text: INTEGRAL GROUP RINGS WITH TRIVIAL CENTRAL UNITS 11 generator of the Frobenius complement, and B (y) is an epimorphic image G of H. But this is a contradiction since H does not have an epimorphic image that is cyclic of order 21. Hence F has exponent 7. Each 7-element x has to be inverse semi-rational in G, thus it has to have three conjugates in the cyclic subgroup generated by it, so the generator t of a Frobenius (cid:104) (cid:105) complement has to normalize x . Conversely, assume that F is a 7-group of exponent 7 such that it admits a ﬁxed point free automorphism α of order (cid:111) (cid:104) (cid:105) 3 mapping each cyclic subgroup to itself. Then, clearly, F α deﬁnes a (cid:3) cut Frobenius group. The result is proved. Remark 4.5. As by [CD10, Remark 13], groups of odd order are semi-rational if and only if they are cut (= inverse semi-rational), (2b) of the Theorem 1.3 also gives a more precise description of the Frobenius groups appearing in [CD10, Theorem 3], where the semi-rational groups of odd order are classiﬁed. The arguments from the last part of the proof of (1) can also be used to reduce the list of potential semi-rational Frobenius groups in [ADD16, Theorem 1.2], e.g. groups with complement C or Q , and Frobenius kernels with an order divisi- 4 8 ble by two diﬀerent primes can be removed from the list as well as groups with complement Q and kernel being an elementary abelian 17-group. Some of the ar- 16 guments also apply to show that the isomorphism type of semi-rational Frobenius groups of a certain type are unique. In general, a semi-rational Frobenius group is not uniquely determined by its Frobenius kernel and complement. The two non- × (cid:111) isomorphic Frobenius groups (C C ) C are both semi-rational (however, only 5 5 4 one of them is cut). We now describe the cut Frobenius groups with abelian kernel. Corollary 4.6. Let G be a cut Frobenius group with abelian kernel. Then G appears in Theorem 1.3 (1) or is isomorphic to × (cid:48) (cid:111) (cid:111) (cid:48) ∈ Z (cid:48) ∈ Z (Ca Ca ) C or Cd C (a, a 2 ≥ , a + a > 0, d ≥ ). 2 4 3 7 3 0 1 For each of the above structures there is a unique cut Frobenius group of that form. Proof. If the order of the complement is even, then the group was handled in Theorem 1.3 (1). In the case of a complement of odd order, it follows from the proof of Theorem 1.3 (2) that the Frobenius kernel F is an abelian group of exponent 7 or a divisor of 4. If exp F = 7, then we can argue as in (i) in the proof of Theorem 1.3, to see that (cid:111) ∈ Z there is a unique cut Frobenius group of the form and Cd C for all d ≥ . 7 3 × (cid:48) 1 Assume that F is a 2-group. As all groups of the form Ca Ca are cut, the ×2 (cid:48) (cid:111)4 task amounts to ﬁnd all Frobenius groups of the form (Ca Ca ) C . But it is (cid:48)2∈ Z4 3 well-known that there is such a group if and only if a, a 2 ≥ , not both zero, 0 and this group is unique up to isomorphism. This can be extracted, for example, (cid:3) from the more general discussion in [BH98, Section 11]. Remark 4.7. Turning our attention to cut Frobenius groups with non-abelian Frobenius kernels we provide examples in both situations, when the kernel is a 7- group and when it is a 2-group. For the ﬁrst, we refer to [CD10, Example 1]. There it is shown that the Heisenberg group P of order 73 (i.e. the non-abelian group of order 73 and exponent 7) admits a ﬁxed-point free automorphism of order 3, which (cid:111) results in a Frobenius group P C that is semi-rational (recall that in the case of 3 groups of odd order, cut and semi-rational are the same [CD10, Remark 13]).

text: INTEGRAL

text: GROUP

text: RINGS

text: WITH

text: TRIVIAL

text: CENTRAL

text: UNITS

text: generator

label: list

text: of

text: the

text: Frobenius

text: complement,

text: and

text: B

text: (y)

text: is

text: an

text: epimorphic

text: image

text: G

text: of

text: H.

text: But

text: this

text: is

text: a

text: contradiction

text: since

text: H

text: does

text: not

text: have

text: an

text: epimorphic

text: image

text: that

text: is

text: cyclic

text: of

text: order

text: Hence

text: F

text: has

text: exponent

text: Each

text: x

text: has

text: to

text: be

text: inverse

text: semi-rational

text: in

text: G,

text: thus

text: it

text: has

text: to

text: have

text: three

text: conjugates

text: in

text: the

text: cyclic

text: subgroup

text: generated

text: by

text: it,

text: so

text: the

text: generator

text: t

text: of

text: a

text: Frobenius

text: complement

text: has

text: to

text: normalize

text: x

text: Conversely,

text: assume

text: that

text: F

text: is

text: a

text: of

text: exponent

text: such

text: that

text: it

text: admits

text: a

text: ﬁxed

text: point

text: free

text: automorphism

text: α

text: of

text: order

text: mapping

text: each

text: cyclic

text: subgroup

text: to

text: itself.

text: Then,

text: clearly,

text: F

text: α

text: deﬁnes

text: a

text: cut

text: Frobenius

text: group.

text: The

text: result

text: is

text: proved.

text: Remark

text: As

text: by

text: Remark

text: groups

text: of

text: odd

text: order

text: are

text: semi-rational

text: if

text: and

text: only

text: if

text: they

text: are

text: cut

text: inverse

text: semi-rational),

text: of

text: the

text: Theorem

text: also

text: gives

text: a

text: more

text: precise

text: description

text: of

text: the

text: Frobenius

text: groups

text: appearing

text: in

text: Theorem

text: where

text: the

text: semi-rational

text: groups

text: of

text: odd

text: order

text: are

text: classiﬁed.

text: The

text: arguments

text: from

text: the

text: last

text: part

text: of

text: the

text: proof

text: of

text: can

text: also

text: be

text: used

text: to

text: reduce

text: the

text: list

text: of

text: potential

text: semi-rational

text: Frobenius

text: groups

text: in

text: Theorem

text: e.g.

text: groups

text: with

text: complement

text: C

text: or

text: Q

text: and

text: Frobenius

text: kernels

text: with

text: an

text: order

text: divisi-

text: ble

text: by

text: two

text: diﬀerent

text: primes

text: can

text: be

text: removed

text: from

text: the

text: list

text: as

text: well

text: as

text: groups

text: with

text: complement

text: Q

text: and

text: kernel

text: being

text: an

text: elementary

text: abelian

text: Some

text: of

text: the

text: ar-

text: guments

text: also

text: apply

text: to

text: show

text: that

text: the

text: isomorphism

text: type

text: of

text: semi-rational

text: Frobenius

text: groups

text: of

text: a

text: certain

text: type

text: are

text: unique.

text: In

text: general,

text: a

text: semi-rational

text: Frobenius

text: group

text: is

text: not

text: uniquely

text: determined

text: by

text: its

text: Frobenius

text: kernel

text: and

text: complement.

text: The

text: two

text: non-

text: isomorphic

text: Frobenius

text: groups

text: (C

text: C

text: C

text: are

text: both

text: semi-rational

text: (however,

text: only

text: one

text: of

text: them

text: is

text: cut).

text: We

text: now

text: describe

text: the

text: cut

text: Frobenius

text: groups

text: with

text: abelian

text: kernel.

text: Corollary

text: Let

text: G

text: be

text: a

text: cut

text: Frobenius

text: group

text: with

text: abelian

text: kernel.

text: Then

text: G

text: appears

text: in

text: Theorem

text: or

text: is

text: isomorphic

text: to

text: Z

text: Z

text: (Ca

text: Ca

text: C

text: or

text: Cd

text: C

text: (a,

text: a

text: a

text: a

text: d

text: For

text: each

text: of

text: the

text: above

text: structures

text: there

text: is

text: a

text: unique

text: cut

text: Frobenius

text: group

text: of

text: that

text: form.

text: Proof.

text: If

text: the

text: order

text: of

text: the

text: complement

text: is

text: even,

text: then

text: the

text: group

text: was

text: handled

text: in

text: Theorem

text: In

text: the

text: case

text: of

text: a

text: complement

text: of

text: odd

text: order,

text: it

text: follows

text: from

text: the

text: proof

text: of

text: Theorem

text: that

text: the

text: Frobenius

text: kernel

text: F

text: is

text: an

text: abelian

text: group

text: of

text: exponent

text: or

text: a

text: divisor

text: of

text: If

text: exp

text: F

text: then

text: we

text: can

text: argue

text: as

text: in

text: (i)

text: in

text: the

text: proof

text: of

text: Theorem

text: to

text: see

text: that

text: Z

text: there

text: is

text: a

text: unique

text: cut

text: Frobenius

text: group

text: of

text: the

text: form

text: and

text: Cd

text: C

text: for

text: all

text: d

text: Assume

text: that

text: F

text: is

text: a

text: As

text: all

text: groups

text: of

text: the

text: form

text: Ca

text: Ca

text: are

text: cut,

text: the

text: ×2

text: task

text: amounts

text: to

text: ﬁnd

text: all

text: Frobenius

text: groups

text: of

text: the

text: form

text: (Ca

text: Ca

text: C

text: But

text: it

text: is

text: well-known

text: that

text: there

text: is

text: such

text: a

text: group

text: if

text: and

text: only

text: if

text: a,

text: a

text: not

text: both

text: zero,

text: and

text: this

text: group

text: is

text: unique

text: up

text: to

text: isomorphism.

text: This

text: can

text: be

text: extracted,

text: for

text: example,

text: from

text: the

text: more

text: general

text: discussion

text: in

text: Section

text: Remark

text: Turning

text: our

text: attention

text: to

text: cut

text: Frobenius

text: groups

text: with

text: non-abelian

text: Frobenius

text: kernels

text: we

text: provide

text: examples

text: in

text: both

text: situations,

text: when

text: the

text: kernel

text: is

text: a

text: group

text: and

text: when

text: it

text: is

text: a

text: For

text: the

text: ﬁrst,

text: we

text: refer

text: to

text: Example

text: There

text: it

text: is

text: shown

text: that

text: the

text: Heisenberg

text: group

text: P

text: of

text: order

text: (i.e.

