text: May 28, 2020 10:50 WSPC Proceedings - 9.75in x 6.5in ASCDMG14˙v1.2 page 1 1 Will a nuclear stellar disk form in the galaxy Henize 2-10? ∗ ∗∗ Manuel Arca-Sedda , and Roberto Capuzzo-Dolcetta Dept. of Physics, University of Rome “Sapienza”, P.le A. Moro 5, Rome, I-00185, Italy ∗ e-mail: manuel.arcasedda@uniroma1.it ∗∗ e-mail: roberto.capuzzodolcetta@uniroma1.it We present results of a set of N-body simulations to model the future evolution of the 11 young massive clusters hosted in the central region of the dwarf starburst galaxy Henize (cid:39) × 2-10, which contains at its center a massive black hole with a mass MBH 2 106 M(cid:12). Nuclear star clusters are present in a great quantity of galaxies of mass similar to Henize 2-10. Our results1 show that the orbital decay and merging of the Henize 2-10 clusters (cid:39) − × will likely lead to the formatio(cid:39)n of a nuclear star cluster with mass MNSC 4 6 106 M(cid:12) and eﬀective radius rNSC 4.1 pc. Additionally, we found that this mechanism can lead to the formation of disky structures with global properties similar to those of nuclear stellar disks, which reside in many “middle-weight” galaxies. This work conﬁrms and enlarge recent results2 that indicate how nuclear star clusters and super massive black holes are only partially correlated, since the formation process of nuclear star clusters is poorly aﬀected by a black hole of the size of that in Henize 2-10. A new result is that nuclear star clusters and nuclear stellar disks may share the same formation path. Keywords: galaxies: individual (Henize 2-10); galaxies: nuclei; galaxies: black holes; galaxies: star clusters: general; methods: numerical 1. Introduction ∼ Most of galaxies with masses above 108 M(cid:12) host at their centre compact massive objects (CMOs), whose nature and mass depend on the properties of the host galaxy. In particular, galaxies with masses above 1010 M(cid:12) contain, usually, su- per massive black holes (SMBHs) and/or nuclear stellar disks (NSDs), while lighter galaxies host nuclear star clusters (NSCs)3. Moreover, in many galaxies with masses − in the range 1010 1011 M(cid:12) SMBHs and NSCs (or NSDs) may co-exist in the galactic centre. The existence of scaling relations connecting SMBHs or NSCs to their host properties might help in shedding light on the mechanisms of origin of these objects. For instance, Ferrarese et al. (2006)4 found that the relation between the NSC mass and the velocity dispersion of its host is similar to that for SMBHs. However, more recent studies based on a larger amount of NSC data have pointed out that such re- lation is shallower for NSCs than for SMBHs5,6. These results seem to indicate that SMBHs and NSCs represent two diﬀerent manifestations of CMOs whose particular nature depends on the galactic environment, while their interaction and common evolution is still matter of debate. One possible formation mechanism for NSCs is the “dry-merger” scenario, in which star clusters orbitally decay toward the galactic centre under the action of dynamical friction (DF)7–9. Their subsequent merging lead to the formation of a dense stellar nucleus with characteristics comparable to those of observed galactic nuclei. In this scenario, the presence of a SMBH in the galactic centre may turn on tidal forces strong enough to disrupt the infalling clus- ters, thereby quenching the formation of a nucleus. On the other hand, theoretical ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: May

text: WSPC

text: Proceedings

text: x

text: page

text: Will

label: title

text: a

text: nuclear

text: stellar

text: disk

text: form

text: in

text: the

text: galaxy

text: Henize

label: author

text: Manuel

text: Arca-Sedda

text: and

text: Roberto

text: Capuzzo-Dolcetta

text: Dept.

text: of

text: Physics,

text: University

text: of

text: Rome

text: “Sapienza”,

text: P.le

text: A.

text: Moro

text: Rome,

text: Italy

text: e-mail:

text: e-mail:

text: We

label: abstract

text: present

text: results

text: of

text: a

text: set

text: of

text: N-body

text: simulations

text: to

text: model

text: the

text: future

text: evolution

text: of

text: the

text: young

text: massive

text: clusters

text: hosted

text: in

text: the

text: central

text: region

text: of

text: the

text: dwarf

text: starburst

text: galaxy

text: Henize

text: which

text: contains

text: at

text: its

text: center

text: a

text: massive

text: black

text: hole

text: with

text: a

text: mass

text: MBH

text: Nuclear

text: star

text: clusters

text: are

text: present

text: in

text: a

text: great

text: quantity

text: of

text: galaxies

text: of

text: mass

text: similar

text: to

text: Henize

text: Our

text: show

text: that

text: the

text: orbital

text: decay

text: and

text: merging

text: of

text: the

text: Henize

text: clusters

text: will

text: likely

text: lead

text: to

text: the

text: of

text: a

text: nuclear

text: star

text: cluster

text: with

text: mass

text: MNSC

text: and

text: eﬀective

text: radius

text: rNSC

text: pc.

text: Additionally,

text: we

text: found

text: that

text: this

text: mechanism

text: can

text: lead

text: to

text: the

text: formation

text: of

text: disky

text: structures

text: with

text: global

text: properties

text: similar

text: to

text: those

text: of

text: nuclear

text: stellar

text: disks,

text: which

text: reside

text: in

text: many

text: “middle-weight”

text: galaxies.

text: This

text: work

text: conﬁrms

text: and

text: enlarge

text: recent

text: that

text: indicate

text: how

text: nuclear

text: star

text: clusters

text: and

text: super

text: massive

text: black

text: holes

text: are

text: only

text: partially

text: correlated,

text: since

text: the

text: formation

text: process

text: of

text: nuclear

text: star

text: clusters

text: is

text: poorly

text: aﬀected

text: by

text: a

text: black

text: hole

text: of

text: the

text: size

text: of

text: that

text: in

text: Henize

text: A

text: new

text: result

text: is

text: that

text: nuclear

text: star

text: clusters

text: and

text: nuclear

text: stellar

text: disks

text: may

text: share

text: the

text: same

text: formation

text: path.

text: Keywords:

text: galaxies:

text: individual

text: (Henize

text: galaxies:

text: nuclei;

text: galaxies:

text: black

text: holes;

text: galaxies:

text: star

text: clusters:

text: general;

text: methods:

text: numerical

text: Introduction

text: Most

text: of

text: galaxies

text: with

text: masses

text: above

text: host

text: at

text: their

text: centre

text: compact

text: massive

text: objects

text: (CMOs),

text: whose

text: nature

text: and

text: mass

text: depend

text: on

text: the

text: properties

text: of

text: the

text: host

text: galaxy.

text: In

text: particular,

text: galaxies

text: with

text: masses

text: above

text: contain,

text: usually,

text: su-

text: per

text: massive

text: black

text: holes

text: (SMBHs)

text: and/or

text: nuclear

text: stellar

text: disks

text: (NSDs),

text: while

text: lighter

text: galaxies

text: host

text: nuclear

text: star

text: clusters

text: Moreover,

text: in

text: many

text: galaxies

text: with

text: masses

text: in

text: the

text: range

text: SMBHs

text: and

text: NSCs

text: (or

text: NSDs)

text: may

text: co-exist

text: in

text: the

text: galactic

text: centre.

text: The

text: existence

text: of

text: scaling

text: relations

text: connecting

text: SMBHs

text: or

text: NSCs

text: to

text: their

text: host

text: properties

text: might

text: help

text: in

text: shedding

text: light

text: on

text: the

text: mechanisms

text: of

text: origin

text: of

text: these

text: objects.

text: For

text: instance,

text: Ferrarese

text: et

text: al.

text: found

text: that

text: the

text: relation

text: between

text: the

text: NSC

text: mass

text: and

text: the

text: velocity

text: dispersion

text: of

text: its

text: host

text: is

text: similar

text: to

text: that

text: for

text: SMBHs.

text: However,

text: more

text: recent

text: studies

text: based

text: on

text: a

text: larger

text: amount

text: of

text: NSC

text: data

text: have

text: pointed

text: out

text: that

text: such

text: re-

text: lation

text: is

text: shallower

text: for

text: NSCs

text: than

text: for

text: These

text: results

text: seem

text: to

text: indicate

text: that

text: SMBHs

text: and

text: NSCs

text: represent

text: two

text: diﬀerent

text: manifestations

text: of

text: CMOs

text: whose

text: particular

text: nature

text: depends

text: on

text: the

text: galactic

text: environment,

text: while

text: their

text: interaction

text: and

text: common

text: evolution

text: is

text: still

text: matter

text: of

text: debate.

text: One

text: possible

text: formation

text: mechanism

text: for

text: NSCs

text: is

text: the

text: “dry-merger”

text: scenario,

text: in

text: which

text: star

text: clusters

text: orbitally

text: decay

text: toward

text: the

text: galactic

text: centre

text: under

text: the

text: action

text: of

text: dynamical

text: friction

text: Their

text: subsequent

text: merging

text: lead

text: to

text: the

text: formation

text: of

text: a

text: dense

text: stellar

text: nucleus

text: with

text: characteristics

text: comparable

text: to

text: those

text: of

text: observed

text: galactic

text: nuclei.

