text: 9 as the dispersion relation for the acoustic phonons in In this Appe(cid:90)nd(cid:90)ix we e(cid:90)va(cid:90)luate the four–point func- Abrikosov vortex lattices in conventional superconduc- tion (45) which has the following explicit form: tors. As in the case of usual superconductors, the “super- (cid:48) (cid:48) dxdy dx dy softness” of the transverse phonon mode may lead to an Q = (cid:89) (cid:88) (cid:88) l2,k2,l1,k1 L L L L instability of the ρ vortex lattice against thermal and x y x y quantum ﬂuctuations. Depending on the strength of 2 − (cid:48) these ﬂuctuations, the superconductor’s vortex lattice M2δ(x x )f(y) (A1) ∈Z ∈Z may either withstand the perturbations or melt into ei- i=1 mi ni ther a vortex liquid [38] or a vortex glass [39] (we refer a αh (cid:48) − − − g(x, x ) , reader to the review [24] for further details). The study ∂2 + 4π2[k l + ν(n m )]2 + M2 x 1,y 1,y 1 1 of stability of the vortex lattice is beyond the scope of (cid:48) where x is an arbitrary coordinate and this article. − − V. CONCLUSIONS f(y) = e2πiy(k1,y+k2,y l1,y l2,y) · − − e2πiyν(n1+n2 m1 m2), We have demonstrated the existence of the acoustic (cid:48) − (cid:48)− − − (cid:48)− − phonon modes in a suggested superconducting phase of g(x, x ) = e π(x νn2 k2,y)2 π(x νm2 l2,y)2 (A2) QCD vacuum at strong magnetic ﬁeld. · e−π(x−νn1−k2,y)2−π(x−νm2−l2,y)2, In the mean–ﬁeld approach, the superconducting ground − − h = e2πiν(k1,xn1+k2,xn2 l1,xm1 l2,xm2), state resembles an Abrikosov vortex lattice in the mixed ∗ ∗ phase an ordinary type–II superconductor. The vortices α = αm2αm1αn2αn1. are embedded in a vector quark-antiquark condensate We have assumed that we are working in a large but ﬁnite with carries the quantum numbers of ρ mesons. The × transverse space with dimensions L L . The derivative vortices form a hexagonal lattice in the transverse plane x y(cid:48) in Eq. (A1) acts on the function g(x, x ) only. For the with respect to the axis of the magnetic ﬁeld. The lat- sake of brevity, we do not show explicitly all arguments of tice breaks translational and rotational symmetries of the functions f, g, h and α which are deﬁned in Eq. (A2). the coordinate space and leads to the appearance of the The in(cid:90)tegral in Eq. ((cid:0)A1) over the y coordinate can be Nambu-Goldstone modes (acoustic phonons). taken as follows. Firstly, one gets: We have shown that the acoustic vibrations of the vor- − − (cid:1) tex lattice along the direction of the magnetic ﬁeld is dyf(y) = δ k + k l l 1,y 2,y 1,y 2,y a linear function of momentum. A phonon propagating − − in the transverse plane possesses a quadratic (supersoft) +ν(n + n m m ) . (A3) 1 2 1 2 dispersion relation in the limit of small momenta. The Because of the constraint we put on momenta in the y longitudinal phonons propagate with the speed of light direction (42), one ﬁnds that the absolute value of the while the velocity of their transverse counterparts de- − − sum k + k l l in Eq. (A3) should always be pends on their energy and it may be much smaller than 1,y 2,y 1,y 2,y smaller than 2ν. Moreover, we need only the expressions the speed of light. In the infrared limit the spectrum of with two nonzero momenta, so that the above sum turns acoustic phonons is given by Eq. (77). out to be(cid:90)smaller than ν. The later automatically implies The presence of the supersoft phonon modes is known − − n + n m m = 0, so that to be crucial for the stability of the vortex lattice since 1 2 1 2 these modes make an infrared divergent contribution to − − dyf(y) = δ (k + k l l ) the free energy of the system [24]. As a result, the vortex 1,y 2,y 1,y 2,y lattice may become unstable and melt into a vortex liq- · δ − − . (A4) uid. In our paper we have found the supersoft mode in n1+n2 m1 m2 (cid:48) the phonon spectrum. This ﬁnding is in agreement with The function g(x, x ) in Eq. (A2) can be rewritten as the results of the numerical simulations of lattice QCD follows: in strong magnetic ﬁeld background [25] which indicate g(x, x(cid:48)) = e−π[(x+x(cid:48)−2rν−ay)2+(x−x(cid:48)−2sν−by)2] the presence of the a liquid vortex state rather then an · − ordered vortex lattice. e π[(2tν+cy)2+(2uν+dy)2] , (A5) ACKNOWLEDGMENTS with n + m + n + m k + l + k + l The work of M.N.C. was partially supported by grant r = 1 1 2 2 , a = 1 1 2 2 , No. ANR-10-JCJC-0408 HYPERMAG of Agence Na- −2 − −2 − tionale de la Recherche (France). The work of J.V.D. was s = n1 + m1 n2 m2 , b = k1 + l1 k2 l2 , supported by a grant from La R´egion Centre (France). 2 2 − − − − n m n + m l k l + k t = 1 1 2 2 , c = 1 1 2 2 , 2 2 − − − − Appendix A: Explicit calculation of the quartic term n m + n m l k + l k u = 1 1 2 2 , d = 1 1 2 2 . 2 2 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: as

text: the

text: dispersion

text: relation

text: for

text: the

text: acoustic

text: phonons

text: in

text: In

text: this

text: we

text: the

text: four–point

text: func-

text: Abrikosov

text: vortex

text: lattices

text: in

text: conventional

text: superconduc-

text: tion

text: which

text: has

text: the

text: following

text: explicit

text: form:

text: tors.

text: As

text: in

text: the

text: case

text: of

text: usual

text: superconductors,

text: the

text: “super-

text: dxdy

text: dx

text: dy

text: softness”

text: of

text: the

text: transverse

text: phonon

text: mode

text: may

text: lead

text: to

text: an

text: Q

text: L

text: L

text: L

text: L

text: instability

text: of

text: the

text: ρ

text: vortex

text: lattice

text: against

text: thermal

text: and

text: x

text: y

text: x

text: y

text: quantum

text: ﬂuctuations.

text: Depending

text: on

text: the

text: strength

text: of

text: these

text: ﬂuctuations,

text: the

text: superconductor’s

text: vortex

text: lattice

text: x

text: )f(y)

text: ∈Z

text: ∈Z

text: may

text: either

text: withstand

text: the

text: perturbations

text: or

text: melt

text: into

text: ei-

text: mi

text: ni

text: ther

text: a

text: vortex

text: liquid

text: or

text: a

text: vortex

text: glass

text: (we

text: refer

text: a

text: αh

text: g(x,

text: x

text: reader

text: to

text: the

text: review

text: for

text: further

text: details).

text: The

text: study

text: ∂2

text: l

text: ν(n

text: m

text: x

text: of

text: stability

text: of

text: the

text: vortex

text: lattice

text: is

text: beyond

text: the

text: scope

text: of

text: where

text: x

text: is

text: an

text: arbitrary

text: coordinate

text: and

text: this

text: article.

text: V.

text: CONCLUSIONS

text: f(y)

text: We

text: have

text: demonstrated

text: the

text: existence

text: of

text: the

text: acoustic

text: phonon

text: modes

text: in

text: a

text: suggested

text: superconducting

text: phase

text: of

text: g(x,

text: x

text: e

text: π(x

text: π(x

text: QCD

text: vacuum

text: at

text: strong

text: magnetic

text: ﬁeld.

