text: 22 Charles Curry, Kurusch Ebrahimi–Fard, Simon J.A. Malham and Anke Wiese (cid:0) (cid:1) (cid:0) (cid:1) A (cid:0) (cid:1) (cid:0) (cid:1) any alphabet constructed from at least one deterministic and one non- A∗ ⊂ A∗ deterministic letter, we have π ≤ π ≤ . Thus correspond- gms n A∗ ⊂ gwl n A∗ ingly for their complements π ≥ π ≥ . Second, let Rms gwl n+1 gms n+1 and Rwl denote the remainder endomorphisms associated with truncating the (cid:0) (cid:1) (cid:0) (cid:1) stochastic Taylor expansion by mean-square and word length gradings, re- (cid:104) (cid:105) (cid:104) (cid:105) spectively. Hence for the inner products Rms, Rms and Rwl, Rwl we sum (cid:0) (cid:1) A∗ (cid:0) (cid:1) A∗ over words in π ≥ and π ≥ , respectively. The diﬀerence gms n+1 gwl−n+1 remainder endomorphism Rˆ := Rms Rwl is at leading order non-zero on A∗ ∩ A∗ words in π π ≤ . For any word w in this set we have gms=n+1 gwl n ≤ 2ζ(w) + ξ(w) = n + 1 and ζ(w) + ξ(w) n, which imply ζ(w) > 0 and (cid:104) (cid:105) thus ξ(w) < n + 1. Now consider the inner product Rwl, Rˆ . Any word u ≥ on which Rwl is non-trivial at leading order has length ζ(u) + ξ(u) n + 1. Any word v on which Rˆ is non-trivial at leading order, we have just shown (cid:54) (cid:54) that ξ(v) < n + 1. Hence we must have ξ(u) = ξ(v) and so red(u) = red(v). (cid:104) (cid:105) We deduce that Rwl, Rˆ is zero. The result then follows from the relation (cid:107) (cid:107) (cid:107) (cid:107) (cid:104) (cid:105) (cid:107) (cid:107) Rms 2 = Rwl 2 + 2 Rˆ, Rwl + Rˆ 2. This result holds whether we include ∗ the 0n in ϕˆms or not. t 5 Antisymmetric sign reverse integrator We now present the error analysis of map-truncate-invert schemes, con- cluding with the description of the antisymmetric sign reverse integrator and the main theorem concerning its eﬃciency. We also present the direct map- § truncate-invert schemes alluded to at the end of 3. We begin by introduc- ing the pre-remainder endomorphism associated to any map-truncate-invert scheme, that is the remainder terms associated to the truncation of the se- ries F(cid:63)(id). We relate the pre-remainder and remainder of such schemes and use this relation to compute the inverse step of a direct map-truncate-invert scheme. We follow with the introduction of the antisymmetric sign reverse in- tegrator and its explicit characterization. We then prove our main theorem on the eﬃciency of the antisymmetric sign reverse integrator. R(cid:104)A(cid:105) → Deﬁnition 14 (Pre-remainder endomorphism) Let F : End( ∗) R(cid:104)A(cid:105) End( ∗) be an invertible map and suppose π ≤ is a projection corre- g n sponding to a truncation. The pre-remainder endomorphism Q associated to ◦ π ≤ F(id) is deﬁned by g n − ◦ Q := F(id) π ≤ F(id). g n This deﬁnition allows for more general maps F than power series in the con- volution algebra, which we require presently. The relationship between the pre-remainder and remainder is critical to the error analysis of map-truncate- invert schemes. Hereafter we assume a grading g which is preserved by the quasi-shuﬄe ∈ A∗ product in question. By this we mean that for all words u, v , the poly- ∗ nomial u v is homogeneous of degree g(u) + g(v), i.e. it is a sum of words of ##LTLine##

text: Charles

text: Curry,

text: Kurusch

text: Ebrahimi–Fard,

text: Simon

text: J.A.

text: Malham

text: and

text: Anke

text: Wiese

text: A

text: any

text: alphabet

text: constructed

text: from

text: at

text: least

text: one

text: deterministic

text: and

text: one

text: non-

text: A∗

text: A∗

text: deterministic

text: letter,

text: we

text: have

text: π

text: π

text: Thus

text: correspond-

text: gms

text: n

text: A∗

text: gwl

text: n

text: A∗

text: ingly

text: for

text: their

text: complements

text: π

text: π

text: Second,

text: let

text: Rms

text: gwl

text: gms

text: and

text: Rwl

text: denote

text: the

text: remainder

text: endomorphisms

text: associated

text: with

text: truncating

text: the

text: stochastic

text: Taylor

text: expansion

text: by

text: mean-square

text: and

text: word

text: length

text: gradings,

text: re-

text: spectively.

text: Hence

text: for

text: the

text: inner

text: products

text: Rms,

text: Rms

text: and

text: Rwl,

text: Rwl

text: we

text: sum

text: A∗

text: A∗

text: over

text: words

text: in

text: π

text: and

text: π

text: respectively.

text: The

text: diﬀerence

text: gms

text: remainder

text: endomorphism

text: Rˆ

text: Rms

text: Rwl

text: is

text: at

text: leading

text: order

text: non-zero

text: on

text: A∗

text: A∗

text: words

text: in

text: π

text: π

text: For

text: any

text: word

text: w

text: in

text: this

text: set

text: we

text: have

text: gwl

text: n

text: ξ(w)

text: n

text: and

text: ζ(w)

text: ξ(w)

text: n,

text: which

text: imply

text: ζ(w)

text: and

text: thus

text: ξ(w)

text: n

text: Now

text: consider

text: the

text: inner

text: product

text: Rwl,

text: Rˆ

text: Any

text: word

text: u

text: on

text: which

text: Rwl

text: is

text: non-trivial

text: at

text: leading

text: order

text: has

text: length

text: ζ(u)

text: ξ(u)

text: n

text: Any

text: word

text: v

text: on

text: which

text: Rˆ

text: is

text: non-trivial

text: at

text: leading

text: order,

text: we

text: have

text: just

text: shown

text: that

text: ξ(v)

text: n

text: Hence

text: we

text: must

text: have

text: ξ(u)

text: ξ(v)

text: and

text: so

text: red(u)

text: red(v).

text: We

text: deduce

text: that

text: Rwl,

text: Rˆ

text: is

text: zero.

text: The

text: result

text: then

text: follows

text: from

text: the

text: relation

text: Rms

text: Rwl

text: Rˆ,

text: Rwl

text: Rˆ

text: This

text: result

text: holds

text: whether

text: we

text: include

text: the

text: in

text: ϕˆms

text: or

text: not.

text: t

text: Antisymmetric

text: sign

text: reverse

text: integrator

text: We

text: now

text: present

text: the

text: error

text: analysis

text: of

text: map-truncate-invert

text: schemes,

text: con-

text: cluding

text: with

text: the

text: description

text: of

text: the

text: antisymmetric

text: sign

text: reverse

text: integrator

text: and

text: the

text: main

text: theorem

text: concerning

text: its

text: eﬃciency.

text: We

text: also

text: present

text: the

text: direct

text: map-

text: truncate-invert

text: schemes

text: alluded

text: to

text: at

text: the

text: end

text: of

text: We

text: begin

text: by

text: introduc-

text: ing

text: the

text: pre-remainder

text: endomorphism

text: associated

text: to

text: any

text: map-truncate-invert

text: scheme,

text: that

text: is

text: the

text: remainder

text: terms

text: associated

text: to

text: the

text: truncation

text: of

text: the

text: se-

text: ries

text: We

text: relate

text: the

text: pre-remainder

text: and

text: remainder

text: of

text: such

text: schemes

text: and

text: use

text: this

text: relation

text: to

text: compute

text: the

text: inverse

text: step

text: of

text: a

text: direct

text: map-truncate-invert

text: scheme.

text: We

text: follow

text: with

text: the

text: introduction

text: of

text: the

text: antisymmetric

text: sign

text: reverse

text: in-

text: tegrator

text: and

text: its

text: explicit

text: characterization.

text: We

text: then

text: prove

text: our

text: main

text: theorem

text: on

text: the

text: eﬃciency

text: of

text: the

text: antisymmetric

text: sign

text: reverse

text: integrator.

text: Deﬁnition

text: (Pre-remainder

text: endomorphism)

text: Let

text: F

text: End(

text: End(

text: be

text: an

text: invertible

text: map

text: and

text: suppose

text: π

text: is

text: a

text: projection

text: corre-

text: g

text: n

text: sponding

text: to

text: a

text: truncation.

text: The

text: pre-remainder

text: endomorphism

text: Q

text: associated

text: to

text: π

text: F(id)

text: is

text: deﬁned

text: by

text: g

text: n

text: Q

text: F(id)

text: π

text: F(id).