text: the

text: non-abelian

text: group

text: of

text: order

text: and

text: exponent

text: admits

text: a

text: ﬁxed-point

text: free

text: automorphism

text: of

text: order

text: which

text: results

text: in

text: a

text: Frobenius

text: group

text: P

text: C

text: that

text: is

text: semi-rational

text: (recall

text: that

text: in

text: the

text: case

text: of

text: groups

text: of

text: odd

text: order,

text: cut

text: and

text: semi-rational

text: are

text: the

text: same

text: Remark


text: Mon. Not. R. Astron. Soc. 000, 1–?? (2014) Printed 29 May 2020 (MN LATEX style ﬁle v2.2) Unveiling Vela - Time Variability of Interstellar Lines in the Direction of the Vela Supernova Remnant II. Na D and Ca II (cid:63) N. Kameswara Rao1,2, David L. Lambert2, Arumalla B. S. Reddy2, Ranjan Gupta3, S. Muneer1 & Harinder P. Singh4 † , 1Indian Institute of Astrophysics, Bangalore 560034, India 2The W.J. McDonald Observatory and Department of Astronomy, The University of Texas, Austin, TX 78712-1083, USA 3Inter University Centre for Astronomy and Astrophysics, IUCAA Pune, 411007 India 4Department of Physics & Astrophysics, University of Delhi, Delhi 110007 India 29 May 2020 ABSTRACT In a survey conducted between 2011-12 of interstellar Na i D line proﬁles in the direc- tion of the Vela supernova remnant, a few lines of sight showed dramatic changes in low velocity absorption components with respect to proﬁles from 1993-1994 reported by Cha & Sembach. Three stars - HD 63578, HD 68217 and HD 76161 showed large decrease in strength over the 1993-2012 interval. HD 68217 and HD 76161 are as- sociated with the Vela SNR whereas HD 63578 is associated with γ2 Velorum wind bubble. Here, we present high spectral resolution observations of Ca ii K lines obtained with the Southern African Large Telescope (SALT) towards these three stars along with simultaneous observations of Na i D lines. These new spectra conﬁrm that the Na D interstellar absorption weakened drastically between 1993-1994 and 2011-2012 but show for the ﬁrst time that the Ca II K line is unchanged between 1993-1994 and 2015. This remarkable contrast between the behaviour of Na D and Ca II K line absorption lines is a puzzle concerning gas presumably aﬀected by the outﬂow from the SNR and the wind from γ2 Velorum. Key words: Star: individual: ISM: variable ISM lines: Supernova Remnants :other 1 INTRODUCTION proﬁles of Ca ii K and Na i D lines superimposed on stellar spectra. Vela supernova remnant (SNR) is the closest relic of of ± a stellar explosion to earth, located at a distance of 287 19 Recently, we completed a study of high resolution Na i pc, as infered from the VLBI parallax of its pulsar (Dod- D line proﬁles towards sixty four OB stars in the direction of son et al 2003) and represents a supernova explosion that Vela SNR , mainly carried out during 2011-2012, and com- occurred 11000 years ago (Reichley, Downs & Morris1970). pared them with Ca ii K and Na i D line proﬁles of the same Observational studies of interstellar lines present in the spec- stars earlier obtained by Cha & Sembach (2000) in the pe- tra of stars in the direction of vela SNR provide information riod 1993-1996 at comparable spectral resolution (Rao et al about the interaction of the remnant with local interstellar 2015 -hereafter Paper I). Comparison of the proﬁles of Na i D medium (ISM). Such studies predominantly concentrated on from these two epochs revealed major decreases in low veloc- ity absorption components towards a few stars. When Paper I was written, we did not have Ca ii K proﬁles for compar- ison with K line observations of Cha & Sembach (2000) to constrain the cause of this Na D variability. Subsequently, (cid:63) Based on observations obtained with Southern African Large we could obtain with the high-resolution spectrograph at Telescope (SALT) and The Vainu Bappu Telescope (VBT). † E-mail: nkrao@iiap.res.in (NKR); dll@astro.as.utexas.edu Southern African Large Telescope (SALT) Ca ii K proﬁles (DLL);muneers@iiap.res.in (SM); rag@iucaa.in (RG); for three of the stars exhibiting dramatic weakening of Na singh@associates.iucaa.in (HPS) D absorption. In the present paper, we report on the Ca ii K (cid:13) c 2014 RAS

text: Mon.

label: paragraph

text: Not.

text: R.

text: Astron.

text: Soc.

text: Printed

text: May

text: (MN

text: LATEX

text: style

text: ﬁle

text: Unveiling

text: Vela

text: Time

text: Variability

text: of

text: Interstellar

text: Lines

text: in

text: the

text: Direction

text: of

text: the

text: Vela

text: Supernova

text: Remnant

text: II.

text: D

text: and

text: Ca

text: II

text: N.

text: Kameswara

text: David

text: L.

text: Arumalla

text: B.

text: S.

text: Ranjan

text: S.

text: Harinder

text: P.

text: Institute

text: of

text: Astrophysics,

text: Bangalore

text: India

text: W.J.

text: McDonald

text: Observatory

text: and

text: Department

text: of

text: Astronomy,

text: The

text: University

text: of

text: Texas,

text: Austin,

text: TX

text: USA

text: University

text: Centre

text: for

text: Astronomy

text: and

text: Astrophysics,

text: IUCAA

text: Pune,

text: India

text: of

text: Physics

text: Astrophysics,

text: University

text: of

text: Delhi,

text: Delhi

text: India

text: May

text: ABSTRACT

text: In

text: a

text: survey

text: conducted

text: between

text: of

text: interstellar

text: i

text: D

text: line

text: proﬁles

text: in

text: the

text: direc-

text: tion

text: of

text: the

text: Vela

text: supernova

text: remnant,

text: a

text: few

text: lines

text: of

text: sight

text: showed

text: dramatic

text: changes

text: in

text: low

text: velocity

text: absorption

text: components

text: with

text: respect

text: to

text: proﬁles

text: from

text: reported

text: by

text: Cha

text: Sembach.

text: Three

text: stars

text: HD

text: HD

text: and

text: HD

text: showed

text: large

text: decrease

text: in

text: strength

text: over

text: the

text: interval.

text: HD

text: and

text: HD

text: are

text: as-

text: sociated

text: with

text: the

text: Vela

text: SNR

text: whereas

text: HD

text: is

text: associated

text: with

text: γ2

text: Velorum

text: wind

text: bubble.

text: Here,

text: we

text: present

text: high

text: spectral

text: resolution

text: observations

text: of

text: Ca

text: ii

text: K

text: lines

text: obtained

text: with

text: the

text: Southern

text: African

text: Large

text: Telescope

text: (SALT)

text: towards

text: these

text: three

text: stars

text: along

text: with

text: simultaneous

text: observations

text: of

text: i

text: D

text: lines.

text: These

text: new

text: spectra

text: conﬁrm

text: that

text: the

text: D

text: interstellar

text: absorption

text: weakened

text: drastically

text: between

text: and

text: but

text: show

text: for

text: the

text: ﬁrst

text: time

text: that

text: the

text: Ca

text: II

text: K

text: line

text: is

text: unchanged

text: between

text: and

text: This

text: remarkable

text: contrast

text: between

text: the

text: behaviour

text: of

text: D

text: and

text: Ca

text: II

text: K

text: line

text: absorption

text: lines

text: is

text: a

text: puzzle

text: concerning

text: gas

text: presumably

text: aﬀected

text: by

text: the

text: outﬂow

text: from

text: the

text: SNR

text: and

text: the

text: wind

text: from

text: γ2

text: Velorum.

text: Key

text: words:

text: Star:

text: individual:

text: ISM:

text: variable

text: ISM

text: lines:

text: Supernova

text: Remnants

text: :other

text: INTRODUCTION

text: proﬁles

text: of

text: Ca

text: ii

text: K

text: and

text: i

text: D

text: lines

text: superimposed

text: on

text: stellar

text: spectra.

text: Vela

text: supernova

text: remnant

text: (SNR)

text: is

text: the

text: closest

text: relic

text: of

text: of

text: a

text: stellar

text: explosion

text: to

text: earth,

text: located

text: at

text: a

text: distance

text: of

text: Recently,

text: we

text: completed

text: a

text: study

text: of

text: high

text: resolution

text: i

text: pc,

text: as

text: infered

text: from

text: the

text: VLBI

text: parallax

text: of

text: its

text: pulsar

text: (Dod-

text: D

text: line

text: proﬁles

text: towards

text: sixty

text: four

text: OB

text: stars

text: in

text: the

text: direction

text: of

text: son

text: et

text: al

text: and

text: represents

text: a

text: supernova

text: explosion

text: that

text: Vela

text: SNR

text: mainly

text: carried

text: out

text: during

text: and

text: com-

text: occurred

text: years

text: ago

text: (Reichley,

text: Downs

text: pared

text: them

text: with

text: Ca

text: ii

text: K

text: and

text: i

text: D

text: line

text: proﬁles

text: of

text: the

text: same

text: Observational

text: studies

text: of

text: interstellar

text: lines

text: present

text: in

text: the

text: spec-

text: stars

text: earlier

text: obtained

text: by

text: Cha

text: Sembach

text: in

text: the

text: pe-

text: tra

text: of

text: stars

text: in

text: the

text: direction

text: of

text: vela

text: SNR

text: provide

text: information

text: riod

text: at

text: comparable

text: spectral

text: resolution

text: (Rao

text: et

text: al

text: about

text: the

text: interaction

text: of

text: the

text: remnant

text: with

text: local

text: interstellar

text: -hereafter

text: Paper

text: I).

text: Comparison

text: of

text: the

text: proﬁles

text: of

text: i

text: D

text: medium

text: (ISM).

text: Such

text: studies

text: predominantly

text: concentrated

text: on

text: from

text: these

text: two

text: epochs

text: revealed

text: major

text: decreases

text: in

text: low

text: veloc-

text: ity

text: absorption

text: components

text: towards

text: a

text: few

text: stars.

text: When

text: Paper

text: I

text: was

text: written,

text: we

text: did

text: not

text: have

text: Ca

text: ii

text: K

text: proﬁles

text: for

text: compar-

text: ison

text: with

text: K

text: line

text: observations

text: of

text: Cha

text: Sembach

text: to

text: constrain

text: the

text: cause

text: of

text: this

text: D

text: variability.

text: Subsequently,

text: Based

text: on

text: observations

text: obtained

text: with

text: Southern

text: African

text: Large

text: we

text: could

text: obtain

text: with

text: the

text: high-resolution

text: spectrograph

text: at

text: Telescope

text: (SALT)

text: and

text: The

text: Vainu

text: Bappu

text: Telescope

text: (VBT).

text: E-mail:

text: nkrao@iiap.res.in

text: (NKR);

text: dll@astro.as.utexas.edu

text: Southern

text: African

text: Large

text: Telescope

text: (SALT)

text: Ca

text: ii

text: K

text: proﬁles

text: (DLL);muneers@iiap.res.in

text: (SM);

text: rag@iucaa.in

text: (RG);

text: for

text: three

text: of

text: the

text: stars

text: exhibiting

text: dramatic

text: weakening

text: of

text: singh@associates.iucaa.in

text: (HPS)

text: D

text: absorption.

