text: 21   After the unitary transformation, the block part of the ξ bands in the BdG Hamiltonian is given by +    1 k2 − µ ∆0 + ∆1 k2a2 c1Q21+c2Q22 0 0   ∗ 2m∗ + 2 −2 Qc   ∆0 + ∆1 k2a2 c1Q21+c2Q22 1 k2 + µ 0 0  2 2 Qc 2m+ − 0 0 1 k2 µ ∆0 + ∆1 k2a2 c1Q21+c2Q22 ∗ 2m∗ + 2 −2 Qc 0 0 ∆0 + ∆1 k2a2 c1Q21+c2Q22 1 k2 + µ 2 2 Qc 2m+   while the block for the ξ− bands is    1 k2 − µ ∆0 − ∆1 k2a2 c1Q21+c2Q22 0 0   ∗ − 2m∗ − 2 −2 Qc   ∆0 ∆1 k2a2 c1Q21+c2Q22 1 k2 + µ 0 0  2 2 Qc 2m− − − . 0 0 1 k2 µ ∆0 ∆1 k2a2 c1Q21+c2Q22 ∗ − 2m∗ − 2 −2 Qc 0 0 ∆0 ∆1 k2a2 c1Q21+c2Q22 1 k2 + µ 2 2 Qc 2m− The coupling between diﬀerent ξ± blocks is given by the oﬀ-diagonal terms of Eq. (C19), which is zero in the isotropic limit(c = c ) and can be neglected for small anisotropy. Even if the anisotropy is not small, it can still be neglected 1 2 since the physics related with pairing is only relevant near Fermi surfaces. From the expressions of ξ± blocks, we notice that the d-wave quintet pairing is transformed into s-wave singlet pairing with k2 dependence after the projection. Such form of pairing is normally known as extended s-wave pairing in literature39,40. As a result, it is easy to see that the nodal condition is determined by the vanishing of this extended s-wave gap function at the Fermi surface of each band. This Fermi surface project scheme provides a more clear physical picture of how the singlet-quintet mixing mechanism can induce nodal points or lines in the gap function. For the ξ band, the nodal condition is determined by + 1 − ∆ ∆ c Q2 + c Q2 k2 µ = 0 and 0 + 1 k2a2 1 1 2 2 = 0, 2m 2 2 Q + c which gives (cid:112) | | | | ∆˜ − 1 c Q2 + c Q2 1 + 2mQ > 0 & k = 2m µ & 0 = 1 1 2 2 . (C20) c + ∆˜ 1 1 + 2mQc Qc For the ξ− band, the nodal condition is 1 − ∆ − ∆ c Q2 + c Q2 k2 µ = 0 and 0 1 k2a2 1 1 2 2 = 0, 2m− 2 2 Q c which gives (cid:112) | | | | − ∆˜ 1 c Q2 + c Q2 1 2mQ > 0 & k = 2m−µ & 0 = − 1 1 2 2 . (C21) c ∆˜ 1 1 2mQc Qc Here we have used m < 0, µ < 0 and c c > 0 and the relation between ∆˜ (∆˜ ) and ∆ (∆ ) is deﬁned in the 1 2 0 1 0 1 main text. The nodal condition for the ξ (ξ−) band requires ∆˜ /∆˜ < 0 (∆˜ /∆˜ > 0). According to Eq. (B68), which is + 0 1 0 1 solved from the linearized gap equation, the sign of ∆˜ /∆˜ is determined by y . Therefore, we need to discuss two 0 1 2 cases with diﬀerent signs of y , separately. 2 If y > 0, ∆˜ /∆˜ > 0 and thus nodal points cannot exist on the ξ Fermi surface. In this case, the nodal points 2 0 1 + for the ξ− band require (cid:112) − λ˜ f−y + y 1 2mQ > 0 & k = 2m−µ & 1 = 1 2 & y > 0, (C22) c λ f−2 y + f−y 2 0 2 3 | | | | where f− = − 1 c1 Q21+ c2 Q22 . 1 2mQc Qc ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: After

text: the

text: unitary

text: transformation,

text: the

text: block

text: part

text: of

text: the

text: ξ

text: bands

text: in

text: the

text: BdG

text: Hamiltonian

text: is

text: given

text: by

text: µ

text: ∆0

text: ∆1

text: −2

text: Qc

text: ∆0

text: ∆1

text: µ

text: Qc

text: µ

text: ∆0

text: ∆1

text: −2

text: Qc

text: ∆0

text: ∆1

text: µ

text: Qc

text: while

text: the

text: block

text: for

text: the

text: ξ−

text: bands

text: is

text: µ

text: ∆0

text: ∆1

text: −2

text: Qc

text: ∆0

text: ∆1

text: µ

text: Qc

text: µ

text: ∆0

text: ∆1

text: −2

text: Qc

text: ∆0

text: ∆1

text: µ

text: Qc

text: The

text: coupling

text: between

text: diﬀerent

text: ξ±

text: blocks

text: is

text: given

text: by

text: the

text: oﬀ-diagonal

text: terms

text: of

text: Eq.

text: which

text: is

text: zero

text: in

text: the

text: isotropic

text: limit(c

text: c

text: and

text: can

text: be

text: neglected

text: for

text: small

text: anisotropy.

text: Even

text: if

text: the

text: anisotropy

text: is

text: not

text: small,

text: it

text: can

text: still

text: be

text: neglected

text: since

text: the

text: physics

text: related

text: with

text: pairing

text: is

text: only

text: relevant

text: near

text: Fermi

text: surfaces.

text: From

text: the

text: expressions

text: of

text: ξ±

text: blocks,

text: we

text: notice

text: that

text: the

text: d-wave

text: quintet

text: pairing

text: is

text: transformed

text: into

text: s-wave

text: singlet

text: pairing

text: with

text: dependence

text: after

text: the

text: projection.

text: Such

text: form

text: of

text: pairing

text: is

text: normally

text: known

text: as

text: extended

text: s-wave

text: pairing

text: in

text: As

text: a

text: result,

text: it

text: is

text: easy

text: to

text: see

text: that

text: the

text: nodal

text: condition

text: is

text: determined

text: by

text: the

text: vanishing

text: of

text: this

text: extended

text: s-wave

text: gap

text: function

text: at

text: the

text: Fermi

text: surface

text: of

text: each

text: band.

text: This

text: Fermi

text: surface

text: project

text: scheme

text: provides

text: a

text: more

text: clear

text: physical

text: picture

text: of

text: how

text: the

text: singlet-quintet

text: mixing

text: mechanism

text: can

text: induce

text: nodal

text: points

text: or

text: lines

text: in

text: the

text: gap

text: function.

text: For

text: the

text: ξ

text: band,

text: the

text: nodal

text: condition

text: is

text: determined

text: by

text: c

text: c

text: µ

text: and

text: Q

text: c

text: which

text: gives

text: c

text: c

text: k

text: µ

text: c

text: Qc

text: For

text: the

text: ξ−

text: band,

text: the

text: nodal

text: condition

text: is

text: c

text: c

text: µ

text: and

text: Q

text: c

text: which

text: gives

text: c

text: c

text: k

text: 2m−µ

text: c

text: Qc

text: Here

text: we

text: have

text: used

text: m

text: µ

text: and

text: c

text: c

text: and

text: the

text: relation

text: between

text: and

text: is

text: deﬁned

text: in

text: the

text: main

text: text.

text: The

text: nodal

text: condition

text: for

text: the

text: ξ

text: (ξ−)

text: band

text: requires

text: According

text: to

text: Eq.

text: which

text: is

text: solved

text: from

text: the

text: linearized

text: gap

text: equation,

text: the

text: sign

text: of

text: is

text: determined

text: by

text: y

text: Therefore,

text: we

text: need

text: to

text: discuss

text: two

text: cases

text: with

text: diﬀerent

text: signs

text: of

text: y

text: separately.

text: If

text: y

text: and

text: thus

text: nodal

text: points

text: cannot

text: exist

text: on

text: the

text: ξ

text: Fermi

text: surface.

text: In

text: this

text: case,

text: the

text: nodal

text: points

text: for

text: the

text: ξ−

text: band

text: require

text: λ˜

text: f−y

text: y

text: k

text: 2m−µ

text: y

text: c

text: λ

text: y

text: f−y

text: where

text: f−

text: Qc

text: ##LTLine##

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text: 4 a 𝑖𝑣 b 𝑖𝑖𝑖 (𝑖) (𝑖𝑖) (𝑖𝑖𝑖) (𝑖𝑣) 𝑖𝑖 𝑖 𝛼 𝜆 1 𝜆 0 𝑐1 = 𝑐2 c (𝑖) (𝑖𝑖) (𝑖𝑖𝑖) (𝑖𝑣) 𝑘 |2𝑚𝑐 | 2 1 e (𝑖) (𝑖𝑖) 𝑘 1 𝑘 1 𝑘 1 𝑘 1 d (𝑖) (𝑖𝑖) (𝑖𝑖𝑖) (𝑖𝑣) 20𝐸 (𝑖𝑖𝑖) 𝜇 𝑘 𝑘 𝑘 𝑘 1 1 1 1 FIG. 2. (a) shows the phase diagram in the parameter space spanned by interaction strength ratio λ˜ /λ and symmetric | | 1 0 SOC strength 2mc . In the yellow and red regions, the system are nodal. In the inset, the dashed line indicates the | | −1 path α (2m c = 0.8) with four points i, ..., iv on it. Here λ˜ /λ = 0.4246, 0.4507, 0.4615, 0.4716 for (i), (ii), (iii), (iv), 1 1 0 respectively. (b),(c) and (d) show the bulk nodal line structures (blue lines), zero-energy density of states on (111) surface and energy dispersion along (11¯2) axis on (111) surface for the four points i, ..., iv in the in√set of (a). The red circle in (i) of (b) shows a typical path along which the topological invariant is calculated. k˜ = k / 2mµ are momenta along (11¯2) | | 1,2 1,2 and (¯110), respectively, and c > 0 and ∆˜ / µ = 1 are chosen. (e) shows three typical nodal structures in the red region of 1 | | −0 | | − | | − | | − (a). Parameters are chosen as 2m c = 0.12, 2m c = 0.5 and λ˜ /λ = 1.12 for (i), 2m c = 0.12, 2m c = 0.5 and | | 1− | | 2− 1 0 1 2 λ˜ /λ = 1.155 for (ii), and 2m c = 0.08, 2m c = 0.5 and λ˜ /λ = 1.329 for (iii). 1 0 1 2 1 0 temperatures around 0.8K and 1.6K, respectively, thus providing good candidates for TNLS. YPtBi is a SC with inverted band structure20 and recent ﬁrst principles calculations13,17,22 suggest that its energy dispersion might belong to regime III, although debates still exist13,15. Evidence of TNLS has been found in the penetration depth experiment15 . Previous study attributes the nodal structure to the p-wave septet pairing mixed with subdominant s-wave singlet pairing due to asymmetric SOC13,15. Our work here provides an alternative explanation of the nodal structure as a result of singlet-quintet mixing induced by symmetric SOC h . In realistic half-Heusler compounds, the energy ∼ SOC ∼ scale of symmetric SOC ( 1eV ) is two orders of magnitude larger than anti-symmetric SOC ( 0.01eV )13,29. Thus, anti-symmetric SOC should be regarded as a perturbation and its inﬂuence on nodal-ring structures is discussed in Sec.C6 of SMs. Furthermore, the interaction in s-wave singlet channel is normally the dominant mechanism for superconductivity in weakly correlated materials. Therefore, we expect singlet-quintet mixing should be dominant over singlet-septet mixing and response for the nodal line structure in realistic SCs. Our new pairing mixing mechanism opens up a door to explore other exotic superconducting phenomena in spin-orbit coupled SCs with j = 3/2 electrons. Acknowledgment JY owes a large amount of thanks to Lun-Hui Hu for patiently answering his questions on superconductivity. JY also thanks Rui-Xing Zhang, Yang Ge and Jian-Xiao Zhang for helpful discussion. CXL and JY acknowledge the ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine##

label: paragraph

text: a

font: default

text: 𝑖𝑣

text: b

text: 𝑖𝑖𝑖

text: (𝑖)

text: (𝑖𝑖)

text: (𝑖𝑖𝑖)

text: (𝑖𝑣)

text: 𝑖𝑖

text: 𝑖

text: 𝛼

text: 𝜆

text: 𝜆

text: 𝑐1

text: 𝑐2

text: c

text: (𝑖)

text: (𝑖𝑖)

text: (𝑖𝑖𝑖)

text: (𝑖𝑣)

text: 𝑘

text: |2𝑚𝑐

text: e

text: (𝑖)

text: (𝑖𝑖)

text: 𝑘

text: 𝑘

text: 𝑘

text: 𝑘

text: d

text: (𝑖)

text: (𝑖𝑖)

text: (𝑖𝑖𝑖)

text: (𝑖𝑣)

text: (𝑖𝑖𝑖)

text: 𝜇

text: 𝑘

text: 𝑘

text: 𝑘

text: 𝑘

text: FIG.