text: g

text: n

text: This

text: deﬁnition

text: allows

text: for

text: more

text: general

text: maps

text: F

text: than

text: power

text: series

text: in

text: the

text: con-

text: volution

text: algebra,

text: which

text: we

text: require

text: presently.

text: The

text: relationship

text: between

text: the

text: pre-remainder

text: and

text: remainder

text: is

text: critical

text: to

text: the

text: error

text: analysis

text: of

text: map-truncate-

text: invert

text: schemes.

text: Hereafter

text: we

text: assume

text: a

text: grading

text: g

text: which

text: is

text: preserved

text: by

text: the

text: quasi-shuﬄe

text: A∗

text: product

text: in

text: question.

text: By

text: this

text: we

text: mean

text: that

text: for

text: all

text: words

text: u,

text: v

text: the

text: poly-

text: nomial

text: u

text: v

text: is

text: homogeneous

text: of

text: degree

text: g(u)

text: g(v),

text: i.e.

text: it

text: is

text: a

text: sum

text: of

text: words

text: of

text: ##LTLine##


text: 32 Charles Curry, Kurusch Ebrahimi–Fard, Simon J.A. Malham and Anke Wiese The terms in the second sum with a1 = a2 = a3 are zero and need not be evaluated. The modiﬁed antisymmetric sign reverse integrator includes the (cid:88) (cid:0) (cid:1) following additional terms, 1 − I I I I + I V˜ . 3 aa a [a,a] a [a,a,a] w w=aaa Example 5 We now examine the case of order one half integrators. Here we − − have coshlog(cid:63)(i(cid:88)d) = ν + 12J(cid:63)(cid:88)2 and (cid:0)(S Sˆ)(a1a2) = 12[a2a(cid:1)1], hence the ap- proximation takes the form − I V˜ + 1 I I + I I I V˜ | |≤ w w 2 a1 a2 a2 a1 [a2a1] w w 1 w=a1a2 (cid:0) (cid:1) Consider the equation dy = A y dt + A y dW1 + V (y )dW2 + A y dN˜ , t 0 t 1 t t 2 t t 3 t t w(cid:0) here W1, W2 are Wiener pr(cid:1)ocesses and N˜ is a standard Poisson process t t t with intensity λ. Here V is the nonlinear vector ﬁeld V x , x , x , x = 2 2 1 2 3 4 − T sin(x ), cos(x ), x, sin(x ) , whilst the constant coeﬃcientlinear vector 1 2 4 3 ﬁelds are deﬁned by the following matrices      0.314724 0.132359 0.457507 0.457167  − − 0.405792 0.402460 0.464889 0.014624 A = − − − 0  0.373013 0.221502 0.342387 −0.300280  0.413376 0.046882 0.470593 0.358114   −   0.078239 0.155741 0.178735 0.155478  − − 0.415736 0.464288 0.257740 0.328813 A = 1  0.292207 0.349129 0.243132 0.206046  − − 0.459492 0.433993 0.107773 0.468167   − − −   0.223077 0.194829 0.061256 0.313127 − − − − 0.453829 0.182901 0.118442 0.010236 A = − − . 3 0.402868 0.450222 0.265517 0.054414 − 0.323458 0.465554 0.295200 0.146313 | − | We compare the global mean square error E(sup ≤ ≤ y yˆ 2)1/2, estimated 0 t T t t by sampling 1000 paths, for two approximations of yˆ : the Milstein scheme and t the order 1 modiﬁed antisymmetric sign reverse integrator. The intensity of N˜ was taken to be λ = 50 and the initial condition was y = (1, 0.8, 0.6, 0.4)T . t 0 The code employed is available in the electronic supplementary material to- gether with similar plots for the drift-diﬀusion case, linear vector ﬁelds and so forth. We observe in Figure 1 (left panel) that the antisymmetric sign reverse integrator is more accurate than the Milstein scheme for any given step size, in accordance with the theory. The situation is more nuanced when the mean square error is plotted against the computational time as in Figure 1 (right panel). Generically the Milstein scheme outperforms the modiﬁed antisymmet- ric sign reverse integrator for large step sizes, when the computational eﬀort of the schemes is dominated by the evaluation of the vector ﬁelds. For smaller ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: Charles

label: author

text: Curry,

text: Kurusch

text: Ebrahimi–Fard,

text: Simon

text: J.A.

text: Malham

text: and

text: Anke

text: Wiese

text: The

text: terms

text: in

text: the

text: second

text: sum

text: with

text: are

text: zero

text: and

text: need

text: not

text: be

text: evaluated.

text: The

text: modiﬁed

text: antisymmetric

text: sign

text: reverse

text: integrator

text: includes

text: the

text: following

text: additional

text: terms,

text: I

text: I

text: I

text: I

text: I

text: V˜

text: aa

text: a

text: [a,a]

text: a

text: [a,a,a]

text: w

text: w=aaa

text: Example

text: We

text: now

text: examine

text: the

text: case

text: of

text: order

text: one

text: half

text: integrators.

text: Here

text: we

text: have

text: ν

text: and

text: hence

text: the

text: ap-

text: proximation

text: takes

text: the

text: form

text: I

text: V˜

text: I

text: I

text: I

text: I

text: I

text: V˜

text: w

text: w

text: w

text: w

text: Consider

text: the

text: equation

text: dy

text: A

text: y

text: dt

text: A

text: y

text: V

text: (y

text: A

text: y

text: dN˜

text: t

text: t

text: t

text: t

text: t

text: t

text: t

text: t

text: here

text: are

text: Wiener

text: and

text: N˜

text: is

text: a

text: standard

text: Poisson

text: process

text: t

text: t

text: t

text: with

text: intensity

text: λ.

text: Here

text: V

text: is

text: the

text: nonlinear

text: vector

text: ﬁeld

text: V

text: x

text: x

text: x

text: x

text: T

text: sin(x

text: cos(x

text: x,

text: sin(x

text: whilst

text: the

text: constant

text: coeﬃcientlinear

text: vector

text: ﬁelds

text: are

text: deﬁned

text: by

text: the

text: following

text: matrices

text: A

text: A

text: A

text: We

text: compare

text: the

text: global

text: mean

text: square

text: error

text: E(sup

text: y

text: yˆ

text: estimated

text: t

text: T

text: t

text: t

text: by

text: sampling

text: paths,

text: for

text: two

text: approximations

text: of

text: yˆ

text: the

text: Milstein

text: scheme

text: and

text: t

text: the

text: order

text: modiﬁed

text: antisymmetric

text: sign

text: reverse

text: integrator.

text: The

text: intensity

text: of

text: N˜

text: was

text: taken

text: to

text: be

text: λ

text: and

text: the

text: initial

text: condition

text: was

text: y

text: t

text: The

text: code

text: employed

text: is

text: available

text: in

text: the

text: electronic

text: supplementary

text: material

text: to-

text: gether

text: with

text: similar

text: plots

text: for

text: the

text: drift-diﬀusion

text: case,

text: linear

text: vector

text: ﬁelds

text: and

text: so

text: forth.

text: We

text: observe

text: in

text: Figure

text: (left

text: panel)

text: that

text: the

text: antisymmetric

text: sign

text: reverse

text: integrator

text: is

text: more

text: accurate

text: than

text: the

text: Milstein

text: scheme

text: for

text: any

text: given

text: step

text: size,

text: in

text: accordance

text: with

text: the

text: theory.

text: The

text: situation

text: is

text: more

text: nuanced

text: when

text: the

text: mean

text: square

text: error

text: is

text: plotted

text: against

text: the

text: computational

text: time

text: as

text: in

text: Figure

text: (right

text: panel).

text: Generically

text: the

text: Milstein

text: scheme

text: outperforms

text: the

text: modiﬁed

text: antisymmet-

text: ric

text: sign

text: reverse

text: integrator

text: for

text: large

text: step

text: sizes,

text: when

text: the

text: computational

text: eﬀort

text: of

text: the

text: schemes

text: is

text: dominated

text: by

text: the

text: evaluation

text: of

text: the

text: vector

text: ﬁelds.

