 ϵ ( / ) ϵ ( / ) ϵ ( / ) 6.6MeV 0.66GeV γγ =66G=eV / 6.6TeV 6.6MeV 0.66GeV γγ =66G=eV / 6.6TeV 6.6MeV 0.66GeV γγ =66G=eV / 6.6TeV BNS 1012G 34 BNS 1012G 34 BNS 1012G 34 1, p 1.0 1, p 2.0 1, p 3.0 min min min 1042 3M 1042 3M 1042 3M 6M 6M 6M 1040 10M 1040 10M 1040 10M 1038 1038 1038 1036 1036 1036 1034 1034 1034 ϵ ( / ) ϵ ( / ) ϵ ( / ) 6.6MeV 0.66GeV γ 66GeV / 6.6TeV 6.6MeV 0.66GeV γ 66GeV / 6.6TeV 6.6MeV 0.66GeV γ 66GeV / 6.6TeV BNS 1012G 34 BNS 1012G 34 BNS 1012G 34 FIG. 7: The spectrum of primary curvature radiation at times corresponding to binary separations of 10M, 6M, and 3M (dot-dashed, dashed, solid). Each panel is for the labeled minimum electron Lorentz factor and power law index p of electron energies. γ =RadRx refers to the min radiation reaction limited Lorentz factor at the point of weakest electric ﬁeld in the region connecting NS and BH (of order a few to 10 times smaller than the maximum Lorentz factor near merger). Acknowledgments Appendix A: Parameter Dependence of Curvature Spectra The authors thank Andrei Beloborodov, Brian Metzger, and Sean McWilliams for useful discussions. The authors also Figure 7 plots the curvature radiation spectra, identical to thank the anonymous referee for comments that improved Figure 3, but for different values of the electron-energy power the manuscript. D.J.D. acknowledges support from a Na- law index p, and the minimum electron Lorentz factor in the tional Science Foundation Graduate Research Fellowship un- magnetosphere, γ . We vary p from 1.0 to 3.0. We choose min der Grant No. DGE1144155. J.L. thanks the Tow Foundation minimum Lorentz factors which bracket the range of plausi- for their support. J.L. was also supported by a Guggenheim ble values: γ = 1, and a minimum radiation-reaction lim- min Fellowship and is a Chancellor’s Fellow at Chapman Univer- ited Lorentz factor which we compute with Eq. (14) but with sity. This research was undertaken, in part, thanks to funding electric ﬁeld at the edge of the binary orbital light cylinder − from the Canada Research Chairs program. NWM was sup- (Ω /c) that falls off from its horizon value as r 2 [26]. Near orb ported in part by the Natural Sciences and Engineering Coun- merger this is only a few times smaller than the maximum γ cil of Canada. computed form the horizon electric ﬁelds. ] [1] S. T. McWilliams and J. Levin, ApJ 742, 90 (2011), 1101.1969. R. Bondarescu, Physical Review Letters 108, 011102 (2012), [2] D. J. D’Orazio and J. Levin, PRD 88, 064059 (2013), 1110.0467. 1302.3885. [6] D. Tsang, ApJ 777, 103 (2013), 1307.3554. [3] R. Narayan, B. Paczynski, and T. Piran, ApJL 395, L83 (1992), [7] G. M. Harry and LIGO Scientiﬁc Collaboration, Classical and astro-ph/9204001. Quantum Gravity 27, 084006 (2010). [4] F. Özel, D. Psaltis, R. Narayan, and J. E. McClintock, ApJ 725, [8] C. M. F. Mingarelli, J. Levin, and T. J. W. Lazio, ApJL 814, 1918 (2010), 1006.2834. L20 (2015), 1511.02870. [5] D. Tsang, J. S. Read, T. Hinderer, A. L. Piro, and [9] P. Goldreich and D. Lynden-Bell, ApJ 156, 59 (1969). ] / ) / ([ ω ] / ) / ([ ω s gre 2 G2101 SNB vruCP s gre 2 G2101 SNB vruCP ] / ) / ([ ω ] / ) / ([ ω s gre 2 G2101 SNB vruCP s gre 2 G2101 SNB vruCP ] / ) / ([ ω ] / ) / ([ ω s gre 2 G2101 SNB vruCP s gre 2 G2101 SNB vruCP ##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## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## 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text: A&A proofs: manuscript no. DarkInﬂuencesIIIstructuralcharacterization Fig. 2. Evolution of the inner parts of the dwarf galaxy shown in Fig. 1 around the time