text: In

text: the

text: mean–ﬁeld

text: approach,

text: the

text: superconducting

text: ground

text: h

text: state

text: resembles

text: an

text: Abrikosov

text: vortex

text: lattice

text: in

text: the

text: mixed

text: phase

text: an

text: ordinary

text: type–II

text: superconductor.

text: The

text: vortices

text: α

text: are

text: embedded

text: in

text: a

text: vector

text: quark-antiquark

text: condensate

text: We

text: have

text: assumed

text: that

text: we

text: are

text: working

text: in

text: a

text: large

text: but

text: ﬁnite

text: with

text: carries

text: the

text: quantum

text: numbers

text: of

text: ρ

text: mesons.

text: The

text: transverse

text: space

text: with

text: dimensions

text: L

text: L

text: The

text: derivative

text: vortices

text: form

text: a

text: hexagonal

text: lattice

text: in

text: the

text: transverse

text: plane

text: x

text: in

text: Eq.

text: acts

text: on

text: the

text: function

text: g(x,

text: x

text: only.

text: For

text: the

text: with

text: respect

text: to

text: the

text: axis

text: of

text: the

text: magnetic

text: ﬁeld.

text: The

text: lat-

text: sake

text: of

text: brevity,

text: we

text: do

text: not

text: show

text: explicitly

text: all

text: arguments

text: of

text: tice

text: breaks

text: translational

text: and

text: rotational

text: symmetries

text: of

text: the

text: functions

text: f,

text: g,

text: h

text: and

text: α

text: which

text: are

text: deﬁned

text: in

text: Eq.

text: the

text: coordinate

text: space

text: and

text: leads

text: to

text: the

text: appearance

text: of

text: the

text: The

text: in

text: Eq.

text: over

text: the

text: y

text: coordinate

text: can

text: be

text: Nambu-Goldstone

text: modes

text: (acoustic

text: phonons).

text: taken

text: as

text: follows.

text: Firstly,

text: one

text: gets:

text: We

text: have

text: shown

text: that

text: the

text: acoustic

text: vibrations

text: of

text: the

text: vor-

text: tex

text: lattice

text: along

text: the

text: direction

text: of

text: the

text: magnetic

text: ﬁeld

text: is

text: dyf(y)

text: δ

text: k

text: k

text: l

text: l

text: a

text: linear

text: function

text: of

text: momentum.

text: A

text: phonon

text: propagating

text: in

text: the

text: transverse

text: plane

text: possesses

text: a

text: quadratic

text: (supersoft)

text: +ν(n

text: n

text: m

text: m

text: dispersion

text: relation

text: in

text: the

text: limit

text: of

text: small

text: momenta.

text: The

text: Because

text: of

text: the

text: constraint

text: we

text: put

text: on

text: momenta

text: in

text: the

text: y

text: longitudinal

text: phonons

text: propagate

text: with

text: the

text: speed

text: of

text: light

text: direction

text: one

text: ﬁnds

text: that

text: the

text: absolute

text: value

text: of

text: the

text: while

text: the

text: velocity

text: of

text: their

text: transverse

text: counterparts

text: de-

text: sum

text: k

text: k

text: l

text: l

text: in

text: Eq.

text: should

text: always

text: be

text: pends

text: on

text: their

text: energy

text: and

text: it

text: may

text: be

text: much

text: smaller

text: than

text: smaller

text: than

text: Moreover,

text: we

text: need

text: only

text: the

text: expressions

text: the

text: speed

text: of

text: light.

text: In

text: the

text: infrared

text: limit

text: the

text: spectrum

text: of

text: with

text: two

text: nonzero

text: momenta,

text: so

text: that

text: the

text: above

text: sum

text: turns

text: acoustic

text: phonons

text: is

text: given

text: by

text: Eq.

text: out

text: to

text: than

text: ν.

text: The

text: later

text: automatically

text: implies

text: The

text: presence

text: of

text: the

text: supersoft

text: phonon

text: modes

text: is

text: known

text: n

text: n

text: m

text: m

text: so

text: that

text: to

text: be

text: crucial

text: for

text: the

text: stability

text: of

text: the

text: vortex

text: lattice

text: since

text: these

text: modes

text: make

text: an

text: infrared

text: divergent

text: contribution

text: to

text: dyf(y)

text: δ

text: (k

text: k

text: l

text: l

text: the

text: free

text: energy

text: of

text: the

text: system

text: As

text: a

text: result,

text: the

text: vortex

text: lattice

text: may

text: become

text: unstable

text: and

text: melt

text: into

text: a

text: vortex

text: liq-

text: δ

text: uid.

text: In

text: our

text: paper

text: we

text: have

text: found

text: the

text: supersoft

text: mode

text: in

text: the

text: phonon

text: spectrum.

text: This

text: ﬁnding

text: is

text: in

text: agreement

text: with

text: The

text: function

text: g(x,

text: x

text: in

text: Eq.

text: can

text: be

text: rewritten

text: as

text: the

text: results

text: of

text: the

text: numerical

text: simulations

text: of

text: lattice

text: QCD

text: follows:

text: in

text: strong

text: magnetic

text: ﬁeld

text: background

text: which

text: indicate

text: g(x,

text: the

text: presence

text: of

text: the

text: a

text: liquid

text: vortex

text: state

text: rather

text: then

text: an

text: ordered

text: vortex

text: lattice.

text: e

text: ACKNOWLEDGMENTS

text: with

text: n

text: m

text: n

text: m

text: k

text: l

text: k

text: l

text: The

text: work

text: of

text: M.N.C.

text: was

text: partially

text: supported

text: by

text: grant

text: r

text: a

text: No.

text: HYPERMAG

text: of

text: Agence

text: Na-

text: −2

text: −2

text: tionale

text: de

text: la

text: Recherche

text: (France).

text: The

text: work

text: of

text: J.V.D.

text: was

text: s

text: b

text: supported

text: by

text: a

text: grant

text: from

text: La

text: R´egion

text: Centre

text: (France).

text: n

text: m

text: n

text: m

text: l

text: k

text: l

text: k

text: t

text: c

text: Appendix

text: A:

text: Explicit

text: calculation

text: of

text: the

text: quartic

text: term

text: n

text: m

text: n

text: m

text: l

text: k

text: l

text: k

text: u

text: d

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 1 Introduction Consider the regression of a response variable Y on a random vector X = (X , . . . , X )T 1 p of predictors. In full generality the goal is to describe the dependence on X of the condi- tional distribution of Y given X. As remarked by Li (1991), lowering dimensionality prior to running a regression is practically important and in many cases crucial for further analysis: after projecting the data onto a smaller space, we are then in a better position to graph- ical displays, model building, curve ﬁtting, model checking, and so on. For this purpose, linear suﬃcient dimension reduction (Cook 1998) focuses on ﬁnding a few linear combina- tions βT X, . . . , βT X that can replace X without loss of information and without requiring 1 d × a parametric model. If β denotes the p d matrix with columns β , . . . , β , we require that 1 d ⊥⊥ | Y X βT X, ⊥⊥ where the notation indicates independence. The statement is thus that Y is indepen- dent of X given any value assumed by βT X, or equivalently, the conditional distribution | | of Y X equals that of Y βT X. If β satisﬁes this relation, then its column space is called a dimension-reduction subspace (Li 1991). Under mild assumptions, the intersection of all dimension-reduction subspaces is itself a dimension-reduction subspace (Yin, Li and Cook 2008); in these cases it is called the central subspace for the regression of Y on X, denoted by S | , and its dimension, d | = dim(S | ), is called the structural dimension. An- Y X Y X Y X other important and closely related concept is that of central mean subspace (Cook and Li | 2002), which is concerned with dimension reduction for the conditional mean E(Y X). The conditional independence setting is 2

label: section

text: Introduction

text: Consider

label: paragraph

text: the

text: regression

text: of

text: a

text: response

text: variable

text: Y

text: on

text: a

text: random

text: vector

text: X

text: (X

text: X

text: )T

text: p

text: of

text: predictors.