text: (a)

text: shows

text: the

text: phase

text: diagram

text: in

text: the

text: parameter

text: space

text: spanned

text: by

text: interaction

text: strength

text: ratio

text: λ˜

text: /λ

text: and

text: symmetric

text: SOC

text: strength

text: In

text: the

text: yellow

text: and

text: red

text: regions,

text: the

text: system

text: are

text: nodal.

text: In

text: the

text: inset,

text: the

text: dashed

text: line

text: indicates

text: the

text: −1

text: path

text: α

text: c

text: with

text: four

text: points

text: i,

text: iv

text: on

text: it.

text: Here

text: λ˜

text: /λ

text: for

text: (i),

text: (ii),

text: (iii),

text: (iv),

text: respectively.

text: (b),(c)

text: and

text: (d)

text: show

text: the

text: bulk

text: nodal

text: line

text: structures

text: (blue

text: lines),

text: zero-energy

text: density

text: of

text: states

text: on

text: surface

text: and

text: energy

text: dispersion

text: along

text: axis

text: on

text: surface

text: for

text: the

text: four

text: points

text: i,

text: iv

text: in

text: the

text: in√set

text: of

text: (a).

text: The

text: red

text: circle

text: in

text: (i)

text: of

text: (b)

text: shows

text: a

text: typical

text: path

text: along

text: which

text: the

text: topological

text: invariant

text: is

text: calculated.

text: k˜

text: k

text: are

text: momenta

text: along

text: and

text: respectively,

text: and

text: c

text: and

text: µ

text: are

text: chosen.

text: (e)

text: shows

text: three

text: typical

text: nodal

text: structures

text: in

text: the

text: red

text: region

text: of

text: −0

text: (a).

text: Parameters

text: are

text: chosen

text: as

text: c

text: c

text: and

text: λ˜

text: /λ

text: for

text: (i),

text: c

text: c

text: and

text: 1−

text: 2−

text: λ˜

text: /λ

text: for

text: (ii),

text: and

text: c

text: c

text: and

text: λ˜

text: /λ

text: for

text: (iii).

text: temperatures

text: around

text: and

text: respectively,

text: thus

text: providing

text: good

text: candidates

text: for

text: TNLS.

text: YPtBi

text: is

text: a

text: SC

text: with

text: inverted

text: band

text: and

text: recent

text: ﬁrst

text: principles

text: suggest

text: that

text: its

text: energy

text: dispersion

text: might

text: belong

text: to

text: regime

text: III,

text: although

text: debates

text: still

text: Evidence

text: of

text: TNLS

text: has

text: been

text: found

text: in

text: the

text: penetration

text: depth

text: Previous

text: study

text: attributes

text: the

text: nodal

text: structure

text: to

text: the

text: p-wave

text: septet

text: pairing

text: mixed

text: with

text: subdominant

text: s-wave

text: singlet

text: pairing

text: due

text: to

text: asymmetric

text: Our

text: work

text: here

text: provides

text: an

text: alternative

text: explanation

text: of

text: the

text: nodal

text: structure

text: as

text: a

text: result

text: of

text: singlet-quintet

text: mixing

text: induced

text: by

text: symmetric

text: SOC

text: h

text: In

text: realistic

text: half-Heusler

text: compounds,

text: the

text: energy

text: SOC

text: scale

text: of

text: symmetric

text: SOC

text: is

text: two

text: orders

text: of

text: magnitude

text: larger

text: than

text: anti-symmetric

text: SOC

text: Thus,

text: anti-symmetric

text: SOC

text: should

text: be

text: regarded

text: as

text: a

text: perturbation

text: and

text: its

text: inﬂuence

text: on

text: nodal-ring

text: structures

text: is

text: discussed

text: in

text: of

text: SMs.

text: Furthermore,

text: the

text: interaction

text: in

text: s-wave

text: singlet

text: channel

text: is

text: normally

text: the

text: dominant

text: mechanism

text: for

text: superconductivity

text: in

text: weakly

text: correlated

text: materials.

text: Therefore,

text: we

text: expect

text: singlet-quintet

text: mixing

text: should

text: be

text: dominant

text: over

text: singlet-septet

text: mixing

text: and

text: response

text: for

text: the

text: nodal

text: line

text: structure

text: in

text: realistic

text: SCs.

text: Our

text: new

text: pairing

text: mixing

text: mechanism

text: opens

text: up

text: a

text: door

text: to

text: explore

text: other

text: exotic

text: superconducting

text: phenomena

text: in

text: spin-orbit

text: coupled

text: SCs

text: with

text: j

text: electrons.

text: Acknowledgment

text: JY

text: owes

text: a

text: large

text: amount

text: of

text: thanks

text: to

text: Lun-Hui

text: Hu

text: for

text: patiently

text: answering

text: his

text: questions

text: on

text: superconductivity.

text: JY

text: also

text: thanks

text: Rui-Xing

text: Zhang,

text: Yang

text: Ge

text: and

text: Jian-Xiao

text: Zhang

text: for

text: helpful

text: discussion.

text: CXL

text: and

text: JY

text: acknowledge

text: the

text: ##LTFigure##

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text: ##LTFigure##

text: ##LTLine##


text: 5 support from Oﬃce of Naval Research (Grant No. N00014-15-1-2675). 1 E. Bauer and M. Sigrist, Non-centrosymmetric superconductors: introduction and overview, Vol. 847 (Springer Science & Business Media, 2012). 2 L. P. Gor’kov and E. I. Rashba, Phys. Rev. Lett. 87, 037004 (2001). 3 P. A. Frigeri, D. F. Agterberg, A. Koga, and M. Sigrist, Phys. Rev. Lett. 92, 097001 (2004). 4 T. Yasuda, H. Shishido, T. Ueda, S. Hashimoto, R. Settai, T. Takeuchi, T. D Matsuda, Y. Haga, and Y. O¯nuki, Journal of the Physical Society of Japan 73, 1657 (2004). 5 T. Takeuchi, T. Yasuda, M. Tsujino, H. Shishido, R. Settai, H. Harima, and Y. O¯nuki, Journal of the Physical Society of Japan 76, 014702 (2006). 6 R. Settai, Y. Miyauchi, T. Takeuchi, F. L´evy, I. Sheikin, and Y. O¯nuki, Journal of the Physical Society of Japan 77, 073705 (2008). 7 H. Mukuda, T. Ohara, M. Yashima, Y. Kitaoka, R. Settai, Y. O¯nuki, K. Itoh, and E. Haller, Physical review letters 104, 017002 (2010). 8 S. K. Yip, Phys. Rev. B 65, 144508 (2002). 9 V. M. Edelstein, Phys. Rev. Lett. 75, 2004 (1995). 10 S. Fujimoto, Phys. Rev. B 72, 024515 (2005). 11 M. Sato and S. Fujimoto, Phys. Rev. B 79, 094504 (2009). 12 Y. Tanaka, T. Yokoyama, A. V. Balatsky, and N. Nagaosa, Phys. Rev. B 79, 060505 (2009). 13 P. M. R. Brydon, L. Wang, M. Weinert, and D. F. Agterberg, Phys. Rev. Lett. 116, 177001 (2016). 14 N. P. Butch, P. Syers, K. Kirshenbaum, A. P. Hope, and J. Paglione, Phys. Rev. B 84, 220504 (2011). 15 H. Kim, K. Wang, Y. Nakajima, R. Hu, S. Ziemak, P. Syers, L. Wang, H. Hodovanets, J. D. Denlinger, P. M. Brydon, et al., arXiv preprint arXiv:1603.03375 (2016). 16 T. V. Bay, T. Naka, Y. K. Huang, and A. de Visser, Phys. Rev. B 86, 064515 (2012). 17 M. Meinert, Phys. Rev. Lett. 116, 137001 (2016). 18 C. Wu, Modern Physics Letters B 20, 1707 (2006). 19 R. Winkler, S. Papadakis, E. De Poortere, and M. Shayegan, Spin-Orbit Coupling in Two-Dimensional Electron and Hole Systems, Vol. 41 (Springer, 2003) pp. 211–223. 20 S. Chadov, X. Qi, J. Ku¨bler, G. H. Fecher, C. Felser, and S. C. Zhang, Nature materials 9, 541 (2010). 21 J. M. Luttinger, Phys. Rev. 102, 1030 (1956). 22 H. Yang, J. Yu, S. S. P. Parkin, C. Felser, C.-X. Liu, and B. Yan, Phys. Rev. Lett. 119, 136401 (2017). 23 W. Yang, T. Xiang, and C. Wu, Phys. Rev. B 96, 144514 (2017). 24 C. Timm, A. P. Schnyder, D. F. Agterberg, and P. M. R. Brydon, Phys. Rev. B 96, 094526 (2017). 25 B. Roy, S. A. A. Ghorashi, M. S. Foster, and A. H. Nevidomskyy, arXiv preprint arXiv:1708.07825 (2017). 26 I. Boettcher and I. F. Herbut, arXiv preprint arXiv:1707.03444 (2017). 27 W. Yang, Y. Li, and C. Wu, Phys. Rev. Lett. 117, 075301 (2016). 28 J. W. Venderbos, L. Savary, J. Ruhman, P. A. Lee, and L. Fu, arXiv preprint arXiv:1709.04487 (2017). 29 L. Savary, J. Ruhman, J. W. F. Venderbos, L. Fu, and P. A. Lee, Phys. Rev. B 96, 214514 (2017). 30 A. P. Schnyder, P. M. R. Brydon, and C. Timm, Phys. Rev. B 85, 024522 (2012). 31 P. M. R. Brydon, A. P. Schnyder, and C. Timm, Phys. Rev. B 84, 020501 (2011). 32 K. Yada, M. Sato, Y. Tanaka, and T. Yokoyama, Phys. Rev. B 83, 064505 (2011). 33 A. P. Schnyder and S. Ryu, Phys. Rev. B 84, 060504 (2011). 34 M. L. Sancho, J. L. Sancho, J. L. Sancho, and J. Rubio, Journal of Physics F: Metal Physics 15, 851 (1985). 35 Y. Nakajima, R. Hu, K. Kirshenbaum, A. Hughes, P. Syers, X. Wang, K. Wang, R. Wang, S. R. Saha, D. Pratt, et al., Science advances 1, e1500242 (2015). 36 S. Murakami, N. Nagosa, and S.-C. Zhang, Phys. Rev. B 69, 235206 (2004). 37 P. A. Frigeri, D. F. Agterberg, A. Koga, and M. Sigrist, Phys. Rev. Lett. 92, 097001 (2004). 38 X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Phys. Rev. B 81, 134508 (2010). 39 G.-m. Zhao, Phys. Rev. B 64, 024503 (2001). 40 B. H. Brandow, Phys. Rev. B 65, 054503 (2002). A. Expressions and Conventions The ﬁve d-orbital cubic harmonics are given by36 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: K.

text: Kirshenbaum,

text: A.