text: In

text: this

text: scenario,

text: the

text: presence

text: of

text: a

text: SMBH

text: in

text: the

text: galactic

text: centre

text: may

text: turn

text: on

text: tidal

text: forces

text: strong

text: enough

text: to

text: disrupt

text: the

text: infalling

text: clus-

text: ters,

text: thereby

text: quenching

text: the

text: formation

text: of

text: a

text: nucleus.

text: On

text: the

text: other

text: hand,

text: theoretical

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: May 28, 2020 10:50 WSPC Proceedings - 9.75in x 6.5in ASCDMG14˙v1.2 page 2 2 and numerical modelling of the inspiral of star clusters have clearly shown that the environmental tidal forces are signiﬁcant only when the central SMBH mass exceeds 108 M(cid:12)2,6,10. In this framework, galaxies hosting several star clusters near their photometric centres represent an ideal laboratory to test the dry-merger scenario. One example is the Henize 2-10 starburst dwarf galaxy11. Indeed, this galaxy hosts ± an SMBH candidate of mass Log(M/M(cid:12)) = 6.3 1.112,13, and 11 young super-star (cid:46) clusters (SSCs) with masses above 105 M(cid:12), orbiting at projected distances 140 pc from the galactic centre14. Detailed information available on the projected mass proﬁle of the galaxy and on the SSC masses and sizes make Henize 2-10 an interest- ing case study for testing the dry-merger scenario. In particular, Arca-Sedda et al. (2015)1 used several direct N-body simulations to investigate whether the future evolution of the star cluster system (SCS) observed in Henize 2-10 will lead to the formation of a central stellar nucleus. The set of initial conditions chosen for the SSCs are consistent with the high-quality data available for their orbits and internal properties. 2. Modelling Henize 2-10 and its star cluster system We shortly resume here the methods and results of the Arca-Sedda et al. (2015)1 paper. The galaxy density proﬁle (cid:18)was (cid:19)mode(cid:18)lled as:(cid:19) − − − (3 γ)M r γ r γ 4 1 ρ (r) = H + 1 , (1) γ 4πr3 r r cosh(r/r ) s s s tr where M is the total mass of the galaxy, r its scale radius, γ is the inner density H s proﬁle slope, cosh(r/r ) is the usual hyperbolic cosine function, and r = 150 pc tr (cid:28) tr is the truncation radius of the model. At radii r r the model converges to a tr Dehnen’s density proﬁle15. This density proﬁle represents satisfactorily the inner 150 pc of Henize 2-10 × under the choice of parameters: M = 1.6 109 M(cid:12), r = 110 pc and γ = 1/2. In H s particular, our model has an eﬀective radius R = 269 pc and a velocity dispersion − e σ = 41 km s 1 averaged in the region between 40 and 260 pc from the center, values very close to those inferred from observations. To highlight the role played by the SMBH in the formation of the NSC, we compared the results in models containing an SMBH with those obtained without an SMBH in the galaxy center. To represent the eleven SSCs (hereafter labelled with the letter C followed by a number from 1 to 11) we adopted King models tuned on data provided by Nguyen et al. (2014)14. Furthermore, we chose three diﬀerent sets of ICs for the SSCs motion, referred to as S1, S2 and S3 in the following. In conﬁguration S1, the SSCs ICs are sampled from the same distribution function of the background galaxy, with the constraint that the clusters projected positions correspond to their actually observed positions. In S2, instead, all the SSCs move on circular orbits on the same plane. This scenario is motivated by recent observations claiming that the clusters move on the same disk of the molecular gas that resides in the Henize 2-10 nucleus. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: May

label: paragraph

text: WSPC

text: Proceedings

text: x

text: page

text: and

text: numerical

text: modelling

text: of

text: the

text: inspiral

text: of

text: star

text: clusters

text: have

text: clearly

text: shown

text: that

text: the

text: environmental

text: tidal

text: forces

text: are

text: signiﬁcant

text: only

text: when

text: the

text: central

text: SMBH

text: mass

text: exceeds

text: In

text: this

text: framework,

text: galaxies

text: hosting

text: several

text: star

text: clusters

text: near

text: their

text: photometric

text: centres

text: represent

text: an

text: ideal

text: laboratory

text: to

text: test

text: the

text: dry-merger

text: scenario.

text: One

text: example

text: is

text: the

text: Henize

text: starburst

text: dwarf

text: Indeed,

text: this

text: galaxy

text: hosts

text: an

text: SMBH

text: candidate

text: of

text: mass

text: and

text: young

text: super-star

text: clusters

text: (SSCs)

text: with

text: masses

text: above

text: orbiting

text: at

text: projected

text: distances

text: pc

text: from

text: the

text: galactic

text: Detailed

text: information

text: available

text: on

text: the

text: projected

text: mass

text: proﬁle

text: of

text: the

text: galaxy

text: and

text: on

text: the

text: SSC

text: masses

text: and

text: sizes

text: make

text: Henize

text: an

text: interest-

text: ing

text: case

text: study

text: for

text: testing

text: the

text: dry-merger

text: scenario.

text: In

text: particular,

text: Arca-Sedda

text: et

text: al.

text: used

text: several

text: direct

text: N-body

text: simulations

text: to

text: investigate

text: whether

text: the

text: future

text: evolution

text: of

text: the

text: star

text: cluster

text: system

text: (SCS)

text: observed

text: in

text: Henize

text: will

text: lead

text: to

text: the

text: formation

text: of

text: a

text: central

text: stellar

text: nucleus.

text: The

text: set

text: of

text: initial

text: conditions

text: chosen

text: for

text: the

text: SSCs

text: are

text: consistent

text: with

text: the

text: high-quality

text: data

text: available

text: for

text: their

text: orbits

text: and

text: internal

text: properties.

text: Modelling

text: Henize

text: and

text: its

text: star

text: cluster

text: system

text: We

text: shortly

text: resume

text: here

text: the

text: methods

text: and

text: results

text: of

text: the

text: Arca-Sedda

text: et

text: al.

text: paper.

text: The

text: galaxy

text: density

text: proﬁle

text: γ)M

text: r

text: γ

text: r

text: γ

text: ρ

text: (r)

text: H

text: γ

text: r

text: r

text: cosh(r/r

text: s

text: s

text: s

text: tr

text: where

text: M

text: is

text: the

text: total

text: mass

text: of

text: the

text: galaxy,

text: r

text: its

text: scale

text: radius,

text: γ

text: is

text: the

text: inner

text: density

text: H

text: s

text: proﬁle

text: slope,

text: cosh(r/r

text: is

text: the

text: usual

text: hyperbolic

text: cosine

text: function,

text: and

text: r

text: pc

text: tr

text: tr

text: is

text: the

text: truncation

text: radius

text: of

text: the

text: model.

text: At

text: radii

text: r

text: r

text: the

text: model

text: converges

text: to

text: a

text: tr

text: Dehnen’s

text: density

text: This

text: density

text: proﬁle

text: represents

text: satisfactorily

text: the

text: inner

text: pc

text: of

text: Henize

text: under

text: the

text: choice

text: of

text: parameters:

text: M

text: r

text: pc

text: and

text: γ

text: In

text: H

text: s

text: particular,

text: our

text: model

text: has

text: an

text: eﬀective

text: radius

text: R

text: pc

text: and

text: a

text: velocity

text: dispersion

text: e

text: σ

text: km

text: s

text: averaged

text: in

text: the

text: region

text: between

text: and

text: pc

text: from

text: the

text: center,

text: values

text: very

text: close

text: to

text: those

text: inferred

text: from

text: observations.

text: To

text: highlight

text: the

text: role

text: played

text: by

text: the

text: SMBH

text: in

text: the

text: formation

text: of

text: the

text: NSC,

text: we

text: compared

text: the

text: results

text: in

text: models

text: containing

text: an

text: SMBH

text: with

text: those

text: obtained

text: without

text: an

text: SMBH

text: in

text: the

text: galaxy

text: center.

text: To

text: represent

text: the

text: eleven

text: SSCs

text: (hereafter

text: labelled

text: with

text: the

text: letter

text: C

text: followed

text: by

text: a

text: number

text: from

text: to

text: we

text: adopted

text: King

text: models

text: tuned

text: on

text: data

text: provided

text: by

text: Nguyen

text: et

text: al.

text: Furthermore,

text: we

text: chose

text: three

text: diﬀerent

text: sets

text: of

text: ICs

text: for

text: the

text: SSCs

text: motion,

text: referred

text: to

text: as

text: and

text: in

text: the

text: following.

text: In

text: conﬁguration

text: the

text: SSCs

text: ICs

text: are

text: sampled

text: from

text: the

text: same

text: distribution

text: function

text: of

text: the

text: background

text: galaxy,

text: with

text: the

text: constraint

text: that

text: the

text: clusters

text: projected

text: positions

text: correspond

text: to

text: their

text: actually

text: observed

text: positions.

text: In

text: instead,

text: all

text: the

text: SSCs

text: move

text: on

text: circular

text: orbits

text: on

text: the

text: same

text: plane.

text: This

text: scenario

text: is

text: motivated

text: by

text: recent

text: observations

text: claiming

text: that

text: the

text: clusters

text: move

text: on

text: the

text: same

text: disk

text: of

text: the

text: molecular

text: gas

text: that

text: resides

text: in

text: the

text: Henize

text: nucleus.