text: In

text: full

text: generality

text: the

text: goal

text: is

text: to

text: describe

text: the

text: dependence

text: on

text: X

text: of

text: the

text: condi-

text: tional

text: distribution

text: of

text: Y

text: given

text: X.

text: As

text: remarked

text: by

text: Li

text: lowering

text: dimensionality

text: prior

text: to

text: running

text: a

text: regression

text: is

text: practically

text: important

text: and

text: in

text: many

text: cases

text: crucial

text: for

text: further

text: analysis:

text: after

text: projecting

text: the

text: data

text: onto

text: a

text: smaller

text: space,

text: we

text: are

text: then

text: in

text: a

text: better

text: position

text: to

text: graph-

text: ical

text: displays,

text: model

text: building,

text: curve

text: ﬁtting,

text: model

text: checking,

text: and

text: so

text: on.

text: For

text: this

text: purpose,

text: linear

text: suﬃcient

text: dimension

text: reduction

text: (Cook

text: focuses

text: on

text: ﬁnding

text: a

text: few

text: linear

text: combina-

text: tions

text: βT

text: X,

text: βT

text: X

text: that

text: can

text: replace

text: X

text: without

text: loss

text: of

text: information

text: and

text: without

text: requiring

text: d

text: a

text: parametric

text: model.

text: If

text: β

text: denotes

text: the

text: p

text: d

text: matrix

text: with

text: columns

text: β

text: β

text: we

text: require

text: that

text: d

text: Y

text: X

text: βT

text: X,

text: where

text: the

text: notation

text: indicates

text: independence.

text: The

text: statement

text: is

text: thus

text: that

text: Y

text: is

text: indepen-

text: dent

text: of

text: X

text: given

text: any

text: value

text: assumed

text: by

text: βT

text: X,

text: or

text: equivalently,

text: the

text: conditional

text: distribution

text: of

text: Y

text: X

text: equals

text: that

text: of

text: Y

text: βT

text: X.

text: If

text: β

text: satisﬁes

text: this

text: relation,

text: then

text: its

text: column

text: space

text: is

text: called

text: a

text: dimension-reduction

text: subspace

text: (Li

text: Under

text: mild

text: assumptions,

text: the

text: intersection

text: of

text: all

text: dimension-reduction

text: subspaces

text: is

text: itself

text: a

text: dimension-reduction

text: subspace

text: (Yin,

text: Li

text: and

text: Cook

text: in

text: these

text: cases

text: it

text: is

text: called

text: the

text: central

text: subspace

text: for

text: the

text: regression

text: of

text: Y

text: on

text: X,

text: denoted

text: by

text: S

text: and

text: its

text: dimension,

text: d

text: dim(S

text: is

text: called

text: the

text: structural

text: dimension.

text: An-

text: Y

text: X

text: Y

text: X

text: Y

text: X

text: other

text: important

text: and

text: closely

text: related

text: concept

text: is

text: that

text: of

text: central

text: mean

text: subspace

text: (Cook

text: and

text: Li

text: which

text: is

text: concerned

text: with

text: dimension

text: reduction

text: for

text: the

text: conditional

text: mean

text: E(Y

text: X).

text: The

text: conditional

text: independence

text: setting

text: is


text: case the structural dimension is d | = 4. The resulting estimators are denoted respectively Y X by f-SIR, T-SIR, YJ-SIR and SIR. Ten slices are used for all four methods considered here; it is well-known that the performance of SIR is not very sensitive to the number of slices, although how to tune the number of slices remains a diﬃcult open problem. Both the vector correlation coeﬃcient (VCC) and the trace correlation coeﬃcient (TCC) (cid:81) (cid:80) ˆ are employed to evaluate the estimation accuracy. For an estimator B of B, VCC is deﬁned − ≥ ≥ · · · ≥ ≥ to be ( d φ2)1/2 and TCC is deﬁned to be (d 1 d φ2)1/2, where 1 φ2 φ2 0 l=1 l l=1 l 1 d are the eigenvalues of the matrix BˆT B BT Bˆ with Bˆ and B being the orthonormalized o o o o o o ˆ versions of B and B respectively. A correlation coeﬃcient closer to unity means better esti- | mation of the (transformed) central subspace. Here, for Y f(X) we have B = (ηf, ηf) and 1 2 | d = 2, while for Y X we have B = (e , e , e , e ) and d = 4, where ηf = (1, 1, 0, . . . , 0)T , ηf = 1 2 3 4 1 2 (0, 0, 1, 1, 0, . . . , 0)T and e is a vector of length 10 whose i-th element is 1 and all other i elements are 0. The means and standard deviations of VCC and TCC, based on 200 repeti- tions, are presented in Tables 1 and 2. Several observations can be made as follows. First, we can see that, somewhat surprisingly, T-SIR performs slightly better than f-SIR; that is, using the estimated transformations yields a more accurate estimate than using the true ones. Second, we observe that, except for Case 3, probability-integral-transformed SIR is the best performer, followed by f-SIR and YJ-SIR. As we can see, the performance of SIR after the Yeo-Johnson transformation is very sensitive to the marginal distributions of X; YJ-SIR performs poorly in Cases 5 and 6. Third, using SIR directly for the regression of Y on X leads to very poor estimates with alarmingly low vector correlation coeﬃcient. This is not unexpected, because d | = 4 is much larger than d | = 2, making the estimation Y X Y f(X) problem considerably more diﬃcult. Finally, as the sample size increases, the performance of f-SIR, T-SIR and YJ-SIR improves greatly, but that of SIR is not much aﬀected. The estimation accuracy of each method relies on the selection of the structural dimension which is assumed to be known in the above discussion. We now study numerical aspects 12

text: case

text: the

text: structural

text: dimension

text: is

text: d

text: The

text: resulting

text: estimators

text: are

text: denoted

text: respectively

text: Y

text: X

text: by

text: f-SIR,

text: T-SIR,

text: YJ-SIR

text: and

text: SIR.

text: Ten

text: slices

text: are

text: used

text: for

text: all

text: four

text: methods

text: considered

text: here;

text: it

text: is

text: well-known

text: that

text: the

text: performance

text: of

text: SIR

text: is

text: not

text: very

text: sensitive

text: to

text: the

text: number

text: of

text: slices,

text: although

text: how

text: to

text: tune

text: the

text: number

text: of

text: slices

text: remains

text: a

text: diﬃcult

text: open

text: problem.

text: Both

text: the

text: vector

text: correlation

text: coeﬃcient

text: (VCC)

text: and

text: the

text: trace

text: correlation

text: coeﬃcient

text: (TCC)

text: ˆ

text: are

text: employed

text: to

text: evaluate

text: the

text: estimation

text: accuracy.

