text: Finally, notice that the possible term c = D can appear only if Q = N because (D, N) = 1, and we apply (30) to it hence the δ terms in the Lemma. Q=N To conclude this paper, we prove how Theorem 1 implies Corollary 4. Proof of Corollary 4. ∈ (a) By Theorem 1 in case χ = 1, there is an eigenform f S2(Γ0(dp2))+p2 such that L(f, 1) is nonzero. By the famous result of Kolyvagin and Logachev ([KL90], Theorem 0.3), this implies that the abelian variety A associated to f obtained as a quotient of J(d, p) is of algebraic rank f zero. ∈ ⊗ (b) By Theorem 1, there is an eigenform f −S2(Γ0(dp2))+p2,new such that L(f χ, 1) is nonzero. Such an eigenform necessarily satisﬁes f| = χ(d)f because if f| = χ(d)f, L(f, 1) = 0 (Lemma 6 (c) because χ(−1) = 1 here). By Theowredm 0.3 of [KL90] appliedwtdo f ⊗χ, the abelian variety A obtained as a quotient of J(d, p) then has its twist by K/Q relatively to [−1] of algebraic rank zerfo (because this twist is isogenous over Q to the abelian variety A ⊗ ), and the canonical quotient → ◦ − f χ− morphism J(d, p) A satisﬁes π w = χ(d)π because f| = χ(d)f, therefore this twist is a rational quotient of Jf(d, p, χ). Indeedd, let i : J(d, p) → J(d,wpd, χ) be the twist isomorphism (i.e. ◦ → ⊗ deﬁned over K and such that jσ = χ(d)j w ) and i : A A χ (i.e. deﬁned over K such that − (cid:48) → d f f iσ = i, and consider π : J(d, p, χ) A ⊗ the natural quotient morphism making the diagram f f χ below commutative J(d, p) πf (cid:47)(cid:47) A . f j i (cid:15)(cid:15) (cid:48) (cid:15)(cid:15) J(d, p, χ) πf (cid:47)(cid:47) A ⊗ χ f (cid:48) Therefore, π is deﬁned over K and f (cid:48) ◦ ◦ − − ◦ ◦ ◦ − − ◦ ◦ − (cid:48) π σ = iσ πσ (j 1)σ = i π (χ(d)w j 1) = ( χ(d))2i π j 1 = π , f f f d f f ⊗ which proves that A χ is a rational quotient of J(d, p, χ). f (c) This is a consequence of (a) and (b) coming from the fact that for d prime to p and squarefree, there is a rational isogeny (cid:48) → ⊕ J (d, p) J(d, p) J (d) (32) 0 equivariant under the action of the Hecke algebra generated by the T , (cid:96) prime not dividing ∈ { (cid:96) } dp, and w such as it can be deﬁned naturally on both sides. For d 2, 3, 5, 7, 13 , J (d) = 0 d 0 hence the result. This fact is due to [Che00] for d = 1 and cited in [DM97], and its principle has been generalised to any d by [dSE00]. For the details, one proof can be found in [LF15a] (Lemma I.6.2). References [Akb97] Amir Akbary, Non-vanishing of modular L-functions with large level, PhD Thesis, ∼ http://www.cs.uleth.ca/ akbary/publications.html, 1997. [AL70] A. Atkin and J. Lehner, Hecke operators on Γ (m), Math. Ann. 185 (1970), 134–160. 0 [Bum96] Daniel Bump, Automorphic forms and representations, Cambridge University Press, 1996. [Che00] Imin Chen, On Relations between Jacobians of Certain Modular Curves, Journal of Al- gebra 231 (2000), no. 1, 414–448. 19 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Finally,

text: notice

text: that

text: the

text: possible

text: term

text: c

text: D

text: can

text: appear

text: only

text: if

text: Q

text: N

text: because

text: (D,

text: N)

text: and

text: we

text: apply

text: to

text: it

text: hence

text: the

text: δ

text: terms

text: in

text: the

text: Lemma.

text: Q=N

text: To

text: conclude

text: this

text: paper,

text: we

text: prove

text: how

text: Theorem

text: implies

text: Corollary

text: Proof

text: of

text: Corollary

text: (a)

text: By

text: Theorem

text: in

text: case

text: χ

text: there

text: is

text: an

text: eigenform

text: f

text: such

text: that

text: L(f,

text: is

text: nonzero.

text: By

text: the

text: famous

text: result

text: of

text: Kolyvagin

text: and

text: Logachev

text: Theorem

text: this

text: implies

text: that

text: the

text: abelian

text: variety

text: A

text: associated

text: to

text: f

text: obtained

text: as

text: a

text: quotient

text: of

text: J(d,

text: p)

text: is

text: of

text: algebraic

text: rank

text: f

text: zero.

text: (b)

text: By

text: Theorem

text: there

text: is

text: an

text: eigenform

text: f

text: such

text: that

text: L(f

text: χ,

text: is

text: nonzero.

text: Such

text: an

text: eigenform

text: necessarily

text: satisﬁes

text: f|

text: χ(d)f

text: because

text: if

text: f|

text: χ(d)f,

text: L(f,

text: (Lemma

text: (c)

text: because

text: here).

text: By

text: Theowredm

text: of

text: appliedwtdo

text: f

text: ⊗χ,

text: the

text: abelian

text: variety

text: A

text: obtained

text: as

text: a

text: quotient

text: of

text: J(d,

text: p)

text: then

text: has

text: its

text: twist

text: by

text: K/Q

text: relatively

text: to

text: of

text: algebraic

text: rank

text: zerfo

text: (because

text: this

text: twist

text: is

text: isogenous

text: over

text: Q

text: to

text: the

text: abelian

text: variety

text: A

text: and

text: the

text: canonical

text: quotient

text: f

text: χ−

text: morphism

text: J(d,

text: p)

text: A

text: satisﬁes

text: π

text: w

text: χ(d)π

text: because

text: f|

text: χ(d)f,

text: therefore

text: this

text: twist

text: is

text: a

text: rational

text: quotient

text: of

text: Jf(d,

text: p,

text: χ).

text: Indeedd,

text: let

text: i

text: J(d,

text: p)

text: J(d,wpd,

text: χ)

text: be

text: the

text: twist

text: isomorphism

text: (i.e.

text: deﬁned

text: over

text: K

text: and

text: such

text: that

text: jσ

text: χ(d)j

text: w

text: and

text: i

text: A

text: A

text: χ

text: (i.e.

text: deﬁned

text: over

text: K

text: such

text: that

text: d

text: f

text: f

text: iσ

text: i,

text: and

text: consider

text: π

text: J(d,

text: p,

text: χ)

text: A

text: the

text: natural

text: quotient

text: morphism

text: making

text: the

text: diagram

text: f

text: f

text: χ

text: below

text: commutative

text: J(d,

text: p)

text: πf

text: A

text: f

text: j

text: i

text: J(d,

text: p,

text: χ)

text: πf

text: A

text: χ

text: f

text: Therefore,

text: π

text: is

text: deﬁned

text: over

text: K

text: and

text: f

text: π

text: σ

text: iσ

text: πσ

text: (j

text: i

text: π

text: (χ(d)w

text: j

text: π

text: j

text: π

text: f

text: f

text: f

text: d

text: f

text: f

text: which

text: proves

text: that

text: A

text: χ

text: is

text: a

text: rational

text: quotient

text: of

text: J(d,

text: p,

text: χ).

text: f

text: (c)

text: This

text: is

text: a

text: consequence

text: of

text: (a)

text: and

text: (b)

text: coming

text: from

text: the

text: fact

text: that

text: for

text: d

text: prime

text: to

text: p

text: and

text: squarefree,

text: there

text: is

text: a

text: rational

text: isogeny

text: J

text: (d,

text: p)

text: J(d,

text: p)

text: J

text: (d)

text: equivariant

text: under

text: the

text: action

text: of

text: the

text: Hecke

text: algebra

text: generated

text: by

text: the

text: T

text: prime

text: not

text: dividing

text: dp,

text: and

text: w

text: such

text: as

text: it

text: can

text: be

text: deﬁned

text: naturally

text: on

text: both

text: sides.

text: For

text: d

text: J

text: (d)

text: d

text: hence

text: the

text: result.

text: This

text: fact

text: is

text: due

text: to

text: for

text: d

text: and

text: cited

text: in

text: and

text: its

text: principle

text: has

text: been

text: generalised

text: to

text: any

text: d

text: by

text: For

text: the

text: details,

text: one

text: proof

text: can

text: be

text: found

text: in

text: (Lemma

text: References

text: Amir

text: Akbary,

text: Non-vanishing

text: of

text: modular

text: L-functions

text: with

text: large

text: level,

text: PhD

text: Thesis,

text: akbary/publications.html,

text: A.

text: Atkin

text: and

text: J.

text: Lehner,

text: Hecke

text: operators

text: on

text: Γ

text: (m),

text: Math.

text: Ann.

text: Daniel

text: Bump,

text: Automorphic

text: forms

text: and

text: representations,

text: Cambridge

text: University

text: Press,

text: Imin

text: Chen,

text: On

text: Relations

text: between

text: Jacobians

text: of

text: Certain

text: Modular

text: Curves,

text: Journal

text: of

text: Al-

text: gebra

text: no.

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: The Bessel function of the ﬁrst kind a(cid:88)nd order 1 is (cid:16)the(cid:17)entire function J deﬁned by 1 ∞ − + ( 1)n z 2n+1 J (z) = . 1 n!(n + 1)! 2 n=0 It has the following integral representation ([W(cid:90)at22], 6.21, Formula 8) J (z) = z x+i∞ ew−4zw2 dw (23) 1 4iπ − ∞ w2 x i for all z ∈ C and all x > 0. The goal of this subsection is to prove Propositions 3 and 14. With our notations, the left-hand term of (1) is exactly √1 (a , a )εQ. 2π mn m n N Before the proof, let us make some remarks about Proposition 3. Remark 10. − (a) Summing for any Q the formulas for ε = 1 and ε = 1, we recover the original Petersson ≥ trace formula ([IK04], Proposition 14.5), which generalises to every weight k 2. However, its proof for k = 2 is more involved because the Poincaré series cannot be deﬁned as uniformly convergent series so we will focus on this case (it is also the only one we need), but it is very likely to be generalised to k > 2 as well. The trace formula above has been originally proved for Q = N in the Chapter 3 of [Akb97], but to our knowledge, not for any other Q. (b) For Q = N prime, formula (1) can be found (in a diﬀerent form) in ([IK04], Proposition 14.25). Notice that there is a mistake in one of the arguments of J in that book, as it should 1≥ actually be rewritten (with our notations) for prim(cid:88)e level q and m,(cid:18)n 1 : (cid:19) √ √ (2aπm√, amnn)εq = δmn − 2π qδm,nq − 2π | S(cid:88)(mc, n; c)J1 4π (cid:18)cmn (cid:19) (24) q c √ √ S(m, nq; c) 4π mnq + 2πε q J . | c 1 c q c The proof of this result is simply based on the natural system of formulas (which crucially uses Theorem 3 of [AL70] hence the hypothesis q prime) − (a , a ) = (a , a )+ + (a , a ) m n q m n q m n q − − (a , a ) = (a , a )+ + (a , a ) m nq q m n q m n q combined with the original Petersson trace formula. Notice there is a δ appearing here m,nq but not in (1) so even under this form, the fact that formulas (1) and (24) are the same is not ≥ || obvious. To see it, we use that for c 1 such that q c, − − S(m, nq; c) = S(m, nq 1; c/q) ≥ (e.g. by Theorem 68 of [HW08]), and that for all m, n 1, we(cid:18)can check(cid:19)(separating between (cid:88) oldforms and newforms) that √ − S(m, nq; c) 4π mnq 0 = (a , a ) = δ 2π J , m nq q2 m,nq | c 1 c q2 c and we readily obtain the equivalence using these two results. 10 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: The

label: paragraph

text: Bessel

text: function

text: of

text: the

text: ﬁrst

text: kind

text: order

text: is

text: function

text: J

text: deﬁned

text: by

text: z

text: J

text: (z)

text: n!(n

text: It

text: has

text: the

text: following

text: integral

text: representation

label: equation

text: Formula

text: J

text: (z)

text: z

text: x+i∞

text: dw

text: x

text: i

text: for

text: all

text: z

text: C

text: and

text: all

text: x

text: The

text: goal

text: of

text: this

text: subsection

text: is

text: to

text: prove

text: Propositions

text: and

text: With

text: our

text: notations,

text: the

text: left-hand

text: term

text: of

text: is

text: exactly

text: √1

text: (a

text: a

text: )εQ.

