text: 10 terns of 95,97,98,100Mo found in these grains. We expect (cid:20)(iMo/96Mo) (cid:21) that better agreement with the data, e.g., elimination g = f (cid:12)(iMo) + (3) of the discrepancy for 98Mo in the grain 209-1, may be (iMo/96Mo)(cid:12) s, achieved by varying δ 100Mo within the measurement − P(iMo) (100Mo/96Mo) errors or using diﬀerent wind conditions to optimize the (1 fw)fr,(cid:12)(iMo) + fw P(100Mo) (100Mo/96Mo)(cid:12)g , wind production factors, or both. However, we consider that the results shown in Fig. 9 are suﬃcient as a proof − where f (cid:12)(iMo) and f (cid:12)(iMo) = 1 f (cid:12)(iMo) are the of concept. s, r, s, 4. CONCLUSIONS s-process and r-process fractions of the solar iMo abun- dance, respectively. We take f (cid:12)(iMo) = 0.50, 0.59, We have presented a detailed parametric study on the s, production of Mo and Ru isotopes in neutrino-driven 0.75, and 0 for 95Mo, 97Mo, 98Mo, and 100Mo, respec- winds. Our results can be generalized if conditions sim- tively. These values are consistent with both the esti- ilar to those studied here are also obtained for other mates of Arlandini et al. (1999) and the s-process pat- types of ejecta in either CCSNe or neutron star merg- terns found in mainstream SiC grains (e.g., Pellin et al. − ers. Consequently, our study is complementary to post- 1999). Note that f (cid:12)(iMo) = 1 f (cid:12)(iMo) is valid r, s, processing studies based on speciﬁc simulations of these when the wind makes negligible contributions to the so- events. With regard to the p-isotopes, we ﬁnd that lar abundances of the rele(cid:20)vant Mo isotopes. (cid:21) proton-rich winds can make dominant contributions to In terms of the meteoritic notation the solar abundance of 98Ru, signiﬁcant contributions (cid:46) (cid:46) to those of 96Ru ( 40%) and 92Mo ( 27%), and rel- ≡ × (iMo/96Mo) − (cid:46) δ iMo 1000 g 1 , (4) atively minor contributions to that of 94Mo ( 14%). (iMo/96Mo)(cid:12) In contrast, neutron-rich winds make negligible contri- (cid:20) (cid:21) butions to the solar abundances of 92,94Mo and cannot Eq. (3) can be written as produce 96,98Ru. However, we have shown that some neutron-rich winds can account for the peculiar patterns (cid:20) P(iMo) − (cid:21) of 95,97,98,100Mo in SiC X grains. δ iMo = 103fw P(100Mo) fr,(cid:12)(iMo) + (5) In conclusion, our results strongly suggest that the so- lar abundances of 92,94Mo are dominantly produced by (1 − f )f (cid:12)(iMo) + f P(iMo) δ 100Mo. sources other than the neutrino-driven wind. In addi- w r, w P(100Mo) tion, while the wind can be a dominant source for the 98Ru in the solar system, other sources are also required Using the central value of δ 100Mo for a grain along to account for the 96Ru. with the wind production factor P(iMo) relative to P(100Mo) and the solar r-process fraction f (cid:12)(iMo) r, given above, we ﬁnd fw for which Eq. (5) gives δ iMo in This work was supported in part by the Helmholtz- good agreement with the data on 95,97,98Mo in the same University Young Investigator grant No. VH-NG-825, grain. These results are shown in Fig. 9 with f = 0.23, Deutsche Forschungsgemeinschaft through SFB 1245, w 0.3, 0.7, 0.45, and 1 for ﬁve SiC X grains 113-2, 113-3, BMBF under grant No. 05P15RDFN1, ERC 677912 209-1, 100-2, and B2-05 (Pellin et al. 2006), respectively. EUROPIUM, and the US DOE grant DE-FG02-87ER40328. It can be seen that the above simple model including the We acknowledge useful discussions with Camilla J. Hansen, wind contributions can explain the overall peculiar pat- Maria Lugaro, Marco Pignatari, and Claudia Travaglio. REFERENCES Angulo, C., Arnould, M., Rayet, M., Descouvemont, P., Baye, D., Burbidge, E. M., Burbidge, G. R., Fowler, W. A., & Hoyle, F. Leclercq-Willain, C., Coc, A., Barhoumi, S., Aguer, P., Rolfs, 1957, Rev. Mod. Phys., 29, 547 C., Kunz, R., Hammer, J. W., Mayer, A., Paradellis, T., Cameron, A. G. W. 1957, AJ, 62, 9 Kossionides, S., Chronidou, C., Spyrou, K., degl’Innocenti, S., Davis, A. M., Pellin, M. J., Lewis, R. S., Amari, S., & Clayton, Fiorentini, G., Ricci, B., Zavatarelli, S., Providencia, C., R. N. 1999, Meteoritics & Planet. Sci. Suppl., 34, 30 Wolters, H., Soares, J., Grama, C., Rahighi, J., Shotter, A., & Farouqi, K., Kratz, K.-L., Mashonkina, L. I., Pfeiﬀer, B., Cowan, Lamehi Rachti, M. 1999, Nuclear Physics A, 656, 3 J. J., Thielemann, F.-K., & Truran, J. W. 2009, ApJ, 694, L49 Arcones, A. & Bliss, J. 2014, J. Phys. G: Nucl. Phys., 41, 044005 Fisker, J. L., Hoﬀman, R. D., & Pruet, J. 2009, ApJ, 690, L135 Arcones, A., Fr¨ohlich, C., & Mart´ınez-Pinedo, G. 2012, ApJ, 750, Freiburghaus, C., Rembges, J.-F., Rauscher, T., Kolbe, E., 18 Thielemann, F.-K., Kratz, K.-L., Pfeiﬀer, B., & Cowan, J. J. Arcones, A. & Janka, H.-T. 2011, A&A, 526, A160+ 1999a, ApJ, 516, 381 Arcones, A., Janka, H.-T., & Scheck, L. 2007, A&A, 467, 1227 Freiburghaus, C., Rosswog, S., & Thielemann, F.-K. 1999b, ApJ, Arcones, A. & Montes, F. 2011, ApJ, 731, 5 525, L121 Arcones, A. & Thielemann, F.-K. 2013, J. Phys. G: Nucl. Phys., Fr¨ohlich, C., Mart´ınez-Pinedo, G., Liebend¨orfer, M., Thielemann, 40, 013201 F.-K., Bravo, E., Hix, W. R., Langanke, K., & Zinner, N. T. Arlandini, C., K¨appeler, F., Wisshak, K., Gallino, R., Lugaro, M., 2006, Phys. Rev. Lett., 96, 142502 Busso, M., & Straniero, O. 1999, ApJ, 525, 886 Goriely, S., Bauswein, A., & Janka, H.-T. 2011, ApJ, 738, L32 Arnould, M. & Goriely, S. 2003, Phys. Rep., 384, 1 Hansen, C. J., Montes, F., & Arcones, A. 2014, ApJ, 797, 123 Banerjee, P., Haxton, W. C., & Qian, Y.-Z. 2011, Physical Hoﬀman, R. D., Woosley, S. E., Fuller, G. M., & Meyer, B. S. Review Letters, 106, 201104 1996, ApJ, 460, 478 Bliss, J., Arcones, A., Montes, F., & Pereira, J. 2017, J. Phys. G: Hoﬀman, R. D., Woosley, S. E., & Qian, Y.-Z. 1997, ApJ, 482, Nucl. Phys., 44, 054003 951 Bliss, J., Witt, M., Arcones, A., Montes, F., & Pereira, J. 2018, Howard, W. M., Meyer, B. S., & Clayton, D. D. 1992, ApJ, 855, 135 Meteoritics, 27, 404 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: terns

text: of

text: found

text: in

text: these

text: grains.

text: We

text: expect

label: equation

text: that

text: better

text: agreement

text: with

text: the

text: data,

text: e.g.,

text: elimination

text: g

text: f

text: of

text: the

text: discrepancy

text: for

text: in

text: the

text: grain

text: may

text: be

text: s,

text: achieved

text: by

text: varying

text: δ

text: within

text: the

text: measurement

text: P(iMo)

text: errors

text: or

text: using

text: diﬀerent

text: wind

text: conditions

text: to

text: optimize

text: the

text: fw

text: wind

text: production

text: factors,

text: or

text: both.

text: However,

text: we

text: consider

text: that

text: the

text: results

text: shown

text: in

text: Fig.

text: are

text: suﬃcient

text: as

text: a

text: proof

text: where

text: f

text: and

text: f

text: f

text: are

text: the

text: of

text: concept.

text: s,

text: r,

text: s,

text: CONCLUSIONS

text: s-process

text: and

text: r-process

text: fractions

text: of

text: the

text: solar

text: iMo

text: abun-

text: dance,

text: respectively.

text: We

text: take

text: f

text: We

text: have

text: presented

text: a

text: detailed

text: parametric

text: study

text: on

text: the

text: s,

text: production

text: of

text: Mo

text: and

text: Ru

text: isotopes

text: in

text: neutrino-driven

text: and

text: for

text: and

text: respec-

text: winds.

