text: 3 (1) (2) (3) depleted by T = 0 decay, in which the Higgs vacuum has a lifetime that is shorter than the age of the Universe. To Boundary T = 0 Instablity TR Instability Stability explain why a point lying in the experimentally allowed (cid:39) Vacuum Quantum Thermal Inﬂationary ellipse at yt 0.95 corresponds to the mo∼st probable anthropically allowed vacuum, we need δλ 0.05, the Selection T = 0 decay decay decay ∼ ∼ (cid:28) approximate width of the metastable region.1 Scenario δλ 0.05 0.02 0.01 (1) could take place for any value of the top mass, within ± (cid:39) Mt/GeV 173.34 2.28 173.34+−12..3248 171 the experimentally preferre±d re×gion, which we take to be the 3σ range M = 173.34 3 0.67 GeV [18]. t Table I: Characteristics of the three cases we consider in the In case (2) we end in a vacuum near the instability text, regarding the critical boundary, vacuum selection mech- boundary for decay by thermal ﬂuctuations with a high anism, step in λ needed and range of top mass (inside the reheating temperature, that reduce the region of metasta- experimental 3σ band) required. bility. As concrete examples we illustrate the cases of T = 1014 and T = 1016 GeV. The boundary of the R R reduced region is shown as the dashed lines in Fig. 2 possibly an extra dimension decompactiﬁes [16]. In clock- (see refs. [19, 20]), and a possible trajectory illustrating (cid:39) work constructions the number of vacua is also ﬁnite as this case is shown along y 0.934. A smaller step size ∼ t the ﬁeld range is compact. δλ 0.02 is suggested for naturally explaining the dis- We as(cid:16)sume that in ad(cid:17)dition to(cid:16) V , there i(cid:17)s a coupling tance of the SM point from the dashed boundary. This of a to the Higgs potential: mechanism, for such large T favors the lower range of (cid:39) R V = − µ2 + c η aΛ |H|2 + λ + c η a |H|4 . (5) the top mass, with Mt 173.34+−12..3248 GeV. h h h λ Λ In case (3) we end very close to the stability boundary √ beyond which the Higgs vacuum is unstable against decay H Such couplings also break the shift symmetry and so we during inﬂation, for suﬃciently large values of N . I e assign a factor η to them. The a-terms in (5) could be This case is illustrated by the trajectory passing through regarded as→arising frOom a| ge| ne|ral|ization of eq. (3) by the bottom of the experimental ellipse. Here the most taking M4 M4 + (Λ2 H 2, H 4) or from expanded probable state would be the one closest to the boundary cos(a/F) potentials in the clockwork re(cid:104)a(cid:105)lization(cid:39). In the in the absolute stability region, and it would require a landscape of vacua of the a ﬁeld, where a = an 2πnf, very small step size δλ to be naturally close to the ex- this shifts the bare values (i.e., the values at the UV scale perimental ellipse. Although this possibility is currently Λ) of the Higgs parameters to disfavored, it is not excluded and provides another possi- − ble regime for explaining near-critical stability, if the top µ2 = µ2 n c η (2πf)Λ , (cid:39) n h h mass is very close to its lowest 3σ value, M 171 GeV. t 2πf ≡ Once δλ is ﬁxed, (6) can be used to to eliminate the λ = λ + n c η λ + n δλ . (6) n λ Λ unknown parameter η in terms of f and δλ. We introduce ≡ the ratio δ δλ/0.05 [which is of order unity in case Here we assume that some other mechanism solves the λ (1)] to allow for the possibility of any of the three cases. weak scale hierarchy problem (e.g. a relaxion mechanism Hence [14]) so that µ is of electroweak size and focus on the n shift in the Higgs coupling. For reasons detailed below Λ c η = 0.05 δ . (7) we also choose cλ > 0. Likewise we must assume there is λ λ 2πf another mechanism for solving the cosmological constant problem, since the vacuum energy varies between a-vacua 3. PROBABILITY DISTRIBUTION OF VACUA due to the nonperiodic part of the potential V . We consider three possible scenarios, each associated to A key ingredient of our scenario is the process by which one of the three critical boundaries shown in ﬁg. 2; these the vacua get populated by quantum ﬂuctuations during are the boundaries of instability and metastability at zero inﬂation, and the resulting probability distribution func- temperature, and the boundary of high-temperature in- tion P(t, a ) for the diﬀerent vacua. It is governed by n stability that depends upon the assumed reheating tem- perature (dashed lines). Our mechanism explains why we would observe (λ, y ) to be near (and to the right t of) one of these boundaries. The characteristics of the 1 This number can be estimated as follows. The vacuum decay three categories are summarized in table I. Fig. 2 shows rate per unit volume is Γ ∼ h4te−8π2/(3|λ(ht)|), where ht is the trajectories of successive vacua that exemplify each case. preferred value for tunneling. The decay probability is Γ times ∼ Which one of the three is actually realized depends upon the 4D volume of our past light-cone∼ −(e140/mP)4. Decay cosmological parameters, as we will discuss in more detail probabilities of order one require λ(ht) 0.05 and this number is conﬁrmed by a more sophisticated calculation (see e.g. [17]). in the next section. ∈ {− } Thus the metastable region is approximately λ(ht) 0.05,0 . In case (1) we end close to the instability boundary and This translates to the region shown in Fig. 1 after running the the probability to live in vacua beyond that boundary is couplings down to the weak scale. ##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##

label: table

text: depleted

text: by

text: T

text: decay,

text: in

text: which

text: the

text: Higgs

text: vacuum

text: has

text: a

text: lifetime

text: that

text: is

text: shorter

text: than

text: the

text: age

text: of

text: the

text: Universe.

text: To

text: Boundary

text: T

text: Instablity

text: TR

text: Instability

text: Stability

text: explain

text: why

text: a

text: point

text: lying

text: in

text: the

text: experimentally

text: allowed

text: Vacuum

text: Quantum

text: Thermal

text: Inﬂationary

text: ellipse

text: at

text: yt

text: corresponds

text: to

text: the

text: mo∼st

text: probable

text: anthropically

text: allowed

text: vacuum,

text: we

text: need

text: δλ

text: the

text: Selection

text: T

text: decay

text: decay

text: decay

text: approximate

text: width

text: of

text: the

text: metastable

text: Scenario

text: δλ

text: could

text: take

text: place

text: for

text: any

text: value

text: of

text: the

text: top

text: mass,

text: within

text: Mt/GeV

text: the

text: experimentally

text: preferre±d

text: re×gion,

text: which

text: we

text: take

text: to

text: be

text: the

text: 3σ

text: range

text: M

text: GeV

text: t

text: Table

text: I:

text: Characteristics

text: of

text: the

text: three

text: cases

text: we

text: consider

text: in

text: the

text: In

text: case

text: we

text: end

text: in

text: a

text: vacuum

text: near

text: the

text: instability

text: text,

text: regarding

text: the

text: critical

text: boundary,

text: vacuum

text: selection

text: mech-

text: boundary

text: for

text: decay

text: by

text: thermal

text: ﬂuctuations

text: with

text: a

text: high

text: anism,

text: step

text: in

text: λ

text: needed

text: and

text: range

text: of

text: top

text: mass

text: (inside

text: the

text: reheating

text: temperature,

text: that

text: reduce

text: the

text: region

text: of

text: metasta-

text: experimental

text: 3σ

text: band)

text: required.

text: bility.

text: As

text: concrete

text: examples

text: we

text: illustrate

text: the

text: cases

text: of

text: T

text: and

text: T

text: GeV.

text: The

text: boundary

text: of

text: the

text: R

text: R

text: reduced

text: region

text: is

text: shown

text: as

text: the

text: dashed

text: lines

text: in

text: Fig.

text: possibly

text: an

text: extra

text: dimension

text: decompactiﬁes

text: In

text: clock-

text: (see

text: refs.

text: and

text: a

text: possible

text: trajectory

text: illustrating

text: work

text: constructions

text: the

text: number

text: of

text: vacua

text: is

text: also

text: ﬁnite

text: as

text: this

text: case

text: is

text: shown

text: along

text: y

text: A

text: smaller

text: step

text: size

text: t

text: the

text: ﬁeld

text: range

text: is

text: compact.

text: δλ

text: is

text: suggested

text: for

text: naturally

text: explaining

text: the

text: dis-

text: We

text: that

text: in

text: V

text: there

text: a

text: coupling

text: tance

text: of

text: the

text: SM

text: point

text: from

text: the

text: dashed

text: boundary.

