text: 7 1.0 1.0 f a, N GOE N 16000 Α (cid:61) 0.8 D2 0.8 (cid:72) (cid:76) W (cid:61)12.5 W (cid:61)16.25 W (cid:61)22.5 0.6 W (cid:61)7.5 0.6 0.4 0.4 0.2 0.2 0.0 a 0 5 10 15 20 0.0 0 1 2 3 4 W FIG. 7: (Color online) Numerical results for f(α, N) on the FIG. 5: (Color online) Exponents D2(W) and α(W) for the Bethe lattice with K = 2 at N = 16000 for diﬀerent values of Bethe lattice with K = 2 and N = 16000. The strong in- disorder W. The dashed straight lines show the initial slope equality α > D2 that follows from the multifractal picture β for the localized and close-to-critical states which is smaller is well fulﬁlled. At the same time, α = D2 in the ins→ula∞tor, than (for localized states with αmin = 0) or equal to (for near critical, and extended ergodic phase in the limit ln N . critical states with the smallest non-zero α ) 1/2. min 3.5 1.0 2(cid:45)Α 3.0 f 1(cid:45)x, N (cid:43)x Γ f(cid:72)1(cid:43)x, N(cid:76) 0.8 2.5 (cid:72) (cid:76) 2.0 0.6 1.5 0.4 W (cid:61) 10 1.0 N (cid:61) 32000 0.2 0.5 5 10 15 20 x W (cid:45)1.0 (cid:45)0.5 0.0 0.5 1.0 − FIG. 8: (Color online) Veriﬁcation of the symmetry Eq.(6) FIG. 6: (Color online) Numerical results for 2 α(W) an×d for f(α, N) with N = 32000 at disorder strength W = 10. γ(W) on the Bethe lattice with K = 2 for N = 2, 4, 8, 16, 32 − 103. In the region of extended states the relation γ = 2 α is ≈ fairly well fulﬁlled with the smooth crossover from γ 2 at → and N = 32000. Albeit not perfect, the coincidence of the localization transition to γ 1 at weak disorder. the curves cannot be accidental. Finally, Fig. 9 shows the dependence of f(α, N) on the principle parameters α and α of f(α). In the region of size of the system N for disorder strength W = 10 in the 1 0 extended states they should be connected by a simple re- region of extended states. Remarkably, in all set of plots − lation α = 2 α that follows from the symmetry Eq.(6). there are two apparent ﬁxed points which positions de- 0 1 Thus if the description in terms of the multifractal ansatz Eq.(4) is valid we expect: − 1.0 f a,N γ(W) = 2 α(W), W < Wc. (38) (cid:72) (cid:76) 0.8 Remarkably, this relation is fulﬁlled within the error bars 0.6 (see Fig. 6): W (cid:61)10 Finally, in Fig. 7 and Fig. 9 we present 0.4 N (cid:61)32000 blue 16000 green 8000(cid:72) yell(cid:76)ow f(α, N) = ln[NxP (x)]/ ln N (39) (cid:72) (cid:76) en 0.2 4000(cid:72)red (cid:76) − 2000(cid:72)blac(cid:76)k as a function of α = 1 ln x/ ln N. If the multifrac- (cid:72) (cid:76) a → 0.0 tal ansatz Eq.(4) applies, f(α, N) f(α) in the limit 0.0 0.5 1.0 1.5 2.0 → ∞ ln N . Indeed, Fig. 7 obtained on the Bethe lattice FIG. 9: (Color online) N-dependence of f(α, N) on the Bethe with K = 2 is visibly similar to (cid:29)the sketch Fig. 3 drawn lattice with K = 2 for N = 2, 4, 8, 16, 32 × 103 (from black to using the arguments valid at K 1. blue in ascending order) in the extended phase W = 10. The ≈ As a quantitative argument we show how the non- plots for diﬀerent N show apparent ﬁxed points at α 0.5 ≈ trivial symmetry Eq.(6) is fulﬁlled for f(α, N) in the re- and α 1.6 indicated by arrows. Similar ﬁxed points with ∼ gion of extended states despite ln N 10 is clearly not W-dependent positions are seen at any strength of disorder suﬃcient to identify f(α, N) =−f(α). In Fig.8 we plot studied. This ma≈kes it un≈likely that f→(α∞, N) approaches the f(1 + x, N) together with f(1 x, N) + x for W = 10 GOE limit αmin αmax 1 at ln N . ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## 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text: 6 P Since both V o and V are assumed to belong to , we authors proposed a solution for a stochastic HJB. It is shown ∈ P 0 conclude that V . that this method gives a control Lyapunov function for the 1 ∈ P ii) Suppose u − is globally stabilizing, and V − for deterministic system. i 1 i 1 ∈ P i > 1. Let us show that u is globally stabilizing, and V . Remark 4.5: Notice that the control policies considered i i Indeed, along the solutions of the closed-loop system com- in the proposed algorithm can be extended to polynomial L posed of (1) and u = u , it follows that fractions. Indeed, instead of requiring (V , u ) to be SOS i 0 1 in Assumption 4.1, let us assume ∇ V˙ − = V T− (f + gu ) i 1 = −Li(V1− , u ) −i r(x, u ) α2(x)L(V0, u1) is SOS (36) i 1 i i ≤ − q(x). where α(x) > 0 is an arbitrary polynomial. Then, the initial − control policy u can take the form of u = α(x) 1v(x) , with Therefore, ui is globally stabilizing, since Vi−1 is a well- v(x) a column v1ector of polynomials. T1hen, we can relax the deﬁned Lyapunov function for the system. In addition, we constraint (25) to the following: have (cid:90) L ∞ α2(x) (V, u ) is SOS (37) ≥ ∀ (cid:54) i V (x ) r(x, u )dt > 0, x = 0. (33) i 0 i 0 0 and it is easy to see the SOS-based policy iteration algorithm Similarly as in (32), we can show can still proceed with this new constraint. Remark 4.6: In addition to Remark 4.5, if R(x) is not ≤ ≤ ∀ ∈ R constant, by deﬁnition, there exists a symmetric matrix of V o(x ) V (x ) V − (x ), x n, (34) ∗ 0 i 0 i 1 0 0 polynomials R (x), such that ∈ P and conclude that Vi . R(x)R∗(x) = det(R(x))I (38) 3) The two inequalities have been proved in (34). m ∈ R { }∞ 4) By 3), for each x n, the sequence V (x) is i i=0 monotonically decreasing with 0 as its lower bound. There- with det(R(x)) > 0. ∗ fore, the limit exists, i.e., there exists V (x), such that As a result, the policy improvement step (27) gives ∗ { }∞ lim V (x) = V (x). Let p be the sequence such that →∞ i i i=1 iVi = pTi m(cid:126) 2,2r(x∗). Th∗en, we know il→im∞ pi = p∗ ∈ Rn2r, ui+1 = −2det(1R(x))R∗(x)gT (x)∇Vi(x). (39) and therefore V = p T m(cid:126) (x). Also, it is easy to show ≤ ∗ ≤ ∗ 2∈,2rR ∩ P V o V V . Hence, V [x] . 0 2,2r which is a vector of polynomial fractions. 5) By 4), we know Now, if we select α(x) in (37) such that det(R(x)) divides L α(x), we see α2(x) (V, u ) is a polynomial, and the proposed H(V ∗) = −L(V ∗, u∗) ≤ 0, (35) SOS-based policy iteratioin algorithm can still proceed with (25) replaced by (37). Following the same reasoning as in the proof of Theorem 4.3, it is easy to show that the solvability ∗ which implies that V is a feasible solution to Problem 3.1. and the convergence properties of the proposed policy iteration Then, the inequalities in (5) can be obtained by the fourth algorithm will be affected with the relaxed constraint (37). property in Theorem 3.3. The proof is thus complete. V. GLOBAL ADAPTIVE DYNAMIC PROGRAMMING FOR Remark 4.4: Notice that Assumption 4.1 holds for any UNCERTAIN POLYNOMIAL SYSTEMS controllable linear systems. For general nonlinear systems, The proposed policy iteration method requires the perfect such a pair (V , u ) satisfying Assumption 4.1 may not always 0 1 knowledge of f and g. In practice, precise system knowledge exist, and in this paper we focus on polynomial systems that may be difﬁcult to obtain. Hence, in this section, we develop satisfy Assumption 4.1. For this class of systems, we assume an online learning method based on the idea of ADP to that both V and u have to be determined before executing 0 1 implement the iterative scheme with real-time data, instead the proposed algorithm. The search of (V , u ) is not trivial, 0 1 of identifying ﬁrst the system dynamics. For simplicity, here because it amounts to solving some bilinear matrix inequalities again we assume R(x) is a constant matrix. The results can (BMI) [40]. However, this problem has been actively studied be straightforwardly extended to non-constant R(x), using the in recent years, and several applicable approaches have been same idea as discussed in Remark 4.6. developed. For example, a Lyapunov based approach utilizing To begin with, consider the system state-dependent linear matrix inequalities has been studied in [40]. This method has been generalized to uncertain nonlinear polynomial systems in [61] and [20]. In [19] and [28], the x˙ = f + g(u + e) (40) i ##LTLine##

label: paragraph

text: P

text: Since

text: both

text: V

text: o

text: and

text: V

text: are

text: assumed

text: to

text: belong

text: to

text: we

text: authors

text: proposed

text: a

text: solution

text: for

text: a

text: stochastic

text: HJB.