text: Our

text: results

text: can

text: be

text: generalized

text: if

text: conditions

text: sim-

text: tively.

text: These

text: values

text: are

text: consistent

text: with

text: both

text: the

text: esti-

text: ilar

text: to

text: those

text: studied

text: here

text: are

text: also

text: obtained

text: for

text: other

text: mates

text: of

text: Arlandini

text: et

text: al.

text: and

text: the

text: s-process

text: pat-

text: types

text: of

text: ejecta

text: in

text: either

text: CCSNe

text: or

text: neutron

text: star

text: merg-

text: terns

text: found

text: in

text: mainstream

text: SiC

text: grains

text: (e.g.,

text: Pellin

text: et

text: al.

text: ers.

text: Consequently,

text: our

text: study

text: is

text: complementary

text: to

text: post-

text: Note

text: that

text: f

text: f

text: is

text: valid

text: r,

text: s,

text: processing

text: studies

text: based

text: on

text: speciﬁc

text: simulations

text: of

text: these

text: when

text: the

text: wind

text: makes

text: negligible

text: contributions

text: to

text: the

text: so-

text: events.

text: With

text: regard

text: to

text: the

text: p-isotopes,

text: we

text: ﬁnd

text: that

text: lar

text: abundances

text: of

text: the

text: Mo

text: isotopes.

text: proton-rich

text: winds

text: can

text: make

text: dominant

text: contributions

text: to

text: In

text: terms

text: of

text: the

text: meteoritic

text: notation

text: the

text: solar

text: abundance

text: of

text: signiﬁcant

text: contributions

text: to

text: those

text: of

text: and

text: and

text: rel-

text: δ

text: iMo

text: g

text: atively

text: minor

text: contributions

text: to

text: that

text: of

text: In

text: contrast,

text: neutron-rich

text: winds

text: make

text: negligible

text: contri-

text: butions

text: to

text: the

text: solar

text: abundances

text: of

text: and

text: cannot

text: Eq.

text: can

text: be

text: written

text: as

text: produce

text: However,

text: we

text: have

text: shown

text: that

text: some

text: neutron-rich

text: winds

text: can

text: account

text: for

text: the

text: peculiar

text: patterns

text: P(iMo)

text: of

text: in

text: SiC

text: X

text: grains.

text: δ

text: iMo

text: In

text: conclusion,

text: our

text: results

text: strongly

text: suggest

text: that

text: the

text: so-

text: lar

text: abundances

text: of

text: are

text: dominantly

text: produced

text: by

text: f

text: )f

text: f

text: P(iMo)

text: δ

text: sources

text: other

text: than

text: the

text: neutrino-driven

text: wind.

text: In

text: addi-

text: w

text: r,

text: w

text: tion,

text: while

text: the

text: wind

text: can

text: be

text: a

text: dominant

text: source

text: for

text: the

text: in

text: the

text: solar

text: system,

text: other

text: sources

text: are

text: also

text: required

text: Using

text: the

text: central

text: value

text: of

text: δ

text: for

text: a

text: grain

text: along

text: to

text: account

text: for

text: the

text: with

text: the

text: wind

text: production

text: factor

text: P(iMo)

text: relative

text: to

text: and

text: the

text: solar

text: r-process

text: fraction

text: f

text: r,

text: given

text: above,

text: we

text: ﬁnd

text: fw

text: for

text: which

text: Eq.

text: gives

text: δ

text: iMo

text: in

text: This

text: work

text: was

text: supported

text: in

text: part

text: by

text: the

text: Helmholtz-

text: good

text: agreement

text: with

text: the

text: data

text: on

text: in

text: the

text: same

text: University

text: Young

text: Investigator

text: grant

text: No.

text: grain.

text: These

text: results

text: are

text: shown

text: in

text: Fig.

text: with

text: f

text: Deutsche

text: Forschungsgemeinschaft

text: through

text: SFB

text: w

text: and

text: for

text: ﬁve

text: SiC

text: X

text: grains

text: BMBF

text: under

text: grant

text: No.

text: ERC

text: and

text: (Pellin

text: et

text: al.

text: respectively.

text: EUROPIUM,

text: and

text: the

text: US

text: DOE

text: grant

text: It

text: can

text: be

text: seen

text: that

text: the

text: above

text: simple

text: model

text: including

text: the

text: We

text: acknowledge

text: useful

text: discussions

text: with

text: Camilla

text: J.

text: Hansen,

text: wind

text: contributions

text: can

text: explain

text: the

text: overall

text: peculiar

text: pat-

text: Maria

text: Lugaro,

text: Marco

text: Pignatari,

text: and

text: Claudia

text: Travaglio.

text: REFERENCES

text: Angulo,

text: C.,

text: Arnould,

text: M.,

text: Rayet,

text: M.,

text: Descouvemont,

text: P.,

text: Baye,

text: D.,

text: Burbidge,

text: E.

text: M.,

text: Burbidge,

text: G.

text: R.,

text: Fowler,

text: W.

text: A.,

text: Hoyle,

text: F.

text: Leclercq-Willain,

text: C.,

text: Coc,

text: A.,

text: Barhoumi,

text: S.,

text: Aguer,

text: P.,

text: Rolfs,

text: Rev.

text: Mod.

text: Phys.,

text: C.,

text: Kunz,

text: R.,

text: Hammer,

text: J.

text: W.,

text: Mayer,

text: A.,

text: Paradellis,

text: T.,

text: Cameron,

text: A.

text: G.

text: W.

text: AJ,

text: Kossionides,

text: S.,

text: Chronidou,

text: C.,

text: Spyrou,

text: K.,

text: degl’Innocenti,

text: S.,

text: Davis,

text: A.

text: M.,

text: Pellin,

text: M.

text: J.,

text: Lewis,

text: R.

text: S.,

text: Amari,

text: S.,

text: Clayton,

text: Fiorentini,

text: G.,

text: Ricci,

text: B.,

text: Zavatarelli,

text: S.,

text: Providencia,

text: C.,

text: R.

text: N.

text: Meteoritics

text: Planet.

text: Sci.

text: Suppl.,

text: Wolters,

text: H.,

text: Soares,

text: J.,

text: Grama,

text: C.,

text: Rahighi,

text: J.,

text: Shotter,

text: A.,

text: Farouqi,

text: K.,

text: Kratz,

text: K.-L.,

text: Mashonkina,

text: L.

text: I.,

text: Pfeiﬀer,

text: B.,

text: Cowan,

text: Lamehi

text: Rachti,

text: M.

text: Nuclear

text: Physics

text: A,

text: J.

text: J.,

text: Thielemann,

text: F.-K.,

text: Truran,

text: J.

text: W.

text: ApJ,

text: Arcones,

text: A.

text: Bliss,

text: J.

text: J.

text: Phys.

text: G:

text: Nucl.

text: Phys.,

text: Fisker,

text: J.

text: L.,

text: Hoﬀman,

text: R.

text: D.,

text: Pruet,

text: J.

text: ApJ,

text: Arcones,

text: A.,

text: Fr¨ohlich,

text: C.,

text: Mart´ınez-Pinedo,

text: G.

text: ApJ,

text: Freiburghaus,

text: C.,

text: Rembges,

text: J.-F.,

text: Rauscher,

text: T.,

text: Kolbe,

text: E.,

text: Thielemann,

text: F.-K.,

text: Kratz,

text: K.-L.,

text: Pfeiﬀer,

text: B.,

text: Cowan,

text: J.

text: J.

text: Arcones,

text: A.

text: Janka,

text: H.-T.

text: A&A,

text: ApJ,

text: Arcones,

text: A.,

text: Janka,

text: H.-T.,

text: Scheck,

text: L.

text: A&A,

text: Freiburghaus,

text: C.,

text: Rosswog,

text: S.,

text: Thielemann,

text: F.-K.

text: ApJ,

text: Arcones,

text: A.

text: Montes,

text: F.

text: ApJ,

text: Arcones,

text: A.

text: Thielemann,

text: F.-K.

text: J.

text: Phys.

text: G:

text: Nucl.

text: Phys.,

text: Fr¨ohlich,

text: C.,

text: Mart´ınez-Pinedo,

text: G.,

text: Liebend¨orfer,

text: M.,

text: Thielemann,

text: F.-K.,

text: Bravo,

text: E.,

text: Hix,

text: W.

text: R.,

text: Langanke,

text: K.,

text: Zinner,

text: N.

text: T.

text: Arlandini,

text: C.,

text: K¨appeler,

text: F.,

text: Wisshak,

text: K.,

text: Gallino,

text: R.,

text: Lugaro,

text: M.,

text: Phys.

text: Rev.

text: Lett.,

text: Busso,

text: M.,

text: Straniero,

text: O.

text: ApJ,

text: Goriely,

text: S.,

text: Bauswein,

text: A.,

text: Janka,

text: H.-T.

text: ApJ,

text: Arnould,

text: M.

text: Goriely,

text: S.

text: Phys.

text: Rep.,

text: Hansen,

text: C.

text: J.,

text: Montes,

text: F.,

text: Arcones,

text: A.

text: ApJ,

text: Banerjee,

text: P.,

text: Haxton,

text: W.

text: C.,

text: Qian,

text: Y.-Z.

text: Physical

text: Hoﬀman,

text: R.