text: ##LTLine##

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text: May 28, 2020 10:50 WSPC Proceedings - 9.75in x 6.5in ASCDMG14˙v1.2 page 6 6 starburst galaxy Henize 2-10 as input for highly detailed, direct, N-body simula- tions. We found that not all the clusters are able to reach the centre, because the lightest of them are shattered by the tidal ﬁeld of the parent galaxy, while the tidal eﬀect caused by the central MBH slightly aﬀects the quantity of mass deposited within the growing NSC. Therefore, it turns out that a MBH does not necessar- ily determine the properties of the NSC that surrounds it unless it is very massive. Furthermore, our results seem to suggest that the decay and merging of star clusters can represent also a satisfactory formation channel for nuclear stellar disk, though this depend strongly on their initial conditions. References 1. M. Arca-Sedda, R. Capuzzo-Dolcetta, F. Antonini and A. Seth, Henize 2-10: The Ongoing Formation of a Nuclear Star Cluster around a Massive Black Hole, ApJ 806, p. 220 (June 2015). 2. M. Arca-Sedda, R. Capuzzo-Dolcetta and M. Spera, The dearth of nuclear star clusters in the centre of the brightest galaxies, MNRAS 456, 2457 (December 2015). 3. N. Scott and A. W. Graham, Updated Mass Scaling Relations for Nuclear Star Clusters and a Comparison to Supermassive Black Holes, ApJ 763, p. 76 (February 2013). 4. L. Ferrarese, P. Cˆot´e, E. Dalla Bont`a, E. W. Peng, D. Merritt, A. Jorda´n, J. P. Blakeslee, M. Ha¸segan, S. Mei, S. Piatek, J. L. Tonry and M. J. West, A Fundamental Relation between Compact Stellar Nuclei, Supermassive Black Holes, and Their Host Galaxies, ApJL 644, L21 (June 2006). 5. N. Leigh, T. B¨oker and C. Knigge, Nuclear star clusters and the stellar spheroids of their host galaxies, MNRAS 424, 2130 (August 2012). 6. M. Arca-Sedda and R. Capuzzo-Dolcetta, The globular cluster migratory origin of nuclear star clusters, MNRAS 444, 3738 (November 2014). 7. S. D. Tremaine, J. P. Ostriker and L. Spitzer, Jr., The formation of the nuclei of galaxies. I - M31, ApJ 196, 407 (March 1975). 8. R. Capuzzo-Dolcetta, The Evolution of the Globular Cluster System in a Triaxial Galaxy: Can a Galactic Nucleus Form by Globular Cluster Capture?, ApJ 415, p. 616 (October 1993). 9. M. Arca-Sedda and R. Capuzzo-Dolcetta, Dynamical Friction in Cuspy Galaxies, ApJ 785, p. 51 (April 2014). 10. F. Antonini, Origin and Growth of Nuclear Star Clusters around Massive Black Holes, ApJ 763, p. 62 (January 2013). 11. H. A. Kobulnicky, J. M. Dickey, A. I. Sargent, D. E. Hogg and P. S. Conti, Aperture Synthesis Observations of Molecular and Atomic Gas in the Wolf- Rayet Starburst Galaxy, AJ 110, p. 116 (July 1995). 12. A. Merloni, S. Heinz and T. di Matteo, A Fundamental Plane of black hole activity, MNRAS 345, 1057 (November 2003). 13. A. E. Reines, G. R. Sivakoﬀ, K. E. Johnson and C. L. Brogan, An actively accreting massive black hole in the dwarf starburst galaxy Henize2-10, Nature 470, 66 (February 2011). 14. D. D. Nguyen, A. C. Seth, A. E. Reines, M. den Brok, D. Sand and B. McLeod, Extended Structure and Fate of the Nucleus in Henize 2-10, ApJ 794, p. 34 (October 2014). 15. W. Dehnen, A Family of Potential-Density Pairs for Spherical Galaxies and Bulges, MNRAS 265, 250 (November 1993). 16. R. Capuzzo-Dolcetta, M. Spera and D. Punzo, A fully parallel, high precision, N-body code running on hybrid computing platforms, Journal of Computational Physics 236, 580 (March 2013). ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: May

text: WSPC

text: Proceedings

text: x

text: page

text: starburst

text: galaxy

text: Henize

text: as

text: input

text: for

text: highly

text: detailed,

text: direct,

text: N-body

text: simula-

text: tions.

text: We

text: found

text: that

text: not

text: all

text: the

text: clusters

text: are

text: able

text: to

text: reach

text: the

text: centre,

text: because

text: the

text: lightest

text: of

text: them

text: are

text: shattered

text: by

text: the

text: tidal

text: ﬁeld

text: of

text: the

text: parent

text: galaxy,

text: while

text: the

text: tidal

text: eﬀect

text: caused

text: by

text: the

text: central

text: MBH

text: slightly

text: aﬀects

text: the

text: quantity

text: of

text: mass

text: deposited

text: within

text: the

text: growing

text: NSC.

text: Therefore,

text: it

text: turns

text: out

text: that

text: a

text: MBH

text: does

text: not

text: necessar-

text: ily

text: determine

text: the

text: properties

text: of

text: the

text: NSC

text: that

text: surrounds

text: it

text: unless

text: it

text: is

text: very

text: massive.

text: Furthermore,

text: our

text: results

text: seem

text: to

text: suggest

text: that

text: the

text: decay

text: and

text: merging

text: of

text: star

text: clusters

text: can

text: represent

text: also

text: a

text: satisfactory

text: formation

text: channel

text: for

text: nuclear

text: stellar

text: disk,

text: though

text: this

text: depend

text: strongly

text: on

text: their

text: initial

text: conditions.

text: References

text: M.

text: Arca-Sedda,

text: R.

text: Capuzzo-Dolcetta,

text: F.

text: Antonini

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text: A.

text: Seth,

text: Henize

text: The

text: Ongoing

text: Formation

text: of

text: a

text: Nuclear

text: Star

text: Cluster

text: around

text: a

text: Massive

text: Black

text: Hole,

text: ApJ

text: p.

text: (June

text: M.

text: Arca-Sedda,

text: R.

text: Capuzzo-Dolcetta

text: and

text: M.

text: Spera,

text: The

text: dearth

text: of

text: nuclear

text: star

text: clusters

text: in

text: the

text: centre

text: of

text: the

text: brightest

text: galaxies,

text: MNRAS

text: (December

text: N.

text: Scott

text: and

text: A.

text: W.

text: Graham,

text: Updated

text: Mass

text: Scaling

text: Relations

text: for

text: Nuclear

text: Star

text: Clusters

text: and

text: a

text: Comparison

text: to

text: Supermassive

text: Black

text: Holes,

text: ApJ

text: p.

text: (February

text: L.

text: Ferrarese,

text: P.

text: Cˆot´e,

text: E.

text: Dalla

text: Bont`a,

text: E.

text: W.

text: Peng,

text: D.

text: Merritt,

text: A.

text: Jorda´n,

text: J.

text: P.

text: Blakeslee,

text: M.

text: Ha¸segan,

text: S.

text: Mei,

text: S.

text: Piatek,

text: J.

text: L.

text: Tonry

text: and

text: M.

text: J.

text: West,

text: A

text: Fundamental

text: Relation

text: between

text: Compact

text: Stellar

text: Nuclei,

text: Supermassive

text: Black

text: Holes,

text: and

text: Their

text: Host

text: Galaxies,

text: ApJL

text: (June

text: N.

text: Leigh,

text: T.

text: B¨oker

text: and

text: C.

text: Knigge,

text: Nuclear

text: star

text: clusters

text: and

text: the

text: stellar

text: spheroids

text: of

text: their

text: host

text: galaxies,

text: MNRAS

text: (August

text: M.

text: Arca-Sedda

text: and

text: R.

text: Capuzzo-Dolcetta,

text: The

text: globular

text: cluster

text: migratory

text: origin

text: of

text: nuclear

text: star

text: clusters,

text: MNRAS

text: (November

text: S.

text: D.

text: Tremaine,

text: J.

text: P.

text: Ostriker

text: and

text: L.

text: Spitzer,

text: Jr.,

text: The

text: formation

text: of

text: the

text: nuclei

text: of

text: galaxies.

text: I

text: ApJ

text: (March

text: R.

text: Capuzzo-Dolcetta,

text: The

text: Evolution

text: of

text: the

text: Globular

text: Cluster

text: System

text: in

text: a

text: Triaxial

text: Galaxy:

text: Can

text: a

text: Galactic

text: Nucleus

text: Form

text: by

text: Globular

text: Cluster

text: Capture?,

text: ApJ

text: p.

text: (October

text: M.

text: Arca-Sedda

text: and

text: R.

text: Capuzzo-Dolcetta,

text: Dynamical

text: Friction

text: in

text: Cuspy

text: Galaxies,

text: ApJ

text: p.

text: (April

text: F.

text: Antonini,

text: Origin

text: and

text: Growth

text: of

text: Nuclear

text: Star

text: Clusters

text: around

text: Massive

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text: ApJ

text: p.