text: It

text: is

text: shown

text: P

text: conclude

text: that

text: V

text: that

text: this

text: method

text: gives

text: a

text: control

text: Lyapunov

text: function

text: for

text: the

text: P

text: ii)

text: Suppose

text: u

text: is

text: globally

text: stabilizing,

text: and

text: V

text: for

text: deterministic

text: system.

text: i

text: i

text: P

text: i

text: Let

text: us

text: show

text: that

text: u

text: is

text: globally

text: stabilizing,

text: and

text: V

text: Remark

text: Notice

text: that

text: the

text: control

text: policies

text: considered

text: i

text: i

text: Indeed,

text: along

text: the

text: solutions

text: of

text: the

text: closed-loop

text: system

text: com-

text: in

text: the

text: proposed

text: algorithm

text: can

text: be

text: extended

text: to

text: polynomial

text: L

text: posed

text: of

text: and

text: u

text: u

text: it

text: follows

text: that

text: fractions.

text: Indeed,

text: instead

text: of

text: requiring

text: (V

text: u

text: to

text: be

text: SOS

text: i

text: in

text: Assumption

text: let

text: us

text: assume

text: V˙

text: V

text: T−

text: (f

text: gu

text: i

text: u

text: −i

text: r(x,

text: u

text: is

text: SOS

text: i

text: i

text: i

text: q(x).

text: where

text: α(x)

text: is

text: an

text: arbitrary

text: polynomial.

text: Then,

text: the

text: initial

text: control

text: policy

text: u

text: can

text: take

text: the

text: form

text: of

text: u

text: α(x)

text: with

text: Therefore,

text: ui

text: is

text: globally

text: stabilizing,

text: since

text: is

text: a

text: well-

text: v(x)

text: a

text: column

text: of

text: polynomials.

text: we

text: can

text: relax

text: the

text: deﬁned

text: Lyapunov

text: function

text: for

text: the

text: system.

text: In

text: addition,

text: we

text: constraint

text: to

text: the

text: following:

text: have

text: L

text: (V,

text: u

text: is

text: SOS

text: i

text: V

text: (x

text: r(x,

text: u

text: )dt

text: x

text: i

text: i

text: and

text: it

text: is

text: easy

text: to

text: see

text: the

text: SOS-based

text: policy

text: iteration

text: algorithm

text: Similarly

text: as

text: in

text: we

text: can

text: show

text: can

text: still

text: proceed

text: with

text: this

text: new

text: constraint.

text: Remark

text: In

text: addition

text: to

text: Remark

text: if

text: R(x)

text: is

text: not

text: R

text: constant,

text: by

text: deﬁnition,

text: there

text: exists

text: a

text: symmetric

text: matrix

text: of

text: V

text: o(x

text: V

text: (x

text: V

text: (x

text: x

text: n,

text: i

text: i

text: polynomials

text: R

text: (x),

text: such

text: that

text: P

text: and

text: conclude

text: that

text: Vi

text: R(x)R∗(x)

text: det(R(x))I

text: The

text: two

text: inequalities

text: have

text: been

text: proved

text: in

text: m

text: R

text: By

text: for

text: each

text: x

text: n,

text: the

text: sequence

text: V

text: (x)

text: is

text: i

text: monotonically

text: decreasing

text: with

text: as

text: its

text: lower

text: bound.

text: There-

text: with

text: det(R(x))

text: fore,

text: the

text: limit

text: exists,

text: i.e.,

text: there

text: exists

text: V

text: (x),

text: such

text: that

text: As

text: a

text: result,

text: the

text: policy

text: improvement

text: step

text: gives

text: lim

text: V

text: (x)

text: V

text: (x).

text: Let

text: p

text: be

text: the

text: sequence

text: such

text: that

text: i

text: i

text: iVi

text: pTi

text: Th∗en,

text: we

text: know

text: il→im∞

text: pi

text: p∗

text: (x)∇Vi(x).

text: and

text: therefore

text: V

text: p

text: T

text: (x).

text: Also,

text: it

text: is

text: easy

text: to

text: show

text: P

text: V

text: o

text: V

text: V

text: Hence,

text: V

text: [x]

text: which

text: is

text: a

text: vector

text: of

text: polynomial

text: fractions.

text: By

text: we

text: know

text: Now,

text: if

text: we

text: select

text: α(x)

text: in

text: such

text: that

text: det(R(x))

text: divides

text: L

text: α(x),

text: we

text: see

text: (V,

text: u

text: is

text: a

text: polynomial,

text: and

text: the

text: proposed

text: H(V

text: −L(V

text: u∗)

text: SOS-based

text: policy

text: iteratioin

text: algorithm

text: can

text: still

text: proceed

text: with

text: replaced

text: by

text: Following

text: the

text: same

text: reasoning

text: as

text: in

text: the

text: proof

text: of

text: Theorem

text: it

text: is

text: easy

text: to

text: show

text: that

text: the

text: solvability

text: which

text: implies

text: that

text: V

text: is

text: a

text: feasible

text: solution

text: to

text: Problem

text: and

text: the

text: convergence

text: properties

text: of

text: the

text: proposed

text: policy

text: iteration

text: Then,

text: the

text: inequalities

text: in

text: can

text: be

text: obtained

text: by

text: the

text: fourth

text: algorithm

text: will

text: be

text: affected

text: with

text: the

text: relaxed

text: constraint

text: property

text: in

text: Theorem

text: The

text: proof

text: is

text: thus

text: complete.

text: V.

text: GLOBAL

text: ADAPTIVE

text: DYNAMIC

text: PROGRAMMING

text: FOR

text: Remark

text: Notice

text: that

text: Assumption

text: holds

text: for

text: any

text: UNCERTAIN

text: POLYNOMIAL

text: SYSTEMS

text: controllable

text: linear

text: systems.

text: For

text: general

text: nonlinear

text: systems,

text: The

text: proposed

text: policy

text: iteration

text: method

text: requires

text: the

text: perfect

text: such

text: a

text: pair

text: (V

text: u

text: satisfying

text: Assumption

text: may

text: not

text: always

text: knowledge

text: of

text: f

text: and

text: g.

text: In

text: practice,

text: precise

text: system

text: knowledge

text: exist,

text: and

text: in

text: this

text: paper

text: we

text: focus

text: on

text: polynomial

text: systems

text: that

text: may

text: be

text: difﬁcult

text: to

text: obtain.

text: Hence,

text: in

text: this

text: section,

text: we

text: develop

text: satisfy

text: Assumption

text: For

text: this

text: class

text: of

text: systems,

text: we

text: assume

text: an

text: online

text: learning

text: method

text: based

text: on

text: the

text: idea

text: of

text: ADP

text: to

text: that

text: both

text: V

text: and

text: u

text: have

text: to

text: be

text: determined

text: before

text: executing

text: implement

text: the

text: iterative

text: scheme

text: with

text: real-time

text: data,

text: instead

text: the

text: proposed

text: algorithm.

text: The

text: search

text: of

text: (V

text: u

text: is

text: not

text: trivial,

text: of

text: identifying

text: ﬁrst

text: the

text: system

text: dynamics.

text: For

text: simplicity,

text: here

text: because

text: it

text: amounts

text: to

text: solving

text: some

text: bilinear

text: matrix

text: inequalities

text: again

text: we

text: assume

text: R(x)

text: is

text: a

text: constant

text: matrix.

text: The

text: results

text: can

text: (BMI)

text: However,

text: this

text: problem

text: has

text: been

text: actively

text: studied

text: be

text: straightforwardly

text: extended

text: to

text: non-constant

text: R(x),

text: using

text: the

text: in

text: recent

text: years,

text: and

text: several

text: applicable

text: approaches

text: have

text: been

text: same

text: idea

text: as

text: discussed

text: in

text: Remark

text: developed.