text: In

text: the

text: present

text: paper,

text: we

text: report

text: on

text: the

text: Ca

text: ii

text: K

text: c

text: RAS


text: Unveiling Vela-SALT 7 − s 1 feature weaker than it was in 1994 but stronger than in binary HD 72127 (Hobbs, Wallerstein & Hu 1982; Hobbs VBT and SALT spectra which suggests that the feature’s et al. 1991; Welty. Simon & Hobbs 2008). The magnitude evolution is not a simple one. of the weakening reported by us earlier is unprecedented in HD 76161: This star which like HD 68217 lies near published studies of interstellar Na D lines (see, for exam- the edge of the ROSAT image of the Vela SNR (paper I) ple, McEvoy et al. 2015) and is plausibly attributable to in- provides the third and the extreme example in which low teractions between ambient diﬀuse interstellar gas and the velocity strong Na D absorptions present in 1993 had almost high-velocity gas in and around the Vela nebula from the disappeared by 2011 - see Figure 5(left) : the saturated ab- Vela supernova and the winds from massive stars. It seems − sorption component at V +9km s 1 has vanished almost highly pertinent to note that out of forty one stars with Na LSR completely by 2011. A redward weaker absorption compo- D lines observed previously by Cha & Sembach and again − nent at +27 km s 1 present in 1993 also weakened dras- at the VBT none showed a comparable strengthening of the tically by 2012. However a weak absorption component at Na D lines between 1993–1994 and 2011–2012. (As noted for − − about 9 km s 1 seems unchanged over the period of 1993 HD 68217, A low-velocity component Na D did strengthen to 2012 baseline. The Na D proﬁle remained unchanged in between 1998 and 1994. Also, HD 73882, a star behind the the period 2011 to 2015 (Figure 6). Vela SNR showed a strengthening of the Ca i 4226 ˚A line The 1993 Ca ii K proﬁles of 1993 and 2015 (Figure 5- between 2006 and 2012 (Galazutdinov et al. 2013) and pre- − − right) appear with components at 8 and +2 km s 1 (Table sumably the Na D lines also strengthened over this interval.) 3). The Ca ii K proﬁle obtained between 1971-77 period by The better established signature of these interactions Wallerstein, Silk & Jenkins (1980) appears to be similar to between high-velocity gas and the ambient interstellar | ∼ | the proﬁle observed by Cha & Sembach (2000) in 1993. m−edium is the presence of high-velocity ( VLSR 100 km Jenkins, Wallerstein & Silk (1984) measured LSR ra- s 1) absorption components seen in stars with distances − − − dial velocity of S ii and Si ii+ Fe ii as 3 km s 1 and 7 greater than about 500 pc. High-velocity gas was ﬁrst re- − km s 1 respectively from their IUE spectrum obtained in ported by Wallerstein & Silk (1971) and Thackeray & War- 1979 September 26, very similar values to that of Ca ii K ren (1972) from observations of the Ca ii K line with exten- components measured bu Cha & Sembach (2000) in 1993. sive follow-up at the Ca ii K line and Na D lines by Cha & Later, Nichols & Slavin (2004) detected C i lines at the Sembach (2000). (High-velocity Na D gas associated with LSR velocity of 8 km s−1 in the same IUE spectrum ob- the SNR Monoceros Loop has been shown to be variable tained in 1979 . This velocity corresponds to the strong Na i (Dirks & Meyer 2016).) High-velocity gas associated with Vela has been studied in the ultraviolet (Jenkins, Waller- component that disappeared after 1993. It appears that a strong neutral cloud was present at +9km s−1 for at least stein & Silk 1976; Wallerstein, Silk & Jenkins 1980; Jenkins et al. 1981; Jenkins, Wallerstein & Silk 1984). With respect a period of 14 years prior to 1993 and then vanished some to the Na D and Ca K line, a characteristic of the high- time in the next 18 years. Nichols & Slavin (2004) investigated C i lines in 54 stars velocity gas is the several 100-fold increase with respect to the value for low velocity gas of the column density ratio of in Vela SNR region including HD 76161, and noted the red- shift of C i lines with respect to ionized lines and suggested Ca ii to Na i which is attributed to the destruction of inter- that C i lines from the ground-state of the atom are a combi- stellar grains and release of substantial amounts of Ca (and other elements) but not Na which is little depleted by grains nation of two absorption components: one near zero velocity (Spitzer 1978; Danks & Sembach 1995; Sembach & Danks which might have its origins in the clouds in the spiral arm 1994). along the sight lines to the star and the other component at a Surely, the outstanding characteristic of the low- higher velocity arising from cloud that might have interacted velocity interstellar lines of the three sightlines where the with the Vela SNR. They also suggest that redshift observed in C i ground-state lines might have its origins in a H i shell Na D lines are strikingly weakened between 1993-1994 and 2012-2015 is that the Ca K line shows no detectable change on the back side of the remnant. Dubner et al. (1998) discov- ered a shell of H i through 21 cm emission around Vela SNR between 1993-1994 and 2015 in two cases (HD 63578 and HD 76161) and a possible mild weakening in the third case that follows closely the ROSAT’s outer X-ray bright shell. This 30 km s−1 expanding The H i shell is thought to be a (HD 68217) where the central depth of the component coin- cident in velocity with the varying Na D feature drops from result of recombined postshocked gas behind the advancing 77% in 1994 to 88% in 2015, a drop only slightly larger than shock front. There is the possibility that the cold neutral the the amplitude of the noise in the SALT spectrum. In cloud that disappeared in the direction of HD 76161 might short, the Ca K line is eﬀectively unaﬀected by the factors have been a shocked cloud. responsible for the large Na D changes and, in particular, 4 CONCLUDING REMARKS the Ca K’s velocity component coincident with the variable The focus of the paper has been three stars in the di- Na D component are unaﬀected. This contrast between Na rection of the Vela SNR in which a severe weakening of low- D and Ca K constrains the factors driving the weakening of velocity absorption components of interstellar Na D lines oc- the Na D line. curred between 1993-1994 and 2011-2012 (Rao et al. 2016). Estimation of the Na i column density for two of the Such low-velocity components are assumed to belong to the three examples is an uncertain procedure for the 1993-1994 local interstellar medium and not to be an immediate prod- spectra because the dominant components are saturated. uct of the Vela SNR or other energetic sources in the vicinity For the VBT and SALT spectra, the variable component of of the Vela nebula. A few earlier reports for stars behind the the Na D lines is little saturated; the equivalent widths of Vela SNR have noted modest variations in the Na D lines the D and D lines approach the 2:1 weak line limit. For HD 2 −1 − at low velocity - see, for example, observations of the visual 68217, the 8 km s 1 component appears to provide both (cid:13) c 2014 RAS, MNRAS 000, 1–??

text: Unveiling

text: Vela-SALT

text: s

text: feature

text: weaker

text: than

text: it

text: was

text: in

text: but

text: stronger

text: than

text: in

text: binary

text: HD

text: (Hobbs,

text: Wallerstein

text: Hu

text: Hobbs

text: VBT

text: and

text: SALT

text: spectra

text: which

text: suggests

text: that

text: the

text: feature’s

text: et

text: al.

text: Welty.

text: Simon

text: Hobbs

text: The

text: magnitude

text: evolution

text: is

text: not

text: a

text: simple

text: one.

text: of

text: the

text: weakening

text: reported

text: by

text: us

text: earlier

text: is

text: unprecedented

text: in

text: HD

text: This

text: star

text: which

text: like

text: HD

text: lies

text: near

text: published

text: studies

text: of

text: interstellar

text: D

text: lines

text: (see,

text: for

text: exam-

text: the

text: edge

text: of

text: the

text: ROSAT

text: image

text: of

text: the

text: Vela

text: SNR

text: (paper

text: I)

text: ple,

text: McEvoy

text: et

text: al.

text: and

text: is

text: plausibly

text: attributable

text: to

text: in-

text: provides

text: the

text: third

text: and

text: the

text: extreme

text: example

text: in

text: which

text: low

text: teractions

text: between

text: ambient

text: diﬀuse

text: interstellar

text: gas

text: and

text: the

text: velocity

text: strong

text: D

text: absorptions

text: present

text: in

text: had

text: almost

text: high-velocity

text: gas

text: in

text: and

text: around

text: the

text: Vela

text: nebula

text: from

text: the

text: disappeared

text: by

text: see

text: Figure

text: the

text: saturated

text: ab-

text: Vela

text: supernova

text: and

text: the

text: winds

text: from

text: massive

text: stars.

text: It

text: seems

text: sorption

text: component

text: at

text: V

text: s

text: has

text: vanished

text: almost

text: highly

text: pertinent

text: to

text: note

text: that

text: out

text: of

text: forty

text: one

text: stars

text: with

text: LSR

text: completely

text: by

text: A

text: redward

text: weaker

text: absorption

text: compo-

text: D

text: lines

text: observed

text: previously

text: by

text: Cha

text: Sembach

text: and

text: again

text: nent

text: at

text: km

text: s

text: present

text: in

text: also

text: weakened

text: dras-

text: at

text: the

text: VBT

text: showed

text: a

text: comparable

text: strengthening

text: of

text: the

text: tically

text: by

text: However

text: a

text: weak

text: absorption

text: component

text: at

text: D

text: lines

text: between

text: and

text: (As

text: noted

text: for

text: about

text: km

text: s

text: seems

text: unchanged

text: over

text: the

text: period

text: of

text: HD

text: A

text: low-velocity

text: component

text: D

text: did

text: strengthen

text: to

text: baseline.

text: The

text: D

text: proﬁle

text: remained

text: unchanged

text: in

text: between

text: and

text: Also,

text: HD

text: a

text: star

text: behind

text: the

text: the

text: period

text: to

text: (Figure

text: Vela

text: SNR

text: showed

text: a

text: strengthening

text: of

text: the

text: Ca

text: i

text: ˚A

text: line

text: The

text: Ca

text: ii

text: K

text: proﬁles

text: of

text: and

text: (Figure

text: between

text: and

text: (Galazutdinov

text: et

text: al.

text: and

text: pre-

text: right)

text: appear

text: with

text: components

text: at

text: and

text: km

text: s

text: (Table

text: sumably

text: the

text: D

text: lines

text: also

text: strengthened

text: over

text: this

text: interval.)

text: The

text: Ca

text: ii

text: K

text: proﬁle

text: obtained

text: between

text: period

text: by

text: The

text: better

text: established

text: signature

text: of

text: these

text: interactions

text: Wallerstein,

text: Silk

text: Jenkins

text: appears

text: to

text: be

text: similar

text: to

text: between

text: high-velocity

text: gas

text: and

text: the

text: ambient

text: interstellar

text: the

text: proﬁle

text: observed

text: by

text: Cha

text: Sembach

text: in

text: m−edium

text: is

text: the

text: presence

text: of

text: high-velocity

text: VLSR

text: km

text: Jenkins,

text: Wallerstein

text: Silk

text: measured

text: LSR

text: ra-

text: s

text: absorption

text: components

text: seen

text: in

text: stars

text: with

text: distances

text: dial

text: velocity

text: of

text: S

text: ii

text: and

text: Si

text: ii+

text: Fe

text: ii

text: as

text: km

text: s

text: and

text: greater

text: than

text: about

text: pc.