text: This

text: of

text: a

text: to

text: the

text: Higgs

text: potential:

text: mechanism,

text: for

text: such

text: large

text: T

text: favors

text: the

text: lower

text: range

text: of

text: R

text: V

text: µ2

text: c

text: η

text: aΛ

text: λ

text: c

text: η

text: a

text: the

text: top

text: mass,

text: with

text: Mt

text: GeV.

text: h

text: h

text: h

text: λ

text: Λ

text: In

text: case

text: we

text: end

text: very

text: close

text: to

text: the

text: stability

text: boundary

text: beyond

text: which

text: the

text: Higgs

text: vacuum

text: is

text: unstable

text: against

text: decay

text: H

text: Such

text: couplings

text: also

text: break

text: the

text: shift

text: symmetry

text: and

text: so

text: we

text: during

text: inﬂation,

text: for

text: suﬃciently

text: large

text: values

text: of

text: N

text: I

text: e

text: assign

text: a

text: factor

text: η

text: to

text: them.

text: The

text: a-terms

text: in

text: could

text: be

text: This

text: case

text: is

text: illustrated

text: by

text: the

text: trajectory

text: passing

text: through

text: regarded

text: as→arising

text: frOom

text: a|

text: ge|

text: ne|ral|ization

text: of

text: eq.

text: by

text: the

text: bottom

text: of

text: the

text: experimental

text: ellipse.

text: Here

text: the

text: most

text: taking

text: H

text: H

text: or

text: from

text: expanded

text: probable

text: state

text: would

text: be

text: the

text: one

text: closest

text: to

text: the

text: boundary

text: cos(a/F)

text: potentials

text: in

text: the

text: clockwork

text: In

text: the

text: in

text: the

text: absolute

text: stability

text: region,

text: and

text: it

text: would

text: require

text: a

text: landscape

text: of

text: vacua

text: of

text: the

text: a

text: ﬁeld,

text: where

text: a

text: an

text: very

text: small

text: step

text: size

text: δλ

text: to

text: be

text: naturally

text: close

text: to

text: the

text: ex-

text: this

text: shifts

text: the

text: bare

text: values

text: (i.e.,

text: the

text: values

text: at

text: the

text: UV

text: scale

text: perimental

text: ellipse.

text: Although

text: this

text: possibility

text: is

text: currently

text: Λ)

text: of

text: the

text: Higgs

text: parameters

text: to

text: disfavored,

text: it

text: is

text: not

text: excluded

text: and

text: provides

text: another

text: possi-

text: ble

text: regime

text: for

text: explaining

text: near-critical

text: stability,

text: if

text: the

text: top

text: µ2

text: µ2

text: n

text: c

text: η

text: n

text: h

text: h

text: mass

text: is

text: very

text: close

text: to

text: its

text: lowest

text: 3σ

text: value,

text: M

text: GeV.

text: t

text: Once

text: δλ

text: is

text: ﬁxed,

text: can

text: be

text: used

text: to

text: to

text: eliminate

text: the

text: λ

text: λ

text: n

text: c

text: η

text: λ

text: n

text: δλ

text: n

text: λ

text: Λ

text: unknown

text: parameter

text: η

text: in

text: terms

text: of

text: f

text: and

text: δλ.

text: We

text: introduce

text: the

text: ratio

text: δ

text: [which

text: is

text: of

text: order

text: unity

text: in

text: case

text: Here

text: we

text: assume

text: that

text: some

text: other

text: mechanism

text: solves

text: the

text: λ

text: to

text: allow

text: for

text: the

text: possibility

text: of

text: any

text: of

text: the

text: three

text: cases.

text: weak

text: scale

text: hierarchy

text: problem

text: (e.g.

text: a

text: relaxion

text: mechanism

text: Hence

text: so

text: that

text: µ

text: is

text: of

text: electroweak

text: size

text: and

text: focus

text: on

text: the

text: n

text: shift

text: in

text: the

text: Higgs

text: coupling.

text: For

text: reasons

text: detailed

text: below

text: Λ

text: c

text: η

text: δ

text: we

text: also

text: choose

text: cλ

text: Likewise

text: we

text: must

text: assume

text: there

text: is

text: λ

text: λ

text: another

text: mechanism

text: for

text: solving

text: the

text: cosmological

text: constant

text: problem,

text: since

text: the

text: vacuum

text: energy

text: varies

text: between

text: a-vacua

text: PROBABILITY

text: DISTRIBUTION

text: OF

text: VACUA

text: due

text: to

text: the

text: nonperiodic

text: part

text: of

text: the

text: potential

text: V

text: We

text: consider

text: three

text: possible

text: scenarios,

text: each

text: associated

text: to

text: A

text: key

text: ingredient

text: of

text: our

text: scenario

text: is

text: the

text: process

text: by

text: which

text: one

text: of

text: the

text: three

text: critical

text: boundaries

text: shown

text: in

text: ﬁg.

text: these

text: the

text: vacua

text: get

text: populated

text: by

text: quantum

text: ﬂuctuations

text: during

text: are

text: the

text: boundaries

text: of

text: instability

text: and

text: metastability

text: at

text: zero

text: inﬂation,

text: and

text: the

text: resulting

text: probability

text: distribution

text: func-

text: temperature,

text: and

text: the

text: boundary

text: of

text: high-temperature

text: in-

text: tion

text: P(t,

text: a

text: for

text: the

text: diﬀerent

text: vacua.

text: It

text: is

text: governed

text: by

text: n

text: stability

text: that

text: depends

text: upon

text: the

text: assumed

text: reheating

text: tem-

text: perature

text: (dashed

text: lines).

text: Our

text: mechanism

text: explains

text: why

text: we

text: would

text: observe

text: (λ,

text: y

text: to

text: be

text: near

text: (and

text: to

text: the

text: right

text: t

text: of)

text: one

text: of

text: these

text: boundaries.

text: The

text: characteristics

text: of

text: the

text: This

text: number

text: can

text: be

text: estimated

text: as

text: follows.

text: The

text: vacuum

text: decay

text: three

text: categories

text: are

text: summarized

text: in

text: table

text: I.

text: Fig.

text: shows

text: rate

text: per

text: unit

text: volume

text: is

text: Γ

text: where

text: ht

text: is

text: the

text: trajectories

text: of

text: successive

text: vacua

text: that

text: exemplify

text: each

text: case.

text: preferred

text: value

text: for

text: tunneling.

text: The

text: decay

text: probability

text: is

text: Γ

text: times

text: Which

text: one

text: of

text: the

text: three

text: is

text: actually

text: realized

text: depends

text: upon

text: the

text: volume

text: of

text: our

text: past

text: light-cone∼

text: Decay

text: cosmological

text: parameters,

text: as

text: we

text: will

text: discuss

text: in

text: more

text: detail

text: probabilities

text: of

text: order

text: one

text: require

text: λ(ht)

text: and

text: this

text: number

text: is

text: conﬁrmed

text: by

text: a

text: more

text: sophisticated

text: calculation

text: (see

text: e.g.

text: in

text: the

text: next

text: section.

text: Thus

text: the

text: metastable

text: region

text: is

text: approximately

text: λ(ht)

text: In

text: case

text: we

text: end

text: close

text: to

text: the

text: instability

text: boundary

text: and

text: This

text: translates

text: to

text: the

text: region

text: shown

text: in

text: Fig.

text: after

text: running

text: the

text: the

text: probability

text: to

text: live

text: in

text: vacua

text: beyond

text: that

text: boundary

text: is

text: couplings

text: down

text: to

text: the

text: weak

text: scale.