text: 2π

text: mn

text: m

text: n

text: N

text: Before

text: the

text: proof,

text: let

text: us

text: make

text: some

text: remarks

text: about

text: Proposition

text: Remark

text: (a)

text: Summing

text: for

text: any

text: Q

text: the

text: formulas

text: for

text: ε

text: and

text: ε

text: we

text: recover

text: the

text: original

text: Petersson

text: trace

text: formula

text: Proposition

text: which

text: generalises

text: to

text: every

text: weight

text: k

text: However,

text: its

text: proof

text: for

text: k

text: is

text: more

text: involved

text: because

text: the

text: Poincaré

text: series

text: cannot

text: be

text: deﬁned

text: as

text: uniformly

text: convergent

text: series

text: so

text: we

text: will

text: focus

text: on

text: this

text: case

text: (it

text: is

text: also

text: the

text: only

text: one

text: we

text: need),

text: but

text: it

text: is

text: very

text: likely

text: to

text: be

text: generalised

text: to

text: k

text: as

text: well.

text: The

text: trace

text: formula

text: above

text: has

text: been

text: originally

text: proved

text: for

text: Q

text: N

text: in

text: the

text: Chapter

text: of

text: but

text: to

text: our

text: knowledge,

text: not

text: for

text: any

text: other

text: Q.

text: (b)

text: For

text: Q

text: N

text: prime,

text: formula

text: can

text: be

text: found

text: (in

text: a

text: diﬀerent

text: form)

text: in

text: Proposition

text: Notice

text: that

text: there

text: is

text: a

text: mistake

text: in

text: one

text: of

text: the

text: arguments

text: of

text: J

text: in

text: that

text: book,

text: as

text: it

text: should

text: 1≥

text: actually

text: be

text: rewritten

text: (with

text: our

text: notations)

text: for

text: level

text: q

text: and

text: amnn)εq

text: δmn

text: 2π

text: qδm,nq

text: 2π

text: n;

text: 4π

text: q

text: c

text: S(m,

text: nq;

text: c)

text: 4π

text: mnq

text: 2πε

text: q

text: J

text: c

text: c

text: q

text: c

text: The

text: proof

text: of

text: this

text: result

text: is

text: simply

text: based

text: on

text: the

text: natural

text: system

text: of

text: formulas

text: (which

text: crucially

text: uses

text: Theorem

text: of

text: hence

text: the

text: hypothesis

text: q

text: prime)

text: (a

text: a

text: (a

text: a

text: (a

text: a

text: m

text: n

text: q

text: m

text: n

text: q

text: m

text: n

text: q

text: (a

text: a

text: (a

text: a

text: (a

text: a

text: m

text: nq

text: q

text: m

text: n

text: q

text: m

text: n

text: q

text: combined

text: with

text: the

text: original

text: Petersson

text: trace

text: formula.

text: Notice

text: there

text: is

text: a

text: δ

text: appearing

text: here

text: m,nq

text: but

text: not

text: in

text: so

text: even

text: under

text: this

text: form,

text: the

text: fact

text: that

text: formulas

text: and

text: are

text: the

text: same

text: is

text: not

text: obvious.

text: To

text: see

text: it,

text: we

text: use

text: that

text: for

text: c

text: such

text: that

text: q

text: c,

text: S(m,

text: nq;

text: c)

text: S(m,

text: nq

text: c/q)

text: (e.g.

text: by

text: Theorem

text: of

text: and

text: that

text: for

text: all

text: m,

text: n

text: between

text: oldforms

text: and

text: newforms)

text: that

text: S(m,

text: nq;

text: c)

text: 4π

text: mnq

text: (a

text: a

text: δ

text: 2π

text: J

text: m

text: nq

text: m,nq

text: c

text: c

text: c

text: and

text: we

text: readily

text: obtain

text: the

text: equivalence

text: using

text: these

text: two

text: results.

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##

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text: Journal of Machine Learning Research 1 (2000) 1-48 Submitted 4/00; Published 10/00 Sharp oracle bounds for monotone and convex regression through aggregation Pierre C. Bellec pierre.bellec@ensae.fr and Alexandre B. Tsybakov alexandre.tsybakov@ensae.fr ENSAE, 3 avenue Pierre Larousse 92240 Malakoﬀ, France Editor: Vladimir Vapnik, Alex Gammerman, and Vladimir Vovk Abstract We derive oracle inequalities for the problems of isotonic and convex regression using the combination of -aggregation procedure and sparsity pattern aggregation. This improves Q upon the previous results including the oracle inequalities for the constrained least squares estimator. One of the improvements is that our oracle inequalities are sharp, i.e., with leading constant 1. It allows us to obtain bounds for the minimax regret thus accounting for model misspeciﬁcation, which was not possible based on the previous results. Another improvement is that we obtain oracle inequalities both with high probability and in expectation. 1. Introduction Assume that we have the observations = + = 1 (1.1) Y µ ξ , i , ..., n, i i i where µ = (µ1, ..., µ )T ∈ Rn is unknown, ξ = (ξ1, ..., ξ )T is a noise vector with n- dimensional Gaussian disntribution N (0, σ2I × ) where σ > 0.nWe observe y = (Y1, ..., Y )T n n n and we want to estimate . We can interpret as the values ( ) of an unknown regression µ µ f X function : X → R at given non-random pointis ∈ X , = 1 i , where X is an abstract f X i , . . . , n i set. Then, the equivalent setting is that we observe y along with ( ) but the values X1, . . . , X n of are of no interest and can be replaced by their indices if we measure the loss in a X discriete norm. Namely, for any u ∈ Rn we consider the scaled (or the empirical) norm k · k deﬁned by X k k2 = 1 n 2 (1.2) u u . n =1 i i We will measure the error of an estimator ˆ of by the distance kˆ − k. Let S↑ be µ µ µ µ the set of all non-decreasing sequences: S↑ := {u = (u1, ..., u ) ∈ Rn : u ≤ u +1, i = 1, ..., n − 1}. (1.3) n i i ©2000 Pierre C. Bellec and Alexandre B. Tsybakov. ##LTLine##

text: Journal

text: of

text: Machine

text: Learning

text: Research

text: Submitted

text: Published

text: Sharp

label: title

text: oracle

text: bounds

text: for

text: monotone

text: and

text: convex

text: regression

text: through

text: aggregation

text: Pierre

label: author

text: C.

text: Bellec

text: pierre.bellec@ensae.fr

text: and

text: Alexandre

text: B.

text: Tsybakov

text: alexandre.tsybakov@ensae.fr

text: ENSAE,

text: avenue

text: Pierre

text: Larousse

text: Malakoﬀ,

text: France

text: Editor:

text: Vladimir

text: Vapnik,

text: Alex

text: Gammerman,

text: and

text: Vladimir

text: Vovk

text: Abstract

text: We

label: abstract

text: derive

text: oracle

text: inequalities

text: for

text: the

text: problems

text: of

text: isotonic

text: and

text: convex

text: regression

text: using

text: the

text: combination

text: of

text: -aggregation

text: procedure

text: and

text: sparsity

text: pattern

text: aggregation.

text: This

text: improves

text: Q

text: upon

text: the

text: previous

text: results

text: including

text: the

text: oracle

text: inequalities

text: for

text: the

text: constrained

text: least

text: squares

text: estimator.

text: One

text: of

text: the

text: improvements

text: is

text: that

text: our

text: oracle

text: inequalities

text: are

text: sharp,

text: i.e.,

text: with

text: leading

text: constant

text: It

text: allows

text: us

text: to

text: obtain

text: bounds

text: for

text: the

text: minimax

text: regret

text: thus

text: accounting

text: for

text: model

text: misspeciﬁcation,

text: which

text: was

text: not

text: possible

text: based

text: on

text: the

text: previous

text: results.

text: Another

text: improvement

text: is

text: that

text: we

text: obtain

text: oracle

text: inequalities

text: both

text: with

text: high

text: probability

text: and

text: in

text: expectation.

text: Introduction

text: Assume

text: that

text: we

text: have

text: the

text: observations

text: Y

text: µ

text: ξ

text: i

text: n,

text: i

text: i

text: i

text: where

text: µ

text: µ

text: )T

text: Rn

text: is

text: unknown,

text: ξ

text: ξ

text: )T

text: is

text: a

text: noise

text: vector

text: with

text: n-

text: dimensional

text: Gaussian

text: disntribution

text: N

text: where

text: σ

text: observe

text: y

text: Y

text: )T

text: n

text: n

text: n

text: and

text: we

text: want

text: to

text: estimate

text: We

text: can

text: interpret

text: as

text: the

text: values

text: of

text: an

text: unknown

text: regression

text: µ

text: µ

text: f

text: X

text: function

text: X

text: R

text: at

text: given

text: non-random

text: pointis

text: X

text: i

text: where

text: X

text: is

text: an

text: abstract

text: f

text: X

text: i

text: n

text: i

text: set.

text: Then,

text: the

text: equivalent

text: setting

text: is

text: that

text: we

text: observe

text: y

text: along

text: with

text: but

text: the

text: values

text: X

text: n

text: of

text: are

text: of

text: no

text: interest

text: and

text: can

text: be

text: replaced

text: by

text: their

text: indices

text: if

text: we

text: measure

text: the

text: loss

text: in

text: a

text: X

text: discriete

text: norm.

text: Namely,

text: for

text: any

text: u

text: Rn

text: we

text: consider

text: the

text: scaled

text: (or

text: the

text: empirical)

text: norm

text: k

text: k

text: deﬁned

text: by

text: X

text: k

text: n

text: u

text: u

text: n

text: i

text: i

text: We

text: will

text: measure

text: the

text: error

text: of

text: an

text: estimator

text: ˆ

text: of

text: by

text: the

text: distance

text: kˆ

text: k.

text: Let

text: S↑

text: be

text: µ

text: µ

text: µ

text: µ

text: the

text: set

text: of

text: all

text: non-decreasing

text: sequences:

text: S↑

text: {u

text: u

text: Rn

text: u

text: u

text: i

text: n

text: n

text: i

text: i

text: Pierre

text: C.

text: Bellec

text: and

text: Alexandre

text: B.

text: Tsybakov.

text: ##LTLine##


text: Bellec and Tsybakov for all µ ∈ Rn. It follows from (1.9) that if the tuning parameter is chosen correctly, the estimator µˆTV achieves, up to a logarithmic factor, the minimax rate n−2/3 in probability on the class S↑(V ). Also, (1.9) implies a bound for the excess losses kµˆTV −µki−min ∈S↑( ) ku− µki, i = 1, 2, corresponding to the class S↑p(V ). However, (1.9) does not allow usu to eVvaluate the expected regrets R (µˆTV , µ) since µˆTV depends on δ. It is also shown in (Dalalyan et al., 2014, Prop(cid:13)ositioni 4)(cid:13) that if λ = 2σ (2/n) log(n/δ), the estimator!µˆTV satisﬁes (cid:13) (cid:13) (cid:13) (cid:13) 2 4 2 ( ) log( ) µˆTV − µ ≤ min ku − µk2 + σ k u n/δ r (u) (1.10) u∈Rn n n with probability greater than 1 − 2δ, where k(u) − 1 for u ∈ Rn is the number of { ∈ { − } 6 } jumps of , i.e., the cardinality of the set 1 1 : = , ( ) = u i , ..., n u u +1 r u i i n 3 + 256(log( ) + ( ∆( ))) and ∆( ) is the minimum distance between two jumps in the n n/ u u sequence : u { ≥ ∃ ∈ { } 6 6 } ∆( ) = min 1 : 1 with = and = u d k , ..., n u +1 u u + +1 u + . k k k d k d The expressions on the right hand sides of (1.6) and (1.10) are small if the unknown sequence is well approximated by a piecewise constant sequence with not too many µ pieces. In this regard, the two bounds have some similarity to sparsity oracle inequalities in high-dimensional linear regression, cf. Rigollet and Tsybakov (2011, 2012); Tsybakov (2014). This similarity can be easily explained as follows. Write (1.1) in the equivalent form y = X ∗ + β ξ, with the matrix X = ( ) where = 1 if ≤ and = 0 otherwise, and X =1 =1 X j i X β∗ = (β1∗, . . . , β∗) whereijβi1∗ =,...µ,n1, jand,...β,n∗ = µ − µij−1 for i = 2, . . . , n.ijWith this notation, ∈ {| ∗| n | ∗| } | ∗| i i i ∗ ( ) 1+ , where denotes the number of non-zero components of . The k µ β 0, β 0 β 0 β ∗ value ( ) is small when is sparse. Thus, the problem of estimation of piecewise constant k µ β sequence with small number of pieces can be considered as the problem of prediction in µ sparse linear regression with a speciﬁc design matrix X. Similarly, we may write = X , u β − for β with components β1 = u1 and β = u u −1 for i = 2, . . . , n. These remarks suggest i i i that we can apply the theory of sparsity oracle inequalities, in particular, sparsity pattern aggregation (cf. Rigollet and Tsybakov (2011, 2012); Tsybakov (2014)) in the context of monotone estimation described above. Similar observation is valid for estimation under convexity constraints (see Section below). In the present paper, we develop this argument using as a building block the -aggregation procedures Rigollet (2012); Dai et al. (2012, Q 2014); Bellec (2014). In particula r, we construct an estimator ˆ!such that µ 2 ( ) Eµ kµˆ − µk2 ≤ um∈iSn↑ kµ − uk2 + cσ nk u log ke(nu) , ∀ µ ∈ Rn, (1.11) for some absolute constant 0. Note that (1.11) is a sharp oracle inequality (i.e., an c > inequality with leading constant 1). It improves upon the oracle inequality (1.6) for the least squares estimator where the leading constant C∗ is noticeably greater than 1 and the 4 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Bellec

text: and

text: Tsybakov

text: for

text: all

text: µ

text: Rn.