text: D.,

text: Woosley,

text: S.

text: E.,

text: Fuller,

text: G.

text: M.,

text: Meyer,

text: B.

text: S.

text: Review

text: Letters,

text: ApJ,

text: Bliss,

text: J.,

text: Arcones,

text: A.,

text: Montes,

text: F.,

text: Pereira,

text: J.

text: J.

text: Phys.

text: G:

text: Hoﬀman,

text: R.

text: D.,

text: Woosley,

text: S.

text: E.,

text: Qian,

text: Y.-Z.

text: ApJ,

text: Nucl.

text: Phys.,

text: Bliss,

text: J.,

text: Witt,

text: M.,

text: Arcones,

text: A.,

text: Montes,

text: F.,

text: Pereira,

text: J.

text: Howard,

text: W.

text: M.,

text: Meyer,

text: B.

text: S.,

text: Clayton,

text: D.

text: D.

text: ApJ,

text: Meteoritics,

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: the canonical (Elson et al. 1987, EFF) proﬁle, r2 µ(r) = µ0(1+ a2 )−γ/2, (1) where µ0 is the central surface brightness. The EFF model’s core radius, a, and the power-law index γ are linked to the core radius of the standard King model, r , through c rc = a(22/γ −1)1/2. (2) The parameters pertaining to the best-ﬁtting EFF proﬁle and the linked core radius of the King proﬁle are also included in the bottom left-hand corner of Fig. 2. Our ﬁt implies that ± the NGC 1850 core radius is 2.76 0.12 pc, which is consis- tent within one sigma with the value of 2.69+0.13 pc derived by −0.17 McLaughlin & van der Marel (2005, their Table 11). Through ∼ visual inspection, we determined that at a radius R 20 pc ∼ (cid:48)(cid:48) ( 83 ; indicated by the vertical red dashed line in Fig. 2) the cluster’s surface brightness proﬁle becomes indistinguishable from the ﬁeld level. We therefore adopted this radius as that encompassing the typical cluster region (shown as the area Figure 1. Spatial distributions of all stars (grey), stars within the adopted containing the black points in Fig. 1). Note that in reality the cluster region (black), reference ﬁeld stars (red), and number density con- surface brightness continues to decrease beyond the cluster tours in the NGC 1850 ﬁeld. The red pentagram indicates the cluster center ◦ (cid:48) (cid:48)(cid:48) region, however, so that the adopted value of 20 pc does not at αJ2000 = 05h08m44.34s, δJ2000 = −68 45 45.60 . Objects located in the represent the full extent of the cluster. This is consistent with white cDirRcAleFtToVthEeRlSeIfOtNofMthAeYre3d1p,e2n0ta2g0ram were removed so as to minimize contamPirnepartiinotntybpyesNetGuCsin1g8LA5T0EBXssttyalres.emulateapj v. 01/23/15 the conclusion of McLaughlin & van der Marel (2005), who derived a tidal radius for NGC 1850 of log(rt pc−1) = 2.16+−00..0097. NEW INSIGHTS INTO THE FORMATION OF THE BLUE MAIN SEQUENCE IN NGC 1850 YUJIAO YANG1,2, CHENGYUAN LI3, LICAI DENG2, RICHARD DE GRIJS3,1,4, AND ANTONINO P. MILONE5,6 Keywords: globular clusters: individual: NGC 1850 — Hertzsprung-Russell and C-M diagrams — Magellanic Clouds 1 Kavli Institute for Astronomy & Astrophysics and Department of As- tronomy, Peking University, Yi He Yuan Lu 5, Hai Dian District, Beijing 100871, China 2 Key Laboratory for Optical Astronomy, National Astronomical Ob- servatories, Chinese Academy of Sciences, 20A Datun Road, Chaoyang District, Beijing 100012, China 3 Department of Physics and Astronomy, Macquarie University, Bala- clava Road, Sydney, NSW 2109, Australia 4 International Space Science Institute–Beijing, 1 Nanertiao, Zhong- guancun, Hai Dian District, Beijing 100190, China 5 Dipartimento di Fisica e Astronomia ‘Galileo Galilei’, Università di Padova, Vicolo dell’Osservatorio 3, Padova, IT-35122, Italy 6 Research School of Astronomy & Astrophysics, Australian National University, Canberra, ACT 2611, Australia ##LTFigure## ##LTFigure## ##LTLine##

text: the

label: paragraph

text: canonical

text: (Elson

text: et

text: al.

text: EFF)

text: proﬁle,

text: µ(r)

text: where

text: µ0

text: is

text: the

text: central

text: surface

text: brightness.

text: The

text: EFF

text: model’s

text: core

text: radius,

text: a,

text: and

text: the

text: power-law

text: index

text: γ

text: are

text: linked

text: to

text: the

text: core

text: radius

text: of

text: the

text: standard

text: King

text: model,

text: r

text: through

text: c

text: rc

text: The

text: parameters

text: pertaining

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text: Running Author / xx (2020) 1–24 15 truck and, then, inserting in new drone trip. First, sub-routes are formed envisioning not to violate the { } { } prohibitions. For example, in Figure 9 the sub-routes are 0, 5 and 1, 6, 4, 2, 8 . Afterwards, an evaluation is made to determine the launch, the new drone customer, and the returning node. If a triple combination of nodes is feasible, it is then accepted, and the node which is going to be visited by the drone node is removed from the route of the truck and inserted in the route of the drone between the launch and return node. In the example, the customers 1, 4 and 6 were chosen to be the launch, visit and return of the drone, respectively. Fig. 9: Move Relocate Customer. 5. Computational Experiments A series of computational experiments was performed to test the effectiveness of the HGVNS. The (cid:13) algorithm was coded in C++ (g++ 5.3.1) and executed on an Intel R CoreTM i7 Processor 3.6 GHz with 16 GB of RAM running Ubuntu Linux 16.04. The MIP solver used to ﬁnd a optimal TSP solution was Concorde1 3.12.19 with CPLEX 12.6.3. HGVNS was tested in three benchmark sets. Two sets found in literature, advised by Ponza (2016) and by Agatz et al. (2016). A new set of instances was developed based on the TSPLIB2 well-known instances for the TSP. In the tables presented hereafter, the column Inst.denotes the instance name and n represents the − number of nodes (the depot and n 1 customers). Moreover, BKS indicates the best-known solution value found in the literature. Columns s and T ime denote, respectively, the best solution value FSTSP and the computational time in seconds associated to the MIP running time plus the GVNS algorithm, ∗ gap states the difference between s and BKS. Column s describes the optimal tour obtained FSTSP TSP with the TSP Solver Concorde. Finally, s is the average solution cost of ten runs and gap is the FSTSP gap between the average solutions and the BKS. A negative gap indicates an improvement. 5.1. Results of HGVNS in Ponza (2016) instances The ﬁrst set of instances was provided by Ponza (2016) based on the original formulation of Murray × and Chu (2015). The instances were generated over a map of 32km 32km, to reﬂect the idea of some companies about a 10 km endurance of the drone and thus be able to perform some feasible routes. In the map, the depot is always at coordinate (0, 0) and the locations of the customers are randomly generated. The parameters previously deﬁned in this work assumed different values: the service time to the launch (sl) and return (sr) of the drone is 0.6 minutes and 0.5 minutes, respectively. The drone speed is 80.47 km/h (50 mph), and the truck speed is 56.32 km/h (35 mph). The endurance of the drone is 24 minutes. Additionally, the travel time matrix of both vehicles is calculated based on the same road network, and, 1Concorde: http://www.math.uwaterloo.ca/tsp/concorde.html 2TSPLIB: https://www.iwr.uniheidelberg.de/groups/comopt/software/TSPLIB95/ ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: 22 Running Author / xx (2020) 1–24 12000 11000 10000 9000 8000 7000 6000 5000 4000 9000 10000 11000 12000 13000 14000 15000 16000 Fig. 10: Delivery route with truck only. 12000 11000 10000 9000 8000 7000 6000 5000 4000 9000 10000 11000 12000 13000 14000 15000 16000 Fig. 11: Delivery route with truck and drone. ##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## 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text: Running Author / xx (2020) 1–24 9 the optimal TSP route. The solver requires the truck travel time matrix τ and the entire set of customers to ∗ generate the solution (line 1). The solution s is the optimal solution for the TSP, as it is also a feasible TSP ∗ solution for the FSTSP, but not including any drone trip. s is given to CreateInitialSolution() TSP procedure to add the drone trips to the route (line 2). Finally, the initial solution s is given to the GVNS ∗ procedure (line 3), to obtain the improved solution s , which is then returned by HGVNS. Algorithm 1 HGVNS Framework (cid:48) (cid:48) Require: cost matrices (τ, τ ); customers (C, C ) ∗ ← 1: s MIP Solver(τ, C) T←SP ∗ (cid:48) (cid:48) 2: s CreateInitialSolution(s , τ, τ , C, C ) ∗ ← (cid:48) (cid:48)TSP 3: s GVNS(s, τ, τ , C, C ) ∗ return 4: s The CreateInitialSolution() procedure, based on the heuristic presented by Murray and Chu (2015), removes some of the truck’s customers and makes them drone customers. Algorithm 2 illus- trates the procedure. The algorithm initializes variable maxSavings and truckSubRoutes. Following, for each eligible drone customer j is computed the savings of removing this customer from the truck route (line 5). Algorithm 2 CreateInitialSolution (cid:48) (cid:48) Require: cost matrices (τ, τ ); customers (C, C ) Ensure: s ← 1: maxSavings 0 ← { ∗ } 2: truckSubRoutes s TSP repeat 3: ∈ (cid:48) for all do 4: j C ← 5: savings computeSavings(truckSubRoutes, j) ∈ for all do 6: subroute truckSubRoutes 7: if (isPairedUAV(subroute) then ← 8: maxSavings computeCostTruck(subRoute, j) else 9: ← 10: maxSavings computeCostUAV(subRoute, j) end if 11: end for 12: end for 13: if then 14: maxSavings > 0 15: Update the route ← 16: Reinitialize maxSavings 0 end if 17: 18: until Stop Next, for each sub-route, it is veriﬁed if there is a drone trip (lines 4 – 13). If the answer is positive, customer j may be relocated in the truck route. If the answer is negative, customer j may be visited by ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Running

text: Author

text: xx

text: the

text: optimal

text: TSP

text: route.