the satellite merges with the host. The threshold is × − × − 0.4 1020 N cm 2 and the contours indicate 1, 4, and 10 1020 N cm 2; see colourbar for the relative values. these dominate the light, the associated local star-forming re- gions may well be the analogues of what is seen in XMP galaxies (Filho et al. 2013, 2015). Figure 3 shows the star formation rate (SFR) during the encounter. The blue curve shows that the total SFR has pro- nounced peaks during the pericentric passages of the satellite. Interestingly, during the later passages, a signiﬁcant fraction × − − (4 10 3 M(cid:12) yr 1) of the total star formation takes place in the tidal tail at the location of the satellite (red curve). Subsequently the total SFR increases, due to the gas that is channeled to the centre, and reaches a plateau around a value that is more than a factor 10 higher than for the equivalent dwarf in isolation. In summary, besides the characteristic starburst, signatures of the merger can be found in the morphology of the old stel- lar disk, the distribution and morphology of the young stellar population that is formed during the encounter, and in the mor- phology and kinematics of the gas. The gaseous and stellar disks show distinguishable eﬀects both in the outskirts and in the in- Fig. 3. The blue curve shows the evolution of the star formation rate of ner parts. Interestingly, in the simulated system shown here star- the dwarf galaxy shown in Figures 1 and 2, while the black curve is for forming regions outside the centre pinpoint the location of the the same model-A dwarf run in isolation. The star formation rate in the merging dark satellite. tidal tail within 1 kpc of the position of the satellite is shown with the red dashed curve, and constitutes a signiﬁcant fraction of the total SFR 4. Quantitative measures of structural properties in the system around the time of the merger. The satellite is completely Although clear eﬀects can be seen in the simulated dwarf disrupted after 2.5 Gyr, but its eﬀects on the SFR are longlasting. The pericentre passages of the satellite are indicated by the black arrows. galaxy which are due to the minor merger, it is important to quantify these in order to make comparisons to observations. A variety of quantitative structural descriptions of the morphol- ply the morphological indicators to our simulations and present ogy and kinematics of galaxies have been put forward in the lit- the results in Sect. , while we focus on the results of the kine- erature. Morphologically the structure is often characterised by matic indicators of the stellar and gas disks in Sect. . the CAS (concentration, asymmetry, and smoothness; Conselice 4.1. Morphological parameters 2003) and GM (Gini and M20; Lotz et al. 2004) indicators. Ad- ditionally for dwarf irregulars and BCDs, the diﬀerence in cen- We estimate the morphological CAS and GM parameters, tral surface brightness obtained by ﬁtting the inner and outer re- including modiﬁcations by Lelli et al. (2014); Holwerda et al. gions is also used (Hunter & Elmegreen 2006; Papaderos et al. (2011d), on a grid with initial size of 20 by 20 kpc which 2008). For XMP galaxies, the often used indicators include dif- is cropped to the regions above a ﬁxed threshold, and a de- ferences in the average velocity, and in the position angles of the fault bin size of 0.2 kpc. The thresholds adopted are close to HI gas and stellar components (Filho et al. 2015, 2013). We ap- those reported for observational studies in the literature, NHI > Article number, page 4 of 10page.10 ##LTFigure## ##LTFigure##