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text: 12 Substructuring preconditioners for elliptic-type equations subject to certain primal constraints across the subdomain interface. How to choose suitable constraints, which heavily depends on the considered models, is the key technique in the BDDC method, as in FETI-DP method. The BDDC algorithms solve linear systems of primal unknowns, in contrast to FETI-DP algorithms. Although the proposed method and the BDDC method were designed based on diﬀerent ideas, they possess the following common merits: (1) the coarse matrix has a favorable sparsity pattern since two coarse dofs are coupled in the coarse stiﬀness matrix only if both dofs appear together in at least one substructure; (2) all the subproblems to be solved preserve the positive deﬁniteness of the original problem; (3) the coarse stiﬀness matrix can be preconditioned more easily since it has almost the same structure as the original stiﬀness matrix. The proposed method and the BDDC method have two respective advantages: for the proposed method, the designs of the coarse solver and local solvers are uniﬁed and do not depend on the considered model, and the basis functions in the coarse space can be directly obtained, without solving minimization problems; for the BDDC method, no coarse mesh is involved so it is particularly attractive if any coarse mesh cannot directly generate a subspace of the solution, also the BDDC preconditioner in general has very fast convergence. 4. Analysis on the convergence of the preconditioner It is known that, when the simplest coarse solver A is used to the construction of d substucturing preconditioners, the condition number of the resulting preconditioned system is not nearly optimal except for some particular cases (the coeﬃcients in the considered equation have no large jump or the subdomains have no internal cross- point). Fortunately, this unsatisfactory condition number has no large inﬂuence on the convergence rate of the PCG iteration for solving the underlying system, provided that the number of the “bad” eigenvalues of the preconditioned system is small (refer to [52]). − Let V˜ (Ω) be a subspace of V (Ω), and let m = dim(V (Ω)) dim(V˜ (Ω)). h h 0 h h Assume that the number m is small and is independent of the mesh size h and 0 − the subdomain size d. We use λ (B 1A) to denote the minimal eigenvalue of − m0+1 − the restriction of B 1A on the subspace V˜ (Ω), and deﬁne κ (B 1A) as the − h m0+1 reduced condition number of B 1A associated with the subspace V˜ (Ω). Namely, h − − λ (B 1A) κm0+1(B 1A) = λ max (B−1A). m0+1 Then the convergence rate of the PCG method with the preconditioner B for solving − the system (2.3) is determined by the reduced condition number κ (B 1A) (see m0+1 ##LTLine##

label: paragraph

text: Substructuring

text: preconditioners

text: for

text: elliptic-type

text: equations

text: subject

text: to

text: certain

text: primal

text: constraints

text: across

text: the

text: subdomain

text: interface.

text: How

text: to

text: choose

text: suitable

text: constraints,

text: which

text: heavily

text: depends

text: on

text: the

text: considered

text: models,

text: is

text: the

text: key

text: technique

text: in

text: the

text: BDDC

text: method,

text: as

text: in

text: FETI-DP

text: method.

text: The

text: BDDC

text: algorithms

text: solve

text: linear

text: systems

text: of

text: primal

text: unknowns,

text: in

text: contrast

text: to

text: FETI-DP

text: algorithms.

text: Although

text: the

text: proposed

text: method

text: and

text: the

text: BDDC

text: method

text: were

text: designed

text: based

text: on

text: diﬀerent

text: ideas,

text: they

text: possess

text: the

text: following

text: common

text: merits:

text: the

text: coarse

text: matrix

text: has

text: a

text: favorable

text: sparsity

text: pattern

text: since

text: two

text: coarse

text: dofs

text: are

text: coupled

text: in

text: the

text: coarse

text: stiﬀness

text: matrix

text: only

text: if

text: both

text: dofs

text: appear

text: together

text: in

text: at

text: least

text: one

text: substructure;

text: all

text: the

text: subproblems

text: to

text: be

text: solved

text: preserve

text: the

text: positive

text: deﬁniteness

text: of

text: the

text: original

text: problem;

text: the

text: coarse

text: stiﬀness

text: matrix

text: can

text: be

text: preconditioned

text: more

text: easily

text: since

text: it

text: has

text: almost

text: the

text: same

text: structure

text: as

text: the

text: original

text: stiﬀness

text: matrix.

text: The

text: proposed

text: method

text: and

text: the

text: BDDC

text: method

text: have

text: two

text: respective

text: advantages:

text: for

text: the

text: proposed

text: method,

text: the

text: designs

text: of

text: the

text: coarse

text: solver

text: and

text: local

text: solvers

text: are

text: uniﬁed

text: and

text: do

text: not

text: depend

text: on

text: the

text: considered

text: model,

text: and

text: the

text: basis

text: functions

text: in

text: the

text: coarse

text: space

text: can

text: be

text: directly

text: obtained,

text: without

text: solving

text: minimization

text: problems;

text: for

text: the

text: BDDC

text: method,

text: no

text: coarse

text: mesh

text: is

text: involved

text: so

text: it

text: is

text: particularly

text: attractive

text: if

text: any

text: coarse

text: mesh

text: cannot

text: directly

text: generate

text: a

text: subspace

text: of

text: the

text: solution,

text: also

text: the

text: BDDC

text: preconditioner

text: in

text: general

text: has

text: very

text: fast

text: convergence.

text: Analysis

text: on

text: the

text: convergence

text: of

text: the

text: preconditioner

text: It

text: is

text: known

text: that,

text: when

text: the

text: simplest

text: coarse

text: solver

text: A

text: is

text: used

text: to

text: the

text: construction

text: of

text: d

text: substucturing

text: preconditioners,

text: the

text: condition

text: number

text: of

text: the

text: resulting

text: preconditioned

text: system

text: is

text: not

text: nearly

text: optimal

text: except

text: for

text: some

text: particular

text: cases

text: (the

text: coeﬃcients

text: in

text: the

text: considered

text: equation

text: have

text: no

text: large

text: jump

text: or

text: the

text: subdomains

text: have

text: no

text: internal

text: cross-

text: point).

text: Fortunately,

text: this

text: unsatisfactory

text: condition

text: number

text: has

text: no

text: large

text: inﬂuence

text: on

text: the

text: convergence

text: rate

text: of

text: the

text: PCG

text: iteration

text: for

text: solving

text: the

text: underlying

text: system,

text: provided

text: that

text: the

text: number

text: of

text: the

text: “bad”

text: eigenvalues

text: of

text: the

text: preconditioned

text: system

text: is

text: small

text: (refer

text: to

text: Let

text: V˜

text: (Ω)

text: be

text: a

text: subspace

text: of

text: V

text: (Ω),

text: and

text: let

text: m

text: dim(V

text: (Ω))

text: dim(V˜

text: (Ω)).

text: h

text: h

text: h

text: h

text: Assume

text: that

text: the

text: number

text: m

text: is

text: small

text: and

text: is

text: independent

text: of

text: the

text: mesh

text: size

text: h

text: and

text: the

text: subdomain

text: size

text: d.

text: We

text: use

text: λ

text: (B

text: to

text: denote

text: the

text: minimal

text: eigenvalue

text: of

text: the

text: restriction

text: of

text: B

text: on

text: the

text: subspace

text: V˜

text: (Ω),

text: and

text: deﬁne

text: κ

text: (B

text: as

text: the

text: h

text: reduced

text: condition

text: number

text: of

text: B

text: associated

text: with

text: the

text: subspace

text: V˜

text: (Ω).