text: For

text: example,

text: a

text: Lyapunov

text: based

text: approach

text: utilizing

text: To

text: begin

text: with,

text: consider

text: the

text: system

text: state-dependent

text: linear

text: matrix

text: inequalities

text: has

text: been

text: studied

text: in

text: This

text: method

text: has

text: been

text: generalized

text: to

text: uncertain

text: nonlinear

text: polynomial

text: systems

text: in

text: and

text: In

text: and

text: the

text: x˙

text: f

text: g(u

text: e)

text: i

text: ##LTLine##


text: 7 − n 1, such that V = pT m(cid:126) (x). Then, along the solutions 2,2r where u is a feedback control policy and e is a bounded of the system (40), it follows that i time-varying function, known as the exploration noise, added for the learning purpose. ∇ ∇ A. Forward completeness V˙ = V T (f + gui) + V T ge − − L ∇ Due to the existence of the exploration noise, we need to = r(x, ui) (V, ui) + V T ge − − L ﬁrst show if solutions of system (40) exist globally, for positive = r(x, u ) (V, u ) i i − ∇ time. For this purpose, we will show that the system (40) is +(R 1gT V )T Re (42) forward complete [1]. Deﬁnition 5.1 ([1]): Consider system (40) with e as the L − ∇ input. The system is called forward complete if, for any initial Notice that the two terms (V, ui) and R 1gT V rely condition x ∈ Rn and every input signal e, the corresponding on f and g. Here, we are interested in solving them without solution of 0system (40) is deﬁned for all t ≥ 0. identifying f and g. athllaLyt es∇mtaVbmii−lai1z(5inf.2g+:coCgnuotnir)osil+dpeuorTiliscRyysutaeinm≤d (t04h.0eTr)e.hSeenxu,ipsptthsoesVesi−yus1ite∈imsPa(,4g0slu)ocbihs- a(nb2+oTdv2ode,)th−wisede−cnad1n,, ﬁannnoddticaaeccotohnnasstttaafnontrtmvtheaectrtiosxarmKlpep∈∈paRRirnm2(d×V,n,wud,ii)tshudcnehﬁ2dtnhe=adt forward complete. Proof: Under Assumptions 2.2 and 4.1, by Theorem 4.3 L ∈ P (V, u ) = lT m(cid:126) (x) (43) we know V − . Then, by completing the squares, it i p 2,2d i 1 −1 − ∇ follows that R 1gT V = K m(cid:126) (x) (44) 2 p 1,d ∇ ≤ − − L − ∇ ViT−1(f + gui + ge) −u| Ti Rui| 2u|Ti |Re Therefore, calculating (V, ui) and R 1gT V amounts to =≤ | |ui + e 2R + e 2R ﬁnSdiunbgstliptutainndg K(4p3.) and (44) in (42), we have e 2 ≤ | |R e 2 + V − . R i 1 − − − V˙ = r(x, u ) lT m(cid:126) (x) 2m(cid:126) T (x)KT Re (45) i p 2,2d 1,d p According to [1, Corollary 2.11], the system (40) is forward complete. Now, integrating the terms in (45) over the interval [t, t+δt], By Lemma 5.2 and Theorem 4.3, we immediately have the we have following Proposition. (cid:90) Proposition 5.3: Under Assumptions 2.2 and 4.1, let ui be (cid:0) − a feedback control policy obtained at the i-th iteration step in pT [m(cid:126) 2,2r(x(t)) m(cid:126) 2,2r(x(t + δt))] the proposed policy iteration algorithm (24)-(27) and e be a t+δt (cid:1) = r(x, u ) + lT m(cid:126) (x) bounded time-varying function. Then, the closed-loop system i p 2,2d t (1) with u = ui + e is forward complete. +2m(cid:126) T (x)KT Re dt (46) 1,d p B. Online implementation Eq. (46) implies that, l and K can be directly calculated Under Assumption 4.1, in the SOS-based policy iteration, p p by using real-time online data, without knowing the precise we have knowledge of f and g. To see how it works, let us deﬁne the following matrices: L(Vi, ui) ∈ R[x]2,2d, ∀i > 1, (41) σReqi×∈n2Rr,ns2udc+h(cid:2)mtnhda,t Φi ∈ Rqi×(n2d+m(cid:3)nd), Ξi ∈ Rqi, Θi ∈ if the integer d satisﬁes −(cid:104) (cid:82) (cid:82) ⊗ (cid:82) (cid:105) σ = m(cid:126) T 2m(cid:126) T eT R T , d ≥ 12max {deg(f−) + 2r − 1, deg(}g) + 2(2r − 1), Φei = (cid:104)(cid:82)(cid:82)tt01,,ii2σ,2eddt tt112,,,dii σ(cid:82)(cid:105)edt · · · ttqqii−,i1·,·i·σedt T , deg(Q), 2(2r 1) + 2deg(g) . Ξ = t1,i r(x, u )dt t2,i r(x, u )dt i t0,i i t1,i i (cid:104) (cid:105) rithAHmlesnosc,aetu,isitﬁhoeesb−retuaiienx∈eidsRtsf[rxoa]m1c,dot.nhsetapnrtopmoasterdixpKoliicy∈itRerma×tionnd, awlgioth- Θi = m(cid:126)ttqqi2i−,,i21r,i|ttr10(,,iix, um(cid:126)i)2d,2tr|ttT21,,ii, · · · m(cid:126) 2,2r|ttqqii,−i1,i T . n = (n+d) 1, such that u = K m(cid:126) (x). Also, suppose d d i ∈ R i 1,d − there exists a constant vector p n2r, with n2r = (n2+r2r) Then, (46) implies ##LTLine## ##LTLine##

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text: 12 Abramson & Kenney TABLE 4 NGC 4402 CLOUDS - OBSERVED AND DERIVED PROPERTIES Feature d (kpc) Measured Probable length (pc) width (pc) avg a min a PA disk v v M (M(cid:12)) M (M(cid:12)) H× H× ◦ ± ◦ 1 8.3* 2.4 104 2.4 105 90 50 0.50** 0.29 37 5 × × ◦ ± ◦ 2 7.2* 3.7 104 3.7 105 170 40 0.54 0.17 40 4 × × ◦ ± ◦ 3 6.4* 5.6 104 5.6 105 130 100 0.45 0.15 57 20 × × 4 6.4* 3.7 104 3.7 105 140 110 0.33 0.12 - × × 5 5.5 2.1 105 2.1 106 260 220 0.50 0.15 - × × 6 5.6 3.5 103 3.5 104 40 20 0.31 0.17 - × × 7 5.6 9.6 103 9.6 104 60 50 0.43 0.15 - × × 8 5.5 2.2 104 2.2 105 100 90 0.33 0.13 - × × 9 5.1 5.4 105 5.4 106 420 210 0.56 0.10 - × × ◦ ± ◦ 10 4.5 8.9 105 8.9 106 616 210 0.51 0.13 331 10 × × 11 5.3 1.1 104 1.1 105 60 50 0.35 0.16 - × × 12 7.5 1.1 104 1.1 105 90 30 0.39 0.25 - × × 13 5.7 2.9 104 2.9 105 80 70 0.49 0.16 - × × 14 5.2 1.1 104 1.1 105 50 50 0.41 0.24 - × × 15 4.6 3.9 103 3.9 104 50 30 0.33 0.13 - × × 16 3.6 6.4 104 6.4 105 140 90 0.52 0.12 - × × 17 3.9 6.5 103 6.5 104 40 30 0.39 0.19 - × × 18 5.9 3.6 103 3.6 104 50 30 0.25 0.14 - × × 19 5.8 2.9 104 2.9 105 100 80 0.40 0.14 - × × ◦ ± ◦ 20 4.5 7.9 105 7.9 106 900 250 0.35 0.12 325 5 Measurements are of pixels with a signiﬁcance level of at least 2σ above background, and include position angles for elongated clouds. The “probable” mass column includes an additional factor of 10 based on CO measurements of similarly-sized clouds. The projected galactocentric radii d of clouds projected disk against the far side of the galaxy are marked with a *. The a values derived from smoothed images (then multiplied by 1.25 to correct for reductions due to the V smoothing process) are marked with a **. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Abramson

text: Kenney

text: TABLE

text: NGC

text: CLOUDS

text: OBSERVED

text: AND

text: DERIVED

text: PROPERTIES

text: Feature

label: table

text: d

text: (kpc)

text: Measured

text: Probable

text: length

text: (pc)

text: width

text: (pc)

text: avg

text: a

text: min

text: a

text: PA

text: disk

text: v

text: v

text: M

text: M

text: H×

text: H×

text: Measurements

text: are

text: of

text: pixels

text: with

text: a

text: signiﬁcance

text: level

text: of

text: at

text: least

text: 2σ

text: above

text: background,

text: and

text: include

text: position

text: angles

text: for

text: elongated

text: clouds.