text: The

text: solver

text: requires

text: the

text: truck

text: travel

text: time

text: matrix

text: τ

text: and

text: the

text: entire

text: set

text: of

text: customers

text: to

text: generate

text: the

text: solution

text: (line

text: The

text: solution

text: s

text: is

text: the

text: optimal

text: solution

text: for

text: the

text: TSP,

text: as

text: it

text: is

text: also

text: a

text: feasible

text: TSP

text: solution

text: for

text: the

text: FSTSP,

text: but

text: not

text: including

text: any

text: drone

text: trip.

text: s

text: is

text: given

text: to

text: CreateInitialSolution()

text: TSP

text: procedure

text: to

text: add

text: the

text: drone

text: trips

text: to

text: the

text: route

text: (line

text: Finally,

text: the

text: initial

text: solution

text: s

text: is

text: given

text: to

text: the

text: GVNS

text: procedure

text: (line

text: to

text: obtain

text: the

text: improved

text: solution

text: s

text: which

text: is

text: then

text: returned

text: by

text: HGVNS.

text: Algorithm

text: HGVNS

text: Framework

text: Require:

text: cost

text: matrices

text: (τ,

text: τ

text: customers

text: (C,

text: C

text: s

text: MIP

text: Solver(τ,

text: C)

text: T←SP

text: s

text: CreateInitialSolution(s

text: τ,

text: τ

text: C,

text: C

text: s

text: GVNS(s,

text: τ,

text: τ

text: C,

text: C

text: return

text: s

text: The

text: CreateInitialSolution()

text: procedure,

text: based

text: on

text: the

text: heuristic

text: presented

text: by

text: Murray

text: and

text: Chu

text: removes

text: some

text: of

text: the

text: truck’s

text: customers

text: and

text: makes

text: them

text: drone

text: customers.

text: Algorithm

text: illus-

text: trates

text: the

text: procedure.

text: The

text: algorithm

text: initializes

text: variable

text: maxSavings

text: and

text: truckSubRoutes.

text: Following,

text: for

text: each

text: eligible

text: drone

text: customer

text: j

text: is

text: computed

text: the

text: savings

text: of

text: removing

text: this

text: customer

text: from

text: the

text: truck

text: route

text: (line

text: Algorithm

text: CreateInitialSolution

text: Require:

text: cost

text: matrices

text: (τ,

text: τ

text: customers

text: (C,

text: C

text: Ensure:

text: s

text: maxSavings

text: truckSubRoutes

text: s

text: TSP

text: repeat

text: for

text: all

text: do

text: j

text: C

text: savings

text: computeSavings(truckSubRoutes,

text: j)

text: for

text: all

text: do

text: subroute

text: truckSubRoutes

text: if

text: (isPairedUAV(subroute)

text: then

text: maxSavings

text: computeCostTruck(subRoute,

text: j)

text: else

text: maxSavings

text: computeCostUAV(subRoute,

text: j)

text: end

text: if

text: end

text: for

text: end

text: for

text: if

text: then

text: maxSavings

text: Update

text: the

text: route

text: Reinitialize

text: maxSavings

text: end

text: if

text: until

text: Stop

text: Next,

text: for

text: each

text: sub-route,

text: it

text: is

text: veriﬁed

text: if

text: there

text: is

text: a

text: drone

text: trip

text: (lines

text: If

text: the

text: answer

text: is

text: positive,

text: customer

text: j

text: may

text: be

text: relocated

text: in

text: the

text: truck

text: route.

text: If

text: the

text: answer

text: is

text: negative,

text: customer

text: j

text: may

text: be

text: visited

text: by

text: ##LTLine##

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text: Projection image-to-image translation in hybrid X-ray/MR imaging Bernhard Stimpel1,2, Christopher Syben1,2, Tobias Wu¨rﬂ1, Katharina Breininger1, Katrin Mentl1, Jonathan Lommen1,2, Arnd Do¨rﬂer2, Andreas Maier1 1 Pattern Recognition Lab, Friedrich-Alexander-Universita¨t Erlangen-Nu¨rnberg 2 Department of Neuroradiology, Friedrich-Alexander-Universita¨t Erlangen-Nu¨rnberg Abstract The potential beneﬁt of hybrid X-ray and MR imaging in the interventional en- vironment is enormous. However, a vast amount of existing image enhancement methods requires the image information to be present in the same domain. To un- lock this potential, we present a solution to image-to-image translation from MR projections to corresponding X-ray projection images. The approach is based on a state-of-the-art image generator network that is modiﬁed to ﬁt the speciﬁc appli- cation. Furthermore, we propose the inclusion of a gradient map to the perceptual loss to emphasize high frequency details. The proposed approach is capable of creating X-ray projection images with natural appearance. Additionally, our ex- tensions show clear improvement compared to the baseline method. 1 Introduction Hybrid imaging exhibits high potential in diagnostic and interventional applications. Future ad- vances in research may leverage the combination of Computed Tomography (CT) and Magnetic Resonance Imaging (MRI) to clinical applicability. Especially for interventional purposes, the gain from simultaneously acquiring soft- and dense-tissue information would yield great opportunities. Assuming the information of both modalities is present at the same time, numerous existing post- processing methods would become applicable. Image fusion techniques, e.g., image overlays, have proven useful in the past. Additionally, one can think about image enhancement techniques, e.g., image de-noising or super-resolution. To enable the latter methods, it is beneﬁcial to have the data available in the same domain. Different solutions to generate CT images from corresponding MRI data were presented in the past [1, 2]. However, all of these are applied to volumetric data. In contrast, interventional procedures rely heavily on line integral data from X-ray projection imaging. Projection images which exhibit the same perspective distortion can also be acquired directly using an MR device [3]. This avoids time-consuming volumetric acquisition and subsequent forward pro- jection. To solve this image-to-image translation task, we investigate a deep learning-based solution to generate X-ray projections from corresponding MRI views. 2 Methods Network architecture: Our network’s design is based on the architecture proposed by Johnson et al. [4], which was also adopted for various other applications, e.g., [5]. In the original manuscript, multiple residual blocks are introduced at the lowest resolution level of the network. However, the underlying variance in medical projection images is only small compared to natural image scenes. Additionally, during interventional treatments, valuable information is largely drawn from high- frequency details such as contrast and clear edges. To this end, we distribute the residual blocks at higher resolution levels instead. Furthermore, bilinear upscaling is used in place of the transposed ##LTLine##

text: Projection

label: title

text: image-to-image

text: translation

text: in

text: hybrid

text: X-ray/MR

text: imaging

text: Bernhard

label: author

text: Christopher

text: Tobias

text: Katharina

text: Katrin

text: Jonathan

text: Arnd

text: Andreas

text: Pattern

text: Recognition

text: Lab,

text: Friedrich-Alexander-Universita¨t

text: Erlangen-Nu¨rnberg

text: Department

text: of

text: Neuroradiology,

text: Friedrich-Alexander-Universita¨t

text: Erlangen-Nu¨rnberg

text: Abstract

text: The

label: abstract

text: potential

text: beneﬁt

text: of

text: hybrid

text: X-ray

text: and

text: MR

text: imaging

text: in

text: the

text: interventional

text: en-

text: vironment

text: is

text: enormous.

text: However,

text: a

text: vast

text: amount

text: of

text: existing

text: image

text: enhancement

text: methods

text: requires

text: the

text: image

text: information

text: to

text: be

text: present

text: in

text: the

text: same

text: domain.

text: To

text: un-

text: lock

text: this

text: potential,

text: we

text: present

text: a

text: solution

text: to

text: image-to-image

text: translation

text: from

text: MR

text: projections

text: to

text: corresponding

text: X-ray

text: projection

text: images.

text: The

text: approach

text: is

text: based

text: on

text: a

text: state-of-the-art

text: image

text: generator

text: network

text: that

text: is

text: modiﬁed

text: to

text: ﬁt

text: the

text: speciﬁc

text: appli-

text: cation.

text: Furthermore,

text: we

text: propose

text: the

text: inclusion

text: of

text: a

text: gradient

text: map

text: to

text: the

text: perceptual

text: loss

text: to

text: emphasize

text: high

text: frequency

text: details.