text: Namely,

text: h

text: λ

text: (B

text: λ

text: max

text: Then

text: the

text: convergence

text: rate

text: of

text: the

text: PCG

text: method

text: with

text: the

text: preconditioner

text: B

text: for

text: solving

text: the

text: system

text: is

text: determined

text: by

text: the

text: reduced

text: condition

text: number

text: κ

text: (B

text: (see

text: ##LTLine##


text: 36 Substructuring preconditioners for elliptic-type equations

text: Substructuring

text: preconditioners

text: for

text: elliptic-type

text: equations


text: 8 Td,(cid:96),(cid:96)−1 + (cid:96)1 H(U((cid:96)1,(cid:96)+1)) Finally, let τ0 := {at : t ∈ [1 : r]} and d (cid:96) { ∈ − − − } (≥k) H(U((cid:96)1+1,(cid:96)+1)) + Td,d−,(cid:96)+1 H(U((cid:96)1,(cid:96)+1))+ τq :=∈ at : t [r + 1 + (q 1)(d (cid:96)) : r + q(d (cid:96))] d (cid:96) for an(cid:83)y q [1 : s]. It is straightforward to verify that: (≥l) (TTdd,d,(cid:96)d,,(cid:96)(cid:96)−1++(cid:96)Td(cid:96),1(cid:96),H(cid:96)1(+U((cid:96)(cid:96)11,)(cid:96)+(cid:88)1d)) Td−,j1,(cid:96)Bj+ ••• ττqssq∩=−=10τ[q(cid:96)τ(cid:48)q+==2∅[:1fdo:+r(cid:96)1a1]n]∪.y [q2(cid:96)(cid:54)=1 +q(cid:48);1 : (cid:96)]; j=(cid:96)+1 Consider a symmetrical (n = d+1, d, N1, . . . , Nd, T, S) code Td,d,(cid:96) +dT−d,(cid:96)(cid:96),(cid:96)1(cid:88)+ (cid:96)1 H(U((cid:96)1,(cid:96))) tshhaotwsabtyisﬁinedsucthtieonnothdaet froegreanneyraptio∈n[1re: qsu],irweme ehnatve(3). Let us (m=) (Td,d,(cid:96)1 + (cid:96)1) d Td−,j1,(cid:96)Bj + Td,dd,(cid:96)−1 +(cid:96) (cid:96)1 H(U((cid:96)1,(cid:96))) pH(U((cid:96)1,(cid:96))|M((cid:96))) + H(U((cid:96)1,(cid:96)1+1)|M((cid:96))) j=(cid:96)+1 where (a) follows from the fact that T +T +d− ≥ pH(U((cid:96)1,(cid:96)+1)|M((cid:96)))+ (cid:83) (cid:96)1 + dT−d,(cid:96)(cid:96)−,=(cid:96)1+Td1,dH,(cid:96)(1U;(((cid:96)b1),(cid:96))f)ol+loHws(Uf(r(cid:96)o1m,(cid:96))dC),(cid:96)o≥,(cid:96)r1o+ll1ary 3d,tdh,(cid:96)a+t 1 H(W[(cid:96)1+1:2(cid:96)1], S(cid:96)1→[(cid:96)1+1:2(cid:96)1], S sq−=p0 τq→(cid:96)+1|M((cid:96))()3.1) d (cid:96) Td,(cid:96)−,(cid:96)1+1 H(U((cid:96)1,(cid:96)+1)) + H(U((cid:96)1,(cid:96)1+1)) To prove the base case of p = 1, ﬁrst note that d (cid:96) | (cid:48) H(U((cid:96)1,(cid:96)) M((cid:96))) by setting j1 = (cid:96), j2 = (cid:96)1 + 1 and i = i = (cid:96)1 in 12; (c) follows from (17) in Lemma 5; (d) follows from the fact (=a) H(U((cid:96)1,(cid:96)), W , S→ |M((cid:96))) tthhaatt THd(,W(cid:96),(cid:96)(cid:96)1++11)+≤dα−; (cid:96)(1f)=anTdd,(cid:96)(,h(cid:96)1);fo(ell)owfosllofrwosmfrtohme ftahcet tfhaactt = H(W[1:(cid:96)1], S→[[1(cid:96):1(cid:96)1+]1:(cid:96)]|M[(cid:96)(1(cid:96)+))1:(cid:96)] Sfa(cid:96)c1t+t1h→at[1H:(cid:96)(1W] is a, Sfuncti→on of )W+(cid:96)H1+(U1;((cid:96)1(g,(cid:96))1)f,oSllows→from )th≥e (=b) H(W[1:(cid:96)], S→[(cid:96)1+1:(cid:96)], S[1:(cid:96)]→(cid:96)+1|M((cid:96))) (cid:96)+1 (cid:96)1+1 [1:(cid:96)1] (cid:96)1+1 [1:(cid:96)1] where (a) follows from the fact that (W , S→ ) is a H(S → )+H(U((cid:96)1+1,(cid:96)1+1), S → ) due to sub- [(cid:96)1:(cid:96)] [(cid:96)1+1:(cid:96)] mod(cid:96)u1l+ar1ity;[1(:i(cid:96))1]follows from (22) in P(cid:96)r1o+po1sit[i1o:(cid:96)n1]2; (j) follows function of U((cid:96)1,(cid:96)) by Lemma 1, and (b) follows from the fact from the fact that T = T + T ; (k) that S[1:(cid:96)]→(cid:96)+1 is a function of W[1:(cid:96)]. Furthermore, follows−from ≥the facdt,sd,(cid:96)t1h+a1t Td,(cid:96)+d1,d,(cid:96),(cid:96)1++11β ≥d,(cid:96)S+→1,[(cid:96)(cid:96)11++12:(cid:96)+1] H(U((cid:96)1,(cid:96)1+1)|M((cid:96))) and (d (cid:96))β S→ ; (l) follows from (19) and (27) of Proposition 1 and 3(cid:96)r+e1spectively; (m) follows from the fact (=a) H(U((cid:96)1,(cid:96)1+1), S→(cid:96)1+1|M((cid:96))) tfhroamt(cid:80)Tdb,odt,h(cid:96)+siTdde,s(cid:96),(cid:96)o1f=thTedi,nd,e(cid:96)q1u.aCliatnycaenlldingnoTrmd,dad,l−(cid:96)i1z(cid:96)+in(cid:96)g1 Hbo(tUh((cid:96)s1id,(cid:96)e)s) = H(W[1:(cid:96)1], S→(cid:96)1+1|M((cid:96))) by dt=(cid:96)+1 Bt complete the proof of (9). (=b) H(W[(cid:96)1+1:2(cid:96)1], S→2(cid:96)1+1|M((cid:96))) (=c) H(W , S→ |M((cid:96))) [(cid:96)1+1:2(cid:96)1] (cid:96)+1 V. CONCLUDING REMARKS (d) = H(W , S → , S → , [(cid:96)1+1:2(cid:96)1] [1:(cid:96)1] (cid:96)+1 [2(cid:96)1+1:(cid:96)] (cid:96)+1 This paper considered the problem of MDC-SR with a S → , S → |M((cid:96))) generalized eavesdropping model. It was shown that the MBR [(cid:96)+2:d+1] (cid:96)+1 (cid:96)+1 [(cid:96)1+1:2(cid:96)1] point of the achievable normalized storage-capacity repair- where (a) follows from the fact that S→(cid:96)1+1 is a function bandwidth tradeoff region depends on the numbers of type I of U((cid:96)1,(cid:96)1+1) by Lemma 1, and (b) and (c) follow from and type II compromised storage nodes only via their total, as the symmetrical code that we consider; (d) follows that long as the number of type I compromised nodes is less than S → is a function of S→ . It follows that (cid:96)+1 [(cid:96)1+1:2(cid:96)1] (cid:96)+1 or equal to the number of type II compromised nodes. Moving | | H(U((cid:96)1,(cid:96)) M((cid:96))) + H(U(cid:96)1,(cid:96)1+1 M((cid:96))) forward, it would be interesting to see whether this result ≥ | extends to the entire achievable normalized storage-capacity H(W , S→ , S → M((cid:96)))+ [1:(cid:96)] [(cid:96)1+1:(cid:96)] [1:(cid:96)] (cid:96)+1 repair-bandwidth tradeoff region. H(W , S → , S → , APPENDIX [(cid:96)1+1:2(cid:96)1] [1:(cid:96)1] (cid:96)+1 [2(cid:96)1:|(cid:96)] (cid:96)+1 PROOF OF THE EXCHANGE LEMMA II S[(cid:96)+2:d+1]→(cid:96)+1, S(cid:96)+1→[(cid:96)1+1:2(cid:96)1] M((cid:96))) s(dF−irs(cid:96)t)+nortefotrhastomde−in(cid:96)1teg>erds−≥(cid:96),1 saondwre∈m[a1y: dw−rit(cid:96)e]. dN−ex(cid:96)t1, le=t (≥a) HS(W[1→:(cid:96)], S→, S[(cid:96)1+1:(cid:96)], → |M((cid:96)))+ ∈ [1:(cid:96)] (cid:96)+1 [(cid:96)+2:d+1] (cid:96)+1 at :=  ttt,++ (cid:96)j,1, ttt ∈∈ [[[1(cid:96)(cid:96)1−:+(cid:96)(cid:96)111] +: (cid:96)1−: (cid:96)d1−] (cid:96)]. HS(cid:96)(+W1→[(cid:96)1[(cid:96)+11+:21(cid:96):21(cid:96)]1, S]|[M1:((cid:96)(cid:96)1)])→(cid:96)+1, S[2(cid:96)1:(cid:96)]→(cid:96)+1, ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Finally,

text: let

text: τ0

text: {at

text: t

text: r]}

text: and

text: d

text: (≥k)

text: τq

text: at

text: t

text: [r

text: (q

text: r

text: q(d

text: d

text: for

label: list

text: q

text: s].