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text: P.

text: Syers,

text: X.

text: Wang,

text: K.

text: Wang,

text: R.

text: Wang,

text: S.

text: R.

text: Saha,

text: D.

text: Pratt,

text: et

text: al.,

text: Science

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text: Murakami,

text: N.

text: Nagosa,

text: and

text: S.-C.

text: Zhang,

text: Phys.

text: Rev.

text: B

text: P.

text: A.

text: Frigeri,

text: D.

text: F.

text: Agterberg,

text: A.

text: Koga,

text: and

text: M.

text: Sigrist,

text: Phys.

text: Rev.

text: Lett.

text: X.-L.

text: Qi,

text: T.

text: L.

text: Hughes,

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text: S.-C.

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text: Expressions

text: and

text: Conventions

text: The

text: ﬁve

text: d-orbital

text: cubic

text: harmonics

text: are

text: given

text: ##LTLine##

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text: ##LTLine##


text: quency ω via the timing of muon decays to positrons using 24 electromag- a netic calorimeters [11,12]. Measurements of the precession frequency and mag- − netic ﬁeld lead to a . The anomalous precession frequency ω = ω ω , µ a s c where the Cyclotron frequency ω = e B and the spin precession frequency c mγ ω = e B(1 + γa ). s mγ µ To reduce the eﬀect of electric ﬁelds, the muons are injected at a magic momentum with γ = 29.3, which cancels the second term in Eq. 1. × −Qe − − mc β(cid:126) E(cid:126) ω(cid:126) = [a B(cid:126) (a ( )2) ], (1) a m µ µ p c which leaves −Qe ω(cid:126) = a B(cid:126). (2) a m µ The proton precession frequency ω is measured as a proxy for B(cid:126) leading to p an expression for the anomalous moment as ω /ω a = a− p . (3) µ µ /µ ω /ω µ p a p The Measurement of ω is performed using NMR. Fixed NMR probes mea- p sure time variations of the ﬁeld during data taking. A trolley with mounted NMR probes periodically circumnavigates the interior of the ring to perform precision measurements of the ﬁeld in the muon storage region, performing 6000 magnetic ﬁeld measurements per trolley run. Probes are calibrated to provide measurement to 35 ppb. We plan to collect 21 times the BNL statistics, which will reduce our sta- tistical uncertainty by a factor of four. To reduce systematic uncertainty, ac- celerator facilities will have p closer to magic momentum, utilize a longer π decay channel, and increase injection eﬃciency. Systematics on ω will be de- a creased from 180 ppb in E821 to 70 ppb by using an improved laser calibration, a segmented calorimeter, better collimator in the ring, and improved tracker. Systematics on ω will be decreased from 170 ppb in E821 to 70 ppb by improv- p ing the uniformity and monitoring of the magnetic ﬁeld, increasing accuracy of position determination of trolly, better temperature stability of the magnet, and providing active feedback to external ﬁelds. 3 Current status Muon g-2 performed a 5 week engineering run during June-July 2017. The ﬁrst beam was injected into the ring on May 31, 2017. The accelerator per- formed well during the run [13], though the beam was mostly protons with ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: quency

text: ω

text: via

text: the

text: timing

text: of

text: muon

text: decays

text: to

text: positrons

text: using

text: electromag-

text: a

text: netic

text: calorimeters

text: Measurements

text: of

text: the

text: precession

text: frequency

text: and

text: mag-

text: netic

text: ﬁeld

text: lead

text: to

text: a

text: The

text: anomalous

text: precession

text: frequency

text: ω

text: ω

text: ω

text: µ

text: a

text: s

text: c

text: where

text: the

text: Cyclotron

text: frequency

text: ω

text: e

text: B

text: and

text: the

text: spin

text: precession

text: frequency

text: c

text: mγ

text: ω

text: e

text: γa

text: s

text: mγ

text: µ

text: To

text: reduce

text: the

text: eﬀect

text: of

text: electric

text: ﬁelds,

text: the

text: muons

text: are

text: injected

text: at

text: a

text: magic

text: momentum

text: with

text: γ

text: which

text: cancels

text: the

text: second

text: term

text: in

text: Eq.

label: equation

text: −Qe

text: mc

text: [a

text: (a

text: a

text: m

text: µ

text: µ

text: p

text: c

text: which

text: leaves

text: −Qe

text: a

text: a

text: m

text: µ

text: The

text: proton

text: precession

text: frequency

text: ω

text: is

text: measured

text: as

text: a

text: proxy

text: for

text: leading

text: to

text: p

text: an

text: expression

text: for

text: the

text: anomalous

text: moment

text: as

text: ω

text: /ω

text: a

text: a−

text: p

text: µ

text: µ

text: /µ

text: ω

text: /ω

text: µ

text: p

text: a

text: p

text: The

text: Measurement

text: of

text: ω

text: is

text: performed

text: using

text: NMR.

text: Fixed

text: NMR

text: probes

text: mea-

text: p

text: sure

text: time

text: variations

text: of

text: the

text: ﬁeld

text: during

text: data

text: taking.

text: A

text: trolley

text: with

text: mounted

text: NMR

text: probes

text: periodically

text: circumnavigates

text: the

text: interior

text: of

text: the

text: ring

text: to

text: perform

text: precision

text: measurements

text: of

text: the

text: ﬁeld

text: in

text: the

text: muon

text: storage

text: region,

text: performing

text: magnetic

text: ﬁeld

text: measurements

text: per

text: trolley

text: run.

text: Probes

text: are

text: calibrated

text: to

text: provide

text: measurement

text: to

text: ppb.

text: We

text: plan

text: to

text: collect

text: times

text: the

text: BNL

text: statistics,

text: which

text: will

text: reduce

text: our

text: sta-

text: tistical

text: uncertainty

text: by

text: a

text: factor

text: of

text: four.

text: To

text: reduce

text: systematic

text: uncertainty,

text: ac-

text: celerator

text: facilities

text: will

text: have

text: p

text: closer

text: to

text: magic

text: momentum,

text: utilize

text: a

text: longer

text: π

text: decay

text: channel,

text: and

text: increase

text: injection

text: eﬃciency.

text: Systematics

text: on

text: ω

text: will

text: be

text: de-

text: a

text: creased

text: from

text: ppb

text: in

text: to

text: ppb

text: by

text: using

text: an

text: improved

text: laser

text: calibration,

text: a

text: segmented

text: calorimeter,

text: better

text: collimator

text: in

text: the

text: ring,

text: and

text: improved

text: tracker.

text: Systematics

text: on

text: ω

text: will

text: be

text: decreased

text: from

text: ppb

text: in

text: to

text: ppb

text: by

text: improv-

text: p

text: ing

text: the

text: uniformity

text: and

text: monitoring

text: of

text: the

text: magnetic

text: ﬁeld,

text: increasing

text: accuracy

text: of

text: position

text: determination

text: of

text: trolly,

text: better

text: temperature

text: stability

text: of

text: the

text: magnet,

text: and

text: providing

text: active

text: feedback

text: to

text: external

text: ﬁelds.

text: Current

text: status

text: Muon

text: performed

text: a

text: week

text: engineering

text: run

text: during

text: June-July

text: The

text: ﬁrst

text: beam

text: was

text: injected

text: into

text: the

text: ring

text: on

text: May

text: The

text: accelerator

text: per-

text: formed

text: well

text: during

text: the

text: run

text: though

text: the

text: beam

text: was

text: mostly

text: protons

text: with

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 14 YUQIU FU AND DANIEL TATARU estimate (3.13) proved later. One may also prove (3.12) directly by modifying our proof for (3.1). By the atomic structure of Up spaces, and embedding (2.1), we obtain the following. ≤ Corollary 1. For every k M + 1, we have (cid:107) (cid:107) (cid:46) − (cid:107) (cid:107) P ≤·≤ u 2 kβ/8 P ≤·≤ u , k M+1 L4 k M+1 U4 t,x h(D) and (cid:107) (cid:107) (cid:46) − (cid:107) (cid:107) P ≤·≤ u 2 kβ/8 P ≤·≤ u . k M+1 L4 k M+1 V 2 t,x h(D) The second part of the section is devoted to bilinear L2 estimates, which exploit transver- sality in the bilinea interaction of waves, rather than dispersion. As a starting point for our analysis, we utilize the following well-known bilinear estimate. The proof given here is adapted from [Bou98]. (cid:48) R Proposition 4. Suppose h(ξ), h (ξ) are smooth functions on some open subset Ω of 2 and (cid:48) (cid:98) (cid:48) Ω Ω. Let Ω , Ω be two open subset of Ω . Suppose 1 2 |∇ − ∇ (cid:48) | θ := inf h(ξ) h (η) > 0 ∈ ∈ ξ Ω1,η Ω2 and { ∈ − ∈ − (cid:48) − } ∞ l = sup meas (h(ξ), ξ) : ξ Ω , ξ ξ Ω , τ h(ξ) = h (ξ ξ) < .1 ∈R×R 1 1 0 2 0 0 (τ0,ξ0) 2 Then we have (cid:107) (cid:48) (cid:107) ≤ − (cid:107) (cid:107) (cid:107) (cid:107) (eith(D)u)(eith (D)v) θ 1/2l1/2 u v L2 L2 L2 ∈ ⊂ ⊂t,x x x for every u, v L2 with supp uˆ Ω , supp vˆ Ω . x 1 2 (cid:90) Proof. By Plancherel’s theorem and the Cauchy-Schwartz inequality (applied to the measure − (cid:48) − (cid:48) − (cid:48) (cid:48) δ (τ h (ξ ξ ) h(ξ ))dξ ), we have 0 (cid:107) (cid:18)(cid:90) (cid:90) (cid:90) (cid:48) (cid:107) (cid:107) − (cid:48) − (cid:48) − (cid:48) −(cid:19)(cid:48) (cid:48) (cid:48)(cid:107) (eith(D)u)(eith (D)v) 2 = δ (τ h (ξ ξ ) h(ξ ))uˆ(ξ ξ )vˆ(ξ )dξ 2 L2 0 L2 t,x τ,ξ ≤ (cid:18)|(cid:90) − (cid:48) | | (cid:48) | − (cid:48) − (cid:48) − (cid:48) (cid:48) (cid:19) uˆ(ξ ξ ) 2 vˆ(ξ ) 2δ (τ h (ξ ξ ) h(ξ ))dξ dξdτ 0 × (cid:32)(cid:90)(cid:48) − (cid:48) − (cid:48) − (cid:48) − (cid:48) (cid:48) (cid:33) sup 1 (ξ )1 (ξ ξ )δ (τ h (ξ ξ ) h(ξ ))dξ Ω1 Ω2 0 τ,ξ (cid:48) ≤ (cid:107) (cid:107) (cid:107) (cid:107) (cid:48) − (cid:48) dµ(ξ ) u 2L2x(cid:32)(cid:90)v 2L2x sτu,ξp h(cid:48)(ξ−ξ(cid:48))+h(ξ(cid:48))=τ 1Ω1(ξ )1Ω2(ξ ξ )|∇h(cid:48)(ξ − ξ(cid:48))(cid:33)− ∇h(ξ(cid:48))| . By our assumption we have (cid:48) (cid:48) − (cid:48) dµ(ξ ) ≤ − sup (cid:48) − (cid:48) (cid:48) 1Ω1(ξ )1Ω2(ξ ξ )|∇h(cid:48)(ξ − ξ(cid:48)) − ∇h(ξ(cid:48))| θ 1l. τ,ξ h (ξ ξ )+h(ξ )=τ (cid:3) Therefore we obtain the claimed estimate. 1The previous condition gurantees that the set being measured, which is the intersection of the graph of (cid:48) R h over Ω1 and an inverse translate of the graph of h over Ω2, is a 1-dimensional submanifold in 3. So here the 1-dimensional Haussdorﬀ measure meas1 just means the length of the curve. ##LTLine## ##LTLine## ##LTLine##

