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text: (cid:16) (cid:17) (cid:21) (cid:16) (cid:17) + K sin θ cos θ + K sin θ sin θ cos φ 3sc (cid:96) (cid:96) 3s (cid:96) Λ + K sin θ cos θ + K sin θ sin θ sin φ , (3.17) 4sc (cid:96) (cid:96) 4s (cid:96) Λ where the angular coeﬃcients K , with n = 1, . . . , 4 and λ = s, c, ss, cc, sc, are functions nλ → of q2, and can be expressed in terms of eight transversity amplitudes for Λ Λ transition b → and the parity-violating decay parameter α in the secondary decay Λ pπ−; their explicit Λ expressions could be found in Ref. [38]. Starting with Eq. (3.17) and in terms of the angular coeﬃcients K , we can then construct nλ the following observables [38–40]: • The diﬀerential decay rate and diﬀerential branching fraction B dΓ d dΓ = 2K + K , = τ , (3.18) dq2 1ss 1cc dq2 Λb dq2 where τ is the Λ lifetime. Λb b • The longitudinal polarization fraction of the dimuon system − ˆ ˆ F = 2K K , (3.19) L 1ss 1cc where we introduce the normalized angular observables Kˆ = Knλ . nλ dΓ/dq2 • The lepton-, hadron- and combined lepton-hadron-side forward-backward asymmetries 3 1 3 A(cid:96) = Kˆ , AΛ = Kˆ + Kˆ , A(cid:96)Λ = Kˆ , (3.20) FB 2 1c FB 2ss 2 2cc FB 4 2c which have characteristic q2 behaviours: within the SM, both A(cid:96) and A(cid:96)Λ have the same FB FB (cid:39) zero-crossing points, q2(A(cid:96) ) q2(A(cid:96)Λ ), to the ﬁrst approximation, while AΛ does not 0 FB 0 FB FB cross zero [38]. Note that, as observed in the mesonic case [47, 94–96], the zero-crossing points are nearly free of hadronic uncertainties [38–40]. 9 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: equation

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text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##


text: v (cid:34)(cid:20) (cid:21) (cid:20) (cid:21)(cid:35) (cid:89) (cid:0) (cid:1) (cid:89) zj(0→,1i),(0,1) = 1−(cid:34)(cid:20) (cid:89) (cid:0)1 − φu[kα→] j − (1 − φ)zk[α→] j(cid:1)(cid:21) + (1 − φ)kj[1] 1(cid:20)− (cid:89)(1 − zk[α→] j) (cid:21)−(cid:35) ∈ \ ∈ \ k ∂j i k ∂j i − (cid:34)(cid:20) (cid:89) (cid:0)1 − φu[kβ→] j − (1 − φ)zk[β→] j (cid:1)(cid:21)+ (1 − φ)kj[2] 1 1(cid:20)− (cid:89)(1 − zj[β→] i) (cid:21)+(cid:35) ∈ \ ∈ \ k ∂j i k ∂j i − (cid:34)(cid:20) (cid:89) (cid:0)1 − φu[kα→,βj] − (1 − φ)zk[α→,βj(cid:1)] + (1 − φ)kj[1,2(cid:89)] 1 1 − ((cid:21)1 − zj[α→,βi]) + ∈ \ ∈ \ k ∂j i k ∂j i 1 −(cid:20)φu(cid:89)[kα→] j − (1 − φ)zk[α→] j (cid:89)− (1 − φ)kj[α] (cid:89)(1 − zk[α→] j) × (cid:21)(cid:35) ∈ ∈ k ∂[α] k ∂[α] j j − ((cid:34)1(cid:20)− φ(cid:89))kj[2] (cid:0)1 ∈ (1 − zk[α→] j) − ∈(cid:1) \ (1 − zk[β→] j) ∈ (cid:89) (1 − zk[α→,βj]) (cid:21) + k ∂[α,β] k ∂[β] i k ∂[α,β] j j j − (cid:20)1 −(cid:89)φu[kβ→] j − (1 − φ)zk[(cid:89)β→] j − (1 − φ)kj[β](cid:89)1 (1 − zk(cid:21)[β→] j) × ∈ \ ∈ \ k ∂[β] i k ∂[β] i j j (1 − φ)kj[1] (1 − z[β→] ) − (1 − z[β→] ) (1 − z[α→,β]) (SM15) k j k j k j ∈ ∈ ∈ k ∂[α,β] k ∂[α] k ∂[α,β] j j j

text: v

text: φu[kα→]

text: j

text: φ)zk[α→]

text: zk[α→]

text: j)

text: k

text: ∂j

text: i

text: k

text: ∂j

text: i

text: φu[kβ→]

text: j

text: φ)zk[β→]

text: j

text: zj[β→]

text: i)

text: k

text: ∂j

text: i

text: k

text: ∂j

text: i

text: φu[kα→,βj]

text: zj[α→,βi])

text: k

text: ∂j

text: i

text: k

text: ∂j

text: i

text: j

text: φ)zk[α→]

text: j

text: φ)kj[α]

text: zk[α→]

text: j)

text: k

text: ∂[α]

text: k

text: ∂[α]

text: j

text: j

text: zk[α→]

text: j)

text: zk[β→]

text: j)

text: zk[α→,βj])

text: k

text: ∂[α,β]

text: k

text: ∂[β]

text: i

text: k

text: ∂[α,β]

text: j

text: j

text: j

text: j

text: j

text: j)

text: k

text: ∂[β]

text: i

text: k

text: ∂[β]

text: i

text: j

text: j

text: z[β→]

text: z[β→]

text: z[α→,β])

text: k

text: j

text: k

text: j

text: k

text: j

text: k

text: ∂[α,β]

text: k

text: ∂[α]

text: k

text: ∂[α,β]

text: j

text: j

text: j


text: On the Eﬀects of Advection and Vortex Stretching Tarek Mohamed Elgindi and In-Jee Jeong May 29, 2020 Abstract We prove ﬁnite-time singularity formation for De Gregorio’s model of the three-dimensional ∩ R ∞ vorticity equation in the class of Lp Cα( ) vorticities for some α > 0 and p < . We also prove ﬁnite-time singularity formation from smooth initial data for the Okamoto-Sakajo- Wunsch models in a new range of parameter values. As a consequence, we have ﬁnite-time singularity for certain inﬁnite-energy solutions of the surface quasi-geostrophic equation which are Cα-regular. One of the diﬃculties in the models we consider is that there are competing nonlocal stabilizing eﬀects (advection) and destabilizing eﬀects (vortex stretch- ing) which are of the same size in terms of scaling. Hence, it is diﬃcult to establish the domination of one eﬀect over the other without having strong control of the solution. We conjecture that strong solutions to the De Gregorio model exhibit the following be- ∈ R havior: for each 0 < α < 1 there exists an initial ω Cα( ) which is compactly supported 0 for which the solution becomes singular in ﬁnite-time; on the other hand, solutions to De ∈ ∩ R ∞ Gregorio’s equation are global whenever ω Lp C1( ) for some p < . Such a di- 0 chotomy seems to be a genuinely non-linear eﬀect which cannot be explained merely by scaling considerations since Cα spaces are scaling subcritical for each α > 0. Contents 1 Introduction 1 1.1 Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 The incompressible Euler equation . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 A short history of 1D Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3.1 The Starting Point: Constantin, Lax, and Majda . . . . . . . . . . . . . . 3 1.3.2 The De Gregorio Model: The Regularity Conjecture . . . . . . . . . . . . 4 1 1.3.3 The Cordoba-Cordoba-Fontelos Model: The C 2 -conjecture. . . . . . . . . 4 1.3.4 The models of Okamoto, Sakajo, and Wunsch . . . . . . . . . . . . . . . . 5 1.3.5 Connections with the SQG Equation . . . . . . . . . . . . . . . . . . . . . 6 1.4 The Setting of the Present Paper and the Main Results . . . . . . . . . . . . . . 6 1.5 A toy model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.6 The Self-Similar Ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.6.1 The a = 0 case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.6.2 The case a > 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.7 The main steps in the proofs of Theorems 1 and 3 . . . . . . . . . . . . . . . . . 11 1.8 Organization of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1 ##LTLine##