text: It

text: follows

text: from

text: that

text: if

text: the

text: tuning

text: parameter

text: is

text: chosen

text: correctly,

text: the

text: estimator

text: µˆTV

text: achieves,

text: up

text: to

text: a

text: logarithmic

text: factor,

text: the

text: minimax

text: rate

text: in

text: probability

text: on

text: the

text: class

text: S↑(V

text: Also,

text: implies

text: a

text: bound

text: for

text: the

text: excess

text: losses

text: kµˆTV

text: −µki−min

text: ∈S↑(

text: ku−

text: µki,

text: i

text: corresponding

text: to

text: the

text: class

text: S↑p(V

text: However,

text: does

text: not

text: allow

text: usu

text: to

text: eVvaluate

text: the

text: expected

text: regrets

text: R

text: (µˆTV

text: µ)

text: since

text: µˆTV

text: depends

text: on

text: δ.

text: It

text: is

text: also

text: shown

text: in

text: (Dalalyan

text: et

text: al.,

text: that

text: if

text: λ

text: 2σ

text: log(n/δ),

text: the

text: estimator!µˆTV

text: satisﬁes

text: log(

text: µˆTV

text: µ

text: min

text: ku

text: σ

text: k

text: u

text: n/δ

text: r

text: (u)

text: u∈Rn

text: n

text: n

text: with

text: probability

text: greater

text: than

text: where

text: k(u)

text: for

text: u

text: Rn

text: is

text: the

text: number

text: of

text: jumps

text: of

text: i.e.,

text: the

text: cardinality

text: of

text: the

text: set

text: u

text: i

text: n

text: u

text: u

text: r

text: u

text: i

text: i

text: n

text: and

text: is

text: the

text: minimum

text: distance

text: between

text: two

text: jumps

text: in

text: the

text: n

text: n/

text: u

text: u

text: sequence

text: u

text: min

text: with

text: and

text: u

text: d

text: k

text: n

text: u

text: u

text: u

text: u

text: k

text: k

text: k

text: d

text: k

text: d

text: The

text: expressions

text: on

text: the

text: right

text: hand

text: sides

text: of

text: and

text: are

text: small

text: if

text: the

text: unknown

text: sequence

text: is

text: well

text: approximated

text: by

text: a

text: piecewise

text: constant

text: sequence

text: with

text: not

text: too

text: many

text: µ

text: pieces.

text: In

text: this

text: regard,

text: the

text: two

text: bounds

text: have

text: some

text: similarity

text: to

text: sparsity

text: oracle

text: inequalities

text: in

text: high-dimensional

text: linear

text: regression,

text: cf.

text: Rigollet

text: and

text: Tsybakov

text: Tsybakov

text: This

text: similarity

text: can

text: be

text: easily

text: explained

text: as

text: follows.

text: Write

text: in

text: the

text: equivalent

text: form

text: y

text: X

text: β

text: ξ,

text: with

text: the

text: matrix

text: X

text: where

text: if

text: and

text: otherwise,

text: and

text: X

text: X

text: j

text: i

text: X

text: β∗

text: β∗)

text: jand,...β,n∗

text: µ

text: for

text: i

text: n.ijWith

text: this

text: notation,

text: n

text: i

text: i

text: i

text: where

text: denotes

text: the

text: number

text: of

text: non-zero

text: components

text: of

text: The

text: k

text: µ

text: β

text: β

text: β

text: β

text: value

text: is

text: small

text: when

text: is

text: sparse.

text: Thus,

text: the

text: problem

text: of

text: estimation

text: of

text: piecewise

text: constant

text: k

text: µ

text: β

text: sequence

text: with

text: small

text: number

text: of

text: pieces

text: can

text: be

text: considered

text: as

text: the

text: problem

text: of

text: prediction

text: in

text: µ

text: sparse

text: linear

text: regression

text: with

text: a

text: speciﬁc

text: design

text: matrix

text: X.

text: Similarly,

text: we

text: may

text: write

text: X

text: u

text: β

text: for

text: β

text: with

text: components

text: β1

text: and

text: β

text: u

text: u

text: −1

text: for

text: i

text: n.

text: These

text: remarks

text: suggest

text: i

text: i

text: i

text: that

text: we

text: can

text: apply

text: the

text: theory

text: of

text: sparsity

text: oracle

text: inequalities,

text: in

text: particular,

text: sparsity

text: pattern

text: aggregation

text: (cf.

text: Rigollet

text: and

text: Tsybakov

text: Tsybakov

text: in

text: the

text: context

text: of

text: monotone

text: estimation

text: described

text: above.

text: Similar

text: observation

text: is

text: valid

text: for

text: estimation

text: under

text: convexity

text: constraints

text: (see

text: Section

text: below).

text: In

text: the

text: present

text: paper,

text: we

text: develop

text: this

text: argument

text: using

text: as

text: a

text: building

text: block

text: the

text: -aggregation

text: procedures

text: Rigollet

text: Dai

text: et

text: al.

text: Q

text: Bellec

text: In

text: particula

text: r,

text: we

text: construct

text: an

text: estimator

text: ˆ!such

text: that

text: µ

text: Eµ

text: kµˆ

text: um∈iSn↑

text: kµ

text: cσ

text: nk

text: u

text: log

text: ke(nu)

text: µ

text: Rn,

text: for

text: some

text: absolute

text: constant

text: Note

text: that

text: is

text: a

text: sharp

text: oracle

text: inequality

text: (i.e.,

text: an

text: c

text: inequality

text: with

text: leading

text: constant

text: It

text: improves

text: upon

text: the

text: oracle

text: inequality

text: for

text: the

text: least

text: squares

text: estimator

text: where

text: the

text: leading

text: constant

text: C∗

text: is

text: noticeably

text: greater

text: than

text: and

text: the

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Monotone and convex regression through aggregation bound is valid only for ∈ S↑. The advantage of having leading constant 1 and arbitrary µ in (1.11) is that it allows us to derive bounds on the excess risk and on the minimax µ regret, which was not possible based on the previous results. We also obtain sharp oracle inequalities with high probability for the same estimator. In addition, we show that it satisﬁes stronger sharp inequalities with the minimum min on the right hand side of ∈S↑ u (1.11) replaced by min ∈R . This implies that our results are invariant to the direction of u n monotonicity; they remain valid if we replace everywhere monotone increasing by monotone decreasing functions. Finally, we derive similar results for the problem of estimation under the convexity constraints improving an oracle inequality obtained in Guntuboyina and Sen (2013). 2. Sparsity pattern aggregation for piecewise constant sequences ⊆ { − } | | For any non-empty set 1 1 , let denote the cardinality of and deﬁne J , ..., n J X J (cid:0) (cid:1) −| | −1 := exp( J ) := n exp(− ) (2.1) πJ H n|−|1 , H =0 i . J i Let P ∈ Rn×n be thenprojector on the linear subspace V of Rn doeﬁned by J J V := u ∈ Rn : ∀i ∈ {1, ..., n − 1} \ J, u +1 = u . (2.2) J i i In words, is the space of all piecewise constant sequences that have jumps only at V points in . GiJven a vector y of observations and = ( ) where each ∈ R, J θ θ ⊆{1 −1} θ J J ,...,n J let X = y (2.3) µ θ P . θ J J ⊆{1 −1} J ,...,n Finally, let µˆQ = µˆ (2.4) θ where ˆ is the solution of the optimiz ation problem ! θ X 2 2| | 1 46 2 1 min k − yk2 + σ J + k − yk2 + σ log θ∈Λ µθ ⊆{1 −1} θJ n 2 µθ PJ n πJ  J ,...,n   X  where   ≥ ⊆ { − } Λ = : 0 for all 1 1 and = 1 θ θ J , ..., n , θ . J J ⊆{1 −1} J ,...,n This optimization problem is a convex quadratic program with a simplex constraint. It performs aggregation of the linear estimators ( y) using the -aggregation P ⊆{1 −1} Q J J ,...,n procedure Dai et al. (2012, 2014); Bellec (2014) with the prior weights (2.1). As the size of 5 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Monotone

text: and

text: convex

text: regression

text: through

text: aggregation

text: bound

text: is

text: valid

text: only

text: for

text: S↑.

text: The

text: advantage

text: of

text: having

text: leading

text: constant

text: and

text: arbitrary

text: µ

text: in

text: is

text: that

text: it

text: allows

text: us

text: to

text: derive

text: bounds

text: on

text: the

text: excess

text: risk

text: and

text: on

text: the

text: minimax

text: µ

text: regret,

text: which

text: was

text: not

text: possible

text: based

text: on

text: the

text: previous

text: results.

text: We

text: also

text: obtain

text: sharp

text: oracle

text: inequalities

text: with

text: high

text: probability

text: for

text: the

text: same

text: estimator.

text: In

text: addition,

text: we

text: show

text: that

text: it

text: satisﬁes

text: stronger

text: sharp

text: inequalities

text: with

text: the

text: minimum

text: min

text: on

text: the

text: right

text: hand

text: side

text: of

text: ∈S↑

text: u

text: replaced

text: by

text: min

text: ∈R

text: This

text: implies

text: that

text: our

text: results

text: are

text: invariant

text: to

text: the

text: direction

text: of

text: u

text: n

text: monotonicity;

text: they

text: remain

text: valid

text: if

text: we

text: replace

text: everywhere

text: monotone

text: increasing

text: by

text: monotone

text: decreasing

text: functions.

text: Finally,

text: we

text: derive

text: similar

text: results

text: for

text: the

text: problem

text: of

text: estimation

text: under

text: the

text: convexity

text: constraints

text: improving

text: an

text: oracle

text: inequality

text: obtained

text: in

text: Guntuboyina

text: and

text: Sen

text: Sparsity

text: pattern

text: aggregation

text: for

text: piecewise

text: constant

text: sequences

text: For

text: any

text: non-empty

text: set

text: let

text: denote

text: the

text: cardinality

text: of

text: and

text: deﬁne

text: J

text: n

text: J

text: X

text: J

text: −1

text: exp(

text: J

text: n

text: exp(−

text: πJ

text: H

text: H

text: i

text: J

text: i

text: Let

text: P

text: Rn×n

text: be

text: thenprojector

text: on

text: the

text: linear

text: subspace

text: V

text: of

text: Rn

text: doeﬁned

text: by

text: J

text: J

text: V

text: u

text: Rn

text: ∀i

text: n

text: J,

text: u

text: u

text: J

text: i

text: i

text: In

text: words,

text: is

text: the

text: space

text: of

text: all

text: piecewise

text: constant

text: sequences

text: that

text: have

text: jumps

text: only

text: at

text: V

text: points

text: in

text: GiJven

text: a

text: vector

text: y

text: of

text: observations

text: and

text: where

text: each

text: R,

text: J

text: θ

text: θ

text: θ

text: J

text: J

text: ,...,n

text: J

text: let

text: X

text: y

text: µ

text: θ

text: P

text: θ

text: J

text: J

text: J

text: ,...,n

text: Finally,

text: let

text: µˆQ

text: µˆ

text: θ

text: where

text: ˆ

text: is

text: the

text: solution

text: of

text: the

text: optimiz

text: ation

text: problem

text: θ

text: X

text: min

text: k

text: σ

text: J

text: k

text: σ

text: log

text: θ∈Λ

text: µθ

text: θJ

text: n

text: µθ

text: PJ

text: n

text: πJ

text: J

text: ,...,n

text: X

text: where

text: Λ

text: for

text: all

text: and

text: θ

text: θ

text: J

text: n

text: θ

text: J

text: J

text: J

text: ,...,n

text: This

text: optimization

text: problem

text: is

text: a

text: convex

text: quadratic

text: program

text: with

text: a

text: simplex

text: constraint.

text: It

text: performs

text: aggregation

text: of

text: the

text: linear

text: estimators

text: y)

text: using

text: the

text: -aggregation

text: P

text: Q

text: J

text: J

text: ,...,n

text: procedure

text: Dai

text: et

text: al.

text: Bellec

text: with

text: the

text: prior

text: weights

text: As

text: the

text: size

text: of

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Bellec and Tsybakov this quadratic program is of order 2n, it is a computationally hard problem. The estimator µˆQ satisﬁes the following sharp oracle inequalities. Theorem 1 Let µ ∈ Rn, n ≥ 2, and assume that the noise vector ξ has distribution N (0, σ2I × ). There exist absolute constants c, c0 > 0 such that for all δ ∈ (0, 1/3), the estimatornµˆn(cid:13)Q satisﬁe(cid:13)s with pro bability at least 1 − 3δ, ! (cid:13) (cid:13) (cid:13) (cid:13) 2 2 ( )) 2 log(1 ) µˆQ − µ ≤ um∈iRnn kµ − uk2 + cσ kn u log ke(nu) + cσ n /δ , (2.5) and ! (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) 2 0 2 ( ) Eµ µˆQ − µ ≤ um∈iRnn kµ − uk2 + c σ nk u log ke(nu) . (2.6) ⊆ { − } | | Proof Let 1 1 . Denote by = + 1 the dimension of the subspace . J , ..., n d J V − J Then, the projection estimator y satisﬁes with probability at least 1 (see, for example, P δ J Hsu et al. (2012)): p + 2 log(1 ) + 2 log(1 ) k y − k2 ≤ k − k2 + d d /δ /δ P µ P µ µ J J n | | 2( + 1) + 3 log(1 ) ≤ min k − k2 + J /δ (2.7) u µ . ∈ u VJ n − The sharp oracle inequality from Bellec (2014) yields that with probability at least 1 2 δ ⊆ { − } for all 1 1 we have J , ..., n (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) 2 1 µˆQ − µ ≤ kP y − µk2 + Cσ2 log + Cσ2 log(1/δ), (2.8) J π J for some absolute constant 0. Combining (2.7) and (2.8) with the union bound and C > ≤ | | | | the inequality (cf. (Rigollet and Tsybakov, 2012, (5.4))) log(1 ) 2( + 1) log( ( + /π J en/ J 1)) + 1/2,(cid:13)we ﬁnd (cid:13)that with probability at least 1 − 3δ, J (cid:18) (cid:19)! (cid:13) (cid:13) (cid:13) (cid:13) 2 2(| | + 1) µˆQ − µ ≤ J⊆{1m,..i.n,n−1} um∈iVnJ kµ − uk2 + cσ Jn log |J|en+ 1 + 2 log(1 ) cσ /δ | | ∈ where c > 0 is an absolute constant. Since J +1 = k(u) for all u V and min ⊆{1 −1} min ∈ = min ∈R , the bound (2.5) follows. Finally, (2.6) is obtained from (J2.5) by inJtegra,.t..i,non. u VJ u n We now discuss some corollaries of Theorem 1. First, it follows that (1.11) is satisﬁed for µˆ = µˆQ, so the remarks after (1.11) apply. Next, in view of (2.6), for the class of monotone 6 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Bellec

text: and

text: Tsybakov

text: this

text: quadratic

text: program

text: is

text: of

text: order

text: it

text: is

text: a

text: computationally

text: hard

text: problem.

text: The

text: estimator

text: µˆQ

text: satisﬁes

text: the

text: following

text: sharp

text: oracle

text: inequalities.

text: Theorem

text: Let

text: µ

text: Rn,

text: n

text: and

text: assume

text: that

text: the

text: noise

text: vector

text: ξ

text: has

text: distribution

text: N

text: There

text: exist

text: absolute

text: constants

text: c,

text: such

text: that

text: for

text: all

text: δ

text: the

text: with

text: pro

text: bability

text: at

text: least

text: µˆQ

text: µ

text: um∈iRnn

text: kµ

text: cσ

text: kn

text: u

text: log

text: ke(nu)

text: cσ

text: n

text: /δ

text: and

text: Eµ

text: µˆQ

text: µ

text: um∈iRnn

text: kµ

text: c

text: σ

text: nk

text: u

text: log

text: ke(nu)

text: Proof

text: Let

text: Denote

text: by

text: the

text: dimension

text: of

text: the

text: subspace

text: J

text: n

text: d

text: J

text: V

text: J

text: Then,

text: the

text: projection

text: estimator

text: y

text: satisﬁes

text: with

text: probability

text: at

text: least

text: (see,

text: for

text: example,

text: P

text: δ

text: J

text: Hsu

text: et

text: al.

text: p

text: k

text: y

text: k

text: d

text: d

text: /δ

text: /δ

text: P

text: µ

text: P

text: µ

text: µ

text: J

text: J

text: n

text: min

text: k

text: J

text: /δ

text: u

text: µ

text: u

text: VJ

text: n

text: The

text: sharp

text: oracle

text: inequality

text: from

text: Bellec

text: yields

text: that

text: with

text: probability

text: at

text: least

text: δ

text: for

text: all

text: we

text: have

text: J

text: n

text: µˆQ

text: µ

text: kP

text: y

text: log

text: J

text: π

text: J

text: for

text: some

text: absolute

text: constant

text: Combining

text: and

text: with

text: the

text: union

text: bound

text: and

text: C

text: the

text: inequality

text: (cf.

text: (Rigollet

text: and

text: Tsybakov,

text: log(

text: /π

text: J

text: en/

text: J

text: ﬁnd

text: with

text: probability

text: at

text: least

text: J

text: µˆQ

text: µ

text: um∈iVnJ

text: kµ

text: cσ

text: Jn

text: log

text: |J|en+

text: cσ

text: /δ

text: where

text: c

text: is

text: an

text: absolute

text: constant.

text: Since

text: J

text: k(u)

text: for

text: all

text: u

text: V

text: and

text: min

text: min

text: min

text: ∈R

text: the

text: bound

text: follows.

text: Finally,

text: is

text: obtained

text: from

text: by

text: inJtegra,.t..i,non.

text: u

text: VJ

text: u

text: n

text: We

text: now

text: discuss

text: some

text: corollaries

text: of

text: Theorem

text: First,

text: it

text: follows

text: that

text: is

text: satisﬁed

text: for

text: µˆ

text: µˆQ,

text: so

text: the

text: remarks

text: after

text: apply.

text: Next,

text: in

text: view

text: of

text: for

text: the

text: class

text: of

text: monotone

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ﬁeld SO(2, 6) USp(4) dim 7! a 7! Ec − − scalar Dij,k(cid:96) (4; 0, 0, 0) [0, 2] 14 5 31 72 12 − fermion χij (9; 1, 1, 1) [1, 1] 16 39 1021 k 2 2 2 2 32 32 − 3-form Tij (5; 1, 1, 1) [0, 1] 5 166 475 µνλ 9 3 − vector Vij (5; 1, 0, 0) (6; 0, 0, 0) [2, 0] 10 275 1755 µ 8 4 − − − − − gravitino ψi (11; 3, 1, 1) (13; 1, 1, 1) [1, 0] 4 4643 137637 µ 2 2 2 2 2 2 2 2 16 16 − graviton h (6; 2, 0, 0) (7; 1, 0, 0) [0, 0] 1 3005 37287 µν 2 Table 2. Representations, a-anomaly and Casimir energy for ﬁelds of (2,0) 6d conformal super- gravity. As in Tabl−e 1 the SO(2, 6) representations (∆; h1, h2, h3) refer to dual ﬁelds of AdS7 super- gravity, i.e. ∆ = 6 ∆− where ∆− is canonical dimension of the 6d conformal ﬁeld. To compute the associated a-anomaly coefﬁcient and Casimir energy we may use again the general expressions (2.5) and (2.6). This was essentially done already in [11]. The relevant ﬁeld representation content is readily determined, using, e.g., the relation to the maximal gauged 7d supergravity (with AdS vacuum) which is the bottom level of the 7 Kaluza-Klein tower of multiplets corresponding to 11d supergravity compactiﬁed on S4 (for translation between ﬁelds of 7d gauged and 6d conformal supergravities see [20, 50]). The resulting data is presented in Table 2. 12 The number of derivatives in kinetic terms is determined by the dimensions of the − ﬁelds. Dij,k(cid:96) is a conformal scalar with canonical dimension ∆− = 6 4 = 2, i.e. is the standard scalar ϕ discussed in Sec. . χij is a conformal spin 1 fermion spinor with ∆− = 3, k 2 2 i.e. it is the 3-derivative fermion ψ(3) in (3.1). Tij is a non gauge invariant (anti) selfdual µνλ antisymmetric 3rd rank tensor with ∆− = 1, i.e. it should have kinetic term T∂4T. The remaining three ﬁelds in Table 2 have maximal gauge invariance, i.e. they are members of conformal higher spin family. The conformal vector Vij with ∆− = 1 is of µ V(4) type in (3.1). The 4 conformal gravitini13 ψi have ∆− = 1 and thus the Lagrangian ∼ µ 2 ψ /∂5ψ . The graviton has ∆− = 0, i.e. its kinetic term has 6 derivatives as appropriate µ µ ∼ for the conformal gravity in 6d (L C ∂2 C + ..., see, e.g., [47]). To summarize, the symbolic form of the linearized Lagrangian of the maximal (2,0) 6d conformal supergravity is (here ψ and F are the gravitino and the vector ﬁeld µν µν 12Here i, j,... = 1,...,4 are USp(4) indices and µ, ν,... are 6d indices. Let us note that the same values of the Casimir energy for these ﬁelds were previously found in [51] (see also [11]). 13We consider all 6d spinors as Majorana-Weyl, i.e. there are 2 Weyl gravitini corresponding to (2,0) super- symmetry. – 10 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: ﬁeld

font: VVXGPG+URWPalladioL-Roma

font: TCYEMI+URWPalladioL-Ital

text: dim

text: a

text: Ec

text: scalar

text: fermion

text: χij

text: k

text: Tij

text: µνλ

text: vector

text: Vij

text: µ

text: gravitino

text: ψi

text: µ

text: graviton

text: h

text: µν

text: Table

text: Representations,

text: a-anomaly

text: and

text: Casimir

text: energy

text: for

text: ﬁelds

text: of

text: conformal

text: super-

text: gravity.

text: As

text: in

text: Tabl−e

text: the

text: representations

text: refer

text: to

text: dual

text: ﬁelds

text: of

text: super-

text: gravity,

text: i.e.

text: where

text: is

text: canonical

text: dimension

text: of

text: the

text: conformal

text: ﬁeld.

text: To

text: compute

text: the

text: associated

text: a-anomaly

text: coefﬁcient

text: and

text: Casimir

text: energy

text: we

text: may

text: use

text: again

text: the

text: general

text: expressions

text: and

text: This

text: was

text: essentially

text: done

text: already

text: in

text: The

text: relevant

text: ﬁeld

text: representation

text: content

text: is

text: readily

text: determined,

text: using,

text: e.g.,

text: the

text: relation

text: to

text: the

text: maximal

text: gauged

text: supergravity

text: (with

text: AdS

text: vacuum)

text: which

text: is

text: the

text: bottom

text: level

text: of

text: the

text: Kaluza-Klein

text: tower

text: of

text: multiplets

text: corresponding

text: to

text: supergravity

text: compactiﬁed

text: on

text: (for

text: translation

text: between

text: ﬁelds

text: of

text: gauged

text: and

text: conformal

text: supergravities

text: see

text: The

text: resulting

text: data

text: is

text: presented

text: in

text: Table

text: The

text: number

text: of

text: derivatives

text: in

text: kinetic

text: terms

text: is

text: determined

text: by

text: the

text: dimensions

text: of

text: the

text: ﬁelds.

text: is

text: a

text: conformal

text: scalar

text: with

text: canonical

text: dimension

text: i.e.

text: is

text: the

text: standard

text: scalar

text: ϕ

text: discussed

text: in

text: Sec.

text: χij

text: is

text: a

text: conformal

text: spin

text: fermion

text: spinor

text: with

text: k

text: i.e.

text: it

text: is

text: the

text: fermion

text: in

text: Tij

text: is

text: a

text: non

text: gauge

text: invariant

text: (anti)

text: selfdual

text: µνλ

text: antisymmetric

text: rank

text: tensor

text: with

text: i.e.

text: it

text: should

text: have

text: kinetic

text: term

text: The

text: remaining

text: three

text: ﬁelds

text: in

text: Table

text: have

text: maximal

text: gauge

text: invariance,

text: i.e.

text: they

text: are

text: members

text: of

text: conformal

text: higher

text: spin

text: family.

text: The

text: conformal

text: vector

text: Vij

text: with

text: is

text: of

text: µ

text: type

text: in

text: The

text: conformal

text: ψi

text: have

text: and

text: thus

text: the

text: Lagrangian

text: µ

text: ψ

text: /∂5ψ

text: The

text: graviton

text: has

text: i.e.

text: its

text: kinetic

text: term

text: has

text: derivatives

text: as

text: appropriate

text: µ

text: µ

text: for

text: the

text: conformal

text: gravity

text: in

text: (L

text: C

text: ∂2

text: C

text: see,

text: e.g.,

text: To

text: summarize,

text: the

text: symbolic

text: form

text: of

text: the

text: linearized

text: Lagrangian

text: of

text: the

text: maximal

text: conformal

text: supergravity

text: is

text: (here

text: ψ

text: and

text: F

text: are

text: the

text: gravitino

text: and

text: the

text: vector

text: ﬁeld

text: µν

text: µν

text: i,

text: j,...

text: are

text: indices

text: and

text: µ,

text: ν,...

text: are

text: indices.

text: Let

text: us

text: note

text: that

text: the

text: same

text: values

text: of

text: the

text: Casimir

text: energy

text: for

text: these

text: ﬁelds

text: were

text: previously

text: found

text: in

text: (see

text: also

text: consider

text: all

text: spinors

text: as

text: Majorana-Weyl,

text: i.e.

text: there

text: are

text: Weyl

text: gravitini

text: corresponding

text: to

text: super-

text: symmetry.

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##


text: Generic representations are non-degenerate (or “massive”), while representations at the unitarity bounds are maximally degenerate (they correspond, in particular, to conformal higher spins associated with massless higher spin ﬁelds in AdS [23]).4 There are also 7 intermediate cases of non-unitary conformal ﬁelds of non-maximal depth related to par- tially massles(cid:82)s ﬁelds in AdS (see [25–27] for general discussions).5 7 Assuming a free conformal ﬁeld ϕ with a local ﬁeld theory description with a free − action S = ddxϕ∂nϕ , its canonical dimension should be ∆ = d n. For example, a 2 conformal higher spin ﬁeld represented by a symmetric rank s tensor (i.e. h = (s, 0, 0)) − with action that has maximal gauge invariance has n = 2s + d 4, i.e. has canonical − ∆ = 2 s and thus is unitary in d = 6 only for s = 0 (cf. (2.1)). To ﬁnd the corresponding conformal anomaly, one may couple ϕ to a background metric g (getting a classically Weyl-invariant action with ϕ transforming with an appro- µν priate Weyl weight) and compute the trace of the variation of the 1-loop effective action − Γ = log Z[g] over g . Equivalently, to extract the a-coefﬁcient it is sufﬁcient to ﬁnd the µν Γ logarithmic UV divergent part of computed on 6-sphere − − Γ = B log(rΛ) + ... , B = 1 Ω(S6) b = 1 b = 96a . (2.2) 6 6 (4π)3 6 60 6 Here Λ is a UV cutoff and r is the radius of S6, see [7] for details. This direct 6d computation of the a-coefﬁcient appears to require a case-by-case anal- ysis, but there exist a remarkably universal method of computing a-anomaly using the AdS/CFT motivated relation between the determinants of a 2nd order operator in AdS d+1 space and of the associated conformal ﬁeld operator on d-dimensional boundary.6 While being a just a technical device (leading to the same results as the 6d computation, as one can check on particular examples), this method makes full use of the underlying confor- mal symmetry and allows one to compute the a-coefﬁcient for a generic representation (∆; h , h , h ). 1 2 3 A conformal ﬁeld ϕ in Rd of canonical dimension ∆− may be interpreted as a shadow (or source) ﬁeld associated to another conformal ﬁeld J (or current) of dimension ∆ = − + d ∆− that has the same SO(d) representation labels h. For example, in the context of vectorial AdS/CFT, the current J may be interpreted as a bilinear in complex scalars which is dual to a massless higher spin ﬁeld φ in AdSd+1 transforming in the same representation of SO(2, d) (the isometry group of AdS ) as J, i.e. (∆ ; h). d+1 + (cid:98) 4 Their characters can be written by suitable subtractions in terms of the massive (generic) representation character Z(∆; h1, h2, h3) = d(h) (1−q∆q)6 . For example, in the(cid:98)∆ = h1 + 4 case (i),(cid:98)we have the following Z Z − Z − massless character (see [11] for details) (h1 + 4; h1, h2, h3) = (h1 + 4; h1, h2, h3) (h1 + 5; h1 1, h2, h3) . 5In general, the origin of non-unitary of a free conformal theory may be due to higher derivative kinetic term (as in conformal higher spin ﬁeld case) and/or reduced gauge invariance that does not allow to eliminate all ghost-like components. 6This relation has a kinematic origin and belongs to a general class of bulk-boundary relations discussed in [28, 29]. Its AdS/CFT interpretation involves the bulk counterpart of a “double trace” deformation of the boundary CFT (see, in particular, [30–32] for scalar operators). In [8, 7, 9] it was applied to the computation of the a-coefﬁcient of totally-symmetric higher-spin conformal ﬁelds and in [10, 11] it was generalized to arbitrary conformal representations in 4d and 6d (see also [33, 34, 10] for related general discussions). – 4 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Generic

text: representations

text: are

text: non-degenerate

text: (or

text: “massive”),

text: while

text: representations

text: at

text: the

text: unitarity

text: bounds

text: are

text: maximally

text: degenerate

text: (they

text: correspond,

text: in

text: particular,

text: to

text: conformal

text: higher

text: spins

text: associated

text: with

text: massless

text: higher

text: spin

text: ﬁelds

text: in

text: AdS

text: There

text: are

text: also

text: intermediate

text: cases

text: of

text: non-unitary

text: conformal

text: ﬁelds

text: of

text: non-maximal

text: depth

text: related

text: to

text: par-

text: tially

text: ﬁelds

text: in

text: AdS

text: (see

text: for

text: general

text: Assuming

text: a

text: free

text: conformal

text: ﬁeld

text: ϕ

text: with

text: a

text: local

text: ﬁeld

text: theory

text: description

text: with

text: a

text: free

text: action

text: S

text: ddxϕ∂nϕ

text: its

text: canonical

text: dimension

text: should

text: be

text: d

text: n.

text: For

text: example,

text: a

text: conformal

text: higher

text: spin

text: ﬁeld

text: represented

text: by

text: a

text: symmetric

text: rank

text: s

text: tensor

text: (i.e.

text: h

text: (s,

text: with

text: action

text: that

text: has

text: maximal

text: gauge

text: invariance

text: has

text: n

text: d

text: i.e.

text: has

text: canonical

text: s

text: and

text: thus

text: is

text: unitary

text: in

text: d

text: only

text: for

text: s

text: (cf.

text: To

text: ﬁnd

text: the

text: corresponding

text: conformal

text: anomaly,

text: one

text: may

text: couple

text: ϕ

text: to

text: a

text: background

text: metric

text: g

text: (getting

text: a

text: classically

text: Weyl-invariant

text: action

text: with

text: ϕ

text: transforming

text: with

text: an

text: appro-

text: µν

text: priate

text: Weyl

text: weight)

text: and

text: compute

text: the

text: trace

text: of

text: the

text: variation

text: of

text: the

text: effective

text: action

text: Γ

text: log

text: Z[g]

text: over

text: g

text: Equivalently,

text: to

text: extract

text: the

text: a-coefﬁcient

text: it

text: is

text: sufﬁcient

text: to

text: ﬁnd

text: the

text: µν

text: Γ

text: logarithmic

text: UV

text: divergent

text: part

text: of

text: computed

text: on

text: Γ

text: B

text: log(rΛ)

text: B

text: b

text: b

text: Here

text: Λ

text: is

text: a

text: UV

text: cutoff

text: and

text: r

text: is

text: the

text: radius

text: of

text: see

text: for

text: details.

text: This

text: direct

text: computation

text: of

text: the

text: a-coefﬁcient

text: appears

text: to

text: require

text: a

text: case-by-case

text: anal-

text: ysis,

text: but

text: there

text: exist

text: a

text: remarkably

text: universal

text: method

text: of

text: computing

text: a-anomaly

text: using

text: the

text: AdS/CFT

text: motivated

text: relation

text: between

text: the

text: determinants

text: of

text: a

text: order

text: operator

text: in

text: AdS

text: space

text: and

text: of

text: the

text: associated

text: conformal

text: ﬁeld

text: operator

text: on

text: d-dimensional

text: While

text: being

text: a

text: just

text: a

text: technical

text: device

text: (leading

text: to

text: the

text: same

text: results

text: as

text: the

text: computation,

text: as

text: one

text: can

text: check

text: on

text: particular

text: examples),

text: this

text: method

text: makes

text: full

text: use

text: of

text: the

text: underlying

text: confor-

text: mal

text: symmetry

text: and

text: allows

text: one

text: to

text: compute

text: the

text: a-coefﬁcient

text: for

text: a

text: generic

text: representation

text: h

text: h

text: h

text: A

text: conformal

text: ﬁeld

text: ϕ

text: in

text: Rd

text: of

text: canonical

text: dimension

text: may

text: be

text: interpreted

text: as

text: a

text: shadow

text: (or

text: source)

text: ﬁeld

text: associated

text: to

text: another

text: conformal

text: ﬁeld

text: J

text: (or

text: current)

text: of

text: dimension

text: d

text: that

text: has

text: the

text: same

text: SO(d)

text: representation

text: labels

text: h.

text: For

text: example,

text: in

text: the

text: context

text: of

text: vectorial

text: AdS/CFT,

text: the

text: current

text: J

text: may

text: be

text: interpreted

text: as

text: a

text: bilinear

text: in

text: complex

text: scalars

text: which

text: is

text: dual

text: to

text: a

text: massless

text: higher

text: spin

text: ﬁeld

text: φ

text: in

text: transforming

text: in

text: the

text: same

text: representation

text: of

text: d)

text: (the

text: isometry

text: group

text: of

text: AdS

text: as

text: J,

text: i.e.

text: h).

text: Their

text: characters

text: can

text: be

text: written

text: by

text: suitable

text: subtractions

text: in

text: terms

text: of

text: the

text: massive

text: (generic)

text: representation

text: character

text: Z(∆;

text: d(h)

text: For

text: example,

text: in

text: case

text: have

text: the

text: following

text: Z

text: Z

text: Z

text: massless

text: character

text: (see

text: for

text: details)

text: general,

text: the

text: origin

text: of

text: non-unitary

text: of

text: a

text: free

text: conformal

text: theory

text: may

text: be

text: due

text: to

text: higher

text: derivative

text: kinetic

text: term

text: (as

text: in

text: conformal

text: higher

text: spin

text: ﬁeld

text: case)

text: and/or

text: reduced

text: gauge

text: invariance

text: that

text: does

text: not

text: allow

text: to

text: eliminate

text: all

text: ghost-like

text: components.

text: relation

text: has

text: a

text: kinematic

text: origin

text: and

text: belongs

text: to

text: a

text: general

text: class

text: of

text: bulk-boundary

text: relations

text: discussed

text: in

text: Its

text: AdS/CFT

text: interpretation

text: involves

text: the

text: bulk

text: counterpart

text: of

text: a

text: “double

text: trace”

text: deformation

text: of

text: the

text: boundary

text: CFT

text: (see,

text: in

text: particular,

text: for

text: scalar

text: operators).

text: In

text: it

text: was

text: applied

text: to

text: the

text: computation

text: of

text: the

text: a-coefﬁcient

text: of

text: totally-symmetric

text: higher-spin

text: conformal

text: ﬁelds

text: and

text: in

text: it

text: was

text: generalized

text: to

text: arbitrary

text: conformal

text: representations

text: in

text: and

text: (see

text: also

text: for

text: related

text: general

text: discussions).

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 13 (cid:80) (cid:80) { } − ρ(r, φk, pk ) = e β[V0(r)+m q ωq2φ2q/2+ q p2q/2m]/Z(r), (55) meaning expectation values may be calculated using standard identities such as Wick’s theorem43. This truncation ∇ ≥ clearly neglects any thermal expansion arising from the purely vibrational anharmonicities n 3V . In a previous Φ publication2 we systematically included these terms to produce an expression for γ up to linear order in temperature. It was shown that our main result, the anomalous temperature independent drag coeﬃcient γ in equation (59) below, 0 is unaﬀected by these additional terms. To analytically evaluate the defect force autocorrelation and hence the memory kernel C(s), we evolve the vibrational coordinates from a ﬁxed r. This is justiﬁed by the time-scale separation, as C(s) is expected to decay to incoherence before r changes appreciably, an assumption we will test in MD simulation. To do this we truncate the vibrational force to linear order in (15), diﬀerentiate (43) and invoke the timescale separation to neglect terms of order v, obtaining an approximate vibrational equation of motion of − φ¨ = mω2φ . (56) q q q We emphasize that this equation of motion only holds over short time periods where the defect co-ordinate r is eﬀectively stationary and the defect force autocorrelation C(s) is expected to decay to incoherence, an assumption { } we will test explicitly in section . Over longer periods of time r, and hence the vibrational basis e (r) , will change; k (cid:104) (cid:105) nevertheless, on the short timescale of present interest (56) may be solved with initial conditions φ (0)φ (0) = (cid:104) (cid:105) q r{ } (k T/mω2)δ , φ˙ (0)φ (0) = 0, allowing us to explicitly evaluate time dependent expectation values of φ in B q qr q r k terms of the two point correlation functions43 (cid:104) (cid:105) k T φ (t)φ (0) = B δ cos(ω t). (57) q r mω2 qr q q As appropriate for non-conservative dynamics, the correlation function is evaluated using only initial conditions and consequently is closely related to the retarded Green’s function44 (cid:104) (cid:105) Θ(t) G (t) = Θ(t)β φ (t)φ (0) = cos(ω t), (58) q q q mω2 q q where Θ(t) is the Heaviside step function. Finally, using Wick’s theorem43 to simplify high order expectation values (cid:104) (cid:105) (cid:104) (cid:105) φ φ φ , φ φ φ φ in the force autocorrelation, we perform the elementary Gaussian integrations to give our main q r s q r s t result (cid:90) ∞ (cid:90) ∇ γ = (∂ V )2G (t)dt r φq q 0 ∞ k T (cid:90) ∇ ∇ + B ∂ 2 V G (t)G (t)∂ 2 V dt 2 r φqφr r s r φsφq 0∞ k T ∇ ∇ + B ∂ V G (t)G (t)∂ 3 V dt (59) 2 r φq q r r φqφrφr 0 We see that the friction coeﬃcient takes the f(cid:90)orm γ = γ + k Tγ , with a temperature independent component 0 B w ∞ ∇ (∂ V )2 γ = r φq cos(ω t)dt. (60) 0 mω2 q 0 q The temperature independent drag coeﬃcient γ couples the defect to thermal vibrations via a mixed quadratic 0 derivative ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ρ(r,

text: φk,

text: pk

text: e

text: q

text: q

text: meaning

text: expectation

text: values

text: may

text: be

text: calculated

text: using

text: standard

text: identities

text: such

text: as

text: Wick’s

text: This

text: truncation

text: clearly

text: neglects

text: any

text: thermal

text: expansion

text: arising

text: from

text: the

text: purely

text: vibrational

text: anharmonicities

text: n

text: In

text: a

text: previous

text: Φ

text: we

text: systematically

text: included

text: these

text: terms

text: to

text: produce

text: an

text: expression

text: for

text: γ

text: up

text: to

text: linear

text: order

text: in

text: temperature.

text: It

text: was

text: shown

text: that

text: our

text: main

text: result,

text: the

text: anomalous

text: temperature

text: independent

text: drag

text: coeﬃcient

text: γ

text: in

text: equation

text: below,

text: is

text: unaﬀected

text: by

text: these

text: additional

text: terms.

text: To

text: analytically

text: evaluate

text: the

text: defect

text: force

text: autocorrelation

text: and

text: hence

text: the

text: memory

text: kernel

text: C(s),

text: we

text: evolve

text: the

text: vibrational

text: coordinates

text: from

text: a

text: ﬁxed

text: r.

text: This

text: is

text: justiﬁed

text: by

text: the

text: time-scale

text: separation,

text: as

text: C(s)

text: is

text: expected

text: to

text: decay

text: to

text: incoherence

text: before

text: r

text: changes

text: appreciably,

text: an

text: assumption

text: we

text: will

text: test

text: in

text: MD

text: simulation.

text: To

text: do

text: this

text: we

text: truncate

text: the

text: vibrational

text: force

text: to

text: linear

text: order

text: in

text: diﬀerentiate

text: and

text: invoke

text: the

text: timescale

text: separation

text: to

text: neglect

text: terms

text: of

text: order

text: v,

text: obtaining

text: an

text: approximate

text: vibrational

text: equation

text: of

text: motion

text: of

text: φ¨

text: mω2φ

text: q

text: q

text: q

text: We

text: emphasize

text: that

text: this

text: equation

text: of

text: motion

text: only

text: holds

text: over

text: short

text: time

text: periods

text: where

text: the

text: defect

text: co-ordinate

text: r

text: is

text: eﬀectively

text: stationary

text: and

text: the

text: defect

text: force

text: autocorrelation

text: C(s)

text: is

text: expected

text: to

text: decay

text: to

text: incoherence,

text: an

text: assumption

text: we

text: will

text: test

text: explicitly

text: in

text: section

text: Over

text: longer

text: periods

text: of

text: time

text: r,

text: and

text: hence

text: the

text: vibrational

text: basis

text: e

text: (r)

text: will

text: change;

text: k

text: nevertheless,

text: on

text: the

text: short

text: timescale

text: of

text: present

text: interest

text: may

text: be

text: solved

text: with

text: initial

text: conditions

text: φ

text: q

text: r{

text: (k

text: φ˙

text: allowing

text: us

text: to

text: explicitly

text: evaluate

text: time

text: dependent

text: expectation

text: values

text: of

text: φ

text: in

text: B

text: q

text: qr

text: q

text: r

text: k

text: terms

text: of

text: the

text: two

text: point

text: correlation

text: k

text: T

text: φ

text: (t)φ

text: B

text: δ

text: cos(ω

text: t).

text: q

text: r

text: qr

text: q

text: q

text: As

text: appropriate

text: for

text: non-conservative

text: dynamics,

text: the

text: correlation

text: function

text: is

text: evaluated

text: using

text: only

text: initial

text: conditions

text: and

text: consequently

text: is

text: closely

text: related

text: to

text: the

text: retarded

text: Green’s

text: Θ(t)

text: G

text: (t)

text: Θ(t)β

text: φ

text: (t)φ

text: cos(ω

text: t),

text: q

text: q

text: q

text: q

text: q

text: where

text: Θ(t)

text: is

text: the

text: Heaviside

text: step

text: function.

text: Finally,

text: using

text: Wick’s

text: to

text: simplify

text: high

text: order

text: expectation

text: values

text: φ

text: φ

text: φ

text: φ

text: φ

text: φ

text: φ

text: in

text: the

text: force

text: autocorrelation,

text: we

text: perform

text: the

text: elementary

text: Gaussian

text: integrations

text: to

text: give

text: our

text: main

text: q

text: r

text: s

text: q

text: r

text: s

text: t

text: result

text: γ

text: V

text: (t)dt

text: r

text: φq

text: q

text: k

text: T

text: B

text: V

text: G

text: (t)G

text: (t)∂

text: V

text: dt

text: r

text: φqφr

text: r

text: s

text: r

text: φsφq

text: 0∞

text: k

text: T

text: B

text: V

text: G

text: (t)G

text: (t)∂

text: V

text: dt

text: r

text: φq

text: q

text: r

text: r

text: φqφrφr

text: We

text: see

text: that

text: the

text: friction

text: coeﬃcient

text: takes

text: the

text: γ

text: γ

text: k

text: Tγ

text: with

text: a

text: temperature

text: independent

text: component

text: B

text: w

text: V

text: γ

text: r

text: φq

text: cos(ω

text: t)dt.

text: q

text: q

text: The

text: temperature

text: independent

text: drag

text: coeﬃcient

text: γ

text: couples

text: the

text: defect

text: to

text: thermal

text: vibrations

text: via

text: a

text: mixed

text: quadratic

text: derivative

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 17 valid for all values of the defect position. As the vibrational modes are now modes of the full Hessian rather than just the Φ-Hessian, the amplitude φ of each vibrational modes can be considered as an isolated harmonic oscillator, in- q dependent of any defect before the inclusion of anharmonic vibrational terms, with a conserved energy and momentum. Phonon scattering calculations also assume that phonons are harmonic oscillators with a conserved energy and momentum, interacting with defects only after the inclusion of anharmonic vibrational terms. But we have just seen that for these assumptions to hold in our general approach we are forced to consider only integrable systems, rare non-linear models that typically only exist in one dimension47. It is therefore very interesting that only in this some- what abstract limit does our general approach come into agreement with the central tenet of any phonon scattering calculation. This highlights the ‘artiﬁcial integrability’ that scattering calculations must impose on a problem in order to have a well deﬁned phase space for phonon scattering. In general we do not expect these assumptions to hold and (cid:54) thus we anticipate γ = 0 to be a general feature of defects in any realistic crystalline system. 0 These considerations promote the Frenkel-Kontorova (62) model as an interesting test case to calculate γ , as by 0 better approximating the continuum limit we reach the Sine-Gordon model, where by equation (80) γ should vanish. 0 To this end we have performed static calculations on the Frenkel-Kontorova model (62), which in appropriate regions of (cid:112) (cid:112) parameter space also supports soliton-like ‘kink’ deformations under boundary conditions X = X +b. To investi- n+N n → gate the continuum limit we sc√ale the line tension κ and lattice barrier ν in inverse proportion, i.e. (κ, ν) (ακ, ν/α) ∼ ∼ → such that the kink energy κν is constant and the kink width κ/ν α κ/ν increases; in this way the deformations of the chain become smoother and better approximate the continuum ﬁeld of the sine-Gordon model, whilst as the kink energy is roughly constant we can compare results taken for diﬀerent values of the kink width. (cid:38) It is well known5,47 that for kink widths greater than a few node spacings (i.e. w/a 2 the system state is very − − well approximated by the discretised soliton kink solution X = U (r) = na + U(na r), where U(na r) is given by n n (65). However, when we substitute this solution into the form of the total energy (62) the discrete summation retains a dependence on kink position, giving a system energy of approximately5 √ 2πw2ν − E(r) = (2b/π) 2κν + exp( πw/a) sin2(πr/a), (81) a √ being the continuum result (2b/π) 2κν plus a periodic term, of period a, that decays exponentially fast with the kink width w. The presence of this periodic migration barrier is an accepted signature of discreteness eﬀects in crystalline systems, with the exponential decay with defect width famously derived by Peierls39 to explain the low critical shear stress of dislocations. The kink migration barrier decays rapidly with increasing kink width as a broader kink proﬁle causes the summa- tion in the total energy (62) to better approximate the integral (64) of the continuum limit. As this integral has no dependence on kink position the kink migration barrier vanishes. It is interesting to ask whether we expect a similar dependence for the temperature independent drag parameter γ . 0 Whilst the kink migration barrier arises from the kink formation energy being a function of defect position in a static system, the expression (60) for γ , in contrast, is a sum across all the ﬂuctuations in the system, weighted by · ∇ · 0 (cid:82) a coupling term (∂ U 2V e )2. r q In the continuum limit of the Sine-Gordon model we(cid:82)have seen (equation (80)) that this coupling term vanishes, as { } ∇ − (cid:48) ∇ (cid:48) (cid:48) the e (x, r) are eigenmodes of the full Hessian 2V (x)δ(x x ), namely that L 2V (x, x )e (x , r) = µω2e (x, r). q ∇ (cid:48) (cid:48) 0 q ∇ · q q In this case the discrete summation that approximates L 2V (x, x )e (x , r) is the inner product 2V e , where the 0 q q summation is over all elements [(e ) , (e ) , ...(e ) ] of the vector e . However, unlike the kink proﬁle, the ﬂuctuations q 1 q 2 q N q e in a discrete system have no requirement to vary slowly between lattice sites, regardless of the kink width or any q defects the same system. As a result we cannot say that an increasing kink width suppresses discreteness eﬀects in the ﬂuctuations, meaning we expect γ to decay much more slowly than the migration barrier as we attempt to 0 approximate the continuum limit. We emphasize that this is a general feature; even if discreteness eﬀects are negligible for the migration of stable defect structures that possess a broad displacement ﬁeld, the ﬂuctuations around a crystal defect of any size will always be sensitive to discreteness as they have no requirement vary slowly across lattice sites and thus approximate a continuum solution. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: valid

text: for

text: all

text: values

text: of

text: the

text: defect

text: position.

text: As

text: the

text: vibrational

text: modes

text: are

text: now

text: modes

text: of

text: the

text: full

text: Hessian

text: rather

text: than

text: just

text: the

text: Φ-Hessian,

text: the

text: amplitude

text: φ

text: of

text: each

text: vibrational

text: modes

text: can

text: be

text: considered

text: as

text: an

text: isolated

text: harmonic

text: oscillator,

text: in-

text: q

text: dependent

text: of

text: any

text: defect

text: before

text: the

text: inclusion

text: of

text: anharmonic

text: vibrational

text: terms,

text: with

text: a

text: conserved

text: energy

text: and

text: momentum.

text: Phonon

text: scattering

text: calculations

text: also

text: assume

text: that

text: phonons

text: are

text: harmonic

text: oscillators

text: with

text: a

text: conserved

text: energy

text: and

text: momentum,

text: interacting

text: with

text: defects

text: only

text: after

text: the

text: inclusion

text: of

text: anharmonic

text: vibrational

text: terms.

text: But

text: we

text: have

text: just

text: seen

text: that

text: for

text: these

text: assumptions

text: to

text: hold

text: in

text: our

text: general

text: approach

text: we

text: are

text: forced

text: to

text: consider

text: only

text: integrable

text: systems,

text: rare

text: non-linear

text: models

text: that

text: typically

text: only

text: exist

text: in

text: one

text: It

text: is

text: therefore

text: very

text: interesting

text: that

text: only

text: in

text: this

text: some-

text: what

text: abstract

text: limit

text: does

text: our

text: general

text: approach

text: come

text: into

text: agreement

text: with

text: the

text: central

text: tenet

text: of

text: any

text: phonon

text: scattering

text: calculation.

text: This

text: highlights

text: the

text: ‘artiﬁcial

text: integrability’

text: that

text: scattering

text: calculations

text: must

text: impose

text: on

text: a

text: problem

text: in

text: order

text: to

text: have

text: a

text: well

text: deﬁned

text: phase

text: space

text: for

text: phonon

text: scattering.

text: In

text: general

text: we

text: do

text: not

text: expect

text: these

text: assumptions

text: to

text: hold

text: and

text: thus

text: we

text: anticipate

text: γ

text: to

text: be

text: a

text: general

text: feature

text: of

text: defects

text: in

text: any

text: realistic

text: crystalline

text: system.

text: These

text: considerations

text: promote

text: the

text: Frenkel-Kontorova

text: model

text: as

text: an

text: interesting

text: test

text: case

text: to

text: calculate

text: γ

text: as

text: by

text: better

text: approximating

text: the

text: continuum

text: limit

text: we

text: reach

text: the

text: Sine-Gordon

text: model,

text: where

text: by

text: equation

text: γ

text: should

text: vanish.

text: To

text: this

text: end

text: we

text: have

text: performed

text: static

text: calculations

text: on

text: the

text: Frenkel-Kontorova

text: model

text: which

text: in

text: appropriate

text: regions

text: of

text: parameter

text: space

text: also

text: supports

text: soliton-like

text: ‘kink’

text: deformations

text: under

text: boundary

text: conditions

text: X

text: X

text: +b.

text: To

text: investi-

text: n+N

text: n

text: gate

text: the

text: continuum

text: limit

text: we

text: sc√ale

text: the

text: line

text: tension

text: κ

text: and

text: lattice

text: barrier

text: ν

text: in

text: inverse

text: proportion,

text: i.e.

text: (κ,

text: ν)

text: (ακ,

text: ν/α)

text: such

text: that

text: the

text: kink

text: energy

text: κν

text: is

text: constant

text: and

text: the

text: kink

text: width

text: κ/ν

text: α

text: κ/ν

text: increases;

text: in

text: this

text: way

text: the

text: deformations

text: of

text: the

text: chain

text: become

text: smoother

text: and

text: better

text: approximate

text: the

text: continuum

text: ﬁeld

text: of

text: the

text: sine-Gordon

text: model,

text: whilst

text: as

text: the

text: kink

text: energy

text: is

text: roughly

text: constant

text: we

text: can

text: compare

text: results

text: taken

text: for

text: diﬀerent

text: values

text: of

text: the

text: kink

text: width.

text: It

text: is

text: well

text: that

text: for

text: kink

text: widths

text: greater

text: than

text: a

text: few

text: node

text: spacings

text: (i.e.

text: w/a

text: the

text: system

text: state

text: is

text: very

text: well

text: approximated

text: by

text: the

text: discretised

text: soliton

text: kink

text: solution

text: X

text: U

text: (r)

text: U(na

text: r),

text: where

text: U(na

text: r)

text: is

text: given

text: by

text: n

text: n

text: However,

text: when

text: we

text: substitute

text: this

text: solution

text: into

text: the

text: form

text: of

text: the

text: total

text: energy

text: the

text: discrete

text: summation

text: retains

text: a

text: dependence

text: on

text: kink

text: position,

text: giving

text: a

text: system

text: energy

text: of

text: E(r)

text: 2κν

text: exp(

text: πw/a)

text: a

text: being

text: the

text: continuum

text: result

text: 2κν

text: plus

text: a

text: periodic

text: term,

text: of

text: period

text: a,

text: that

text: decays

text: exponentially

text: fast

text: with

text: the

text: kink

text: width

text: w.

text: The

text: presence

text: of

text: this

text: periodic

text: migration

text: barrier

text: is

text: an

text: accepted

text: signature

text: of

text: discreteness

text: eﬀects

text: in

text: crystalline

text: systems,

text: with

text: the

text: exponential

text: decay

text: with

text: defect

text: width

text: famously

text: derived

text: by

text: to

text: explain

text: the

text: low

text: critical

text: shear

text: stress

text: of

text: dislocations.

text: The

text: kink

text: migration

text: barrier

text: decays

text: rapidly

text: with

text: increasing

text: kink

text: width

text: as

text: a

text: broader

text: kink

text: proﬁle

text: causes

text: the

text: summa-

text: tion

text: in

text: the

text: total

text: energy

text: to

text: better

text: approximate

text: the

text: integral

text: of

text: the

text: continuum

text: limit.

text: As

text: this

text: integral

text: has

text: no

text: dependence

text: on

text: kink

text: position

text: the

text: kink

text: migration

text: barrier

text: vanishes.

text: It

text: is

text: interesting

text: to

text: ask

text: whether

text: we

text: expect

text: a

text: similar

text: dependence

text: for

text: the

text: temperature

text: independent

text: drag

text: parameter

text: γ

text: Whilst

text: the

text: kink

text: migration

text: barrier

text: arises

text: from

text: the

text: kink

text: formation

text: energy

text: being

text: a

text: function

text: of

text: defect

text: position

text: in

text: a

text: static

text: system,

text: the

text: expression

text: for

text: γ

text: in

text: contrast,

text: is

text: a

text: sum

text: across

text: all

text: the

text: ﬂuctuations

text: in

text: the

text: system,

text: weighted

text: by

text: a

text: coupling

text: term

text: U

text: e

text: r

text: q

text: In

text: the

text: continuum

text: limit

text: of

text: the

text: Sine-Gordon

text: model

text: seen

text: (equation

text: that

text: this

text: coupling

text: term

text: vanishes,

text: as

text: the

text: e

text: (x,

text: r)

text: are

text: eigenmodes

text: of

text: the

text: full

text: Hessian

text: (x)δ(x

text: x

text: namely

text: that

text: L

text: (x,

text: x

text: )e

text: (x

text: r)

text: µω2e

text: (x,

text: r).

text: q

text: q

text: q

text: q

text: In

text: this

text: case

text: the

text: discrete

text: summation

text: that

text: approximates

text: L

text: (x,

text: x

text: )e

text: (x

text: r)

text: is

text: the

text: inner

text: product

text: e

text: where

text: the

text: q

text: q

text: summation

text: is

text: over

text: all

text: elements

text: [(e

text: (e

text: ...(e

text: of

text: the

text: vector

text: e

text: However,

text: unlike

text: the

text: kink

text: proﬁle,

text: the

text: ﬂuctuations

text: q

text: q

text: q

text: N

text: q

text: e

text: in

text: a

text: discrete

text: system

text: have

text: no

text: requirement

text: to

text: vary

text: slowly

text: between

text: lattice

text: sites,

text: regardless

text: of

text: the

text: kink

text: width

text: or

text: any

text: q

text: defects

text: the

text: same

text: system.

text: As

text: a

text: result

text: we

text: cannot

text: say

text: that

text: an

text: increasing

text: kink

text: width

text: suppresses

text: discreteness

text: eﬀects

text: in

text: the

text: ﬂuctuations,

text: meaning

text: we

text: expect

text: γ

text: to

text: decay

text: much

text: more

text: slowly

text: than

text: the

text: migration

text: barrier

text: as

text: we

text: attempt

text: to

text: approximate

text: the

text: continuum

text: limit.