text: (January

text: H.

text: A.

text: Kobulnicky,

text: J.

text: M.

text: Dickey,

text: A.

text: I.

text: Sargent,

text: D.

text: E.

text: Hogg

text: and

text: P.

text: S.

text: Conti,

text: Aperture

text: Synthesis

text: Observations

text: of

text: Molecular

text: and

text: Atomic

text: Gas

text: in

text: the

text: Wolf-

text: Rayet

text: Starburst

text: Galaxy,

text: AJ

text: p.

text: (July

text: A.

text: Merloni,

text: S.

text: Heinz

text: and

text: T.

text: di

text: Matteo,

text: A

text: Fundamental

text: Plane

text: of

text: black

text: hole

text: activity,

text: MNRAS

text: (November

text: A.

text: E.

text: Reines,

text: G.

text: R.

text: Sivakoﬀ,

text: K.

text: E.

text: Johnson

text: and

text: C.

text: L.

text: Brogan,

text: An

text: actively

text: accreting

text: massive

text: black

text: hole

text: in

text: the

text: dwarf

text: starburst

text: galaxy

text: Nature

text: (February

text: D.

text: D.

text: Nguyen,

text: A.

text: C.

text: Seth,

text: A.

text: E.

text: Reines,

text: M.

text: den

text: Brok,

text: D.

text: Sand

text: and

text: B.

text: McLeod,

text: Extended

text: Structure

text: and

text: Fate

text: of

text: the

text: Nucleus

text: in

text: Henize

text: ApJ

text: p.

text: (October

text: W.

text: Dehnen,

text: A

text: Family

text: of

text: Potential-Density

text: Pairs

text: for

text: Spherical

text: Galaxies

text: and

text: Bulges,

text: MNRAS

text: (November

text: R.

text: Capuzzo-Dolcetta,

text: M.

text: Spera

text: and

text: D.

text: Punzo,

text: A

text: fully

text: parallel,

text: high

text: precision,

text: N-body

text: code

text: running

text: on

text: hybrid

text: computing

text: platforms,

text: Journal

text: of

text: Computational

text: Physics

text: (March

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##


text: RICHTER et al.: SCALABILITY IN NEURAL CONTROL OF MUSCULOSKELETAL ROBOTS (IEEE Robotics & Automation Magazine, accepted 2015-12-31) 1 Scalability in Neural Control of Musculoskeletal Robots Christoph Richter, So¨ren Jentzsch, Rafael Hostettler, Jesu´s A. Garrido, Eduardo Ros, Alois Knoll, Florian Ro¨hrbein, Patrick van der Smagt, and Jo¨rg Conradt Abstract— Anthropomimetic robots are robots that sense, behave, interact and feel like humans. By this deﬁnition, an- thropomimetic robots require human-like physical hardware and actuation, but also brain-like control and sensing. The most self- evident realization to meet those requirements would be a human- 9 muscles assembly like musculoskeletal robot with a brain-like neural controller. While both musculoskeletal robotic hardware and neural control ball-in-socket joint software have existed for decades, a scalable approach that could be used to build and control an anthropomimetic human- scale robot has not been demonstrated yet. Combining My- orobotics, a framework for musculoskeletal robot development, with SpiNNaker, a neuromorphic computing platform, we present the proof-of-principle of a system that can scale to dozens of neurally-controlled, physically compliant joints. At its core, it implements a closed-loop cerebellar model which provides real- time low-level neural control at minimal power consumption and maximal extensibility: higher-order (e.g., cortical) neural networks and neuromorphic sensors like silicon-retinae or - cochleae can naturally be incorporated. Index Terms—adaptive control, neurocontrollers, anthropo- morphism, human-robot interaction, distributed processing, large-scale systems, parallel architectures, neural network hard- 2 elbow muscles ware, biological neural networks, neurorobotics, learning (artiﬁ- cial intelligence) MyoGanglion I. INTRODUCTION A major challenge and vision for articulated robots is to behave and interact with humans in a safe and natural asymmetric hinge manner. Robots that mimic the mechanical properties of the human build strive toward both attributes simultaneously [1], [2] as they possess built-in compliance and relatively natural, i.e., human-like, mass-distribution and dynamics by design. Musculoskeletal robots in particular offer lightweight, low- inertia end-effectors, since the main actuators, the skeletal muscles, can be kept at rest. Figure 1 shows such a design, which coarsely mirrors a human arm. Most of the muscle light-weight mass is rigidly attached to the torso. Muscles connect to the carbon-fiber bone distal bone only via tendons, which have a negligible weight. In this way, two passive safety aspects, which minimize the head injury criterion [1], are intrinsic to the anthropomimetic C. Richter and J. Conradt: Neuroscientiﬁc System Theory, Department of Electrical and Computer Engineering, Technische Universita¨t Mu¨nchen, Germany. Correspondence should be addressed to c.richter@tum.de. C. Richter: Bernstein Center for Computational Neuroscience Munich, Fig. 1. Complex Myorobotics arm mimicking the complexity of a human Germany arm without spatula. 9 muscles cooperate to control the ball-in-socket-joint. S. Jentzsch and P. van der Smagt: fortiss GmbH, Associate Institute of the One of these muscles, relating to the biceps, is biarticular, as it is attached so Technische Universita¨t Mu¨nchen, Germany. that it affects the motion of two joints, effectively coupling the shoulder and R. Hostettler, F. Ro¨hrbein, A. Knoll, and P. van der Smagt: Robotics elbow joint. and Embedded Systems, Department of Informatics, Technische Universita¨t Mu¨nchen, Germany. J.A. Garrido, E. Ros: Department of Computer Architecture and Technol- ogy, CITIC, University of Granada, Granada, Spain. ##LTFigure## ##LTFigure## ##LTFigure##

text: RICHTER

text: et

text: al.:

text: SCALABILITY

text: IN

text: NEURAL

text: CONTROL

text: OF

text: MUSCULOSKELETAL

text: ROBOTS

text: (IEEE

text: Robotics

text: Automation

text: Magazine,

text: accepted

text: Scalability

text: in

text: Neural

text: Control

text: of

text: Musculoskeletal

text: Robots

text: Christoph

text: Richter,

text: So¨ren

text: Jentzsch,

text: Rafael

text: Hostettler,

text: Jesu´s

text: A.

text: Garrido,

text: Eduardo

text: Ros,

text: Alois

text: Knoll,

text: Florian

text: Ro¨hrbein,

text: Patrick

text: van

text: der

text: Smagt,

text: and

text: Jo¨rg

text: Conradt

text: Abstract—

text: Anthropomimetic

text: robots

text: are

text: robots

text: that

text: sense,

text: behave,

text: interact

text: and

text: feel

text: like

text: humans.

text: By

text: this

text: deﬁnition,

text: an-

text: thropomimetic

text: robots

text: require

text: human-like

text: physical

text: hardware

text: and

text: actuation,

text: but

text: also

text: brain-like

text: control

text: and

text: sensing.

text: The

text: most

text: self-

text: evident

text: realization

text: to

text: meet

text: those

text: requirements

text: would

text: be

text: a

text: human-

font: default

text: muscles

text: assembly

text: like

text: musculoskeletal

text: robot

text: with

text: a

text: brain-like

text: neural

text: controller.

text: While

text: both

text: musculoskeletal

text: robotic

text: hardware

text: and

text: neural

text: control

text: ball-in-socket

text: joint

text: software

text: have

text: existed

text: for

text: decades,

text: a

text: scalable

text: approach

text: that

text: could

text: be

text: used

text: to

text: build

text: and

text: control

text: an

text: anthropomimetic

text: human-

text: scale

text: robot

text: has

text: not

text: been

text: demonstrated

text: yet.

text: Combining

text: My-

text: orobotics,

text: a

text: framework

text: for

text: musculoskeletal

text: robot

text: development,

text: with

text: SpiNNaker,

text: a

text: neuromorphic

text: computing

text: platform,

text: we

text: present

text: the

text: proof-of-principle

text: of

text: a

text: system

text: that

text: can

text: scale

text: to

text: dozens

text: of

text: neurally-controlled,

text: physically

text: compliant

text: joints.

text: At

text: its

text: core,

text: it

text: implements

text: a

text: closed-loop

text: cerebellar

text: model

text: which

text: provides

text: real-

text: time

text: low-level

text: neural

text: control

text: at

text: minimal

text: power

text: consumption

text: and

text: maximal

text: extensibility:

text: higher-order

text: (e.g.,

text: cortical)

text: neural

text: networks

text: and

text: neuromorphic

text: sensors

text: like

text: silicon-retinae

text: or

text: cochleae

text: can

text: naturally

text: be

text: incorporated.

text: Index

text: Terms—adaptive

text: control,

text: neurocontrollers,

text: anthropo-

text: morphism,

text: human-robot

text: interaction,

text: distributed

text: processing,

text: large-scale

text: systems,

text: parallel

text: architectures,

text: neural

text: network

text: hard-

text: elbow

text: muscles

text: ware,

text: biological

text: neural

text: networks,

text: neurorobotics,

text: learning

text: (artiﬁ-

text: cial

text: intelligence)

text: MyoGanglion

text: I.