text: The

text: “probable”

text: mass

text: column

text: includes

text: an

text: additional

text: factor

text: of

text: based

text: on

text: CO

text: measurements

text: of

text: similarly-sized

text: clouds.

text: The

text: projected

text: galactocentric

text: radii

text: d

text: of

text: clouds

text: projected

text: disk

text: against

text: the

text: far

text: side

text: of

text: the

text: galaxy

text: are

text: marked

text: with

text: a

text: The

text: a

text: values

text: derived

text: from

text: smoothed

text: images

text: (then

text: multiplied

text: by

text: to

text: correct

text: for

text: reductions

text: due

text: to

text: the

text: V

text: smoothing

text: process)

text: are

text: marked

text: with

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text: Decoupled Clouds in Stripped Virgo Spirals 17 in these calculations we include them (Cloud 9 in NGC 4402 and Cloud 1 in NGC 4522). FIG. 19.— Viewing geometry of NGC 4402 and NGC 4522, showing an edge-on view of the galaxy where the observer is located to the right of the frame. The inclination of the stellar disk is i. The estimated pre- and post-stripping surface densities for NGC 4402 and NGC 4522 are shown in Figures 20 and 21. Figure 20 shows all four sectors of NGC 4402 with their mod- eled pre-stripping and observed post-stripping ISM surface densities and includes the factor of 10 mass correction dis- cussed in Section . The observed surface density in the tran- sition zone, as indicated by a dotted line connecting open di- amonds, is the mass of all the small clouds in a 1-kpc radial bin of the sector divided by the surface area of the bin. The solid and dashed lines represent the surface density predicted by the exponential disk model described above. The solid line shows points at radii smaller than the transition zone, where most or all of the original ISM is still in the disk. The dashed line shows what the original exponential distribution would look like if it extended into the transition zone. The bound- aries of the transition zone are marked. Figure 21 shows the same information for NGC 4522. Sector 2 of NGC 4522 does prFeI-Gst.ri2p0p.i—ng GmaassssuinrfNacGeCde4n4s0i2tyaΣziHmI+uHt2haolfstehcetorresm. TaihneinogbsISerMveadngdasmsoudrefalecde not lend itself to this technique since there is a large dust com- density in the transition zone is indicated with diamonds, and the predicted plex rather than discrete clouds with a well-deﬁned truncation exponential distribution is shown by the dashed lines. This ﬁgure is discussed boundary. in Section . We then sum the masses of each galaxy’s sectors to get the total estimated original mass in its near-side transition zone. We divide the remaining mass by the estimated original The remaining mass in each sector is the sum of the small mass to get the fraction of mass remaining in the transition clouds and/or large dust plumes contained in the sector. There zone at the current stage of stripping. Both the measured and is some ambiguity as to whether clouds located just above a likely values, the latter incorporating the factor-of-10 correc- galaxy’s major axis are located near the disk plane or not, and tion, are given in Table 5. Even with the larger cloud mass ##LTFigure## ##LTFigure##

text: Decoupled

text: Clouds

text: in

text: Stripped

text: Virgo

text: Spirals

text: in

text: these

text: calculations

text: we

text: include

text: them

text: (Cloud

text: in

text: NGC

text: and

text: Cloud

text: in

text: NGC

text: FIG.

text: Viewing

text: geometry

text: of

text: NGC

text: and

text: NGC

text: showing

text: an

text: edge-on

text: view

text: of

text: the

text: galaxy

text: where

text: the

text: observer

text: is

text: located

text: to

text: the

text: right

text: of

text: the

text: frame.

text: The

text: inclination

text: of

text: the

text: stellar

text: disk

text: is

text: i.

text: The

text: estimated

text: pre-

text: and

text: post-stripping

text: surface

text: densities

text: for

text: NGC

text: and

text: NGC

text: are

text: shown

text: in

text: Figures

text: and

text: Figure

text: shows

text: all

text: four

text: sectors

text: of

text: NGC

text: with

text: their

text: mod-

text: eled

text: pre-stripping

text: and

text: observed

text: post-stripping

text: ISM

text: surface

text: densities

text: and

text: includes

text: the

text: factor

text: of

text: mass

text: correction

text: dis-

text: cussed

text: in

text: Section

text: The

text: observed

text: surface

text: density

text: in

text: the

text: tran-

text: sition

text: zone,

text: as

text: indicated

text: by

text: a

text: dotted

text: line

text: connecting

text: open

text: di-

text: amonds,

text: is

text: the

text: mass

text: of

text: all

text: the

text: small

text: clouds

text: in

text: a

text: radial

text: bin

text: of

text: the

text: sector

text: divided

text: by

text: the

text: surface

text: area

text: of

text: the

text: bin.

text: The

text: solid

text: and

text: dashed

text: lines

text: represent

text: the

text: surface

text: density

text: predicted

text: by

text: the

text: exponential

text: disk

text: model

text: described

text: above.

text: The

text: solid

text: line

text: shows

text: points

text: at

text: radii

text: smaller

text: than

text: the

text: transition

text: zone,

text: where

text: most

text: or

text: all

text: of

text: the

text: original

text: ISM

text: is

text: still

text: in

text: the

text: disk.

text: The

text: dashed

text: line

text: shows

text: what

text: the

text: original

text: exponential

text: distribution

text: would

text: look

text: like

text: if

text: it

text: extended

text: into

text: the

text: transition

text: zone.

text: The

text: bound-

text: aries

text: of

text: the

text: transition

text: zone

text: are

text: marked.

text: Figure

text: shows

text: the

text: same

text: information

text: for

text: NGC

text: Sector

text: of

text: NGC

text: does

text: not

text: lend

text: itself

text: to

text: this

text: technique

text: since

text: there

text: is

text: a

text: large

text: dust

text: com-

text: density

text: in

text: the

text: transition

text: zone

text: is

text: indicated

text: with

text: diamonds,

text: and

text: the

text: predicted

text: plex

text: rather

text: than

text: discrete

text: clouds

text: with

text: a

text: well-deﬁned

text: truncation

text: exponential

text: distribution

text: is

text: shown

text: by

text: the

text: dashed

text: lines.

text: This

text: ﬁgure

text: is

text: discussed

text: boundary.

text: in

text: Section

text: We

text: then

text: sum

text: the

text: masses

text: of

text: each

text: galaxy’s

text: sectors

text: to

text: get

text: the

text: total

text: estimated

text: original

text: mass

text: in

text: its

text: near-side

text: transition

text: zone.

text: We

text: divide

text: the

text: remaining

text: mass

text: by

text: the

text: estimated

text: original

text: The

text: remaining

text: mass

text: in

text: each

text: sector

text: is

text: the

text: sum

text: of

text: the

text: small

text: mass

text: to

text: get

text: the

text: fraction

text: of

text: mass

text: remaining

text: in

text: the

text: transition

text: clouds

text: and/or

text: large

text: dust

text: plumes

text: contained

text: in

text: the

text: sector.

text: There

text: zone

text: at

text: the

text: current

text: stage

text: of

text: stripping.