text: It

text: is

text: straightforward

text: to

text: verify

text: that:

text: (≥l)

text: Consider

text: a

text: symmetrical

text: (n

text: d,

text: Nd,

text: T,

text: S)

text: code

text: tshhaotwsabtyisﬁinedsucthtieonnothdaet

text: qsu],irweme

text: Let

text: us

text: (m=)

text: d

text: where

text: (a)

text: follows

text: from

text: the

text: fact

text: that

text: T

text: +T

text: +d−

text: S

text: d

text: To

text: prove

text: the

text: base

text: case

text: of

text: p

text: ﬁrst

text: note

text: that

text: d

text: by

text: setting

text: and

text: i

text: i

text: in

text: (c)

text: follows

text: from

text: in

text: Lemma

text: (d)

text: follows

text: from

text: the

text: fact

text: (=a)

text: W

text: S→

text: tthhaatt

text: ftahcet

text: tfhaactt

text: is

text: a,

text: Sfuncti→on

text: of

text: )th≥e

text: (=b)

text: where

text: (a)

text: follows

text: from

text: the

text: fact

text: that

text: (W

text: S→

text: is

text: a

text: H(S

text: S

text: due

text: to

text: sub-

text: from

text: in

text: (j)

text: follows

text: function

text: of

text: by

text: Lemma

text: and

text: (b)

text: follows

text: from

text: the

text: fact

text: from

text: the

text: fact

text: that

text: T

text: T

text: T

text: (k)

text: that

text: is

text: a

text: function

text: of

text: Furthermore,

text: follows−from

text: ≥the

text: and

text: (d

text: S→

text: (l)

text: follows

text: from

text: and

text: of

text: Proposition

text: and

text: (m)

text: follows

text: from

text: the

text: fact

text: (=a)

text: by

text: Bt

text: complete

text: the

text: proof

text: of

text: (=b)

text: (=c)

text: H(W

text: S→

text: V.

text: CONCLUDING

text: REMARKS

text: (d)

text: H(W

text: S

text: S

text: This

text: paper

text: considered

text: the

text: problem

text: of

text: MDC-SR

text: with

text: a

text: S

text: S

text: generalized

text: eavesdropping

text: model.

text: It

text: was

text: shown

text: that

text: the

text: MBR

text: point

text: of

text: the

text: achievable

text: normalized

text: storage-capacity

text: repair-

text: where

text: (a)

text: follows

text: from

text: the

text: fact

text: that

text: is

text: a

text: function

text: bandwidth

text: tradeoff

text: region

text: depends

text: on

text: the

text: numbers

text: of

text: type

text: I

text: of

text: by

text: Lemma

text: and

text: (b)

text: and

text: (c)

text: follow

text: from

text: and

text: type

text: II

text: compromised

text: storage

text: nodes

text: only

text: via

text: their

text: total,

text: as

text: the

text: symmetrical

text: code

text: that

text: we

text: consider;

text: (d)

text: follows

text: that

text: long

text: as

text: the

text: number

text: of

text: type

text: I

text: compromised

text: nodes

text: is

text: less

text: than

text: S

text: is

text: a

text: function

text: of

text: S→

text: It

text: follows

text: that

text: or

text: equal

text: to

text: the

text: number

text: of

text: type

text: II

text: compromised

text: nodes.

text: Moving

text: forward,

text: it

text: would

text: be

text: interesting

text: to

text: see

text: whether

text: this

text: result

text: extends

text: to

text: the

text: entire

text: achievable

text: normalized

text: storage-capacity

text: H(W

text: S→

text: S

text: repair-bandwidth

text: tradeoff

text: region.

text: H(W

text: S

text: S

text: APPENDIX

text: PROOF

text: OF

text: THE

text: EXCHANGE

text: LEMMA

text: II

text: le=t

text: (≥a)

text: S→,

text: at

text: ttt,++

text: ttt

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 10 which is equivalent to (11) by noting that − − s + −r = s(d −(cid:96)) + r = d −(cid:96)1 . d (cid:96) d (cid:96) d (cid:96) This completes the proof of the exchange lemma. REFERENCES [1] S. Shao, T. Liu, C. Tian and C. Shen, ”Multilevel diversity coding with secure regeneration: Separate coding achieves the MBR point,” Preprint. [Online] https://arxiv.org/pdf/1712.03326.pdf [2] C. Tian and T. Liu, “Multilevel diversity coding with regeneration,” IEEE Trans. Inf. Theory, vol. 62, no. 9, pp. 4833–4847, Sep. 2016. [3] S. Shao, T. Liu, and C. Tian, “Multilevel diversity coding with regen- eration: Separate coding achieves the MBR point,” in Proc. Ann. Conf. Inf. Sci. Systems (CISS), Princeton, NJ, USA, Mar. 2016, pp. 602–607. [4] S. Pawar, S. El Rouayheb, and K. Ramchandran, “On secure distributed data storage under repair dynamics,” in Proc. IEEE Int. Sym. Inf. Theory (ISIT), Austin, TX, USA, Jun. 2010, pp. 2543–2547. [5] S. Pawar, S. El Rouayheb, and K. Ramchandran, “Securing dynamic dis- tributed storage systems against eavesdropping and adversarial Attacks,” IEEE Trans. Inf. Theory, vol. 57, no. 10, pp. 6734–6753, Oct. 2011. [6] N. B. Shah, K. V. Rashmi, and P. V. Kumar, “Information-theoretically secure regenerating codes for distributed storage,” in Proc. IEEE Glo. Tel. Conference (GLOBECOM), Houston, TX, USA, Dec. 2011, pp. 1–5. [7] S. Goparaju, S. El Rouayheb, R. Calderbank, and H. V. Poor, “Data secrecy in distributed storage systems under exact repair,” in Proc. IEEE Int. Sym. Net. Coding (NetCod), Calgary, AB, Canada, Jun. 2013, pp. 1– 6. [8] A. S. Rawat, O. O. Koyluoglu, N. Silberstein, and S. Vishwanath, “Opti- mal locally repairable and secure codes for distributed storage systems”, IEEE Trans. Inf. Theory. vol. 60, no. 1, pp. 212–236, Jan. 2014. [9] R. Tandon, S. Amuru, T. C. Clancy, and R. M. Buehrer, “Towards optimal secure distributed storage systems with exact repair,” IEEE Trans. Inf. Theory, vol. 62, no. 6, pp. 3477–3492, Jun. 2016. [10] F. Ye, K. W. Shum, and R. W. Yeung, ”The Rate Region for Secure Distributed Storage Systems,” in IEEE Trans. on Inf. Theory, vol. 63, no. 11, pp. 7038-7051, Nov. 2017. [11] S. Shao, T. Liu, C. Tian, and C. Shen, “On the tradeoff region of secure exact-repair regenerating codes,” IEEE Trans. Inf. Theory, vol. 63, no. 11, pp. 7253–7266, Nov. 2017. [12] C. Tian, “Characterizing the rate region of the (4,3,3) exact-repair regenerating codes,” IEEE J. Sel. Are. Communications, vol. 32, no. 5, pp. 967–975, May 2014. [13] T. S. Han, “Nonnegative entropy measures of multivariate symmetric correlations,” Inf. Control, vol. 36, no. 2, pp. 133–156, Feb. 1978. ##LTLine## ##LTLine## ##LTLine##

text: which

text: is

text: equivalent

text: to

text: by

text: noting

text: that

text: s

text: −r

text: s(d

text: r

text: d

text: d

text: d

text: d

text: This

text: completes

text: the

text: proof

text: of

text: the

text: exchange

text: lemma.

text: REFERENCES

label: reference

text: S.

text: Shao,

text: T.

text: Liu,

text: C.

text: Tian

text: and

text: C.

text: Shen,

text: ”Multilevel

text: diversity

text: coding

text: with

text: secure

text: regeneration:

text: Separate

text: coding

text: achieves

text: the

text: MBR

text: point,”

text: Preprint.

text: [Online]

text: C.

text: Tian

text: and

text: T.

text: Liu,

text: “Multilevel

text: diversity

text: coding

text: with

text: regeneration,”

text: IEEE

text: Trans.

text: Inf.

text: Theory,

text: vol.

text: no.

text: pp.

text: Sep.

text: S.

text: Shao,

text: T.

text: Liu,

text: and

text: C.

text: Tian,

text: “Multilevel

text: diversity

text: coding

text: with

text: regen-

text: eration:

text: Separate

text: coding

text: achieves

text: the

text: MBR

text: point,”

text: in

text: Proc.

text: Ann.

text: Conf.

text: Inf.

text: Sci.

text: Systems

text: (CISS),

text: Princeton,

text: NJ,

text: USA,

text: Mar.

text: pp.

text: S.

text: Pawar,

text: S.