text: INTRODUCTION

text: A

text: major

text: challenge

text: and

text: vision

text: for

text: articulated

text: robots

text: is

text: to

text: behave

text: and

text: interact

text: with

text: humans

text: in

text: a

text: safe

text: and

text: natural

text: asymmetric

text: hinge

text: manner.

text: Robots

text: that

text: mimic

text: the

text: mechanical

text: properties

text: of

text: the

text: human

text: build

text: strive

text: toward

text: both

text: attributes

text: simultaneously

text: as

text: they

text: possess

text: built-in

text: compliance

text: and

text: relatively

text: natural,

text: i.e.,

text: human-like,

text: mass-distribution

text: and

text: dynamics

text: by

text: design.

text: Musculoskeletal

text: robots

text: in

text: particular

text: offer

text: lightweight,

text: low-

text: inertia

text: end-effectors,

text: since

text: the

text: main

text: actuators,

text: the

text: skeletal

text: muscles,

text: can

text: be

text: kept

text: at

text: rest.

text: Figure

text: shows

text: such

text: a

text: design,

text: which

text: coarsely

text: mirrors

text: a

text: human

text: arm.

text: Most

text: of

text: the

text: muscle

text: light-weight

text: mass

text: is

text: rigidly

text: attached

text: to

text: the

text: torso.

text: Muscles

text: connect

text: to

text: the

text: carbon-fiber

text: bone

text: distal

text: bone

text: only

text: via

text: tendons,

text: which

text: have

text: a

text: negligible

text: weight.

text: In

text: this

text: way,

text: two

text: passive

text: safety

text: aspects,

text: which

text: minimize

text: the

text: head

text: injury

text: criterion

text: are

text: intrinsic

text: to

text: the

text: anthropomimetic

text: C.

text: Richter

text: and

text: J.

text: Conradt:

text: Neuroscientiﬁc

text: System

text: Theory,

text: Department

text: of

text: Electrical

text: and

text: Computer

text: Engineering,

text: Technische

text: Universita¨t

text: Mu¨nchen,

text: Germany.

text: Correspondence

text: should

text: be

text: addressed

text: to

text: c.richter@tum.de.

text: C.

text: Richter:

text: Bernstein

text: Center

text: for

text: Computational

text: Neuroscience

text: Munich,

text: Fig.

text: Complex

text: Myorobotics

text: arm

text: mimicking

text: the

text: complexity

text: of

text: a

text: human

text: Germany

text: arm

text: without

text: spatula.

text: muscles

text: cooperate

text: to

text: control

text: the

text: ball-in-socket-joint.

text: S.

text: Jentzsch

text: and

text: P.

text: van

text: der

text: Smagt:

text: fortiss

text: GmbH,

text: Associate

text: Institute

text: of

text: the

text: One

text: of

text: these

text: muscles,

text: relating

text: to

text: the

text: biceps,

text: is

text: biarticular,

text: as

text: it

text: is

text: attached

text: so

text: Technische

text: Universita¨t

text: Mu¨nchen,

text: Germany.

text: that

text: it

text: affects

text: the

text: motion

text: of

text: two

text: joints,

text: effectively

text: coupling

text: the

text: shoulder

text: and

text: R.

text: Hostettler,

text: F.

text: Ro¨hrbein,

text: A.

text: Knoll,

text: and

text: P.

text: van

text: der

text: Smagt:

text: Robotics

text: elbow

text: joint.

text: and

text: Embedded

text: Systems,

text: Department

text: of

text: Informatics,

text: Technische

text: Universita¨t

text: Mu¨nchen,

text: Germany.

text: J.A.

text: Garrido,

text: E.

text: Ros:

text: Department

text: of

text: Computer

text: Architecture

text: and

text: Technol-

text: ogy,

text: CITIC,

text: University

text: of

text: Granada,

text: Granada,

text: Spain.

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##


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text: RICHTER

text: et

text: al.:

text: SCALABILITY

text: IN

text: NEURAL

text: CONTROL

text: OF

text: MUSCULOSKELETAL

text: ROBOTS

text: (IEEE

text: Robotics

text: Automation

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text: accepted

text: and

text: J.C.

text: acknowledge

text: funding

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text: B.

text: Andersen,

text: L.

text: Korbo,

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text: B.

text: Pakkenberg,

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text: study

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text: techniques,”

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text: Neurology,

text: vol.

text: no.

text: pp.

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text: Center

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text: Computational

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text: A.

text: Grubl,

text: M.

text: Hock,

text: K.

text: Meier,

text: and

text: S.

text: Millner,

text: and

text: D.

text: Rasmussen,

text: “A

text: large-scale

text: model

text: of

text: the

text: functioning

text: brain,”

text: “A

text: wafer-scale

text: neuromorphic

text: hardware

text: system

text: for

text: large-scale

text: neural

text: Science,

text: vol.

text: no.

text: pp.

text: [Online].

text: Available:

text: modeling,”

text: in

text: Circuits

text: and

text: Systems

text: (ISCAS),

text: Proceedings

text: of

text: IEEE

text: International

text: Symposium

text: on.

text: IEEE,

text: pp.

text: J.

text: L.

text: Krichmar,

text: “Design

text: principles

text: for

text: biologically

text: inspired

text: cognitive

text: [Online].

text: Available:

text: all.jsp?

text: robotics,”

text: Biologically

text: Inspired

text: Cognitive

text: Architectures,

text: vol.

text: pp.

text: P.

text: A.

text: Merolla,

text: J.

text: V.

text: Arthur,

text: R.

text: Alvarez-Icaza,

text: A.

text: S.

text: Cassidy,

text: J.

text: Sawada,

text: F.

text: Akopyan,

text: B.

text: L.

text: Jackson,

text: N.

text: Imam,

text: C.

text: Guo,

text: Y.

text: Nakamura,

text: B.

text: Brezzo,

text: I.

text: Vo,

text: S.

text: K.

text: Esser,

text: R.

text: Appuswamy,

text: B.

text: Taba,

text: A.

text: Amir,

text: M.

text: D.

text: Flickner,

text: W.

text: P.

text: Risk,

text: R.

text: Manohar,

text: and

text: D.

text: S.

text: Modha,

text: “A

text: million

text: spiking-neuron

text: integrated

text: circuit

text: with

text: a

text: scalable

text: communication

text: network

text: and

text: interface,”

text: Science,

text: vol.

text: no.

text: pp.

text: [Online].

text: Available:

text: ##LTLine##

text: ##LTLine##


text: this section is that (cid:96) sparse regularisation can give similar accuracy to Tikhonov 1 regularisation for noisy data, but with a small fraction of the number of charge points required to produce the reconstruction. 4.2 Higher and irregular frequencies We now investigate the behavior of the method for some potentially problem- atic choices of the wavenumber k. First we look at the case when the frequency is increased, including when the Nyquist frequency is exceeded. Since our mea- surements are taken at triangle centroids then the resulting measurement grid is irregular and so the Nyquist frequency is not well-deﬁned. We therefore choose the Nyquist frequency associated with the regular grid given by the triangle vertices, as a value approximately representative of the Nyquist frequency. For the discretisation considered in the previous section with 576 triangles, the grid spacing is ∆x = 0.04667, meaning that the wavenumber corresponding to the Nyquist frequency is k = π/∆x = 67.32. We also investigate the perfor- nyq mance of the method close to other typical threshold frequencies for numerical solution approaches, such as the six grid points per wavelength rule of thumb for ﬁnite and boundary element methods, which gives a maximum wavenumber of k = 22.44 for the grid described above. The performance of the method at irregular frequencies will also be investigated. For the method of superposition these irregular frequencies are the resonances of the region enclosed by the in- terior source surface [5]. Numerical studies indicate that one such frequency is ∗ close to k = 17.54/α, which here is k = 52.62. Table 2 gives the reconstruction error for a range of wavenumbers k using (cid:96) 1 reconstruction techniques and compares the accuracy of the reconstruction using all 576 charge points, and using two diﬀerent values of the sparsity parameter β. In particular, we compare the default choice used in the last section of β = 0.5 with the choice β = 0.9, which uses fewer charge points but at the potential cost of poorer accuracy. The maximum wavenumber studied corresponds to the wavelength being close to (but still greater than) the exterior measurement distance δ. The results show that both irregular and high frequencies lead to a degradation in the accuracy of the reconstruction, and lead to a loss of sparsity in the reconstructions. Accurate and reasonably sparse reconstructions can be generated provided there are at least 3 data points per wavelength since for up to k = 44.88 we can reconstruct the solution with a smaller error than the level of added noise (15%) and with at least an order of magnitude reduction from the total number of charge points (576). These results are consistent with the ﬁndings of Ref. [18], where it is also suggested that a superposition method will give accurate results provided there are at least 3 degrees of freedom per wavelength. We note that if the surface of interior charge points includes the monopole generating the acoustic ﬁeld then one would obtain exact representa- tions for arbitrarily high frequencies. In addition to the general trend of increased errors for higher frequencies, one also observes a local peak in the error at the characteristic wavenumber k = 52.62. Here we see that the error is particularly poor for the sparse re- 12

text: this

text: section

text: is

text: that

text: sparse

text: regularisation

text: can

text: give

text: similar

text: accuracy

text: to

text: Tikhonov

text: regularisation

text: for

text: noisy

text: data,

text: but

text: with

text: a

text: small

text: fraction

text: of

text: the

text: number

text: of

text: charge

text: points

text: required

text: to

text: produce

text: the

text: reconstruction.