text: On

label: title

text: the

text: Eﬀects

text: of

text: Advection

text: and

text: Vortex

text: Stretching

text: Tarek

label: author

text: Mohamed

text: Elgindi

text: and

text: In-Jee

text: Jeong

text: May

label: date

text: Abstract

text: We

label: abstract

text: prove

text: ﬁnite-time

text: singularity

text: formation

text: for

text: De

text: Gregorio’s

text: model

text: of

text: the

text: three-dimensional

text: R

text: vorticity

text: equation

text: in

text: the

text: class

text: of

text: Lp

text: Cα(

text: vorticities

text: for

text: some

text: α

text: and

text: p

text: We

text: also

text: prove

text: ﬁnite-time

text: singularity

text: formation

text: from

text: smooth

text: initial

text: data

text: for

text: the

text: Okamoto-Sakajo-

text: Wunsch

text: models

text: in

text: a

text: new

text: range

text: of

text: parameter

text: values.

text: As

text: a

text: consequence,

text: we

text: have

text: ﬁnite-time

text: singularity

text: for

text: certain

text: inﬁnite-energy

text: solutions

text: of

text: the

text: surface

text: quasi-geostrophic

text: equation

text: which

text: are

text: Cα-regular.

text: One

text: of

text: the

text: diﬃculties

text: in

text: the

text: models

text: we

text: consider

text: is

text: that

text: there

text: are

text: competing

text: nonlocal

text: stabilizing

text: eﬀects

text: (advection)

text: and

text: destabilizing

text: eﬀects

text: (vortex

text: stretch-

text: ing)

text: which

text: are

text: of

text: the

text: same

text: size

text: in

text: terms

text: of

text: scaling.

text: Hence,

text: it

text: is

text: diﬃcult

text: to

text: establish

text: the

text: domination

text: of

text: one

text: eﬀect

text: over

text: the

text: other

text: without

text: having

text: strong

text: control

text: of

text: the

text: solution.

text: We

text: conjecture

text: that

text: strong

text: solutions

text: to

text: the

text: De

text: Gregorio

text: model

text: exhibit

text: the

text: following

text: be-

text: R

text: havior:

text: for

text: each

text: α

text: there

text: exists

text: an

text: initial

text: ω

text: Cα(

text: which

text: is

text: compactly

text: supported

text: for

text: which

text: the

text: solution

text: becomes

text: singular

text: in

text: ﬁnite-time;

text: on

text: the

text: other

text: hand,

text: solutions

text: to

text: De

text: R

text: Gregorio’s

text: equation

text: are

text: global

text: whenever

text: ω

text: Lp

text: for

text: some

text: p

text: Such

text: a

text: di-

text: chotomy

text: seems

text: to

text: be

text: a

text: genuinely

text: non-linear

text: eﬀect

text: which

text: cannot

text: be

text: explained

text: merely

text: by

text: scaling

text: considerations

text: since

text: Cα

text: spaces

text: are

text: scaling

text: subcritical

text: for

text: each

text: α

text: Contents

text: Introduction

text: Synopsis

text: The

text: incompressible

text: Euler

text: equation

text: A

text: short

text: history

text: of

text: Models

text: The

text: Starting

text: Point:

text: Constantin,

text: Lax,

text: and

text: Majda

text: The

text: De

text: Gregorio

text: Model:

text: The

text: Regularity

text: Conjecture

text: The

text: Cordoba-Cordoba-Fontelos

text: Model:

text: The

text: C

text: -conjecture.

text: The

text: models

text: of

text: Okamoto,

text: Sakajo,

text: and

text: Wunsch

text: Connections

text: with

text: the

text: SQG

text: Equation

text: The

text: Setting

text: of

text: the

text: Present

text: Paper

text: and

text: the

text: Main

text: Results

text: A

text: toy

text: model

text: The

text: Self-Similar

text: Ansatz

text: The

text: a

text: case.

text: The

text: case

text: a

text: The

text: main

text: steps

text: in

text: the

text: proofs

text: of

text: Theorems

text: and

text: Organization

text: of

text: the

text: paper

text: ##LTLine##


text: with F (z) = z . As mentioned in the footnote, a simple scaling argument will then imply that z2+(cid:48)1 ∞ ∈ whenever 0 < ω (0) < , and ω C2, ω is asymptotically equal to a self similar proﬁle. In 0 0 fact, this is the only such self-similar proﬁle. (cid:18) (cid:19) It is straightforward to check that with the self-similar ansatz 1 x ω(t, x) = − F − , 0 1 t 1 t the Constantin-Lax-Madja equation (1.3) reduces to the following: (cid:48) F + zF (z) + 2F H(F ) = 0. (2.1) 0 0 0 0 We have already seen that F (z) = Cz/(1 + C2z2) is a solution to this equation. We also have: 0 Proposition 2.1. Assuming that the proﬁle F (z) in (2.1) is odd, smooth, and decays at inﬁnity, 0 we have z F (z) = , 0 1 + z2 up to a scaling of the variable z. Proof. Taking the Hilbert transform of both sides of the equation (2.1), we see: (cid:48) − H(F ) + zH(F ) F 2 + H(F )2 = 0. (2.2) 0 0 0 0 Now let F + iH(F ) = V. Then, since 0 0 − − − 2F H(F ) + i(H(F )2 F 2) = i(F + iH(F ))2 = iV 2, 0 0 0 0 0 the system of equations (2.1), (2.2) can be rewritten as (cid:48) − V + zV iV 2 = 0. Integrating the system gives 1 V = , i + Cz and it is easy to see that the real and imaginary parts of V are odd and even respectively precisely when C is a real constant. Thus, Cz i V = + 1 + C2z2 1 + C2z2 and in particular, Cz F = . 0 1 + C2z2 15 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: with

text: F

text: (z)

text: z

text: As

text: mentioned

text: in

text: the

text: footnote,

text: a

text: simple

text: scaling

text: argument

text: will

text: then

text: imply

text: that

text: whenever

text: ω

text: and

text: ω

text: ω

text: is

text: asymptotically

text: equal

text: to

text: a

text: self

text: similar

text: proﬁle.

text: In

text: fact,

text: this

text: is

text: the

text: only

text: such

text: self-similar

text: proﬁle.

text: It

text: is

text: straightforward

text: to

text: check

text: that

text: with

text: the

text: self-similar

text: ansatz

text: x

text: ω(t,

text: x)

text: F

text: t

text: t

text: the

text: Constantin-Lax-Madja

text: equation

text: reduces

text: to

text: the

text: following:

text: F

text: zF

text: (z)

text: H(F

text: We

text: have

text: already

text: seen

text: that

text: F

text: (z)

text: is

text: a

text: solution

text: to

text: this

text: equation.