text: High-velocity

text: gas

text: was

text: ﬁrst

text: re-

text: km

text: s

text: respectively

text: from

text: their

text: IUE

text: spectrum

text: obtained

text: in

text: ported

text: by

text: Wallerstein

text: Silk

text: and

text: Thackeray

text: War-

text: September

text: very

text: similar

text: values

text: to

text: that

text: of

text: Ca

text: ii

text: K

text: ren

text: from

text: observations

text: of

text: the

text: Ca

text: ii

text: K

text: line

text: with

text: exten-

text: components

text: measured

text: bu

text: Cha

text: Sembach

text: in

text: sive

text: follow-up

text: at

text: the

text: Ca

text: ii

text: K

text: line

text: and

text: D

text: lines

text: by

text: Cha

text: Later,

text: Nichols

text: Slavin

text: detected

text: C

text: i

text: lines

text: at

text: the

text: Sembach

text: (High-velocity

text: D

text: gas

text: associated

text: with

text: LSR

text: velocity

text: of

text: km

text: in

text: the

text: same

text: IUE

text: spectrum

text: ob-

text: the

text: SNR

text: Monoceros

text: Loop

text: has

text: been

text: shown

text: to

text: be

text: variable

text: tained

text: in

text: This

text: velocity

text: corresponds

text: to

text: the

text: strong

text: i

text: (Dirks

text: Meyer

text: High-velocity

text: gas

text: associated

text: with

text: Vela

text: has

text: been

text: studied

text: in

text: the

text: ultraviolet

text: (Jenkins,

text: Waller-

text: component

text: that

text: disappeared

text: after

text: It

text: appears

text: that

text: a

text: strong

text: neutral

text: cloud

text: was

text: present

text: at

text: for

text: at

text: least

text: stein

text: Silk

text: Wallerstein,

text: Silk

text: Jenkins

text: Jenkins

text: et

text: al.

text: Jenkins,

text: Wallerstein

text: Silk

text: With

text: respect

text: a

text: period

text: of

text: years

text: prior

text: to

text: and

text: then

text: vanished

text: some

text: to

text: the

text: D

text: and

text: Ca

text: K

text: line,

text: a

text: characteristic

text: of

text: the

text: high-

text: time

text: in

text: the

text: next

text: years.

text: Nichols

text: Slavin

text: investigated

text: C

text: i

text: lines

text: in

text: stars

text: velocity

text: gas

text: is

text: the

text: several

text: increase

text: with

text: respect

text: to

text: the

text: value

text: for

text: low

text: velocity

text: gas

text: of

text: the

text: column

text: density

text: ratio

text: of

text: in

text: Vela

text: SNR

text: region

text: including

text: HD

text: and

text: noted

text: the

text: red-

text: shift

text: of

text: C

text: i

text: lines

text: with

text: respect

text: to

text: ionized

text: lines

text: and

text: suggested

text: Ca

text: ii

text: to

text: i

text: which

text: is

text: attributed

text: to

text: the

text: destruction

text: of

text: inter-

text: that

text: C

text: i

text: lines

text: from

text: the

text: ground-state

text: of

text: the

text: atom

text: are

text: a

text: combi-

text: stellar

text: grains

text: and

text: release

text: of

text: substantial

text: amounts

text: of

text: Ca

text: (and

text: other

text: elements)

text: but

text: not

text: which

text: is

text: little

text: depleted

text: by

text: grains

text: nation

text: of

text: two

text: absorption

text: components:

text: one

text: near

text: zero

text: velocity

text: (Spitzer

text: Danks

text: Sembach

text: Sembach

text: Danks

text: which

text: might

text: have

text: its

text: origins

text: in

text: the

text: clouds

text: in

text: the

text: spiral

text: arm

text: along

text: the

text: sight

text: lines

text: to

text: the

text: star

text: and

text: the

text: other

text: component

text: at

text: a

text: Surely,

text: the

text: outstanding

text: characteristic

text: of

text: the

text: low-

text: higher

text: velocity

text: arising

text: from

text: cloud

text: that

text: might

text: have

text: interacted

text: velocity

text: interstellar

text: lines

text: of

text: the

text: three

text: sightlines

text: where

text: the

text: with

text: the

text: Vela

text: SNR.

text: They

text: also

text: suggest

text: that

text: redshift

text: observed

text: in

text: C

text: i

text: ground-state

text: lines

text: might

text: have

text: its

text: origins

text: in

text: a

text: H

text: i

text: shell

text: D

text: lines

text: are

text: strikingly

text: weakened

text: between

text: and

text: is

text: that

text: the

text: Ca

text: K

text: line

text: shows

text: no

text: detectable

text: change

text: on

text: the

text: back

text: side

text: of

text: the

text: remnant.

text: Dubner

text: et

text: al.

text: discov-

text: ered

text: a

text: shell

text: of

text: H

text: i

text: through

text: cm

text: emission

text: around

text: Vela

text: SNR

text: between

text: and

text: in

text: two

text: cases

text: (HD

text: and

text: HD

text: and

text: a

text: possible

text: mild

text: weakening

text: in

text: the

text: third

text: case

text: that

text: follows

text: closely

text: the

text: ROSAT’s

text: outer

text: X-ray

text: bright

text: shell.

text: This

text: km

text: expanding

text: The

text: H

text: i

text: shell

text: is

text: thought

text: to

text: be

text: a

text: (HD

text: where

text: the

text: central

text: depth

text: of

text: the

text: component

text: coin-

text: cident

text: in

text: velocity

text: with

text: the

text: varying

text: D

text: feature

text: drops

text: from

text: result

text: of

text: recombined

text: postshocked

text: gas

text: behind

text: the

text: advancing

text: in

text: to

text: in

text: a

text: drop

text: only

text: slightly

text: larger

text: than

text: shock

text: front.

text: There

text: is

text: the

text: possibility

text: that

text: the

text: cold

text: neutral

text: the

text: the

text: amplitude

text: of

text: the

text: noise

text: in

text: the

text: SALT

text: spectrum.

text: In

text: cloud

text: that

text: disappeared

text: in

text: the

text: direction

text: of

text: HD

text: might

text: short,

text: the

text: Ca

text: K

text: line

text: is

text: eﬀectively

text: unaﬀected

text: by

text: the

text: factors

text: have

text: been

text: a

text: shocked

text: cloud.

text: responsible

text: for

text: the

text: large

text: D

text: changes

text: and,

text: in

text: particular,

text: CONCLUDING

text: REMARKS

text: the

text: Ca

text: K’s

text: velocity

text: component

text: coincident

text: with

text: the

text: variable

text: The

text: focus

text: of

text: the

text: paper

text: has

text: been

text: three

text: stars

text: in

text: the

text: di-

text: D

text: component

text: are

text: unaﬀected.

text: This

text: contrast

text: between

text: rection

text: of

text: the

text: Vela

text: SNR

text: in

text: which

text: a

text: severe

text: weakening

text: of

text: low-

text: D

text: and

text: Ca

text: K

text: constrains

text: the

text: factors

text: driving

text: the

text: weakening

text: of

text: velocity

text: absorption

text: components

text: of

text: interstellar

text: D

text: lines

text: oc-

text: the

text: D

text: line.

text: curred

text: between

text: and

text: (Rao

text: et

text: al.

text: Estimation

text: of

text: the

text: i

text: column

text: density

text: for

text: two

text: of

text: the

text: Such

text: low-velocity

text: components

text: are

text: assumed

text: to

text: belong

text: to

text: the

text: three

text: examples

text: is

text: an

text: uncertain

text: procedure

text: for

text: the

text: local

text: interstellar

text: medium

text: and

text: not

text: to

text: be

text: an

text: immediate

text: prod-

text: spectra

text: because

text: the

text: dominant

text: components

text: are

text: saturated.

text: uct

text: of

text: the

text: Vela

text: SNR

text: or

text: other

text: energetic

text: sources

text: in

text: the

text: vicinity

text: For

text: the

text: VBT

text: and

text: SALT

text: spectra,

text: the

text: variable

text: component

text: of

text: of

text: the

text: Vela

text: nebula.

text: A

text: few

text: earlier

text: reports

text: for

text: stars

text: behind

text: the

text: the

text: D

text: lines

text: is

text: little

text: saturated;

text: the

text: equivalent

text: widths

text: of

text: Vela

text: SNR

text: have

text: noted

text: modest

text: variations

text: in

text: the

text: D

text: lines

text: the

text: D

text: and

text: D

text: lines

text: approach

text: the

text: weak

text: line

text: limit.

text: For

text: HD

text: −1

text: at

text: low

text: velocity

text: see,

text: for

text: example,

text: observations

text: of

text: the

text: visual

text: the

text: km

text: s

text: component

text: appears

text: to

text: provide

text: both

text: c

text: RAS,

text: MNRAS


text: For the hyper-parameters where there is a “correct” answer, such as the learning rate (0.001) and the presence of data augmentation, the guided search quickly converges and most models inherit their values. We should remove them on further analysis and instead ex- plore a wider range of architectures by adding hyper-parameters con- trolling the fully-connected layers, such as the number of units, the number of layers, adding new types of layers such as dropout and batch normalization placed across the network or even explore other learning method such as RMSProp or Adadelta. One limitation of the current system is that some hyper- parameters depend on the value of others. For example we are unable to choose a different number of ﬁlters per layer as the number of layers is not ﬁxed. We also limited the range of some hyper-parameters such as the ﬁlter size so as to have networks that would ﬁt in memory and be trained in a reasonable amount of time. Only a subset of values combinations would cause a failure. Further work could incorporate constraints on time and memory either by precise estimations when possible or by measures during training to produce estimations with a GP. Results on volume classiﬁcation were very satisfactory. Since it is done at slice level, we have obtained a decent localization tool Fig. 5. Confusion matrices on the test set, for the baseline network which shows the robustness of our ensemble. Further work will fo- (left) and the ensemble (right), computed on slices (ﬁrst row) and on cus on adding more anatomical regions, which might require split- volumes (second row). ting slices in patches to identify smaller regions such as organs. 5. REFERENCES [1] “Special Issue on Deep Learning in Medical Imaging,” IEEE Transactions on Medical Imaging, vol. 35, no. 5, 2016. [2] A. Krizhevsky, I. Sutskever, and G. E. Hinton, “ImageNet Clas- siﬁcation with Deep Convolutional Neural Networks,” in Ad- vances in Neural Information Processing Systems 25, pp. 1097– 1105. 2012. [3] K. Simonyan and A. Zisserman, “Very Deep Convolutional Net- works for Large-Scale Image Recognition,” Sept. 2014. [4] D. R. Jones, “A Taxonomy of Global Optimization Methods Based on Response Surfaces,” Journal of Global Optimization, vol. 21, no. 4, pp. 345–383, 2001. Fig. 6. Slice by slice classiﬁcation on a full body volume. Top: [5] I. Goodfellow, Y. Bengio, and A. Courville, “Deep learning,” Class probabilities. Filled areas correspond to decision made when MIT Press, 2016. the probability is higher than 0.7. Bottom: Volume and ground truth. [6] J. Bergstra and Y. Bengio, “Random Search for Hyper- parameter Optimization,” The Journal of Machine Learning Re- search, vol. 13, pp. 281–305, 2012. 0.7, and if no class meets this criterion, then we do not choose any. As we can see in Figure 6, the network is doing well at identifying [7] Jasper Snoek, Hugo Larochelle, and Ryan P. Adams, “Practi- the abdomen and the head. It also identiﬁes correctly the pelvis, cal Bayesian Optimization of Machine Learning Algorithms,” with some uncertainty. No class is dominant for most of the legs, NIPS, 2012. however the feet are considered as spine with high probability. It [8] R. L. Anderson, “Recent advances in ﬁnding best operating con- also mistakenly identiﬁes the neck as pelvis. ditions,” Journal of the American Statistical Association, vol. Those mistakes could be corrected by using a more complex de- 48, no. 264, pp. 789–798, 1953. cision criterion than a simple probability cutoff. We also expect that [9] C. E. Rasmussen and C. K. I. Williams, Gaussian Processes for adding more anatomical regions to our dataset will allow for a better Machine Learning, The MIT Press, 2005. localization of the present regions. 4. DISCUSSION AND CONCLUSION To the problem of ﬁnding more accurate networks than hand- crafted ones, we have answered with two viable strategies. Random search is as easy to implement as grid search and quickly improves on the baseline, which makes it perfect for time-constrained situa- tions. Without this constraint and at a higher cost in implementation, an adaptive search based on Gaussian Processes explores a range of highly accurate models suited for ensembling. ##LTFigure## ##LTFigure##