text: For

text: smaller

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: SDEs driven by L´evy processes 33 stepsizes, the computational cost of simulating iterated integrals dominates, upon which the modiﬁed antisymmetric sign reverse integrator outperforms the Milstein scheme. Acknowledgement C.C. would like to thank Michael Tretyakov and Seva Shneer for useful com- ments. We are very grateful to the referees whose comments and suggestions signiﬁcantly helped to improve the presentation of the original manuscript. References 1. Abe, E. 1980 Hopf algebras, Cambridge University Press. 2. Applebaum, D. 2004 L´evy Processes and Stochastic Calculus, Cambridge University Press. 3. Azencott, R. 1982 Formule de Taylor stochastique et d´eveloppement asymptotique d’int´egrales de Feynman, Seminar on Probability XVI, Lecture Notes in Math., 921, Springer, 237–285. 4. Barndorﬀ–Nielsen, O., Mikosch, T., Resnick, S., ed. 2001 L´evy processes: theory and applications, Springer, New York. 5. Barski, M. 2015 Approximations for solutions of L´evy-type stochastic diﬀerential equa- tions, arXiv:1512.06572v1 6. Baudoin, F. 2004 An introduction to the geometry of stochastic ﬂows, Imperial College Press. 7. Ben Arous, G. 1989 Flots et series de Taylor stochastiques, Probab. Theory Relat. Fields, 81, 29–77. 8. Brouder, C., 2000, Runge–Kutta methods and renormalization, Eur. Phys. J. C 12, 521–534. 9. Burrage, K., Burrage, P. M. 1999 High strong order methods for non-commutative stochastic ordinary diﬀerential equation systems and the Magnus formula, Phys. D 133, 34–48. 10. Butcher, J.C., 1972 An algebraic theory of integration methods, Math. Comp. 26, 79– 106. 11. Castell, F. 1993 Asymptotic expansion of stochastic ﬂows, Probab. Theory Related Fields, 96, 225–239. 12. Castell, F., Gaines, J., 1995 An eﬃcient approximation method for stochastic diﬀerential equations by means of the exponential Lie series, Math. Comp. Simulation 38, 13–19. (doi:10.1016/0378-4754(93)E0062-A) 13. Castell, F., Gaines, J., 1996 The ordinary diﬀerential equation approach to asymptot- ically eﬃcient schemes for solutions of stochastic diﬀerential equations, Ann. Inst. H. Poincar´e Probab. Statist. 32, 231–250. 14. Chen, K.-T. 1968 Algebraic paths., Journal of Algebra, 10, 8–36. 15. Clark, J.M.C. 1982 An eﬃcient approximation for a class of stochastic diﬀerential equations, in Advances in Filtering and Optimal Stochastic Control, Cocoyoc, Mex- ico, 1982, Lecture Notes in Control and Information Sciences, 42, Berlin etc: Springer. (doi:10.1007/BFb0004526) 16. Connes, A., Kreimer, D. 1998 Hopf algebras, renormalization and noncommutative ge- ometry, Commun. Math. Phys. 199, 203–242 17. Cont, R., Tankov, P. 2004 Financial Modelling with Jump Processes, Chapman & Hal- l/CRC. 18. Curry, C., 2014 Algebraic structures in stochastic diﬀerential equations, PhD thesis, Heriot–Watt University. 19. Curry, C., Ebrahimi–Fard, K., Malham, S.J.A., Wiese, A. 2014 L´evy processes and quasi-shuﬄe algebras, Stochastics 86(4), 632–642. (doi:10.1080/17442508.2013.865131) ##LTLine##

text: SDEs

text: driven

text: by

text: L´evy

text: processes

text: stepsizes,

text: the

text: computational

text: cost

text: of

text: simulating

text: iterated

text: integrals

text: dominates,

text: upon

text: which

text: the

text: modiﬁed

text: antisymmetric

text: sign

text: reverse

text: integrator

text: outperforms

text: the

text: Milstein

text: scheme.

text: Acknowledgement

label: section

text: C.C.

text: would

text: like

text: to

text: thank

text: Michael

text: Tretyakov

text: and

text: Seva

text: Shneer

text: for

text: useful

text: com-

text: ments.

text: We

text: are

text: very

text: grateful

text: to

text: the

text: referees

text: whose

text: comments

text: and

text: suggestions

text: signiﬁcantly

text: helped

text: to

text: improve

text: the

text: presentation

text: of

text: the

text: original

text: manuscript.

text: References

label: reference

text: Abe,

text: E.

text: Hopf

text: algebras,

text: Cambridge

text: University

text: Press.

text: Applebaum,

text: D.

text: L´evy

text: Processes

text: and

text: Stochastic

text: Calculus,

text: Cambridge

text: University

text: Press.

text: Azencott,

text: R.

text: Formule

text: de

text: Taylor

text: stochastique

text: et

text: d´eveloppement

text: asymptotique

text: d’int´egrales

text: de

text: Feynman,

text: Seminar

text: on

text: Probability

text: XVI,

text: Lecture

text: Notes

text: in

text: Math.,

text: Springer,

text: Barndorﬀ–Nielsen,

text: O.,

text: Mikosch,

text: T.,

text: Resnick,

text: S.,

text: ed.

text: L´evy

text: processes:

text: theory

text: and

text: applications,

text: Springer,

text: New

text: York.

text: Barski,

text: M.

text: Approximations

text: for

text: solutions

text: of

text: L´evy-type

text: stochastic

text: diﬀerential

text: equa-

text: tions,

text: Baudoin,

text: F.

text: An

text: introduction

text: to

text: the

text: geometry

text: of

text: stochastic

text: ﬂows,

text: Imperial

text: College

text: Press.

text: Ben

text: Arous,

text: G.

text: Flots

text: et

text: series

text: de

text: Taylor

text: stochastiques,

text: Probab.

text: Theory

text: Relat.

text: Fields,

text: Brouder,

text: C.,

text: Runge–Kutta

text: methods

text: and

text: renormalization,

text: Eur.

text: Phys.

text: J.

text: C

text: Burrage,

text: K.,

text: Burrage,

text: P.

text: M.

text: High

text: strong

text: order

text: methods

text: for

text: non-commutative

text: stochastic

text: ordinary

text: diﬀerential

text: equation

text: systems

text: and

text: the

text: Magnus

text: formula,

text: Phys.

text: D

text: Butcher,

text: J.C.,

text: An

text: algebraic

text: theory

text: of

text: integration

text: methods,

text: Math.

text: Comp.

text: Castell,

text: F.

text: Asymptotic

text: expansion

text: of

text: stochastic

text: ﬂows,

text: Probab.

text: Theory

text: Related

text: Fields,

text: Castell,

text: F.,

text: Gaines,

text: J.,

text: An

text: eﬃcient

text: approximation

text: method

text: for

text: stochastic

text: diﬀerential

text: equations

text: by

text: means

text: of

text: the

text: exponential

text: Lie

text: series,

text: Math.

text: Comp.

text: Simulation

text: Castell,

text: F.,

text: Gaines,

text: J.,

text: The

text: ordinary

text: diﬀerential

text: equation

text: approach

text: to

text: asymptot-

text: ically

text: eﬃcient

text: schemes

text: for

text: solutions

text: of

text: stochastic

text: diﬀerential

text: equations,

text: Ann.

text: Inst.

text: H.

text: Poincar´e

text: Probab.

text: Statist.

text: Chen,

text: K.-T.

text: Algebraic

text: paths.,

text: Journal

text: of

text: Algebra,

text: Clark,

text: J.M.C.

text: An

text: eﬃcient

text: approximation

text: for

text: a

text: class

text: of

text: stochastic

text: diﬀerential

text: equations,

text: in

text: Advances

text: in

text: Filtering

text: and

text: Optimal

text: Stochastic

text: Control,

text: Cocoyoc,

text: Mex-

text: ico,

text: Lecture

text: Notes

text: in

text: Control

text: and

text: Information

text: Sciences,

text: Berlin

text: etc:

text: Springer.

text: Connes,

text: A.,

text: Kreimer,

text: D.

text: Hopf

text: algebras,

text: renormalization

text: and

text: noncommutative

text: ge-

text: ometry,

text: Commun.

text: Math.

text: Phys.

text: Cont,

text: R.,

text: Tankov,

text: P.

text: Financial

text: Modelling

text: with

text: Jump

text: Processes,

text: Chapman

text: Hal-

text: l/CRC.

text: Curry,

text: C.,

text: Algebraic

text: structures

text: in

text: stochastic

text: diﬀerential

text: equations,

text: PhD

text: thesis,

text: Heriot–Watt

text: University.

text: Curry,

text: C.,

text: Ebrahimi–Fard,

text: K.,

text: Malham,

text: S.J.A.,

text: Wiese,

text: A.