text: YUQIU

text: FU

text: AND

text: DANIEL

text: TATARU

text: estimate

text: proved

text: later.

text: One

text: may

text: also

text: prove

text: directly

text: by

text: modifying

text: our

text: proof

text: for

text: By

text: the

text: atomic

text: structure

text: of

text: Up

text: spaces,

text: and

text: embedding

text: we

text: obtain

text: the

text: following.

text: Corollary

text: For

text: every

text: k

text: M

text: we

text: have

text: P

text: u

text: P

text: u

text: k

text: k

text: t,x

text: h(D)

text: and

text: P

text: u

text: P

text: u

text: k

text: k

text: V

text: t,x

text: h(D)

text: The

text: second

text: part

text: of

text: the

text: section

text: is

text: devoted

text: to

text: bilinear

text: estimates,

text: which

text: exploit

text: transver-

text: sality

text: in

text: the

text: bilinea

text: interaction

text: of

text: waves,

text: rather

text: than

text: dispersion.

text: As

text: a

text: starting

text: point

text: for

text: our

text: analysis,

text: we

text: utilize

text: the

text: following

text: well-known

text: bilinear

text: estimate.

text: The

text: proof

text: given

text: here

text: is

text: adapted

text: from

text: R

text: Proposition

text: Suppose

text: h(ξ),

text: h

text: (ξ)

text: are

text: smooth

text: functions

text: on

text: some

text: open

text: subset

text: Ω

text: of

text: and

text: Ω

text: Ω.

text: Let

text: Ω

text: Ω

text: be

text: two

text: open

text: subset

text: of

text: Ω

text: Suppose

text: θ

text: inf

text: h(ξ)

text: h

text: (η)

text: ξ

text: Ω1,η

text: Ω2

text: and

text: l

text: sup

text: meas

text: (h(ξ),

text: ξ)

text: ξ

text: Ω

text: ξ

text: ξ

text: Ω

text: τ

text: h(ξ)

text: h

text: (ξ

text: ξ)

text: ∈R×R

text: Then

text: we

text: have

text: (eith(D)u)(eith

text: (D)v)

text: θ

text: u

text: v

text: ⊂t,x

text: x

text: x

text: for

text: every

text: u,

text: v

text: with

text: supp

text: uˆ

text: Ω

text: supp

text: vˆ

text: Ω

text: x

text: Proof.

text: By

text: Plancherel’s

text: theorem

text: and

text: the

text: Cauchy-Schwartz

text: inequality

text: (applied

text: to

text: the

text: measure

text: δ

text: (τ

text: h

text: (ξ

text: ξ

text: h(ξ

text: ))dξ

text: we

text: have

text: (eith(D)u)(eith

text: (D)v)

text: δ

text: (τ

text: h

text: (ξ

text: ξ

text: h(ξ

text: ))uˆ(ξ

text: ξ

text: )vˆ(ξ

text: )dξ

text: t,x

text: τ,ξ

text: uˆ(ξ

text: ξ

text: vˆ(ξ

text: 2δ

text: (τ

text: h

text: (ξ

text: ξ

text: h(ξ

text: ))dξ

text: dξdτ

text: sup

text: (ξ

text: (ξ

text: ξ

text: )δ

text: (τ

text: h

text: (ξ

text: ξ

text: h(ξ

text: ))dξ

text: Ω1

text: Ω2

text: τ,ξ

text: dµ(ξ

text: u

text: sτu,ξp

text: ξ

text: By

text: our

text: assumption

text: we

text: have

text: dµ(ξ

text: sup

text: ξ

text: θ

text: τ,ξ

text: h

text: (ξ

text: ξ

text: )+h(ξ

text: )=τ

text: Therefore

text: we

text: obtain

text: the

text: claimed

text: estimate.

text: previous

text: condition

text: gurantees

text: that

text: the

text: set

text: being

text: measured,

text: which

text: is

text: the

text: intersection

text: of

text: the

text: graph

text: of

text: R

text: h

text: over

text: Ω1

text: and

text: an

text: inverse

text: translate

text: of

text: the

text: graph

text: of

text: h

text: over

text: is

text: a

text: submanifold

text: in

text: So

text: here

text: the

text: Haussdorﬀ

text: measure

text: just

text: means

text: the

text: length

text: of

text: the

text: curve.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 16 YUQIU FU AND DANIEL TATARU ≥ Case 1 k (β + 1)k . 1 2 ≤ ≤ − We let L = (β + 1)k . For L m 2 we let Q be the Fourier multiplier given by the 2 m,j symbol 1 ≤ | | (ξ), j2m∼2π ξ/ ξ <(j+1)2m2π S R Z where we(cid:13) used the identiﬁcation(cid:13) 1 = /2π . Put another way, Q (ξ) = 1 if and only if | | (cid:13) ∈ (cid:13) m,j ξ = ξ eiθ(cid:13)for some θ [j2m2π, ((cid:13)j + 1)2m2π). Then by the triangle inequality we have (3.16) (eit(cid:88)h(D)u(cid:88))(eith(D)R v) (cid:88) k L2 − − − ≤ 2 2 L 1 (cid:107) (cid:107) (eith(D)Q u)(eith(D)Q (cid:48)R v) (cid:88) (cid:88) m,j m,j k L2 (cid:48) | (cid:48)− | { − − } m=L j=0 j :1< j j <min 6,2 m 1 − − 2 L 1 (cid:107) (cid:107) + (eith(D)Q u)(eith(D)Q (cid:48)R v) m,j m,j k L2 (cid:48) | (cid:48)− |≤ j=0 j : j j 1 We deﬁne (cid:8) (cid:9) { ∈ − | | } U := ξ Ω : (j 1/3)2m2π < ξ/ ξ < (j + 4/3)2m2π m,j 1 ∈ (cid:48) − | | (cid:48) V (cid:48) := ξ Ωk : (j 1/3)2m2π < ξ/ ξ < (j + 4/3)2m2π . m,j Then U , V (cid:48) contain the supp(cid:12)ort of Q (ξ)uˆ(ξ) and(cid:12)Q (cid:48)(ξ)R (ξ)vˆ(ξ) respectively. m,j m,j ∈ (cid:48) ∈(cid:12) m,j (cid:12) m,j k Note that when ξ Ω1 and ξ (cid:12)(cid:12) Ωk we have (cid:12)(cid:12) (cid:48) |∇ − ∇ (cid:48) | (cid:12)(cid:12) (cid:48) | | (cid:18)ξ − (cid:48) | (cid:48)|(cid:19)ξ (cid:12)(cid:12) h(ξ) h(ξ ) = (cid:12)γ ( ξ )| | γ ( ξ )| (cid:48)| (cid:12) (cid:12) ξ ξ (cid:12) (cid:48) (cid:48) (cid:12)(cid:12) (cid:48) | | ξ(cid:12)(cid:12) − ξ (cid:48) | | − (cid:48) | (cid:48)| ξ = (cid:12)γ ( ξ ) |(cid:12)| | (cid:48)| + (γ ( ξ ) γ ( ξ )) | (cid:48)| (cid:12) ξ(cid:12) ξ ξ (cid:48) (cid:38) ξ − ξ | | | (cid:48)| + 2(β+1)k2 ξ ξ (cid:12) (cid:12) (cid:48) | | (cid:48) | (cid:48)| ∼ (cid:12)(cid:90) (cid:12) because γ ( ξ ), γ ( ξ ) 1 and (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) | | (cid:12) | (cid:48) | | − (cid:48) | (cid:48)| | ξ (cid:48)(cid:48) ∼ γ ( ξ ) γ ( ξ ) = γ (r)dr 2(β+1)k2. | (cid:48)| ξ (cid:12) (cid:12) | − (cid:48)| So if we let θ be the parameter in Propos(cid:12)ition 4 for(cid:12) U and V (cid:48) then when 1 < j j < 6 ≤ ≤ − (cid:12) (cid:12) m,j m,j and L m 4, (cid:12) (cid:12) (cid:48) (cid:38) ξ − ξ (cid:38) θ | | | (cid:48)| 2m ξ ξ | (cid:48) − | ≤ and when j j 1 and m = L, (cid:38) θ 2(β+1)k2 = 2L. We let l be the parameter in Proposition 4 for U and V (cid:48). Since h has bounded − || | − | ≤ m,j m,j C3 norm on the annulus ξ 1 2M+3, we have ∼ { ∈ − ∈ − } l sup meas ξ U : η ξ V (cid:48), h(η ξ) + h(ξ) = τ . 1 m,j m,j τ,η We let γ be the curve τ,η { ∈ − ∈ − } ξ U : η ξ V (cid:48), h(η ξ) + h(ξ) = τ . m,j m,j ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: YUQIU

text: FU

text: AND

text: DANIEL

text: TATARU

text: Case

text: k

text: (β

text: We

text: let

text: L

text: (β

text: For

text: L

text: m

text: we

text: let

text: Q

text: be

text: the

text: Fourier

text: multiplier

text: given

text: by

text: the

text: m,j

text: symbol

text: (ξ),

text: ξ/

text: ξ

text: S

text: R

text: Z

text: where

text: used

text: the

text: Put

text: another

text: way,

text: Q

text: (ξ)

text: if

text: and

text: only

text: if

text: m,j

text: ξ

text: ξ

text: some

text: θ

text: Then

text: by

text: the

text: triangle

text: inequality

text: we

text: have

text: v)

text: k

text: L

text: (eith(D)Q

text: u)(eith(D)Q

text: v)

text: m,j

text: m,j

text: k

text: m=L

text: j

text: j

text: j

text: <min

text: m

text: L

text: (eith(D)Q

text: u)(eith(D)Q

text: v)

text: m,j

text: m,j

text: k

text: j

text: j

text: j

text: We

text: deﬁne

text: U

text: ξ

text: Ω

text: (j

text: ξ/

text: ξ

text: (j

text: m,j

text: V

text: ξ

text: Ωk

text: (j

text: ξ/

text: ξ

text: (j

text: m,j

text: Then

text: U

text: V

text: contain

text: the

text: of

text: Q

text: (ξ)uˆ(ξ)

text: (ξ)vˆ(ξ)

text: respectively.