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text: answer,

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text: rate

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text: most

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text: inherit

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text: values.

text: We

text: should

text: remove

text: them

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text: GP.

text: Results

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text: were

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text: Since

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text: at

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text: we

text: have

text: obtained

text: a

text: decent

text: localization

text: tool

text: Fig.

text: Confusion

text: matrices

text: on

text: the

text: test

text: set,

text: for

text: the

text: baseline

text: network

text: which

text: shows

text: the

text: robustness

text: of

text: our

text: ensemble.

text: Further

text: work

text: will

text: fo-

text: (left)

text: and

text: the

text: ensemble

text: (right),

text: computed

text: on

text: slices

text: (ﬁrst

text: row)

text: and

text: on

text: cus

text: on

text: adding

text: more

text: anatomical

text: regions,

text: which

text: might

text: require

text: split-

text: volumes

text: (second

text: row).

text: ting

text: slices

text: in

text: patches

text: to

text: identify

text: smaller

text: regions

text: such

text: as

text: organs.

text: REFERENCES

text: “Special

text: Issue

text: on

text: Deep

text: Learning

text: in

text: Medical

text: Imaging,”

text: IEEE

text: Transactions

text: on

text: Medical

text: Imaging,

text: vol.

text: no.

text: A.

text: Krizhevsky,

text: I.

text: Sutskever,

text: and

text: G.

text: E.

text: Hinton,

text: “ImageNet

text: Clas-

text: siﬁcation

text: with

text: Deep

text: Convolutional

text: Neural

text: Networks,”

text: in

text: Ad-

text: vances

text: in

text: Neural

text: Information

text: Processing

text: Systems

text: pp.

text: K.

text: Simonyan

text: and

text: A.

text: Zisserman,

text: “Very

text: Deep

text: Convolutional

text: Net-

text: works

text: for

text: Large-Scale

text: Image

text: Recognition,”

text: Sept.

text: D.

text: R.

text: Jones,

text: “A

text: Taxonomy

text: of

text: Global

text: Optimization

text: Methods

text: Based

text: on

text: Response

text: Surfaces,”

text: Journal

text: of

text: Global

text: Optimization,

text: vol.

text: no.

text: pp.

text: Fig.

text: Slice

text: by

text: slice

text: classiﬁcation

text: on

text: a

text: full

text: body

text: volume.

text: Top:

text: I.

text: Goodfellow,

text: Y.

text: Bengio,

text: and

text: A.

text: Courville,

text: “Deep

text: learning,”

text: Class

text: probabilities.

text: Filled

text: areas

text: correspond

text: to

text: decision

text: made

text: when

text: MIT

text: Press,

text: the

text: probability

text: is

text: higher

text: than

text: Bottom:

text: Volume

text: and

text: ground

text: truth.

text: J.

text: Bergstra

text: and

text: Y.

text: Bengio,

text: “Random

text: Search

text: for

text: Hyper-

text: parameter

text: Optimization,”

text: The

text: Journal

text: of

text: Machine

text: Learning

text: Re-

text: search,

text: vol.

text: pp.

text: and

text: if

text: no

text: class

text: meets

text: this

text: criterion,

text: then

text: we

text: do

text: not

text: choose

text: any.

text: As

text: we

text: can

text: see

text: in

text: Figure

text: the

text: network

text: is

text: doing

text: well

text: at

text: identifying

text: Jasper

text: Snoek,

text: Hugo

text: Larochelle,

text: and

text: Ryan

text: P.

text: Adams,

text: “Practi-

text: the

text: abdomen

text: and

text: the

text: head.

text: It

text: also

text: identiﬁes

text: correctly

text: the

text: pelvis,

text: cal

text: Bayesian

text: Optimization

text: of

text: Machine

text: Learning

text: Algorithms,”

text: with

text: some

text: uncertainty.

text: No

text: class

text: is

text: dominant

text: for

text: most

text: of

text: the

text: legs,

text: NIPS,

text: however

text: the

text: feet

text: are

text: considered

text: as

text: spine

text: with

text: high

text: probability.

text: It

text: R.

text: L.

text: Anderson,

text: “Recent

text: advances

text: in

text: ﬁnding

text: best

text: operating

text: con-

text: also

text: mistakenly

text: identiﬁes

text: the

text: neck

text: as

text: pelvis.

text: ditions,”

text: Journal

text: of

text: the

text: American

text: Statistical

text: Association,

text: vol.

text: Those

text: mistakes

text: could

text: be

text: corrected

text: by

text: using

text: a

text: more

text: complex

text: de-

text: no.

text: pp.

text: cision

text: criterion

text: than

text: a

text: simple

text: probability

text: cutoff.

text: We

text: also

text: expect

text: that

text: C.

text: E.

text: Rasmussen

text: and

text: C.

text: K.

text: I.

text: Williams,

text: Gaussian

text: Processes

text: for

text: adding

text: more

text: anatomical

text: regions

text: to

text: our

text: dataset

text: will

text: allow

text: for

text: a

text: better

text: Machine

text: Learning,

text: The

text: MIT

text: Press,

text: localization

text: of

text: the

text: present

text: regions.

text: DISCUSSION

text: AND

text: CONCLUSION

text: To

text: the

text: problem

text: of

text: ﬁnding

text: more

text: accurate

text: networks

text: than

text: hand-

text: crafted

text: ones,

text: we

text: have

text: answered

text: with

text: two

text: viable

text: strategies.

text: Random

text: search

text: is

text: as

text: easy

text: to

text: implement

text: as

text: grid

text: search

text: and

text: quickly

text: improves

text: on

text: the

text: baseline,

text: which

text: makes

text: it

text: perfect

text: for

text: time-constrained

text: situa-

text: tions.

text: Without

text: this

text: constraint

text: and

text: at

text: a

text: higher

text: cost

text: in

text: implementation,

text: an

text: adaptive

text: search

text: based

text: on

text: Gaussian

text: Processes

text: explores

text: a

text: range

text: of

text: highly

text: accurate

text: models

text: suited

text: for

text: ensembling.

text: ##LTFigure##

text: ##LTFigure##


text: 4 (cid:0) (cid:1) (cid:16) (cid:17) (cid:90) The contribution of n one-loop vacuum polarization insertions is obtained from Eq.(17) with αq2I (q) n α α 1 v2 1 − v32 and the appropriate combinatoric factor. Tπ he 1i,nβtegral π I1,β(q) = π dv (1 − v2)q2 + 4 . (19) over v can be performed analytically, and the resulting 0 β q2 integral is   (cid:113) (cid:16)(cid:113) (cid:17)  (cid:16) (cid:17) (cid:90)        −  − − n 1 ∆E =mα5 α n−1 n ∞ dq2f (q) (12 5q2β) + 6(q2β 2) q2qβ2+β4arctanh q2qβ2+β4 1VPX π (cid:16)π (cid:17) 3 p 9q2β 0 − mα5 α n 1 = C , (20) π π n0,VPX where the factor of n reﬂects the combinatorics. In the case of β = 1, we can analytically solve this to guide estimates of higher order corrections. (cid:16) (cid:17) equation for n = 2, 3, 4. Our results agree with previ- IV. CONCLUSION ous calculations for C = 5/9 [49] and C = 20,VPX 30,VPX − 1 6π2 8 [50, 51]. These terms have already been in- In conclusion, we have computed several critical cor- 3 35 9 (cid:0) (cid:1) rections to the hyperﬁne splitting of true muonium nec- cluded in the hfs of true muonium. While experimental essary to reach 100 MHz precision. The recalculation precision even for positronium has not reach reached this − − of the VPCT, VPCA, and VPT diagrams has reduced level, we found C = 1 5255 2016π2 + 51840ζ(3) . 40,VPX 127575 the error budget by 200 MHz by removing the error The contribution from this term is 0.000614 MHz in true from ﬁnite precision and in the case of VPT has yielded muonium and 0.00297 kHz in positronium. analytic results. We have further computed C n0,VPX for n = 2, 3, 4 for values of β that are physically rel- evant. The current theoretical prediction is ∆E1s = hfs For values of β < 1, we have been unable to ﬁnd an 42329435(16) (700) MHz. In light of this work, the had miss O analytic expression, and have instead numerically inte- largest uncertainty arises from uncalculated (m α6) µ grated Eq. 20 for n = 2, 3, 4. The results for physi- electron loops. cally relevant cases are found in Tab. I. The contribu- ACKNOWLEDGMENTS tion from τ-loop terms in true muonium is 2.225430 + 0.001023 + 0.000002 = 2.226455 MHz. HL would like to express thanks to Michael Eides for pointing out errors in an earlier draft of this work. HL is supported by the National Science Foundation un- We have ﬁt a large number of numerical values to ﬁnd der Grant Nos. PHY-1068286 and PHY-1403891 and the β scaling for n = 2, 3, 4. The coeﬃcients are increas- by the U.S. Department of Energy under Contract No. ∝ ingly well-approximated by C β with increasing DE-FG02-93ER-40762. YJ acknowledges the Deutsche ∝ n0,VPX n. The C β behavior at large n agrees with Forschungsgemeinschaft for support under grant BR n0,VPX naive scaling expectations inside loops, and can be used 2021/7-1. [1] S. J. Brodsky and R. F. Lebed, Phys. Rev. Lett. 102, 213401[1(02]0J0.9)J,aaecrkXeilva:0n9d04S..22R2o5y,[hPehpy-ps.hR].ev. D82, 125020 (2010), arXiv:1008.3536 [hep-ph]. [2] H. Lamm, (2016), arXiv:1611.04258 [physics.atom-ph]. [11] B. Batell, D. McKeen, and M. Pospelov, Phys. Rev. Lett. 107, 011803 (2011), arXiv:1103.0721 [hep-ph]. [3] H. Lamm, Phys. Rev. D 91, 073008 (2015). [12] V. Barger, C.-W. Chiang, W.-Y. Keung, and D. Marfatia, Phys.Rev.Lett. 108, 081802 (2012), arXiv:1109.6652 [hep-ph]. [4] G. Bennett et al. (Muon G-2 Collaboration), Phys. Rev. D73[1,30]7S2.00G3. (K20a0rs6h)e,nabrXoiimv:,hPeph-yesx./R06ev0.20A3853[h, e0p6-2e1x1]9. (2011), arXiv:1005.4875 [hep-ph]. [5] A. Antognini et al., Science 339, 417 (2013). [14] S. Karshenboim, Phys. Rev. Lett. 104, 220406 (2010), arXiv:1005.4859 [hep-ph]. [6] R. Aaij et al. (LHCb collaboration), Phys. Rev. Lett. 113, 15[1156]01S.(K20a1r4s)h,eanrbXoiivm:1, 4P0h6y.6s4. 8R2ev[h.eDp-8e2x],.113013 (2010), arXiv:1005.4868 [hep-ph]. [7] R. Aaij et al. (LHCb), Phys. Rev. Lett. 115, 111803 (201[156)], S[A. Kddaernshdeunmb:oimP,hyPsh.yRs.eRv.evL.eDtt8.121,50,n7o3.01053,1(529091001)(,2a0r1X5i)v],:1005.4872 [hep-ph]. arXiv:1506.08614 [hep-ex]. [17] S. Karshenboim and V. Flambaum, Phys. Rev. A84, 064502 (2011), arXiv:1110.6259 [physics.atom-ph]. [8] R. Pohl et al. (CREMA), Science 353, 669 (2016). [18] S. G. Karshenboim, D. McKeen, and M. Pospelov, Phys. Rev. D90, 073004 (2014), arXiv:1401.6154 [hep-ph]. [9] D. Tucker-Smith and I. Yavin, Phys. Rev. D83, 101702 (201[11)9,]aCrX.iEv:.1C01a1rl.s4o9n22an[hdepB-.pCh].. Rislow, Phys.Rev. D86, 035013 (2012), arXiv:1206.3587 [hep-ph]. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: equation