text: L´evy

text: processes

text: and

text: quasi-shuﬄe

text: algebras,

text: Stochastics

text: ##LTLine##


text: 6 Charles Curry, Kurusch Ebrahimi–Fard, Simon J.A. Malham and Anke Wiese − equal to the true remainder, i.e. equal to the remainder after applying f 1 to the truncated f(ϕ ). Herein lies the room-for-manouver of which we take ad- t vantage. Naturally the map-truncate-invert scheme must emulate the scheme based on the corresponding stochastic Taylor expansion truncated according to the same grading. The question is thus, is the pre-remainder (and thus re- mainder) associated with the map-truncate-invert scheme at hand less than the remainder associated with the corresponding stochastic Taylor approxima- tion to leading order in the mean-square measure? The answer is “yes” for the antisymmetric sign reverse integrator. This integrator is constructed by taking half the diﬀerence of the stochastic Taylor expansion and the stochastic Taylor expansion with all the words indexing the multiple integrals reversed in order and signed according to the lengt(cid:88)h of the(cid:0) words. Hen(cid:1)ce the pre-remainder and thus remainder are − RASRI(y ) = 1 I I V (y ). t 0 2 w S(w) w 0 | | w =n+1 | | Here w represents the length of the word w and S is the signed reversal (cid:55)→ − map deﬁned as S : a a . . . a ( 1)n a a − . . . a for any word consisting 1 2(cid:13) n A (cid:13) n(cid:13) n 1 (cid:13) 1 ≡ − of letters a , a , . . . , a (cid:13)from . W(cid:13)e natu(cid:13)rally assu(cid:13)me I− I . Our main 1 2 n w w result is thus RASRI(y ) 2 (cid:54) Rwl(y ) 2 , t 0 L2 t 0 L2 for all y , all L´evy-driven systems, to all orders n. Using our result above this 0 means the antisymmetric sign reverse integrator is more accurate than the corresponding stochastic Taylor approximation truncated according to mean- square grading. Furthermore we show that when n is odd, the inequality just above is optimal in the following sense. Suppose we perturb the antisymmet- § ric sign reverse integrator by a general class of perturbations—see for more details. Then when n is odd we show that the diﬀerence between the two mean-square error measures for Rwl(y ) and RASRI(y ) above are minimized t 0 t 0 when the perturbation is zero. One further technical practical aspect comes − into play. In the Castell–Gaines case we compute f 1 = exp by solving an or- − dinary diﬀerential system. If all the underlying vector ﬁelds are linear f 1 may be computed as a matrix-valued compositional inverse function; see Malham & Wiese [51]. However we cannot do this in general for map-truncate-invert schemes. To circumvent this issue we have developed direct-map-truncate- invert schemes with a special case being the direct antisymmetric sign reverse integrator. In direct-map-truncate-invert schemes we utilize the observation above that map-truncate-invert schemes naturally emulate the scheme based on the corresponding stochastic Taylor approximation to the same order, but have a diﬀerent remainder at leading order as diﬀerent terms at leading order in the remainder are included in the integrator. Thus in order to simulate a direct-map-truncate-invert scheme we simulate the corresponding stochas- tic Taylor approximation to the same order as well as the terms that lie in the diﬀerence between the stochastic Taylor approximation remainder and map-truncate-invert approximation remainder at leading order. These are ad- ditional terms indexed by words of length n + 1, the terms concerned will not ##LTLine## ##LTLine##

text: Charles

text: Curry,

text: Kurusch

text: Ebrahimi–Fard,

text: Simon

text: J.A.

text: Malham

text: and

text: Anke

text: Wiese

text: equal

text: to

text: the

text: true

text: remainder,

text: i.e.

text: equal

text: to

text: the

text: remainder

text: after

text: applying

text: f

text: to

text: the

text: truncated

text: f(ϕ

text: Herein

text: lies

text: the

text: room-for-manouver

text: of

text: which

text: we

text: take

text: ad-

text: t

text: vantage.

text: Naturally

text: the

text: map-truncate-invert

text: scheme

text: must

text: emulate

text: the

text: scheme

text: based

text: on

text: the

text: corresponding

text: stochastic

text: Taylor

text: expansion

text: truncated

text: according

text: to

text: the

text: same

text: grading.

text: The

text: question

text: is

text: thus,

text: is

text: the

text: pre-remainder

text: (and

text: thus

text: re-

text: mainder)

text: associated

text: with

text: the

text: map-truncate-invert

text: scheme

text: at

text: hand

text: less

text: than

text: the

text: remainder

text: associated

text: with

text: the

text: corresponding

text: stochastic

text: Taylor

text: approxima-

text: tion

text: to

text: leading

text: order

text: in

text: the

text: mean-square

text: measure?

text: The

text: answer

text: is

text: “yes”

text: for

text: the

text: antisymmetric

text: sign

text: reverse

text: integrator.

text: This

text: integrator

text: is

text: constructed

text: by

text: taking

text: half

text: the

text: diﬀerence

text: of

text: the

text: stochastic

text: Taylor

text: expansion

text: and

text: the

text: stochastic

text: Taylor

text: expansion

text: with

text: all

text: the

text: words

text: indexing

text: the

text: multiple

text: integrals

text: reversed

text: in

text: order

text: and

text: signed

text: according

text: to

text: the

text: of

text: words.

text: the

text: pre-remainder

text: and

text: thus

text: remainder

text: are

text: RASRI(y

text: I

text: I

text: V

text: (y

text: t

text: w

text: S(w)

text: w

text: w

text: Here

text: w

text: represents

text: the

text: length

text: of

text: the

text: word

text: w

text: and

text: S

text: is

text: the

text: signed

text: reversal

text: map

text: deﬁned

text: as

text: S

text: a

text: a

text: a

text: a

text: a

text: a

text: for

text: any

text: word

text: consisting

text: n

text: A

text: n

text: of

text: letters

text: a

text: a

text: a

text: I−

text: I

text: Our

text: main

text: n

text: w

text: w

text: result

text: is

text: thus

text: RASRI(y

text: Rwl(y

text: t

text: t

text: for

text: all

text: y

text: all

text: L´evy-driven

text: systems,

text: to

text: all

text: orders

text: n.

text: Using

text: our

text: result

text: above

text: this

text: means

text: the

text: antisymmetric

text: sign

text: reverse

text: integrator

text: is

text: more

text: accurate

text: than

text: the

text: corresponding

text: stochastic

text: Taylor

text: approximation

text: truncated

text: according

text: to

text: mean-

text: square

text: grading.

text: Furthermore

text: we

text: show

text: that

text: when

text: n

text: is

text: odd,

text: the

text: inequality

text: just

text: above

text: is

text: optimal

text: in

text: the

text: following

text: sense.

text: Suppose

text: we

text: perturb

text: the

text: antisymmet-

text: ric

text: sign

text: reverse

text: integrator

text: by

text: a

text: general

text: class

text: of

text: perturbations—see

text: for

text: more

text: details.

text: Then

text: when

text: n

text: is

text: odd

text: we

text: show

text: that

text: the

text: diﬀerence

text: between

text: the

text: two

text: mean-square

text: error

text: measures

text: for

text: Rwl(y

text: and

text: RASRI(y

text: above

text: are

text: minimized

text: t

text: t

text: when

text: the

text: perturbation

text: is

text: zero.

text: One

text: further

text: technical

text: practical

text: aspect

text: comes

text: into

text: play.

text: In

text: the

text: Castell–Gaines

text: case

text: we

text: compute

text: f

text: exp

text: by

text: solving

text: an

text: or-

text: dinary

text: diﬀerential

text: system.

text: If

text: all

text: the

text: underlying

text: vector

text: ﬁelds

text: are

text: linear

text: f

text: may

text: be

text: computed

text: as

text: a

text: matrix-valued

text: compositional

text: inverse

text: function;

text: see

text: Malham

text: Wiese

text: However

text: we

text: cannot

text: do

text: this

text: in

text: general

text: for

text: map-truncate-invert

text: schemes.

text: To

text: circumvent

text: this

text: issue

text: we

text: have

text: developed

text: direct-map-truncate-

text: invert

text: schemes

text: with

text: a

text: special

text: case

text: being

text: the

text: direct

text: antisymmetric

text: sign

text: reverse

text: integrator.

text: In

text: direct-map-truncate-invert

text: schemes

text: we

text: utilize

text: the

text: observation

text: above

text: that

text: map-truncate-invert

text: schemes

text: naturally

text: emulate

text: the

text: scheme

text: based

text: on

text: the

text: corresponding

text: stochastic

text: Taylor

text: approximation

text: to

text: the

text: same

text: order,

text: but

text: have

text: a

text: diﬀerent

text: remainder

text: at

text: leading

text: order

text: as

text: diﬀerent

text: terms

text: at

text: leading

text: order

text: in

text: the

text: remainder

text: are

text: included

text: in

text: the

text: integrator.

text: Thus

text: in

text: order

text: to

text: simulate

text: a

text: direct-map-truncate-invert

text: scheme

text: we

text: simulate

text: the

text: corresponding

text: stochas-

text: tic

text: Taylor

text: approximation

text: to

text: the

text: same

text: order

text: as

text: well

text: as

text: the

text: terms

text: that

text: lie

text: in

text: the

text: diﬀerence

text: between

text: the

text: stochastic

text: Taylor

text: approximation

text: remainder

text: and

text: map-truncate-invert

text: approximation

text: remainder

text: at

text: leading

text: order.