text: For

text: an

text: estimator

text: B

text: of

text: B,

text: VCC

text: is

text: deﬁned

text: to

text: be

text: d

text: and

text: TCC

text: is

text: deﬁned

text: to

text: be

text: (d

text: d

text: where

text: φ2

text: φ2

text: l

text: l

text: d

text: are

text: the

text: eigenvalues

text: of

text: the

text: matrix

text: BˆT

text: B

text: BT

text: Bˆ

text: with

text: Bˆ

text: and

text: B

text: being

text: the

text: orthonormalized

text: o

text: o

text: o

text: o

text: o

text: o

text: ˆ

text: versions

text: of

text: B

text: and

text: B

text: respectively.

text: A

text: correlation

text: coeﬃcient

text: closer

text: to

text: unity

text: means

text: better

text: esti-

text: mation

text: of

text: the

text: (transformed)

text: central

text: subspace.

text: Here,

text: for

text: Y

text: f(X)

text: we

text: have

text: B

text: (ηf,

text: ηf)

text: and

text: d

text: while

text: for

text: Y

text: X

text: we

text: have

text: B

text: (e

text: e

text: e

text: e

text: and

text: d

text: where

text: ηf

text: ηf

text: and

text: e

text: is

text: a

text: vector

text: of

text: length

text: whose

text: i-th

text: element

text: is

text: and

text: all

text: other

text: i

text: elements

text: are

text: The

text: means

text: and

text: standard

text: deviations

text: of

text: VCC

text: and

text: TCC,

text: based

text: on

text: repeti-

text: tions,

text: are

text: presented

text: in

text: Tables

text: and

text: Several

text: observations

text: can

text: be

text: made

text: as

text: follows.

text: First,

text: we

text: can

text: see

text: that,

text: somewhat

text: surprisingly,

text: T-SIR

text: performs

text: slightly

text: better

text: than

text: f-SIR;

text: that

text: is,

text: using

text: the

text: estimated

text: transformations

text: yields

text: a

text: more

text: accurate

text: estimate

text: than

text: using

text: the

text: true

text: ones.

text: Second,

text: we

text: observe

text: that,

text: except

text: for

text: Case

text: probability-integral-transformed

text: SIR

text: is

text: the

text: best

text: performer,

text: followed

text: by

text: f-SIR

text: and

text: YJ-SIR.

text: As

text: we

text: can

text: see,

text: the

text: performance

text: of

text: SIR

text: after

text: the

text: Yeo-Johnson

text: transformation

text: is

text: very

text: sensitive

text: to

text: the

text: marginal

text: distributions

text: of

text: X;

text: YJ-SIR

text: performs

text: poorly

text: in

text: Cases

text: and

text: Third,

text: using

text: SIR

text: directly

text: for

text: the

text: regression

text: of

text: Y

text: on

text: X

text: leads

text: to

text: very

text: poor

text: estimates

text: with

text: alarmingly

text: low

text: vector

text: correlation

text: coeﬃcient.

text: This

text: is

text: not

text: unexpected,

text: because

text: d

text: is

text: much

text: larger

text: than

text: d

text: making

text: the

text: estimation

text: Y

text: X

text: Y

text: f(X)

text: problem

text: considerably

text: more

text: diﬃcult.

text: Finally,

text: as

text: the

text: sample

text: size

text: increases,

text: the

text: performance

text: of

text: f-SIR,

text: T-SIR

text: and

text: YJ-SIR

text: improves

text: greatly,

text: but

text: that

text: of

text: SIR

text: is

text: not

text: much

text: aﬀected.

text: The

text: estimation

text: accuracy

text: of

text: each

text: method

text: relies

text: on

text: the

text: selection

text: of

text: the

text: structural

text: dimension

text: which

text: is

text: assumed

text: to

text: be

text: known

text: in

text: the

text: above

text: discussion.

text: We

text: now

text: study

text: numerical

text: aspects


text: Table 6: The frequencies of estimated structural dimension out of 200 repetitions by the BIC-type criterion n = 200 n = 400 dˆ | < 2 dˆ | = 2 dˆ | > 2 dˆ | < 2 dˆ | = 2 dˆ | > 2 Y f(X) Y f(X) Y f(X) Y f(X) Y f(X) Y f(X) T-SIR Case 7 (k = 5) 2 192 6 0 199 1 Case 7 (k = 10) 4 192 4 1 199 0 Case 7 (k = 20) 0 197 3 0 200 0 Case 8 2 195 3 3 200 0 f-SIR Case 7 (k = 5) 4 186 10 1 198 1 Case 7 (k = 10) 6 190 4 2 197 1 Case 7 (k = 20) 1 196 3 0 200 0 Case 8 1 197 2 0 200 0 4 Estimation: data-driven transformations As mentioned in Remark 3.2, the above estimation procedure, which makes use of the well-known probability integral transformation, suﬀers from misspeciﬁcation of transforma- tions. One strategy for dealing with this problem is to estimate the monotone transformations fully nonparametrically, as shown below. Since a monotonically increasing function has a positive ﬁrst derivative, it is reasonable to express f as j (cid:90) t { } f (t) = C + exp s (t) dt, (4.1) j j j tj1 where C is a constant, s is an unconstrained function, and t is the ﬁxed origin for j j j1 the range of t-values for which the data are being ﬁt. For simplicity, we assume that for each j, s (t) can be well approximated by a linear combination of basis functions j 16 ##LTLine## ##LTLine##

text: Table

text: The

label: caption

text: frequencies

text: of

text: estimated

text: structural

text: dimension

text: out

text: of

text: repetitions

text: by

text: the

text: BIC-type

text: criterion

text: n

label: table

text: n

text: dˆ

text: dˆ

text: dˆ

text: dˆ

text: dˆ

text: dˆ

text: Y

text: f(X)

text: Y

text: f(X)

text: Y

text: f(X)

text: Y

text: f(X)

text: Y

text: f(X)

text: Y

text: f(X)

text: T-SIR

text: Case

text: (k

text: Case

text: (k

text: Case

text: (k

text: Case

text: f-SIR

text: Case

text: (k

text: Case

text: (k

text: Case

text: (k

text: Case

text: Estimation:

text: data-driven

text: transformations

text: As

text: mentioned

text: in

text: Remark

text: the

text: above

text: estimation

text: procedure,

text: which

text: makes

text: use

text: of

text: the

text: well-known

text: probability

text: integral

text: transformation,

text: suﬀers

text: from

text: misspeciﬁcation

text: of

text: transforma-

text: tions.

text: One

text: strategy

text: for

text: dealing

text: with

text: this

text: problem

text: is

text: to

text: estimate

text: the

text: monotone

text: transformations

text: fully

text: nonparametrically,

text: as

text: shown

text: below.

text: Since

text: a

text: monotonically

text: increasing

text: function

text: has

text: a

text: positive

text: ﬁrst

text: derivative,

text: it

text: is

text: reasonable

text: to

text: express

text: f

text: as

text: j

text: t

text: f

text: (t)

text: C

text: exp

text: s

text: (t)

text: dt,

text: j

text: j

text: j

text: where

text: C

text: is

text: a

text: constant,

text: s

text: is

text: an

text: unconstrained

text: function,

text: and

text: t

text: is

text: the

text: ﬁxed

text: origin

text: for

text: j

text: j

text: the

text: range

text: of

text: t-values

text: for

text: which

text: the

text: data

text: are

text: being

text: ﬁt.