text: The

text: contribution

text: of

text: n

text: one-loop

text: vacuum

text: polarization

text: insertions

text: is

text: obtained

text: from

text: with

text: (q)

text: n

text: α

text: α

text: and

text: the

text: appropriate

text: combinatoric

text: factor.

text: Tπ

text: he

text: π

text: π

text: dv

text: over

text: v

text: can

text: be

text: performed

text: analytically,

text: and

text: the

text: resulting

text: β

text: integral

text: is

text: n

text: ∆E

text: α

text: n

text: (q)

text: π

text: p

text: α

text: n

text: C

text: π

text: π

text: where

text: the

text: factor

text: of

text: n

text: reﬂects

text: the

text: combinatorics.

text: In

text: the

text: case

text: of

text: β

text: we

text: can

text: analytically

text: solve

text: this

text: to

text: guide

text: estimates

text: of

text: higher

text: order

text: corrections.

text: equation

text: for

text: n

text: Our

text: results

text: agree

text: with

text: previ-

text: IV.

text: CONCLUSION

text: ous

text: calculations

text: for

text: C

text: and

text: C

text: These

text: terms

text: have

text: already

text: been

text: in-

text: In

text: conclusion,

text: we

text: have

text: computed

text: several

text: critical

text: cor-

text: rections

text: to

text: the

text: hyperﬁne

text: splitting

text: of

text: true

text: muonium

text: nec-

text: cluded

text: in

text: the

text: hfs

text: of

text: true

text: muonium.

text: While

text: experimental

text: essary

text: to

text: reach

text: MHz

text: precision.

text: The

text: recalculation

text: precision

text: even

text: for

text: positronium

text: has

text: not

text: reach

text: reached

text: this

text: of

text: the

text: VPCT,

text: VPCA,

text: and

text: VPT

text: diagrams

text: has

text: reduced

text: level,

text: we

text: found

text: C

text: the

text: error

text: budget

text: by

text: MHz

text: by

text: removing

text: the

text: error

text: The

text: contribution

text: from

text: this

text: term

text: is

text: MHz

text: in

text: true

text: from

text: ﬁnite

text: precision

text: and

text: in

text: the

text: case

text: of

text: VPT

text: has

text: yielded

text: muonium

text: and

text: kHz

text: in

text: positronium.

text: analytic

text: results.

text: We

text: have

text: further

text: computed

text: C

text: for

text: n

text: for

text: values

text: of

text: β

text: that

text: are

text: physically

text: rel-

text: evant.

text: The

text: current

text: theoretical

text: prediction

text: is

text: hfs

text: For

text: values

text: of

text: β

text: we

text: have

text: been

text: unable

text: to

text: ﬁnd

text: an

text: MHz.

text: In

text: light

text: of

text: this

text: work,

text: the

text: had

text: miss

text: O

text: analytic

text: expression,

text: and

text: have

text: instead

text: numerically

text: inte-

text: largest

text: uncertainty

text: arises

text: from

text: uncalculated

text: (m

text: µ

text: grated

text: Eq.

text: for

text: n

text: The

text: results

text: for

text: physi-

text: electron

text: loops.

text: cally

text: relevant

text: cases

text: are

text: found

text: in

text: Tab.

text: I.

text: The

text: contribu-

text: ACKNOWLEDGMENTS

text: tion

text: from

text: τ-loop

text: terms

text: in

text: true

text: muonium

text: is

text: MHz.

text: HL

text: would

text: like

text: to

text: express

text: thanks

text: to

text: Michael

text: Eides

text: for

text: pointing

text: out

text: errors

text: in

text: an

text: earlier

text: draft

text: of

text: this

text: work.

text: HL

text: is

text: supported

text: by

text: the

text: National

text: Science

text: Foundation

text: un-

text: We

text: have

text: ﬁt

text: a

text: large

text: number

text: of

text: numerical

text: values

text: to

text: ﬁnd

text: der

text: Grant

text: Nos.

text: and

text: and

text: the

text: β

text: scaling

text: for

text: n

text: The

text: coeﬃcients

text: are

text: increas-

text: by

text: the

text: U.S.

text: Department

text: of

text: Energy

text: under

text: Contract

text: No.

text: ingly

text: well-approximated

text: by

text: C

text: β

text: with

text: increasing

text: YJ

text: acknowledges

text: the

text: Deutsche

text: n.

text: The

text: C

text: β

text: behavior

text: at

text: large

text: n

text: agrees

text: with

text: Forschungsgemeinschaft

text: for

text: support

text: under

text: grant

text: BR

text: naive

text: scaling

text: expectations

text: inside

text: loops,

text: and

text: can

text: be

text: used

text: S.

text: J.

text: Brodsky

text: and

text: R.

text: F.

text: Lebed,

text: Phys.

text: Rev.

text: Lett.

text: [hep-ph].

text: H.

text: Lamm,

text: [physics.atom-ph].

text: B.

text: Batell,

text: D.

text: McKeen,

text: and

text: M.

text: Pospelov,

text: Phys.

text: Rev.

text: Lett.

text: [hep-ph].

text: H.

text: Lamm,

text: Phys.

text: Rev.

text: D

text: V.

text: Barger,

text: C.-W.

text: Chiang,

text: W.-Y.

text: Keung,

text: and

text: D.

text: Marfatia,

text: Phys.Rev.Lett.

text: [hep-ph].

text: G.

text: Bennett

text: et

text: al.

text: (Muon

text: Collaboration),

text: Phys.

text: Rev.

text: [hep-ph].

text: A.

text: Antognini

text: et

text: al.,

text: Science

text: S.

text: Karshenboim,

text: Phys.

text: Rev.

text: Lett.

text: [hep-ph].

text: R.

text: Aaij

text: et

text: al.

text: (LHCb

text: collaboration),

text: Phys.

text: Rev.

text: Lett.

text: [hep-ph].

text: R.

text: Aaij

text: et

text: al.

text: (LHCb),

text: Phys.

text: Rev.

text: Lett.

text: S[A.

text: [hep-ph].

text: [hep-ex].

text: S.

text: Karshenboim

text: and

text: V.

text: Flambaum,

text: Phys.

text: Rev.

text: [physics.atom-ph].

text: R.

text: Pohl

text: et

text: al.

text: (CREMA),

text: Science

text: S.

text: G.

text: Karshenboim,

text: D.

text: McKeen,

text: and

text: M.

text: Pospelov,

text: Phys.

text: Rev.

text: [hep-ph].

text: D.

text: Tucker-Smith

text: and

text: I.

text: Yavin,

text: Phys.

text: Rev.

text: Rislow,

text: Phys.Rev.

text: [hep-ph].

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text: Throughout, we assume that any such function is deﬁned by an evaluation procedure consisting of a sequence of elemental functions v = ϕ (v ) ≺ , where i i j j i ϕ is either smooth or the absolute value function. The data dependence relation ≺i generates a partial ordering, which yields a directed acyclic graph. In other words, we assume that we have a straight line program without any loops or jumps in the control ﬂow. In the above reference a piecewise linearization ∆F(˚x; ∆x) of F about the point ˚x is constructed such that, for arbitrary x ∈ Rn with F˚ = F(˚x), it holds − O (cid:107) − (cid:107) F(x) = F˚ + ∆F(˚x; x ˚x) + ( x ˚x 2) . Since ∆F(˚x; ∆x) is constructed by replacing smooth elementals by their tangent lines at ˚x, we will refer to it as piecewise tangent linearization. An impor- tant property of these piecewise linearizations is that they vary continuously with respect to the sample point or points at which they are developed. As a generalization of ∆F(˚x; ∆x) we construct a piecewise secant lin- earization ∆F(xˇ, xˆ; ∆x) of F at the pair xˇ and xˆ such that for the midpoints ˚x = (xˇ + xˆ)/2 and F˚ = (Fˇ + Fˆ)/2 with Fˇ = F(xˇ) and Fˆ = F(xˆ) it holds − O (cid:107) − (cid:107)(cid:107) − (cid:107) F(x) = F˚ + ∆F(xˇ, xˆ; x ˚x) + ( x xˇ x xˆ ) . The new ∆F(xˇ, xˆ; ∆x) reduces to ∆F(˚x; ∆x) when the two sample points xˇ and xˆ coalesce at ˚x. Our results in this article are twofold: • Firstly, we present approximation properties and Lipschitz continuity es- timates. Here the two major points are that the piecewise linearization is a second order approximation to the underlying function. Moreover, we do not only prove the existence of Lipschitz constants, but provide ex- plicit estimates. These results immediately yield additional statement on conditions for perturbation-stable surjectivity of a piecewise linear model. • Secondly, we develop an application for the piecewise linearization. Namely, we introduce, for each linearization mode, a Newton’s method based on successive piecewise linearization and give suﬃcient conditions for conver- gence, as well as statements on the convergence rates. In an appended section we moreover provide formulas for singularity free and thus numerically stable implementations of the piecewise secant linearizations. One signiﬁcant advantage of the piecewise linearization-approach is that, once the piecewise linearization is generated, we also know where its kinks are located. This means that we are liberated of the complications of event han- dling that the nondiﬀerentiabilities of piecewise smooth functions usually bring about, which was one of the motivations for the development of the techniques presented in [Gri13]. In said reference it is proved that the variations of the piecewise linear model are Lipschitz continuous with respect to perturbations of the development point. The results mentioned in the ﬁrst bullet point represent a signiﬁcant improvement over those given in [Gri13] in so far as they are not only sharper, but also provide explicit bounds for the Lipschitz constants for the variations of the model with respect to perturbations of the base point. 2

text: Throughout,

text: we

text: assume

text: that

text: any

text: such

text: function

text: is

text: deﬁned

text: by

text: an

text: evaluation

text: procedure

text: consisting

text: of

text: a

text: sequence

text: of

text: elemental

text: functions

text: v

text: ϕ

text: (v

text: where

text: i

text: i

text: j

text: j

text: i

text: ϕ

text: is

text: either

text: smooth

text: or

text: the

text: absolute

text: value

text: function.