text: These

text: are

text: ad-

text: ditional

text: terms

text: indexed

text: by

text: words

text: of

text: length

text: n

text: the

text: terms

text: concerned

text: will

text: not

text: ##LTLine##

text: ##LTLine##


text: SDEs driven by L´evy processes 9 Runge–Kutta methods by Butcher [10] in the late sixties and early seven- ties; see Hairer, Lubich and Wanner [34]. Hopf algebraic structures are now a natural lexicon in the study of (to name a few): (i) numerical methods for deterministic systems, such as Runge–Kutta methods and geometric and sym- plectic integrators—see for example Munthe–Kaas & Wright [57], Lundervold & Munthe–Kaas [46] and McLachlan, Modin, Munthe–Kaas & Verdier [53]; (ii) approximations for stochastic diﬀerential equations—see for example Castell & Gaines [12,13], Malham & Wiese [51], Ebrahimi–Fard, et al. [20], Ebrahimi– Fard, Malham, Patras & Wiese [21,22]; (iii) rough paths—see for example Hairer & Kelly [33] and Gubinelli & Tindel [31] and (iv) regularity structures for stochastic partial diﬀerential equations—see for example Gubinelli [30] and Hairer [32]. In summary in this paper, what is new is that we: 1. Show the stochastic Taylor series expansion for the ﬂowmap can be written in separated form, i.e. as a series of terms, each of which is decomposable into a multiple Itˆo integral and a composition of associated diﬀerential § operators (see ); 2. Describe the class of map-truncate-invert schemes for L´evy-driven equa- tions and give an algebraic framework for encoding and comparing such schemes. These results generalize those in Malham & Wiese [51] and Ebrahimi– § Fard et al. [20] (see ); 3. Prove truncations according to the word length grading give more accurate schemes than approximations of the same order of convergence obtained § by truncations according to the mean-square grading (see ); 4. Introduce the antisymmetric sign reverse integrator, a new integration scheme for L´evy-driven equations, represented as half the diﬀerence of the identity and sign reverse endomorphisms on the vector space generated by words indexing multiple integrals. This scheme is eﬃcient in the sense that its leading order mean-square error is less than that of the corresponding stochastic Taylor scheme, independent of the governing vector ﬁelds (see § ); 5. Prove the antisymmetric sign reverse integrator is optimal in the sense out- lined above at half-integer global root mean-square orders of convergence § (also see ); 6. Show how to compute map-truncate-invert schemes in practice, in particu- lar, how to deal with the inversion stage. We call these direct map-truncate- § invert schemes (again see ). We round oﬀ this paper by establishing global convergence results from local § error estimates in and providing some explicit antisymmetric sign reverse § integrators as well as numerical experiments demonstrating our results in . (cid:0) (cid:1) 2 Separated stochastic Taylor expansions F F Our setting is a complete, ﬁltered probability space Ω, , ( ) ≥ , P , t t 0 assumed to satisfy the usual hypotheses—see Protter [64, p.3]. For the Itˆo ##LTLine##

text: SDEs

text: driven

text: by

text: L´evy

text: processes

text: Runge–Kutta

text: methods

text: by

text: Butcher

text: in

text: the

text: late

text: sixties

text: and

text: early

text: seven-

text: ties;

text: see

text: Hairer,

text: Lubich

text: and

text: Wanner

text: Hopf

text: algebraic

text: structures

text: are

text: now

text: a

text: natural

text: lexicon

text: in

text: the

text: study

text: of

text: (to

text: name

text: a

text: few):

text: (i)

text: numerical

text: methods

text: for

text: deterministic

text: systems,

text: such

text: as

text: Runge–Kutta

text: methods

text: and

text: geometric

text: and

text: sym-

text: plectic

text: integrators—see

text: for

text: example

text: Munthe–Kaas

text: Wright

text: Lundervold

text: Munthe–Kaas

text: and

text: McLachlan,

text: Modin,

text: Munthe–Kaas

text: Verdier

text: (ii)

text: approximations

text: for

text: stochastic

text: diﬀerential

text: equations—see

text: for

text: example

text: Castell

text: Gaines

text: Malham

text: Wiese

text: Ebrahimi–Fard,

text: et

text: al.

text: Ebrahimi–

text: Fard,

text: Malham,

text: Patras

text: Wiese

text: (iii)

text: rough

text: paths—see

text: for

text: example

text: Hairer

text: Kelly

text: and

text: Gubinelli

text: Tindel

text: and

text: (iv)

text: regularity

text: structures

text: for

text: stochastic

text: partial

text: diﬀerential

text: equations—see

text: for

text: example

text: Gubinelli

text: and

text: Hairer

text: In

text: summary

text: in

text: this

text: paper,

text: what

text: is

text: new

text: is

text: that

text: we:

text: Show

text: the

text: stochastic

text: Taylor

text: series

text: expansion

text: for

text: the

text: ﬂowmap

text: can

text: be

text: written

text: in

text: separated

text: form,

text: i.e.

text: as

text: a

text: series

text: of

text: terms,

text: each

text: of

text: which

text: is

text: decomposable

text: into

text: a

text: multiple

text: Itˆo

text: integral

text: and

text: a

text: composition

text: of

text: associated

text: diﬀerential

text: operators

text: (see

text: Describe

text: the

text: class

text: of

text: map-truncate-invert

text: schemes

text: for

text: L´evy-driven

text: equa-

text: tions

text: and

text: give

text: an

text: algebraic

text: framework

text: for

text: encoding

text: and

text: comparing

text: such

text: schemes.

text: These

text: results

text: generalize

text: those

text: in

text: Malham

text: Wiese

text: and

text: Ebrahimi–

text: Fard

text: et

text: al.

text: (see

text: Prove

text: truncations

text: according

text: to

text: the

text: word

text: length

text: grading

text: give

text: more

text: accurate

text: schemes

text: than

text: approximations

text: of

text: the

text: same

text: order

text: of

text: convergence

text: obtained

text: by

text: truncations

text: according

text: to

text: the

text: mean-square

text: grading

text: (see

text: Introduce

text: the

text: antisymmetric

text: sign

text: reverse

text: integrator,

text: a

text: new

text: integration

text: scheme

text: for

text: L´evy-driven

text: equations,

text: represented

text: as

text: half

text: the

text: diﬀerence

text: of

text: the

text: identity

text: and

text: sign

text: reverse

text: endomorphisms

text: on

text: the

text: vector

text: space

text: generated

text: by

text: words

text: indexing

text: multiple

text: integrals.

text: This

text: scheme

text: is

text: eﬃcient

text: in

text: the

text: sense

text: that

text: its

text: leading

text: order

text: mean-square

text: error

text: is

text: less

text: than

text: that

text: of

text: the

text: corresponding

text: stochastic

text: Taylor

text: scheme,

text: independent

text: of

text: the

text: governing

text: vector

text: ﬁelds

text: (see

text: Prove

text: the

text: antisymmetric

text: sign

text: reverse

text: integrator

text: is

text: optimal

text: in

text: the

text: sense

text: out-

text: lined

text: above

text: at

text: half-integer

text: global

text: root

text: mean-square

text: orders

text: of

text: convergence

text: (also

text: see

text: Show

text: how

text: to

text: compute

text: map-truncate-invert

text: schemes

text: in

text: practice,

text: in

text: particu-

text: lar,

text: how

text: to

text: deal

text: with

text: the

text: inversion

text: stage.