text: El

text: Rouayheb,

text: and

text: K.

text: Ramchandran,

text: “On

text: secure

text: distributed

text: data

text: storage

text: under

text: repair

text: dynamics,”

text: in

text: Proc.

text: IEEE

text: Int.

text: Sym.

text: Inf.

text: Theory

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text: S.

text: Naoz

text: et

text: al.

text: ate

text: from

text: the

text: physical

text: one,

text: but

text: also

text: the

text: disk’s

text: members

text: Eisenhauer,

text: F.,

text: Genzel,

text: R.,

text: Alexander,

text: T.,

text: Abuter,

text: R.,

text: Paumard,

text: semi-major

text: axis.

text: Ignoring

text: the

text: contribution

text: from

text: binary

text: T.,

text: Ott,

text: T.,

text: Gilbert,

text: A.,

text: Gillessen,

text: S.,

text: Horrobin,

text: M.,

text: Trippe,

text: S.,

text: motion

text: will

text: cause

text: many

text: disk

text: members

text: to

text: appear

text: more

text: Bonnet,

text: H.,

text: Dumas,

text: C.,

text: Hubin,

text: N.,

text: Kaufer,

text: A.,

text: Kissler-Patig,

text: M.,

text: Monnet,

text: G.,

text: Stro¨bele,

text: S.,

text: Szeifert,

text: T.,

text: Eckart,

text: A.,

text: Sch¨odel,

text: distant

text: from

text: the

text: SMBH

text: than

text: they

text: actually

text: are

text: (see

text: Fig-

text: R.,

text: Zucker,

text: S.

text: ApJ,

text: ure

text: left

text: panel).

text: ApJ,

text: We

text: deﬁned

text: a

text: disk

text: membership

text: condition

text: using

text: a

text: quan-

text: Feldmeier-Krause,

text: A.,

text: Neumayer,

text: N.,

text: Sch¨odel,

text: R.,

text: Seth,

text: A.,

text: tity

text: that

text: roughly

text: describes

text: an

text: angular

text: distance

text: from

text: the

text: Hilker,

text: M.,

text: de

text: Zeeuw,

text: P.

text: T.,

text: Kuntschner,

text: H.,

text: Walcher,

text: C.

text: J.,

text: disk

text: location

text: on

text: the

text: sky,

text: β,

text: in

text: the

text: longitude

text: of

text: ascend-

text: Lu¨tzgendorf,

text: N.,

text: Kissler-Patig,

text: M.

text: A&A,

text: Genzel,

text: R.,

text: Sch¨odel,

text: R.,

text: Ott,

text: T.,

text: Eisenhauer,

text: F.,

text: Hofmann,

text: R.,

text: ing

text: nodes,

text: Ω

text: and

text: the

text: inclination,

text: i,

text: plane.

text: We

text: estimate

text: Lehnert,

text: M.,

text: Eckart,

text: A.,

text: Alexander,

text: T.,

text: Sternberg,

text: A.,

text: Lenzen,

text: the

text: disk

text: membership

text: by

text: considering

text: the

text: systems

text: that

text: lay

text: R.,

text: Cl´enet,

text: Y.,

text: Lacombe,

text: F.,

text: Rouan,

text: D.,

text: Renzini,

text: A.,

text: within

text: (i.e.,

text: corresponding

text: to

text: of

text: systems

text: around

text: Tacconi-Garman,

text: L.

text: E.

text: ApJ,

text: the

text: initial

text: location

text: of

text: the

text: disk).

text: As

text: expected,

text: we

text: ﬁnd

text: that

text: Ghez,

text: A.

text: M.,

text: Duchˆene,

text: G.,

text: Matthews,

text: K.,

text: Hornstein,

text: S.

text: D.,

text: the

text: disk

text: fraction

text: decreases

text: as

text: a

text: function

text: of

text: the

text: projected

text: Tanner,

text: A.,

text: Larkin,

text: J.,

text: Morris,

text: M.,

text: Becklin,

text: E.

text: E.,

text: Salim,

text: S.,

text: Kremenek,

text: T.,

text: Thompson,

text: D.,

text: Soifer,

text: B.

text: T.,

text: Neugebauer,

text: G.,

text: distance

text: (e.g.,

text: Figure

text: bottom

text: panel).

text: This

text: trend

text: is

text: McLean,

text: I.

text: ApJ,

text: consistent

text: with

text: the

text: observations,

text: as

text: shown

text: in

text: that

text: Fig-

text: Ghez,

text: A.

text: M.,

text: Salim,

text: S.,

text: Hornstein,

text: S.

text: D.,

text: Tanner,

text: A.,

text: Lu,

text: J.

text: R.,

text: ure,

text: black

text: line.

text: This

text: functional

text: agreement,

text: suggests

text: that

text: Morris,

text: M.,

text: Becklin,

text: E.

text: E.,

text: Duchˆene,

text: G.

text: ApJ,

text: binaries

text: are

text: more

text: prevalent

text: in

text: the

text: Galactic

text: Center

text: and

text: Ghez,

text: A.

text: M.,

text: Salim,

text: S.,

text: Weinberg,

text: N.

text: N.,

text: Lu,

text: J.

text: R.,

text: Do,

text: T.,

text: Dunn,

text: may

text: result

text: in

text: reducing

text: the

text: disk

text: membership.

text: Taking

text: on

text: J.

text: K.,

text: Matthews,

text: K.,

text: Morris,

text: M.

text: R.,

text: Yelda,

text: S.,

text: Becklin,

text: E.

text: E.,

text: Kremenek,

text: T.,

text: Milosavljevic,

text: M.,

text: Naiman,

text: J.

text: ApJ,

text: face

text: value

text: our

text: initial

text: conditions

text: and

text: comparing

text: our

text: re-

text: sults

text: to

text: observatio∼ns,

text: may

text: suggest

text: that

text: disk

text: membership

text: Gillessen,

text: S.,

text: Eisenhauer,

text: F.,

text: Trippe,

text: S.,

text: Alexander,

text: T.,

text: Genzel,

text: can

text: be

text: as

text: high

text: as

text: R.,

text: Martins,

text: F.,

text: Ott,

text: T.

text: ApJ,

text: We

text: conclude

text: that

text: the

text: possibility

text: of

text: existing

text: binaries

text: Gillessen,

text: S.,

text: Plewa,

text: P.

text: M.,

text: Eisenhauer,

text: F.,

text: Sari,

text: R.,

text: Waisberg,

text: I.,

text: Habibi,

text: M.,

text: Pfuhl,

text: O.,

text: George,

text: E.,

text: Dexter,

text: J.,

text: von

text: Fellenberg,

text: in

text: the

text: measurements

text: of

text: the

text: stellar

text: disk

text: properties

text: can-

text: S.,

text: Ott,

text: T.,

text: Genzel,

text: R.

text: ApJ,

text: not

text: be

text: ignored.

text: As

text: we

text: showed,

text: many

text: of

text: the

text: puzzles

text: and

text: Ginsburg,

text: I.

text: Loeb,

text: A.

text: MNRAS,

text: controversies

text: involved

text: the

text: characteristics

text: of

text: the

text: disk

text: may

text: Hills,

text: J.

text: G.

text: Nature,

text: be

text: explained

text: by

text: the

text: existence

text: of

text: binaries.

text: We

text: encour-

text: Hoang,

text: B.-M.,

text: Naoz,

text: S.,

text: Kocsis,

text: B.,

text: Rasio,

text: F.

text: A.,

text: Dosopoulou,

text: age

text: the

text: community

text: to

text: take

text: more

text: measurements

text: of

text: the

text: F.

text: ArXiv

text: e-prints

text: Hobbs,

text: A.

text: Nayakshin,

text: S.

text: MNRAS,

text: z-component

text: of

text: the

text: stars’

text: velocities.

text: Kouwenhoven,

text: M.

text: B.

text: N.,

text: de

text: Grijs,

text: R.

text: A&A,

text: Levin,

text: Y.

text: MNRAS,

text: Levin,

text: Y.

text: Beloborodov,

text: A.

text: M.

text: ApJ,

text: We

text: thank

text: the

text: referee

text: for

text: his/her

text: useful

text: comments.

text: S.N

text: Li,

text: G.,

text: Ginsburg,

text: I.,

text: Naoz,

text: S.,

text: Loeb,

text: A.

text: ApJ,

text: and

text: A.M.G

text: thank

text: the

text: Keck

text: foundation

text: for

text: their

text: partial

text: Lu,

text: J.