text: Both

text: the

text: measured

text: and

text: is

text: some

text: ambiguity

text: as

text: to

text: whether

text: clouds

text: located

text: just

text: above

text: a

text: likely

text: values,

text: the

text: latter

text: incorporating

text: the

text: correc-

text: galaxy’s

text: major

text: axis

text: are

text: located

text: near

text: the

text: disk

text: plane

text: or

text: not,

text: and

text: tion,

text: are

text: given

text: in

text: Table

text: Even

text: with

text: the

text: larger

text: cloud

text: mass

text: ##LTFigure##

text: ##LTFigure##


text: 6 Abramson & Kenney 4. RESULTS front of the cloud, as 4.1. Dust Cloud Identiﬁcation and Measurement − − Both galaxies have many small, discrete dust clouds beyond Sdc,λ/Sbg,λ = x+(1 x)e τλ (2) their main dust lanes, and we seek to identify the clouds and measure their properties in a self-consistent way. Color im- where τ is the extinction optical depth through the cloud λ ages of NGC 4522 and NGC 4402 are shown in Figures 3 and at a given wavelength. 4 respectively, with insets showing the dust clouds. Although Combining Equations 1 and 2 with x=0, we see that the true the unsharp masked images aided in the identiﬁcation of some extinction Aλ = 1.086τλ. We can then ﬁnd the relationship of the isolated dust clouds, our detection criteria are based on between the observed and true extinctions, which is the V-band images, and those are the images we measured to − − − determine cloud sizes and masses unless otherwise noted. We aλ = 2.5log(x+(1 x)e Aλ/1.086) (3) deﬁne a “cloud” as an area of the image beyond the main dust truncation boundary (where the dust goes from a continuous distribution to a few discrete clouds, discussed further in Sec- We see that foreground light can have a signiﬁcant effect on tion ), where at least one pixel has a surface brightness > 4σ the observed extinction - if 50% of the assumed background below the local background, and where at least 4 contiguous light is actually in the foreground, the observed extinction aλ pixels are > 3σ below the local background. Our detection will be about 50% lower than the actual extinction Aλ. limit is roughly a cloud diameter of 2-3 pixels, or 8-12 pc, In this work, we assume that all galactic light is behind the and most of the detected clouds range in width from 20 - 80 dust features, meaning aλ = Aλ, recognizing that the resulting pc. Each cloud is located in a “region” that is comprised of column density estimates will be lower limits. We can then the cloud and the local background. use the correlation between dust extinction and gas column For each region, we measure the local background in sev- density in the Milky Way (Bohlin et al. 1978), eral spots near the dust cloud that appear to be free from dust × − extinction or individual bright stars. Because there is a large NH = 5.8 1021 E(B V) (4) gradient in the background surface brightness for some of the clouds, we interpolate linearly between the measured back- ground points to generate an estimate of the background sur- where NH = N(HI) + 2N(H2), to estimate a lower limit for the total hydrogen column density of the gas in a given face brightness value for each pixel in the cloud (i.e., the sur- cloud. We can use a Cardelli et al. (1989) extinction law, face brightness we would observe from the galaxy if the ob- which describes interstellar extinction with the parameter R scuring cloud were not along our line of sight). Each region V ≡ with its simulated background and detected dust pixels are A−V . Observed lines of sight in the Milky Way are well- E(B V) shown in Figures 5, 6, and 7. described by this single-parameter relationship with a mean In order to determine which pixels should be considered value of RV = 3.1, with variations related to the size of the part of each cloud, we calculate σ, the standard deviation of dust grains in the cloud. Dust grains scatter wavelengths of the local background surface brightness. The value of σ is light smaller than their diameters, so if a dust grain is large higher in NGC 4522 than in NGC 4402 due to the fact that enough, it will scatter red and blue light equally well, result- the outer stellar disk in NGC 4522 is younger and the stellar ing in “grey” extinction rather than extinction with reddening. light distribution is less smooth. In order to calculate cloud Large dust grains are found preferentially in dense regions mass and radius consistently, we measure the clouds down to (e.g., Cardelli et al. 1989). similar minimum V-band dust extinction values (Tables 3 and We assume a Milky Way average R value rather than us- V 4). To achieve this, we consider all of the contiguous pixels ing the R we measure for each cloud in order to be consistent V with surface brightness > 1σ below the local background to with previous works using this technique (e.g., Howk & Sav- be part of the cloud for NGC 4522, and > 2σ below the local age 1997; Thompson et al. 2004), and because the R values V background for NGC 4402. we measure in the galaxies are roughly consistent with this We can calculate a lower limit on the mass of each cloud us- value. We measure average R values in NGC 4522 and NGC V ing a technique similar to Howk & Savage (1997) and Thomp- 4402 of 3.3 and 2.3, respectively, and if we had used those val- son et al. (2004). They use the relationship between dust ex- ues, our N values would have differed by up to 30%, which H tinction and gas column density in the Milky Way (Bohlin is within the expected errors of this work. Additionally, our et al. 1978) to estimate the masses of dust clouds in nearby measured R values may differ somewhat from the actual val- V galaxies. The apparent dust extinction at a given wavelength ues due to the effects of foreground light (discussed above). λbeiscaelxcpurleastseedduassintghe quantity aλ. The apparent extinction can 1.9A×ss1u0m2i1nAgV[acmM−i2lk],ywWhiacyh wRVe ugsieveisn othuer creallactuiloantisohnips. NVHary=- ing R among the range of observed values in the Milky Way − V aλ = 2.5log(Sdc,λ/Sbg,λ) (1) (thReV t=wo1 g-a5la, xwiehsi)chcaisn scihmainlagrettohethNe ranesgteimoaftveabriyataiofnacwtoitrhoinf ∼ H 2. The relationship between A in magnitudes and the ra- V where S is the surface brightness towards the dust cloud, tio of cloud to background ﬂux is approximately linear for the dc,λ ∼ and S is the local background surface brightness. The ap- ﬂux ratios in the observed clouds, for A values from 0-0.7, bg,λ V parent extinction a does not necessarily match the true ex- meaning that the relationship between A and N is approxi- λ V H tinction, A because some of the light we interpret as “back- mately linear as well. λ ground” may actually be located in front of the cloud. The This approach provides a lower limit on the cloud mass for measured cloud-to-background surface brightness ratio can be three reasons. The ﬁrst reason is that it is unlikely that all expressed using the quantity x, the fraction of light emitted in of the galaxy’s light in the direction of the cloud is actually ##LTLine##

text: Abramson

text: Kenney

text: RESULTS

text: front

text: of

text: the

text: cloud,

text: as

text: Dust

text: Cloud

text: Identiﬁcation

text: and

text: Measurement

text: Both

text: galaxies

text: have

text: many

text: small,

text: discrete

text: dust

text: clouds

text: beyond

text: Sdc,λ/Sbg,λ

text: x)e

text: τλ

font: OCVPXG+StandardSymL

text: their

text: main

text: dust

text: lanes,

text: and

text: we

text: seek

text: to

text: identify

text: the

text: clouds

text: and

text: measure

text: their

text: properties

text: in

text: a

text: self-consistent

text: way.

text: Color

text: im-

text: where

text: τ

text: is

text: the

text: extinction

text: optical

text: depth

text: through

text: the

text: cloud

text: λ

text: ages

text: of

text: NGC

text: and

text: NGC

text: are

text: shown

text: in

text: Figures

text: and

text: at

text: a

text: given

text: wavelength.

text: respectively,

text: with

text: insets

text: showing

text: the

text: dust

text: clouds.

text: Although

text: Combining

text: Equations

text: and

text: with

text: we

text: see

text: that

text: the

text: true

text: the

text: unsharp

text: masked

text: images

text: aided

text: in

text: the

text: identiﬁcation

text: of

text: some

text: extinction

text: Aλ

text: We

text: can

text: then

text: ﬁnd

text: the

text: relationship

text: of

text: the

text: isolated

text: dust

text: clouds,

text: our

text: detection

text: criteria

text: are

text: based

text: on

text: between

text: the

text: observed

text: and

text: true

text: extinctions,

text: which

text: is

text: the

text: V-band

text: images,

text: and

text: those

text: are

text: the

text: images

text: we

text: measured

text: to

text: determine

text: cloud

text: sizes

text: and

text: masses

text: unless

text: otherwise

text: noted.

text: We

text: aλ

text: x)e

text: deﬁne

text: a

text: “cloud”

text: as

text: an

text: area

text: of

text: the

text: image

text: beyond

text: the

text: main

text: dust

text: truncation

text: boundary

text: (where

text: the

text: dust

text: goes

text: from

text: a

text: continuous

text: distribution

text: to

text: a

text: few

text: discrete

text: clouds,

text: discussed

text: further

text: in

text: Sec-

text: We

text: see

text: that

text: foreground

text: light

text: can

text: have

text: a

text: signiﬁcant

text: effect

text: on

text: tion

text: where

text: at

text: least

text: one

text: pixel

text: has

text: a

text: surface

text: brightness

text: 4σ

text: the

text: observed

text: extinction

text: if

text: of

text: the

text: assumed

text: background

text: below

text: the

text: local

text: background,

text: and

text: where

text: at

text: least

text: contiguous

text: light

text: is

text: actually

text: in

text: the

text: foreground,

text: the

text: observed

text: extinction

text: aλ

text: pixels

text: are

text: 3σ

text: below

text: the

text: local

text: background.

text: Our

text: detection

text: will

text: be

text: about

text: lower

text: than

text: the

text: actual

text: extinction

text: Aλ.

text: limit

text: is

text: roughly

text: a

text: cloud

text: diameter

text: of

text: pixels,

text: or

text: pc,

text: In

text: this

text: work,

text: we

text: assume

text: that

text: all

text: galactic

text: light

text: is

text: behind

text: the

text: and

text: most

text: of

text: the

text: detected

text: clouds

text: range

text: in

text: width

text: from

text: dust

text: features,

text: meaning

text: aλ

text: Aλ,

text: recognizing

text: that

text: the

text: resulting

text: pc.

text: Each

text: cloud

text: is

text: located

text: in

text: a

text: “region”

text: that

text: is

text: comprised

text: of

text: column

text: density

text: estimates

text: will

text: be

text: lower

text: limits.