label: section

text: Higher

text: and

text: irregular

text: frequencies

text: We

text: now

text: investigate

text: the

text: behavior

text: of

text: the

text: method

text: for

text: some

text: potentially

text: problem-

text: atic

text: choices

text: of

text: the

text: wavenumber

text: k.

text: First

text: we

text: look

text: at

text: the

text: case

text: when

text: the

text: frequency

text: is

text: increased,

text: including

text: when

text: the

text: Nyquist

text: frequency

text: is

text: exceeded.

text: Since

text: our

text: mea-

text: surements

text: are

text: taken

text: at

text: triangle

text: centroids

text: then

text: the

text: resulting

text: measurement

text: grid

text: is

text: irregular

text: and

text: so

text: the

text: Nyquist

text: frequency

text: is

text: not

text: well-deﬁned.

text: We

text: therefore

text: choose

text: the

text: Nyquist

text: frequency

text: associated

text: with

text: the

text: regular

text: grid

text: given

text: by

text: the

text: triangle

text: vertices,

text: as

text: a

text: value

text: approximately

text: representative

text: of

text: the

text: Nyquist

text: frequency.

text: For

text: the

text: discretisation

text: considered

text: in

text: the

text: previous

text: section

text: with

text: triangles,

text: the

text: grid

text: spacing

text: is

text: ∆x

text: meaning

text: that

text: the

text: wavenumber

text: corresponding

text: to

text: the

text: Nyquist

text: frequency

text: is

text: k

text: π/∆x

text: We

text: also

text: investigate

text: the

text: perfor-

text: nyq

text: mance

text: of

text: the

text: method

text: close

text: to

text: other

text: typical

text: threshold

text: frequencies

text: for

text: numerical

text: solution

text: approaches,

text: such

text: as

text: the

text: six

text: grid

text: points

text: per

text: wavelength

text: rule

text: of

text: thumb

text: for

text: ﬁnite

text: and

text: boundary

text: element

text: methods,

text: which

text: gives

text: a

text: maximum

text: wavenumber

text: of

text: k

text: for

text: the

text: grid

text: described

text: above.

text: The

text: performance

text: of

text: the

text: method

text: at

text: irregular

text: frequencies

text: will

text: also

text: be

text: investigated.

text: For

text: the

text: method

text: of

text: superposition

text: these

text: irregular

text: frequencies

text: are

text: the

text: resonances

text: of

text: the

text: region

text: enclosed

text: by

text: the

text: in-

text: terior

text: source

text: surface

text: Numerical

text: studies

text: indicate

text: that

text: one

text: such

text: frequency

text: is

text: close

text: to

text: k

text: which

text: here

text: is

text: k

text: Table

text: gives

text: the

text: reconstruction

text: error

text: for

text: a

text: range

text: of

text: wavenumbers

text: k

text: using

text: reconstruction

text: techniques

text: and

text: compares

text: the

text: accuracy

text: of

text: the

text: reconstruction

text: using

text: all

text: charge

text: points,

text: and

text: using

text: two

text: diﬀerent

text: values

text: of

text: the

text: sparsity

text: parameter

text: β.

text: In

text: particular,

text: we

text: compare

text: the

text: default

text: choice

text: used

text: in

text: the

text: last

text: section

text: of

text: β

text: with

text: the

text: choice

text: β

text: which

text: uses

text: fewer

text: charge

text: points

text: but

text: at

text: the

text: potential

text: cost

text: of

text: poorer

text: accuracy.

text: The

text: maximum

text: wavenumber

text: studied

text: corresponds

text: to

text: the

text: wavelength

text: being

text: close

text: to

text: (but

text: still

text: greater

text: than)

text: the

text: exterior

text: measurement

text: distance

text: δ.

text: The

text: results

text: show

text: that

text: both

text: irregular

text: and

text: high

text: frequencies

text: lead

text: to

text: a

text: degradation

text: in

text: the

text: accuracy

text: of

text: the

text: reconstruction,

text: and

text: lead

text: to

text: a

text: loss

text: of

text: sparsity

text: in

text: the

text: reconstructions.

text: Accurate

text: and

text: reasonably

text: sparse

text: reconstructions

text: can

text: be

text: generated

text: provided

text: there

text: are

text: at

text: least

text: data

text: points

text: per

text: wavelength

text: since

text: for

text: up

text: to

text: k

text: we

text: can

text: reconstruct

text: the

text: solution

text: with

text: a

text: smaller

text: error

text: than

text: the

text: level

text: of

text: added

text: noise

text: and

text: with

text: at

text: least

text: an

text: order

text: of

text: magnitude

text: reduction

text: from

text: the

text: total

text: number

text: of

text: charge

text: points

text: These

text: results

text: are

text: consistent

text: with

text: the

text: ﬁndings

text: of

text: Ref.

text: where

text: it

text: is

text: also

text: suggested

text: that

text: a

text: superposition

text: method

text: will

text: give

text: accurate

text: results

text: provided

text: there

text: are

text: at

text: least

text: degrees

text: of

text: freedom

text: per

text: wavelength.

text: We

text: note

text: that

text: if

text: the

text: surface

text: of

text: interior

text: charge

text: points

text: includes

text: the

text: monopole

text: generating

text: the

text: acoustic

text: ﬁeld

text: then

text: one

text: would

text: obtain

text: exact

text: representa-

text: tions

text: for

text: arbitrarily

text: high

text: frequencies.

text: In

text: addition

text: to

text: the

text: general

text: trend

text: of

text: increased

text: errors

text: for

text: higher

text: frequencies,

text: one

text: also

text: observes

text: a

text: local

text: peak

text: in

text: the

text: error

text: at

text: the

text: characteristic

text: wavenumber

text: k

text: Here

text: we

text: see

text: that

text: the

text: error

text: is

text: particularly

text: poor

text: for

text: the

text: sparse

text: re-


text: Initially we will study test problems where the exterior acoustic ﬁeld is gen- erated by a monopole point source inside the structure. In this case the solution could be represented exactly for arbitrarily high frequencies using the method of superposition if one of the points on the surface of interior charge points coincides with the monopole generating the acoustic ﬁeld. However, in general such information would not be known a-priori when sampling an acoustic ﬁeld in NAH. We will therefore examine the dependence of the reconstruction on the locations of the charge points and the monopole generating the acoustic ﬁeld. Finally, we will consider a case where the acoustic ﬁeld is not generated by an interior monopole, but rather by a relatively small vibrating patch on an otherwise rigid structure. This leads to a problem where the method of su- perposition should return a sparse expansion for interior charge point surfaces very close to the physical boundary, since the weighting of sources close to the vibrating region will be dominant compared to those close to the rigid part of the structure. The latter (locally radiating) case also has parallels with the situation that arises when modelling acoustic radiation from loudspeakers. 2 Method of Superposition for Near-ﬁeld Acous- tic Holography Let Ω ⊂ R3 be a ﬁnite domain with boundary surface Γ. Let Ω = R3 \ Ω¯ + denote the unbounded exterior domain, which is assumed to be ﬁlled with a homogeneous compressible acoustic medium with density ρ and speed of sound c. For a time-harmonic disturbance of frequency ω, the sound pressure p satisﬁes the homogeneous Helmholtz equation in Ω + ∆p + k2p = 0, (1) where k = ω/c is the wave-number. Since this work considers an unbounded exterior domain, then p must also satisfy the Sommerfeld radiation condition. ∈ The superposition method approximates p at some point x Ω¯ using a basis + (cid:88) expansion of the form ≈ n p(x) σ G (x, y ), (2) j k j j=1 where G is the free space Green’s function for Helmholtz equation in three k dimensions given by | − | eik x y G (x, y) = | − |. (3) k 4π x y ∈ Here y Ω, i = 1, ..., n are the source locations and σ are the source i i strengths, which are determined by application of the method. In the NAH problem we are given values of the acoustic pressure p at a discrete set of points the acoustic near ﬁeld within Ω . We will assume that + 4 ##LTLine##

text: Initially

text: we

text: will

text: study

text: test

text: problems

text: where

text: the

text: exterior

text: acoustic

text: ﬁeld

text: is

text: gen-

text: erated

text: by

text: a

text: monopole

text: point

text: source

text: inside

text: the

text: structure.

text: In

text: this

text: case

text: the

text: solution

text: could

text: be

text: represented

text: exactly

text: for

text: arbitrarily

text: high

text: frequencies

text: using

text: the

text: method

text: of

text: superposition

text: if

text: one

text: of

text: the

text: points

text: on

text: the

text: surface

text: of

text: interior

text: charge

text: points

text: coincides

text: with

text: the

text: monopole

text: generating

text: the

text: acoustic

text: ﬁeld.

text: However,

text: in

text: general

text: such

text: information

text: would

text: not

text: be

text: known

text: a-priori

text: when

text: sampling

text: an

text: acoustic

text: ﬁeld

text: in

text: NAH.