text: We

text: call

text: these

text: direct

text: map-truncate-

text: invert

text: schemes

text: (again

text: see

text: We

text: round

text: oﬀ

text: this

text: paper

text: by

text: establishing

text: global

text: convergence

text: results

text: from

text: local

text: error

text: estimates

text: in

text: and

text: providing

text: some

text: explicit

text: antisymmetric

text: sign

text: reverse

text: integrators

text: as

text: well

text: as

text: numerical

text: experiments

text: demonstrating

text: our

text: results

text: in

text: Separated

text: stochastic

text: Taylor

text: expansions

text: F

text: F

text: Our

text: setting

text: is

text: a

text: complete,

text: ﬁltered

text: probability

text: space

text: Ω,

text: P

text: t

text: t

text: assumed

text: to

text: satisfy

text: the

text: usual

text: hypotheses—see

text: Protter

text: For

text: the

text: Itˆo

text: ##LTLine##


text: Variational integrators of fractional Lagrangian systems in the framework of discrete embeddings 3 − − Deﬁnition 2 (case σ = ). We call backward ﬁnite diﬀe(cid:16)ren(cid:17)ces embedding denoted by FDE the deﬁnition of the following elements: the application disc : C0([a, b], Rd) −→ Rd N+1 , (cid:55)−→ q (q(t )) k k=0,...,N (cid:16) (cid:17) (cid:16) (cid:17) and the discrete operator (cid:16) (cid:17) ∆− : Rd N+1 −→ Rd N (cid:55)−→ − Q = (Qk)k=0,...,N Qk hQk−1 k=1,..,N . Deﬁnition 3 (case σ = +). We call f(cid:16)orw(cid:17)ard ﬁnite diﬀ(cid:16)ere(cid:17)nces embedding denoted by FDE+ the deﬁnition of the following elements: the application disc and the discrete operator (cid:16) (cid:17) ∆ : Rd N+1 −→ Rd N + (cid:55)−→ − Q = (Qk)k=0,...,N Qk hQk+1 k=0,..,N−1 . − Let us notice that the discrete analogous of d/dt in FDEσ is then σ∆ . We use these σ notations in order to be uniform with the fractional notations (see Section ). 1.2. Direct discrete embeddings Deﬁning a discrete embedding allows us to deﬁne a direct discrete version of a given diﬀerential equation: Deﬁnition 4. Let be ﬁxed a discrete embedding as deﬁned in Deﬁnition 1 and let (E) be an ∈ C ordinary diﬀerential equation of unknown q 0([a, b], Rd) given by: O(q) = 0, (cid:80) (E) · ◦ · where O is a diﬀerential operator shaped as O = i fi( )(d/dt)i gi( ) where fi, gi are functions. Then, the direct discrete embedding of (E) is (E ) the system of equations of ∈ h unknown qh (Rd)m1 given by: Oh(qh) = 0, (cid:80) (Eh) · ◦ · where Oh is the discretized operator of O given by Oh = i fi( )∆i gi( ). ∈ C As an example, we consider the Newton’s equation with friction of unknown q 0([a, b], Rd) given by: ∀ ∈ t [a, b], q¨(t) + q˙(t) + q(t) = 0. (NE) − Then, the direct discrete embedding of (NE) with respect to FDE is (NE ) the system ∈ h of equations of unknown Q (Rd)N+1 given by: ##LTLine## ##LTLine##

text: Variational

label: title

text: integrators

text: of

text: fractional

text: Lagrangian

text: systems

text: in

text: the

text: framework

text: of

text: discrete

text: embeddings

font: EKDKPI+txsy

text: Deﬁnition

text: (case

text: σ

font: GWQIQB+rtxmi

font: HIUMXN+rtxr

text: We

text: call

text: backward

text: ﬁnite

text: embedding

text: denoted

text: by

text: FDE

text: the

text: deﬁnition

text: of

text: the

text: following

text: elements:

text: the

text: application

text: disc

text: b],

text: Rd)

text: Rd

text: q

text: (q(t

text: k

font: GBEWSZ+txex

text: and

text: the

text: discrete

text: operator

text: Rd

text: Rd

text: N

text: Q

text: Qk

text: Deﬁnition

text: (case

text: σ

text: We

text: call

text: ﬁnite

text: embedding

text: denoted

text: by

text: FDE+

text: the

text: deﬁnition

text: of

text: the

text: following

text: elements:

text: the

text: application

text: disc

text: and

text: the

text: discrete

text: operator

text: Rd

text: Rd

text: N

text: Q

text: Qk

text: Let

text: us

text: notice

text: that

text: the

text: discrete

text: analogous

text: of

text: d/dt

text: in

text: FDEσ

text: is

text: then

text: σ∆

text: We

text: use

text: these

text: σ

text: notations

text: in

text: order

text: to

text: be

text: uniform

text: with

text: the

text: fractional

text: notations

text: (see

text: Section

text: Direct

text: discrete

text: embeddings

text: Deﬁning

text: a

text: discrete

text: embedding

text: allows

text: us

text: to

text: deﬁne

text: a

text: direct

text: discrete

text: version

text: of

text: a

text: given

text: diﬀerential

text: equation:

text: Deﬁnition

text: Let

text: be

text: ﬁxed

text: a

text: discrete

text: embedding

text: as

text: deﬁned

text: in

text: Deﬁnition

text: and

text: let

text: (E)

text: be

text: an

text: C

text: ordinary

text: diﬀerential

text: equation

text: of

text: unknown

text: q

text: b],

text: Rd)

text: given

text: by:

text: O(q)

text: (E)

text: where

text: O

text: is

text: a

text: diﬀerential

text: operator

text: shaped

text: as

text: O

text: i

text: fi(

text: )(d/dt)i

text: gi(

text: where

text: fi,

text: gi

text: are

text: functions.

text: Then,

text: the

text: direct

text: discrete

text: embedding

text: of

text: (E)

text: is

text: (E

text: the

text: system

text: of

text: equations

text: of

text: h

text: unknown

text: qh

text: given

text: by:

text: Oh(qh)

text: (Eh)

text: where

text: Oh

text: is

text: the

text: discretized

text: operator

text: of

text: O

text: given

text: by

text: Oh

text: i

text: fi(

text: )∆i

text: gi(

text: C

text: As

text: an

text: example,

text: we

text: consider

text: the

text: Newton’s

text: equation

text: with

text: friction

text: of

text: unknown

text: q

text: b],

text: Rd)

text: given

text: by:

text: t

text: [a,

text: b],

text: q¨(t)

text: q˙(t)

text: q(t)

text: (NE)

text: Then,

text: the

text: direct

text: discrete

text: embedding

text: of

text: (NE)

text: with

text: respect

text: to

text: FDE

text: is

text: (NE

text: the

text: system

text: h

text: of

text: equations

text: of

text: unknown

text: Q

text: given

text: by:

text: ##LTLine##

text: ##LTLine##


text: F. Nehlig, B. Vollmer, J. Braine : Environmental gas compression is observed northwest of the galaxy center. We thus con- 6.2. NGC 4567/68 dynamical simulation clude that the detailed observed gas distribution depends For simulations of colliding galaxies the parameter space on galactic structure, i.e., whether a spiral arm is located (impact parameter, relative velocity, disk inclinations, pro- in the compressed region. grade/retrograde encounter) is huge and degenerated. Our The simulated gas surface density of simulation A re- aim was to simulate NGC 4568 in a ﬁrst place. produces (i) the well-deﬁned compressed western edge, (ii) the three low surface density spiral arms in the eastern part The size and rotation velocity of the model galaxies of the disk extending to the north, (iii) the regions of high were adapted to the observed properties of NGC 4567 and gas surface density in the northern part of the disk, and (vi) NGC 4568. For the search of the orbit of NGC 4567 with the high molecular fraction in the western compressed side respect to NGC 4568 we made several dozens of test simula- of the disk. The SFE map is quantitatively reproduced, tions without a collisional component. We aimed at repro- the higher SFE in tHh2e northwestern quadrant of the disk ducing the position of NGC 4567 with respect to NGC 4568 is present in theHs2imulation as is the U-shaped low SFE together with a western compression ridge in NGC 4568. region in the south. Moreover, the modeled SFE is lowHe2r We have chosen a retrograde encounter with respect to in the north of the compressed region as has beenHo2bserved. NGC 4568. To produce a visible eﬀect, the impact param- However, the model does not reproduce the observed higher eter is small (∆x = 6 kpc). The relative velocity at clos- − SFE west of the center. est approach is ∆v = 460 km s 1. NGC 4567 approaches WHi2thin the western compression region, the reason for NGC 4568 from the south/southwest. The time of inter- the low molecular star formation eﬃciency in the low sur- est is 10 Myr after the closest approach. In this stage the stellar disks overlap and a compression region forms in the face density gas is the following. In a quiet model without western half of the disk of NGC 4568 and in the eastern interaction, the velocity dispersion of the gas particles is half of the disk of NGC 4567. To avoid the overlap of the proportional to the square root of the gas density. Once the gas disk, we truncated the gas disk of NGC 4567. We are gas is compressed, it adapts its velocity dispersion to the aware that this simulation does not perfectly reproduce the new situation within a gas particle collision timescale. In the observed properties of the system. However, it can serve as compressed regions of low surface density the particle den- a guideline to interpret observations. sity and thus the cloud collision frequency is smaller th√an the compression timescale and the equilibrium v ∝ ρ The 3D simulations have the advantage that the two is not yet reached at the time of interest. The tmuroblecular galaxies can be separated properly avoiding superposition star formation eﬃciency is thus lower. On the other hand, eﬀects. The maps of the gas distribution, molecular frac- in the high surface density region the collision timescale is tion, R /P , and molecular SFE are presented in short and the equilibrium is reached rapidly, i.e., the molec- Figs. 28maonl d 2to9t. H2 ular star formation eﬃciency is unchanged. We believe that, In the following we name the model galaxies in reality, the molecular star formation eﬃciency decreases NGC 4567m and NGC 4568m. NGC 4568m shows a south- once the compression becomes adiabatic, i.e., once the tur- ern spiral arm curved to the west and an east-west asym- ∼ bulent dissipation timescale t l /v is longer metry within the southern half of the disk; the southeast- turb driv turb than the compression timescale t = ρ/(∂ρ/∂t). Even ern part of the disk shows lower surface densities than the comp if the physics in the model and in reality that lead to a southwestern part. Thus, the southern part of the galactic decreasing molecular star formation eﬃciency are quite dif- disk of NGC 4568 (Fig. 8) is reproduced well by the model. ferent, the mechanisms are similar because in a collisional The northern spiral arm of NGC 4568 is also reproduced by model the dissipation timescale is proportional to the col- the model, but located somewhat farther to the west. The lision timescale. Therefore, our simple dynamical model is simulated gas surface density in the center of NGC 4568m ∼ able to predict regions of decreased molecular star forma- is lower than the observed value by a factor of 2. The tion eﬃciency due to adiabatic compression. In the future, total gas distribution of NGC 4567m does not show the we will test this prediction with simulations of NGC 4438 observed spiral arms. The highest gas surface densities of and the Taﬀy galaxies, where adiabatic compression de- NGC 4567m are not found in the center as in NGC 4567, creases SFE by a factor of four to ﬁve (Braine et al. but east of the center, where the gas is compressed by the mol 2003, Vollmer et al. 2009, 2012). tidal ﬁeld. The overall gas surface density of NGC 4567m The global model R /P is higher than observed. is signiﬁcantly higher than that of NGC 4567. mol tot Nevertheless, we reproduce the main observational charac- The simulated overall distribution of the molecular frac- teristics: (i) a low R /P in the northwestern quadrant tion of NGC 4568m (middle right panel of Fig. 28) is in mol tot and in the compressed region and (ii) a high R /P in rough agreement with observations (Fig. 10): it is symmet- mol tot the northwestern quadrant. ric in the inner parts of the galactic disks. In addition, there A pixel-by-pixel analysis was made for the NGC 4501 is an excess of the molecular fraction in the southwestern simulation A. The ﬁtting results and the Spearman rank part of the disk of NGC 4568m. The region between the correlation coeﬃcients ρ are presented in Table 8. galaxies shows a higher molecular fraction in the simula- We ﬁnd slopes in agreement with observations for the tion than in the observations. The molecular fractions of following relations: SFR - Σ , SFR - Σ , R -P , R - NGC 4567m and NGC 4568m do not show the observed Σ , R - Σ , and R /P H2-Q. Howevger, tmhoel sltoopte ofmtohl e high values in the galaxy centers. This is due to a gas hole (cid:63) mol g mol tot R - P relation is lower in our simulation of NGC 4501 in the simulated galaxies that we introduced for computa- mol tot (0.7) than the observed slope of 1.0. We found a correlation tional reasons. between R /P - Q (Spearman ρ = 0.9) with a slope of The simulated overall distribution of the molecular ∼ mol tot ± 0.7 in agreement with the observed slope 0.84 0.36. star formation eﬃciency of NGC 4568m (lower left panel We thus conclude that the dynamical simulation of of Fig. 29) is also in rough agreement with observations NGC 4501 is in good agreement with observations. (Fig. 17). Only the model SFE of the galaxy centers is H2 Article number, page 15 of 44 ##LTLine##