text: We

text: also

text: have:

text: Proposition

text: Assuming

text: that

text: the

text: proﬁle

text: F

text: (z)

text: in

text: is

text: odd,

text: smooth,

text: and

text: decays

text: at

text: inﬁnity,

text: we

text: have

text: z

text: F

text: (z)

text: up

text: to

text: a

text: scaling

text: of

text: the

text: variable

text: z.

text: Proof.

text: Taking

text: the

text: Hilbert

text: transform

text: of

text: both

text: sides

text: of

text: the

text: equation

text: we

text: see:

text: H(F

text: zH(F

text: F

text: H(F

text: Now

text: let

text: F

text: iH(F

text: V.

text: Then,

text: since

text: H(F

text: i(H(F

text: F

text: i(F

text: iH(F

text: iV

text: the

text: system

text: of

text: equations

text: can

text: be

text: rewritten

text: as

text: V

text: zV

text: iV

text: Integrating

text: the

text: system

text: gives

text: V

text: i

text: Cz

text: and

text: it

text: is

text: easy

text: to

text: see

text: that

text: the

text: real

text: and

text: imaginary

text: parts

text: of

text: V

text: are

text: odd

text: and

text: even

text: respectively

text: precisely

text: when

text: C

text: is

text: a

text: real

text: constant.

text: Thus,

text: Cz

text: i

text: V

text: and

text: in

text: particular,

text: Cz

text: F

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: by an explicit integral formula; see below (3.6). The purpose of the following lemma is mainly to extract a representation formula for the − L 1, supposing that the right hand side is suﬃciently smooth and decays fast at inﬁnity. Then in the next lemma, we carry out precise norm estimates for F . n Lemma 3.1. Consider the linear equation Lf = g ∈ ∞ R ∈ R where g C ( ) is a given odd function. Then a solution f H3( ) (which is necessarily c (cid:48) (cid:48) odd) exists if and only if g (0) + 2Hg(0) =(cid:90) 0 and once we further require that f (0) = 0, it is uniquely determined by the expression − − − z(1 z2) · z 1(cid:90) w2 (L 1g)(z) = gˆ(w) + 2hˆ(w)dw (1 + z2)2 w − 0 − (3.6) 2z2 · z − 1 w2 + 2gˆ(w) + hˆ(w)dw, (1 + z2)2 w 0 with − g(z) − 1 (cid:48) Hg(z) Hg(0) 2z gˆ(z) := g (0), hˆ(z) := + Hg(0). z 1 + z2 z 1 + z2 ∈ R Proof. Assuming that we are given a solution f H3( ) to Lf = g, one may take the Hilbert transform of both sides to obtain (cid:48) − 2z − 2 Hf + z(Hf) f Hf = Hg, (3.7) 1 + z2 1 + z2 where we have used t(cid:18)he (cid:19)Tricom(cid:18)i (cid:19)identity. Set(cid:18)ting d =(cid:19)H(cid:18)f (cid:19)for s(cid:18)imp(cid:19)licity, we observe that a solution f of Lf = g must solve the following linear system of ODEs (cid:48) − f f 2 1 z f g + z + − − = , (3.8) d d 1 + z2 z 1 d h with d = Hf and h = Hg. Since we are interested in the case where f is odd (and therefore d is necessarily even), it suﬃces to solve the initial value problem with given f(0) and d(0) on the positive reals. Due to the presence of the coeﬃcient z, it is not clear whether the initial value (cid:48) problem is well-posed, and the condition g (0) + 2h(0) = 0 comes out naturally in this setting. Indeed, evaluating both sides of the second component of (3.8) at z = 0, one obtains − d(0) 2d(0) = h(0), (cid:18) (cid:19) → while dividing both sides of the ﬁrst component of (3.8) by z and taking the limit z 0+, f(z) (cid:48) − 2 f(z) 2 − g(z) → − (cid:48) 0 = lim + f (z) + d(z) 2d(0) g (0). z→0+ z 1 + z2 z 1 + z2 z Therefore we need − 1 (cid:48) d(0) = h(0) = g (0) 2 19 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: by

text: an

text: explicit

text: integral

text: formula;

text: see

text: below

text: The

text: purpose

text: of

text: the

text: following

text: lemma

text: is

text: mainly

text: to

text: extract

text: a

text: representation

text: formula

text: for

text: the

text: L

text: supposing

text: that

text: the

text: right

text: hand

text: side

text: is

text: suﬃciently

text: smooth

text: and

text: decays

text: fast

text: at

text: inﬁnity.

text: Then

text: in

text: the

text: next

text: lemma,

text: we

text: carry

text: out

text: precise

text: norm

text: estimates

text: for

text: F

text: n

text: Lemma

text: Consider

text: the

text: linear

text: equation

text: Lf

text: g

text: R

text: R

text: where

text: g

text: C

text: is

text: a

text: given

text: odd

text: function.

text: Then

text: a

text: solution

text: f

text: (which

text: is

text: necessarily

text: c

text: odd)

text: exists

text: if

text: and

text: only

text: if

text: g

text: and

text: once

text: we

text: further

text: require

text: that

text: f

text: it

text: is

text: uniquely

text: determined

text: by

text: the

text: expression

text: z

text: (L

text: gˆ(w)

text: w

text: z

text: hˆ(w)dw,

text: w

text: with

text: g(z)

text: Hg(z)

text: gˆ(z)

text: g

text: hˆ(z)

text: z

text: z

text: R

text: Proof.

text: Assuming

text: that

text: we

text: are

text: given

text: a

text: solution

text: f

text: to

text: Lf

text: g,

text: one

text: may

text: take

text: the

text: Hilbert

text: transform

text: of

text: both

text: sides

text: to

text: obtain

text: Hf

text: z(Hf)

text: f

text: Hf

text: Hg,

text: where

text: we

text: have

text: used

text: d

text: we

text: observe

text: that

text: a

text: solution

text: f

text: of

text: Lf

text: g

text: must

text: solve

text: the

text: following

text: linear

text: system

text: of

text: ODEs

text: f

text: f

text: z

text: f

text: g

text: z

text: d

text: d

text: z

text: d

text: h

text: with

text: d

text: Hf

text: and

text: h

text: Hg.

text: Since

text: we

text: are

text: interested

text: in

text: the

text: case

text: where

text: f

text: is

text: odd

text: (and

text: therefore

text: d

text: is

text: necessarily

text: even),

text: it

text: suﬃces

text: to

text: solve

text: the

text: initial

text: value

text: problem

text: with

text: given

text: and

text: on

text: the

text: positive

text: reals.

text: Due

text: to

text: the

text: presence

text: of

text: the

text: coeﬃcient

text: z,

text: it

text: is

text: not

text: clear

text: whether

text: the

text: initial

text: value

text: problem

text: is

text: well-posed,

text: and

text: the

text: condition

text: g

text: comes

text: out

text: naturally

text: in

text: this

text: setting.