text: We

text: will

text: therefore

text: examine

text: the

text: dependence

text: of

text: the

text: reconstruction

text: on

text: the

text: locations

text: of

text: the

text: charge

text: points

text: and

text: the

text: monopole

text: generating

text: the

text: acoustic

text: ﬁeld.

text: Finally,

text: we

text: will

text: consider

text: a

text: case

text: where

text: the

text: acoustic

text: ﬁeld

text: is

text: not

text: generated

text: by

text: an

text: interior

text: monopole,

text: but

text: rather

text: by

text: a

text: relatively

text: small

text: vibrating

text: patch

text: on

text: an

text: otherwise

text: rigid

text: structure.

text: This

text: leads

text: to

text: a

text: problem

text: where

text: the

text: method

text: of

text: su-

text: perposition

text: should

text: return

text: a

text: sparse

text: expansion

text: for

text: interior

text: charge

text: point

text: surfaces

text: very

text: close

text: to

text: the

text: physical

text: boundary,

text: since

text: the

text: weighting

text: of

text: sources

text: close

text: to

text: the

text: vibrating

text: region

text: will

text: be

text: dominant

text: compared

text: to

text: those

text: close

text: to

text: the

text: rigid

text: part

text: of

text: the

text: structure.

text: The

text: latter

text: (locally

text: radiating)

text: case

text: also

text: has

text: parallels

text: with

text: the

text: situation

text: that

text: arises

text: when

text: modelling

text: acoustic

text: radiation

text: from

text: loudspeakers.

text: Method

text: of

text: Superposition

text: for

text: Near-ﬁeld

text: Acous-

text: tic

text: Holography

text: Let

text: Ω

text: be

text: a

text: ﬁnite

text: domain

text: with

text: boundary

text: surface

text: Γ.

text: Let

text: Ω

text: Ω¯

text: denote

text: the

text: unbounded

text: exterior

text: domain,

text: which

text: is

text: assumed

text: to

text: be

text: ﬁlled

text: with

text: a

text: homogeneous

text: compressible

text: acoustic

text: medium

text: with

text: density

text: ρ

text: and

text: speed

text: of

text: sound

text: c.

text: For

text: a

text: time-harmonic

text: disturbance

text: of

text: frequency

text: ω,

text: the

text: sound

text: pressure

text: p

text: satisﬁes

text: the

text: homogeneous

text: Helmholtz

text: equation

text: in

text: Ω

text: ∆p

text: where

text: k

text: ω/c

text: is

text: the

text: wave-number.

text: Since

text: this

text: work

text: considers

text: an

text: unbounded

text: exterior

text: domain,

text: then

text: p

text: must

text: also

text: satisfy

text: the

text: Sommerfeld

text: radiation

text: condition.

text: The

text: superposition

text: method

text: approximates

text: p

text: at

text: some

text: point

text: x

text: Ω¯

text: using

text: a

text: basis

text: expansion

text: of

text: the

text: form

text: n

text: p(x)

text: σ

text: G

text: (x,

text: y

text: j

text: k

text: j

text: where

text: G

text: is

text: the

text: free

text: space

text: Green’s

text: function

text: for

text: Helmholtz

text: equation

text: in

text: three

text: k

text: dimensions

text: given

text: by

text: eik

text: x

text: y

text: G

text: (x,

text: y)

text: k

text: 4π

text: x

text: y

text: Here

text: y

text: Ω,

text: i

text: n

text: are

text: the

text: source

text: locations

text: and

text: σ

text: are

text: the

text: source

text: i

text: i

text: strengths,

text: which

text: are

text: determined

text: by

text: application

text: of

text: the

text: method.

text: In

text: the

text: NAH

text: problem

text: we

text: are

text: given

text: values

text: of

text: the

text: acoustic

text: pressure

text: p

text: at

text: a

text: discrete

text: set

text: of

text: points

text: the

text: acoustic

text: near

text: ﬁeld

text: within

text: Ω

text: We

text: will

text: assume

text: that

text: ##LTLine##


text: 2 mentioned in the literature. Ultimately, it may be that the Monster group and its structure ﬁts our quantum world, as in Dyson’s words [11]. More cautiously, in Sec. of the present paper, we brieﬂy account for the group concepts involved in our approach P − D − G P by deﬁning a tripod . One leg is a desired two-generator permutation D representation of a ﬁnite group P [8]. Another leg signs the coset structure of the (cid:48) used subgroup H of the two-generator free group G (or of a subgroup G of G with D relations), whose ﬁnite index [G, H] = n is the number edges of , and at the same P G time the size of the set on which acts, as in [10]. Finally, is the geometry with n D vertices that is deﬁned/stabilized by [9]. Then, in Sec. , we organize the relevant P −D −G tripods taken from the classes of the Atlas and ﬁnd that many of them reﬂect quantum commutation, speciﬁcally the symplectic, unitary and orthogonal classes. The geometries of other (classical and sporadic) classes are investigated similarly with the goal to recognize their possible physical signiﬁcance. A physically oriented survey of simple groups is [12]. P − D − G 2. Group concepts for the puzzle 2.1. Groups, dessins and ﬁnite geometries Following the impetus given by Grothendieck [14], it is now known that there are various ways to dress a group P generated by two permutations, (i) as a connected graph drawn on a compact oriented two-dimensional surface -a bicolored map (or hypermap) with n edges, B black points, W white points, F faces, genus g and Euler characteristic − − 2 2g = B + W + F n [15], (ii) as a Riemann surface X of the same genus equipped C ¯ with a meromorphic function f from X to the Riemann sphere unramiﬁed outside the { ∞} critical set 0, 1, -the pair (X, f) called a Belyi pair and, in this context, hypermaps (cid:104) (cid:105) are called dessins d’enfants [13, 14], (iii) as a subgroup H of the free group G = a, b where P encodes the action of (right) cosets of H on the two generators a and b -the Coxeter-Todd algorithm does the job [10] and ﬁnally (iv), when P is of rank at least three, that is of point stabilizer with at least three orbits, as a non-trivial ﬁnite geometry [9]-[11]. Finite simple groups are generated by two of their elements [16] so that it is useful to characterize them as members of the categories just described. There are many mathematical papers featuring the correspondence between items (i) and (ii) in view of a better understanding of the action of the absolute Galois Q Q Q ¯ ¯ group Gal( / ) -the automorphism group of the ﬁeld of algebraic numbers- on the hypermaps [14, 15, 17]. Coset enumeration featured in item (iii) is at work in the permutation representations of ﬁnite groups found in the Atlas [8]. Item (i) in conjunction to (iii) and (iv) allowed us to arrive at the concept of geometric contextuality as a lack of commutativity of cosets on the lines of the ﬁnite geometry stabilized by P [10]. Item (iv) may be further clariﬁed thanks to the concept of rank of a permutation group P . First it is expected that P acts faithfully and transitively on the set { · · · } Ω = 1, 2, , n as a subgroup of the symmetric group S . The action of P on a pair n∈ (cid:54) of distinct elements of Ω is deﬁned as (α, β)p = (αp, βp), p P , α = β. The orbits of P

text: mentioned

text: in

text: the

text: literature.

text: Ultimately,

text: it

text: may

text: be

text: that

text: the

text: Monster

text: group

text: and

text: its

text: structure

text: ﬁts

text: our

text: quantum

text: world,

text: as

text: in

text: Dyson’s

text: words

text: More

text: cautiously,

text: in

text: Sec.

text: of

text: the

text: present

text: paper,

text: we

text: brieﬂy

text: account

text: for

text: the

text: group

text: concepts

text: involved

text: in

text: our

text: approach

text: P

text: D

text: G

text: P

text: by

text: deﬁning

text: a

text: tripod

text: One

text: leg

text: is

text: a

text: desired

text: two-generator

text: permutation

text: D

text: representation

text: of

text: a

text: ﬁnite

text: group

text: P

text: Another

text: leg

text: signs

text: the

text: coset

text: structure

text: of

text: the

text: used

text: subgroup

text: H

text: of

text: the

text: two-generator

text: free

text: group

text: G

text: (or

text: of

text: a

text: subgroup

text: G

text: of

text: G

text: with

text: D

text: relations),

text: whose

text: ﬁnite

text: index

text: [G,

text: H]

text: n

text: is

text: the

text: number

text: edges

text: of

text: and

text: at

text: the

text: same

text: P

text: G

text: time

text: the

text: size

text: of

text: the

text: set

text: on

text: which

text: acts,

text: as

text: in

text: Finally,

text: is

text: the

text: geometry

text: with

text: n

text: D

text: vertices

text: that

text: is

text: deﬁned/stabilized

text: by

text: Then,

text: in

text: Sec.

text: we

text: organize

text: the

text: relevant

text: P

text: −D

text: −G

text: tripods

text: taken

text: from

text: the

text: classes

text: of

text: the

text: Atlas

text: and

text: ﬁnd

text: that

text: many

text: of

text: them

text: reﬂect

text: quantum

text: commutation,

text: speciﬁcally

text: the

text: symplectic,

text: unitary

text: and

text: orthogonal

text: classes.

text: The

text: geometries

text: of

text: other

text: (classical

text: and

text: sporadic)

text: classes

text: are

text: investigated

text: similarly

text: with

text: the

text: goal

text: to

text: recognize

text: their

text: possible

text: physical

text: signiﬁcance.