text: The

text: proposed

text: approach

text: is

text: capable

text: of

text: creating

text: X-ray

text: projection

text: images

text: with

text: natural

text: appearance.

text: Additionally,

text: our

text: ex-

text: tensions

text: show

text: clear

text: improvement

text: compared

text: to

text: the

text: baseline

text: method.

text: Introduction

text: Hybrid

text: imaging

text: exhibits

text: high

text: potential

text: in

text: diagnostic

text: and

text: interventional

text: applications.

text: Future

text: ad-

text: vances

text: in

text: research

text: may

text: leverage

text: the

text: combination

text: of

text: Computed

text: Tomography

text: (CT)

text: and

text: Magnetic

text: Resonance

text: Imaging

text: (MRI)

text: to

text: clinical

text: applicability.

text: Especially

text: for

text: interventional

text: purposes,

text: the

text: gain

text: from

text: simultaneously

text: acquiring

text: soft-

text: and

text: dense-tissue

text: information

text: would

text: yield

text: great

text: opportunities.

text: Assuming

text: the

text: information

text: of

text: both

text: modalities

text: is

text: present

text: at

text: the

text: same

text: time,

text: numerous

text: existing

text: post-

text: processing

text: methods

text: would

text: become

text: applicable.

text: Image

text: fusion

text: techniques,

text: e.g.,

text: image

text: overlays,

text: have

text: proven

text: useful

text: in

text: the

text: past.

text: Additionally,

text: one

text: can

text: think

text: about

text: image

text: enhancement

text: techniques,

text: e.g.,

text: image

text: de-noising

text: or

text: super-resolution.

text: To

text: enable

text: the

text: latter

text: methods,

text: it

text: is

text: beneﬁcial

text: to

text: have

text: the

text: data

text: available

text: in

text: the

text: same

text: domain.

text: Different

text: solutions

text: to

text: generate

text: CT

text: images

text: from

text: corresponding

text: MRI

text: data

text: were

text: presented

text: in

text: the

text: past

text: However,

text: all

text: of

text: these

text: are

text: applied

text: to

text: volumetric

text: data.

text: In

text: contrast,

text: interventional

text: procedures

text: rely

text: heavily

text: on

text: line

text: integral

text: data

text: from

text: X-ray

text: projection

text: imaging.

text: Projection

text: images

text: which

text: exhibit

text: the

text: same

text: perspective

text: distortion

text: can

text: also

text: be

text: acquired

text: directly

text: using

text: an

text: MR

text: device

text: This

text: avoids

text: time-consuming

text: volumetric

text: acquisition

text: and

text: subsequent

text: forward

text: pro-

text: jection.

text: To

text: solve

text: this

text: image-to-image

text: translation

text: task,

text: we

text: investigate

text: a

text: deep

text: learning-based

text: solution

text: to

text: generate

text: X-ray

text: projections

text: from

text: corresponding

text: MRI

text: views.

text: Methods

text: Network

text: architecture:

text: Our

text: network’s

text: design

text: is

text: based

text: on

text: the

text: architecture

text: proposed

text: by

text: Johnson

text: et

text: al.

text: which

text: was

text: also

text: adopted

text: for

text: various

text: other

text: applications,

text: e.g.,

text: In

text: the

text: original

text: manuscript,

text: multiple

text: residual

text: blocks

text: are

text: introduced

text: at

text: the

text: lowest

text: resolution

text: level

text: of

text: the

text: network.

text: However,

text: the

text: underlying

text: variance

text: in

text: medical

text: projection

text: images

text: is

text: only

text: small

text: compared

text: to

text: natural

text: image

text: scenes.

text: Additionally,

text: during

text: interventional

text: treatments,

text: valuable

text: information

text: is

text: largely

text: drawn

text: from

text: high-

text: frequency

text: details

text: such

text: as

text: contrast

text: and

text: clear

text: edges.

text: To

text: this

text: end,

text: we

text: distribute

text: the

text: residual

text: blocks

text: at

text: higher

text: resolution

text: levels

text: instead.

text: Furthermore,

text: bilinear

text: upscaling

text: is

text: used

text: in

text: place

text: of

text: the

text: transposed

text: ##LTLine##


text: 4 AICKE HINRICHS AND JOSCHA PROCHNO AND MARIO ULLRICH (cid:110) (cid:111) Then the curse of dimensionality holds for the class C 1(A , B , K ) if √ d d d d lim sup min A d L , B d L2 > 0 . →∞ d Kd d Kd d In addition to these lower bounds on the complexity of numerical integration, ∈ we will present lower bounds for the more general class of ψ -bodies with α [1, 2) α (see Section ). These bounds are, however, no longer exponential in d, but in dα/2. We shall present this result in Proposition 4.1. √ Remark 1.2. In Asymptotic Geometric Analysis terms, our radial condition (1.2) simply means that such a body K is of “small diameter” (i.e., rad(K ) = α d L ∈ ∞ d d √ Kd for some α [1, )) with a constant strictly smaller than 2. It follows directly ⊆ R ≤ from the isotropic condition that, for any isotropic convex body K d, d L K rad(K). Additionally, as shown by Kannan, Lov´asz and Simonovits in [14], we have ≤ the upper bound rad(K) (d + 1)L . We refer to the monograph [8] for more K information. Remark 1.3. If we compare the two conditions (1.1) and (1.2) on the radii, then we see that, contrary to condition (1.1) considered in [13], our condition (1.2) involves the isotropic constant L of the integral domain K . Using the well-known Kd d √ fact that among all isotropic convex bodies the (volume normalized) Euclidean D ≥ ball, we write d, minimizes the isotropic constant with LD 1/ 2πe (see, e.g., 2 d 2 [8, Proposition 3.3.1]), we obtain from our assumption (1.2) exactly the radius condition (1.1). The observation that this universal lower bound on the isotropic constant of the convex body under consideration, which appears in (1.1) only in disguise, can be substituted by the isotropic constant of the body itself, is the essential step that allows us to extend the results from [13]. At the same time it sheds light on the rˆole of isotropicity in this problem. Remark 1.4. As will become apparent, Theorem 1.1 is essentially Proposition 4.1 below in the case α = 2. As a matter of fact, in the statement of Theorem 1.1 the isotropic constants appearing in the two conditions involving the Lipschitz parameters (A ) and (B ) could have been omitted. The reason is that Bourgain d d d d ≥ proved in [7] that if K is a symmetric ψ -body with constant b 1, then its 2 ≤ isotropic constant is bounded, even more precisely, L Cb log(b + 1) with an ∈ ∞ K absolute constant C (0, ). However, we chose to include the constants (L ) Kd d anyways to underline where the isotropicity enters the scene. Whether or not the isotropic constant is uniformly bounded above by an absolute constant in general is a famous open problem (see Remark 2.3). We complement our lower bounds by the following concentration result, which easily implies an upper bound on the complexity of numerical integration in the space C 1(A , B , K ). We stress that this result, as the last one, heavily relies on d d d d the thin-shell estimate of Gu´edon and Milman [11]. ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: AICKE

text: HINRICHS

text: AND

text: JOSCHA

text: PROCHNO

text: AND

text: MARIO

text: ULLRICH

text: Then

text: the

text: curse

text: of

text: dimensionality

text: holds

text: for

text: the

text: class

text: C

text: B

text: K

text: if

text: d

text: d

text: d

text: d

text: lim

text: sup

text: min

text: A

text: d

text: L

text: B

text: d

text: d

text: Kd

text: d

text: Kd

text: d

text: In

text: addition

text: to

text: these

text: lower

text: bounds

text: on

text: the

text: complexity

text: of

text: numerical

text: integration,

text: we

text: will

text: present

text: lower

text: bounds

text: for

text: the

text: more

text: general

text: class

text: of

text: ψ

text: -bodies

text: with

text: α

text: α

text: (see

text: Section

text: These

text: bounds

text: are,

text: however,

text: no

text: longer

text: exponential

text: in

text: d,

text: but

text: in

text: We

text: shall

text: present

text: this

text: result

text: in

text: Proposition

text: Remark

text: In

text: Asymptotic

text: Geometric

text: Analysis

text: terms,

text: our

text: radial

text: condition

text: simply

text: means

text: that

text: such

text: a

text: body

text: K

text: is

text: of

text: “small

text: diameter”

text: (i.e.,

text: rad(K

text: α

text: d

text: L

text: d

text: d

text: Kd

text: for

text: some

text: α

text: with

text: a

text: constant

text: strictly

text: smaller

text: than

text: It

text: follows

text: directly

text: R

text: from

text: the

text: isotropic

text: condition

text: that,

text: for

text: any

text: isotropic

text: convex

text: body

text: K

text: d,

text: d

text: L

text: K

text: rad(K).

text: Additionally,

text: as

text: shown

text: by

text: Kannan,

text: Lov´asz

text: and

text: Simonovits

text: in

text: we

text: have

text: the

text: upper

text: bound

text: rad(K)

text: (d

text: We

text: refer

text: to

text: the

text: monograph

text: for

text: more

text: K

text: information.