text: For

text: simplicity,

text: we

text: assume

text: that

text: for

text: each

text: j,

text: s

text: (t)

text: can

text: be

text: well

text: approximated

text: by

text: a

text: linear

text: combination

text: of

text: basis

text: functions

text: j

text: ##LTLine##

text: ##LTLine##


text: 410–428. [9] Cook, R. D. and Weisberg, S. (1991) Comment on “Sliced inverse regression for dimension reduction”. Journal of the American Statistical Association, 86, 328–332. [10] Cook, R. D. and Weisberg, S. (1999) Applied Regression Including Computing and Graphics. John Wiley & Sons, New York. [11] Li, B., Artemiou, A. and Li, L. (2011) Principle support vector machines for linear and nonlinear suﬃcient dimension reduction. The Annals of Statistics, 39, 3182–3210. [12] Li, B. and Dong, Y. (2009) Dimension reduction for nonelliptically distributed predic- tors. The Annals of Statistics, 37, 1272–1298. [13] Li, B. and Wang, S. (2007) On directional regression for dimension reduction. Journal of the American Statistical Association, 102, 997–1008. [14] Li, B., Wen, S. and Zhu, L. X. (2008) On a projective resampling method for dimension reduction with multivariate responses. Journal of the American Statistical Association, 103, 1177–1186. [15] Li, K. C. (1991) Sliced inverse regression for dimension reduction. Journal of the Amer- ican Statistical Association, 86, 316–327. [16] Li, K. C. (1992) On principal Hessian directions for data visualization and dimension reduction: Another application of Stein’s lemma. Journal of the American Statistical As- sociation, 87, 1025–1039. [17] Ma, Y. and Zhu, L. P. (2012) A semiparametric approach to dimension reduction. Journal of the American Statistical Association, 107, 168–179. [18] Marron, J. S. and Wand, M. P. (1992) Exact mean integrated squared error. The Annals of Statistics, 20, 712–736. 28

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text: R.

text: D.

text: and

text: Weisberg,

text: S.

text: Comment

text: on

text: “Sliced

text: inverse

text: regression

text: for

text: dimension

text: reduction”.

text: Journal

text: of

text: the

text: American

text: Statistical

text: Association,

text: Cook,

text: R.

text: D.

text: and

text: Weisberg,

text: S.

text: Applied

text: Regression

text: Including

text: Computing

text: and

text: Graphics.

text: John

text: Wiley

text: Sons,

text: New

text: York.

text: Li,

text: B.,

text: Artemiou,

text: A.

text: and

text: Li,

text: L.

text: Principle

text: support

text: vector

text: machines

text: for

text: linear

text: and

text: nonlinear

text: suﬃcient

text: dimension

text: reduction.

text: The

text: Annals

text: of

text: Statistics,

text: Li,

text: B.

text: and

text: Dong,

text: Y.

text: Dimension

text: reduction

text: for

text: nonelliptically

text: distributed

text: predic-

text: tors.

text: The

text: Annals

text: of

text: Statistics,

text: Li,

text: B.

text: and

text: Wang,

text: S.

text: On

text: directional

text: regression

text: for

text: dimension

text: reduction.

text: Journal

text: of

text: the

text: American

text: Statistical

text: Association,

text: Li,

text: B.,

text: Wen,

text: S.

text: and

text: Zhu,

text: L.

text: X.

text: On

text: a

text: projective

text: resampling

text: method

text: for

text: dimension

text: reduction

text: with

text: multivariate

text: responses.

text: Journal

text: of

text: the

text: American

text: Statistical

text: Association,

text: Li,

text: K.

text: C.

text: Sliced

text: inverse

text: regression

text: for

text: dimension

text: reduction.

text: Journal

text: of

text: the

text: Amer-

text: ican

text: Statistical

text: Association,

text: Li,

text: K.

text: C.

text: On

text: principal

text: Hessian

text: directions

text: for

text: data

text: visualization

text: and

text: dimension

text: reduction:

text: Another

text: application

text: of

text: Stein’s

text: lemma.

text: Journal

text: of

text: the

text: American

text: Statistical

text: As-

text: sociation,

text: Ma,

text: Y.

text: and

text: Zhu,

text: L.

text: P.

text: A

text: semiparametric

text: approach

text: to

text: dimension

text: reduction.

text: Journal

text: of

text: the

text: American

text: Statistical

text: Association,

text: Marron,

text: J.

text: S.

text: and

text: Wand,

text: M.

text: P.

text: Exact

text: mean

text: integrated

text: squared

text: error.

text: The

text: Annals

text: of

text: Statistics,


text: A SYMPLECTIC PROLEGOMENON 13 of J will be regular for simple (non-multiple-cover) curves. For discs, viewed as maps of the zt˘ u t u strip D2 1 which extend smoothly over the boundary punctures, a generic path J Pr s t t 0,1 of complex structures will again be regular for solutions which have an injective point x, i.e. ´ p p qq “ t u one where u 1 u x x . Example 3.3. There are also transversality issues not related to multiple covers. Suppose X carries a symplectic involution ι with ﬁxed locus Xι, and one has ι-invariant Lagrangians L with “ X P X i Lι L Xι Lagrangian in Xι. If for x, y Lι Lι the virtual dimension of the moduli space i i 1 2 containing a Floer trajectory u inside Xι is greater than the corresponding virtual dimension of u in X, then one cannot simultaneously have that the curve is regular in both spaces: the dense locus of regular J may be disjoint from the inﬁnite-codimension locus of ι-invariant J. (If there are no trajectories in the ﬁxed locus, one can achieve equivariant transversality [57].) Moduli spaces of solutions, even when regular, will typically not be compact. Gromov compact- ness [51] produces a compactiﬁcation which includes nodal curves (trees of spheres and discs). An important point is that although the Floer equation is a deformed version of the Cauchy- Riemann equation, the bubbles arise from a rescaling process which means that they satisfy B p q “ the undeformed equation u 0. In consequence, one cannot typically hope to overcome J transversality problems with multiply covered bubbles by generic choices of inhomogeneous terms or domain-dependent choices of almost complex structures. Typically, boundary strata for compactiﬁed spaces of holomorphic curves may have dimension far in excess of that of the open stratum one is trying to compactify. There are no easy general solutions to this issue (the solutions that exist [44, 53], which involve perturbing moduli spaces of solutions inside ambient Banach spaces of maps rather than perturbing the equations or other geometric data themselves, go by the general name of virtual perturbation theory). “ Concretely, to deﬁne Floer cohomology, one needs to know that d2 0 in the Floer complex. As in Morse theory, the proof of this relation comes from studying boundaries of one-dimensional (modulo translation) moduli spaces of solutions. Along with the boundary strata which con- tribute to d2, there is a problematic kind of breaking, in which a holomorphic disc with boundary 1 on either L or L bubbles oﬀ, cf. Figure 2. (The degenerate case in which the boundary values of the disc are mapped by a constant map to the Lagrangian, giving a rational curve inside X which passes through L, is also problematic.) “ Figure 2. Holomorphic disc bubbles can obstruct d2 0 in the Floer complex. There are three basic routes to avoiding the problems caused by disc bubbling. ##LTFigure## ##LTLine##

text: A

text: SYMPLECTIC

text: PROLEGOMENON

text: of

text: J

text: will

text: be

text: regular

text: for

text: simple

text: (non-multiple-cover)

text: curves.