text: Indeed,

text: evaluating

text: both

text: sides

text: of

text: the

text: second

text: component

text: of

text: at

text: z

text: one

text: obtains

text: while

text: dividing

text: both

text: sides

text: of

text: the

text: ﬁrst

text: component

text: of

text: by

text: z

text: and

text: taking

text: the

text: limit

text: z

text: f(z)

text: f(z)

text: g(z)

text: lim

text: f

text: (z)

text: d(z)

text: g

text: z

text: z

text: z

text: Therefore

text: we

text: need

text: g

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##

text: ##LTLine##

text: ##LTLine##


text: (cid:107) (cid:107) and then clearly both terms can be bonded in H3 by C F˜ . H3 The corresponding estimate for the function Sσ(F˜) can be carried out in a similar way. Let us only point out that we lose by a factor of 1/α simply because of the loss in H3-estimate from Lemma A.5: (cid:107) (cid:107) ≤ C (cid:107) (cid:107) H˜ F˜ F˜ . H3 α H3 (ii) proof of (4.1(cid:32)4) (cid:33) (cid:90) − (cid:48) We proceed to the proof of (4.14), sta(cid:0)rtin(cid:1)g with the case of T 1(wF˜ ). We need to estimate: P (w) w 1 F˜(cid:32)(cid:48)((cid:90)s) − 1 F˜(cid:48)(0(cid:90)) ds (cid:0) (cid:1) (cid:33) 0 s 1 + 2 cos α2π s + s2 (cid:0) (cid:1) (cid:48) − (cid:48) · w F˜ (s) F˜ (0) (cid:48) · w 2 cos απ + s = P (w) ds + F˜ (0) 2 ds , s 1 + 2 cos απ s + s2 0 0 2 but the corresponding proof from the smooth case carries over to this one, which bounds the above in H3 by a constant multiple of F˜ in H3. The cases σ = 0, 1 are simpler and can be done as in the case of (4.13). (cid:48) (cid:48) Given the desired bound for T σ(wF˜(cid:16)), the(cid:17)corresponding result for Sσ(wF˜ ) similarly follows, using the convenient fact that (cid:48) (cid:48) H˜ wF˜ = w(H˜ (F˜)) , see (A.11) in the Appendix. (iii) proof of (4.15) To begin with, it is(cid:101) necessary to note that (cid:101) − · (cid:48) 1 − · (cid:48) − − · (cid:48) (Λ 1E F )(w) = ( (Λ 1E)(z) zF (z))(w) = w α1 Λ˜ 1(E˜)(w) αwF˜ (w) z and that its derivative at w = 0 equals · · (cid:48) α H˜ (E˜)(0) F˜ (0). (cid:101) Let us restrict ourselves to obtain(cid:90)ing an(cid:16)H3 estimate for T −1((Λ−1E · F (cid:48))). We(cid:17) need to bound − − · (cid:48) · w 1 − − (cid:90)· (cid:48) − (cid:0) (cid:1) · (cid:48) T 1(Λ˜ 1(E˜) F˜ )(w) = P (w) α s s α1 Λ˜ 1(E˜)(s) F˜ (s) H˜ ((cid:0)E˜)((cid:1)0) F˜ (0) ds 0 · · (cid:48) · w 2 cos απ + s + P (w) αH˜ (E˜)(0) F˜ (0) 2 ds. 1 + 2 cos απ s + s2 (cid:0) (cid:1) (cid:0) (cid:1) 0 2 To estimate the latter term, it suﬃces to observe that (cid:90) (cid:18)(cid:90) (cid:19) − · (cid:48) − · (cid:48) H˜ Λ 1(E) F ˜(0) = H Λ 1(E) F (0) ∞ − 2 (cid:90) (cid:18)1 z(cid:90) (cid:48) (cid:19) = H(E)(s)ds F (z)dz π z 0∞ 0 − 2 − w · 1 − · (cid:48) = w α1 H˜ (E˜)(t) t1αα dt F˜ (w)dw π α 0 0 39 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: and

text: then

text: clearly

text: both

text: terms

text: can

text: be

text: bonded

text: in

text: by

text: C

text: F˜

text: The

text: corresponding

text: estimate

text: for

text: the

text: function

text: Sσ(F˜)

text: can

text: be

text: carried

text: out

text: in

text: a

text: similar

text: way.

text: Let

text: us

text: only

text: point

text: out

text: that

text: we

text: lose

text: by

text: a

text: factor

text: of

text: simply

text: because

text: of

text: the

text: loss

text: in

text: from

text: Lemma

text: C

text: H˜

text: F˜

text: F˜

text: α

text: (ii)

text: proof

text: of

text: We

text: proceed

text: to

text: the

text: proof

text: of

text: with

text: the

text: case

text: of

text: T

text: We

text: need

text: to

text: estimate:

text: P

text: (w)

text: w

text: ds

text: s

text: cos

text: α2π

text: s

text: w

text: F˜

text: (s)

text: F˜

text: w

text: cos

text: απ

text: s

text: P

text: (w)

text: ds

text: F˜

text: ds

text: s

text: cos

text: απ

text: s

text: but

text: the

text: corresponding

text: proof

text: from

text: the

text: smooth

text: case

text: carries

text: over

text: to

text: this

text: one,

text: which

text: bounds

text: the

text: above

text: in

text: by

text: a

text: constant

text: multiple

text: of

text: F˜

text: in

text: The

text: cases

text: σ

text: are

text: simpler

text: and

text: can

text: be

text: done

text: as

text: in

text: the

text: case

text: of

text: Given

text: the

text: desired

text: bound

text: for

text: T

text: result

text: for

text: Sσ(wF˜

text: similarly

text: follows,

text: using

text: the

text: convenient

text: fact

text: that

text: H˜

text: wF˜

text: w(H˜

text: (F˜))

text: see

text: in

text: the

text: Appendix.

text: (iii)

text: proof

text: of

text: To

text: begin

text: with,

text: it

text: necessary

text: to

text: note

text: that

text: (Λ

text: F

text: )(w)

text: (Λ

text: zF

text: (z))(w)

text: w

text: α1

text: Λ˜

text: αwF˜

text: (w)

text: z

text: and

text: that

text: its

text: derivative

text: at

text: w

text: equals

text: α

text: H˜

text: F˜

text: Let

text: us

text: restrict

text: ourselves

text: to

text: estimate

text: for

text: T

text: F

text: need

text: to

text: bound

text: w

text: T

text: 1(Λ˜

text: F˜

text: )(w)

text: P

text: (w)

text: α

text: s

text: s

text: α1

text: Λ˜

text: F˜

text: (s)

text: H˜

text: F˜

text: ds

text: w

text: cos

text: απ

text: s

text: P

text: (w)

text: αH˜

text: F˜

text: ds.

text: cos

text: απ

text: s

text: To

text: estimate

text: the

text: latter

text: term,

text: it

text: suﬃces

text: to

text: observe

text: that

text: H˜

text: Λ

text: F

text: H

text: Λ

text: F

text: H(E)(s)ds

text: F

text: (z)dz

text: π

text: z

text: 0∞

text: w

text: w

text: α1

text: H˜

text: (E˜)(t)