text: R.,

text: Do,

text: T.,

text: Ghez,

text: A.

text: M.,

text: Morris,

text: M.

text: R.,

text: Yelda,

text: S.,

text: Matthews,

text: K.

text: ApJ,

text: support

text: of

text: the

text: NStarsOrbits

text: Project.

text: S.N.

text: acknowledges

text: Lu,

text: J.

text: R.,

text: Ghez,

text: A.

text: M.,

text: Hornstein,

text: S.

text: D.,

text: Morris,

text: M.

text: R.,

text: Becklin,

text: partial

text: support

text: from

text: a

text: Sloan

text: Foundation

text: Fellowship.

text: E.

text: E.,

text: Matthews,

text: K.

text: ApJ,

text: A.M.G

text: thanks

text: the

text: NSF

text: grant

text: number

text: Martins,

text: F.,

text: Trippe,

text: S.,

text: Paumard,

text: T.,

text: Ott,

text: T.,

text: Genzel,

text: R.,

text: Rauw,

text: G.,

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text: The kinematics of OB associations 3 Table 1. Distances (D) and measured dispersions for each cluster in the ﬁve dimensions used in the selection criteria of Section 2.1: dispersions in the plane of the sky (σX,σY), dispersion in the line-of-sight distance derived from TGAS parallaxes (σZ), and velocity dispersions in the plane of the sky (σvX , σvY ). OB assoc. kDpc σpXc σpYc σpcZ kσmvsX−1 kσmvsY−1 CamOB1 0.82 15 9 12 8 8 CasOB14 0.85 12 9 27 13 5 CepOB2 0.71 19 8 11 10 6 CepOB3 0.68 35 14 14 11 6 CepOB4 0.72 7 16 6 12 7 Coll140 0.35 45 7 28 7 7 Figure 1. Relative uncertainties in the derived distances against distance Coll359 0.20 22 7 9 8 7 CygOB7 0.68 14 12 46 11 6 based on parallax measurements for the entire TGAS catalogue. The dashed CygOB9 0.94 15 22 13 10 3 lines mark the 3σ distance uncertainty cut-oﬀ imposed in this work, moti- MonOB1 0.60 3 14 32 7 6 vating a maximum distance cut of 1.6kpc. Only those stars that fall in the OriOB1 0.40 14 28 22 4 5 lowerleftquadrantofthisFigureareconsideredinthisstudy.Thisdeﬁnesthe PerOB3 0.18 23 8 31 12 10 parent sample from which OB associations will be extracted based on their ScoOB2 0.15 13 10 12 3 3 clustering in 5-dimensional phase space, yielding much smaller numbers of SctOB3 1.25 26 67 45 9 7 stars (N∗ = 100-400). SgrOB1 1.27 38 87 29 7 4 Tr27 0.90 10 15 18 8 10 sky from the median position of the OB stars in the sample in pc. VelOB2 0.38 30 14 42 6 6 Likewise, the same distances are used to convert observed proper VulOB4 0.80 30 13 16 8 10 − motions from mas/yr to physical velocities in kms 1. We deﬁne over-abundances of OB stars as regions with peak number densities of at least three times the observed background level of O- and B- on a reasonable determination of the centres of the OB associations. type stars measured across the 150pc radius region with bin sizes The centres of mass of the associations require known masses for all of 2pc. We then determine the position and size of the associations association members, which are not available. Therefore we deﬁne in each axis by ﬁtting Gaussian proﬁles to the over-densities in each two association centres and use both in the subsequent analysis. The o| f|theA5ssdoicmiaetniosinonmseamv≡baielrasbal−er:eXse,lYec,tDed, vaxs≡,saantidsf−vyyin. g |x| < σX and ﬁporsstitcieonntroefitshdeeOteBrmsitnaresd-tothbise itsheofmceoaunrs4e-dbiimaseendsitoonwaalr(dXs,Yth,evxm,voys)t y < σY, where x X µX and y Y µY. A strict 1σ massive and least numerous sources in the association. The prop- criterion is imposed where possible in X andY in order to minimise erties of the OB associations using the mean position and velocity the likelihood of selecting contaminating ﬁeld stars as association of the OB-type stars are given in Table 2. The second centre is set members. In cases where the 1 criterion did not yield at least 100 σ as the mean 4-dimensional position of all stars in the association, stars including at least 10 O- or B- type stars, a 3 criterion was σ biased towards the more numerous low mass stars. Table 3 shows imposed and if insuﬃcient numbers of stars were selected within the properties of the OB associations when using the mean position 3 the association was excluded from the sample. Similarly, we σ | | ( and velocity of all stars. Given that one centre is biased towards imp)o/se a line-of-s≡ight d−istance selection criterion of Z < ∆D2 + low mass stars and the other is biased towards high mass stars, it fσroZ2m1t2hewuhnecreertZaintyDin thµeDTGanAdS∆pDarailslatxhemuenacseurrteaminetnyt.aTs hdiesrtihveedn is expected that the centre of mass of each association lies between these two points. removes the outliers in line-of-sight distance from the sample. The Once the velocities are in the association centre frame, the dispersions used to carry out the above selection criteria are listed radial velocities are determined by taking the dot product of the vr in Table 1. velocity vector with the radial unit vector. The tangential velocity sociaAtiosnwmeeamrebeirnst,ekreinsteemdaitnictohuetlkieinrsemwahtiicchparroepnerottietsruoefatshseocaias-- is then obtained by subtraction as vt = (v2 − vr2)1/2. We deﬁne a positive radial velocity as moving away from the centre of the OB tion members are potentially a serious problem as they can artiﬁ- association and a negative value as moving towards the centre of the cially skew the diagnostics employed in Section 3. For this reason, | | | | association. Similarly, we (arbitrarily) deﬁne the tangential velocity we impose a velocity cut of 3 and 3 with ≡ − and ≡ v−x < , σrevmXoving vthye m<ostσevxYtreme as being positive if the motion is anti-clockwise with respect to the vvexlocitvyXsourµcveXs whilevymainvtaYininµgvtYhe vast majority of the sample. previously determined association centre and negative if the motion is clockwise with respect to the association centre. OB associations were rejected if they did not contain at least 100 2.2 Model distributions stars including at least 10 O- or B-type stars within the 3 limit. σ This yields a ﬁnal sample of 18 OB associations. The number of se- In order to assess whether the observed kinematics are con- lected members and the number of dispersions in X,Y and distance sistent with a net radial expansion, we compare our results for used to select the members of each association are listed in Tables the observed OB associations with those of randomly generated 2 and 3. These kinematic criteria typically ﬁlter out less than 1% of velocity distributions. Six sets of models will be calculated: two the stars in each association, implying that it has a limited inﬂuence distributions representing random velocity ﬁelds, two distributions on the diagnostics presented in Section 3. representing localised expansion within associations, and two dis- The position-velocity analysis employed in this work depends tributions representing globally expanding velocity ﬁelds. These MNRAS 000, 1–18 (2017) ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine##

text: The

font: WPGKXI+TeXGyreTermes-Italic

text: kinematics

text: of

text: OB

text: associations

font: LPOWXB+TeXGyreTermes-Regular

text: Table

font: VMXCSC+TeXGyreTermes-Bold

text: Distances

label: caption

text: (D)

text: and

text: measured

text: dispersions

text: for

text: each

text: cluster

text: in

text: the

text: ﬁve

text: dimensions

text: used

text: in

text: the

text: selection

text: criteria

text: of

text: Section

text: dispersions

text: in

text: the

text: plane

text: of

text: the

text: sky

text: (σX,σY),

text: dispersion

text: in

text: the

text: line-of-sight

text: distance

text: derived

text: from

text: TGAS

text: parallaxes

text: (σZ),

text: and

text: velocity

text: dispersions

text: in

text: the

text: plane

text: of

text: the

text: sky

text: (σvX

text: σvY

text: OB

text: assoc.

text: kDpc

text: σpXc

text: σpYc

text: σpcZ

text: Figure

text: Relative

text: uncertainties

text: in

text: the

text: derived

text: distances

text: against

text: distance

text: based

text: on

text: parallax

text: measurements

text: for

text: the

text: entire

text: TGAS

text: catalogue.