text: We

text: can

text: then

text: the

text: cloud

text: and

text: the

text: local

text: background.

text: use

text: the

text: correlation

text: between

text: dust

text: extinction

text: and

text: gas

text: column

text: For

text: each

text: region,

text: we

text: measure

text: the

text: local

text: background

text: in

text: sev-

text: density

text: in

text: the

text: Milky

text: Way

text: (Bohlin

text: et

text: al.

text: eral

text: spots

text: near

text: the

text: dust

text: cloud

text: that

text: appear

text: to

text: be

text: free

text: from

text: dust

text: extinction

text: or

text: individual

text: bright

text: stars.

text: Because

text: there

text: is

text: a

text: large

text: NH

text: E(B

text: V)

text: gradient

text: in

text: the

text: background

text: surface

text: brightness

text: for

text: some

text: of

text: the

text: clouds,

text: we

text: interpolate

text: linearly

text: between

text: the

text: measured

text: back-

text: ground

text: points

text: to

text: generate

text: an

text: estimate

text: of

text: the

text: background

text: sur-

text: where

text: NH

text: N(HI)

text: to

text: estimate

text: a

text: lower

text: limit

text: for

text: the

text: total

text: hydrogen

text: column

text: density

text: of

text: the

text: gas

text: in

text: a

text: given

text: face

text: brightness

text: value

text: for

text: each

text: pixel

text: in

text: the

text: cloud

text: (i.e.,

text: the

text: sur-

text: cloud.

text: We

text: can

text: use

text: a

text: Cardelli

text: et

text: al.

text: extinction

text: law,

text: face

text: brightness

text: we

text: would

text: observe

text: from

text: the

text: galaxy

text: if

text: the

text: ob-

text: which

text: describes

text: interstellar

text: extinction

text: with

text: the

text: parameter

text: R

text: scuring

text: cloud

text: were

text: not

text: along

text: our

text: line

text: of

text: sight).

text: Each

text: region

text: V

text: with

text: its

text: simulated

text: background

text: and

text: detected

text: dust

text: pixels

text: are

text: A−V

text: Observed

text: lines

text: of

text: sight

text: in

text: the

text: Milky

text: Way

text: are

text: well-

text: E(B

text: V)

text: shown

text: in

text: Figures

text: and

text: described

text: by

text: this

text: single-parameter

text: relationship

text: with

text: a

text: mean

text: In

text: order

text: to

text: determine

text: which

text: pixels

text: should

text: be

text: considered

text: value

text: of

text: RV

text: with

text: variations

text: related

text: to

text: the

text: size

text: of

text: the

text: part

text: of

text: each

text: cloud,

text: we

text: calculate

text: σ,

text: the

text: standard

text: deviation

text: of

text: dust

text: grains

text: in

text: the

text: cloud.

text: Dust

text: grains

text: scatter

text: wavelengths

text: of

text: the

text: local

text: background

text: surface

text: brightness.

text: The

text: value

text: of

text: σ

text: is

text: light

text: smaller

text: than

text: their

text: diameters,

text: so

text: if

text: a

text: dust

text: grain

text: is

text: large

text: higher

text: in

text: NGC

text: than

text: in

text: NGC

text: due

text: to

text: the

text: fact

text: that

text: enough,

text: it

text: will

text: scatter

text: red

text: and

text: blue

text: light

text: equally

text: well,

text: result-

text: the

text: outer

text: stellar

text: disk

text: in

text: NGC

text: is

text: younger

text: and

text: the

text: stellar

text: ing

text: in

text: “grey”

text: extinction

text: rather

text: than

text: extinction

text: with

text: reddening.

text: light

text: distribution

text: is

text: less

text: smooth.

text: In

text: order

text: to

text: calculate

text: cloud

text: Large

text: dust

text: grains

text: are

text: found

text: preferentially

text: in

text: dense

text: regions

text: mass

text: and

text: radius

text: consistently,

text: we

text: measure

text: the

text: clouds

text: down

text: to

text: (e.g.,

text: Cardelli

text: et

text: al.

text: similar

text: minimum

text: V-band

text: dust

text: extinction

text: values

text: (Tables

text: and

text: We

text: assume

text: a

text: Milky

text: Way

text: average

text: R

text: value

text: rather

text: than

text: us-

text: V

text: To

text: achieve

text: this,

text: we

text: consider

text: all

text: of

text: the

text: contiguous

text: pixels

text: ing

text: the

text: R

text: we

text: measure

text: for

text: each

text: cloud

text: in

text: order

text: to

text: be

text: consistent

text: V

text: with

text: surface

text: brightness

text: 1σ

text: below

text: the

text: local

text: background

text: to

text: with

text: previous

text: works

text: using

text: this

text: technique

text: (e.g.,

text: Howk

text: Sav-

text: be

text: part

text: of

text: the

text: cloud

text: for

text: NGC

text: and

text: 2σ

text: below

text: the

text: local

text: age

text: Thompson

text: et

text: al.

text: and

text: because

text: the

text: R

text: values

text: V

text: background

text: for

text: NGC

text: we

text: measure

text: in

text: the

text: galaxies

text: are

text: roughly

text: consistent

text: with

text: this

text: We

text: can

text: calculate

text: a

text: lower

text: limit

text: on

text: the

text: mass

text: of

text: each

text: cloud

text: us-

text: value.

text: We

text: measure

text: average

text: R

text: values

text: in

text: NGC

text: and

text: NGC

text: V

text: ing

text: a

text: technique

text: similar

text: to

text: Howk

text: Savage

text: and

text: Thomp-

text: of

text: and

text: respectively,

text: and

text: if

text: we

text: had

text: used

text: those

text: val-

text: son

text: et

text: al.

text: They

text: use

text: the

text: relationship

text: between

text: dust

text: ex-

text: ues,

text: our

text: N

text: values

text: would

text: have

text: differed

text: by

text: up

text: to

text: which

text: H

text: tinction

text: and

text: gas

text: column

text: density

text: in

text: the

text: Milky

text: Way

text: (Bohlin

text: is

text: within

text: the

text: expected

text: errors

text: of

text: this

text: work.

text: Additionally,

text: our

text: et

text: al.

text: to

text: estimate

text: the

text: masses

text: of

text: dust

text: clouds

text: in

text: nearby

text: measured

text: R

text: values

text: may

text: differ

text: somewhat

text: from

text: the

text: actual

text: val-

text: V

text: galaxies.

text: The

text: apparent

text: dust

text: extinction

text: at

text: a

text: given

text: wavelength

text: ues

text: due

text: to

text: the

text: effects

text: of

text: foreground

text: light

text: (discussed

text: above).

text: λbeiscaelxcpurleastseedduassintghe

text: quantity

text: aλ.

text: The

text: apparent

text: extinction

text: can

text: wRVe

text: ugsieveisn

text: othuer

text: creallactuiloantisohnips.

text: NVHary=-

text: ing

text: R

text: among

text: the

text: range

text: of

text: observed

text: values

text: in

text: the

text: Milky

text: Way

text: V

text: aλ

text: (thReV

text: xwiehsi)chcaisn

text: scihmainlagrettohethNe

text: H

text: The

text: relationship

text: between

text: A

text: in

text: magnitudes

text: and

text: the

text: ra-

text: V

text: where

text: S

text: is

text: the

text: surface

text: brightness

text: towards

text: the

text: dust

text: cloud,

text: tio

text: of

text: cloud

text: to

text: background

text: ﬂux

text: is

text: approximately

text: linear

text: for

text: the

text: dc,λ

text: and

text: S

text: is

text: the

text: local

text: background

text: surface

text: brightness.

text: The

text: ap-

text: ﬂux

text: ratios

text: in

text: the

text: observed

text: clouds,

text: for

text: A

text: values

text: from

text: bg,λ

text: V

text: parent

text: extinction

text: a

text: does

text: not

text: necessarily

text: match

text: the

text: true

text: ex-

text: meaning

text: that

text: the

text: relationship

text: between

text: A

text: and

text: N

text: is

text: approxi-

text: λ

text: V

text: H

text: tinction,

text: A

text: because

text: some

text: of

text: the

text: light

text: we

text: interpret

text: as

text: “back-

text: mately

text: linear

text: as

text: well.

text: λ

text: ground”

text: may

text: actually

text: be

text: located

text: in

text: front

text: of

text: the

text: cloud.

text: The

text: This

text: approach

text: provides

text: a

text: lower

text: limit

text: on

text: the

text: cloud

text: mass

text: for

text: measured

text: cloud-to-background

text: surface

text: brightness

text: ratio

text: can

text: be

text: three

text: reasons.