text: t1αα

text: dt

text: F˜

text: (w)dw

text: π

text: α

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##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: { } { } Now that the form of the sequence of inequalities for l , µ are equivalent with that of {| |} { } n n λ , µ from the smooth case, we can run the exact same argument to deduce the desired n n bounds. (cid:88) (cid:88) This shows that the series ∞ ∞ F (α)(z) = F (α)(z) + anF (α)(z), λ(α)(a) = anλ(α) 0 n n n=1 n=1 ∈ − are convergent for some interval a ( 1/(c α), 1/(c α)) on which it can be also guaranteed − 0 0 that λ(α)(a) > 1. R Next, it is easy to see that the self-similar proﬁle F (α) indeed belongs to Cα( ): it suﬃces ∩ ∞ to observe that F˜(α) belongs to C˙ 1 L . To show decay of the function F (α)(z), we simply argue using the equation (4.4) just as we did in the proof of Theorem 2. Indeed, note that if F solves 1 + λ (cid:48) − (cid:48) F + zF + 2F H(F ) + aΛ 1(F )F = 0 α ∈ and if F = F˜(zα) with F˜ H3 (as we have shown above) then F˜ solves (cid:48) · − − · (cid:48) F˜(w) + (1 + λ)wF˜ (w) + 2F˜ H˜ (F˜) + aw α1 Λ˜ 1(F˜) wF˜ (w) = 0. − Proceeding a−s in the proof of Theorem 2, we obtain that F˜ decays like w 1+1λ , and therefore F α decays like z 1+λ . Now we are done. 5 A method to get singularity formation in the full range To close the paper, we would like to mention another method which likely could lead to ﬁnite-time singularity formation for Cα solutions to the De Gregorio model which are periodic. Consider De Gregorio’s model: ∂ ω + 2u∂ ω = 2∂ uω. t x x Assume that the data is odd in x, 2π periodic, and positive in (0, π). Suppose further that | | ≥ ω (x) C(sin(x))α 0 (cid:90) on (0, π) for some α < 1 and some C > 0. Now deﬁne x dy f (x) := . 0 ω (y) 0 0 (cid:90) Since ω is bounded from below as above and since α < 1, f is well-deﬁned. In general, if we 0 0 deﬁne x dy f(t, x) = ω(t, y) 0 we see that f is a 2π periodic even function satisfying a transport equation: 42 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Now

text: that

text: the

text: form

text: of

text: the

text: sequence

text: of

text: inequalities

text: for

text: l

text: µ

text: are

text: equivalent

text: with

text: that

text: of

text: n

text: n

text: λ

text: µ

text: from

text: the

text: smooth

text: case,

text: we

text: can

text: run

text: the

text: exact

text: same

text: argument

text: to

text: deduce

text: the

text: desired

text: n

text: n

text: bounds.

text: This

text: shows

text: that

text: the

text: series

text: F

text: (α)(z)

text: F

text: (α)(z)

text: anF

text: (α)(z),

text: λ(α)(a)

text: anλ(α)

text: n

text: n

text: are

text: convergent

text: for

text: some

text: interval

text: a

text: α),

text: α))

text: on

text: which

text: it

text: can

text: be

text: also

text: guaranteed

text: that

text: λ(α)(a)

text: R

text: Next,

text: it

text: is

text: easy

text: to

text: see

text: that

text: the

text: self-similar

text: proﬁle

text: F

text: (α)

text: indeed

text: belongs

text: to

text: Cα(

text: it

text: suﬃces

text: to

text: observe

text: that

text: F˜(α)

text: belongs

text: to

text: C˙

text: L

text: To

text: show

text: decay

text: of

text: the

text: function

text: F

text: (α)(z),

text: we

text: simply

text: argue

text: using

text: the

text: equation

text: just

text: as

text: we

text: did

text: in

text: the

text: proof

text: of

text: Theorem

text: Indeed,

text: note

text: that

text: if

text: F

text: solves

text: λ

text: F

text: zF

text: H(F

text: aΛ

text: )F

text: α

text: and

text: if

text: F

text: F˜(zα)

text: with

text: F˜

text: (as

text: we

text: have

text: shown

text: above)

text: then

text: F˜

text: solves

text: F˜(w)

text: λ)wF˜

text: (w)

text: H˜

text: (F˜)

text: aw

text: α1

text: Λ˜

text: wF˜

text: (w)

text: Proceeding

text: a−s

text: in

text: the

text: proof

text: of

text: Theorem

text: we

text: obtain

text: that

text: F˜

text: decays

text: like

text: w

text: and

text: therefore

text: F

text: α

text: decays

text: like

text: z

text: Now

text: we

text: are

text: done.

text: A

text: method

text: to

text: get

text: singularity

text: formation

text: in

text: the

text: full

text: range

text: To

text: close

text: the

text: paper,

text: we

text: would

text: like

text: to

text: mention

text: another

text: method

text: which

text: likely

text: could

text: lead

text: to

text: ﬁnite-time

text: singularity

text: formation

text: for

text: Cα

text: solutions

text: to

text: the

text: De

text: Gregorio

text: model

text: which

text: are

text: periodic.

text: Consider

text: De

text: Gregorio’s

text: model:

text: ω

text: ω

text: 2∂

text: uω.

text: t

text: x

text: x

text: Assume

text: that

text: the

text: data

text: is

text: odd

text: in

text: x,

text: 2π

text: periodic,

text: and

text: positive

text: in

text: π).

text: Suppose

text: further

text: that

text: ω

text: (x)

text: C(sin(x))α

text: on

text: π)

text: for

text: some

text: α

text: and

text: some

text: C

text: Now

text: deﬁne

text: x

text: dy

text: f

text: (x)

text: ω

text: (y)

text: Since

text: ω

text: is

text: bounded

text: from

text: below

text: as

text: above

text: and

text: since

text: α

text: f

text: is

text: well-deﬁned.

text: In

text: general,

text: if

text: we

text: deﬁne

text: x

text: dy

text: f(t,

text: x)

text: ω(t,

text: y)

text: we

text: see

text: that

text: f

text: is

text: a

text: 2π

text: periodic

text: even

text: function

text: satisfying

text: a

text: transport

text: equation:

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Proof. This is a purely algebraic statement which can be checked directly. (cid:90) We set 1 2t H˜ (n)(f)(w) = p.v. − f(t)dt (A.5) 0 π R w2 t2 + ∈ {± · · · ± − } (cid:90) and for j 1, , (n 1) − 1 ζj H˜ (n)(f)(w) = − 2n f(t)dt. (A.6) j π R+ w ζj t 2n | | Note that H˜ (n) is exactly the Hilbert transform of the odd function f( w )sgn(w) restricted R 0 onto +. In particular, (cid:107) (cid:107) ≤ (cid:107) (cid:107) H˜ (n)(f) f . 0 L2 L2 The following lemma follows directly from the deﬁnition of the operators H˜ (n). j ≤ | | (cid:90) Lemma A.4. For 1 j < n, we have the identities − 1 · − 1 H˜ (n) (tf(t)) (w) = wH˜ (n)(f)(w) f(t)dt, (A.7) j ζj j π R+ 2n (cid:18) (cid:19) f(t) − f(0) · H˜ (n)f(w) − H˜ (n)f(0) H˜ (n) (w) = ζj j j , (A.8) j t 2n w and (cid:48) − · (cid:48) H˜ (n)(f )(w) = ζj H˜ (n)(f) . (A.9) j 2n j As a consequence, we obtain the identity (cid:48) (cid:48) H˜ (n)(tf (t))(w) = w(H˜ (n)f) (w), (A.10) j j which in particular implies that (cid:48) (cid:48) H˜ (n)(tf (t))(w) = w(H˜ (n)f) (w). (A.11) ≤ | | ≤   Lemma A.5. For each 1 j n, we have the estimate  (cid:16) (cid:17) (cid:107) (cid:107) ≤ 1 (cid:107) (cid:107) H˜ (n)(f) C ln 1 + f j L2 L2 sin2 jπ n 45 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Proof.

text: This

text: is

text: a

text: purely

text: algebraic

text: statement

text: which

text: can

text: be

text: checked

text: directly.