text: A&A

text: proofs:

text: manuscript

text: no.

text: Fig.

text: Evolution

text: of

text: the

text: inner

text: parts

text: of

text: the

text: dwarf

text: galaxy

text: shown

text: in

text: Fig.

text: around

text: the

text: time

text: the

text: satellite

text: merges

text: with

text: the

text: host.

text: The

text: threshold

text: is

font: GJEUYM+txsy

text: N

text: cm

text: and

text: the

text: contours

text: indicate

text: and

text: N

text: cm

text: see

text: colourbar

text: for

text: the

text: relative

text: values.

text: these

text: dominate

text: the

text: light,

text: the

text: associated

text: local

text: star-forming

text: re-

text: gions

text: may

text: well

text: be

text: the

text: analogues

text: of

text: what

text: is

text: seen

text: in

text: XMP

text: galaxies

text: (Filho

text: et

text: al.

text: Figure

text: shows

text: the

text: star

text: formation

text: rate

text: (SFR)

text: during

text: the

text: encounter.

text: The

text: blue

text: curve

text: shows

text: that

text: the

text: total

text: SFR

text: has

text: pro-

text: nounced

text: peaks

text: during

text: the

text: pericentric

text: passages

text: of

text: the

text: satellite.

text: Interestingly,

text: during

text: the

text: later

text: passages,

text: a

text: signiﬁcant

text: fraction

text: yr

text: of

text: the

text: total

text: star

text: formation

text: takes

text: place

text: in

text: the

text: tidal

text: tail

text: at

text: the

text: location

text: of

text: the

text: satellite

text: (red

text: curve).

text: Subsequently

text: the

text: total

text: SFR

text: increases,

text: due

text: to

text: the

text: gas

text: that

text: is

text: channeled

text: to

text: the

text: centre,

text: and

text: reaches

text: a

text: plateau

text: around

text: a

text: value

text: that

text: is

text: more

text: than

text: a

text: factor

text: higher

text: than

text: for

text: the

text: equivalent

text: dwarf

text: in

text: isolation.

text: In

text: summary,

text: besides

text: the

text: characteristic

text: starburst,

text: signatures

text: of

text: the

text: merger

text: can

text: be

text: found

text: in

text: the

text: morphology

text: of

text: the

text: old

text: stel-

text: lar

text: disk,

text: the

text: distribution

text: and

text: morphology

text: of

text: the

text: young

text: stellar

text: population

text: that

text: is

text: formed

text: during

text: the

text: encounter,

text: and

text: in

text: the

text: mor-

text: phology

text: and

text: kinematics

text: of

text: the

text: gas.

text: The

text: gaseous

text: and

text: stellar

text: disks

text: show

text: distinguishable

text: eﬀects

text: both

text: in

text: the

text: outskirts

text: and

text: in

text: the

text: in-

text: Fig.

text: The

text: blue

text: curve

text: shows

text: the

text: evolution

text: of

text: the

text: star

text: formation

text: rate

text: of

text: ner

text: parts.

text: Interestingly,

text: in

text: the

text: simulated

text: system

text: shown

text: here

text: star-

text: the

text: dwarf

text: galaxy

text: shown

text: in

text: Figures

text: and

text: while

text: the

text: black

text: curve

text: is

text: for

text: forming

text: regions

text: outside

text: the

text: centre

text: pinpoint

text: the

text: location

text: of

text: the

text: the

text: same

text: model-A

text: dwarf

text: run

text: in

text: isolation.

text: The

text: star

text: formation

text: rate

text: in

text: the

text: merging

text: dark

text: satellite.

text: tidal

text: tail

text: within

text: kpc

text: of

text: the

text: position

text: of

text: the

text: satellite

text: is

text: shown

text: with

text: the

text: red

text: dashed

text: curve,

text: and

text: constitutes

text: a

text: signiﬁcant

text: fraction

text: of

text: the

text: total

text: SFR

text: Quantitative

text: measures

text: of

text: structural

text: properties

text: in

text: the

text: system

text: around

text: the

text: time

text: of

text: the

text: merger.

text: The

text: satellite

text: is

text: completely

text: Although

text: clear

text: eﬀects

text: can

text: be

text: seen

text: in

text: the

text: simulated

text: dwarf

text: disrupted

text: after

text: Gyr,

text: but

text: its

text: eﬀects

text: on

text: the

text: SFR

text: are

text: longlasting.

text: The

text: pericentre

text: passages

text: of

text: the

text: satellite

text: are

text: indicated

text: by

text: the

text: black

text: arrows.