text: For

text: discs,

text: viewed

text: as

text: maps

text: of

text: the

text: zt˘

text: u

text: t

text: u

text: strip

text: which

text: extend

text: smoothly

text: over

text: the

text: boundary

text: punctures,

text: a

text: generic

text: path

text: J

text: Pr

text: s

text: t

text: t

text: of

text: complex

text: structures

text: will

text: again

text: be

text: regular

text: for

text: solutions

text: which

text: have

text: an

text: injective

text: point

text: x,

text: i.e.

text: p

text: p

text: qq

text: t

text: u

text: one

text: where

text: u

text: u

text: x

text: x

text: Example

text: There

text: are

text: also

text: transversality

text: issues

text: not

text: related

text: to

text: multiple

text: covers.

text: Suppose

text: X

text: carries

text: a

text: symplectic

text: involution

text: ι

text: with

text: ﬁxed

text: locus

text: Xι,

text: and

text: one

text: has

text: ι-invariant

text: Lagrangians

text: L

text: with

text: X

text: P

text: X

text: i

text: Lι

text: L

text: Xι

text: Lagrangian

text: in

text: Xι.

text: If

text: for

text: x,

text: y

text: Lι

text: Lι

text: the

text: virtual

text: dimension

text: of

text: the

text: moduli

text: space

text: i

text: i

text: containing

text: a

text: Floer

text: trajectory

text: u

text: inside

text: Xι

text: is

text: greater

text: than

text: the

text: corresponding

text: virtual

text: dimension

text: of

text: u

text: in

text: X,

text: then

text: one

text: cannot

text: simultaneously

text: have

text: that

text: the

text: curve

text: is

text: regular

text: in

text: both

text: spaces:

text: the

text: dense

text: locus

text: of

text: regular

text: J

text: may

text: be

text: disjoint

text: from

text: the

text: inﬁnite-codimension

text: locus

text: of

text: ι-invariant

text: J.

text: (If

text: there

text: are

text: no

text: trajectories

text: in

text: the

text: ﬁxed

text: locus,

text: one

text: can

text: achieve

text: equivariant

text: transversality

text: Moduli

text: spaces

text: of

text: solutions,

text: even

text: when

text: regular,

text: will

text: typically

text: not

text: be

text: compact.

text: Gromov

text: compact-

text: ness

text: produces

text: a

text: compactiﬁcation

text: which

text: includes

text: nodal

text: curves

text: (trees

text: of

text: spheres

text: and

text: discs).

text: An

text: important

text: point

text: is

text: that

text: although

text: the

text: Floer

text: equation

text: is

text: a

text: deformed

text: version

text: of

text: the

text: Cauchy-

text: Riemann

text: equation,

text: the

text: bubbles

text: arise

text: from

text: a

text: rescaling

text: process

text: which

text: means

text: that

text: they

text: satisfy

text: B

text: p

text: q

text: the

text: undeformed

text: equation

text: u

text: In

text: consequence,

text: one

text: cannot

text: typically

text: hope

text: to

text: overcome

text: J

text: transversality

text: problems

text: with

text: multiply

text: covered

text: bubbles

text: by

text: generic

text: choices

text: of

text: inhomogeneous

text: terms

text: or

text: domain-dependent

text: choices

text: of

text: almost

text: complex

text: structures.

text: Typically,

text: boundary

text: strata

text: for

text: compactiﬁed

text: spaces

text: of

text: holomorphic

text: curves

text: may

text: have

text: dimension

text: far

text: in

text: excess

text: of

text: that

text: of

text: the

text: open

text: stratum

text: one

text: is

text: trying

text: to

text: compactify.

text: There

text: are

text: no

text: easy

text: general

text: solutions

text: to

text: this

text: issue

text: (the

text: solutions

text: that

text: exist

text: which

text: involve

text: perturbing

text: moduli

text: spaces

text: of

text: solutions

text: inside

text: ambient

text: Banach

text: spaces

text: of

text: maps

text: rather

text: than

text: perturbing

text: the

text: equations

text: or

text: other

text: geometric

text: data

text: themselves,

text: go

text: by

text: the

text: general

text: name

text: of

text: virtual

text: perturbation

text: theory).

text: Concretely,

text: to

text: deﬁne

text: Floer

text: cohomology,

text: one

text: needs

text: to

text: know

text: that

text: in

text: the

text: Floer

text: complex.

text: As

text: in

text: Morse

text: theory,

text: the

text: proof

text: of

text: this

text: relation

text: comes

text: from

text: studying

text: boundaries

text: of

text: one-dimensional

text: (modulo

text: translation)

text: moduli

text: spaces

text: of

text: solutions.

text: Along

text: with

text: the

text: boundary

text: strata

text: which

text: con-

text: tribute

text: to

text: there

text: is

text: a

text: problematic

text: kind

text: of

text: breaking,

text: in

text: which

text: a

text: holomorphic

text: disc

text: with

text: boundary

text: on

text: either

text: L

text: or

text: L

text: bubbles

text: oﬀ,

text: cf.

text: Figure

text: (The

text: degenerate

text: case

text: in

text: which

text: the

text: boundary

text: values

text: of

text: the

text: disc

text: are

text: mapped

text: by

text: a

text: constant

text: map

text: to

text: the

text: Lagrangian,

text: giving

text: a

text: rational

text: curve

text: inside

text: X

text: which

text: passes

text: through

text: L,

text: is

text: also

text: problematic.)

text: Figure

text: Holomorphic

text: disc

text: bubbles

text: can

text: obstruct

text: in

text: the

text: Floer

text: complex.

text: There

text: are

text: three

text: basic

text: routes

text: to

text: avoiding

text: the

text: problems

text: caused

text: by

text: disc

text: bubbling.

text: ##LTFigure##

text: ##LTLine##


text: A SYMPLECTIC PROLEGOMENON 3 alternative worldview, one could say that any symplectic manifold is locally as complicated as Euclidean space, about which a vast array of questions remain open. Two especially important classes of symplectic manifolds are: Ă P (1) (quasi-)projective varieties X N, equipped with the restriction of the Fubini-Study form ω ; in particular, any ﬁnite type oriented two-dimensional surface, where the FS symplectic form is just a way of measuring area; ˚ “ (2) cotangent bundles T Q, with the canonical form ω dθ, θ the Liouville 1-form, and more generally Stein manifolds, i.e. closed properly embedded holomorphic submanifolds C of N, equivalently complex manifolds equipped with an exhausting strictly plurisub- “ ´ harmonic function h, with ω ddch. Symplectic manifolds are ﬂexible enough to admit cut-and-paste type surgeries [48]; many such in dimension four yield examples which are provably not algebraic, and one dominant theme t u Ă t u Ă t u in symplectic topology is to understand the inclusions K¨ahler Symplectic Smooth . For open manifolds, there is a good existence theory for symplectic structures, but questions of uniqueness remain subtle: there are countably many symplectically distinct (ﬁnite type) Stein structures on Euclidean space, and recognising the standard structure is algorithmically undecidable [68]. A second essential feature of symplectic geometry is that any symplectic manifold admits a contractible space J of almost complex structures which tame the symplectic form, in the sense p q ą that ω v, Jv 0 for any non-zero tangent vector v. Following a familiar pattern, discrete (e.g. enumerative) invariants deﬁned with respect to an auxiliary choice drawn from a contractible space reﬂect intrinsic properties of the underlying symplectic structure. Ă | “ A submanifold L M is Lagrangian if it is isotropic, ω 0, and half-dimensional (maximal L dimension for an isotropic submanifold). In the classes given above, Lagrangian submanifolds R include (1) the real locus of an algebraic variety deﬁned over , (2) the zero-section and cotan- ˚ gent ﬁbre in T Q, respectively. Gromov’s fundamental insight was that, for a generic taming almost complex structure, spaces of non-multiply-covered holomorphic curves in M (perhaps with Lagrangian boundary conditions) are ﬁnite-dimensional smooth manifolds, which moreover admit geometrically meaningful compactiﬁcations by nodal curves. It is the resulting invariants which count “pseudoholomorphic” curves (maps from Riemann surfaces which are J-complex linear for some taming but not necessarily integrable J) that dominate the ﬁeld. Discrete invariants of symplectic manifolds and Lagrangian submanifolds include characteristic classes which determine the virtual, i.e. expected, dimensions of moduli spaces of holomorphic curves (when non-empty), via index theory. Any choice of taming J makes T M a complex p q P p q p q{ p q Z bundle, which has Chern classes c M H2i M; . Since the Grassmannian U n O n of i C Z Lagrangian subspaces of n has fundamental group , any Lagrangian submanifold has a Maslov p q Ñ Z class µ : π M, L , which measures the twisting of the Lagrangian tangent planes around L 2 | p q Ñ p q Z Z the boundary of a disc D relative to a trivialisation of T M ; under H2 M, L; H2 M; , p q D Ñ P µ maps to 2c M . The moduli space of parametrised rational curves 1 M2n representing L 1 x p q y ` p q P C a homology class A has real dimension 2 c M , A 2n, hence dividing by SL the space 1 2