text: We

text: set

text: H˜

text: (n)(f)(w)

text: p.v.

text: f(t)dt

text: π

text: R

text: and

text: for

text: j

text: (n

text: ζj

text: H˜

text: (n)(f)(w)

text: f(t)dt.

text: j

text: π

text: R+

text: w

text: ζj

text: t

text: Note

text: that

text: H˜

text: (n)

text: is

text: exactly

text: the

text: Hilbert

text: transform

text: of

text: the

text: odd

text: function

text: f(

text: w

text: )sgn(w)

text: restricted

text: R

text: onto

text: In

text: particular,

text: H˜

text: (n)(f)

text: f

text: The

text: following

text: lemma

text: follows

text: directly

text: from

text: the

text: deﬁnition

text: of

text: the

text: operators

text: H˜

text: (n).

text: j

text: Lemma

text: For

text: j

text: n,

text: we

text: have

text: the

text: identities

text: H˜

text: (n)

text: (tf(t))

text: (w)

text: wH˜

text: (n)(f)(w)

text: f(t)dt,

text: j

text: ζj

text: j

text: π

text: R+

text: f(t)

text: H˜

text: (n)f(w)

text: H˜

text: H˜

text: (n)

text: (w)

text: ζj

text: j

text: j

text: j

text: t

text: w

text: and

text: H˜

text: (n)(f

text: )(w)

text: ζj

text: H˜

text: (n)(f)

text: j

text: j

text: As

text: a

text: consequence,

text: we

text: obtain

text: the

text: identity

text: H˜

text: (n)(tf

text: (t))(w)

text: w(H˜

text: (n)f)

text: (w),

text: j

text: j

text: which

text: in

text: particular

text: implies

text: that

text: H˜

text: (n)(tf

text: (t))(w)

text: w(H˜

text: (n)f)

text: (w).

text: Lemma

text: For

text: each

text: j

text: n,

text: we

text: have

text: the

text: estimate

text: H˜

text: (n)(f)

text: C

text: ln

text: f

text: j

text: jπ

text: n

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: which is the bound we wanted. The imaginary part can be treated in a similar way and results in a better estimate, without the logarithmic factor. Given this L2-bound for H˜ (n), the bound (A.12) followssimply from summing the estimates ≤ | | ≤ j over 1 j n: it(cid:88)only suﬃces to observe that (cid:89)  (cid:16) (cid:17)   (cid:16) (cid:17) 1 1 ln 1 + = ln  1 +  (cid:18) (cid:19) ≤| |≤ sin2 jπ ≤(cid:89)| |≤ sin2 jπ 1 j n n 1 j n  n n ≤ n ≤ (cid:48) C ln 1 + C n, j j=1 (cid:18) (cid:19) (cid:89) where the last inequality is a consequence of the fact that n n ≤ (2n)! ≤ 1 + C exp(Cn) j (n!)2 j=1 for some constant C > 0 using Sterling’s approximation. Finally, the Hm-bound follows similarly, using the identity (A.9). A.2 Functional inequalities We state and prove the Hardy(cid:90) inequalities. ∈ R ≥ Lemma A.6 (Hardy inequalities). For any f Hs( ) with s 0, we have (cid:107)1 z (cid:107) ≤ (cid:107) (cid:107) f(w)dw R C f R . z Hs( ) s Hs( ) (cid:18) (cid:90) 0 (cid:19) ≤ ≤ More precisely, for each 0 σ s, we have (cid:107) 1 z (cid:107) 2 (cid:107) (cid:107) ∂σ f(w)dw R < ∂σf . (A.14) z z L2( ) 2σ + 1 z L2 0 ∈ ∞ ∞ Proof. In the proof,(cid:90)without loss of genera(cid:90)lity we shall(cid:18)a(cid:90)ssume that(cid:19)f C ([0, )). c We ﬁrst consider the case s = 0: we have ∞ (cid:107)1 z (cid:107) (cid:90) − − (cid:18) (cid:90)z (cid:19)2 f(w)dw 2 ∞ = ∂ z 1 f(w)dw dz z L2(0, ) z 0 0 ∞ 0 (cid:90) · 1 z = 2 f(z) f(w)dw dz z 0 0 (cid:107)1 z (cid:107) · (cid:107) (cid:107) < 2 f(w)dw ∞ f ∞ , z L2(0, ) L2(0, ) 0 which establishes the sta(cid:18)teme(cid:90)nt in this ca(cid:19)se, notin(cid:18)g that th(cid:90)e strict equa(cid:19)lity forces f(z) = c1zc2 R for some constants c , c , which never belongs to L2( ). Next, take s = σ = 1, and from 1 2 − z − − z ∂ z 1 f(w)dw = z 2 zf(z) f(w)dw , z 0 0 47 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: 6 message and the dummy message) and uses it to re-encode his own message. In addition, the noiseless feedback ∗ ∗ ∗ is used to generate a pair of secret keys (K , K ), and K (j = 1, 2) is used to encrypt the sub-message W . 1 2 j j,1 C C • Comparing the above code construction of in with that of DF , the encoding and decoding schemes of these s s ∗ two bounds are almost the same, except that the sub-message W (j = 1, 2) is encrypted by a secret key K . j,1 j Thus the secrecy sum rate R + R is bounded by two part: the ﬁrst part is the upper bound on the sum rate 1 2 C ∗ ∗ of DF , and the second part is the upper bound on the rate of the secret keys K and K . Using the balanced s 1 2 ∗ ∗ coloring lemma introduced by Ahlswede and Cai [8], we conclude that the rate of the secret keys K and K 1 2 { | } C is bounded by min H(Y X , X , Z), I(X , X ; Z) . Thus, the hybrid inner bound in is obtained. 1 2 1 2 s The details of the proof are in Appendix A. Remark 2: C C Comparing the DF inner bound DF and the partial DF inner bound PDF with our hybrid new inner bound s s C C C C in, it is easy to see that our new inner bound in is strictly larger than DF and PDF . s s s s C Theorem 3: For the discrete memoryless MAC-WT with noiseless feedback, an outer bound out on the secrecy s C capacity region is given by s C { ≥ ≥ ≤ | } out = (R 0, R 0) : R + R H(Y Z) , s 1 2 1 2 for some distribution | · P | (z, y x , x ) P (x , x ). (2.7) Z,Y X1,X2 1 2 X1X2 1 2 C Proof: The outer bound out is a simple sato-type outer bound, and the proof is in Appendix B. s III. GAUSSIAN EXAMPLE A. Capacity Results on the Gaussian MAC-WT with Noiseless Feedback For the Gaussian case of the model of Figure 1, the channel inputs and outputs satisfy Y = X + X + N Z = X + X + N , (3.1) 1 2 1 1 2 2 ∼ N where the channel noises N and N are independent and Gaussian distributed, i.e., N (0, σ2), and 1 2 1 1 ∼ N N (0, σ2). The average power constraint of the transmitted signal X (j = 1, 2) is given by 2 2 j (cid:88) 1 N ≤ E[X2 ] P , j = 1, 2. (3.2) N ji j i=1 ##LTLine##