text: Remark

text: If

text: we

text: compare

text: the

text: two

text: conditions

text: and

text: on

text: the

text: radii,

text: then

text: we

text: see

text: that,

text: contrary

text: to

text: condition

text: considered

text: in

text: our

text: condition

text: involves

text: the

text: isotropic

text: constant

text: L

text: of

text: the

text: integral

text: domain

text: K

text: Using

text: the

text: well-known

text: Kd

text: d

text: fact

text: that

text: among

text: all

text: isotropic

text: convex

text: bodies

text: the

text: (volume

text: normalized)

text: Euclidean

text: D

text: ball,

text: we

text: write

text: d,

text: minimizes

text: the

text: isotropic

text: constant

text: with

text: LD

text: (see,

text: e.g.,

text: d

text: Proposition

text: we

text: obtain

text: from

text: our

text: assumption

text: exactly

text: the

text: radius

text: condition

text: The

text: observation

text: that

text: this

text: universal

text: lower

text: bound

text: on

text: the

text: isotropic

text: constant

text: of

text: the

text: convex

text: body

text: under

text: consideration,

text: which

text: appears

text: in

text: only

text: in

text: disguise,

text: can

text: be

text: substituted

text: by

text: the

text: isotropic

text: constant

text: of

text: the

text: body

text: itself,

text: is

text: the

text: essential

text: step

text: that

text: allows

text: us

text: to

text: extend

text: the

text: results

text: from

text: At

text: the

text: same

text: time

text: it

text: sheds

text: light

text: on

text: the

text: rˆole

text: of

text: isotropicity

text: in

text: this

text: problem.

text: Remark

text: As

text: will

text: become

text: apparent,

text: Theorem

text: is

text: essentially

text: Proposition

text: below

text: in

text: the

text: case

text: α

text: As

text: a

text: matter

text: of

text: fact,

text: in

text: the

text: statement

text: of

text: Theorem

text: the

text: isotropic

text: constants

text: appearing

text: in

text: the

text: two

text: conditions

text: involving

text: the

text: Lipschitz

text: parameters

text: (A

text: and

text: (B

text: could

text: have

text: been

text: omitted.

text: The

text: reason

text: is

text: that

text: Bourgain

text: d

text: d

text: d

text: d

text: proved

text: in

text: that

text: if

text: K

text: is

text: a

text: symmetric

text: ψ

text: -body

text: with

text: constant

text: b

text: then

text: its

text: isotropic

text: constant

text: is

text: bounded,

text: even

text: more

text: precisely,

text: L

text: Cb

text: log(b

text: with

text: an

text: K

text: absolute

text: constant

text: C

text: However,

text: we

text: chose

text: to

text: include

text: the

text: constants

text: (L

text: Kd

text: d

text: anyways

text: to

text: underline

text: where

text: the

text: isotropicity

text: enters

text: the

text: scene.

text: Whether

text: or

text: not

text: the

text: isotropic

text: constant

text: is

text: uniformly

text: bounded

text: above

text: by

text: an

text: absolute

text: constant

text: in

text: general

text: is

text: a

text: famous

text: open

text: problem

text: (see

text: Remark

text: We

text: complement

text: our

text: lower

text: bounds

text: by

text: the

text: following

text: concentration

text: result,

text: which

text: easily

text: implies

text: an

text: upper

text: bound

text: on

text: the

text: complexity

text: of

text: numerical

text: integration

text: in

text: the

text: space

text: C

text: B

text: K

text: We

text: stress

text: that

text: this

text: result,

text: as

text: the

text: last

text: one,

text: heavily

text: relies

text: on

text: d

text: d

text: d

text: d

text: the

text: thin-shell

text: estimate

text: of

text: Gu´edon

text: and

text: Milman

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Table 2: LDA Notation Name Description Model Notation T Number of topics W Dictionary size m Number of paths in a multipath model η Hyperparameter for words in a topic α Hyperparameter for topics in a document ≤ β Topics, i T. i Corpus Notation N Total number of tokens in a corpus D Total number of documents in a corpus ≤ ≤ ∀ ≤ w The i-th token in a corpus. i N and w W i N Di i ≤ D ≤ ∀ ≤ The document of the i-th token in a corpus. i N and D i N i i Sam(cid:80)pler N(cid:12)(cid:12)(cid:110)otation (cid:111)(cid:12)(cid:12) (cid:12) ≤ ≤ (cid:12) ≤ ∀ pj Topic assignment to the i-th w(cid:80)ord in a corpus. i N,j m and pj T i, j Ci C ≤ | ∧ i TW Topic-word counter, TW = ≤ i N w = w pj = t Ctw tw C j (cid:12)m(cid:110)C i i (cid:111)(cid:12) T Topic occurrence count, T = (cid:12) TW (cid:12) Ct t w(cid:12) tw C |{ ≤(cid:12) | D }| D Document occurrence count (size of document d), D = i N = d Cd C ≤ | D ∧d i≤ DT Document-topic counter, DT = i N = d pj = t with j m jdt jdt i i 20 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Table

text: LDA

text: Notation

text: Name

label: table

text: Description

text: Model

text: Notation

text: T

text: Number

text: of

text: topics

text: W

text: Dictionary

text: size

text: m

text: Number

text: of

text: paths

text: in

text: a

text: multipath

text: model

text: η

text: Hyperparameter

text: for

text: words

text: in

text: a

text: topic

text: α

text: Hyperparameter

text: for

text: topics

text: in

text: a

text: document

text: β

text: Topics,

text: i

text: T.

text: i

text: Corpus

text: Notation

text: N

text: Total

text: number

text: of

text: tokens

text: in

text: a

text: corpus

text: D

text: Total

text: number

text: of

text: documents

text: in

text: a

text: corpus

text: w

text: The

text: i-th

text: token

text: in

text: a

text: corpus.

text: i

text: N

text: and

text: w

text: W

text: i

text: N

text: Di

text: i

text: D

text: The

text: document

text: of

text: the

text: i-th

text: token

text: in

text: a

text: corpus.

text: i

text: N

text: and

text: D

text: i

text: N

text: i

text: i

text: pj

text: Topic

text: assignment

text: to

text: the

text: i-th

text: in

text: a

text: corpus.

text: i

text: N,j

text: m

text: and

text: pj

text: T

text: i,

text: j

text: Ci

text: C

text: i

text: TW

text: Topic-word

text: counter,

text: TW

text: i

text: N

text: w

text: w

text: pj

text: t

text: Ctw

text: tw

text: C

text: j

text: i

text: i

text: T

text: Topic

text: occurrence

text: count,

text: T

text: TW

text: Ct

text: t

text: tw

text: C

text: D

text: D

text: Document

text: occurrence

text: count

text: (size

text: of

text: document

text: d),

text: D

text: i

text: N

text: d

text: Cd

text: C

text: D

text: ∧d

text: i≤

text: DT

text: Document-topic

text: counter,

text: DT

text: i

text: N

text: d

text: pj

text: t

text: with

text: j

text: m

text: jdt

text: jdt

text: i

text: i

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: directly approximates EM as m grows. 4 Experiments In this Section we describe the HMM and LDA experiments. In addition, in Section , after describing the HMM experiments we discuss Figure 1, referred to in Section , which demonstrates the local maxima behavior of the Gibbs sampler. 4.1 HMMs The HMM experiments use synthetic data. The data was generated by an HMM with two states, such that the transition probability between the states is 0.45, and the emissions are distributions on W = 10 symbols. The emissions are show in Figure 3 as the ground truth. All the experiments were conducted using the same ﬁxed random sample of size N = 200000 from this HMM. These particular parameters were chosen in order to make the HMM hard to learn. Otherwise, all methods would succeed all the time, and would be diﬃcult to compare. This corresponds to realistic situations where diﬀerent emissions are hard to distinguish. Note that while N = 200000 may appear as a high number of samples, in this case, due to transitions probabilities which are close to 0.5, this high number of samples is in fact necessary. As we discuss later, even with this number of samples, both the EM algorithm and some runs of the Gibbs sampler manage to slightly overﬁt the solution. To put the HMM learning problem in the Bayesian setting, we endow the set of parameters with a prior where transition and emission distributions are sampled from a uniform Dirchlet, Dir(1). In other words, transition distribu- tions are sampled uniformly from a 3-simplex, and emission distributions from an 11-simplex. The data w is the given emissions of the HMM, and the path p corresponds to the latent sequence of the states of the HMM. Figure 1 demonstrates the local maxima behavior. It shows the data log- likelihoods of a solution along a run of a collapsed Gibbs sampler, for 16 inde- pendent runs of a collapsed Gibbs sampler. Each run had 200000 iterations, with data log-likelihoods computed every 100 iterations. As discussed in the Introduction, each run converges fairly quickly to an area of ﬁxed likelihood, and stays there for extremely long time intervals. Figure 2 shows the data log-likelihoods of the parameters φ for multiple runs of diﬀerent algorithms after 200000 iterations. Each algorithm was run independently 16 times (all on the same data), and the resulting data log- likelihoods are plotted in ascending order. The algorithms included are the Collapsed Gibbs sampler for 1 and ﬁve paths (C 1, 5) and a non-collapsed sampler (PC 1, 5). For reference, the constant line (BW) corresponds to a solution found by the Baum-Welch algorithm, and Ground line corresponds to the data likelihood 10

text: directly

text: approximates

text: EM

text: as

text: m

text: grows.