text: F.

text: Nehlig,

text: B.

text: Vollmer,

text: J.

text: Braine

text: Environmental

text: gas

text: compression

text: is

text: observed

text: northwest

text: of

text: the

text: galaxy

text: center.

text: We

text: thus

text: con-

text: NGC

text: dynamical

text: simulation

text: clude

text: that

text: the

text: detailed

text: observed

text: gas

text: distribution

text: depends

text: For

text: simulations

text: of

text: colliding

text: galaxies

text: the

text: parameter

text: space

text: on

text: galactic

text: structure,

text: i.e.,

text: whether

text: a

text: spiral

text: arm

text: is

text: located

text: (impact

text: parameter,

text: relative

text: velocity,

text: disk

text: inclinations,

text: pro-

text: in

text: the

text: compressed

text: region.

text: grade/retrograde

text: encounter)

text: is

text: huge

text: and

text: degenerated.

text: Our

text: The

text: simulated

text: gas

text: surface

text: density

text: of

text: simulation

text: A

text: re-

text: aim

text: was

text: to

text: simulate

text: NGC

text: in

text: a

text: ﬁrst

text: place.

text: produces

text: (i)

text: the

text: well-deﬁned

text: compressed

text: western

text: edge,

text: (ii)

text: the

text: three

text: low

text: surface

text: density

text: spiral

text: arms

text: in

text: the

text: eastern

text: part

text: The

text: size

text: and

text: rotation

text: velocity

text: of

text: the

text: model

text: galaxies

text: of

text: the

text: disk

text: extending

text: to

text: the

text: north,

text: (iii)

text: the

text: regions

text: of

text: high

text: were

text: adapted

text: to

text: the

text: observed

text: properties

text: of

text: NGC

text: and

text: gas

text: surface

text: density

text: in

text: the

text: northern

text: part

text: of

text: the

text: disk,

text: and

text: (vi)

text: NGC

text: For

text: the

text: search

text: of

text: the

text: orbit

text: of

text: NGC

text: with

text: the

text: high

text: molecular

text: fraction

text: in

text: the

text: western

text: compressed

text: side

text: respect

text: to

text: NGC

text: we

text: made

text: several

text: dozens

text: of

text: test

text: simula-

text: of

text: the

text: disk.

text: The

text: SFE

text: map

text: is

text: quantitatively

text: reproduced,

text: tions

text: without

text: a

text: collisional

text: component.

text: We

text: aimed

text: at

text: repro-

text: the

text: higher

text: SFE

text: in

text: northwestern

text: quadrant

text: of

text: the

text: disk

text: ducing

text: the

text: position

text: of

text: NGC

text: with

text: respect

text: to

text: NGC

text: is

text: present

text: in

text: as

text: is

text: the

text: U-shaped

text: low

text: SFE

text: together

text: with

text: a

text: western

text: compression

text: ridge

text: in

text: NGC

text: region

text: in

text: the

text: south.

text: Moreover,

text: the

text: modeled

text: SFE

text: is

text: We

text: have

text: chosen

text: a

text: retrograde

text: encounter

text: with

text: respect

text: to

text: in

text: the

text: north

text: of

text: the

text: compressed

text: region

text: as

text: has

text: NGC

text: To

text: produce

text: a

text: visible

text: eﬀect,

text: the

text: impact

text: param-

text: However,

text: the

text: model

text: does

text: not

text: reproduce

text: the

text: observed

text: higher

text: eter

text: is

text: small

text: (∆x

text: kpc).

text: The

text: relative

text: velocity

text: at

text: clos-

text: SFE

text: west

text: of

text: the

text: center.

text: est

text: approach

text: is

text: ∆v

text: km

text: s

text: NGC

text: approaches

text: the

text: western

text: compression

text: region,

text: the

text: reason

text: for

text: NGC

text: from

text: the

text: south/southwest.

text: The

text: time

text: of

text: inter-

text: the

text: low

text: molecular

text: star

text: formation

text: eﬃciency

text: in

text: the

text: low

text: sur-

text: est

text: is

text: Myr

text: after

text: the

text: closest

text: approach.

text: In

text: this

text: stage

text: the

text: stellar

text: disks

text: overlap

text: and

text: a

text: compression

text: region

text: forms

text: in

text: the

text: face

text: density

text: gas

text: is

text: the

text: following.

text: In

text: a

text: quiet

text: model

text: without

text: western

text: half

text: of

text: the

text: disk

text: of

text: NGC

text: and

text: in

text: the

text: eastern

text: interaction,

text: the

text: velocity

text: dispersion

text: of

text: the

text: gas

text: particles

text: is

text: half

text: of

text: the

text: disk

text: of

text: NGC

text: To

text: avoid

text: the

text: overlap

text: of

text: the

text: proportional

text: to

text: the

text: square

text: root

text: of

text: the

text: gas

text: density.

text: Once

text: the

text: gas

text: disk,

text: we

text: truncated

text: the

text: gas

text: disk

text: of

text: NGC

text: We

text: are

text: gas

text: is

text: compressed,

text: it

text: adapts

text: its

text: velocity

text: dispersion

text: to

text: the

text: aware

text: that

text: this

text: simulation

text: does

text: not

text: perfectly

text: reproduce

text: the

text: new

text: situation

text: within

text: a

text: gas

text: particle

text: collision

text: timescale.

text: In

text: the

text: observed

text: properties

text: of

text: the

text: system.

text: However,

text: it

text: can

text: serve

text: as

text: compressed

text: regions

text: of

text: low

text: surface

text: density

text: the

text: particle

text: den-

text: a

text: guideline

text: to

text: interpret

text: observations.

text: sity

text: and

text: thus

text: the

text: cloud

text: collision

text: frequency

text: is

text: smaller

text: th√an

text: the

text: compression

text: timescale

text: and

text: the

text: equilibrium

text: v

text: ρ

text: The

text: simulations

text: have

text: the

text: advantage

text: that

text: the

text: two

text: is

text: not

text: yet

text: reached

text: at

text: the

text: time

text: of

text: interest.