text: The

text: ﬁrst

text: reason

text: is

text: that

text: it

text: is

text: unlikely

text: that

text: all

text: expressed

text: using

text: the

text: quantity

text: x,

text: the

text: fraction

text: of

text: light

text: emitted

text: in

text: of

text: the

text: galaxy’s

text: light

text: in

text: the

text: direction

text: of

text: the

text: cloud

text: is

text: actually

text: ##LTLine##


text: 2 written as (cid:34) (cid:18) (cid:19)(cid:35) where Nobs and Nno-osc are, respectively, the num- νµ νµ ber of observed ν events and the number of expected ν − 1 − − ∆m2 µ µ PC1→ = 1 sin2(2θ) 1 e γ0x cos x (1) events in the absence of oscillations. From the muon neu- νµ νµ 2 2E trino beam, we assume the ratio to no oscillation that can be obtained by mean of the article [12] in which we take and (cid:40) (cid:18) (cid:19) the superior error bar as the probability uncertainty. The ratio to no oscillation from the muon antineutrino beam are obtained in the article [10], where using the Eq. (5) P˜νCµ7→νµ = 12 + e−γ0x 12 − sin2(2θ)(cid:18)sin2 ∆4(cid:19)mE(cid:41)2 x wase ﬁnd this ratio and deﬁne(cid:113)the probability uncertainty γ E ∆m2 + 0 sin φ sin(4θ) sin x , 2∆m2 2E ∆P → = Nνoµbs + α . (6) (2) νµ νµ Nνnµo-osc In this case, we assume also that the superior error − where ∆m2 = m2 m2 is the mass square diﬀerence, θ bars are Nobs uncertainty. So, α is a factor that reﬂects 3 3 is the mixing angle, γ0 is the dissipative eﬀect and x is the the systematic uncertainty obtained by mean of the dif- distance between the source and the detector. Note that ference between Nobs data uncertainty and Nobs statistic the γ0 parameter has diﬀerent meaning in Eqs. (1) and uncertainty. νµ νµ (2). In Eq. (1), the γ0 describes decoherence and in Eq. In order to improve our analysis, we calculate the mean (2) describes a more general quantum dissipative eﬀect, value of the survival probabilities, Eq. (1),(2) and (4), as it was discussed in the reference [1]. Furthermore, we in each range energy where was it deﬁned a bin energy. are following the same notation of [1], where superscript Furthermore, we consider, for sake of simplicity, the fol- Case 1 and Case 7 refers to Case 1 and Case 7 which lowing deﬁnition for χ2 fu(cid:0)nction (cid:1) were analysed in Ref. [1] and P˜ means that survival (cid:88) → ≥ probability is obtained when γn 0 to n 2. − 0 Pi Pi 2 As it is usual, we will assume an energy dependence of χ2 = exp theo (7) γ by means of a power-law(cid:18)writt(cid:19)en as σ2 0 i i E n where Peixp is the data obtained using the Eq. (6), γ0 = γ E0 , (3) Ptiheo is the theoretical survival probability and σ is the uncertainty deﬁned in Eq. (6). We also deﬁne the global ± ± χ2 as where n = 0, 1, 2. The energy scale, E , modulates glob 0 the magnitude expected for the dissipation eﬀects. This procedure is performed because the eﬀects are included χ2 = χ2 + χ2 , (8) glob ν ν¯ phenomenologically and, in the present moment, it is not possible to determine if these eﬀects are due to quantum gravity [23, 24] or to a hypothetic medium with reser- once that we will take into account in our analysis that voir behavior as it is thought via open quantum system neutrinos can be equivalent to antineutrino and the dis- approach [5, 6, 25–27]. sipative eﬀect must happen in both channel. We also consider de usual survival probability that can III. RESULTS AND DISCUSSIONS be obtained directly from Eq. (1) and (2) when we lead → γ 0, thus, it is written as (cid:18) (cid:19) We start the analysis considering the standard oscil- 0 lation model to verify if the approach introduced before yields results compatible to the MINOS result [12, 29]. − ∆m2 P → = 1 sin2(2θ) sin2 x . (4) νµ νµ 4E As we can see in the Table I, our results agree at 68% C.L. with the values obtained from MINOS collaboration In order to clearly see the dissipative eﬀects acting in to both neutrino and antineutrino parameters. MINOS the neutrino propagation, we will use a very simple ap- collaboration indicates that the oscillation parameter val- proach to perform the analysis. We will use only the ratio ues are: ∆m2 = 2.32+−0.12, sin2(2θ) = 1.00− ,∆m¯ 2 = to no oscillation that can be obtained supposing [28] 0.08 0.06 2.62+−0.40, sin2(2θ¯) = 0.95+−0.11[12, 29]. 0.37 0.12 The analysis from the neutrinos and antineutrinos Nobs P → = νµ , (5) show consistency [29], but the values for each parame- νµ νµ Nno-osc ters are diﬀerent from each other. Then, we perform νµ ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: written

text: as

text: where

text: Nobs

text: and

text: Nno-osc

text: are,

text: respectively,

text: the

text: num-

text: νµ

text: νµ

text: ber

text: of

text: observed

text: ν

text: events

text: and

text: the

text: number

text: of

text: expected

text: ν

text: µ

text: µ

text: e

text: cos

text: x

text: events

text: in

text: the

text: absence

text: of

text: oscillations.

text: From

text: the

text: muon

text: neu-

text: νµ

text: νµ

text: trino

text: beam,

text: we

text: assume

text: the

text: ratio

text: to

text: no

text: oscillation

text: that

text: can

text: be

text: obtained

text: by

text: mean

text: of

text: the

text: article

text: in

text: which

text: we

text: take

text: and

text: the

text: superior

text: error

text: bar

text: as

text: the

text: probability

text: uncertainty.

text: The

text: ratio

text: to

text: no

text: oscillation

text: from

text: the

text: muon

text: antineutrino

text: beam

text: are

text: obtained

text: in

text: the

text: article

text: where

text: using

text: the

text: Eq.

text: P˜νCµ7→νµ

text: x

text: wase

text: ﬁnd

text: this

text: ratio

text: and

text: probability

text: uncertainty

text: γ

text: E

text: sin

text: φ

text: sin

text: x

text: ∆P

text: Nνoµbs

text: α

text: νµ

text: νµ

text: Nνnµo-osc

text: In

text: this

text: case,

text: we

text: assume

text: also

text: that

text: the

text: superior

text: error

text: where

text: is

text: the

text: mass

text: square

text: diﬀerence,

text: θ

text: bars

text: are

text: Nobs

text: uncertainty.

text: So,

text: α

text: is

text: a

text: factor

text: that

text: reﬂects

text: is

text: the

text: mixing

text: angle,

text: γ0

text: is

text: the

text: dissipative

text: eﬀect

text: and

text: x

text: is

text: the

text: the

text: systematic

text: uncertainty

text: obtained

text: by

text: mean

text: of

text: the

text: dif-

text: distance

text: between

text: the

text: source

text: and

text: the

text: detector.

text: Note

text: that

text: ference

text: between

text: Nobs

text: data

text: uncertainty

text: and

text: Nobs

text: statistic

text: the

text: γ0

text: parameter

text: has

text: diﬀerent

text: meaning

text: in

text: Eqs.

text: and

text: uncertainty.

text: νµ

text: νµ

text: In

text: Eq.

text: the

text: γ0

text: describes

text: decoherence

text: and

text: in

text: Eq.

text: In

text: order

text: to

text: improve

text: our

text: analysis,

text: we

text: calculate

text: the

text: mean

text: describes

text: a

text: more

text: general

text: quantum

text: dissipative

text: eﬀect,

text: value

text: of

text: the

text: survival

text: probabilities,

text: Eq.

text: and

text: as

text: it

text: was

text: discussed

text: in

text: the

text: reference

text: Furthermore,

text: we

text: in

text: each

text: range

text: energy

text: where

text: was

text: it

text: deﬁned

text: a

text: bin

text: energy.

text: are

text: following

text: the

text: same

text: notation

text: of

text: where

text: superscript

text: Furthermore,

text: we

text: consider,

text: for

text: sake

text: of

text: simplicity,

text: the

text: fol-

text: Case

text: and

text: Case

text: refers

text: to

text: Case

text: and

text: Case

text: which

text: lowing

text: deﬁnition

text: for

text: χ2

text: were

text: analysed

text: in

text: Ref.

text: and

text: P˜

text: means

text: that

text: survival

text: probability

text: is

text: obtained

text: when

text: γn

text: to

text: n

text: Pi

text: Pi

text: As

text: it

text: is

text: usual,

text: we

text: will

text: assume

text: an

text: energy

text: dependence

text: of

text: χ2

text: exp

text: theo

text: γ

text: by

text: means

text: of

text: a

text: as

text: σ2

text: i

text: i

text: E

text: n

text: where

text: Peixp

text: is

text: the

text: data

text: obtained

text: using

text: the

text: Eq.