text: galaxy

text: which

text: are

text: due

text: to

text: the

text: minor

text: merger,

text: it

text: is

text: important

text: to

text: quantify

text: these

text: in

text: order

text: to

text: make

text: comparisons

text: to

text: observations.

text: A

text: variety

text: of

text: quantitative

text: structural

text: descriptions

text: of

text: the

text: morphol-

text: ply

text: the

text: morphological

text: indicators

text: to

text: our

text: simulations

text: and

text: present

text: ogy

text: and

text: kinematics

text: of

text: galaxies

text: have

text: been

text: put

text: forward

text: in

text: the

text: lit-

text: the

text: results

text: in

text: Sect.

text: while

text: we

text: focus

text: on

text: the

text: results

text: of

text: the

text: kine-

text: erature.

text: Morphologically

text: the

text: structure

text: is

text: often

text: characterised

text: by

text: matic

text: indicators

text: of

text: the

text: stellar

text: and

text: gas

text: disks

text: in

text: Sect.

text: the

text: CAS

text: (concentration,

text: asymmetry,

text: and

text: smoothness;

text: Conselice

text: Morphological

text: parameters

text: and

text: GM

text: (Gini

text: and

text: Lotz

text: et

text: al.

text: indicators.

text: Ad-

text: ditionally

text: for

text: dwarf

text: irregulars

text: and

text: BCDs,

text: the

text: diﬀerence

text: in

text: cen-

text: We

text: estimate

text: the

text: morphological

text: CAS

text: and

text: GM

text: parameters,

text: tral

text: surface

text: brightness

text: obtained

text: by

text: ﬁtting

text: the

text: inner

text: and

text: outer

text: re-

text: including

text: modiﬁcations

text: by

text: Lelli

text: et

text: al.

text: Holwerda

text: et

text: al.

text: gions

text: is

text: also

text: used

text: (Hunter

text: Elmegreen

text: Papaderos

text: et

text: al.

text: on

text: a

text: grid

text: with

text: initial

text: size

text: of

text: by

text: kpc

text: which

text: For

text: XMP

text: galaxies,

text: the

text: often

text: used

text: indicators

text: include

text: dif-

text: is

text: cropped

text: to

text: the

text: regions

text: above

text: a

text: ﬁxed

text: threshold,

text: and

text: a

text: de-

text: ferences

text: in

text: the

text: average

text: velocity,

text: and

text: in

text: the

text: position

text: angles

text: of

text: the

text: fault

text: bin

text: size

text: of

text: kpc.

text: The

text: thresholds

text: adopted

text: are

text: close

text: to

text: HI

text: gas

text: and

text: stellar

text: components

text: (Filho

text: et

text: al.