text: The

text: data

text: dependence

text: relation

text: ≺i

text: generates

text: a

text: partial

text: ordering,

text: which

text: yields

text: a

text: directed

text: acyclic

text: graph.

text: In

text: other

text: words,

text: we

text: assume

text: that

text: we

text: have

text: a

text: straight

text: line

text: program

text: without

text: any

text: loops

text: or

text: jumps

text: in

text: the

text: control

text: ﬂow.

text: In

text: the

text: above

text: reference

text: a

text: piecewise

text: linearization

text: ∆F(˚x;

text: ∆x)

text: of

text: F

text: about

text: the

text: point

text: ˚x

text: is

text: constructed

text: such

text: that,

text: for

text: arbitrary

text: x

text: Rn

text: with

text: F˚

text: F(˚x),

text: it

text: holds

text: O

text: F(x)

text: F˚

text: ∆F(˚x;

text: x

text: ˚x)

text: x

text: ˚x

text: Since

text: ∆F(˚x;

text: ∆x)

text: is

text: constructed

text: by

text: replacing

text: smooth

text: elementals

text: by

text: their

text: tangent

text: lines

text: at

text: ˚x,

text: we

text: will

text: refer

text: to

text: it

text: as

text: piecewise

text: tangent

text: linearization.

text: An

text: impor-

text: tant

text: property

text: of

text: these

text: piecewise

text: linearizations

text: is

text: that

text: they

text: vary

text: continuously

text: with

text: respect

text: to

text: the

text: sample

text: point

text: or

text: points

text: at

text: which

text: they

text: are

text: developed.

text: As

text: a

text: generalization

text: of

text: ∆F(˚x;

text: ∆x)

text: we

text: construct

text: a

text: piecewise

text: secant

text: lin-

text: earization

text: ∆F(xˇ,

text: xˆ;

text: ∆x)

text: of

text: F

text: at

text: the

text: pair

text: xˇ

text: and

text: xˆ

text: such

text: that

text: for

text: the

text: midpoints

text: ˚x

text: (xˇ

text: and

text: F˚

text: (Fˇ

text: with

text: Fˇ

text: F(xˇ)

text: and

text: Fˆ

text: F(xˆ)

text: it

text: holds

text: O

text: F(x)

text: F˚

text: ∆F(xˇ,

text: xˆ;

text: x

text: ˚x)

text: x

text: xˇ

text: x

text: xˆ

text: The

text: new

text: ∆F(xˇ,

text: xˆ;

text: ∆x)

text: reduces

text: to

text: ∆F(˚x;

text: ∆x)

text: when

text: the

text: two

text: sample

text: points

text: xˇ

text: and

text: xˆ

text: coalesce

text: at

text: ˚x.

text: Our

text: results

text: in

text: this

text: article

text: are

text: twofold:

text: Firstly,

text: we

text: present

text: approximation

text: properties

text: and

text: Lipschitz

text: continuity

text: es-

text: timates.

text: Here

text: the

text: two

text: major

text: points

text: are

text: that

text: the

text: piecewise

text: linearization

text: is

text: a

text: second

text: order

text: approximation

text: to

text: the

text: underlying

text: function.

text: Moreover,

text: we

text: do

text: not

text: only

text: prove

text: the

text: existence

text: of

text: Lipschitz

text: constants,

text: but

text: provide

text: ex-

text: plicit

text: estimates.

text: These

text: results

text: immediately

text: yield

text: additional

text: statement

text: on

text: conditions

text: for

text: perturbation-stable

text: surjectivity

text: of

text: a

text: piecewise

text: linear

text: model.

text: Secondly,

text: we

text: develop

text: an

text: application

text: for

text: the

text: piecewise

text: linearization.

text: Namely,

text: we

text: introduce,

text: for

text: each

text: linearization

text: mode,

text: a

text: Newton’s

text: method

text: based

text: on

text: successive

text: piecewise

text: linearization

text: and

text: give

text: suﬃcient

text: conditions

text: for

text: conver-

text: gence,

text: as

text: well

text: as

text: statements

text: on

text: the

text: convergence

text: rates.

text: In

text: an

text: appended

text: section

text: we

text: moreover

text: provide

text: formulas

text: for

text: singularity

text: free

text: and

text: thus

text: numerically

text: stable

text: implementations

text: of

text: the

text: piecewise

text: secant

text: linearizations.

text: One

text: signiﬁcant

text: advantage

text: of

text: the

text: piecewise

text: linearization-approach

text: is

text: that,

text: once

text: the

text: piecewise

text: linearization

text: is

text: generated,

text: we

text: also

text: know

text: where

text: its

text: kinks

text: are

text: located.

text: This

text: means

text: that

text: we

text: are

text: liberated

text: of

text: the

text: complications

text: of

text: event

text: han-

text: dling

text: that

text: the

text: nondiﬀerentiabilities

text: of

text: piecewise

text: smooth

text: functions

text: usually

text: bring

text: about,

text: which

text: was

text: one

text: of

text: the

text: motivations

text: for

text: the

text: development

text: of

text: the

text: techniques

text: presented

text: in

text: In

text: said

text: reference

text: it

text: is

text: proved

text: that

text: the

text: variations

text: of

text: the

text: piecewise

text: linear

text: model

text: are

text: Lipschitz

text: continuous

text: with

text: respect

text: to

text: perturbations

text: of

text: the

text: development

text: point.

text: The

text: results

text: mentioned

text: in

text: the

text: ﬁrst

text: bullet

text: point

text: represent

text: a

text: signiﬁcant

text: improvement

text: over

text: those

text: given

text: in

text: in

text: so

text: far

text: as

text: they

text: are

text: not

text: only

text: sharper,

text: but

text: also

text: provide

text: explicit

text: bounds

text: for

text: the

text: Lipschitz

text: constants

text: for

text: the

text: variations

text: of

text: the

text: model

text: with

text: respect

text: to

text: perturbations

text: of

text: the

text: base

text: point.


text: We investigated the behavior of the tangent and secant mode Newton on F both with and without noise. That is, we investigated F and F˜, where · sin(5000 [x + x ]) F˜(x) = F(x) + 1 2 . 104 − − (cid:62) The secant mode was started with the initial values xˇ = [ 3.7, 2.05] and (cid:62) xˆ = [7.0, 8.0] . The tangent mode was started with the mean value of the latter points. (cid:12) (cid:12) We recall the well known formula(cid:12)for appr(cid:12)oximating the convergence rate (cid:12) (cid:12) numerically: (cid:12) (cid:12) (cid:12) − (cid:12) γ ≈ log(cid:12) xxnn+−−1xnx−n1 (cid:12) . log xn −xn−1 xn−1 xn−2 The ﬁrst table shows both modes’ residuals in the iterations without noise. iteration residual with tangent mode res. with secant mode 0 13.3919956235 5.81435555868 1 5.65630249881 19.6157765738 2 3.39957297287 4.99712831052 3 0.00920821188601 1.2635478817 − 4 2.65403135735e 06 0.0882985228824 − 5 2.13162820728e 13 0.00192152171278 − − 6 8.881784197e 15 5.19844852542e 06 − 7 3.0607116841e 10 − 8 8.881784197e 16 √ ≈ ≈ γ 2.07814399547 2 1.64753467681 1+ 5 2 The next table shows the residuals of the iterations with noise. iteration residual with tangent mode res. with secant mode 0 13.3920216719 5.81445152008 1 60.308012713 19.6157077846 2 81.8554424857 4.99709320593 3 8.54532016753 1.26352999658 4 5.69986744799 0.0883961689187 5 1.92933721639 0.00178884970844 6 0.425650504358 0.00013247524387 − 7 0.127098157087 3.54530493811e 05 − 8 0.0253173333077 2.27749388494e 06 − 9 0.00248354296218 3.65673925216e 08 − 10 0.000950425306504 3.71373256312e 11 − − 11 6.728080753e 05 4.95408564432e 15 − 12 1.87922375892e 06 − 13 1.57356337999e 09 − 14 2.89904818901e 15 √ ≈ ≈ γ 2.01181918489 2 1.66153582947 1+ 5 2 20 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: We

text: investigated

text: the

text: behavior

text: of

text: the

text: tangent

text: and

text: secant

text: mode

text: Newton

text: on

text: F

text: both

text: with

text: and

text: without

text: noise.

text: That

text: is,

text: we

text: investigated

text: F

text: and

text: F˜,

text: where

text: [x

text: x

text: F˜(x)

text: F(x)

text: The

text: secant

text: mode

text: was

text: started

text: with

text: the

text: initial

text: values

text: xˇ

text: and

text: xˆ

text: The

text: tangent

text: mode

text: was

text: started

text: with

text: the

text: mean

text: value

text: of

text: the

text: latter

text: points.

text: We

text: recall

text: the

text: well

text: known

text: the

text: convergence

text: rate

text: numerically:

text: γ

text: log

text: xn

text: The

text: ﬁrst

text: table

text: shows

text: both

text: modes’

text: residuals

text: in

text: the

text: iterations

text: without

text: noise.

text: iteration

text: residual

text: with

text: tangent

text: mode

text: res.

text: with

text: secant

text: mode

text: γ

text: The

text: next

text: table

text: shows

text: the

text: residuals

text: of

text: the

text: iterations

text: with

text: noise.

text: iteration

text: residual

text: with

text: tangent

text: mode

text: res.