text: γ0

text: γ

text: Ptiheo

text: is

text: the

text: theoretical

text: survival

text: probability

text: and

text: σ

text: is

text: the

text: uncertainty

text: deﬁned

text: in

text: Eq.

text: We

text: also

text: deﬁne

text: the

text: global

text: χ2

text: as

text: where

text: n

text: The

text: energy

text: scale,

text: E

text: modulates

text: glob

text: the

text: magnitude

text: expected

text: for

text: the

text: dissipation

text: eﬀects.

text: This

text: procedure

text: is

text: performed

text: because

text: the

text: eﬀects

text: are

text: included

text: χ2

text: χ2

text: χ2

text: glob

text: ν

text: ν¯

text: phenomenologically

text: and,

text: in

text: the

text: present

text: moment,

text: it

text: is

text: not

text: possible

text: to

text: determine

text: if

text: these

text: eﬀects

text: are

text: due

text: to

text: quantum

text: gravity

text: or

text: to

text: a

text: hypothetic

text: medium

text: with

text: reser-

text: once

text: that

text: we

text: will

text: take

text: into

text: account

text: in

text: our

text: analysis

text: that

text: voir

text: behavior

text: as

text: it

text: is

text: thought

text: via

text: open

text: quantum

text: system

text: neutrinos

text: can

text: be

text: equivalent

text: to

text: antineutrino

text: and

text: the

text: dis-

text: approach

text: sipative

text: eﬀect

text: must

text: happen

text: in

text: both

text: channel.

text: We

text: also

text: consider

text: de

text: usual

text: survival

text: probability

text: that

text: can

text: III.

text: RESULTS

text: AND

text: DISCUSSIONS

text: be

text: obtained

text: directly

text: from

text: Eq.

text: and

text: when

text: we

text: lead

text: γ

text: thus,

text: it

text: is

text: written

text: as

text: We

text: start

text: the

text: analysis

text: considering

text: the

text: standard

text: oscil-

text: lation

text: model

text: to

text: verify

text: if

text: the

text: approach

text: introduced

text: before

text: yields

text: results

text: compatible

text: to

text: the

text: MINOS

text: result

text: P

text: x

text: νµ

text: νµ

text: As

text: we

text: can

text: see

text: in

text: the

text: Table

text: I,

text: our

text: results

text: agree

text: at

text: C.L.

text: with

text: the

text: values

text: obtained

text: from

text: MINOS

text: collaboration

text: In

text: order

text: to

text: clearly

text: see

text: the

text: dissipative

text: eﬀects

text: acting

text: in

text: to

text: both

text: neutrino

text: and

text: antineutrino

text: parameters.

text: MINOS

text: the

text: neutrino

text: propagation,

text: we

text: will

text: use

text: a

text: very

text: simple

text: ap-

text: collaboration

text: indicates

text: that

text: the

text: oscillation

text: parameter

text: val-

text: proach

text: to

text: perform

text: the

text: analysis.

text: We

text: will

text: use

text: only

text: the

text: ratio

text: ues

text: are:

text: ,∆m¯

text: to

text: no

text: oscillation

text: that

text: can

text: be

text: obtained

text: supposing

text: The

text: analysis

text: from

text: the

text: neutrinos

text: and

text: antineutrinos

text: Nobs

text: P

text: νµ

text: show

text: consistency

text: but

text: the

text: values

text: for

text: each

text: parame-

text: νµ

text: νµ

text: Nno-osc

text: ters

text: are

text: diﬀerent

text: from

text: each

text: other.

text: Then,

text: we

text: perform

text: νµ

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: v p s (cid:48) u (cid:48) p s (cid:48) v Figure 2: Two interpretations from ?? deﬁnition synchronized at positions depicted by dotted lines. 4. Main result Theorem 6. Any PD0L system that is not weakly circular is repetitive. The two following lemmas will be used to prove this theorem,. The next lemma and its proof is based on the ideas in the proof of Theorem 4.35 in [K˚ur03]. A Lemma 7. Let G = ( , ϕ, w) be a PD0L system. If there exists a sequence ∞ ∈ N (cid:15)(k) with lim (cid:15)(k) = + and if for any k there are two non-empty words → ∞ k + u and v in S(L(G)) containing an unbounded letter such that the following conditions are satisﬁed | | (i) u = k; (ii) there are two integers m and n such that m > n and letters a and b such ∈ { } that for each i m, n the word ϕi(u) is a factor of ϕi(v) and ϕi(v) is a | | | | factor of ϕi(aub), moreover, |ϕi(u)| > (cid:15)(k) or |ϕi(u)| > (cid:15)(k); and ϕi(a) ϕi(b) ∈ { } (iii) for each i m, n the factor ϕi(u) has no synchronizing point: two non- − − synchronized interpretations are (ε, ϕi 1(u), ε) and (p , ϕi 1(v), s ), i i then the D0L system is repetitive. | | Proof. Suppose that |ϕi(u)| > (cid:15)(k) is true in requirement (ii), the other case ϕi(a) | | |ϕi(u)| > (cid:15)(k) is analogous. It holds that ϕi(b) − − − ϕm(v) = p ϕm(u)s = ϕm n(ϕn(v)) = ϕm n(p )ϕm(u)ϕm n(s ). m m n n − − The fact that the interpretations (ε, ϕm 1(u), ε) and (p , ϕm 1(v), s ) are not (cid:54) − m − m synchronized implies that p = ϕm n(p ) (if p = ϕm n(p ), the two in- m n m n terpretations of ϕm(u) are synchronized at position 0, see Figure 3). Since 4 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: v

text: p

text: s

text: u

text: p

text: s

text: v

text: Figure

text: Two

text: interpretations

text: from

text: deﬁnition

text: synchronized

text: at

text: positions

text: depicted

text: by

text: dotted

text: lines.

text: Main

text: result

text: Theorem

text: Any

text: system

text: that

text: is

text: not

text: weakly

text: circular

text: is

text: repetitive.

text: The

text: two

text: following

text: lemmas

text: will

text: be

text: used

text: to

text: prove

text: this

text: theorem,.

text: The

text: next

text: lemma

text: and

text: its

text: proof

text: is

text: based

text: on

text: the

text: ideas

text: in

text: the

text: proof

text: of

text: Theorem

text: in

text: A

text: Lemma

text: Let

text: G

text: ϕ,

text: w)

text: be

text: a

text: system.

text: If

text: there

text: exists

text: a

text: sequence

text: N

text: with

text: lim

text: and

text: if

text: for

text: any

text: k

text: there

text: are

text: two

text: non-empty

text: words

text: k

text: u

text: and

text: v

text: in

text: S(L(G))

text: containing

text: an

text: unbounded

text: letter

text: such

text: that

text: the

text: following

text: conditions

text: are

text: satisﬁed

text: (i)

text: u

text: k;

text: (ii)

text: there

text: are

text: two

text: integers

text: m

text: and

text: n

text: such

text: that

text: m

text: n

text: and

text: letters

text: a

text: and

text: b

text: such

text: that

text: for

text: each

text: i

text: m,

text: n

text: the

text: word

text: ϕi(u)

text: is

text: a

text: factor

text: of

text: ϕi(v)

text: and

text: ϕi(v)

text: is

text: a

text: factor

text: of

text: ϕi(aub),

text: moreover,

text: |ϕi(u)|

text: or

text: |ϕi(u)|

text: and

text: ϕi(a)

text: ϕi(b)

text: (iii)

text: for

text: each

text: i

text: m,

text: n

text: the

text: factor

text: ϕi(u)

text: has

text: no

text: synchronizing

text: point:

text: two

text: non-

text: synchronized

text: interpretations

text: are

text: (ε,

text: ϕi

text: ε)

text: and

text: (p

text: ϕi

text: s

text: i

text: i

text: then

text: the

text: system

text: is

text: repetitive.

text: Proof.

text: Suppose

text: that

text: |ϕi(u)|

text: is

text: true

text: in

text: requirement

text: (ii),

text: the

text: other

text: case

text: ϕi(a)

text: |ϕi(u)|

text: is

text: analogous.

text: It

text: holds

text: that

text: ϕi(b)

text: ϕm(v)

text: p

text: ϕm(u)s

text: ϕm

text: n(ϕn(v))

text: ϕm

text: n(p

text: )ϕm(u)ϕm

text: n(s

text: m

text: m

text: n

text: n

text: The

text: fact

text: that

text: the

text: interpretations

text: (ε,

text: ϕm

text: ε)

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