text: A

text: SYMPLECTIC

text: PROLEGOMENON

text: alternative

text: worldview,

text: one

text: could

text: say

text: that

text: any

text: symplectic

text: manifold

text: is

text: locally

text: as

text: complicated

text: as

text: Euclidean

text: space,

text: about

text: which

text: a

text: vast

text: array

text: of

text: questions

text: remain

text: open.

text: Two

text: especially

text: important

text: classes

text: of

text: symplectic

text: manifolds

text: are:

text: Ă

label: list

text: P

text: (quasi-)projective

text: varieties

text: X

text: N,

text: equipped

text: with

text: the

text: restriction

text: of

text: the

text: Fubini-Study

text: form

text: ω

text: in

text: particular,

text: any

text: ﬁnite

text: type

text: oriented

text: two-dimensional

text: surface,

text: where

text: the

text: FS

text: symplectic

text: form

text: is

text: just

text: a

text: way

text: of

text: measuring

text: area;

text: cotangent

text: bundles

text: T

text: Q,

text: with

text: the

text: canonical

text: form

text: ω

text: dθ,

text: θ

text: the

text: Liouville

text: and

text: more

text: generally

text: Stein

text: manifolds,

text: i.e.

text: closed

text: properly

text: embedded

text: holomorphic

text: submanifolds

text: C

text: of

text: N,

text: equivalently

text: complex

text: manifolds

text: equipped

text: with

text: an

text: exhausting

text: strictly

text: plurisub-

text: harmonic

text: function

text: h,

text: with

text: ω

text: ddch.

text: Symplectic

text: manifolds

text: are

text: ﬂexible

text: enough

text: to

text: admit

text: cut-and-paste

text: type

text: surgeries

text: many

text: such

text: in

text: dimension

text: four

text: yield

text: examples

text: which

text: are

text: provably

text: not

text: algebraic,

text: and

text: one

text: dominant

text: theme

text: t

text: u

text: Ă

text: t

text: u

text: Ă

text: t

text: u

text: in

text: symplectic

text: topology

text: is

text: to

text: understand

text: the

text: inclusions

text: K¨ahler

text: Symplectic

text: Smooth

text: For

text: open

text: manifolds,

text: there

text: is

text: a

text: good

text: existence

text: theory

text: for

text: symplectic

text: structures,

text: but

text: questions

text: of

text: uniqueness

text: remain

text: subtle:

text: there

text: are

text: countably

text: many

text: symplectically

text: distinct

text: (ﬁnite

text: type)

text: Stein

text: structures

text: on

text: Euclidean

text: space,

text: and

text: recognising

text: the

text: standard

text: structure

text: is

text: algorithmically

text: undecidable

text: A

text: second

text: essential

text: feature

text: of

text: symplectic

text: geometry

text: is

text: that

text: any

text: symplectic

text: manifold

text: admits

text: a

text: contractible

text: space

text: J

text: of

text: almost

text: complex

text: structures

text: which

text: tame

text: the

text: symplectic

text: form,

text: in

text: the

text: sense

text: p

text: q

text: ą

text: that

text: ω

text: v,

text: Jv

text: for

text: any

text: non-zero

text: tangent

text: vector

text: v.

text: Following

text: a

text: familiar

text: pattern,

text: discrete

text: (e.g.

text: enumerative)

text: invariants

text: deﬁned

text: with

text: respect

text: to

text: an

text: auxiliary

text: choice

text: drawn

text: from

text: a

text: contractible

text: space

text: reﬂect

text: intrinsic

text: properties

text: of

text: the

text: underlying

text: symplectic

text: structure.

text: Ă

text: A

text: submanifold

text: L

text: M

text: is

text: Lagrangian

text: if

text: it

text: is

text: isotropic,

text: ω

text: and

text: half-dimensional

text: (maximal

text: L

text: dimension

text: for

text: an

text: isotropic

text: submanifold).

text: In

text: the

text: classes

text: given

text: above,

text: Lagrangian

text: submanifolds

text: R

text: include

text: the

text: real

text: locus

text: of

text: an

text: algebraic

text: variety

text: deﬁned

text: over

text: the

text: zero-section

text: and

text: cotan-

text: gent

text: ﬁbre

text: in

text: T

text: Q,

text: respectively.

text: Gromov’s

text: fundamental

text: insight

text: was

text: that,

text: for

text: a

text: generic

text: taming

text: almost

text: complex

text: structure,

text: spaces

text: of

text: non-multiply-covered

text: holomorphic

text: curves

text: in

text: M

text: (perhaps

text: with

text: Lagrangian

text: boundary

text: conditions)

text: are

text: ﬁnite-dimensional

text: smooth

text: manifolds,

text: which

text: moreover

text: admit

text: geometrically

text: meaningful

text: compactiﬁcations

text: by

text: nodal

text: curves.

text: It

text: is

text: the

text: resulting

text: invariants

text: which

text: count

text: “pseudoholomorphic”

text: curves

text: (maps

text: from

text: Riemann

text: surfaces

text: which

text: are

text: J-complex

text: linear

text: for

text: some

text: taming

text: but

text: not

text: necessarily

text: integrable

text: J)

text: that

text: dominate

text: the

text: ﬁeld.

text: Discrete

text: invariants

text: of

text: symplectic

text: manifolds

text: and

text: Lagrangian

text: submanifolds

text: include

text: characteristic

text: classes

text: which

text: determine

text: the

text: virtual,

text: i.e.

text: expected,

text: dimensions

text: of

text: moduli

text: spaces

text: of

text: holomorphic

text: curves

text: (when

text: non-empty),

text: via

text: index

text: theory.