text: with

text: secant

text: mode

text: γ

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text: (cid:32) (cid:114) (cid:33) (cid:16) (cid:17) 1 ± 2 bT Be Λ1±(∆t, ε) = cbT BA˜e cbT BA˜e + 4 . (38) 2 ∆t Next, we show that the above characteristic speeds are limited. Here we need to assume that the scheme is GSA. In fact, (cid:40) → 1, for ζ 0 − bT Be = bT (ζI + A) 1e = ∼ 1 → ∞ (39) , for ζ ζ − where for the ﬁrst case using the ISA property we get bT A 1e = eT e = 1 (here eT = · · · ∈ R s s (0, , 0, 1) s), whereas for the second case we note that for the consistency of the scheme bT e = 1. On the other hand, (cid:40) → 1, for ζ 0 − bT BA˜e = bT (ζI + A) 1A˜e = ∼ 1 → ∞ (40) , for ζ 2ζ − where for the ﬁrst case using the GSA property we get bT A 1A˜e = eT A˜e = ˜bT e = 1, whereas s for the second case we note that the quantity bT A˜e is a number and if we assume that the scheme is a second order accurate one, this(cid:32)gives(cid:114)bT A˜e = 1/(cid:33)2, by (26). As a consequence we have 1 ± 4 ±1 Λ1±(∆t, 0) = c c2 + , Λ1±(0, ε) = , (41) 2 ∆t ε (cid:32) (cid:114) (cid:33) → and therefore the CFL condition, in the limit ε 0, becomes √ − | | 1 ≤ c c2 1 √ 2 √∆t ∆t + + ∆x = | | ∆x. (42) 2 4 ∆t c ∆t + c2∆t + 4 (cid:28) Now, if ∆t 1 we get from √ ≤ √ 2√ ∆t | | ∆x, c ∆t + c2∆t + 4 ∼ the parabolic time step restriction ∆t ∆x2. → Finally, we prove that the scheme (33)-(34) in the limit case ε 0 is a consistent discretiza- tion of the limit equation (6), i.e. the scheme is AP. From (32) we get − − − − − − O V = ζA 1vne A 1(I ζA 1)A˜(f(U) p(U) ) + (ζ2), x and substituting in the numerical solution vn+1 we obtain − − − − − − − − O ζvn+1 = ζ(1 bT A 1e)vn + (bT A 1A˜ ˜bT )(U f(U)) ζbT A 2A˜(f(U) p(U) ) + (ζ2). x x 11 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: bT

text: Be

text: ε)

text: cbT

text: BA˜e

text: cbT

text: BA˜e

text: ∆t

text: Next,

text: we

text: show

text: that

text: the

text: above

text: characteristic

text: speeds

text: are

text: limited.

text: Here

text: we

text: need

text: to

text: assume

text: that

text: the

text: scheme

text: is

text: GSA.

text: In

text: fact,

text: for

text: ζ

text: bT

text: Be

text: bT

text: (ζI

text: A)

text: for

text: ζ

text: ζ

text: where

text: for

text: the

text: ﬁrst

text: case

text: using

text: the

text: ISA

text: property

text: we

text: get

text: bT

text: A

text: eT

text: e

text: (here

text: eT

text: R

text: s

text: s

text: s),

text: whereas

text: for

text: the

text: second

text: case

text: we

text: note

text: that

text: for

text: the

text: consistency

text: of

text: the

text: scheme

text: bT

text: e

text: On

text: the

text: other

text: hand,

text: for

text: ζ

text: bT

text: BA˜e

text: bT

text: (ζI

text: A)

text: for

text: ζ

text: 2ζ

text: where

text: for

text: the

text: ﬁrst

text: case

text: using

text: the

text: GSA

text: property

text: we

text: get

text: bT

text: A

text: eT

text: A˜e

text: ˜bT

text: e

text: whereas

text: s

text: for

text: the

text: second

text: case

text: we

text: note

text: that

text: the

text: quantity

text: bT

text: A˜e

text: is

text: a

text: number

text: and

text: if

text: we

text: assume

text: that

text: the

text: scheme

text: is

text: a

text: second

text: order

text: accurate

text: one,

text: A˜e

text: by

text: As

text: a

text: consequence

text: we

text: have

text: ±1

text: c

text: ε)

text: ∆t

text: ε

text: and

text: therefore

text: the

text: CFL

text: condition,

text: in

text: the

text: limit

text: ε

text: becomes

text: c

text: √∆t

text: ∆t

text: ∆x

text: ∆x.

text: ∆t

text: c

text: ∆t

text: Now,

text: if

text: ∆t

text: we

text: get

text: from

text: 2√

text: ∆t

text: ∆x,

text: c

text: ∆t

text: the

text: parabolic

text: time

text: step

text: restriction

text: ∆t

text: Finally,

text: we

text: prove

text: that

text: the

text: scheme

text: in

text: the

text: limit

text: case

text: ε

text: is

text: a

text: consistent

text: discretiza-

text: tion

text: of

text: the

text: limit

text: equation

text: i.e.

text: the

text: scheme

text: is

text: AP.

text: From

text: we

text: get

text: O

text: V

text: ζA

text: A

text: ζA

text: p(U)

text: x

text: and

text: substituting

text: in

text: the

text: numerical

text: solution

text: we

text: obtain

text: O

text: bT

text: A

text: (bT

text: A

text: ˜bT

text: )(U

text: f(U))

text: ζbT

text: A

text: p(U)

text: x

text: x

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: → As ε 0 we get the explicit Runge-Kutta scheme for system (4) − U une + A˜f(U) = 0 x ∆t − (46) un+1 un + ˜bT f(U) = 0. x ∆t All the previous results can be stated by the following Theorem 1 If the IMEX-RK scheme (30)-(31) applied to () satisﬁes the GSA property then, → as ε 0, it becomes the explicit RK scheme characterized by the pair (A˜, w˜) applied to the limit convection-diﬀusion equation (6) for α = 1 and to the limit scalar conservation law (3) for ∈ α [0, 1). Some remarks are in order. Remark 1 • The advantage of formulating the uniﬁed IMEX-RK approach in the form (33)-(34) is that we can now adopt diﬀerent space discretizations for the various term appearing in the scheme. Typically, we use classical hyperbolic-type schemes, like WENO, for the space derivatives characterizing the hyperbolic part (which now has ﬁnite characteristic speeds) and centered discretization for the second order term characterizing the asymptotic parabolic behavior. • If the IMEX-RK scheme satisﬁes the GSA property the numerical solution is the same as the last stage, then V s = vn+1 and, observing that ˜b = 0, from the second equation of (cid:32) s (cid:33) (34), we have (cid:88) (cid:88) − − − vn+1 vn 1 s 1 − 1 s 1 − + ˜b p(Uj) = (b V i ˜b f(Ui)) + b vn+1 . ∆t ε2 i x ε2 i i s i=1 i=1 (cid:32) (cid:33) Solving for vn+1, after som(cid:88)e algebra, the equation can be(cid:88)written as − − − vn+1 vn 1 s 1 − 1 s 1 − + ˜b p(Uj) = (b V i ˜b f(Ui)) + b vn . ∆t ε2 + b ∆t i x ε2 + b ∆t i i s s s i=1 i=1 (cid:48) O Assuming p (u) = 1 and using the fact that U = une + (∆t) we can now write the hyperbolic part in (33) as O u + ζbT Bev + cbT BA˜eu = (∆t) t x x 1 O (47) v + u = (∆t). t ε2 + a ∆t x ss  (cid:115)  Therefore, the characteristic speeds read (cid:16) (cid:17)   1 ± 2 ζbT Be Λ1±(∆t, ε) = cbT BA˜e cbT BA˜e + 4 , (48) 2 ε2 + a ∆t ss 13 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: As

text: ε

text: we

text: get

text: the

text: explicit

text: Runge-Kutta

text: scheme

text: for

text: system

text: U

text: une

text: A˜f(U)

text: x

text: ∆t

text: un

text: ˜bT

text: f(U)

text: x

text: ∆t

text: All

text: the

text: previous

text: results

text: can

text: be

text: stated

text: by

text: the

text: following

text: Theorem

text: If

text: the

text: IMEX-RK

text: scheme

text: applied

text: to

text: satisﬁes

text: the

text: GSA

text: property

text: then,

text: as

text: ε

text: it

text: becomes

text: the

text: explicit

text: RK

text: scheme

text: characterized

text: by

text: the

text: pair

text: (A˜,

text: w˜)

text: applied

text: to

text: the

text: limit

text: convection-diﬀusion

text: equation

text: for

text: α

text: and

text: to

text: the

text: limit

text: scalar

text: conservation

text: law

text: for

text: α

text: Some

text: remarks

text: are

text: in

text: order.

text: Remark

text: The

text: advantage

text: of

text: formulating

text: the

text: uniﬁed

text: IMEX-RK

text: approach

text: in

text: the

text: form

text: is

text: that

text: we

text: can

text: now

text: adopt

text: diﬀerent

text: space

text: discretizations

text: for

text: the

text: various

text: term

text: appearing

text: in

text: the

text: scheme.

text: Typically,

text: we

text: use

text: classical

text: hyperbolic-type

text: schemes,

text: like

text: WENO,

text: for

text: the

text: space

text: derivatives

text: characterizing

text: the

text: hyperbolic

text: part

text: (which

text: now

text: has

text: ﬁnite

text: characteristic

text: speeds)

text: and

text: centered

text: discretization

text: for

text: the

text: second

text: order

text: term

text: characterizing

text: the

text: asymptotic

text: parabolic

text: behavior.

text: If

text: the

text: IMEX-RK

text: scheme

text: satisﬁes

text: the

text: GSA

text: property

text: the

text: numerical

text: solution

text: is

text: the

text: same

text: as

text: the

text: last

text: stage,

text: then

text: V

text: s

text: and,

text: observing

text: that

text: ˜b

text: from

text: the

text: second

text: equation

text: of

text: s

text: we

text: have

text: vn

text: s

text: s

text: ˜b

text: p(Uj)

text: (b

text: V

text: i

text: ˜b

text: f(Ui))

text: b

text: ∆t

text: ε2

text: i

text: x

text: ε2

text: i

text: i

text: s

text: Solving

text: for

text: after

text: algebra,

text: the

text: equation

text: can

text: as

text: vn

text: s

text: s

text: ˜b

text: p(Uj)

text: (b

text: V

text: i

text: ˜b

text: f(Ui))

text: b

text: vn

text: ∆t

text: ε2

text: b

text: ∆t

text: i

text: x

text: ε2

text: b

text: ∆t

text: i

text: i

text: s

text: s

text: s

text: O

text: Assuming

text: p

text: (u)

text: and

text: using

text: the

text: fact

text: that

text: U

text: une

text: (∆t)

text: we

text: can

text: now

text: write

text: the

text: hyperbolic

text: part

text: in

text: as

text: O

text: u

text: ζbT

text: Bev

text: cbT

text: BA˜eu

text: (∆t)

text: t

text: x

text: x

text: O

text: v

text: u

text: (∆t).

text: t

text: ε2

text: a

text: ∆t

text: x

text: ss

text: Therefore,

text: the

text: characteristic

text: speeds

text: read

text: ζbT

text: Be

text: ε)

text: cbT

text: BA˜e

text: cbT

text: BA˜e

text: ε2

text: a

text: ∆t

text: ss

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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