text: The

text: dashed

text: lines

text: mark

text: the

text: 3σ

text: distance

text: uncertainty

text: cut-oﬀ

text: imposed

text: in

text: this

text: work,

text: moti-

text: vating

text: a

text: maximum

text: distance

text: cut

text: of

text: Only

text: those

text: stars

text: that

text: fall

text: in

text: the

text: parent

text: sample

text: from

text: which

text: OB

text: associations

text: will

text: be

text: extracted

text: based

text: on

text: their

text: clustering

text: in

text: phase

text: space,

text: yielding

text: much

text: smaller

text: numbers

text: of

text: stars

text: (N∗

text: sky

text: from

text: the

text: median

text: position

text: of

text: the

text: OB

text: stars

text: in

text: the

text: sample

text: in

text: pc.

text: Likewise,

text: the

text: same

text: distances

text: are

text: used

text: to

text: convert

text: observed

text: proper

text: motions

text: from

text: mas/yr

text: to

text: physical

text: velocities

text: in

text: kms

text: We

text: deﬁne

text: over-abundances

text: of

text: OB

text: stars

text: as

text: regions

text: with

text: peak

text: number

text: densities

text: of

text: at

text: least

text: three

text: times

text: the

text: observed

text: background

text: level

text: of

text: O-

text: and

text: B-

text: on

text: a

text: reasonable

text: determination

text: of

text: the

text: centres

text: of

text: the

text: OB

text: associations.

text: type

text: stars

text: measured

text: across

text: the

text: radius

text: region

text: with

text: bin

text: sizes

text: The

text: centres

text: of

text: mass

text: of

text: the

text: associations

text: require

text: known

text: masses

text: for

text: all

text: of

text: We

text: then

text: determine

text: the

text: position

text: and

text: size

text: of

text: the

text: associations

text: association

text: members,

text: which

text: are

text: not

text: available.

text: Therefore

text: we

text: deﬁne

text: in

text: each

text: axis

text: by

text: ﬁtting

text: Gaussian

text: proﬁles

text: to

text: the

text: over-densities

text: in

text: each

text: two

text: association

text: centres

text: and

text: use

text: both

text: in

text: the

text: subsequent

text: analysis.

text: The

text: o|

text: vaxs≡,saantidsf−vyyin.

text: g

text: |x|

text: σX

text: and

text: y

text: σY,

text: where

text: x

text: X

text: µX

text: and

text: y

text: Y

text: µY.

text: A

text: strict

text: 1σ

text: massive

text: and

text: least

text: numerous

text: sources

text: in

text: the

text: association.

text: The

text: prop-

text: criterion

text: is

text: imposed

text: where

text: possible

text: in

text: X

text: andY

text: in

text: order

text: to

text: minimise

text: erties

text: of

text: the

text: OB

text: associations

text: using

text: the

text: mean

text: position

text: and

text: velocity

text: the

text: likelihood

text: of

text: selecting

text: contaminating

text: ﬁeld

text: stars

text: as

text: association

text: of

text: the

text: OB-type

text: stars

text: are

text: given

text: in

text: Table

text: The

text: second

text: centre

text: is

text: set

text: members.

text: In

text: cases

text: where

text: the

text: criterion

text: did

text: not

text: yield

text: at

text: least

text: σ

text: as

text: the

text: mean

text: position

text: of

text: all

text: stars

text: in

text: the

text: association,

text: stars

text: including

text: at

text: least

text: O-

text: or

text: B-

text: type

text: stars,

text: a

text: criterion

text: was

text: σ

text: biased

text: towards

text: the

text: more

text: numerous

text: low

text: mass

text: stars.

text: Table

text: shows

text: imposed

text: and

text: if

text: insuﬃcient

text: numbers

text: of

text: stars

text: were

text: selected

text: within

text: the

text: properties

text: of

text: the

text: OB

text: associations

text: when

text: using

text: the

text: mean

text: position

text: the

text: association

text: was

text: excluded

text: from

text: the

text: sample.

text: Similarly,

text: we

text: σ

text: and

text: velocity

text: of

text: all

text: stars.

text: Given

text: that

text: one

text: centre

text: is

text: biased

text: towards

text: imp)o/se

text: a

text: line-of-s≡ight

text: d−istance

text: selection

text: criterion

text: of

text: Z

text: low

text: mass

text: stars

text: and

text: the

text: other

text: is

text: biased

text: towards

text: high

text: mass

text: stars,

text: it

text: hdiesrtihveedn

text: is

text: expected

text: that

text: the

text: centre

text: of

text: mass

text: of

text: each

text: association

text: lies

text: between

text: these

text: two

text: points.

text: removes

text: the

text: outliers

text: in

text: line-of-sight

text: distance

text: from

text: the

text: sample.

text: The

text: Once

text: the

text: velocities

text: are

text: in

text: the

text: association

text: centre

text: frame,

text: the

text: dispersions

text: used

text: to

text: carry

text: out

text: the

text: above

text: selection

text: criteria

text: are

text: listed

text: radial

text: velocities

text: are

text: determined

text: by

text: taking

text: the

text: dot

text: product

text: of

text: the

text: vr

text: in

text: Table

text: velocity

text: vector

text: with

text: the

text: radial

text: unit

text: vector.

text: The

text: tangential

text: velocity

text: is

text: then

text: obtained

text: by

text: subtraction

text: as

text: vt

text: We

text: deﬁne

text: a

text: positive

text: radial

text: velocity

text: as

text: moving

text: away

text: from

text: the

text: centre

text: of

text: the

text: OB

text: tion

text: members

text: are

text: potentially

text: a

text: serious

text: problem

text: as

text: they

text: can

text: artiﬁ-

text: association

text: and

text: a

text: negative

text: value

text: as

text: moving

text: towards

text: the

text: centre

text: of

text: the

text: cially

text: skew

text: the

text: diagnostics

text: employed

text: in

text: Section

text: For

text: this

text: reason,

text: association.

text: Similarly,

text: we

text: (arbitrarily)

text: deﬁne

text: the

text: tangential

text: velocity

text: we

text: impose

text: a

text: velocity

text: cut

text: of

text: and

text: with

text: and

text: v−x

text: σrevmXoving

text: vthye

text: m<ostσevxYtreme

text: as

text: being

text: positive

text: if

text: the

text: motion

text: is

text: anti-clockwise

text: with

text: respect

text: to

text: the

text: vvexlocitvyXsourµcveXs

text: whilevymainvtaYininµgvtYhe

text: vast

text: majority

text: of

text: the

text: sample.

text: previously

text: determined

text: association

text: centre

text: and

text: negative

text: if

text: the

text: motion

text: is

text: clockwise

text: with

text: respect

text: to

text: the

text: association

text: centre.

text: OB

text: associations

text: were

text: rejected

text: if

text: they

text: did

text: not

text: contain

text: at

text: least

text: Model

text: distributions

text: stars

text: including

text: at

text: least

text: O-

text: or

text: B-type

text: stars

text: within

text: the

text: limit.

text: σ

text: This

text: yields

text: a

text: ﬁnal

text: sample

text: of

text: OB

text: associations.

text: The

text: number

text: of

text: se-

text: In

text: order

text: to

text: assess

text: whether

text: the

text: observed

text: kinematics

text: are

text: con-

text: lected

text: members

text: and

text: the

text: number

text: of

text: dispersions

text: in

text: X,Y

text: and

text: distance

text: sistent

text: with

text: a

text: net

text: radial

text: expansion,

text: we

text: compare

text: our

text: results

text: for

text: used

text: to

text: select

text: the

text: members

text: of

text: each

text: association

text: are

text: listed

text: in

text: Tables

text: the

text: observed

text: OB

text: associations

text: with

text: those

text: of

text: randomly

text: generated

text: and

text: These

text: kinematic

text: criteria

text: typically

text: ﬁlter

text: out

text: less

text: than

text: of

text: velocity

text: distributions.

text: Six

text: sets

text: of

text: models

text: will

text: be

text: calculated:

text: two

text: the

text: stars

text: in

text: each

text: association,

text: implying

text: that

text: it

text: has

text: a

text: limited

text: inﬂuence

text: distributions

text: representing

text: random

text: velocity

text: ﬁelds,

text: two

text: distributions

text: on

text: the

text: diagnostics

text: presented

text: in

text: Section

text: representing

text: localised

text: expansion

text: within

text: associations,

text: and

text: two

text: dis-

text: The

text: position-velocity

text: analysis

text: employed

text: in

text: this

text: work

text: depends

text: tributions

text: representing

text: globally

text: expanding

text: velocity

text: ﬁelds.

text: These

text: MNRAS

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##