text: m,j

text: m,j

text: m,j

text: m,j

text: k

text: Note

text: that

text: when

text: ξ

text: Ω1

text: and

text: ξ

text: Ωk

text: we

text: have

text: h(ξ)

text: h(ξ

text: ξ

text: γ

text: ξ

text: ξ

text: ξ

text: ξ

text: ξ

text: ξ

text: (γ

text: ξ

text: γ

text: ξ

text: ξ

text: ξ

text: ξ

text: ξ

text: ξ

text: ξ

text: because

text: γ

text: ξ

text: γ

text: ξ

text: and

text: ξ

text: γ

text: ξ

text: γ

text: ξ

text: γ

text: (r)dr

text: ξ

text: So

text: if

text: we

text: let

text: θ

text: be

text: the

text: parameter

text: in

text: U

text: and

text: V

text: then

text: when

text: j

text: j

text: m,j

text: m,j

text: and

text: L

text: m

text: ξ

text: ξ

text: θ

text: ξ

text: ξ

text: and

text: when

text: j

text: j

text: and

text: m

text: L,

text: θ

text: We

text: let

text: l

text: be

text: the

text: parameter

text: in

text: Proposition

text: for

text: U

text: and

text: V

text: Since

text: h

text: has

text: bounded

text: m,j

text: m,j

text: norm

text: on

text: the

text: annulus

text: ξ

text: we

text: have

text: l

text: sup

text: meas

text: ξ

text: U

text: η

text: ξ

text: V

text: h(η

text: ξ)

text: h(ξ)

text: τ

text: m,j

text: m,j

text: τ,η

text: We

text: let

text: γ

text: be

text: the

text: curve

text: τ,η

text: ξ

text: U

text: η

text: ξ

text: V

text: h(η

text: ξ)

text: h(ξ)

text: τ

text: m,j

text: m,j

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: NULL STRUCTURES AND DEGENERATE DISPERSION RELATIONS IN TWO SPACE DIMENSIONS 17 ∈ R ∈ R Fix η 2 and τ . To estimate the length of γ we examine its deﬁning function τ,η − f(ξ) := h(η ξ) + h(ξ). Then − ∇ (cid:48) | | ξ − (cid:48) | − | η ξ (cid:18) (cid:19) (3.17) f(ξ) = γ ( ξ )| | γ ( η ξ )| − | ξ η ξ − (cid:48) | | − (cid:48) | − | ξ (cid:48) | − | ξ − η ξ = (γ ( ξ ) γ ( η ξ ))| | + γ ( η ξ ) | | | − | . ξ ξ η ξ (cid:18) (cid:19) ∇ ∇ ∇ We write f = f + f where r θ ∇ ∇ · ξ ∇ ∇ − ∇ f(ξ) = f | | ξ, f = f f r ξ θ r ∇ ∈ − ∈ are the radial and tangential parts of f respectively. Note that when ξ U and η ξ m,j V (cid:48), m,j − (cid:48) | | (cid:48) | − | ∼ (cid:48) | − | ∼ γ ( ξ ) + γ ( η ξ ) 2(β+1)k2, γ ( η ξ ) 1  ≥ | − (cid:48)| So by (3.17) and our assumption k (β + 1)k we know that when 1 < j j < 6 1 2 |∇ | (cid:46) ≤ − 1 if k /2 + (β + 1)k /2 < m 4, f(ξ) 1 2 |∇r | (cid:38) 2k2(β+1) ≤ ≤ θf(ξ) 2m if L m k1/2 + (β + 1)k2/2. ∈ − ∈ for every ξ U satisfying η ξ V (cid:48). So if we let l be the parameter in Proposition 4 m,j | −m,j(cid:48)| for U and V (cid:48), then2 when 1 < j j < 6, m,j m,j  ≤ − (cid:46) 2k1 if k1/2 + (β + 1)k2/2 < m 4, l 2m ≤ ≤ 2m if L m k /2 + (β + 1)k /2 2k2(β+1) 1 2 | (cid:48) − | ≤ and when j j 1 and m = L = (β + 1)k , 2 (cid:46) l 2L = 2(β+1)k2. (cid:13) (cid:13) | − (cid:48)| Ap(cid:13)plying Proposition 4 we obtain wh(cid:13)en 1 < j j < 6, (cid:13) (cid:13) (cid:40) (eith(D)Q u)(eith(D)Q (cid:48)R v) m,j m,j k L2 − (cid:107) (cid:107) (cid:107) (cid:107) ≤ − (cid:46) 2k1−/(cid:40)22 m/2 Q−m,ju (cid:107)L2 Qm(cid:107),j(cid:48)R(cid:107)kv L2 (cid:107) if k1≤/2 + ≤(β + 1)k2/2 < m 4, 2m k2(β+1)/22 m/2 Qm,ju L2 Qm,j(cid:48)Rkv L2 if L m k1/2 + (β + 1)k2/2 − (cid:107) (cid:107) (cid:107) (cid:107) ≤ − = 2k1/22−m/2 Qm,(cid:107)ju L2 Q(cid:107) m,(cid:107)j(cid:48)Rkv L2 (cid:107) if k1≤/2 + ≤(β + 1)k2/2 < m 4, 2m/22 k2(β+1)/2 Qm,ju L2 Qm,j(cid:48)Rkv L2 if L m k1/2 + (β + 1)k2/2 2To be completely rigorous we need to show that the number of components of γ is bounded by some τ,ξ admissible constant. See the proof of Lemma 2 below. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: STRUCTURES

text: AND

text: DEGENERATE

text: DISPERSION

text: RELATIONS

text: IN

text: TWO

text: SPACE

text: DIMENSIONS

text: R

text: R

text: Fix

text: η

text: and

text: τ

text: To

text: estimate

text: the

text: length

text: of

text: γ

text: we

text: examine

text: its

text: deﬁning

text: function

text: τ,η

text: f(ξ)

text: h(η

text: ξ)

text: h(ξ).

text: Then

text: ξ

text: η

text: ξ

text: f(ξ)

text: γ

text: ξ

text: γ

text: η

text: ξ

text: ξ

text: η

text: ξ

text: ξ

text: ξ

text: η

text: ξ

text: (γ

text: ξ

text: γ

text: η

text: ξ

text: γ

text: η

text: ξ

text: ξ

text: ξ

text: η

text: ξ

text: We

text: write

text: f

text: f

text: f

text: where

text: r

text: θ

text: ξ

text: f(ξ)

text: f

text: ξ,

text: f

text: f

text: f

text: r

text: ξ

text: θ

text: r

text: are

text: the

text: radial

text: and

text: tangential

text: parts

text: of

text: f

text: respectively.

text: Note

text: that

text: when

text: ξ

text: U

text: and

text: η

text: ξ

text: m,j

text: V

text: m,j

text: γ

text: ξ

text: γ

text: η

text: ξ

text: γ

text: η

text: ξ

text: So

text: by

text: and

text: our

text: assumption

text: k

text: (β

text: we

text: know

text: that

text: when

text: j

text: j

text: if

text: k

text: (β

text: m

text: f(ξ)

text: |∇r

text: θf(ξ)

text: if

text: L

text: m

text: (β

text: for

text: every

text: ξ

text: U

text: satisfying

text: η

text: ξ

text: V

text: So

text: if

text: we

text: let

text: l

text: be

text: the

text: parameter

text: in

text: Proposition

text: m,j

text: for

text: U

text: and

text: V

text: when

text: j

text: j

text: m,j

text: m,j

text: if

text: (β

text: m

text: l

text: if

text: L

text: m

text: k

text: (β

text: and

text: when

text: j

text: j

text: and

text: m

text: L

text: (β

text: l

text: Proposition

text: we

text: obtain

text: j

text: j

text: (eith(D)Q

text: u)(eith(D)Q

text: v)

text: m,j

text: m,j

text: k

text: Q−m,ju

text: if

text: ≤(β

text: m

text: Qm,ju

text: if

text: L

text: m

text: (β

text: if

text: ≤(β

text: m

text: Qm,ju

text: if

text: L

text: m

text: (β

text: be

text: completely

text: rigorous

text: we

text: need

text: to

text: show

text: that

text: the

text: number

text: of

text: components

text: of

text: γ

text: is

text: bounded

text: by

text: some

text: τ,ξ

text: admissible

text: constant.

text: See

text: the

text: proof

text: of

text: Lemma

text: below.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 22 YUQIU FU AND DANIEL TATARU ≤ − ≤ Corollary 2. Suppose k k 10 and k M + 1. Then we have 1 2 2 (cid:107) (cid:107) (cid:46) − − (cid:107) (cid:107) (cid:107) (cid:107) (3.26) P≤ uP ≤·≤ v 2 βk2/42(k1 k2)/4 P≤ u P ≤·≤ v . k1 k2 M+1 L2t,x k1 Uh2(D) k2 M+1 Uh2(D) − The right hand side of (3.26) is evaluated in U2 based spaces, which, as we shall see later, is insuﬃcient for our ﬁxed point argument. However, we can use the following interpolation − − result from [HHK09] to transfer from Up based estimates to V p based estimates, provided − we have an additional Uq based estimate for some q > p. Another way of transferring from Up to V p can be found in [CH16], which is not easy to implement here. → Proposition 6. Suppose 1 < p < q. Let E be a Banach space and T : Uq E be a linear ∈ bounded operator with operator norm C . Su(cid:18)ppose in add(cid:19)ition there exists C (0, C ] such (cid:107) (cid:107) ≤ (cid:107) (cid:107) q ∈ p ∈q that the estimate Tu C u holds for all u Up. Then we have for every u V p E p Up rc (cid:107) (cid:107) ≤ C (cid:107) (cid:107) Tu C C log q + 1 u , E p,q p C V p p where C is a constant depending only on p, q. p,q − In our case we already have a U2 based estimate in Corollary 2. We can also obtain a − U4 based estimate from Proposition 3. Indeed, under the same condition as in Corollary 2, by H¨older’s inequality we have (cid:107) (cid:107) (cid:46) (cid:107) (cid:107) (cid:107) (cid:107) P uP ≤·≤ v P u P ≤·≤ v k1 k2 M+1 L2t,x (cid:46) −k1 L4t,x (cid:107)k2 (cid:107)M+1 (cid:107)L4t,x (cid:107) 2 β(k1+k2)/8 Pk1u Uh4(D) Pk2≤·≤M+1v Uh4(D). ≤ − Therefore applying Proposition 6 twice4, and noticing the assumption that k k 10, we 1 2 obtain (cid:107) (cid:107) (cid:46) − − | − | (cid:107) (cid:107) (cid:107) (cid:107) Pk1uPk2≤·≤M+1v L2t,x 2 βk2/42(k1 k2)/4 k2 k1 2 Pk1u Vh2(D) Pk2≤·≤M+1v Vh2(D) ∈ if P u, P ≤·≤ v V 2 . Summing the above inequality over k and using the triangle k1 k2 M+1 h(D),rc 1 inequality we obtain the following corollary: ≤ − ≤ ∈ Corollary 3. Suppose k k 10 and k M + 1. If P≤ u, P ≤·≤ v V 2 , then 1 2 2 k1 k2 M+1 rc,h(D) (cid:107) (cid:107) (cid:46) − (cid:107) (cid:107) (cid:107) (cid:107) (3.27) P≤k1uPk2≤·≤M+1v L2t,x 2 βk2/4 P≤k1u Vh2(D) Pk2≤·≤M+1v Vh2(D) The localized linear and bilinear estimates discussed above allow us to obtain the following crucial estimate on the nonlinear ter(cid:90)m, whose proof will occupy the next section. We recall that I is the operator given by T t − I (f) = 1 (s)ei(t s)h(D)f(s)ds. T [0,T) 0 4In fact we apply Proposition 6 twice to obtain the estimate for enlarged projections: (cid:107) (cid:107) (cid:46) − − | − | (cid:107) (cid:107) (cid:107) (cid:107) Pk1−2≤·≤k1+2uPk2−2≤·≤M+3v L2t,x 2 βk2/42(k1 k2)/4 k2 k1 2 u Vh2(D) v Vh2(D). Then we substitute u and v by P u and P ≤·≤ v respectively. k1 k2 M+1 ##LTLine## ##LTLine##