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text: DIALOG CONTEXT LANGUAGE MODELING WITH RECURRENT NEURAL NETWORKS Bing Liu1, Ian Lane1,2 1Electrical and Computer Engineering, Carnegie Mellon University 2Language Technologies Institute, Carnegie Mellon University liubing@cmu.edu, lane@cmu.edu ABSTRACT play an important role. Modeling utterances in a dialog as a sequence of inputs might not well capture the pauses, turn- In this work, we propose contextual language models that taking, and grounding phenomena [13] in a dialog. In this incorporate dialog level discourse information into language work, we propose contextual RNN language models that spe- modeling. Previous works on contextual language model cially track the interactions between speakers. We expect treat preceding utterances as a sequence of inputs, without such models to generate better representations of the dialog considering dialog interactions. We design recurrent neu- context. ral network (RNN) based contextual language models that The remainder of the paper is organized as follows. In specially track the interactions between speakers in a dialog. section 2, we introduce the background on contextual lan- Experiment results on Switchboard Dialog Act Corpus show guage modeling. In section 3, we describe the proposed dia- that the proposed model outperforms conventional single turn log context language models. Section 4 discusses the evalua- based RNN language model by 3.3% on perplexity. The pro- tion procedures and results. Section 5 concludes the work. posed models also demonstrate advantageous performance 2. BACKGROUND over other competitive contextual language models. Index Terms— RNNLM, contextual language model, di- 2.1. RNN Language Model alog modeling, dialog act A language model assigns a probability to a sequence of words w = (w , w , ..., w ) following probability distribu- 1. INTRODUCTION 1 2 T tion. Using the chain rule, the likelihood of the word sequence w can be factorized as: Language model plays an important role in many natu- (cid:89) ral language processing systems, such as in automatic speech recognition [1, 2] and machine translation systems [3, 4]. Re- T | current neural network (RNN) based models [5, 6] have re- P(w) = P(w1, w2, ..., wT ) = P(wt w<t) (1) cently shown success in language modeling, outperforming t=1 conventional n-gram based models. Long short-term memory [7, 8] is a widely used RNN variant for language modeling At time step t, the system input is the embedding of the due to its superior performance in capturing longer term de- word at index t, and the system output is the probability dis- pendencies. tribution of the word at index t + 1. The RNN hidden state ht encodes the information of the word sequence up till current Conventional RNN based language model uses a hidden step: state to represent the summary of the preceding words in a sentence without considering context signals. Mikolov et al. proposed a context dependent RNN language model [9] by h = RNN(h − , w ) (2) connecting a contextual vector to the RNN hidden state. This t | t 1 t contextual vector is produced by applying Latent Dirichlet P(wt+1 w<t+1) = softmax(Woht + bo) (3) Allocation [10] on preceding text. Several other contextual language models were later proposed by using bag-of-word where W and b are the output layer weights and biases. [11] and RNN methods [12] to learn larger context represen- o o 2.2. Contextual RNN Language Model tation that beyond the target sentence. The previously proposed contextual language models A number of methods have been proposed to introduce treat preceding sentences as a sequence of inputs, and they contextual information to the language model. Mikolov and are suitable for document level context modeling. In dia- Zweig [9] proposed a topic-conditioned RNNLM by intro- log modeling, however, dialog interactions between speakers ducing a contextual real-valued vector to RNN hidden state.

text: DIALOG

text: CONTEXT

text: LANGUAGE

text: MODELING

text: WITH

text: RECURRENT

text: NEURAL

text: NETWORKS

text: Bing

text: Ian

text: and

text: Computer

text: Engineering,

text: Carnegie

text: Mellon

text: University

text: Technologies

text: Institute,

text: Carnegie

text: Mellon

text: University

text: liubing@cmu.edu,

font: BXTVHX+NimbusMonL-Regu

text: lane@cmu.edu

text: ABSTRACT

text: play

text: an

text: important

text: role.

text: Modeling

text: utterances

text: in

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text: XXV European Cosmic Ray Symposium, Turin, Sept. 4-9 2016 3 TABLE II: Solar Orbiter scientiﬁc payload. Instrument Acronym Type Principal Investigator Energetic Particle Detector suite EPD IS J. Rodr´ıguez-Pacheco (Spain) Magnetometer experiment MAG IS T.S. Horbury (UK) Radio and Plasma Waves experiment RPW IS M. Maksimovic (France) Solar Wind Analyser instrument suite SWA IS C.J. Owen (UK) Extreme Ultraviolet Imager EUI RS P. Rochus (Belgium) Multi Element Telescope for Imaging METIS RS E. Antonucci (Italy) and Spectroscopy (Coronagraph) Polarimetric and Helioseismic Imager (Magnetograph) PHI RS S.K. Solanki (Germany) Solar Orbiter Heliospheric Imager SoloHI RS R.A. Howard (USA) Spectral Imaging of the Coronal Environment SPICE RS European-led (Extreme ultraviolet imaging spectrograph) facility instrument Spectrometer/Telescope for Imaging X-rays STIX RS S. Krucker (Switzerland) Suprathermal particles with energies above the am- bient plasma in the outer corona and the solar wind are known to play an important role as seed popu- lation for acceleration during SEP events. The vari- ability of this seed population may be a key factor to explain the wide range of intensities and compo- sition observed in SEP events. EPD will have the opportunity to perform IS measurements of the com- position and temporal variations of the suprathermal seed population close to the Sun, contributing also to understand the origins of the suprathermal ion pool itself. Energetic particles escaping from the acceleration sites continue their propagation through the turbu- lent interplanetary magnetic ﬁeld. As shown by He- FIG. 2: Energy windows covered by the EPD instruments lios observations [5], SEP events close to the Sun are for diﬀerent species. much less disturbed by interplanetary transport ef- fects compared to 1 AU observations. As the observ- ing s/c goes farther from the Sun, the interplanetary The energy intervals covered by the diﬀerent instru- scattering eﬀects become more important and often ments for various particle species are summarized in multiple injections closely spaced in time cannot be Figure 2. The four EPD sensors share a common In- resolved. For this reason, Solar Orbiter observations strument Control Unit (ICU). EPD instruments and close to the perihelion will be crucial to unveil SEP in- the ICU have signiﬁcant heritage from previous mis- jection, acceleration, trapping, release and transport sions, improved and optimized for the close approach processes. These observations will contribute to solve to the Sun. Figure 3 shows a picture of the whole the controversy about the SEP acceleration sites, dis- EPD instrument suite during the integration tests per- entangling the contribution of acceleration at CME- formed in July 2016. driven shocks and at reconnection sites in solar ﬂares or behind CMEs. The EPD sensors will measure the composition, EPD consists of four instruments measuring ener- spectra and anisotropies of energetic particles with getic electrons, protons and ions, operating at partly suﬃcient temporal, spectral, angular and mass res- overlapping energy ranges covering from few keV to olution to achieve the mission science goals. The ge- 450 MeV/nuc: ometric factors are scaled to avoid saturation by high particle ﬂuxes close to the perihelia. Since Solar Or- • SupraThermal Electrons and Protons (STEP) biter is a three-axis stabilized s/c, EPD uses multiple • apertures and sectoring to cover diﬀerent pointing di- Suprathermal Ion Spectrograph (SIS) rections, providing information about the directional • distribution of energetic particles reaching the s/c. Electron Proton Telescope (EPT) This information combined with the magnetic ﬁeld • High Energy Telescope (HET) data will be used to obtain energetic particle pitch eConf C16-09-04.3 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: XXV

text: European

text: Cosmic

text: Ray

text: Symposium,

text: Turin,

text: Sept.

text: TABLE

text: II:

text: Solar

label: caption

text: Orbiter

text: scientiﬁc

text: payload.