label: section

text: Experiments

text: In

text: this

text: Section

text: we

text: describe

text: the

text: HMM

text: and

text: LDA

text: experiments.

text: In

text: addition,

text: in

text: Section

text: after

text: describing

text: the

text: HMM

text: experiments

text: we

text: discuss

text: Figure

text: referred

text: to

text: in

text: Section

text: which

text: demonstrates

text: the

text: local

text: maxima

text: behavior

text: of

text: the

text: Gibbs

text: sampler.

text: HMMs

text: The

text: HMM

text: experiments

text: use

text: synthetic

text: data.

text: The

text: data

text: was

text: generated

text: by

text: an

text: HMM

text: with

text: two

text: states,

text: such

text: that

text: the

text: transition

text: probability

text: between

text: the

text: states

text: is

text: and

text: the

text: emissions

text: are

text: distributions

text: on

text: W

text: symbols.

text: The

text: emissions

text: are

text: show

text: in

text: Figure

text: as

text: the

text: ground

text: truth.

text: All

text: the

text: experiments

text: were

text: conducted

text: using

text: the

text: same

text: ﬁxed

text: random

text: sample

text: of

text: size

text: N

text: from

text: this

text: HMM.

text: These

text: particular

text: parameters

text: were

text: chosen

text: in

text: order

text: to

text: make

text: the

text: HMM

text: hard

text: to

text: learn.

text: Otherwise,

text: all

text: methods

text: would

text: succeed

text: all

text: the

text: time,

text: and

text: would

text: be

text: diﬃcult

text: to

text: compare.

text: This

text: corresponds

text: to

text: realistic

text: situations

text: where

text: diﬀerent

text: emissions

text: are

text: hard

text: to

text: distinguish.

text: Note

text: that

text: while

text: N

text: may

text: appear

text: as

text: a

text: high

text: number

text: of

text: samples,

text: in

text: this

text: case,

text: due

text: to

text: transitions

text: probabilities

text: which

text: are

text: close

text: to

text: this

text: high

text: number

text: of

text: samples

text: is

text: in

text: fact

text: necessary.

text: As

text: we

text: discuss

text: later,

text: even

text: with

text: this

text: number

text: of

text: samples,

text: both

text: the

text: EM

text: algorithm

text: and

text: some

text: runs

text: of

text: the

text: Gibbs

text: sampler

text: manage

text: to

text: slightly

text: overﬁt

text: the

text: solution.

text: To

text: put

text: the

text: HMM

text: learning

text: problem

text: in

text: the

text: Bayesian

text: setting,

text: we

text: endow

text: the

text: set

text: of

text: parameters

text: with

text: a

text: prior

text: where

text: transition

text: and

text: emission

text: distributions

text: are

text: sampled

text: from

text: a

text: uniform

text: Dirchlet,

text: In

text: other

text: words,

text: transition

text: distribu-

text: tions

text: are

text: sampled

text: uniformly

text: from

text: a

text: and

text: emission

text: distributions

text: from

text: an

text: The

text: data

text: w

text: is

text: the

text: given

text: emissions

text: of

text: the

text: HMM,

text: and

text: the

text: path

text: p

text: corresponds

text: to

text: the

text: latent

text: sequence

text: of

text: the

text: states

text: of

text: the

text: HMM.

text: Figure

text: demonstrates

text: the

text: local

text: maxima

text: behavior.

text: It

text: shows

text: the

text: data

text: log-

text: likelihoods

text: of

text: a

text: solution

text: along

text: a

text: run

text: of

text: a

text: collapsed

text: Gibbs

text: sampler,

text: for

text: inde-

text: pendent

text: runs

text: of

text: a

text: collapsed

text: Gibbs

text: sampler.

text: Each

text: run

text: had

text: iterations,

text: with

text: data

text: log-likelihoods

text: computed

text: every

text: iterations.

text: As

text: discussed

text: in

text: the

text: Introduction,

text: each

text: run

text: converges

text: fairly

text: quickly

text: to

text: an

text: area

text: of

text: ﬁxed

text: likelihood,

text: and

text: stays

text: there

text: for

text: extremely

text: long

text: time

text: intervals.

text: Figure

text: shows

text: the

text: data

text: log-likelihoods

text: of

text: the

text: parameters

text: φ

text: for

text: multiple

text: runs

text: of

text: diﬀerent

text: algorithms

text: after

text: iterations.

text: Each

text: algorithm

text: was

text: run

text: independently

text: times

text: (all

text: on

text: the

text: same

text: data),

text: and

text: the

text: resulting

text: data

text: log-

text: likelihoods

text: are

text: plotted

text: in

text: ascending

text: order.

text: The

text: algorithms

text: included

text: are

text: the

text: Collapsed

text: Gibbs

text: sampler

text: for

text: and

text: ﬁve

text: paths

text: (C

text: and

text: a

text: non-collapsed

text: sampler

text: (PC

text: For

text: reference,

text: the

text: constant

text: line

text: (BW)

text: corresponds

text: to

text: a

text: solution

text: found

text: by

text: the

text: Baum-Welch

text: algorithm,

text: and

text: Ground

text: line

text: corresponds

text: to

text: the

text: data

text: likelihood


text: 5 currents via optical and static electric ﬁelds. [6] L. P. Kadanoﬀ and G. Baym, Quantum Statistical Me- chanics (CRC Press, 1994). [7] P. U. Jepsen, R. H. Jacobsen, and S. R. Keiding, J. Opt. Acknowledgments.- We acknowledge support from Soc. Am. B 13, 2424 (1996). DOE-NERSC contract No. DE-AC02-05CH11231, and [8] T. Morimoto and N. Nagaosa, Science Advances 2 (2016). from the Natural Sciences and Engineering Research [9] C. Aversa and J. E. Sipe, Phys. Rev. B 52, 14636 (1995). Council of Canada. [10] R. W. Boyd, Nonlinear Optics (Academic Press; 2nd edi- tion, San Diego, USA, 2003). [11] D. Xiao, G.-B. Liu, W. Feng, X. Xu, and W. Yao, Phys. Rev. Lett. 108, 196802 (2012). [12] S. Priyadarshi, M. Leidinger, K. Pierz, A. M. Racu, ∗ U. Siegner, M. Bieler, and P. Dawson, Applied Physics [1] R. Karplus and J. M. Luttinger, Phys. Rev. 95, 1154 Letters 95, 151110 (2009). (1954). [13] R. Atanasov, A. Hach´e, J. L. P. Hughes, H. M. van Driel, [2] D. Xiao, M.-C. Chang, and Q. Niu, Rev. Mod. Phys. 82, and J. E. Sipe, Phys. Rev. Lett. 76, 1703 (1996). 1959 (2010). [14] N. Laman, A. I. Shkrebtii, J. E. Sipe, and H. M. van [3] B. I. Sturman and V. M. Fridkin, The Pho- Driel, Applied Physics Letters 75, 2581 (1999). tovoltaic and Photorefractive Eﬀects in Non- [15] M. Bieler, N. Laman, H. M. van Driel, and A. L. Smirl, CentrosymmetricMaterials (Gordon and Breach Science Applied Physics Letters 86, 061102 (2005). Publishers, Philadelphia, 1992). [16] D. A. Bas, K. Vargas-Velez, S. Babakiray, T. A. Johnson, [4] R. von Baltz and W. Kraut, Phys. Rev. B 23, 5590 P. Borisov, T. D. Stanescu, D. Lederman, and A. D. (1981). Bristow, Applied Physics Letters 106, 041109 (2015). [5] J. E. Sipe and A. I. Shkrebtii, Phys. Rev. B 61, 5337 [17] L. Z. Tan, F. Zheng, S. M. Young, F. Wang, S. Liu, and (2000). A. M. Rappe, npj Comput. Mater. 2, 16026 (2016) SUPLEMENTARY INFORMAION Derivation of jerk current from the third order susceptibility divergence − The third order susceptibility χabcd( ω , ω , ω , ω ) for free Bloch fermions has been calculated before [9]. For 3 Σ β σ ∆ completeness we reproduce the ﬁrst fe(cid:18)w terms of th(cid:19)e intraband part of χ (cid:18) (cid:19) (cid:88) (cid:88) 3 − e4 − rb rc f − e4 (va va )rd rb f · · · (cid:15)0χ3i = ω ω2 (cid:126)V (vna vma ) ωnm m−n ωmn ω2 (cid:126)V ω n − ωm +mωn ω nm−mωn + , (19) 2 Σ nmk nm β ;d Σ nmk nm β σ nm β ;c − where va = ∂ω /∂ka, ω = ω + ω , the electric ﬁeld is parametrized as Eb(t) = Eb e iωt + E−b eiωt, and various n n 2 β σ ω ω other quantities have already been deﬁned in the main text. The covariant derivative of a matrix element Oa of nm Bloch bands n, m is ∂Oa − − Oa = nm i(ξb ξb )Oa , (20) nm;b ∂kb nn mm nm (cid:104) |∇ | (cid:105) where ξa = i u a u are the Berry connections and u the periodic part of the Bloch state of band n. The nn n n n (cid:54) standard dipole matrix elements are given by ra = ξa for n = m and vanish for n = m. We also deﬁned the sum nm nm of frequencies as ω = ω + ω + ω . Eq. 19 needs to be symmetrized under pair-wise exchange of any two electric Σ↔ β ↔ σ ∆ ﬁeld indices [10] Eb Ec Ed . We will show that the ﬁrst two terms give rise to ι . Let us label each term as χ , β σ ∆ 3 3i1 and χ and consider them separately. By power counting the frequencies in denominators, it is clear that the ﬁrst 3i2 term in Eq. 19 could lead to a ﬁnite ι but only when one of the frequencies vanishes. Without loss of generality let 3 − ω = 0 (and hence ω = ω ). Integrating by parts χ , symmetrizing and multiplying by ( iω )3 gives, to zeroth ∆ Σ 2 3i1 Σ order in ω (cid:18) (cid:19) Σ (cid:88) − − ie4 ∂2ω rb rc rc rb ( iωΣ)3(cid:15)0χ3i1 = 6(cid:126)3V fmn ∂kd∂nkma ω nm−mωn + ω nm−mωn . (21) nm β nm σ nmk ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: currents

text: via

text: optical

text: and

text: static

text: electric

text: ﬁelds.