text: We

text: ap-

text: those

text: reported

text: for

text: observational

text: studies

text: in

text: the

text: literature,

text: NHI

text: Article

text: number,

text: page

text: of

text: ##LTFigure##

text: ##LTFigure##


text: 12 PIOTR M. HAJAC AND TOMASZ MASZCZYK (cid:16) (cid:0) (cid:1) (cid:17) (cid:16) (cid:17) (cid:48) (cid:0) (cid:1) (cid:16) (cid:0) (cid:1) (cid:104)(cid:17)(cid:105) (cid:16) (cid:17) (cid:104) (cid:105) σ (f(a)) = f(a) f (cid:96) f(a) 1 ϕ (cid:96)(f(a) ) 2 (0) (1) (1) (cid:16) (cid:0) (cid:1) (cid:104)(cid:16)(cid:105) (cid:17)(cid:17) (cid:104) (cid:105) = f a f (cid:96) a 1 ϕ (cid:96)(a ) 2 (0) (1) (1) (cid:104) (cid:105) (cid:104) (cid:105) = f a (cid:96) a 1 ϕ (cid:96)(a ) 2 (0) (1) (1) (1.25) = f(σ(a)). Hence (cid:0) (cid:1) (cid:0) (cid:1) (cid:48) f e = f (σ (r (c )a )) = σ (f (r (c )a )) (µ,i)(ν,j) µ ij ν µ ij ν (cid:48) (cid:48) (cid:48) (cid:48) (1.26) = σ (f (r (c )) f (a )) = σ r (c )a . (cid:0) (cid:1) µ ij ν µ ij ν (cid:54)∈ Note that f e is zero for ν I because then f(a ) = 0. (µ,i)(ν,j) ν (cid:48) Furthermore, applying [8, Theorem 3.1] to the strong connection (cid:96) , the basis { (cid:48) } ∈ ∈ { } a , and the matrix coeﬃcients c , for all µ, ν I, i, j 1, . . . , dim V , we µ µ ij obtain (cid:0) (cid:1) (cid:48) (cid:48) (cid:48) (cid:48) (1.27) σ r (c )a =: e , µ ij ν (µ,i)(ν,j) (cid:48) (cid:48) B(cid:48) (cid:48) (cid:48) ∼ where the e are the entries of an idempotent matrix e such that N e = A(cid:48) C (µB,i(cid:48))(ν,j) (cid:50) V as left -modules. Thus, in the block matrix notation, we arrive at the following crucial equality (cid:18) (cid:19) (cid:48) e 0 (1.28) f(e) = , d 0 where d is unspeciﬁed. Now, taking into account that f(e) is an idempotent (cid:48) matrix, we derive the equality de = d, which allows us to verify that (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:48) (cid:48) −1 e 0 1 0 e 0 1 0 (1.29) = . d 0 d 1 0 0 d 1 B(cid:48) Hence the corresponding ﬁnitely generated projective left -modules are iso- morphic:

text: PIOTR

text: M.

text: HAJAC

text: AND

text: TOMASZ

text: MASZCZYK

text: σ

text: (f(a))

text: f(a)

text: f

text: f(a)

text: ϕ

text: f

text: a

text: f

text: a

text: ϕ

text: f

text: a

text: a

text: ϕ

text: f(σ(a)).

text: Hence

text: f

text: e

text: f

text: (σ

text: (r

text: (c

text: )a

text: σ

text: (f

text: (r

text: (c

text: )a

text: (µ,i)(ν,j)

text: µ

text: ij

text: ν

text: µ

text: ij

text: ν

text: σ

text: (f

text: (r

text: (c

text: f

text: (a

text: σ

text: r

text: (c

text: )a

text: µ

text: ij

text: ν

text: µ

text: ij

text: ν

text: Note

text: that

text: f

text: e

text: is

text: zero

text: for

text: ν

text: I

text: because

text: then

text: f(a

text: (µ,i)(ν,j)

text: ν

text: Furthermore,

text: applying

text: Theorem

text: to

text: the

text: strong

text: connection

text: the

text: basis

text: a

text: and

text: the

text: matrix

text: coeﬃcients

text: c

text: for

text: all

text: µ,

text: ν

text: I,

text: i,

text: j

text: dim

text: V

text: we

text: µ

text: µ

text: ij

text: obtain

text: σ

text: r

text: (c

text: )a

text: e

text: µ

text: ij

text: ν

text: (µ,i)(ν,j)

text: where

text: the

text: e

text: are

text: the

text: entries

text: of

text: an

text: idempotent

text: matrix

text: e

text: such

text: that

text: N

text: e

text: C

text: V

text: as

text: left

text: -modules.

text: Thus,

text: in

text: the

text: block

text: matrix

text: notation,

text: we

text: arrive

text: at

text: the

text: following

text: crucial

text: equality

text: e

text: f(e)

text: d

text: where

text: d

text: is

text: unspeciﬁed.

text: Now,

text: taking

text: into

text: account

text: that

text: f(e)

text: is

text: an

text: idempotent

text: matrix,

text: we

text: derive

text: the

text: equality

text: de

text: d,

text: which

text: allows

text: us

text: to

text: verify

text: that

text: −1

text: e

text: e

text: d

text: d

text: d

text: Hence

text: the

text: corresponding

text: ﬁnitely

text: generated

text: projective

text: left

text: -modules

text: are

text: iso-

text: morphic: