text: 6 IV. GENERAL SOLUTION AND BOUNDARY Let us now consider periodic solutions. Because both − CONDITIONS independent zero-mode solutions, ψ(I), in Eq. (30) and − ψ(II), in Eq. (40), are already periodic, any periodic A ﬁxing of a concrete form of the holomorphic func- set of the coeﬃcient functions (h1, h2) should give us a tions h (z) in the most general zero-mode solution (41) proper periodic solution. Since we already know that 1,2 corresponds to choosing speciﬁc boundary conditions for the function h1 should be analytical, then the require- our solution, and vice versa. The boundary conditions ment of periodicity singles out a constant solution for can be divided roughly into two classes based on their this function. On the other hand, the function h2 is not behavior towards the boundary of the system: the ones restricted to be a constant, and in this case it must have which corresponds to a periodic lattice of zero modes (a square root singularities according to our previous dis- Z relevant example is given in the upper panel of Fig. 2) and cussion. The only ambiguity allowed by the residual 2 the localized solutions which give zero modes with ﬁnite degeneracy appears in the sign of the fermion wave func- overall fermion number (e.g., the one which is is given tion, so that the function h2(z) must be the square root in the lower panel of Fig. 2), and therefore converging of a periodic single-valued function with simple poles. suﬃciently fast to zero at spatial inﬁnity. Thus, in order to describe a periodic zero-mode solu- In addition to the boundary conditions at spatial in- tion, the coeﬃcient functions h1 and (cid:112)h2 should satisfy ﬁnity, one may also impose the conditions at the vortex the following constraints: cores where the gluon condensate is vanishing. Notice that the Z2 part of the gauge group is not ﬁxed by the h1(z) = const. , h2(z) = s(z) , (43) gluon condensate because the gluons are transforming where the poles and simple zeroes of the function s(z) in adjoint representation of the gauge group. However, Z must coincide with vortex cores of the background lat- the fermion degrees of freedom are sensitive to the 2 tice. The function s(z) must therefore be an elliptic func- subgroup, so that each vortex core may correspond – de- tion on a sublattice of the original vortex, and it must pending on our choice of the boundary conditions – to a ≡ have simple poles only. A solution with h (z) const square root branch point exhibiting a multivalued behav- 2 corresponds to a periodic solution found in [13]. An il- ior. We will see that this property is essential for ﬁnding lustration of the periodic solution is given in the upper all possible zero-mode solutions. panel of Fig. 2. At any vortex core (located, for example, at z = 0), a There are also zero-mode solutions which are localized zero-mode solution has the following behavior: around a single vortex or a compact group of vortices. | | | −| ∼ h (z) 2 | | The density of the localized zero modes tends to zero at ψ 2 (. . .) 1| | + (. . .) h (z) 2 (42) ≡ z 2 spatial inﬁnity of the transverse plane, so that h1 0 for the localized modes. In order for the zero mode to where the ellipsis correspond to regular functions at z = be normalizable in the vicinity of the vortex cores, the 0. In order for the mode to be integrable in the transver- function h still needs to possess at least one singular 2 sal plane, a singularity of the function h (z) at z = 0 − 1 square root cut at a vortex position. However, in this should be less severe than z 12 . M−oreover,→the function simplest case the solution turns out to non-normalizable h2(z) must converge faster than z 1 as z 0. An ad- in the infrared sense. Thus, a general localized solution is ditional constraint for the zero mode is coming from the given by Eq. (41) with the follow(cid:114)ing set of the coeﬃcient (cid:89) fact that the nontrivial winding (monodromy) around the functions: Z vortex cores should belong to the (unbroken) center 2 of the gauge group. Therefore, half-integral powers of z ≡ N 1 ≥ h (z) 0 , h (z) = − , N 2 , (44) are allowed, and by consequence the function h1(z) is, in 1 2 z zi fact, analytical at z = 0, contrary to the function h (z), i=1 2 for which the analyticity at z = 0 is not required, in where the z correspond to a certain arbitrary set of vor- i general. tex positions in the transverse plane. The vortex posi- Another important consequence of the integrability of tions coincide with the zeros of the gluon condensate (12). the zero-mode solution is that all singularities of the The simplest example of the loca(cid:18)lized mode is(cid:19)given by zero mode should correspond to nontrivial winding only the following expression: (cid:114) which, in turn, can only occur in the vortex cores. There- (cid:16) (cid:17) − fore, all singularities in a general zero-vortex solution − C− ψ(II), (z, z¯) ψ (z, z¯) = 1 − , (45) are pinned to the background lattice of the vector boson − ψ(II), (z, z¯) condensate. For a ﬁxed background, this requirement z z X1,0 2 2 severely restricts possible zero-mode solutions. In partic- ular, it prohibits a continuous transition of zero modes where C− is a normalization constant. According to localized at diﬀerent vortices. In a vortex liquid of real Eq. (12), the mode i≡s localized at the vor≡tices located dynamical QCD – where the background lattice is by no at points z1 = X0,0 0 and z2 = X1,0 1/ν of the 2 means constrained to the vortex lattice – the mentioned transverse plane. An illustration of the localized zero- restriction is a much less severe constraint. mode solution (45) is given in the lower panel of Fig. 2. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: IV.

text: GENERAL

text: SOLUTION

text: AND

text: BOUNDARY

text: Let

text: us

text: now

text: consider

text: periodic

text: solutions.

text: Because

text: both

text: CONDITIONS

text: independent

text: zero-mode

text: solutions,

text: ψ(I),

text: in

text: Eq.

text: and

text: ψ(II),

text: in

text: Eq.

text: are

text: already

text: periodic,

text: any

text: periodic

text: A

text: ﬁxing

text: of

text: a

text: concrete

text: form

text: of

text: the

text: holomorphic

text: func-

text: set

text: of

text: the

text: coeﬃcient

text: functions

text: should

text: give

text: us

text: a

text: tions

text: h

text: (z)

text: in

text: the

text: most

text: general

text: zero-mode

text: solution

text: proper

text: periodic

text: solution.

text: Since

text: we

text: already

text: know

text: that

text: corresponds

text: to

text: choosing

text: speciﬁc

text: boundary

text: conditions

text: for

text: the

text: function

text: should

text: be

text: analytical,

text: then

text: the

text: require-

text: our

text: solution,

text: and

text: vice

text: versa.

text: The

text: boundary

text: conditions

text: ment

text: of

text: periodicity

text: singles

text: out

text: a

text: constant

text: solution

text: for

text: can

text: be

text: divided

text: roughly

text: into

text: two

text: classes

text: based

text: on

text: their

text: this

text: function.

text: On

text: the

text: other

text: hand,

text: the

text: function

text: is

text: not

text: behavior

text: towards

text: the

text: boundary

text: of

text: the

text: system:

text: the

text: ones

text: restricted

text: to

text: be

text: a

text: constant,

text: and

text: in

text: this

text: case

text: it

text: must

text: have

text: which

text: corresponds

text: to

text: a

text: periodic

text: lattice

text: of

text: zero

text: modes

text: (a

text: square

text: root

text: singularities

text: according

text: to

text: our

text: previous

text: dis-

text: Z

text: relevant

text: example

text: is

text: given

text: in

text: the

text: upper

text: panel

text: of

text: Fig.

text: and

text: cussion.

text: The

text: only

text: ambiguity

text: allowed

text: by

text: the

text: residual

text: the

text: localized

text: solutions

text: which

text: give

text: zero

text: modes

text: with

text: ﬁnite

text: degeneracy

text: appears

text: in

text: the

text: sign

text: of

text: the

text: fermion

text: wave

text: func-

text: overall

text: fermion

text: number

text: (e.g.,

text: the

text: one

text: which

text: is

text: is

text: given

text: tion,

text: so

text: that

text: the

text: function

text: must

text: be

text: the

text: square

text: root

text: in

text: the

text: lower

text: panel

text: of

text: Fig.

text: and

text: therefore

text: converging

text: of

text: a

text: periodic

text: single-valued

text: function

text: with

text: simple

text: poles.

text: suﬃciently

text: fast

text: to

text: zero

text: at

text: spatial

text: inﬁnity.

text: Thus,

text: in

text: order

text: to

text: describe

text: a

text: periodic

text: zero-mode

text: solu-

text: In

text: addition

text: to

text: the

text: boundary

text: conditions

text: at

text: spatial

text: in-

text: tion,

text: the

text: coeﬃcient

text: functions

text: and

text: should

text: satisfy

text: ﬁnity,

text: one

text: may

text: also

text: impose

text: the

text: conditions

text: at

text: the

text: vortex

text: the

text: following

text: constraints:

text: cores

text: where

text: the

text: gluon

text: condensate

text: is

text: vanishing.

text: Notice

text: that

text: the

text: part

text: of

text: the

text: gauge

text: group

text: is

text: not

text: ﬁxed

text: by

text: the

text: const.

text: s(z)

text: gluon

text: condensate

text: because

text: the

text: gluons

text: are

text: transforming

text: where

text: the

text: poles

text: and

text: simple

text: zeroes

text: of

text: the

text: function

text: s(z)

text: in

text: adjoint

text: representation

text: of

text: the

text: gauge

text: group.

text: However,

text: Z

text: must

text: coincide

text: with

text: vortex

text: cores

text: of

text: the

text: background

text: lat-

text: the

text: fermion

text: degrees

text: of

text: freedom

text: are

text: sensitive

text: to

text: the

text: tice.

text: The

text: function

text: s(z)

text: must

text: therefore

text: be

text: an

text: elliptic

text: func-

text: subgroup,

text: so

text: that

text: each

text: vortex

text: core

text: may

text: correspond

text: de-

text: tion

text: on

text: a

text: sublattice

text: of

text: the

text: original

text: vortex,

text: and

text: it

text: must

text: pending

text: on

text: our

text: choice

text: of

text: the

text: boundary

text: conditions

text: to

text: a

text: have

text: simple

text: poles

text: only.

text: A

text: solution

text: with

text: h

text: (z)

text: const

text: square

text: root

text: branch

text: point

text: exhibiting

text: a

text: multivalued

text: behav-

text: corresponds

text: to

text: a

text: periodic

text: solution

text: found

text: in

text: An

text: il-

text: ior.

text: We

text: will

text: see

text: that

text: this

text: property

text: is

text: essential

text: for

text: ﬁnding

text: lustration

text: of

text: the

text: periodic

text: solution

text: is

text: given

text: in

text: the

text: upper

text: all

text: possible

text: zero-mode

text: solutions.

text: panel

text: of

text: Fig.

text: At

text: any

text: vortex

text: core

text: (located,

text: for

text: example,

text: at

text: z

text: a

text: There

text: are

text: also

text: zero-mode

text: solutions

text: which

text: are

text: localized

text: zero-mode

text: solution

text: has

text: the

text: following

text: behavior:

text: around

text: a

text: single

text: vortex

text: or

text: a

text: compact

text: group

text: of

text: vortices.

text: h

text: (z)

text: The

text: density

text: of

text: the

text: localized

text: zero

text: modes

text: tends

text: to

text: zero

text: at

text: ψ

text: h

text: (z)

text: z

text: spatial

text: inﬁnity

text: of

text: the

text: transverse

text: plane,

text: so

text: that

text: for

text: the

text: localized

text: modes.

text: In

text: order

text: for

text: the

text: zero

text: mode

text: to

text: where

text: the

text: ellipsis

text: correspond

text: to

text: regular

text: functions

text: at

text: z

text: be

text: normalizable

text: in

text: the

text: vicinity

text: of

text: the

text: vortex

text: cores,

text: the

text: In

text: order

text: for

text: the

text: mode

text: to

text: be

text: integrable

text: in

text: the

text: transver-

text: function

text: h

text: still

text: needs

text: to

text: possess

text: at

text: least

text: one

text: singular

text: sal

text: plane,

text: a

text: singularity

text: of

text: the

text: function

text: h

text: (z)

text: at

text: z

text: square

text: root

text: cut

text: at

text: a

text: vortex

text: position.

text: However,

text: in

text: this

text: should

text: be

text: less

text: severe

text: than

text: z

text: M−oreover,→the

text: function

text: simplest

text: case

text: the

text: solution

text: turns

text: out

text: to

text: non-normalizable

text: must

text: converge

text: faster

text: than

text: z

text: as

text: z

text: An

text: ad-

text: in

text: the

text: infrared

text: sense.

text: Thus,

text: a

text: general

text: localized

text: solution

text: is

text: ditional

text: constraint

text: for

text: the

text: zero

text: mode

text: is

text: coming

text: from

text: the

text: given

text: by

text: Eq.

text: with

text: the

text: set

text: of

text: the

text: coeﬃcient

text: fact

text: that

text: the

text: nontrivial

text: winding

text: (monodromy)

text: around

text: the

text: functions:

text: Z

text: vortex

text: cores

text: should

text: belong

text: to

text: the

text: (unbroken)

text: center

text: of

text: the

text: gauge

text: group.

text: Therefore,

text: half-integral

text: powers

text: of

text: z

text: N

text: h

text: (z)

text: h

text: (z)

text: N

text: are

text: allowed,

text: and

text: by

text: consequence

text: the

text: function

text: is,

text: in

text: z

text: zi

text: fact,

text: analytical

text: at

text: z

text: contrary

text: to

text: the

text: function

text: h

text: (z),

text: for

text: which

text: the

text: analyticity

text: at

text: z

text: is

text: not

text: required,

text: in

text: where

text: the

text: z

text: correspond

text: to

text: a

text: certain

text: arbitrary

text: set

text: of

text: vor-

text: i

text: general.

text: tex

text: positions

text: in

text: the

text: transverse

text: plane.

text: The

text: vortex

text: posi-

text: Another

text: important

text: consequence

text: of

text: the

text: integrability

text: of

text: tions

text: coincide

text: with

text: the

text: zeros

text: of

text: the

text: gluon

text: condensate

text: the

text: zero-mode

text: solution

text: is

text: that

text: all

text: singularities

text: of

text: the

text: The

text: simplest

text: example

text: of

text: the

text: mode

text: by

text: zero

text: mode

text: should

text: correspond

text: to

text: nontrivial

text: winding

text: only

text: the

text: following

text: expression:

text: which,

text: in

text: turn,

text: can

text: only

text: occur

text: in

text: the

text: vortex

text: cores.

text: There-

text: fore,

text: all

text: singularities

text: in

text: a

text: general

text: zero-vortex

text: solution

text: C−

text: ψ(II),

text: (z,

text: z¯)

text: ψ

text: (z,

text: z¯)

text: are

text: pinned

text: to

text: the

text: background

text: lattice

text: of

text: the

text: vector

text: boson

text: ψ(II),

text: (z,

text: z¯)

text: condensate.

text: For

text: a

text: ﬁxed

text: background,

text: this

text: requirement

text: z

text: z

text: severely

text: restricts

text: possible

text: zero-mode

text: solutions.

text: In

text: partic-

text: ular,

text: it

text: prohibits

text: a

text: continuous

text: transition

text: of

text: zero

text: modes

text: where

text: C−

text: is

text: a

text: normalization

text: constant.

text: According

text: to

text: localized

text: at

text: diﬀerent

text: vortices.

text: In

text: a

text: vortex

text: liquid

text: of

text: real

text: Eq.

text: the

text: mode

text: i≡s

text: localized

text: at

text: the

text: vor≡tices

text: located

text: dynamical

text: QCD

text: where

text: the

text: background

text: lattice

text: is

text: by

text: no

text: at

text: points

text: and

text: of

text: the

text: means

text: constrained

text: to

text: the

text: vortex

text: lattice

text: the

text: mentioned

text: transverse

text: plane.

text: An

text: illustration

text: of

text: the

text: localized

text: zero-

text: restriction

text: is

text: a

text: much

text: less

text: severe

text: constraint.

text: mode

text: solution

text: is

text: given

text: in

text: the

text: lower

text: panel

text: of

text: Fig.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 8 where η is the Dedekind eta function. Then the prefactor is an alternative convenient expansion of the θ function 1 N can be evaluated numerically: which enter the solution in Eq. (35). The validity of Eq. (A7) can also be checked numerically. N = 3.1825 . . . (A6) (cid:12) (cid:12) (cid:88) ≡ Thus, we c(cid:12)onclude that ω˜(z¯) ω(z¯), and (cid:12) − θ1(z¯, eiπτ) = Nei8π (z¯ X¯m,n) (A7) τ=eiπ3 m,n × − | − | − e π2 z Xm,n 2+π2 (z¯Xm,n zX¯m,n), ∗ [13] B. B´eri, D. Tong and K. Wong, J. High Energy Phys. 09 (2013) 025. [arXiv:1305.2414 [hep-th]]. † On leave from ITEP, Moscow, Russia. [14] J. Ambjorn and P. Olesen, Nucl. Phys. B 315, 606 (1989); Phys. Lett. B 218, 67 (1989); Int. J. Mod. Phys. A 5, 4525 [1] N.K. Nielsen and P. Olesen, Nucl. Phys. B144, 376 (1978). (1990). [2] N.K. Nielsen and P. Olesen, Phys. Lett. 79B, 304 (1978). [15] M.N. Chernodub, J. Van Doorsselaere and H. Verschelde, Phys. Rev. D 88, 065006 (2013). [3] H.B. Nielsen and P. Olesen, Nucl. Phys. B160, 380 (1979). [16] M.N. Chernodub, Phys. Rev. D 82, 085011 (2010); Phys. Rev. Lett. 106, 142003 (2011); M. N. Chernodub, J. Van [4] J. Ambjorn and P. Olesen, Nucl. Phys. B170, 60 (1980). Doorsselaere and H. Verschelde, Phys. Rev. D 85, 045002 (2012). [5] J. Ambjorn and P. Olesen, ibid. B170, 265 (1980). [17] V. Skokov, A. Y. Illarionov and V. Toneev, Int. J. Mod. Phys. A 24, 5925 (2009) [6] G. K. Savvidy, Phys. Lett. 71B, 133 (1977). [18] M. Rasolt, Phys. Rev. Lett. 58, 1482 (1987); Z. Teˇsanovi´c, M. Rasolt, and L. Xing. ibid., 63 2425 (1989); M. Rasolt and [7] A.A. Abrikosov, Sov. Phys. JETP 5, 1174 (1957). Z. Teˇsanovi´c, Rev. Mod. Phys. 64, 709 (1992). [8] M.N. Chernodub, J. Van Doorsselaere, T. Kalaydzhyan and H[1.9]VPer.sOchleelsdeen,, ParhXysiv.L:1e3tt1.1B.4571390,. 63 (2014). − {− } [9] R. Jackiw and P. Rossi, Nucl. Phys. B190, 681 (1981). [20] Another solution, ω(z¯) = θ ( iνz¯, q) exp πz¯2/2 , is essentially identical to the solution (8) modulo a translation and 3 [10] E. Witten, Nucl. Phys. B249, 557 (1985). rotation in the transverse plane. [11] L.F. Cugliandolo, E.H. Fradkin and F.A. Schaposnik, Phys. L[2e1t]t.CB.G2.2C4a, l4la0n7,(J1r9.8a9n).d J.A. Harvey, Nucl. Phys. B250, 427 (1985). [12] T. Kalaydzhyan, Nucl. Phys. A913 243 (2013). ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: where

text: η

text: is

text: the

text: Dedekind

text: eta

text: function.

text: Then

text: the

text: prefactor

text: is

text: an

text: alternative

text: convenient

text: expansion

text: of

text: the

text: θ

text: function

text: N

text: can

text: be

text: evaluated

text: numerically:

text: which

text: enter

text: the

text: solution

text: in

text: Eq.

text: The

text: validity

text: of

text: Eq.

text: can

text: also

text: be

text: checked

text: numerically.

text: N

text: Thus,

text: we

text: that

text: ω˜(z¯)

text: ω(z¯),

text: and

text: eiπτ)

text: (z¯

text: X¯m,n)

text: m,n

text: e

text: π2

text: z

text: Xm,n

text: (z¯Xm,n

text: zX¯m,n),

label: reference

text: B.

text: B´eri,

text: D.

text: Tong

text: and

text: K.

text: Wong,

text: J.

text: High

text: Energy

text: Phys.

text: [hep-th]].

text: On

text: leave

text: from

text: ITEP,

text: Moscow,

text: Russia.

text: J.

text: Ambjorn

text: and

text: P.

text: Olesen,

text: Nucl.

text: Phys.

text: B

text: Phys.

text: Lett.

text: B

text: Int.

text: J.

text: Mod.

text: Phys.

text: A

text: N.K.

text: Nielsen

text: and

text: P.

text: Olesen,

text: Nucl.

text: Phys.

text: N.K.

text: Nielsen

text: and

text: P.

text: Olesen,

text: Phys.

text: Lett.

text: M.N.

text: Chernodub,

text: J.

text: Van

text: Doorsselaere

text: and

text: H.

text: Verschelde,

text: Phys.

text: Rev.

text: D

text: H.B.

text: Nielsen

text: and

text: P.

text: Olesen,

text: Nucl.

text: Phys.

text: M.N.

text: Chernodub,

text: Phys.

text: Rev.

text: D

text: Phys.

text: Rev.

text: Lett.

text: M.

text: N.

text: Chernodub,

text: J.

text: Van

text: J.

text: Ambjorn

text: and

text: P.

text: Olesen,

text: Nucl.

text: Phys.

text: Doorsselaere

text: and

text: H.

text: Verschelde,

text: Phys.

text: Rev.

text: D

text: J.

text: Ambjorn

text: and

text: P.

text: Olesen,

text: ibid.

text: V.

text: Skokov,

text: A.

text: Y.

text: Illarionov

text: and

text: V.

text: Toneev,

text: Int.

text: J.

text: Mod.

text: Phys.

text: A

text: G.

text: K.

text: Savvidy,

text: Phys.

text: Lett.

text: M.

text: Rasolt,

text: Phys.

text: Rev.

text: Lett.

text: Z.

text: Teˇsanovi´c,

text: M.

text: Rasolt,

text: and

text: L.

text: Xing.

text: ibid.,

text: M.

text: Rasolt

text: and

text: A.A.

text: Abrikosov,

text: Sov.

text: Phys.

text: JETP

text: Z.

text: Teˇsanovi´c,

text: Rev.

text: Mod.

text: Phys.

text: M.N.

text: Chernodub,

text: J.

text: Van

text: Doorsselaere,

text: T.

text: Kalaydzhyan

text: and

text: R.

text: Jackiw

text: and

text: P.

text: Rossi,

text: Nucl.

text: Phys.

text: Another

text: solution,

text: ω(z¯)

text: θ

text: iνz¯,

text: q)

text: exp

text: is

text: essentially

text: identical

text: to

text: the

text: solution

text: modulo

text: a

text: translation

text: and

text: E.

text: Witten,

text: Nucl.

text: Phys.

text: rotation

text: in

text: the

text: transverse

text: plane.

text: L.F.

text: Cugliandolo,

text: E.H.

text: Fradkin

text: and

text: F.A.

text: Schaposnik,

text: Phys.

text: J.A.

text: Harvey,

text: Nucl.

text: Phys.

text: T.

text: Kalaydzhyan,

text: Nucl.

text: Phys.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Reasoning about Meaning in Natural Language with Compact Closed Categories ∗ and Frobenius Algebras Dimitri Kartsaklis, Mehrnoosh Sadrzadeh, Stephen Pulman, and Bob Coecke Department of Computer Science, University of Oxford Wolfson Building, Parks Road, Oxford, OX1 3QD firstname.lastname@cs.ox.ac.uk Abstract Compact closed categories have found applications in modeling quantum information pro- tocols by Abramsky-Coecke. They also provide semantics for Lambek’s pregroup algebras, applied to formalizing the grammatical structure of natural language, and are implicit in a distributional model of word meaning based on vector spaces. Speciﬁcally, in previous work Coecke-Clark-Sadrzadeh used the product category of pregroups with vector spaces and provided a distributional model of meaning for sentences. We recast this theory in terms of strongly monoidal functors and advance it via Frobenius algebras over vector spaces. The former are used to formalize topological quantum ﬁeld theories by Atiyah and Baez-Dolan, and the latter are used to model classical data in quantum protocols by Coecke-Pavlovic- Vicary. The Frobenius algebras enable us to work in a single space in which meanings of words, phrases, and sentences of any structure live. Hence we can compare meanings of diﬀerent language constructs and enhance the applicability of the theory. We report on experimental results on a number of language tasks and verify the theoretical predictions. 1 Introduction Compact closed categories were ﬁrst introduced by Kelly [21] in early 1970’s. Some thirty years later they found applications in quantum mechanics [1], whereby the vector space foun- dations of quantum mechanics were recasted in a higher order language and quantum protocols such as teleportation found succinct conceptual proofs. Compact closed categories are complete with regard to a pictorial calculus [21, 38]; this calculus is used to depict and reason about information ﬂows in entangled quantum states modeled in tensor spaces, the phenomena that were considered to be mysteries of quantum mechanics and the Achilles heel of quantum logic [5]. The pictorial calculus revealed the multi-linear algebraic level needed for proving quan- tum information protocols and simpliﬁed the reasoning thereof to a great extent, by hiding the underlying additive vector space structure. ∗ To appear in Chubb, J., Eskandarian, A. and Harizanov, V., editors, Logic and Algebraic Structures in Quan- tum Computing and Information (2014), Cambridge University Press. Support by EPSRC grant EP/F042728/1 is acknowledged by the ﬁrst two authors. 1 ##LTLine##

text: Reasoning

label: title

text: about

text: Meaning

text: in

text: Natural

text: Language

text: with

text: Compact

text: Closed

text: Categories

text: and

text: Frobenius

text: Algebras

text: Dimitri

label: author

text: Kartsaklis,

text: Mehrnoosh

text: Sadrzadeh,

text: Stephen

text: Pulman,

text: and

text: Bob

text: Coecke

text: Department

text: of

text: Computer

text: Science,

text: University

text: of

text: Oxford

text: Wolfson

text: Building,

text: Parks

text: Road,

text: Oxford,

text: firstname.lastname@cs.ox.ac.uk

text: Abstract

text: Compact

label: abstract

text: closed

text: categories

text: have

text: found

text: applications

text: in

text: modeling

text: quantum

text: information

text: pro-

text: tocols

text: by

text: Abramsky-Coecke.

text: They

text: also

text: provide

text: semantics

text: for

text: Lambek’s

text: pregroup

text: algebras,

text: applied

text: to

text: formalizing

text: the

text: grammatical

text: structure

text: of

text: natural

text: language,

text: and

text: are

text: implicit

text: in

text: a

text: distributional

text: model

text: of

text: word

text: meaning

text: based

text: on

text: vector

text: spaces.

text: Speciﬁcally,

text: in

text: previous

text: work

text: Coecke-Clark-Sadrzadeh

text: used

text: the

text: product

text: category

text: of

text: pregroups

text: with

text: vector

text: spaces

text: and

text: provided

text: a

text: distributional

text: model

text: of

text: meaning

text: for

text: sentences.

text: We

text: recast

text: this

text: theory

text: in

text: terms

text: of

text: strongly

text: monoidal

text: functors

text: and

text: advance

text: it

text: via

text: Frobenius

text: algebras

text: over

text: vector

text: spaces.

text: The

text: former

text: are

text: used

text: to

text: formalize

text: topological

text: quantum

text: ﬁeld

text: theories

text: by

text: Atiyah

text: and

text: Baez-Dolan,

text: and

text: the

text: latter

text: are

text: used

text: to

text: model

text: classical

text: data

text: in

text: quantum

text: protocols

text: by

text: Coecke-Pavlovic-

text: Vicary.

text: The

text: Frobenius

text: algebras

text: enable

text: us

text: to

text: work

text: in

text: a

text: single

text: space

text: in

text: which

text: meanings

text: of

text: words,

text: phrases,

text: and

text: sentences

text: of

text: any

text: structure

text: live.

text: Hence

text: we

text: can

text: compare

text: meanings

text: of

text: diﬀerent

text: language

text: constructs

text: and

text: enhance

text: the

text: applicability

text: of

text: the

text: theory.

text: We

text: report

text: on

text: experimental

text: results

text: on

text: a

text: number

text: of

text: language

text: tasks

text: and

text: verify

text: the

text: theoretical

text: predictions.

text: Introduction

text: Compact

text: closed

text: categories

text: were

text: ﬁrst

text: introduced

text: by

text: Kelly

text: in

text: early

text: Some

text: thirty

text: years

text: later

text: they

text: found

text: applications

text: in

text: quantum

text: mechanics

text: whereby

text: the

text: vector

text: space

text: foun-

text: dations

text: of

text: quantum

text: mechanics

text: were

text: recasted

text: in

text: a

text: higher

text: order

text: language

text: and

text: quantum

text: protocols

text: such

text: as

text: teleportation

text: found

text: succinct

text: conceptual

text: proofs.

text: Compact

text: closed

text: categories

text: are

text: complete

text: with

text: regard

text: to

text: a

text: pictorial

text: calculus

text: this

text: calculus

text: is

text: used

text: to

text: depict

text: and

text: reason

text: about

text: information

text: ﬂows

text: in

text: entangled

text: quantum

text: states

text: modeled

text: in

text: tensor

text: spaces,

text: the

text: phenomena

text: that

text: were

text: considered

text: to

text: be

text: mysteries

text: of

text: quantum

text: mechanics

text: and

text: the

text: Achilles

text: heel

text: of

text: quantum

text: logic

text: The

text: pictorial

text: calculus

text: revealed

text: the

text: multi-linear

text: algebraic

text: level

text: needed

text: for

text: proving

text: quan-

text: tum

text: information

text: protocols

text: and

text: simpliﬁed

text: the

text: reasoning

text: thereof

text: to

text: a

text: great

text: extent,

text: by

text: hiding

text: the

text: underlying

text: additive

text: vector

text: space

text: structure.

text: To

text: appear

text: in

text: Chubb,

text: J.,

text: Eskandarian,

text: A.

text: and

text: Harizanov,

text: V.,

text: editors,

text: Logic

text: and

text: Algebraic

text: Structures

text: in

text: Quan-

text: tum

text: Computing

text: and

text: Information

text: Cambridge

text: University

text: Press.

text: Support

text: by

text: EPSRC

text: grant

text: is

text: acknowledged

text: by

text: the

text: ﬁrst

text: two

text: authors.

text: ##LTLine##


text: plain to see that th(cid:101)e ﬁeld is Gaussian. It remains to(cid:82) identify the covariance structure E[X(x)X(y)] = E(cid:88)[X(x)X(y)e−α D:X(x)(cid:80)2:eU(x)dx]/Zα ∞ ej(x)ej(y)(cid:115)E[ε2je−α ∞k=1 ε2kλ−k1 ] = (cid:88) λ Z j α j=1 ∞ = (cid:88) ej(xλ)ej(y) λj λ+ 2αe−λαj E[ε2je−αε2jλ−j 1 ] j j j=1 ∞ e (x)e (y) j j = . λ + 2α j j=1 Hence the law of X is that of the Massive Free Field in the metric eU(x)dx2 conditioned to be 0 U × on the boundary of with mass 2α 2π. Proof of Lemma 3.5. (cid:90) The ﬁrst step is to prove that the random variable 1 − − − γ2 U − γ2 H = (eγX(x) 1 γX(x) ln C(x, ) : X(x)2 :)g(x)dx γ γ2 U 2 2 L → converges in 2 towards 0 as(cid:90)γ 0. We (cid:16)have (cid:17) H = 1 (cid:90) : eγX(x) : C(x, U)γ2/2 − 1 − γ2 ln C(x, U)dx γ γ2 U 2 1 − − − γ2 (: eγX(x) : 1 γX(x) : X(x)2 :)g(x)dx γ2 U 2 L It is rather straightforward to check the convergence to zero in of the ﬁrst term. As for the 2 term of the second line, by the expansion(cid:90)(2.13) and the orthogonality of Wick polynomials (2.8), (cid:88) its variance is equal to 1 − γ2n 4 (GnU(x, y))ng(x)g(y)dxdy, (3.20) ≥ n! U2 n 3 and hence is O(γ2). The proof of Lemma 3.5 is complete provided that we prove the following lemma. (cid:104) (cid:105) Lemma 3.8. For any positive (cid:82)bounded function g one has sup E e− U g(x)(eγX(x)−1−γX(x)−γ22 lnC(x,U))dx < +∞. (cid:82) γ>0 ≡ Proof. For notational simplicity we assume in the proof that g 1 and that U dx = 1 but the proof with general g (and in particular g = eU(x)) works just the same. For the same reasons, we U ≤ further assume that C(x, ) 1 on the domain. Our strategy is to introduce ﬁrst the white noise cutoﬀ (X ) for the ﬁeld X. ε ε −Introducing the cutoﬀ. Recall that : eγXε(x) : stands for eγXε(x)−γ22 E[Xε(x)2]. Then as ex − x 1 is a positive function, we have 17 ##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: plain

text: to

text: see

text: that

text: ﬁeld

text: is

text: Gaussian.

text: It

text: remains

text: identify

text: the

text: covariance

text: structure

text: E[X(x)X(y)]

text: λ

text: Z

text: j

text: α

text: ej(xλ)ej(y)

text: λj

text: λ+

text: 2αe−λαj

text: j

text: j

text: e

text: (x)e

text: (y)

text: j

text: j

text: λ

text: 2α

text: j

text: Hence

text: the

text: law

text: of

text: X

text: is

text: that

text: of

text: the

text: Massive

text: Free

text: Field

text: in

text: the

text: metric

text: conditioned

text: to

text: be

text: U

text: on

text: the

text: boundary

text: of

text: with

text: mass

text: 2α

text: Proof

text: of

text: Lemma

text: The

text: ﬁrst

text: step

text: is

text: to

text: prove

text: that

text: the

text: random

text: variable

text: γ2

text: U

text: γ2

text: H

text: (eγX(x)

text: γX(x)

text: ln

text: C(x,

text: :)g(x)dx

text: γ

text: γ2

text: U

text: L

text: converges

text: in

text: towards

text: We

text: H

text: eγX(x)

text: C(x,

text: γ2

text: ln

text: C(x,

text: U)dx

text: γ

text: γ2

text: U

text: γ2

text: eγX(x)

text: γX(x)

text: :)g(x)dx

text: γ2

text: U

text: L

text: It

text: is

text: rather

text: straightforward

text: to

text: check

text: the

text: convergence

text: to

text: zero

text: in

text: of

text: the

text: ﬁrst

text: term.

text: As

text: for

text: the

text: term

text: of

text: the

text: second

text: line,

text: by

text: the

text: and

text: the

text: orthogonality

text: of

text: Wick

text: polynomials

text: its

text: variance

text: is

text: equal

text: to

text: (GnU(x,

text: y))ng(x)g(y)dxdy,

text: n!

text: n

text: and

text: hence

text: is

text: The

text: proof

text: of

text: Lemma

text: is

text: complete

text: provided

text: that

text: we

text: prove

text: the

text: following

text: lemma.

text: Lemma

text: For

text: any

text: positive

text: function

text: g

text: one

text: has

text: sup

text: E

text: e−

text: U

text: lnC(x,U))dx

text: Proof.

text: For

text: notational

text: simplicity

text: we

text: assume

text: in

text: the

text: proof

text: that

text: g

text: and

text: that

text: U

text: dx

text: but

text: the

text: proof

text: with

text: general

text: g

text: (and

text: in

text: particular

text: g

text: eU(x))

text: works

text: just

text: the

text: same.

text: For

text: the

text: same

text: reasons,

text: we

text: U

text: further

text: assume

text: that

text: C(x,

text: on

text: the

text: domain.

text: Our

text: strategy

text: is

text: to

text: introduce

text: ﬁrst

text: the

text: white

text: noise

text: cutoﬀ

text: (X

text: for

text: the

text: ﬁeld

text: X.

text: ε

text: ε

text: −Introducing

text: the

text: cutoﬀ.

text: Recall

text: that

text: eγXε(x)

text: stands

text: for

text: Then

text: as

text: ex

text: x

text: is

text: a

text: positive

text: function,

text: we

text: have

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: S { ∈ U − ∈ U } = h H1( ); h = V U; V solution to (3.6) for some function f H1( ) . (3.43) 0 0 ∈ S − If h , we can choose f such that h = V U where V is a solution to (3.6). Then, by (3.42), ∗ we deduce that f is a critical point of H, which is concave. Therefore I (h) = H(f). Plugging the − ∗ − − relation h = V U into the expression of H(f), we get I (h) = E(V ) E(U) = E(U +h) E(U). ∗ S This provides the expression of I on . S U Now we show that is dense set in H1( ). For this purpose we show that 0 S { ∈ U | ∈ U } = h H1( ) ∆h H1( ) . (3.44) 0 0 ∈ U Indeed if one sets V = U + h H1( ). We have 0 ∆V =∆U + ∆h − =8π2ΛeV + (∆U + ∆h 8π2ΛeV ) − =8π2ΛeV + (8π2ΛeU + ∆h 8π2ΛeV ). − − − U Setting f = (4πΛeV 4πΛeU (2π) 1∆h), it remains to to prove that f belongs to H1( ) if − ∈ U 0 and only if ∆h does. In both cases it suﬃces to prove that eU eV H1( ) but this is easy: − 0 − because it vanishes on the boundary because h does, and ∂(eU eV ) = eU∂U eV ∂V is square ∈ U ∈ U integrable. For this last point, one can check that either ∆h H1( ) (easy) or f H1( ) (using − 0 0 ∆V = 8π2ΛeV 2πf) implies that V is bounded. ∗ U \ S Now we establish the expression of I on H1( ) by density. Note that as a supremum ∗ 0 U of continuous linear functions, I restricted to H1( ) is weakly lower semi-continuous (for the U U0 ∈ H1( ) norm). By continuity of E(.) for the H1( ) norm, approximating h H1 by a sequence 0S 0 0 in we deduce that ∀ ∈ U ∗ ≤ − h H1( ), I (h) E(U + h) E(U). (3.45) 0 ∈ U S U Conversely, if h H1( ), we can ﬁnd a sequence (h ) in converging in H1( ) towards 0 ∈ U n n − 0 h. For each n, let us consider f H1( ) such that h = V U where V is the solution to n 0 n n n (3.6) associated to f . From (3.6), one can check that (f ) is Cauchy and strongly converges in − U n n n H 1( ). Then we get: ∗ − I (h) = sup h(f) f(Λ, f) + f(Λ) ∈ U f H1( ) ≥ 0 − h(f ) f(Λ, f ) + f(Λ) n n − − =(h h )(f ) + h (f ) f(Λ, f ) + f(Λ) n n n n n − − =(h h )(f ) + E(U + h ) E(U). n n n − → − → − We conclude by observing that (h h )(f ) 0 and E(U + h ) E(U) E(U + h) E(U) as → ∞ n n n n . To complete the proof, we show that I∗(h) < +∞ implies h ∈ H1(U). For each f¯ ∈ H1(U), let − 0 0 us consider the associated solution to (3.6) and deﬁne h¯ = V¯ U. Repeating the above argument, we have for each f¯ ∈ H1(U) 0 22

text: S

label: equation

text: U

text: U

text: h

text: h

text: V

text: U;

text: V

text: solution

text: to

text: for

text: some

text: function

text: f

text: S

text: If

text: h

text: we

text: can

text: choose

text: f

text: such

text: that

text: h

text: V

text: U

text: where

text: V

text: is

text: a

text: solution

text: to

text: Then,

text: by

text: we

text: deduce

text: that

text: f

text: is

text: a

text: critical

text: point

text: of

text: H,

text: which

text: is

text: concave.

text: Therefore

text: I

text: (h)

text: H(f).

text: Plugging

text: the

text: relation

text: h

text: V

text: U

text: into

text: the

text: expression

text: of

text: H(f),

text: we

text: get

text: I

text: (h)

text: E(V

text: E(U)

text: E(U

text: +h)

text: E(U).

text: S

text: This

text: provides

text: the

text: expression

text: of

text: I

text: on

text: S

text: U

text: Now

text: we

text: show

text: that

text: is

text: dense

text: set

text: in

text: For

text: this

text: purpose

text: we

text: show

text: that

text: S

text: U

text: U

text: h

text: ∆h

text: U

text: Indeed

text: if

text: one

text: sets

text: V

text: U

text: h

text: We

text: have

text: ∆V

text: =∆U

text: ∆h

text: (∆U

text: ∆h

text: ∆h

text: U

text: Setting

text: f

text: it

text: remains

text: to

text: to

text: prove

text: that

text: f

text: belongs

text: to

text: if

text: U

text: and

text: only

text: if

text: ∆h

text: does.

text: In

text: both

text: cases

text: it

text: suﬃces

text: to

text: prove

text: that

text: eU

text: eV

text: but

text: this

text: is

text: easy:

text: because

text: it

text: vanishes

text: on

text: the

text: boundary

text: because

text: h

text: does,

text: and

text: ∂(eU

text: eV

text: eU∂U

text: eV

text: ∂V

text: is

text: square

text: U

text: U

text: integrable.

text: For

text: this

text: last

text: point,

text: one

text: can

text: check

text: that

text: either

text: ∆h

text: (easy)

text: or

text: f

text: (using

text: ∆V

text: implies

text: that

text: V

text: is

text: bounded.

text: U

text: S

text: Now

text: we

text: establish

text: the

text: expression

text: of

text: I

text: on

text: by

text: density.

text: Note

text: that

text: as

text: a

text: supremum

text: U

text: of

text: continuous

text: linear

text: functions,

text: I

text: restricted

text: to

text: is

text: weakly

text: lower

text: semi-continuous

text: (for

text: the

text: U

text: norm).

text: By

text: continuity

text: of

text: E(.)

text: for

text: the

text: norm,

text: approximating

text: h

text: by

text: a

text: sequence

text: in

text: we

text: deduce

text: that

text: U

text: h

text: I

text: (h)

text: E(U

text: h)

text: E(U).

text: U

text: S

text: U

text: Conversely,

text: if

text: h

text: we

text: can

text: ﬁnd

text: a

text: sequence

text: (h

text: in

text: converging

text: in

text: towards

text: U

text: n

text: n

text: h.

text: For

text: each

text: n,

text: let

text: us

text: consider

text: f

text: such

text: that

text: h

text: V

text: U

text: where

text: V

text: is

text: the

text: solution

text: to

text: n

text: n

text: n

text: n

text: associated

text: to

text: f

text: From

text: one

text: can

text: check

text: that

text: (f

text: is

text: Cauchy

text: and

text: strongly

text: converges

text: in

text: U

text: n

text: n

text: n

text: H

text: Then

text: we

text: get:

text: I

text: (h)

text: sup

text: h(f)

text: f(Λ,

text: f)

text: f(Λ)

text: U

text: f

text: h(f

text: f(Λ,

text: f

text: f(Λ)

text: n

text: n

text: =(h

text: h

text: )(f

text: h

text: (f

text: f(Λ,

text: f

text: f(Λ)

text: n

text: n

text: n

text: n

text: n

text: =(h

text: h

text: )(f

text: E(U

text: h

text: E(U).

text: n

text: n

text: n

text: We

text: conclude

text: by

text: observing

text: that

text: (h

text: h

text: )(f

text: and

text: E(U

text: h

text: E(U)

text: E(U

text: h)

text: E(U)

text: as

text: n

text: n

text: n

text: n

text: To

text: complete

text: the

text: proof,

text: we

text: show

text: that

text: I∗(h)

text: implies

text: h

text: For

text: each

text: f¯

text: let

text: us

text: consider

text: the

text: associated

text: solution

text: to

text: and

text: deﬁne

text: h¯

text: V¯

text: U.

text: Repeating

text: the

text: above

text: argument,

text: we

text: have

text: for

text: each

text: f¯


text: ∗ ≥ − − I (h) (h h¯)(f¯) + E(U + h¯) E(U). (3.46) ∗ ∞ − ≥ As I (h) < + and E(U + h¯) E(U) 0, we deduce that ∀f¯ ∈ H1(U), h(f¯) ≤ C + h¯(f¯), (3.47) 0 for some constant C > 0, which does not depend on f¯. Let us further introduce a function g¯ ∈ H1(U) such that −2πf¯ = ∆g¯. To establish that h ∈ H1(U), it suﬃces to prove that the above 0 (cid:107) (cid:107) 0 ≤ (cid:107) (cid:107) ≤ right-hand side of (3.47) is b(cid:90)ounded uniformly w(cid:90)hen f¯ H−1(cid:90) 1 (or equivalently g¯ H1 1). By integrating (3.6) with respect to V¯ , we get − | | ∂V¯ 2 dx = 8π2Λ eV¯ V¯ dx + ∆g¯V¯ dx, U U (cid:90) (cid:90) (cid:90) (cid:90) which can be rewritten as | | − − (cid:104) (cid:105) ∂V¯ 2 dx + 8π2Λ (eV¯ 1)V¯ dx = 8π2Λ V¯ dx + ∂g¯, ∂V¯ dx. (3.48) U U U U (cid:82) (cid:104) (cid:105) ≤ | | | | (cid:82) Using the elementary inequality a, b 1 a 2 + c b 2 for the two terms for a well chosen c > 0 2c 2 (cid:48) | | (cid:82) (cid:48) we can establish that the right-hand side is less than C + 1 U ∂V¯ 2 dx, for some constant C that dbooeusndneodt duenpifeonrdmolynof¯n. tOhbesseertve{ft¯h∈atH(e01(cid:90)u(U−)1|)(cid:107)uf¯≥(cid:107)H0−f1o≤r a1l}l2u(a∈ndRtshousweUd|e∂dh¯u|c2edxthtaoto)U. F|∂iVn¯a|l2lyd,xwies have (cid:104) (cid:105) ≤ | | | | h¯(f¯) = ∂g¯, ∂h¯ dx g¯ h¯ H1 H1 U so that h¯(f¯) is uniformly bounded on the set {f¯ ∈ H1(U) | (cid:107)f¯(cid:107) − ≤ 1}. This implies that ∈ U 0 H 1 h H1( ). 0 U Also, recall that U is the unique minimum in H1( ) of the functional E. Indeed, a function in U 0 H1( ) is a minimum of this functional if and only if it is a weak solution to (3.3). Furthermore, 0 the weak solution of (3.3) is unique as we have proved that the ﬁeld γX converges in law (and even in probability) towards U as soon as we get a weak solution U to this equation. The limit ∈ U (cid:54) in law being unique, we get uniqueness for (3.3). In particular, if h H1( ) and h = 0, we get ∗ − 0 I (h) = E(U + h) E(U) > 0. Proof of Theorem 3.4. Assume that we can prove that the family (Y ) is exponentially tight and that for each function f ∈ H1(U) (cid:2) (cid:3) γ γ 0 E Yγ(f) − li→m γ2 ln µ,γ e γ2 = f(Λ, f) f(Λ). (3.49) γ 0 ∈ U (cid:55)→ − The mapping f H1( ) f(Λ, f) f(Λ) is Gˆateaux-diﬀerentiable as shown in (3.42) and 0 weakly lower semi-continuous (even weakly continuous) from Proposition A.3. Hence we can apply a standard result from the theory of Large deviation in functional spaces [17, Corollary 4.5.27], which entails the proof of Theorem 3.4. So we focus on establishing (3.49) ﬁrst and then we will prove that the family (Y ) is ex- γ γ ponentially tight. As we already know the asymptotic behavior of the partition function, it is suﬃcient to compute the asymptotic behavior of 23 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: I

text: (h)

text: (h

text: h¯)(f¯)

text: E(U

text: h¯)

text: E(U).

text: As

text: I

text: (h)

text: and

text: E(U

text: h¯)

text: E(U)

text: we

text: deduce

text: that

text: ∀f¯

text: h(f¯)

text: C

text: h¯(f¯),

text: for

text: some

text: constant

text: C

text: which

text: does

text: not

text: depend

text: on

text: f¯.

text: Let

text: us

text: further

text: introduce

text: a

text: function

text: g¯

text: such

text: that

text: −2πf¯

text: ∆g¯.

text: To

text: establish

text: that

text: h

text: it

text: suﬃces

text: to

text: prove

text: that

text: the

text: above

text: right-hand

text: side

text: of

text: is

text: uniformly

text: f¯

text: (or

text: equivalently

text: g¯

text: By

text: integrating

text: with

text: respect

text: to

text: V¯

text: we

text: get

text: ∂V¯

text: dx

text: 8π2Λ

text: eV¯

text: V¯

text: dx

text: ∆g¯V¯

text: dx,

text: U

text: U

text: which

text: can

text: be

text: rewritten

text: as

text: ∂V¯

text: dx

text: 8π2Λ

text: (eV¯

text: dx

text: 8π2Λ

text: V¯

text: dx

text: ∂g¯,

text: ∂V¯

text: dx.

text: U

text: U

text: U

text: U

text: Using

text: the

text: elementary

text: inequality

text: a,

text: b

text: a

text: c

text: b

text: for

text: the

text: two

text: terms

text: for

text: a

text: well

text: chosen

text: c

text: we

text: can

text: establish

text: that

text: the

text: right-hand

text: side

text: is

text: less

text: than

text: C

text: U

text: ∂V¯

text: dx,

text: for

text: some

text: constant

text: C

text: that

text: dbooeusndneodt

text: duenpifeonrdmolynof¯n.

text: have

text: h¯(f¯)

text: ∂g¯,

text: ∂h¯

text: dx

text: g¯

text: h¯

text: U

text: so

text: that

text: h¯(f¯)

text: is

text: uniformly

text: bounded

text: on

text: the

text: set

text: {f¯

text: This

text: implies

text: that

text: U

text: H

text: h

text: U

text: Also,

text: recall

text: that

text: U

text: is

text: the

text: unique

text: minimum

text: in

text: of

text: the

text: functional

text: E.

text: Indeed,

text: a

text: function

text: in

text: U

text: is

text: a

text: minimum

text: of

text: this

text: functional

text: if

text: and

text: only

text: if

text: it

text: is

text: a

text: weak

text: solution

text: to

text: Furthermore,

text: the

text: weak

text: solution

text: of

text: is

text: unique

text: as

text: we

text: have

text: proved

text: that

text: the

text: ﬁeld

text: γX

text: converges

text: in

text: law

text: (and

text: even

text: in

text: probability)

text: towards

text: U

text: as

text: soon

text: as

text: we

text: get

text: a

text: weak

text: solution

text: U

text: to

text: this

text: equation.

text: The

text: limit

text: U

text: in

text: law

text: being

text: unique,

text: we

text: get

text: uniqueness

text: for

text: In

text: particular,

text: if

text: h

text: and

text: h

text: we

text: get

text: I

text: (h)

text: E(U

text: h)

text: E(U)

text: Proof

text: of

text: Theorem

text: Assume

text: that

text: we

text: can

text: prove

text: that

text: the

text: family

text: (Y

text: is

text: exponentially

text: tight

text: and

text: that

text: for

text: each

text: function

text: f

text: γ

text: γ

text: E

text: Yγ(f)

text: li→m

text: γ2

text: ln

text: µ,γ

text: e

text: γ2

text: f(Λ,

text: f)

text: f(Λ).

text: γ

text: U

text: The

text: mapping

text: f

text: f(Λ,

text: f)

text: f(Λ)

text: is

text: Gˆateaux-diﬀerentiable

text: as

text: shown

text: in

text: and

text: weakly

text: lower

text: semi-continuous

text: (even

text: weakly

text: continuous)

text: from

text: Proposition

text: Hence

text: we

text: can

text: apply

text: a

text: standard

text: result

text: from

text: the

text: theory

text: of

text: Large

text: deviation

text: in

text: functional

text: spaces

text: Corollary

text: which

text: entails

text: the

text: proof

text: of

text: Theorem

text: So

text: we

text: focus

text: on

text: establishing

text: ﬁrst

text: and

text: then

text: we

text: will

text: prove

text: that

text: the

text: family

text: (Y

text: is

text: ex-

text: γ

text: γ

text: ponentially

text: tight.

text: As

text: we

text: already

text: know

text: the

text: asymptotic

text: behavior

text: of

text: the

text: partition

text: function,

text: it

text: is

text: suﬃcient

text: to

text: compute

text: the

text: asymptotic

text: behavior

text: of

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: L U First note that eU is in p( ) for so(cid:90)me p > 1. Hence, one can consider the following Hilbert- L Schmidt operator on the space 2(eU(x) dx) (cid:55)→ · f GU( , y)f(y)eU(y) dy U This symmetric operator can be d(cid:80)iagonalized along an orthonormal (in L2(eU(x) dx)) sequence (e ) ≥ with associated eigenvalues ( 1 ) ≥ (decreasing order with repetition to account for mul- j j 1 λj −j(cid:90) 1 ∞ tiple eigenvalues). We stress that λ 2 < + . Therefore we have j j e (x) j = GU(x, y)e (y)eU(y) dy. (4.8) j λ U j (cid:16) (cid:90) (cid:17) (cid:16) (cid:90) (cid:17) By using Cauchy-Schwartz, we get |e (x)| ≤ 1/2 1/2 j GU(x, y)2eU(y) dy e (y)2eU(y) dy . j λ U U j Therefore (4.8) implies that e is a continuous bounded function. One can then diﬀerentiate the j U expression (4.8) and see that the sequence (e ) ≥ is in H1( ); it is then standard to check that (√ej ) ≥ is an orthonormal sequence in H1j(Uj )1. In fact0, the sequence (λ ) is the increasing λj j 1 − − 0 j j sequence of eigenvalues of (2π) 1∆ with Dirichlet boundary conditions where g is the metric g √ tensor eU(x) dx2. It remains to show that the sequence ( ej ) ≥ is a basis of H1(U). Consider a U λj j 1 0 fu(cid:90)nction ϕ in H1( ) which is orthog(cid:90)onal to every (e ) ≥ in H1. T(cid:90)hen as 0 j j 1 0 − − − (cid:104) (cid:105) ϕ(x)e (x)eU(x)dx = 2π(λ ) 1 ϕ(x)∆e (x)dy = 2π(λ ) 1 ∂ϕ(x), ∂e (x) dx = 0. (4.9) j j j j j U U U L it is also orthogonal to all the (e ) ≥ as an element of 2(eU(x) dx) and thus is equal to j j 0 zero. Proof of Theorem 4.2. As in the proof of Theorem 3.4, the ﬁrst task is to compute the Laplace transform of linear forms under the Liouville measure. Let V be the (deterministic) weak solution of (4.6) (see Corollary A.5) and set − T := V H. (4.10) We also consider − θ(x) := f(x) 4πΛeV (x) (4.11) (cid:90) Note our deﬁnition together with (4.6) imply that T (x) := θ(y)GU(x, y)dy (4.12) U We deﬁne Y to be a shifted version of the ﬁeld X, γ 29 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: L

text: U

text: First

text: note

text: that

text: eU

text: is

text: in

text: p(

text: for

text: p

text: Hence,

text: one

text: can

text: consider

text: the

text: following

text: Hilbert-

text: L

text: Schmidt

text: operator

text: on

text: the

text: space

text: dx)

text: f

text: GU(

text: y)f(y)eU(y)

text: dy

text: U

text: This

text: symmetric

text: operator

text: can

text: be

text: along

text: an

text: orthonormal

text: (in

text: dx))

text: sequence

text: (e

text: with

text: associated

text: eigenvalues

text: (decreasing

text: order

text: with

text: repetition

text: to

text: account

text: for

text: mul-

text: j

text: j

text: λj

text: tiple

text: eigenvalues).

text: We

text: stress

text: that

text: λ

text: Therefore

text: we

text: have

text: j

text: j

text: e

text: (x)

text: j

text: GU(x,

text: y)e

text: (y)eU(y)

text: dy.

text: j

text: λ

text: U

text: j

text: By

text: using

text: Cauchy-Schwartz,

text: we

text: get

text: |e

text: (x)|

text: j

text: GU(x,

text: dy

text: e

text: dy

text: j

text: λ

text: U

text: U

text: j

text: Therefore

text: implies

text: that

text: e

text: is

text: a

text: continuous

text: bounded

text: function.

text: One

text: can

text: then

text: diﬀerentiate

text: the

text: j

text: U

text: expression

text: and

text: see

text: that

text: the

text: sequence

text: (e

text: is

text: in

text: it

text: is

text: then

text: standard

text: to

text: check

text: that

text: (√ej

text: is

text: an

text: orthonormal

text: sequence

text: in

text: In

text: the

text: sequence

text: (λ

text: is

text: the

text: increasing

text: λj

text: j

text: j

text: j

text: sequence

text: of

text: eigenvalues

text: of

text: 1∆

text: with

text: Dirichlet

text: boundary

text: conditions

text: where

text: g

text: is

text: the

text: metric

text: g

text: tensor

text: eU(x)

text: It

text: remains

text: to

text: show

text: that

text: the

text: sequence

text: ej

text: is

text: a

text: basis

text: of

text: Consider

text: a

text: U

text: λj

text: j

text: ϕ

text: in

text: which

text: is

text: to

text: every

text: (e

text: in

text: as

text: j

text: j

text: ϕ(x)e

text: (x)eU(x)dx

text: 2π(λ

text: ϕ(x)∆e

text: (x)dy

text: 2π(λ

text: ∂ϕ(x),

text: ∂e

text: (x)

text: dx

text: j

text: j

text: j

text: j

text: j

text: U

text: U

text: U

text: L

text: it

text: is

text: also

text: orthogonal

text: to

text: all

text: the

text: (e

text: as

text: an

text: element

text: of

text: dx)

text: and

text: thus

text: is

text: equal

text: to

text: j

text: j

text: zero.

text: Proof

text: of

text: Theorem

text: As

text: in

text: the

text: proof

text: of

text: Theorem

text: the

text: ﬁrst

text: task

text: is

text: to

text: compute

text: the

text: Laplace

text: transform

text: of

text: linear

text: forms

text: under

text: the

text: Liouville

text: measure.

text: Let

text: V

text: be

text: the

text: (deterministic)

text: weak

text: solution

text: of

text: (see

text: Corollary

text: and

text: set

text: T

text: V

text: H.

text: We

text: also

text: consider

text: θ(x)

text: f(x)

text: (x)

text: Note

text: our

text: deﬁnition

text: together

text: with

text: imply

text: that

text: T

text: (x)

text: θ(y)GU(x,

text: y)dy

text: U

text: We

text: deﬁne

text: Y

text: to

text: be

text: a

text: shifted

text: version

text: of

text: the

text: ﬁeld

text: X,

text: γ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (a) curvature=1 (b) curvature=4 Figure 1: Two surfaces with negative curvature continuum setting which is essential in the problem of establishing exact relations for the three point correlation function, the celebrated DOZZ formula [21, 56] derived on the sphere (see [34] for a recent article on this problem). Now recall that, in the physics literature, the Liouville action enables to describe random metrics in Liouville quantum gravity in the conformal gauge as introduced by Polyakov [43] (studied by David [12] and Distler-Kawai [19], see also the seminal work of Knizhnik-Polyakov- Zamolodchikov [37] in the light cone gauge) or in 2d string theory (see Klebanov’s review [36] for instance). There are many excellent reviews on this topic [34, 41, 43, 53]. The rough idea is to couple the action of a conformal matter ﬁeld (say a planar model of statistical physics at its critical point so as to become conformally invariant) to the action of gravity. This gives a couple of random variables (eγϕgˆ, M), where the random metric eγϕgˆ encodes the structure of the space and M stands for the matter ﬁeld. Liouville quantum gravity in 2d can thus be seen as a toy model to understand in quantum gravity how the interaction with matter inﬂuences the geometry of space-time. In the conformal gauge and up to omitting some details, the law of this couple of random variables tensorizes [43, 12] and the marginal law of the metric eγϕgˆ is given by the Liouville action (1.2). The only way the metric keeps track of its interaction with the matter ﬁeld M is through the parameter γ, called Liouville conformal factor, which can be explicitly expressed in terms of the central charge c of the matter ﬁeld using the celebrated KPZ result [37] √ √ − − − 25 c√ 1 c γ = . (1.4) 6 Therefore, the inﬂuence of the matter is parameterized by γ. In a way, we will see that the geometry of space is encoded in the quantity µγ2. More precisely, we study the so-called → semiclassical limit, meaning the convergence of the ﬁeld γϕ when the parameter γ 0 while → ∞ keeping ﬁxed the quantity Λ = µγ2 (thus µ ). In the case where gˆ is ﬂat or hyperbolic, we prove that the ﬁeld γϕ (resp. the measure eγϕ λ (dx)) converges in law towards the solution U gˆ (resp. eU(x)λ (dx)) to the so-called classical Liouville equation gˆ − ∆ U R = 8π2ΛeU. (1.5) gˆ gˆ Recall that equation (1.5) appears when looking for metrics in the conformal equivalence class − of gˆ with prescribed Ricci scalar curvature 8π2Λ and has for instance the following explicit form 4 ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: (a)

text: (b)

text: Figure

text: Two

label: caption

text: surfaces

text: with

text: negative

text: curvature

text: continuum

text: setting

text: which

text: is

text: essential

text: in

text: the

text: problem

text: of

text: establishing

text: exact

text: relations

text: for

text: the

text: three

text: point

text: correlation

text: function,

text: the

text: celebrated

text: DOZZ

text: formula

text: derived

text: on

text: the

text: sphere

text: (see

text: for

text: a

text: recent

text: article

text: on

text: this

text: problem).

text: Now

text: recall

text: that,

text: in

text: the

text: physics

text: literature,

text: the

text: Liouville

text: action

text: enables

text: to

text: describe

text: random

text: metrics

text: in

text: Liouville

text: quantum

text: gravity

text: in

text: the

text: conformal

text: gauge

text: as

text: introduced

text: by

text: Polyakov

text: (studied

text: by

text: David

text: and

text: Distler-Kawai

text: see

text: also

text: the

text: seminal

text: work

text: of

text: Knizhnik-Polyakov-

text: Zamolodchikov

text: in

text: the

text: light

text: cone

text: gauge)

text: or

text: in

text: string

text: theory

text: (see

text: Klebanov’s

text: review

text: for

text: instance).

text: There

text: are

text: many

text: excellent

text: reviews

text: on

text: this

text: topic

text: The

text: rough

text: idea

text: is

text: to

text: couple

text: the

text: action

text: of

text: a

text: conformal

text: matter

text: ﬁeld

text: (say

text: a

text: planar

text: model

text: of

text: statistical

text: physics

text: at

text: its

text: critical

text: point

text: so

text: as

text: to

text: become

text: conformally

text: invariant)

text: to

text: the

text: action

text: of

text: gravity.

text: This

text: gives

text: a

text: couple

text: of

text: random

text: variables

text: (eγϕgˆ,

text: M),

text: where

text: the

text: random

text: metric

text: eγϕgˆ

text: encodes

text: the

text: structure

text: of

text: the

text: space

text: and

text: M

text: stands

text: for

text: the

text: matter

text: ﬁeld.

text: Liouville

text: quantum

text: gravity

text: in

text: can

text: thus

text: be

text: seen

text: as

text: a

text: toy

text: model

text: to

text: understand

text: in

text: quantum

text: gravity

text: how

text: the

text: interaction

text: with

text: matter

text: inﬂuences

text: the

text: geometry

text: of

text: space-time.

text: In

text: the

text: conformal

text: gauge

text: and

text: up

text: to

text: omitting

text: some

text: details,

text: the

text: law

text: of

text: this

text: couple

text: of

text: random

text: variables

text: tensorizes

text: and

text: the

text: marginal

text: law

text: of

text: the

text: metric

text: eγϕgˆ

text: is

text: given

text: by

text: the

text: Liouville

text: action

text: The

text: only

text: way

text: the

text: metric

text: keeps

text: track

text: of

text: its

text: interaction

text: with

text: the

text: matter

text: ﬁeld

text: M

text: is

text: through

text: the

text: parameter

text: γ,

text: called

text: Liouville

text: conformal

text: factor,

text: which

text: can

text: be

text: explicitly

text: expressed

text: in

text: terms

text: of

text: the

text: central

text: charge

text: c

text: of

text: the

text: matter

text: ﬁeld

text: using

text: the

text: celebrated

text: KPZ

text: result

text: c√

text: c

text: γ

text: Therefore,

text: the

text: inﬂuence

text: of

text: the

text: matter

text: is

text: parameterized

text: by

text: γ.

text: In

text: a

text: way,

text: we

text: will

text: see

text: that

text: the

text: geometry

text: of

text: space

text: is

text: encoded

text: in

text: the

text: quantity

text: µγ2.

text: More

text: precisely,

text: we

text: study

text: the

text: so-called

text: semiclassical

text: limit,

text: meaning

text: the

text: convergence

text: of

text: the

text: ﬁeld

text: γϕ

text: when

text: the

text: parameter

text: γ

text: while

text: keeping

text: ﬁxed

text: the

text: quantity

text: Λ

text: µγ2

text: (thus

text: µ

text: In

text: the

text: case

text: where

text: gˆ

text: is

text: ﬂat

text: or

text: hyperbolic,

text: we

text: prove

text: that

text: the

text: ﬁeld

text: γϕ

text: (resp.

text: the

text: measure

text: eγϕ

text: λ

text: (dx))

text: converges

text: in

text: law

text: towards

text: the

text: solution

text: U

text: gˆ

text: (resp.

text: eU(x)λ

text: (dx))

text: to

text: the

text: so-called

text: classical

text: Liouville

text: equation

text: gˆ

text: U

text: R

text: gˆ

text: gˆ

text: Recall

text: that

text: equation

text: appears

text: when

text: looking

text: for

text: metrics

text: in

text: the

text: conformal

text: equivalence

text: class

text: of

text: gˆ

text: with

text: prescribed

text: Ricci

text: scalar

text: curvature

text: 8π2Λ

text: and

text: has

text: for

text: instance

text: the

text: following

text: explicit

text: form

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: The main d(cid:90)iﬃcult term is the integral con(cid:90)taining the exponential term. W(cid:90)ith the help of (A.5), we have − − − | | | | ≤ | | eU0t 1(eUt U0 1)V dx = eU0 V 2 dx + H , H Dt V 3 dx. t t t t t U U (cid:82) U (cid:82)U → L U By the Rellich-Kondrachov theorem, the embedding H1( ) 2( ) is compact so that the ﬁrst |0 | termUin the right-hand side converges towards U eU0 V 2 dx.| Fu| rthermore∞as Vt is bounded in H01( ) , the Sobol(cid:90)ev embedding entails(cid:90)that supt∈]0,1] U(cid:90)eU0 Vt 3 dx (cid:90)< + . Hence the second term goes to 0. We deduce | | − | | l→im ∂Vt 2 dx = 8π2Λ eU0V 2 dx + 2π hV dx = ∂V 2 dx. t 0 U U U U The proof is complete. U Proposition A.3. Assume that the family (f ) is weakly converging towards f in H1( ) as → t t>0 0 0 t 0. Denote by U the solution of the equation t − ∆U = 8π2ΛeUt 2πf , U | U = 0. (A.8) t t t ∂ U Then the family (Ut)t>0 strongly converges in H01( ) towa(cid:82)rds U0. L U Proof. The key points are ﬁrst to observe that (f ) is strongly converging towards f in 2( ) t t>0 | | ∞ 0 by using the Rellich-Kondrachov Theorem and that supt>0 U ft p dx < + by the Sobolev embeddings. Then the arguments are quite similar to the proof of Proposition A.2: we can prove − that U U converges strongly to the a solution of (A.4) with h = 0. Details are thus left to the t 0 reader. ∈ U ∈ Theorem A.4. Consider z , . . . , z and χ (cid:88), . . . χ ]0, 2[. The equation 1 p 1 p p − ∆U = 8π2Λ(cid:80)eU 2π χ δ , U| U = 0 (A.9) i zi ∂ i=1 (cid:80) admits a solution on U such that U − p χ GU(·, z ) is locally H¨older continuous on U. i=1 i i − · Proof. By using the same trick as in the proof of Theorem A.1, by setting V = U p χ GU( , z ), i=1 i i it suﬃces to solve the equation ∆V (x) = h(x)eV (x), V| U = 0 (A.10) (cid:80) ∂ (cid:90) · with h(x) = 8π2Λe pi=1 χiGU( ,zi). Let us consider the functional | | E(V ) = ∂V (x) 2 + 2h(x)eV (x) dx U U ∈ U deﬁned on H1( ) (observe that h Lq( ) for some q < q = 2/ max ∈{ } χ where q > 0 χ i 1,...,p i χ 1). Therefore, one can use the same arguments than the ones in the proof of Theorem A.1 to deduce from [52, Theorem 1.2] and [52, Theorem 1.6] that the functional E is weakly lower semi- U continuous and achieves its inﬁmum in H1( ). Moreover the reader can check that argmin E is 0 reduced to a point (by convexity) which is a solution of (A.10). Let us call it V . With the help of 41 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: The

text: main

text: term

text: is

text: the

text: integral

text: the

text: exponential

text: term.

text: the

text: help

text: of

text: we

text: have

text: dx

text: V

text: dx

text: H

text: H

text: Dt

text: V

text: dx.

text: t

text: t

text: t

text: t

text: t

text: U

text: U

text: U

text: L

text: U

text: By

text: the

text: Rellich-Kondrachov

text: theorem,

text: the

text: embedding

text: is

text: compact

text: so

text: that

text: the

text: ﬁrst

text: termUin

text: the

text: right-hand

text: side

text: converges

text: towards

text: U

text: V

text: dx.|

text: Fu|

text: rthermore∞as

text: Vt

text: is

text: bounded

text: in

text: the

text: embedding

text: Vt

text: dx

text: Hence

text: the

text: second

text: term

text: goes

text: to

text: We

text: deduce

text: l→im

text: ∂Vt

text: dx

text: 8π2Λ

text: dx

text: 2π

text: hV

text: dx

text: ∂V

text: dx.

text: t

text: U

text: U

text: U

text: U

text: The

text: proof

text: is

text: complete.

text: U

text: Proposition

text: Assume

text: that

text: the

text: family

text: (f

text: is

text: weakly

text: converging

text: towards

text: f

text: in

text: as

text: t

text: t

text: Denote

text: by

text: U

text: the

text: solution

text: of

text: the

text: equation

text: t

text: ∆U

text: U

text: U

text: t

text: t

text: t

text: U

text: Then

text: the

text: family

text: strongly

text: converges

text: in

text: L

text: U

text: Proof.

text: The

text: key

text: points

text: are

text: ﬁrst

text: to

text: observe

text: that

text: (f

text: is

text: strongly

text: converging

text: towards

text: f

text: in

text: t

text: by

text: using

text: the

text: Rellich-Kondrachov

text: Theorem

text: and

text: that

text: U

text: ft

text: p

text: dx

text: by

text: the

text: Sobolev

text: embeddings.

text: Then

text: the

text: arguments

text: are

text: quite

text: similar

text: to

text: the

text: proof

text: of

text: Proposition

text: we

text: can

text: prove

text: that

text: U

text: U

text: converges

text: strongly

text: to

text: the

text: a

text: solution

text: of

text: with

text: h

text: Details

text: are

text: thus

text: left

text: to

text: the

text: t

text: reader.

text: U

text: Theorem

text: Consider

text: z

text: z

text: and

text: χ

text: χ

text: The

text: equation

text: p

text: p

text: p

text: ∆U

text: 2π

text: χ

text: δ

text: U|

text: U

text: i

text: zi

text: admits

text: a

text: solution

text: on

text: U

text: such

text: that

text: U

text: p

text: χ

text: GU(·,

text: z

text: is

text: locally

text: H¨older

text: continuous

text: on

text: U.

text: i

text: i

text: Proof.

text: By

text: using

text: the

text: same

text: trick

text: as

text: in

text: the

text: proof

text: of

text: Theorem

text: by

text: setting

text: V

text: U

text: p

text: χ

text: GU(

text: z

text: i

text: i

text: it

text: suﬃces

text: to

text: solve

text: the

text: equation

text: ∆V

text: (x)

text: h(x)eV

text: (x),

text: V|

text: U

text: with

text: h(x)

text: χiGU(

text: ,zi).

text: Let

text: us

text: consider

text: the

text: functional

text: E(V

text: ∂V

text: (x)

text: (x)

text: dx

text: U

text: U

text: U

text: deﬁned

text: on

text: (observe

text: that

text: h

text: Lq(

text: for

text: some

text: q

text: q

text: max

text: χ

text: where

text: q

text: χ

text: i

text: i

text: χ

text: Therefore,

text: one

text: can

text: use

text: the

text: same

text: arguments

text: than

text: the

text: ones

text: in

text: the

text: proof

text: of

text: Theorem

text: to

text: deduce

text: from

text: Theorem

text: and

text: Theorem

text: that

text: the

text: functional

text: E

text: is

text: weakly

text: lower

text: semi-

text: U

text: continuous

text: and

text: achieves

text: its

text: inﬁmum

text: in

text: Moreover

text: the

text: reader

text: can

text: check

text: that

text: argmin

text: E

text: is

text: reduced

text: to

text: a

text: point

text: (by

text: convexity)

text: which

text: is

text: a

text: solution

text: of

text: Let

text: us

text: call

text: it

text: V

text: With

text: the

text: help

text: of

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 14 MULTIDIMENSIONAL SDES WITH DISTRIBUTIONAL DRIFT Proof. By Theorem 16 and the deﬁnition of the set K(β, q) we have that ∈ v (t) C1,α and (23) holds using the deﬁnition of the norms in C([0, T ]; H1+δ) λ p and C1,α. ∈ ∈ − ∇ ∈ Lemma 10 ensures that P w H1+δ for w H β and so P w Hδ. By t p p ∈ t R p the fractional Morrey inequality (Theorem 16) we have P w C1,α( d) and t for each t > 0 | ∇ | ≤ (cid:107)∇ (cid:107) ≤ (cid:107) (cid:107) ≤ − (cid:107) (cid:107) 1+δ+β (25) xs∈uRpd ( Ptw) (x) c Ptw Hpδ c Ptw Hp1+δ ct 2 w Hp−β, having used (15) in the latter inequality. Notice that the constant c depends only on δ, p and d. If we assume for a moment that the mild solution v of (12) is also a λ solution of (cid:90) t(cid:90) − − − · ∇ (26) v = e λ(t r)P (t r) (b(r) v (r)) dr λ p λ 0 t − − − + e λ(t r)P (t r)b(r)dr, p 0 (cid:90) then diﬀerentiating in x we get ∇ · t(cid:90) − − ∇ − · ∇ v (t, ) = e λ(t r) P (t r) (b(r) v (r)) dr λ p λ 0 t − − ∇ − + e λ(t r) P (t r)b(r)dr. p 0 (cid:90) We take the Hδ-norm and use (25) with Lemma 4 to obtain p (cid:107)∇ (cid:107) ≤ t(cid:90) − − − − (cid:107) (cid:107) (cid:107)∇ (cid:107) vλ(t) Hpδ c 0 e λ(t r)(t r) 1+δ2+β b(r) Hq−β vλ(r) Hpδdr t − − − − (cid:107)(cid:90) (cid:107) + c 0 e λ(t r)(t r) 1+δ2+β b(r) Hp−βdr ≤ (cid:48)(cid:107) (cid:107) (cid:90) (cid:107)∇ (cid:107) t − − − − c b ∞,Hq−β 0s≤urp≤t vλ(r) Hpδ 0 e λ(t r)(t r) 1+δ2+β dr (cid:107) (cid:107) t − − − − + c b ∞,Hp−β 0 e λ(t r)(t r) 1+δ2+β dr, so that by Lemma 12 we get (cid:107)∇ (cid:107) ≤ (cid:48)(cid:107) (cid:107) (cid:107)∇ (cid:107) − δ+β 1 0≤sut≤pT vλ(t) Hpδ c b ∞,Hq−β 0≤sut≤pT vλ(t) Hpδλ 2 (cid:107) (cid:107) − δ+β 1 + c b ∞,Hp−βλ 2 . ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: MULTIDIMENSIONAL

text: SDES

text: WITH

text: DISTRIBUTIONAL

text: DRIFT

text: Proof.

text: By

text: Theorem

text: and

text: the

text: deﬁnition

text: of

text: the

text: set

text: K(β,

text: q)

text: we

text: have

text: that

text: v

text: (t)

text: and

text: holds

text: using

text: the

text: deﬁnition

text: of

text: the

text: norms

text: in

text: T

text: λ

text: p

text: and

text: Lemma

text: ensures

text: that

text: P

text: w

text: for

text: w

text: H

text: β

text: and

text: so

text: P

text: w

text: Hδ.

text: By

text: t

text: p

text: p

text: t

text: R

text: p

text: the

text: fractional

text: Morrey

text: inequality

text: (Theorem

text: we

text: have

text: P

text: w

text: d)

text: and

text: t

text: for

text: each

text: t

text: xs∈uRpd

text: Ptw)

text: (x)

text: c

text: Ptw

text: Hpδ

text: c

text: Ptw

text: ct

text: w

text: Hp−β,

text: having

text: used

text: in

text: the

text: latter

text: inequality.

text: Notice

text: that

text: the

text: constant

text: c

text: depends

text: only

text: on

text: δ,

text: p

text: and

text: d.

text: If

text: we

text: assume

text: for

text: a

text: moment

text: that

text: the

text: mild

text: solution

text: v

text: of

text: is

text: also

text: a

text: λ

text: solution

text: of

text: v

text: e

text: λ(t

text: r)P

text: (t

text: r)

text: (b(r)

text: v

text: (r))

text: dr

text: λ

text: p

text: λ

text: t

text: e

text: λ(t

text: r)P

text: (t

text: r)b(r)dr,

text: p

text: then

text: diﬀerentiating

text: in

text: x

text: we

text: get

text: v

text: (t,

text: e

text: λ(t

text: r)

text: P

text: (t

text: r)

text: (b(r)

text: v

text: (r))

text: dr

text: λ

text: p

text: λ

text: t

text: e

text: λ(t

text: r)

text: P

text: (t

text: r)b(r)dr.

text: p

text: We

text: take

text: the

text: Hδ-norm

text: and

text: use

text: with

text: Lemma

text: to

text: obtain

text: p

text: vλ(t)

text: Hpδ

text: c

text: e

text: λ(t

text: r)(t

text: r)

text: b(r)

text: Hq−β

text: vλ(r)

text: Hpδdr

text: t

text: c

text: e

text: λ(t

text: r)(t

text: r)

text: b(r)

text: Hp−βdr

text: t

text: c

text: b

text: ∞,Hq−β

text: vλ(r)

text: Hpδ

text: e

text: λ(t

text: r)(t

text: r)

text: dr

text: t

text: c

text: b

text: ∞,Hp−β

text: e

text: λ(t

text: r)(t

text: r)

text: dr,

text: so

text: that

text: by

text: Lemma

text: we

text: get

text: δ+β

text: vλ(t)

text: Hpδ

text: c

text: b

text: ∞,Hq−β

text: vλ(t)

text: Hpδλ

text: δ+β

text: c

text: b

text: ∞,Hp−βλ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 6 MULTIDIMENSIONAL SDES WITH DISTRIBUTIONAL DRIFT δ − 1 β β − β 1 1 β =: 1 1 d q d q˜ p Figure 1. The set K(β, q). For the following the reader can also consult [25, Section 2.7.1]. Let us R R R R S consider the spaces C0,0( d; d) and C1,0( d; d) deﬁned as the closure of (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107)∇ (cid:107) with respect to the norm f = f ∞ and f = f ∞ + f ∞, C0,0 L C1,0 L L respectively. For 0 < α < 1 we will consider the Banach spaces { ∈ R R (cid:107) (cid:107) ∞} C0,α = f C0,0( d; d) : f < , C0,α { ∈ R R (cid:107) (cid:107) ∞} C1,α = f C1,0( d; d) : f < , C1,α endowed with the norms | − | (cid:107) (cid:107) (cid:107) (cid:107) f(x) f(y) f := f ∞ + sup | − | C0,α L (cid:54) ∈R x y α x=y d |∇ − ∇ | (cid:107) (cid:107) (cid:107) (cid:107) (cid:107)∇ (cid:107) f(x) f(y) f := f ∞ + f ∞ + sup | − | , C1,α L L (cid:54) ∈R x y α x=y d respectively. From now on, we are going to make the following standing assumption (cid:16) (cid:17) on the drift b and on the p(cid:0)ossib(cid:1)le choice of parameters: ∈ ∈ Assumption 5. Let β 0, 1 , q(cid:16) −d , d (cid:17)and set q˜ := −d . The drift 2 1 β β 1 β b will always be of the type ∈ ∞ − b L [0, T ]; H β . (cid:16) q˜,q (cid:17) (cid:16) (cid:17) ∈ ∞ − ∈ Remark 6. The fact that b L [0, T ]; H β implies, for each p [q˜, q], q˜,q ∈ ∞ − that b L [0, T ]; H β . p (cid:26) (cid:27) Moreover we consider the set − d (10) K(β, q) := κ = (δ, p) : β < δ < 1 β, < p < q , δ which is drawn in Figure 1. Note that K(β, q) is nonempty since β < 1 2 and −d < q < d . 1 β β ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: MULTIDIMENSIONAL

text: SDES

text: WITH

text: DISTRIBUTIONAL

text: DRIFT

text: δ

text: β

text: β

text: β

text: β

text: d

text: q

text: d

text: q˜

text: p

text: Figure

text: The

text: set

text: K(β,

text: q).

text: For

text: the

text: following

text: the

text: reader

text: can

text: also

text: consult

text: Section

text: Let

text: us

text: R

text: R

text: R

text: R

text: S

text: consider

text: the

text: spaces

text: d;

text: d)

text: and

text: d;

text: d)

text: deﬁned

text: as

text: the

text: closure

text: of

text: with

text: respect

text: to

text: the

text: norm

text: f

text: f

text: and

text: f

text: f

text: f

text: L

text: L

text: L

text: respectively.

text: For

text: α

text: we

text: will

text: consider

text: the

text: Banach

text: spaces

text: R

text: R

text: f

text: d;

text: d)

text: f

text: R

text: R

text: f

text: d;

text: d)

text: f

text: endowed

text: with

text: the

text: norms

text: f(x)

text: f(y)

text: f

text: f

text: sup

text: L

text: ∈R

text: x

text: y

text: α

text: x=y

text: d

text: f(x)

text: f(y)

text: f

text: f

text: f

text: sup

text: L

text: L

text: ∈R

text: x

text: y

text: α

text: x=y

text: d

text: respectively.

text: From

text: now

text: on,

text: we

text: are

text: going

text: to

text: make

text: the

text: following

text: standing

text: assumption

text: on

text: the

text: drift

text: b

text: and

text: on

text: the

text: choice

text: of

text: parameters:

text: Assumption

text: Let

text: β

text: −d

text: d

text: set

text: q˜

text: −d

text: The

text: drift

text: β

text: β

text: β

text: b

text: will

text: always

text: be

text: of

text: the

text: type

text: b

text: L

text: T

text: H

text: β

text: q˜,q

text: Remark

text: The

text: fact

text: that

text: b

text: L

text: T

text: H

text: β

text: implies,

text: for

text: each

text: p

text: [q˜,

text: q],

text: q˜,q

text: that

text: b

text: L

text: T

text: H

text: β

text: p

text: Moreover

text: we

text: consider

text: the

text: set

text: d

text: K(β,

text: q)

text: κ

text: (δ,

text: p)

text: β

text: δ

text: β,

text: p

text: q

text: δ

text: which

text: is

text: drawn

text: in

text: Figure

text: Note

text: that

text: K(β,

text: q)

text: is

text: nonempty

text: since

text: β

text: and

text: −d

text: q

text: d

text: β

text: β

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 2 swimmer design that has received much at- x2 3 x2 tention recently, namely, an ensemble of three 3 identical spherical beads connected by extend- able links that do not interact with the ﬂuid x1 x1 [12? ]. Models in this class have been utilized 1 1 to elucidate the properties both of individual swimmers [25, 26] and ensemble of swimmers 2 [27, 28], and have found experimental realiza- z1 z2 tion, with optical tweezers used to drive the 2 beads [7]. FIG. 1: Trimer deformations corresponding to In order to proceed, we shall make a number (z > 0, z = 0) (left) and (z = 0, z > 0) (right). 1 2 1 2 of simplifying assumption on the structure of Empty circles indicate the rest shape. the swimmer, on the stroke amplitude and on the nature of the ﬂuctuations. We shall assume that the swimmer moving parts and the stroke amplitude are small on the scale of the swimmer assume that the trimer can undergo only ax- body, and that the dynamics of the system is isymmetric deformations, as illustrated in Fig. slow. The ﬁrst two assumptions allow a quasi- 1. Let us indicate by xi, i = 1, 2, 3, the bead co- linear description of the dynamics, in which the ordinates in the comoving frame, and separate feedback of the ﬂuid perturbation on the swim- the rest component x(0): √i mer dynamics, is disregarded. This corresponds x(0) = R(1/ 3,√0, 0), to the lowest order in the description based on 1 − − the Kirkwood-Smoluchowski equation, utilized x(0) = R( 1/(2√3), 1/2, 0), 2 in [18, 29]. The last assumption of slow dynam- − x(0) = R( 1/(2 3), 1/2, 0), (1) ics guarantees that the response of the system 3 − to the deformation forces obey standard steady- from the deformation component x(1) = x i i state ﬂuctuation-dissipation relations. x(0): i The paper is organized as follows. In Sec. 2, the main results on the optimization of the de- x(11) = R(z1/2, 0, 0), √ − − terministic three-bead swimmer are presented. x(1) = (R/4)( z , √ 3z , 0), In Sec. 3, the modiﬁcation of the problem in the 2 − 1 2 x(1) = (R/4)( z , 3z , 0). (2) presence of a noisy component in the swimmer 3 1 2 internal dynamics are discussed. In Sec. 4, the The parametrization for x(1) has been chosen in optimization of the noisy swimmer in the weak i such a way that dissipation is diagonal [see Eq. noise limit is carried on. In Sec. 5, an exam- (13) below)]. We shall assume small deforma- ple is provided, of how noise can be exploited tions: to simplify the problem of internal control, by − | | ∼ (cid:28) overcoming some of the limitation of the scallop R 1 x(1) z 1 i theorem. Section 6 is devoted to conclusions. II. DETERMINISTIC CASE and seek an expression for the migration veloc- ity of the trimer to lowest order in z. In creeping ﬂow conditions, forces and parti- The microswimmer design that we are going cle velocities are related through the equation to consider is a variation on the three-sphere model of Najaﬁ & Golestanian [12, 30]. In the x˙ = u˜ (t) + f (t)/Γ, (3) i i i original swimmer the three spheres were put on the line, in the present one, they lie (at rest) at where Γ is the Stokes drag coeﬃcient for the the vertices of an equilateral triangle. A sim- beads and u˜ is the ﬂow perturbation gener- i ilar model was used in [25, 31] to study pas- ated by movement of the trimer, calculated at sive swimming in an external ﬂow. As in [12], x (the ﬂuid is considered quiescent in the ab- i the beads are supposed identical. We imagine sence of the trimer). For spherical beads, the that the device (called a trimer) is constrained drag Γ can be expressed in terms of the solvent to remain with its axis of symmetry along x , kinematic viscosity ν and density ρ by means 1 s s (see Fig. 1), but that it is otherwise free to of the formula Γ = 6πaν ρ [32]. The ﬂow per- s s translate. Consistent with this hypothesis, we turbation is determined by the instantaneous ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: swimmer

text: design

text: that

text: has

text: received

text: much

text: at-

font: default

text: tention

text: recently,

text: namely,

text: an

text: ensemble

text: of

text: three

text: identical

text: spherical

text: beads

text: connected

text: by

text: extend-

text: able

text: links

text: that

text: do

text: not

text: interact

text: with

text: the

text: ﬂuid

text: Models

text: in

text: this

text: class

text: have

text: been

text: utilized

text: to

text: elucidate

text: the

text: properties

text: both

text: of

text: individual

text: swimmers

text: and

text: ensemble

text: of

text: swimmers

text: and

text: have

text: found

text: experimental

text: realiza-

text: tion,

text: with

text: optical

text: tweezers

text: used

text: to

text: drive

text: the

text: beads

text: FIG.

text: Trimer

text: deformations

text: corresponding

text: to

text: In

text: order

text: to

text: proceed,

text: we

text: shall

text: make

text: a

text: number

text: (z

text: z

text: (left)

text: and

text: (z

text: z

text: (right).

text: of

text: simplifying

text: assumption

text: on

text: the

text: structure

text: of

text: Empty

text: circles

text: indicate

text: the

text: rest

text: shape.

text: the

text: swimmer,

text: on

text: the

text: stroke

text: amplitude

text: and

text: on

text: the

text: nature

text: of

text: the

text: ﬂuctuations.

text: We

text: shall

text: assume

text: that

text: the

text: swimmer

text: moving

text: parts

text: and

text: the

text: stroke

text: amplitude

text: are

text: small

text: on

text: the

text: scale

text: of

text: the

text: swimmer

text: assume

text: that

text: the

text: trimer

text: can

text: undergo

text: only

text: ax-

text: body,

text: and

text: that

text: the

text: dynamics

text: of

text: the

text: system

text: is

text: isymmetric

text: deformations,

text: as

text: illustrated

text: in

text: Fig.

text: slow.

text: The

text: ﬁrst

text: two

text: assumptions

text: allow

text: a

text: quasi-

text: Let

text: us

text: indicate

text: by

text: xi,

text: i

text: the

text: bead

text: co-

text: linear

text: description

text: of

text: the

text: dynamics,

text: in

text: which

text: the

text: ordinates

text: in

text: the

text: comoving

text: frame,

text: and

text: separate

text: feedback

text: of

text: the

text: ﬂuid

text: perturbation

text: on

text: the

text: swim-

text: the

text: rest

text: component

text: √i

text: mer

text: dynamics,

text: is

text: disregarded.

text: This

text: corresponds

text: to

text: the

text: lowest

text: order

text: in

text: the

text: description

text: based

text: on

text: the

text: Kirkwood-Smoluchowski

text: equation,

text: utilized

text: R(

text: in

text: The

text: last

text: assumption

text: of

text: slow

text: dynam-

text: R(

text: ics

text: guarantees

text: that

text: the

text: response

text: of

text: the

text: system

text: to

text: the

text: deformation

text: forces

text: obey

text: standard

text: steady-

text: from

text: the

text: deformation

text: component

text: x

text: i

text: i

text: state

text: ﬂuctuation-dissipation

text: relations.

text: i

text: The

text: paper

text: is

text: organized

text: as

text: follows.

text: In

text: Sec.

text: the

text: main

text: results

text: on

text: the

text: optimization

text: of

text: the

text: de-

text: terministic

text: three-bead

text: swimmer

text: are

text: presented.

text: z

text: In

text: Sec.

text: the

text: modiﬁcation

text: of

text: the

text: problem

text: in

text: the

text: z

text: presence

text: of

text: a

text: noisy

text: component

text: in

text: the

text: swimmer

text: internal

text: dynamics

text: are

text: discussed.

text: In

text: Sec.

text: the

text: The

text: parametrization

text: for

text: has

text: been

text: chosen

text: in

text: optimization

text: of

text: the

text: noisy

text: swimmer

text: in

text: the

text: weak

text: i

text: such

text: a

text: way

text: that

text: dissipation

text: is

text: diagonal

text: [see

text: Eq.

text: noise

text: limit

text: is

text: carried

text: on.

text: In

text: Sec.

text: an

text: exam-

text: below)].

text: We

text: shall

text: assume

text: small

text: deforma-

text: ple

text: is

text: provided,

text: of

text: how

text: noise

text: can

text: be

text: exploited

text: tions:

text: to

text: simplify

text: the

text: problem

text: of

text: internal

text: control,

text: by

text: overcoming

text: some

text: of

text: the

text: limitation

text: of

text: the

text: scallop

text: R

text: z

text: i

text: theorem.

text: Section

text: is

text: devoted

text: to

text: conclusions.

text: II.

text: DETERMINISTIC

text: CASE

text: and

text: seek

text: an

text: expression

text: for

text: the

text: migration

text: veloc-

text: ity

text: of

text: the

text: trimer

text: to

text: lowest

text: order

text: in

text: z.

text: In

text: creeping

text: ﬂow

text: conditions,

text: forces

text: and

text: parti-

text: The

text: microswimmer

text: design

text: that

text: we

text: are

text: going

text: cle

text: velocities

text: are

text: related

text: through

text: the

text: equation

text: to

text: consider

text: is

text: a

text: variation

text: on

text: the

text: three-sphere

text: model

text: of

text: Najaﬁ

text: Golestanian

text: In

text: the

text: x˙

text: u˜

text: (t)

text: f

text: (t)/Γ,

text: i

text: i

text: i

text: original

text: swimmer

text: the

text: three

text: spheres

text: were

text: put

text: on

text: the

text: line,

text: in

text: the

text: present

text: one,

text: they

text: lie

text: (at

text: rest)

text: at

text: where

text: Γ

text: is

text: the

text: Stokes

text: drag

text: coeﬃcient

text: for

text: the

text: the

text: vertices

text: of

text: an

text: equilateral

text: triangle.

text: A

text: sim-

text: beads

text: and

text: u˜

text: is

text: the

text: ﬂow

text: perturbation

text: gener-

text: i

text: ilar

text: model

text: was

text: used

text: in

text: to

text: study

text: pas-

text: ated

text: by

text: movement

text: of

text: the

text: trimer,

text: calculated

text: at

text: sive

text: swimming

text: in

text: an

text: external

text: ﬂow.

text: As

text: in

text: x

text: (the

text: ﬂuid

text: is

text: considered

text: quiescent

text: in

text: the

text: ab-

text: i

text: the

text: beads

text: are

text: supposed

text: identical.

text: We

text: imagine

text: sence

text: of

text: the

text: trimer).

text: For

text: spherical

text: beads,

text: the

text: that

text: the

text: device

text: (called

text: a

text: trimer)

text: is

text: constrained

text: drag

text: Γ

text: can

text: be

text: expressed

text: in

text: terms

text: of

text: the

text: solvent

text: to

text: remain

text: with

text: its

text: axis

text: of

text: symmetry

text: along

text: x

text: kinematic

text: viscosity

text: ν

text: and

text: density

text: ρ

text: by

text: means

text: s

text: s

text: (see

text: Fig.

text: but

text: that

text: it

text: is

text: otherwise

text: free

text: to

text: of

text: the

text: formula

text: Γ

text: 6πaν

text: ρ

text: The

text: ﬂow

text: per-

text: s

text: s

text: translate.

text: Consistent

text: with

text: this

text: hypothesis,

text: we

text: turbation

text: is

text: determined

text: by

text: the

text: instantaneous

text: ##LTFigure##

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##