text: YUQIU

text: FU

text: AND

text: DANIEL

text: TATARU

text: Corollary

text: Suppose

text: k

text: k

text: and

text: k

text: M

text: Then

text: we

text: have

text: P≤

text: uP

text: v

text: P≤

text: u

text: P

text: v

text: The

text: right

text: hand

text: side

text: of

text: is

text: evaluated

text: in

text: based

text: spaces,

text: which,

text: as

text: we

text: shall

text: see

text: later,

text: is

text: insuﬃcient

text: for

text: our

text: ﬁxed

text: point

text: argument.

text: However,

text: we

text: can

text: use

text: the

text: following

text: interpolation

text: result

text: from

text: to

text: transfer

text: from

text: Up

text: based

text: estimates

text: to

text: V

text: p

text: based

text: estimates,

text: provided

text: we

text: have

text: an

text: additional

text: Uq

text: based

text: estimate

text: for

text: some

text: q

text: p.

text: Another

text: way

text: of

text: transferring

text: from

text: Up

text: to

text: V

text: p

text: can

text: be

text: found

text: in

text: which

text: is

text: not

text: easy

text: to

text: implement

text: here.

text: Proposition

text: Suppose

text: p

text: q.

text: Let

text: E

text: be

text: a

text: Banach

text: space

text: and

text: T

text: Uq

text: E

text: be

text: a

text: linear

text: bounded

text: operator

text: with

text: operator

text: norm

text: C

text: in

text: there

text: exists

text: C

text: C

text: such

text: q

text: p

text: ∈q

text: that

text: the

text: estimate

text: Tu

text: C

text: u

text: holds

text: for

text: all

text: u

text: Up.

text: Then

text: we

text: have

text: for

text: every

text: u

text: V

text: p

text: E

text: p

text: Up

text: rc

text: C

text: Tu

text: C

text: C

text: log

text: q

text: u

text: E

text: p,q

text: p

text: C

text: V

text: p

text: p

text: where

text: C

text: is

text: a

text: constant

text: depending

text: only

text: on

text: p,

text: q.

text: p,q

text: In

text: our

text: case

text: we

text: already

text: have

text: a

text: based

text: estimate

text: in

text: Corollary

text: We

text: can

text: also

text: obtain

text: a

text: based

text: estimate

text: from

text: Proposition

text: Indeed,

text: under

text: the

text: same

text: condition

text: as

text: in

text: Corollary

text: by

text: H¨older’s

text: inequality

text: we

text: have

text: P

text: uP

text: v

text: P

text: u

text: P

text: v

text: Therefore

text: applying

text: Proposition

text: and

text: noticing

text: the

text: assumption

text: that

text: k

text: k

text: we

text: obtain

text: if

text: P

text: u,

text: P

text: v

text: V

text: Summing

text: the

text: above

text: inequality

text: over

text: k

text: and

text: using

text: the

text: triangle

text: h(D),rc

text: inequality

text: we

text: obtain

text: the

text: following

text: corollary:

text: Corollary

text: Suppose

text: k

text: k

text: and

text: k

text: M

text: If

text: P≤

text: u,

text: P

text: v

text: V

text: then

text: rc,h(D)

text: The

text: localized

text: linear

text: and

text: bilinear

text: estimates

text: discussed

text: above

text: allow

text: us

text: to

text: obtain

text: the

text: following

text: crucial

text: estimate

text: on

text: the

text: nonlinear

text: whose

text: proof

text: will

text: occupy

text: the

text: next

text: section.

text: We

text: recall

text: that

text: I

text: is

text: the

text: operator

text: given

text: by

text: T

text: t

text: I

text: (f)

text: (s)ei(t

text: s)h(D)f(s)ds.

text: T

text: fact

text: we

text: apply

text: Proposition

text: twice

text: to

text: obtain

text: the

text: estimate

text: for

text: enlarged

text: projections:

text: u

text: v

text: Then

text: we

text: substitute

text: u

text: and

text: v

text: by

text: P

text: u

text: and

text: P

text: v

text: respectively.

text: ##LTLine##

text: ##LTLine##


text: NULL STRUCTURES AND DEGENERATE DISPERSION RELATIONS IN TWO SPACE DIMENSIONS 9 ≤ As a consequence(cid:13), for k M + 1 we (cid:13)have (cid:13) (cid:13) (cid:13) (cid:13) (cid:46) − (cid:107) (cid:107) (3.2) eith(D)P ≤·≤ u 2 βk/8 P ≤·≤ u . k M+1 0 L4 k M+1 0 L2x t,x Before proving this proposition, we ﬁrst establish the following localized dispersive estimate. Lemma 1. Let χk(D) be t(cid:13)he Fourier multip(cid:13)lier Pk−2≤·≤k+2. Then we have (cid:13) (cid:13) (cid:13) (cid:13) − (cid:46) 2 βk/2 (cid:107) (cid:107) (3.3) eith(D)χ2(D)u | | u . k 0 L∞ t 0 L1x x − ± Proof. We have χ (D) = χ+(D) + χ (D) where χ (D) are Fourier multipliers with sym- ± k k k k(cid:107) (cid:107) bols P − ≤·≤ (ξ). We will only establish (3.3) for eith(D)(χ+(D))2u ∞. The other part (cid:107) k 2 − k+2 (cid:107) k 0 Lx eith(D)(χ (D))2u ∞ can be estimated in the same way and the triangle inequality will k 0 Lx − give the estimate (3.3), as χ+(D)χ (D) = 0. For simplicity of notation we will still write k k χ (D), omitting the + sign. We ﬁrst observe that k ∗ eith(D)χ2(D(cid:90))u = K u  k 0 0 where · dξ K(t, x) = eith(ξ)χ2(ξ) = eix ξeith(ξ)χ2(ξ) . k R k (2π)2 2 Therefore by Young’s inequality if suﬃces to show that − (cid:107) (cid:107) (cid:46) 2 βk/2 K(t, x) ∞ | | . Lx t Since ∇ · (cid:48) | | ξ (th(ξ) + x ξ) = tγ ( ξ )| | + x, ξ ξ by our assumption (b) on γ in Section , there exists a small admissible constant c > 0 such that |∇ · | ≥ | | (3.4) (th(ξ) + x ξ) c x ξ | | ≤ | | | | ∈ − for every t c x and ξ [1 2M+3, 1 + 2M+3], and |∇ · | ≥ | | (3.5) (th(ξ) + x ξ) c t ξ | | − | | | | ∈ − for every t > c 1 x and ξ [1 2M+3, 1 + 2M+3]. Write x = (x , x ), and ξ = (ξ , ξ ). We consider three cases. 1 2 1 2 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: STRUCTURES

text: AND

text: DEGENERATE

text: DISPERSION

text: RELATIONS

text: IN

text: TWO

text: SPACE

text: DIMENSIONS

text: As

text: a

text: for

text: k

text: M

text: we

text: eith(D)P

text: u

text: P

text: u

text: k

text: k

text: t,x

text: Before

text: proving

text: this

text: proposition,

text: we

text: ﬁrst

text: establish

text: the

text: following

text: localized

text: dispersive

text: estimate.

text: Lemma

text: Let

text: χk(D)

text: be

text: Fourier

text: Then

text: we

text: have

text: u

text: k

text: L∞

text: t

text: x

text: Proof.

text: We

text: have

text: χ

text: (D)

text: χ+(D)

text: χ

text: (D)

text: where

text: χ

text: (D)

text: are

text: Fourier

text: multipliers

text: with

text: sym-

text: k

text: k

text: k

text: bols

text: P

text: (ξ).

text: We

text: will

text: only

text: establish

text: for

text: The

text: other

text: part

text: k

text: k

text: Lx

text: eith(D)(χ

text: can

text: be

text: estimated

text: in

text: the

text: same

text: way

text: and

text: the

text: triangle

text: inequality

text: will

text: k

text: Lx

text: give

text: the

text: estimate

text: as

text: χ+(D)χ

text: (D)

text: For

text: simplicity

text: of

text: notation

text: we

text: will

text: still

text: write

text: k

text: k

text: χ

text: (D),

text: omitting

text: the

text: sign.

text: We

text: ﬁrst

text: observe

text: that

text: k

text: K

text: u

text: k

text: where

text: dξ

text: K(t,

text: x)

text: eix

text: k

text: R

text: k

text: Therefore

text: by

text: Young’s

text: inequality

text: if

text: suﬃces

text: to

text: show

text: that

text: K(t,

text: x)

text: Lx

text: t

text: Since

text: ξ

text: (th(ξ)

text: x

text: ξ)

text: tγ

text: ξ

text: x,

text: ξ

text: ξ

text: by

text: our

text: assumption

text: (b)

text: on

text: γ

text: in

text: Section

text: there

text: exists

text: a

text: small

text: admissible

text: constant

text: c

text: such

text: that

text: (th(ξ)

text: x

text: ξ)

text: c

text: x

text: ξ

text: for

text: every

text: t

text: c

text: x

text: and

text: ξ

text: and

text: (th(ξ)

text: x

text: ξ)