label: reference

text: L.

text: P.

text: Kadanoﬀ

text: and

text: G.

text: Baym,

text: Quantum

text: Statistical

text: Me-

text: chanics

text: (CRC

text: Press,

text: P.

text: U.

text: Jepsen,

text: R.

text: H.

text: Jacobsen,

text: and

text: S.

text: R.

text: Keiding,

text: J.

text: Opt.

text: Acknowledgments.-

text: We

text: acknowledge

text: support

text: from

text: Soc.

text: Am.

text: B

text: DOE-NERSC

text: contract

text: No.

text: and

text: T.

text: Morimoto

text: and

text: N.

text: Nagaosa,

text: Science

text: Advances

text: from

text: the

text: Natural

text: Sciences

text: and

text: Engineering

text: Research

text: C.

text: Aversa

text: and

text: J.

text: E.

text: Sipe,

text: Phys.

text: Rev.

text: B

text: Council

text: of

text: Canada.

text: R.

text: W.

text: Boyd,

text: Nonlinear

text: Optics

text: (Academic

text: Press;

text: edi-

text: tion,

text: San

text: Diego,

text: USA,

text: D.

text: Xiao,

text: G.-B.

text: Liu,

text: W.

text: Feng,

text: X.

text: Xu,

text: and

text: W.

text: Yao,

text: Phys.

text: Rev.

text: Lett.

text: S.

text: Priyadarshi,

text: M.

text: Leidinger,

text: K.

text: Pierz,

text: A.

text: M.

text: Racu,

text: U.

text: Siegner,

text: M.

text: Bieler,

text: and

text: P.

text: Dawson,

text: Applied

text: Physics

text: R.

text: Karplus

text: and

text: J.

text: M.

text: Luttinger,

text: Phys.

text: Rev.

text: Letters

text: R.

text: Atanasov,

text: A.

text: Hach´e,

text: J.

text: L.

text: P.

text: Hughes,

text: H.

text: M.

text: van

text: Driel,

text: D.

text: Xiao,

text: M.-C.

text: Chang,

text: and

text: Q.

text: Niu,

text: Rev.

text: Mod.

text: Phys.

text: and

text: J.

text: E.

text: Sipe,

text: Phys.

text: Rev.

text: Lett.

text: N.

text: Laman,

text: A.

text: I.

text: Shkrebtii,

text: J.

text: E.

text: Sipe,

text: and

text: H.

text: M.

text: van

text: B.

text: I.

text: Sturman

text: and

text: V.

text: M.

text: Fridkin,

text: The

text: Pho-

text: Driel,

text: Applied

text: Physics

text: Letters

text: tovoltaic

text: and

text: Photorefractive

text: Eﬀects

text: in

text: Non-

text: M.

text: Bieler,

text: N.

text: Laman,

text: H.

text: M.

text: van

text: Driel,

text: and

text: A.

text: L.

text: Smirl,

text: CentrosymmetricMaterials

text: (Gordon

text: and

text: Breach

text: Science

text: Applied

text: Physics

text: Letters

text: Publishers,

text: Philadelphia,

text: D.

text: A.

text: Bas,

text: K.

text: Vargas-Velez,

text: S.

text: Babakiray,

text: T.

text: A.

text: Johnson,

text: R.

text: von

text: Baltz

text: and

text: W.

text: Kraut,

text: Phys.

text: Rev.

text: B

text: P.

text: Borisov,

text: T.

text: D.

text: Stanescu,

text: D.

text: Lederman,

text: and

text: A.

text: D.

text: Bristow,

text: Applied

text: Physics

text: Letters

text: J.

text: E.

text: Sipe

text: and

text: A.

text: I.

text: Shkrebtii,

text: Phys.

text: Rev.

text: B

text: L.

text: Z.

text: Tan,

text: F.

text: Zheng,

text: S.

text: M.

text: Young,

text: F.

text: Wang,

text: S.

text: Liu,

text: and

text: A.

text: M.

text: Rappe,

text: npj

text: Comput.

text: Mater.

text: SUPLEMENTARY

text: INFORMAION

text: Derivation

text: of

text: jerk

text: current

text: from

text: the

text: third

text: order

text: susceptibility

text: divergence

text: The

text: third

text: order

text: susceptibility

text: χabcd(

text: ω

text: ω

text: ω

text: ω

text: for

text: free

text: Bloch

text: fermions

text: has

text: been

text: calculated

text: before

text: For

text: Σ

text: β

text: σ

text: completeness

text: we

text: reproduce

text: the

text: ﬁrst

text: terms

text: of

text: intraband

text: part

text: of

text: χ

text: rb

text: rc

text: f

text: (va

text: va

text: )rd

text: rb

text: f

text: ω

text: ω2

text: (vna

text: vma

text: ωnm

text: m−n

text: ωmn

text: ω2

text: ω

text: n

text: ωm

text: +mωn

text: ω

text: nm−mωn

text: Σ

text: nmk

text: nm

text: β

text: ;d

text: Σ

text: nmk

text: nm

text: β

text: σ

text: nm

text: β

text: ;c

text: where

text: va

text: ∂ω

text: /∂ka,

text: ω

text: ω

text: ω

text: the

text: electric

text: ﬁeld

text: is

text: parametrized

text: as

text: Eb(t)

text: Eb

text: e

text: iωt

text: E−b

text: eiωt,

text: and

text: various

text: n

text: n

text: β

text: σ

text: ω

text: ω

text: other

text: quantities

text: have

text: already

text: been

text: deﬁned

text: in

text: the

text: main

text: text.

text: The

text: covariant

text: derivative

text: of

text: a

text: matrix

text: element

text: Oa

text: of

text: nm

text: Bloch

text: bands

text: n,

text: m

text: is

text: ∂Oa

text: Oa

text: nm

text: i(ξb

text: ξb

text: )Oa

text: nm;b

text: ∂kb

text: nn

text: mm

text: nm

text: where

text: ξa

text: i

text: u

text: a

text: u

text: are

text: the

text: Berry

text: connections

text: and

text: u

text: the

text: periodic

text: part

text: of

text: the

text: Bloch

text: state

text: of

text: band

text: n.

text: The

text: nn

text: n

text: n

text: n

text: standard

text: dipole

text: matrix

text: elements

text: are

text: given

text: by

text: ra

text: ξa

text: for

text: n

text: m

text: and

text: vanish

text: for

text: n

text: m.

text: We

text: also

text: deﬁned

text: the

text: sum

text: nm

text: nm

text: of

text: frequencies

text: as

text: ω

text: ω

text: ω

text: ω

text: Eq.

text: needs

text: to

text: be

text: symmetrized

text: under

text: pair-wise

text: exchange

text: of

text: any

text: two

text: electric

text: Σ↔

text: β

text: σ

text: ﬁeld

text: indices

text: Eb

text: Ec

text: Ed

text: We

text: will

text: show

text: that

text: the

text: ﬁrst

text: two

text: terms

text: give

text: rise

text: to

text: ι

text: Let

text: us

text: label

text: each

text: term

text: as

text: χ

text: β

text: σ

text: and

text: χ

text: and

text: consider

text: them

text: separately.

text: By

text: power

text: counting

text: the

text: frequencies

text: in

text: denominators,

text: it

text: is

text: clear

text: that

text: the

text: ﬁrst

text: term

text: in

text: Eq.

text: could

text: lead

text: to

text: a

text: ﬁnite

text: ι

text: but

text: only

text: when

text: one

text: of

text: the

text: frequencies

text: vanishes.

text: Without

text: loss

text: of

text: generality

text: let

text: ω

text: (and

text: hence

text: ω

text: ω

text: Integrating

text: by

text: parts

text: χ

text: symmetrizing

text: and

text: multiplying

text: by

text: iω

text: gives,

text: to

text: zeroth

text: Σ

text: Σ

text: order

text: in

text: ω

text: Σ

text: ∂2ω

text: rb

text: rc

text: rc

text: rb

text: fmn

text: ∂kd∂nkma

text: ω

text: nm−mωn

text: ω

text: nm−mωn

text: nm

text: β

text: nm

text: σ

text: nmk

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##