text: We

text: emphasize

text: that

text: this

text: is

text: a

text: general

text: feature;

text: even

text: if

text: discreteness

text: eﬀects

text: are

text: negligible

text: for

text: the

text: migration

text: of

text: stable

text: defect

text: structures

text: that

text: possess

text: a

text: broad

text: displacement

text: ﬁeld,

text: the

text: ﬂuctuations

text: around

text: a

text: crystal

text: defect

text: of

text: any

text: size

text: will

text: always

text: be

text: sensitive

text: to

text: discreteness

text: as

text: they

text: have

text: no

text: requirement

text: vary

text: slowly

text: across

text: lattice

text: sites

text: and

text: thus

text: approximate

text: a

text: continuum

text: solution.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 24 from the atomic equations of motion. Using the Mori-Zwanzig technique, we then derived, under well deﬁned ap- proximations, a Green-Kubo relation (39) for the defect drag parameter γ that equates γ to the time integral of the defect force autocorrelation divided by temperature. A numerical implementation found that this method agreed well with traditional estimates of the diﬀusivity using the position trajectory, but that high quality data on the force autocorrelation could be obtained, due to the much smaller correlation time, from much smaller simulation supercells, oﬀering an interesting method to extract long time behaviour from a relatively short dynamical simulation. Under the assumption of a well deﬁned system temperature (Gaussian Gibbs distribution) we also derived an analytical result (59) for γ in terms of the system potential energy. We found that in general γ = γ + k Tγ , 0 B w where γ is a temperature independent term that is forbidden to exist in all previous theoretical treatments based 0 on phonon scattering formalism, but has been widely observed in MD simulations. We found that γ arises because 0 the vibrations orthogonal to defect motion are in general not eigenmodes of the crystal. This was shown to violate a founding assumption of phonon scattering calculations, namely that thermal vibrations may be considered as the normal modes of a perfect crystal, which then interact with crystal defects through the introduction of anharmonic terms in the potential energy. In a real crystal, the presence of a defect drastically changes the normal mode structure, and in general this new set of normal modes will not be orthogonal to the direction of defect motion, meaning that the vibrations e will · ∇ · (cid:54) q couple to the defect even to quadratic order, i.e. through the quadratic coupling term e 2V ∂ U = 0. q r To test this conclusion we investigated an extreme case, the integrable sine-Gordon system, where analytical results predict that in the presence of a soliton the normal modes are all orthogonal to the direction of defect motion, meaning in turn that for this integrable model γ = 0. However, when simulating the sine-Gordon system, even with a very 0 ﬁne discretisation, we saw that whilst traditional static discreteness eﬀects such as the Peierls barrier are eﬀectively (cid:54) zero, γ = 0 was still present, as ﬂuctuations persist up to the smallest length scales in any discrete system, meaning 0 they cannot be approximated away. As a result we expect the presence of γ to be a general feature in the stochastic 0 dynamics of any crystal defect. Future work will concern detailing the application of this approach to extended defects such as dislocation lines and glissile stacking faults, and investigating how these techniques can be applied to develop a stochastic equation of motion for defects that posses a large migration barrier. This work has been carried out within the framework of the EUROfusion Consortium and has received funding from the Euratom research and training programme 2014-2018 under grant agreement No 633053. The views and opinions expressed herein do not necessarily reﬂect those of the European Commission. fxundeﬁned [1]ifx1 fnum [1]1ﬁrstoftwo fx [1]secondoftwo noop [0]secondoftworef[1]@startlink1@hrefhref[1]1@endlinkanitize@url [0]startlink[1]endlink[0]rl [1]href 1bib@innerbibempty L. Kubin, Dislocations, Mesoscale Simulations and Plastic Flow, Oxford Series on Materials Modelling (OUP Oxford, 2013). T. D. Swinburne, S. L. Dudarev, and A. P. Sutton, Physical Review Letters 113, 215501 (2014). V. V. Bulatov and W. Cai, noop Computer Simulations Of Dislocations (Oxford University Press, 2003). P. M. Derlet, M. R. Gilbert, and S. L. Dudarev, noop Physical Review B 84, 134109 (2011). T. D. Swinburne, S. L. Dudarev, S. P. Fitzgerald, M. R. Gilbert, and A. P. Sutton, noop Physical Review B 87, 64108 (2013). K. Arakawa, K. Ono, M. Isshiki, K. Mimura, M. Uchikoshi, and H. Mori, noop Science 318, 956 (2007). X. Yi, M. L. Jenkins, M. Briceno, S. G. Rs, Z. Zhou, and M. A. Kirk, noop Philosophical Magazine 93, 1715 (2013). W. Coﬀey, Y. P. Kalmykov, and J. T. Waldron, noop The Langevin Equation: With Applications to Stochastic Prob- lems in Physics, Chemistry, and Electrical Engineering, World Scientiﬁc Series in Contemporary Chemical Physics (World Scientiﬁc, 2004). F. Reif, Fundamentals of Statistical and Thermal Physics, McGraw-Hill series in fundamentals of physics (Waveland Press, 2008). J. P. Hirth and J. Lothe, noop Theory Of Dislocations (Malabar, FL Krieger, 1991). F. R. N. Nabarro, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 209, 278 (1951). ##LTLine## ##LTLine## ##LTLine##

text: from

text: the

text: atomic

text: equations

text: of

text: motion.

text: Using

text: the

text: Mori-Zwanzig

text: technique,

text: we

text: then

text: derived,

text: under

text: well

text: deﬁned

text: ap-

text: proximations,

text: a

text: Green-Kubo

text: relation

text: for

text: the

text: defect

text: drag

text: parameter

text: γ

text: that

text: equates

text: γ

text: to

text: the

text: time

text: integral

text: of

text: the

text: defect

text: force

text: autocorrelation

text: divided

text: by

text: temperature.

text: A

text: numerical

text: implementation

text: found

text: that

text: this

text: method

text: agreed

text: well

text: with

text: traditional

text: estimates

text: of

text: the

text: diﬀusivity

text: using

text: the

text: position

text: trajectory,

text: but

text: that

text: high

text: quality

text: data

text: on

text: the

text: force

text: autocorrelation

text: could

text: be

text: obtained,

text: due

text: to

text: the

text: much

text: smaller

text: correlation

text: time,

text: from

text: much

text: smaller

text: simulation

text: supercells,

text: oﬀering

text: an

text: interesting

text: method

text: to

text: extract

text: long

text: time

text: behaviour

text: from

text: a

text: relatively

text: short

text: dynamical

text: simulation.

text: Under

text: the

text: assumption

text: of

text: a

text: well

text: deﬁned

text: system

text: temperature

text: (Gaussian

text: Gibbs

text: distribution)

text: we

text: also

text: derived

text: an

text: analytical

text: result

text: for

text: γ

text: in

text: terms

text: of

text: the

text: system

text: potential

text: energy.

text: We

text: found

text: that

text: in

text: general

text: γ

text: γ

text: k

text: Tγ

text: B

text: w

text: where

text: γ

text: is

text: a

text: temperature

text: independent

text: term

text: that

text: is

text: forbidden

text: to

text: exist

text: in

text: all

text: previous

text: theoretical

text: treatments

text: based

text: on

text: phonon

text: scattering

text: formalism,

text: but

text: has

text: been

text: widely

text: observed

text: in

text: MD

text: simulations.

text: We

text: found

text: that

text: γ

text: arises

text: because

text: the

text: vibrations

text: orthogonal

text: to

text: defect

text: motion

text: are

text: in

text: general

text: not

text: eigenmodes

text: of

text: the

text: crystal.

text: This

text: was

text: shown

text: to

text: violate

text: a

text: founding

text: assumption

text: of

text: phonon

text: scattering

text: calculations,

text: namely

text: that

text: thermal

text: vibrations

text: may

text: be

text: considered

text: as

text: the

text: normal

text: modes

text: of

text: a

text: perfect

text: crystal,

text: which

text: then

text: interact

text: with

text: crystal

text: defects

text: through

text: the

text: introduction

text: of

text: anharmonic

text: terms

text: in

text: the

text: potential

text: energy.

text: In

text: a

text: real

text: crystal,

text: the

text: presence

text: of

text: a

text: defect

text: drastically

text: changes

text: the

text: normal

text: mode

text: structure,

text: and

text: in

text: general

text: this

text: new

text: set

text: of

text: normal

text: modes

text: will

text: not

text: be

text: orthogonal

text: to

text: the

text: direction

text: of

text: defect

text: motion,

text: meaning

text: that

text: the

text: vibrations

text: e

text: will

text: q

text: couple

text: to

text: the

text: defect

text: even

text: to

text: quadratic

text: order,

text: i.e.

text: through

text: the

text: quadratic

text: coupling

text: term

text: e

text: U

text: q

text: r

text: To

text: test

text: this

text: conclusion

text: we

text: investigated

text: an

text: extreme

text: case,

text: the

text: integrable

text: sine-Gordon

text: system,

text: where

text: analytical

text: results

text: predict

text: that

text: in

text: the

text: presence

text: of

text: a

text: soliton

text: the

text: normal

text: modes

text: are

text: all

text: orthogonal

text: to

text: the

text: direction

text: of

text: defect

text: motion,

text: meaning

text: in

text: turn

text: that

text: for

text: this

text: integrable

text: model

text: γ

text: However,

text: when

text: simulating

text: the

text: sine-Gordon

text: system,

text: even

text: with

text: a

text: very

text: ﬁne

text: discretisation,

text: we

text: saw

text: that

text: whilst

text: traditional

text: static

text: discreteness

text: eﬀects

text: such

text: as

text: the

text: Peierls

text: barrier

text: are

text: eﬀectively

text: zero,

text: γ

text: was

text: still

text: present,

text: as

text: ﬂuctuations

text: persist

text: up

text: to

text: the

text: smallest

text: length

text: scales

text: in

text: any

text: discrete

text: system,

text: meaning

text: they

text: cannot

text: be

text: approximated

text: away.

text: As

text: a

text: result

text: we

text: expect

text: the

text: presence

text: of

text: γ

text: to

text: be

text: a

text: general

text: feature

text: in

text: the

text: stochastic

text: dynamics

text: of

text: any

text: crystal

text: defect.

text: Future

text: work

text: will

text: concern

text: detailing

text: the

text: application

text: of

text: this

text: approach

text: to

text: extended

text: defects

text: such

text: as

text: dislocation

text: lines

text: and

text: glissile

text: stacking

text: faults,

text: and

text: investigating

text: how

text: these

text: techniques

text: can

text: be

text: applied

text: to

text: develop

text: a

text: stochastic

text: equation

text: of

text: motion

text: for

text: defects

text: that

text: posses

text: a

text: large

text: migration

text: barrier.

text: This

text: work

text: has

text: been

text: carried

text: out

text: within

text: the

text: framework

text: of

text: the

text: EUROfusion

text: Consortium

text: and

text: has

text: received

text: funding

text: from

text: the

text: Euratom

text: research

text: and

text: training

text: programme

text: under

text: grant

text: agreement

text: No

text: The

text: views

text: and

text: opinions

text: expressed

text: herein

text: do

text: not

text: necessarily

text: reﬂect

text: those

text: of

text: the

text: European

text: Commission.

text: fxundeﬁned

text: fnum

text: fx

text: noop

text: L.

text: Kubin,

text: Dislocations,

text: Mesoscale

text: Simulations

text: and

text: Plastic

text: Flow,

text: Oxford

text: Series

text: on

text: Materials

text: Modelling

text: (OUP

text: Oxford,

text: T.

text: D.

text: Swinburne,

text: S.

text: L.

text: Dudarev,

text: and

text: A.

text: P.

text: Sutton,

text: Physical

text: Review

text: Letters

text: V.

text: V.

text: Bulatov

text: and

text: W.

text: Cai,

text: noop

text: Computer

text: Simulations

text: Of

text: Dislocations

text: (Oxford

text: University

text: Press,

text: P.

text: M.

text: Derlet,

text: M.

text: R.

text: Gilbert,

text: and

text: S.

text: L.

text: Dudarev,

text: noop

text: Physical

text: Review

text: B

text: T.

text: D.

text: Swinburne,

text: S.

text: L.

text: Dudarev,

text: S.

text: P.

text: Fitzgerald,

text: M.

text: R.

text: Gilbert,

text: and

text: A.

text: P.

text: Sutton,

text: noop

text: Physical

text: Review

text: B

text: K.

text: Arakawa,

text: K.

text: Ono,

text: M.

text: Isshiki,

text: K.

text: Mimura,

text: M.

text: Uchikoshi,

text: and

text: H.

text: Mori,

text: noop

text: Science

text: X.

text: Yi,

text: M.

text: L.

text: Jenkins,

text: M.

text: Briceno,

text: S.

text: G.

text: Rs,

text: Z.

text: Zhou,

text: and

text: M.

text: A.

text: Kirk,

text: noop

text: Philosophical

text: Magazine

text: W.

text: Coﬀey,

text: Y.

text: P.

text: Kalmykov,

text: and

text: J.

text: T.

text: Waldron,

text: noop

text: The

text: Langevin

text: Equation:

text: With

text: Applications

text: to

text: Stochastic

text: Prob-

text: lems

text: in

text: Physics,

text: Chemistry,

text: and

text: Electrical

text: Engineering,

text: World

text: Scientiﬁc

text: Series

text: in

text: Contemporary

text: Chemical

text: Physics

text: (World

text: Scientiﬁc,

text: F.

text: Reif,

text: Fundamentals

text: of

text: Statistical

text: and

text: Thermal

text: Physics,

text: McGraw-Hill

text: series

text: in

text: fundamentals

text: of

text: physics

text: (Waveland

text: Press,

text: J.

text: P.

text: Hirth

text: and

text: J.

text: Lothe,

text: noop

text: Theory

text: Of

text: Dislocations

text: (Malabar,

text: FL

text: Krieger,

text: F.

text: R.

text: N.

text: Nabarro,

text: Proceedings

text: of

text: the

text: Royal

text: Society

text: of

text: London.

text: A.

text: Mathematical

text: and

text: Physical

text: Sciences

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##