text: A

text: physically

text: oriented

text: survey

text: of

text: simple

text: groups

text: is

text: P

text: D

text: G

text: Group

text: concepts

text: for

text: the

text: puzzle

text: Groups,

text: dessins

text: and

text: ﬁnite

text: geometries

text: Following

text: the

text: impetus

text: given

text: by

text: Grothendieck

text: it

text: is

text: now

text: known

text: that

text: there

text: are

text: various

text: ways

text: to

text: dress

text: a

text: group

text: P

text: generated

text: by

text: two

text: permutations,

text: (i)

text: as

text: a

text: connected

text: graph

text: drawn

text: on

text: a

text: compact

text: oriented

text: two-dimensional

text: surface

text: -a

text: bicolored

text: map

text: (or

text: hypermap)

text: with

text: n

text: edges,

text: B

text: black

text: points,

text: W

text: white

text: points,

text: F

text: faces,

text: genus

text: g

text: and

text: Euler

text: characteristic

text: B

text: W

text: F

text: n

text: (ii)

text: as

text: a

text: Riemann

text: surface

text: X

text: of

text: the

text: same

text: genus

text: equipped

text: C

text: with

text: a

text: meromorphic

text: function

text: f

text: from

text: X

text: to

text: the

text: Riemann

text: sphere

text: unramiﬁed

text: outside

text: the

text: critical

text: set

text: -the

text: pair

text: (X,

text: f)

text: called

text: a

text: Belyi

text: pair

text: and,

text: in

text: this

text: context,

text: hypermaps

text: are

text: called

text: dessins

text: d’enfants

text: (iii)

text: as

text: a

text: subgroup

text: H

text: of

text: the

text: free

text: group

text: G

text: a,

text: b

text: where

text: P

text: encodes

text: the

text: action

text: of

text: (right)

text: cosets

text: of

text: H

text: on

text: the

text: two

text: generators

text: a

text: and

text: b

text: -the

text: Coxeter-Todd

text: algorithm

text: does

text: the

text: job

text: and

text: ﬁnally

text: (iv),

text: when

text: P

text: is

text: of

text: rank

text: at

text: least

text: three,

text: that

text: is

text: of

text: point

text: stabilizer

text: with

text: at

text: least

text: three

text: orbits,

text: as

text: a

text: non-trivial

text: ﬁnite

text: geometry

text: Finite

text: simple

text: groups

text: are

text: generated

text: by

text: two

text: of

text: their

text: elements

text: so

text: that

text: it

text: is

text: useful

text: to

text: characterize

text: them

text: as

text: members

text: of

text: the

text: categories

text: just

text: described.

text: There

text: are

text: many

text: mathematical

text: papers

text: featuring

text: the

text: correspondence

text: between

text: items

text: (i)

text: and

text: (ii)

text: in

text: view

text: of

text: a

text: better

text: understanding

text: of

text: the

text: action

text: of

text: the

text: absolute

text: Galois

text: Q

text: Q

text: Q

text: group

text: Gal(

text: -the

text: automorphism

text: group

text: of

text: the

text: ﬁeld

text: of

text: algebraic

text: numbers-

text: on

text: the

text: hypermaps

text: Coset

text: enumeration

text: featured

text: in

text: item

text: (iii)

text: is

text: at

text: work

text: in

text: the

text: permutation

text: representations

text: of

text: ﬁnite

text: groups

text: found

text: in

text: the

text: Atlas

text: Item

text: (i)

text: in

text: conjunction

text: to

text: (iii)

text: and

text: (iv)

text: allowed

text: us

text: to

text: arrive

text: at

text: the

text: concept

text: of

text: geometric

text: contextuality

text: as

text: a

text: lack

text: of

text: commutativity

text: of

text: cosets

text: on

text: the

text: lines

text: of

text: the

text: ﬁnite

text: geometry

text: stabilized

text: by

text: P

text: Item

text: (iv)

text: may

text: be

text: further

text: clariﬁed

text: thanks

text: to

text: the

text: concept

text: of

text: rank

text: of

text: a

text: permutation

text: group

text: P

text: First

text: it

text: is

text: expected

text: that

text: P

text: acts

text: faithfully

text: and

text: transitively

text: on

text: the

text: set

text: Ω

text: n

text: as

text: a

text: subgroup

text: of

text: the

text: symmetric

text: group

text: S

text: The

text: action

text: of

text: P

text: on

text: a

text: pair

text: n∈

text: of

text: distinct

text: elements

text: of

text: Ω

text: is

text: deﬁned

text: as

text: (α,

text: β)p

text: (αp,

text: βp),

text: p

text: P

text: α

text: β.

text: The

text: orbits

text: of

text: P


text: 13 Table 9. It is noticeable that some representations are associated to the non-intersection of maximum commuting sets for three qubits [from O+(2) : 2)] and three qutrits [from 8 O (3) or O+(3)]. These geometries are introduced in [1, Table 2]. The srg’s are identiﬁed 7 8 in [19]. Figure 2. A schematic of the hyperbolic polygon on 765 tiles corresponding to the − permutation group of O (2). The picture is split into two horizontal parts. 8 G Several of the conﬁgurations arising from simple orthogonal groups are of type , ≥ i for some i 1. This includes the conﬁgurations attached to polar (strongly regular) − − − graphs of O (2) (on 119 points), O (3) (on 1066 points) and O (2) (on 495 points). 8 8 10 G − There exists a near polygon (thus of type ) built from O (2) (on 765 points) that 1 8 seems to have been unnoticed. The conﬁguration is of the type [765 , 1071 ] with − 7 5 (4) collinearity graph of spectrum [281, 1184, 1476, 7204] and diameter 3 corresponding to a near hexagon of order (4, 6). Since the permutation representation is a subgroup of the Z D modular group Γ = P SL(2, ), it is possible to see the dessin as an hyperbolic polygon D D D H. As in [10, 11], the genus g of equals that of the hyperbolic polygon H, a face of D D D corresponds to a cusp of H, the number of black points (resp. of white points) of − − − − is B = f +ν 1 (resp. W = n+2 2g B c), where f is the number of fractions, c is 3 the number of cusps, ν and ν are the number of elliptic points of order two and three of 2 3 ##LTFigure## ##LTFigure## ##LTFigure##

text: Table

text: It

text: is

text: noticeable

text: that

text: some

text: representations

text: are

text: associated

text: to

text: the

text: non-intersection

text: of

text: maximum

text: commuting

text: sets

text: for

text: three

text: qubits

text: [from

text: and

text: three

text: qutrits

text: [from

text: O

text: or

text: These

text: geometries

text: are

text: introduced

text: in

text: Table

text: The

text: srg’s

text: are

text: identiﬁed

text: in

text: Figure

text: A

label: caption

text: schematic

text: of

text: the

text: hyperbolic

text: polygon

text: on

text: tiles

text: corresponding

text: to

text: the

text: permutation

text: group

text: of

text: O

text: The

text: picture

text: is

text: split

text: into

text: two

text: horizontal

text: parts.

text: G

text: Several

text: of

text: the

text: conﬁgurations

text: arising

text: from

text: simple

text: orthogonal

text: groups

text: are

text: of

text: type

text: i

text: for

text: some

text: i

text: This

text: includes

text: the

text: conﬁgurations

text: attached

text: to

text: polar

text: (strongly

text: regular)

text: graphs

text: of

text: O

text: (on

text: points),

text: O

text: (on

text: points)

text: and

text: O

text: (on

text: points).

text: G

text: There

text: exists

text: a

text: near

text: polygon

text: (thus

text: of

text: type

text: built

text: from

text: O

text: (on

text: points)

text: that

text: seems

text: to

text: have

text: been

text: unnoticed.

text: The

text: conﬁguration

text: is

text: of

text: the

text: type

text: with

text: collinearity

text: graph

text: of

text: spectrum

text: and

text: diameter

text: corresponding

text: to

text: a

text: near

text: hexagon

text: of

text: order

text: Since

text: the

text: permutation

text: representation

text: is

text: a

text: subgroup

text: of

text: the

text: Z

text: D

text: modular

text: group

text: Γ

text: P

text: it

text: is

text: possible

text: to

text: see

text: the

text: dessin

text: as

text: an

text: hyperbolic

text: polygon

text: D

text: D

text: D

text: H.

text: As

text: in

text: the

text: genus

text: g

text: of

text: equals

text: that

text: of

text: the

text: hyperbolic

text: polygon

text: H,

text: a

text: face

text: of

text: D

text: D

text: D

text: corresponds

text: to

text: a

text: cusp

text: of

text: H,

text: the

text: number

text: of

text: black

text: points

text: (resp.

text: of

text: white

text: points)

text: of

text: is

text: B

text: f

text: +ν

text: (resp.

text: W

text: B

text: c),

text: where

text: f

text: is

text: the

text: number

text: of

text: fractions,

text: c

text: is

text: the

text: number

text: of

text: cusps,

text: ν

text: and

text: ν

text: are

text: the

text: number

text: of

text: elliptic

text: points

text: of

text: order

text: two

text: and

text: three

text: of

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##