text: The

text: tmuroblecular

text: galaxies

text: can

text: be

text: separated

text: properly

text: avoiding

text: superposition

text: star

text: formation

text: eﬃciency

text: is

text: thus

text: lower.

text: On

text: the

text: other

text: hand,

text: eﬀects.

text: The

text: maps

text: of

text: the

text: gas

text: distribution,

text: molecular

text: frac-

text: in

text: the

text: high

text: surface

text: density

text: region

text: the

text: collision

text: timescale

text: is

text: tion,

text: R

text: /P

text: and

text: molecular

text: SFE

text: are

text: presented

text: in

text: short

text: and

text: the

text: equilibrium

text: is

text: reached

text: rapidly,

text: i.e.,

text: the

text: molec-

text: Figs.

text: d

text: ular

text: star

text: formation

text: eﬃciency

text: is

text: unchanged.

text: We

text: believe

text: that,

text: In

text: the

text: following

text: we

text: name

text: the

text: model

text: galaxies

text: in

text: reality,

text: the

text: molecular

text: star

text: formation

text: eﬃciency

text: decreases

text: NGC

text: and

text: NGC

text: NGC

text: shows

text: a

text: south-

text: once

text: the

text: compression

text: becomes

text: adiabatic,

text: i.e.,

text: once

text: the

text: tur-

text: ern

text: spiral

text: arm

text: curved

text: to

text: the

text: west

text: and

text: an

text: east-west

text: asym-

text: bulent

text: dissipation

text: timescale

text: t

text: l

text: /v

text: is

text: longer

text: metry

text: within

text: the

text: southern

text: half

text: of

text: the

text: disk;

text: the

text: southeast-

text: turb

text: driv

text: turb

text: than

text: the

text: compression

text: timescale

text: t

text: ρ/(∂ρ/∂t).

text: Even

text: ern

text: part

text: of

text: the

text: disk

text: shows

text: lower

text: surface

text: densities

text: than

text: the

text: comp

text: if

text: the

text: physics

text: in

text: the

text: model

text: and

text: in

text: reality

text: that

text: lead

text: to

text: a

text: southwestern

text: part.

text: Thus,

text: the

text: southern

text: part

text: of

text: the

text: galactic

text: decreasing

text: molecular

text: star

text: formation

text: eﬃciency

text: are

text: quite

text: dif-

text: disk

text: of

text: NGC

text: (Fig.

text: is

text: reproduced

text: well

text: by

text: the

text: model.

text: ferent,

text: the

text: mechanisms

text: are

text: similar

text: because

text: in

text: a

text: collisional

text: The

text: northern

text: spiral

text: arm

text: of

text: NGC

text: is

text: also

text: reproduced

text: by

text: model

text: the

text: dissipation

text: timescale

text: is

text: proportional

text: to

text: the

text: col-

text: the

text: model,

text: but

text: located

text: somewhat

text: farther

text: to

text: the

text: west.

text: The

text: lision

text: timescale.

text: Therefore,

text: our

text: simple

text: dynamical

text: model

text: is

text: simulated

text: gas

text: surface

text: density

text: in

text: the

text: center

text: of

text: NGC

text: able

text: to

text: predict

text: regions

text: of

text: decreased

text: molecular

text: star

text: forma-

text: is

text: lower

text: than

text: the

text: observed

text: value

text: by

text: a

text: factor

text: of

text: The

text: tion

text: eﬃciency

text: due

text: to

text: adiabatic

text: compression.

text: In

text: the

text: future,

text: total

text: gas

text: distribution

text: of

text: NGC

text: does

text: not

text: show

text: the

text: we

text: will

text: test

text: this

text: prediction

text: with

text: simulations

text: of

text: NGC

text: observed

text: spiral

text: arms.

text: The

text: highest

text: gas

text: surface

text: densities

text: of

text: and

text: the

text: Taﬀy

text: galaxies,

text: where

text: adiabatic

text: compression

text: de-

text: NGC

text: are

text: not

text: found

text: in

text: the

text: center

text: as

text: in

text: NGC

text: creases

text: SFE

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text: a

text: factor

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text: four

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text: ﬁve

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text: al.

text: but

text: east

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text: the

text: gas

text: is

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text: the

text: mol

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text: et

text: al.

text: tidal

text: ﬁeld.

text: The

text: overall

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text: density

text: of

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text: global

text: model

text: R

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text: higher

text: than

text: observed.

text: is

text: signiﬁcantly

text: higher

text: than

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text: mol

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text: we

text: reproduce

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text: The

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text: the

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text: NGC

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text: galaxies

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text: slopes

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text: agreement

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text: tion

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text: The

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text: following

text: relations:

text: SFR

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text: NGC

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text: show

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text: tmhoel

text: sltoopte

text: ofmtohl

text: e

text: high

text: values

text: in

text: the

text: galaxy

text: centers.

text: This

text: is

text: due

text: to

text: a

text: gas

text: hole

text: mol

text: g

text: mol

text: tot

text: R

text: P

text: relation

text: is

text: lower

text: in

text: our

text: simulation

text: of

text: NGC

text: in

text: the

text: simulated

text: galaxies

text: that

text: we

text: introduced

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text: computa-

text: mol

text: tot

text: than

text: the

text: observed

text: slope

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text: We

text: found

text: a

text: correlation

text: tional

text: reasons.

text: between

text: R

text: /P

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text: (Spearman

text: ρ

text: with

text: a

text: slope

text: of

text: The

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text: distribution

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text: the

text: molecular

text: mol

text: tot

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text: the

text: observed

text: slope

text: star

text: formation

text: eﬃciency

text: of

text: NGC

text: (lower

text: left

text: panel

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text: thus

text: conclude

text: that

text: the

text: dynamical

text: simulation

text: of

text: of

text: Fig.

text: is

text: also

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text: rough

text: agreement

text: with

text: observations

text: NGC

text: is

text: in

text: good

text: agreement

text: with

text: observations.

text: (Fig.

text: Only

text: the

text: model

text: SFE

text: of

text: the

text: galaxy

text: centers

text: is

text: Article

text: number,

text: page

text: of

text: ##LTLine##


text: A&A proofs: manuscript no. 27021 the star formation activity, whereas the two timescales are not very diﬀerent in NGC 4501 and NGC 4568. Acknowledgements. Based on IRAM observations. IRAM is sup- ported by INSU/CNRS (France), MPG (Germany), and IGN (Spain). References Bigiel, F., Leroy, A., Walter, F., et al. 2008, AJ, 136, 2846 Binney, J. & Merriﬁeld, M. 1998, Galactic Astronomy Blitz, L. & Rosolowsky, E. 2006, ApJ, 650, 933 Bolatto, A. D., Wolﬁre, M., & Leroy, A. K. 2013, ARA&A, 51, 207 Boquien, M., Calzetti, D., Aalto, S., et al. 2015, A&A, 578, A8 Braine, J., Davoust, E., Zhu, M., et al. 2003, A&A, 408, L13 Cayatte, V., Kotanyi, C., Balkowski, C., & van Gorkom, J. H. 1994, AJ, 107, 1003 Cayatte, V., van Gorkom, J. H., Balkowski, C., & Kotanyi, C. 1990, AJ, 100, 604 Chung, A., van Gorkom, J. H., Kenney, J. D. P., Crowl, H., & Vollmer, B. 2009, AJ, 138, 1741 Dickman, R. L., Snell, R. L., & Schloerb, F. P. 1986, ApJ, 309, 326 Elmegreen, B. 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Vollmer, B., Soida, M., Beck, R., et al. 2007, A&A, 464, L37 Vollmer, B., Soida, M., Chung, A., et al. 2008, A&A, 483, 89 Werner, M. W., Roellig, T. L., Low, F. J., et al. 2004, ApJS, 154, 1 Wolﬁre, M. G., Hollenbach, D., & McKee, C. F. 2010, ApJ, 716, 1191 Wong, T. & Blitz, L. 2002, ApJ, 569, 157 Article number, page 18 of 44 ##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## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: A&A

text: proofs:

text: manuscript

text: no.

text: the

text: star

text: formation

text: activity,

text: whereas

text: the

text: two

text: timescales

text: are

text: not

text: very

text: diﬀerent

text: in

text: NGC

text: and

text: NGC

text: Acknowledgements.

text: Based

text: on

text: IRAM

text: observations.

text: IRAM

text: is

text: sup-

text: ported

text: by

text: INSU/CNRS

text: (France),

text: MPG

text: (Germany),

text: and

text: IGN

text: (Spain).

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text: M.

text: Galactic

text: Astronomy

text: Blitz,

text: L.

text: Rosolowsky,

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text: ApJ,

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