text: c

text: t

text: ξ

text: for

text: every

text: t

text: c

text: x

text: and

text: ξ

text: Write

text: x

text: (x

text: x

text: and

text: ξ

text: (ξ

text: ξ

text: We

text: consider

text: three

text: cases.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ting), the performance of our proposed CAFE results on this MultiNLI to date (as of time of model is extremely competitive. We report the writing). On SciTail, Our proposed CAFE model test accuracies of CAFE at different extent of pa- achieves state-of-the-art performance. The perfor- rameterization, varying the size of the LSTM en- mance gain over strong baselines such as Decom- ≈ − coder, width of the pre-softmax hidden layers and pAtt and ESIM are 10% 13% in terms of ac- ﬁnal pooling layer. CAFE obtains 88.5% accu- curacy. CAFE also outperforms DGEM , which racy on the SNLI test set, an extremely compet- use a graph-based attention for improved perfor- itive score on the extremely popular benchmark. mance, by a signiﬁcant margin of 5%. As such, Notably, competitive results can be also achieved empirical results demonstrate the effectiveness of with a much smaller parameterization. For ex- our proposed CAFE model on the challenging Sc- ample, CAFE also achieves 88.3% and 88.1% iTail dataset. test accuracy with only 3.5M and 1.5M respec- tively. This outperforms state-of-the-art ESIM and MultiNLI Model Match Mismatch SciTail DIIN models with only a fraction of the parameter Majority 36.5 35.6 60.3 cost. Moreover, our lightweight adaption achieves NGRAM# - - 70.6 87.7% with only 750K parameters, which makes CBOW(cid:91) 65.2 64.8 - it extremely performant amongst models having BiLSTM(cid:91) 69.8 69.4 - ESIM#,(cid:91) 72.4 72.1 70.6 the same amount of parameters such as the decom- DecompAtt# - - - 72.3 posable attention model (86.8%). DGEM# - - 70.8 After removing the inter-attention layers from DGEM + Edge# - - 77.3 † CAFE, the performance remains competitive to ESIM 76.3 75.8 - † ESIM + Read 77.8 77.0 - top performing encoder models. CAFE (w/o cross CAFE 78.7 77.9 83.3 attention) achieves a respectible 85.9% accuracy CAFE Ensemble 80.2 79.0 - in this setting. Notably, the best performing Table 2: Performance comparison (accuracy) on MultiNLI model, the Gumbel TreeLSTM (86.0%) has over † and SciTail. Models with , # and (cid:91) are reported from (Weis- 10M parameters. On the other hand, our CAFE senborn, 2017), (Khot et al., 2018) and (Williams et al., 2017) model has a 300% less parameters, yet performs respectively. competitively to the Gumbel TreeLSTM. Finally, an ensemble of 5 CAFE models achieves 89.3% test accuracy, the best test scores 4.5 Ablation Study on the SNLI benchmark to date. Overall, we Table 3 reports ablation studies on the MultiNLI believe that the good performance of our CAFE development sets. Firstly, we replaced all FM can be attributed to (1) the effectiveness of the functions with regular full-connected (FC) layers compare-propagate (i.e., providing word represen- (1). We notice an decline in performance in both tations with global and local knowledge for better development sets. Ablation (2-3) explores the util- representation learning) and (2) the expressiveness ity of using character and syntactic embeddings, of factorization layers that are used to decompose which we ﬁnd to have helped CAFE marginally. and compare word alignments. More details are (4) Removes the Inter attention features, which given at the ablation study. Finally, we emphasize naturally impacts the model performance signiﬁ- that CAFE is also relatively lightweight, efﬁcient cantly. (5-6) explores the effectiveness of the high- and fast to train given its performance. A single way layers (in prediction layers and encoding lay- run on SNLI takes approximately 5 minutes per ers) by replacing them to FC layers. Both high- epoch with a batch size of 256. Overall, a single ≈ way layers have marginally helped overall perfor- run takes 3 hours to get to convergence. mance. Finally, (7-9) removes the alignment fea- 4.4 Results on MultiNLI and Scitail tures based on their composition type. We ob- Table 2 reports our results on the MultiNLI and serve that the Sub feat and Concat compositions SciTail datasets. On MultiNLI, CAFE signiﬁ- were more important than the Mul composition. cantly outperforms ESIM, a strong SoTA models However, removing any of the three will result in on both settings. We also outperform the ESIM some performance degrade. Finally, we replace + Read model (Weissenborn, 2017). An ensem- the LSTM encoder with a BiLSTM, observing ble of CAFE models achieve the best reported that adding bidirectionality did not improve per- ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ting),

text: the

text: performance

text: of

text: our

text: proposed

text: CAFE

text: results

text: on

text: this

text: MultiNLI

text: to

text: date

text: (as

text: of

text: time

text: of

text: model

text: is

text: extremely

text: competitive.

text: We

text: report

text: the

text: writing).

text: On

text: SciTail,

text: Our

text: proposed

text: CAFE

text: model

text: test

text: accuracies

text: of

text: CAFE

text: at

text: different

text: extent

text: of

text: pa-

text: achieves

text: state-of-the-art

text: performance.

text: The

text: perfor-

text: rameterization,

text: varying

text: the

text: size

text: of

text: the

text: LSTM

text: en-

text: mance

text: gain

text: over

text: strong

text: baselines

text: such

text: as

text: Decom-

text: coder,

text: width

text: of

text: the

text: pre-softmax

text: hidden

text: layers

text: and

text: pAtt

text: and

text: ESIM

text: are

text: in

text: terms

text: of

text: ac-

text: ﬁnal

text: pooling

text: layer.

text: CAFE

text: obtains

text: accu-

text: curacy.

text: CAFE

text: also

text: outperforms

text: DGEM

text: which

text: racy

text: on

text: the

text: SNLI

text: test

text: set,

text: an

text: extremely

text: compet-

text: use

text: a

text: graph-based

text: attention

text: for

text: improved

text: perfor-

text: itive

text: score

text: on

text: the

text: extremely

text: popular

text: benchmark.

text: mance,

text: by

text: a

text: signiﬁcant

text: margin

text: of

text: As

text: such,

text: Notably,

text: competitive

text: results

text: can

text: be

text: also

text: achieved

text: empirical

text: results

text: demonstrate

text: the

text: effectiveness

text: of

text: with

text: a

text: much

text: smaller

text: parameterization.

text: For

text: ex-

text: our

text: proposed

text: CAFE

text: model

text: on

text: the

text: challenging

text: Sc-

text: ample,

text: CAFE

text: also

text: achieves

text: and

text: iTail

text: dataset.

text: test

text: accuracy

text: with

text: only

text: and

text: respec-

text: tively.

text: This

text: outperforms

text: state-of-the-art

text: ESIM

text: and

text: MultiNLI

text: Model

text: Match

text: Mismatch

text: SciTail

text: DIIN

text: models

text: with

text: only

text: a

text: fraction

text: of

text: the

text: parameter

text: Majority

text: cost.

text: Moreover,

text: our

text: lightweight

text: adaption

text: achieves

text: NGRAM#

text: with

text: only

text: parameters,

text: which

text: makes

text: it

text: extremely

text: performant

text: amongst

text: models

text: having

text: the

text: same

text: amount

text: of

text: parameters

text: such

text: as

text: the

text: decom-

text: DecompAtt#

text: posable

text: attention

text: model

text: DGEM#

text: After

text: removing

text: the

text: inter-attention

text: layers

text: from

text: DGEM

text: Edge#

text: CAFE,

text: the

text: performance

text: remains

text: competitive

text: to

text: ESIM

text: ESIM

text: Read

text: top

text: performing

text: encoder

text: models.

text: CAFE

text: (w/o

text: cross

text: CAFE

text: attention)

text: achieves

text: a

text: respectible

text: accuracy

text: CAFE

text: Ensemble

text: in

text: this

text: setting.

text: Notably,

text: the

text: best

text: performing

text: Table

text: Performance

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text: Tim Rockta¨schel, Edward Grefenstette, Karl Moritz http://aclanthology.info/papers/E17-1038/neural- Hermann, Toma´sˇ Kocˇisky`, and Phil Blunsom. 2015. semantic-encoders. Reasoning about entailment with neural attention. arXiv preprint arXiv:1509.06664 . Lei Sha, Baobao Chang, Zhifang Sui, and Su- jian Li. 2016. Reading and thinking: Re- read LSTM unit for textual entailment recogni- tion. In COLING 2016, 26th International Con- ference on Computational Linguistics, Proceed- ings of the Conference: Technical Papers, Decem- ber 11-16, 2016, Osaka, Japan. pages 2870–2879. http://aclweb.org/anthology/C/C16/C16-1270.pdf. Tao Shen, Tianyi Zhou, Guodong Long, Jing Jiang, Shirui Pan, and Chengqi Zhang. 2017. Disan: Directional self-attention network for rnn/cnn-free language understanding. CoRR abs/1709.04696. http://arxiv.org/abs/1709.04696. Rupesh Kumar Srivastava, Klaus Greff, and Ju¨rgen Schmidhuber. 2015. Highway networks. CoRR abs/1505.00387. http://arxiv.org/abs/1505.00387. Shuohang Wang and Jing Jiang. 2016a. A compare- aggregate model for matching text sequences. CoRR abs/1611.01747. http://arxiv.org/abs/1611.01747. Shuohang Wang and Jing Jiang. 2016b. Learning natu- ral language inference with LSTM. In NAACL HLT 2016, The 2016 Conference of the North American Chapter of the Association for Computational Lin- guistics: Human Language Technologies, San Diego California, USA, June 12-17, 2016. pages 1442– 1451. http://aclweb.org/anthology/N/N16/N16- 1170.pdf. Zhiguo Wang, Wael Hamza, and Radu Florian. 2017. Bilateral multi-perspective matching for natural lan- guage sentences. In Proceedings of the Twenty- Sixth International Joint Conference on Artiﬁ- cial Intelligence, IJCAI 2017, Melbourne, Aus- tralia, August 19-25, 2017. pages 4144–4150. https://doi.org/10.24963/ijcai.2017/579. Dirk Weissenborn. 2017. Reading twice for natu- ral language understanding. CoRR abs/1706.02596. http://arxiv.org/abs/1706.02596. Adina Williams, Nikita Nangia, and Samuel R. Bow- man. 2017. A broad-coverage challenge corpus for sentence understanding through inference. CoRR abs/1704.05426. http://arxiv.org/abs/1704.05426. Jun Xiao, Hao Ye, Xiangnan He, Hanwang Zhang, Fei Wu, and Tat-Seng Chua. 2017. Attentional factor- ization machines: Learning the weight of feature interactions via attention networks. arXiv preprint arXiv:1708.04617 . Hong Yu and Tsendsuren Munkhdalai. 2017. Neu- ral semantic encoders. In Proceedings of the 15th Conference of the European Chapter of the Association for Computational Linguis- tics, EACL 2017, Valencia, Spain, April 3-7, 2017, Volume 1: Long Papers. pages 397–407.

text: Tim

text: Rockta¨schel,

text: Edward

text: Grefenstette,

text: Karl

text: Moritz

text: Hermann,

text: Toma´sˇ

text: Kocˇisky`,

text: and

text: Phil

text: Blunsom.

text: semantic-encoders.

text: Reasoning

text: about

text: entailment

text: with

text: neural

text: attention.

text: arXiv

text: preprint

text: Lei

text: Sha,

text: Baobao

text: Chang,

text: Zhifang

text: Sui,

text: and

text: Su-

text: jian

text: Li.

text: Reading

text: and

text: thinking:

text: Re-

text: read

text: LSTM

text: unit

text: for

text: textual

text: entailment

text: recogni-

text: tion.

text: In

text: COLING

text: International

text: Con-

text: ference

text: on

text: Computational

text: Linguistics,

text: Proceed-

text: ings

text: of

text: the

text: Conference:

text: Technical

text: Papers,

text: Decem-

text: ber

text: Osaka,

text: Japan.

text: pages

text: Tao

text: Shen,

text: Tianyi

text: Zhou,

text: Guodong

text: Long,

text: Jing

text: Jiang,

text: Shirui

text: Pan,

text: and

text: Chengqi

text: Zhang.

text: Disan:

text: Directional

text: self-attention

text: network

text: for

text: rnn/cnn-free

text: language

text: understanding.

text: CoRR

text: Rupesh

text: Kumar

text: Srivastava,

text: Klaus

text: Greff,

text: and

text: Ju¨rgen

text: Schmidhuber.

text: Highway

text: networks.

text: CoRR

text: Shuohang

text: Wang

text: and

text: Jing

text: Jiang.