text: Instrument

label: table

text: Acronym

text: Type

text: Principal

text: Investigator

text: Energetic

text: Particle

text: Detector

text: suite

text: EPD

text: IS

text: J.

text: Rodr´ıguez-Pacheco

text: (Spain)

text: Magnetometer

text: experiment

text: MAG

text: IS

text: T.S.

text: Horbury

text: (UK)

text: Radio

text: and

text: Plasma

text: Waves

text: experiment

text: RPW

text: IS

text: M.

text: Maksimovic

text: (France)

text: Solar

text: Wind

text: Analyser

text: instrument

text: suite

text: SWA

text: IS

text: C.J.

text: Owen

text: (UK)

text: Extreme

text: Ultraviolet

text: Imager

text: EUI

text: RS

text: P.

text: Rochus

text: (Belgium)

text: Multi

text: Element

text: Telescope

text: for

text: Imaging

text: METIS

text: RS

text: E.

text: Antonucci

text: (Italy)

text: and

text: Spectroscopy

text: (Coronagraph)

text: Polarimetric

text: and

text: Helioseismic

text: Imager

text: (Magnetograph)

text: PHI

text: RS

text: S.K.

text: Solanki

text: (Germany)

text: Solar

text: Orbiter

text: Heliospheric

text: Imager

text: SoloHI

text: RS

text: R.A.

text: Howard

text: (USA)

text: Spectral

text: Imaging

text: of

text: the

text: Coronal

text: Environment

text: SPICE

text: RS

text: European-led

text: (Extreme

text: ultraviolet

text: imaging

text: spectrograph)

text: facility

text: instrument

text: Spectrometer/Telescope

text: for

text: Imaging

text: X-rays

text: STIX

text: RS

text: S.

text: Krucker

text: (Switzerland)

text: Suprathermal

text: particles

text: with

text: energies

text: above

text: the

text: am-

text: bient

text: plasma

text: in

text: the

text: outer

text: corona

text: and

text: the

text: solar

text: wind

text: are

text: known

text: to

text: play

text: an

text: important

text: role

text: as

text: seed

text: popu-

text: lation

text: for

text: acceleration

text: during

text: SEP

text: events.

text: The

text: vari-

text: ability

text: of

text: this

text: seed

text: population

text: may

text: be

text: a

text: key

text: factor

text: to

text: explain

text: the

text: wide

text: range

text: of

text: intensities

text: and

text: compo-

text: sition

text: observed

text: in

text: SEP

text: events.

text: EPD

text: will

text: have

text: the

text: opportunity

text: to

text: perform

text: IS

text: measurements

text: of

text: the

text: com-

text: position

text: and

text: temporal

text: variations

text: of

text: the

text: suprathermal

text: seed

text: population

text: close

text: to

text: the

text: Sun,

text: contributing

text: also

text: to

text: understand

text: the

text: origins

text: of

text: the

text: suprathermal

text: ion

text: pool

text: itself.

text: Energetic

text: particles

text: escaping

text: from

text: the

text: acceleration

text: sites

text: continue

text: their

text: propagation

text: through

text: the

text: turbu-

text: lent

text: interplanetary

text: magnetic

text: ﬁeld.

text: As

text: shown

text: by

text: He-

text: FIG.

text: Energy

text: windows

text: covered

text: by

text: the

text: EPD

text: instruments

text: lios

text: observations

text: SEP

text: events

text: close

text: to

text: the

text: Sun

text: are

text: for

text: diﬀerent

text: species.

text: much

text: less

text: disturbed

text: by

text: interplanetary

text: transport

text: ef-

text: fects

text: compared

text: to

text: AU

text: observations.

text: As

text: the

text: observ-

text: ing

text: s/c

text: goes

text: farther

text: from

text: the

text: Sun,

text: the

text: interplanetary

text: The

text: energy

text: intervals

text: covered

text: by

text: the

text: diﬀerent

text: instru-

text: scattering

text: eﬀects

text: become

text: more

text: important

text: and

text: often

text: ments

text: for

text: various

text: particle

text: species

text: are

text: summarized

text: in

text: multiple

text: injections

text: closely

text: spaced

text: in

text: time

text: cannot

text: be

text: Figure

text: The

text: four

text: EPD

text: sensors

text: share

text: a

text: common

text: In-

text: resolved.

text: For

text: this

text: reason,

text: Solar

text: Orbiter

text: observations

text: strument

text: Control

text: Unit

text: (ICU).

text: EPD

text: instruments

text: and

text: close

text: to

text: the

text: perihelion

text: will

text: be

text: crucial

text: to

text: unveil

text: SEP

text: in-

text: the

text: ICU

text: have

text: signiﬁcant

text: heritage

text: from

text: previous

text: mis-

text: jection,

text: acceleration,

text: trapping,

text: release

text: and

text: transport

text: sions,

text: improved

text: and

text: optimized

text: for

text: the

text: close

text: approach

text: processes.

text: These

text: observations

text: will

text: contribute

text: to

text: solve

text: to

text: the

text: Sun.

text: Figure

text: shows

text: a

text: picture

text: of

text: the

text: whole

text: the

text: controversy

text: about

text: the

text: SEP

text: acceleration

text: sites,

text: dis-

text: EPD

text: instrument

text: suite

text: during

text: the

text: integration

text: tests

text: per-

text: entangling

text: the

text: contribution

text: of

text: acceleration

text: at

text: CME-

text: formed

text: in

text: July

text: driven

text: shocks

text: and

text: at

text: reconnection

text: sites

text: in

text: solar

text: ﬂares

text: or

text: behind

text: CMEs.

text: The

text: EPD

text: sensors

text: will

text: measure

text: the

text: composition,

text: EPD

text: consists

text: of

text: four

text: instruments

text: measuring

text: ener-

text: spectra

text: and

text: anisotropies

text: of

text: energetic

text: particles

text: with

text: getic

text: electrons,

text: protons

text: and

text: ions,

text: operating

text: at

text: partly

text: suﬃcient

text: temporal,

text: spectral,

text: angular

text: and

text: mass

text: res-

text: overlapping

text: energy

text: ranges

text: covering

text: from

text: few

text: keV

text: to

text: olution

text: to

text: achieve

text: the

text: mission

text: science

text: goals.

text: The

text: ge-

text: MeV/nuc:

text: ometric

text: factors

text: are

text: scaled

text: to

text: avoid

text: saturation

text: by

text: high

text: particle

text: ﬂuxes

text: close

text: to

text: the

text: perihelia.

text: Since

text: Solar

text: Or-

text: SupraThermal

text: Electrons

text: and

text: Protons

text: (STEP)

text: biter

text: is

text: a

text: three-axis

text: stabilized

text: s/c,

text: EPD

text: uses

text: multiple

text: apertures

text: and

text: sectoring

text: to

text: cover

text: diﬀerent

text: pointing

text: di-

text: Suprathermal

text: Ion

text: Spectrograph

text: (SIS)

text: rections,

text: providing

text: information

text: about

text: the

text: directional

text: distribution

text: of

text: energetic

text: particles

text: reaching

text: the

text: s/c.

text: Electron

text: Proton

text: Telescope

text: (EPT)

text: This

text: information

text: combined

text: with

text: the

text: magnetic

text: ﬁeld

text: High

text: Energy

text: Telescope

text: (HET)

text: data

text: will

text: be

text: used

text: to

text: obtain

text: energetic

text: particle

text: pitch

text: eConf

text: ##LTFigure##

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