text: A

text: compare-

text: aggregate

text: model

text: for

text: matching

text: text

text: sequences.

text: CoRR

text: Shuohang

text: Wang

text: and

text: Jing

text: Jiang.

text: Learning

text: natu-

text: ral

text: language

text: inference

text: with

text: LSTM.

text: In

text: NAACL

text: HLT

text: The

text: Conference

text: of

text: the

text: North

text: American

text: Chapter

text: of

text: the

text: Association

text: for

text: Computational

text: Lin-

text: guistics:

text: Human

text: Language

text: Technologies,

text: San

text: Diego

text: California,

text: USA,

text: June

text: pages

text: Zhiguo

text: Wang,

text: Wael

text: Hamza,

text: and

text: Radu

text: Florian.

text: Bilateral

text: multi-perspective

text: matching

text: for

text: natural

text: lan-

text: guage

text: sentences.

text: In

text: Proceedings

text: of

text: the

text: Twenty-

text: Sixth

text: International

text: Joint

text: Conference

text: on

text: Artiﬁ-

text: cial

text: Intelligence,

text: IJCAI

text: Melbourne,

text: Aus-

text: tralia,

text: August

text: pages

text: Dirk

text: Weissenborn.

text: Reading

text: twice

text: for

text: natu-

text: ral

text: language

text: understanding.

text: CoRR

text: Adina

text: Williams,

text: Nikita

text: Nangia,

text: and

text: Samuel

text: R.

text: Bow-

text: man.

text: A

text: broad-coverage

text: challenge

text: corpus

text: for

text: sentence

text: understanding

text: through

text: inference.

text: CoRR

text: Jun

text: Xiao,

text: Hao

text: Ye,

text: Xiangnan

text: He,

text: Hanwang

text: Zhang,

text: Fei

text: Wu,

text: and

text: Tat-Seng

text: Chua.

text: Attentional

text: factor-

text: ization

text: machines:

text: Learning

text: the

text: weight

text: of

text: feature

text: interactions

text: via

text: attention

text: networks.

text: arXiv

text: preprint

text: Hong

text: Yu

text: and

text: Tsendsuren

text: Munkhdalai.

text: Neu-

text: ral

text: semantic

text: encoders.

text: In

text: Proceedings

text: of

text: the

text: Conference

text: of

text: the

text: European

text: Chapter

text: of

text: the

text: Association

text: for

text: Computational

text: Linguis-

text: tics,

text: EACL

text: Valencia,

text: Spain,

text: April

text: Volume

text: Long

text: Papers.

text: pages


text: 1.1 Main contributions In this work we develop a new framework for CS PAT that allows to bring sparsity into play. Our approach is rooted in the concept of sparsifying temporal transforms developed for PAT in [30, 20] for two and three spatial dimensions. However, the approach in this present paper extends and simpliﬁes this transform approach considerably. First, it equally applies to any detection surface and arbitrary spatial dimension. Second, the new method can even be applied to heterogenous media. In order to achieve this, we use the second time derivative applied to the pressure data as a sparsifying transform. Opposed to [30, 20], where the transform was used to sparsify the measured signals, in the present work we exploit this for obtaining sparsity in the original imaging domain. Our new approach is based on the following. Consider the second time derivative ∂2p(r, t) of t the PA pressure. We will show, that this transformed pressure again satisﬁes the wave equation, however with the modiﬁed PA source c2∆ f in place of the original PA source f. If the original r PA source consists of smooth parts and jumps, the modiﬁed source consists of smooth parts and sparse structures; see Fig. 2 for an example. This enables the use of eﬃcient CS reconstruction algorithms based on sparse recovery. One possible approach is based on the following two-step (cid:39) procedure. First, recover an approximation h(r) ∆ f(r) via (cid:96)1-minimization. In a second r step, recover an approximation to f by solving the Poisson equation ∆ f = h/c2. r While the above two-stage approach turned out to very well recover the singularities of the PA source, at the same time it shows disturbing low-frequency artifacts. Therefore, in this paper we develop a completely novel strategy for jointly recovering f as well as its Laplacian. It is based on solving the constrained optimization problem (cid:107) (cid:107) (cid:2) (cid:3) (cid:2) (cid:3) min h + I (f) 1 C (f,h) − (cid:48)(cid:48) such that Mf, Mh, c2∆ f h = y, y , 0 . r Here M is the forward operator (including wave propagation and compressed measurements), (cid:107)·(cid:107) denotes the (cid:96)1-norm guaranteeing sparsity of h and I the indicator function of the positive 1 (cid:44) { | ≥ } C cone C f f(r) 0 guaranteeing non-negativity. In the case of noisy data we consider a penalized version that can be eﬃciently solved via various modern optimization techniques, such as forward backward splitting. 1.2 Outline In Section we describe PAT and existing CS approaches. We thereby focus on the role of sparsity in PAT. The proposed framework for CS PAT will be presented in Section . This includes the sparsiﬁcation of the PA source and the joint reconstruction algorithm. Numerical and experimental results are presented in Section . The paper concludes with a summary and outlook on future research presented in Section . 2 Compressed photoacoustic tomography 2.1 Photoacoustic tomography ⊆ R Suppose that V d is a bounded region and let r for i = 1, . . . , n denote admissible detector i locations distributed on the boundary ∂V . Then, for a given source f supported inside V , we deﬁne (cid:44) ∈ (Wf)(r , t) p(r , t) for i = 1, . . . , n and t [0, T ] , (2) i i 3

label: section

text: Main

text: contributions

text: In

text: this

text: work

text: we

text: develop

text: a

text: new

text: framework

text: for

text: CS

text: PAT

text: that

text: allows

text: to

text: bring

text: sparsity

text: into

text: play.

text: Our

text: approach

text: is

text: rooted

text: in

text: the

text: concept

text: of

text: sparsifying

text: temporal

text: transforms

text: developed

text: for

text: PAT

text: in

text: for

text: two

text: and

text: three

text: spatial

text: dimensions.

text: However,

text: the

text: approach

text: in

text: this

text: present

text: paper

text: extends

text: and

text: simpliﬁes

text: this

text: transform

text: approach

text: considerably.

text: First,

text: it

text: equally

text: applies

text: to

text: any

text: detection

text: surface

text: and

text: arbitrary

text: spatial

text: dimension.

text: Second,

text: the

text: new

text: method

text: can

text: even

text: be

text: applied

text: to

text: heterogenous

text: media.

text: In

text: order

text: to

text: achieve

text: this,

text: we

text: use

text: the

text: second

text: time

text: derivative

text: applied

text: to

text: the

text: pressure

text: data

text: as

text: a

text: sparsifying

text: transform.

text: Opposed

text: to

text: where

text: the

text: transform

text: was

text: used

text: to

text: sparsify

text: the

text: measured

text: signals,

text: in

text: the

text: present

text: work

text: we

text: exploit

text: this

text: for

text: obtaining

text: sparsity

text: in

text: the

text: original

text: imaging

text: domain.

text: Our

text: new

text: approach

text: is

text: based

text: on

text: the

text: following.

text: Consider

text: the

text: second

text: time

text: derivative

text: t)

text: of

text: t

text: the

text: PA

text: pressure.

text: We

text: will

text: show,

text: that

text: this

text: transformed

text: pressure

text: again

text: satisﬁes

text: the

text: wave

text: equation,

text: however

text: with

text: the

text: modiﬁed

text: PA

text: source

text: f

text: in

text: place

text: of

text: the

text: original

text: PA

text: source

text: f.

text: If

text: the

text: original

text: r

text: PA

text: source

text: consists

text: of

text: smooth

text: parts

text: and

text: jumps,

text: the

text: modiﬁed

text: source

text: consists

text: of

text: smooth

text: parts

text: and

text: sparse

text: structures;

text: see

text: Fig.

text: for

text: an

text: example.

text: This

text: enables

text: the

text: use

text: of

text: eﬃcient

text: CS

text: reconstruction

text: algorithms

text: based

text: on

text: sparse

text: recovery.

text: One

text: possible

text: approach

text: is

text: based

text: on

text: the

text: following

text: two-step

text: procedure.

text: First,

text: recover

text: an

text: approximation

text: h(r)

text: f(r)

text: via

text: In

text: a

text: second

text: r

text: step,

text: recover

text: an

text: approximation

text: to

text: f

text: by

text: solving

text: the

text: Poisson

text: equation

text: f

text: r

text: While

text: the

text: above

text: two-stage

text: approach

text: turned

text: out

text: to

text: very

text: well

text: recover

text: the

text: singularities

text: of

text: the

text: PA

text: source,

text: at

text: the

text: same

text: time

text: it

text: shows

text: disturbing

text: low-frequency

text: artifacts.

text: Therefore,

text: in

text: this

text: paper

text: we

text: develop

text: a

text: completely

text: novel

text: strategy

text: for

text: jointly

text: recovering

text: f

text: as

text: well

text: as

text: its

text: Laplacian.

text: It

text: is

text: based

text: on

text: solving

text: the

text: constrained

text: optimization

text: problem

text: min

text: h

text: I

text: (f)

text: C

text: (f,h)

text: such

text: that

text: Mf,

text: Mh,

text: f

text: h

text: y,

text: y

text: r

text: Here

text: M

text: is

text: the

text: forward

text: operator

text: (including

text: wave

text: propagation

text: and

text: compressed

text: measurements),

text: denotes

text: the

text: guaranteeing

text: sparsity

text: of

text: h

text: and

text: I

text: the

text: indicator

text: function

text: of

text: the

text: positive

text: C

text: cone

text: C

text: f

text: f(r)

text: guaranteeing

text: non-negativity.

text: In

text: the

text: case

text: of

text: noisy

text: data

text: we

text: consider

text: a

text: penalized

text: version

text: that

text: can

text: be

text: eﬃciently

text: solved

text: via

text: various

text: modern

text: optimization

text: techniques,

text: such

text: as

text: forward

text: backward

text: splitting.

text: Outline

text: In

text: Section

text: we

text: describe

text: PAT

text: and

text: existing

text: CS

text: approaches.

text: We

text: thereby

text: focus

text: on

text: the

text: role

text: of

text: sparsity

text: in

text: PAT.

text: The

text: proposed

text: framework

text: for

text: CS

text: PAT

text: will

text: be

text: presented

text: in

text: Section

text: This

text: includes

text: the

text: sparsiﬁcation

text: of

text: the

text: PA

text: source

text: and

text: the

text: joint

text: reconstruction

text: algorithm.

text: Numerical

text: and

text: experimental

text: results

text: are

text: presented

text: in

text: Section

text: The

text: paper

text: concludes

text: with

text: a

text: summary

text: and

text: outlook

text: on

text: future

text: research

text: presented

text: in

text: Section

text: Compressed

text: photoacoustic

text: tomography

text: Photoacoustic

text: tomography

text: R

text: Suppose

text: that

text: V

text: d

text: is

text: a

text: bounded

text: region

text: and

text: let

text: r

text: for

text: i

text: n

text: denote

text: admissible

text: detector

text: i

text: locations

text: distributed

text: on

text: the

text: boundary

text: ∂V

text: Then,

text: for

text: a

text: given

text: source

text: f

text: supported

text: inside

text: V

text: we

text: deﬁne

text: (Wf)(r

text: t)

text: p(r

text: t)

text: for

text: i

text: n

text: and

text: t

text: T

text: i

text: i