text: Any

text: choice

text: of

text: taming

text: J

text: makes

text: T

text: M

text: a

text: complex

text: p

text: q

text: P

text: p

text: q

text: p

text: q{

text: p

text: q

text: Z

text: bundle,

text: which

text: has

text: Chern

text: classes

text: c

text: M

text: M;

text: Since

text: the

text: Grassmannian

text: U

text: n

text: O

text: n

text: of

text: i

text: C

text: Z

text: Lagrangian

text: subspaces

text: of

text: n

text: has

text: fundamental

text: group

text: any

text: Lagrangian

text: submanifold

text: has

text: a

text: Maslov

text: p

text: q

text: Ñ

text: Z

text: class

text: µ

text: π

text: M,

text: L

text: which

text: measures

text: the

text: twisting

text: of

text: the

text: Lagrangian

text: tangent

text: planes

text: around

text: L

text: p

text: q

text: Ñ

text: p

text: q

text: Z

text: Z

text: the

text: boundary

text: of

text: a

text: disc

text: D

text: relative

text: to

text: a

text: trivialisation

text: of

text: T

text: M

text: under

text: M,

text: L;

text: M;

text: p

text: q

text: D

text: Ñ

text: P

text: µ

text: maps

text: to

text: M

text: The

text: moduli

text: space

text: of

text: parametrised

text: rational

text: curves

text: representing

text: L

text: x

text: p

text: q

text: y

text: p

text: q

text: P

text: C

text: a

text: homology

text: class

text: A

text: has

text: real

text: dimension

text: c

text: M

text: A

text: hence

text: dividing

text: by

text: SL

text: the

text: space


text: 22 WEIWEI AO, MONICA MUSSO, FRANK PACARD, AND JUNCHENG WEI Fig 7 : Example of a network N˚ . Because of dihedral symmetry the weights Pol along all edges can be determined from the weights along [0, 1] and [1, ξ]. We claim that : − Lemma 3.8. The map D F has rank 2k 1. a (Id,a)   (cid:88) Proof. We have   − q p D F (a˙) = a˙ | − | . a (Id,a) [p,q] q p ∈ q V˚p p∈V˚ Assume that D F (a˙) = 0 and also that a˙ = 0. Then, looking at the a (Id,a) [1,ξ] component of D F (a˙) at the vertex ξ, we ﬁnd that a (Id,a) − ξ ξ2 a˙ ξ + a˙ | − | = 0, [ξ,0] [ξ,ξ2] 1 ξ − since the vectors ξ and ξ ξ2 are not R-collinear, we conclude that a˙ = a˙ = [ξ,0] [ξ,ξ2] 0. To proceed, one looks at the component of D F (a˙) at the point ξj, which − a (Id,a) gives (after simpliﬁcation by ξj 1) − − ξ 1 ξ ξ2 a˙ ξ + a˙ − | − | + a˙ | − | = 0. [ξj,0] [ξj,ξj 1] 1 ξ [ξj,ξj+1] 1 ξ One then proves by induction that a˙ = a˙ = 0 for all j = 1, . . . , k, [ξj,0] [ξj,ξj+1] following the arguments given in the case where j = 1. We conclude that D F , restricted to the hyperplane a˙ = 0 is injective, a (Id,a) −[1,ξ] (cid:3) and hence we have proven that this map has rank at least 2k 1. As a consequence, we have the : Corollary 3.3. The balanced network (N˚ , a) is ﬂexible in the sense of Deﬁni- Pol tion 3.8. ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: WEIWEI

text: AO,

text: MONICA

text: MUSSO,

text: FRANK

text: PACARD,

text: AND

text: JUNCHENG

text: WEI

text: Fig

text: Example

text: of

text: a

text: network

text: N˚

text: Because

text: of

text: dihedral

text: symmetry

text: the

text: weights

text: Pol

text: along

text: all

text: edges

text: can

text: be

text: determined

text: from

text: the

text: weights

text: along

text: and

text: ξ].

text: We

text: claim

text: that

text: Lemma

text: The

text: map

text: D

text: F

text: has

text: rank

text: a

text: (Id,a)

text: Proof.

text: We

text: have

text: q

text: p

text: D

text: F

text: (a˙)

text: a˙

text: a

text: (Id,a)

text: [p,q]

text: q

text: p

text: q

text: V˚p

text: p∈V˚

text: Assume

text: that

text: D

text: F

text: (a˙)

text: and

text: also

text: that

text: a˙

text: Then,

text: looking

text: at

text: the

text: a

text: (Id,a)

text: component

text: of

text: D

text: F

text: (a˙)

text: at

text: the

text: vertex

text: ξ,

text: we

text: ﬁnd

text: that

text: a

text: (Id,a)

text: ξ

text: ξ2

text: a˙

text: ξ

text: a˙

text: ξ

text: since

text: the

text: vectors

text: ξ

text: and

text: ξ

text: ξ2

text: are

text: not

text: R-collinear,

text: we

text: conclude

text: that

text: a˙

text: a˙

text: To

text: proceed,

text: one

text: looks

text: at

text: the

text: component

text: of

text: D

text: F

text: (a˙)

text: at

text: the

text: point

text: ξj,

text: which

text: a

text: (Id,a)

text: gives

text: (after

text: simpliﬁcation

text: by

text: ξj

text: ξ

text: ξ

text: ξ2

text: a˙

text: ξ

text: a˙

text: a˙

text: [ξj,ξj

text: ξ

text: ξ

text: One

text: then

text: proves

text: by

text: induction

text: that

text: a˙

text: a˙

text: for

text: all

text: j

text: k,

text: following

text: the

text: arguments

text: given

text: in

text: the

text: case

text: where

text: j

text: We

text: conclude

text: that

text: D

text: F

text: restricted

text: to

text: the

text: hyperplane

text: a˙

text: is

text: injective,

text: a

text: (Id,a)

text: and

text: hence

text: we

text: have

text: proven

text: that

text: this

text: map

text: has

text: rank

text: at

text: least

text: As

text: a

text: consequence,

text: we

text: have

text: the

text: Corollary

text: The

text: balanced

text: network

text: (N˚

text: a)

text: is

text: ﬂexible

text: in

text: the

text: sense

text: of

text: Deﬁni-

text: Pol

text: tion

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 24 WEIWEI AO, MONICA MUSSO, FRANK PACARD, AND JUNCHENG WEI In particular, there exists balanced networks which are ﬂexible and which have no symmetry. In fact, more is true and, in the spirit of the result of Proposition 3.2, we have the : Proposition 3.6. The set of ﬂexible balanced networks in a given homotopy class is Zariski open. We will see in section 10, some explicit networks which are balanced and which have no symmetry. 4. Applications We now explain how the previous framework can be used to construct approxi- mate solutions to (1.1). (cid:90)(cid:90) We deﬁne the interaction function Υ by (cid:0) (cid:1) − − (4.19) Υ(s) := u (z se) div u3(z) e dx dy, 0 0 C ∈ where e C is any unit vector (and z = x + iy). Since the function u is radial, 0 it is easy to check that Υ(s) does not depend on the choice of e. If the exact formula for Υ is not known, its asymptotic behavior as s tends to inﬁnity is well understood (cid:18) (cid:19)∈ and, for example, we know that there exists a constant C R such that − 1 O 1 (4.20) ln Υ(t) = t + ln t + C + , 2 t (cid:18) (cid:19) at inﬁnity and also that − Υ(t) − 1 O 1 (4.21) (cid:48) = 1 + , Υ (t) 2t t2 at inﬁnity. ∈ − { } ∈ For all (cid:96) > 0 and a R 0 , we deﬁne, if it exists, α := α (a) R by the (cid:96) identity − | | (4.22) Υ ((cid:96)(1 α)) = a Υ((cid:96)). The asymptotic behavior of the function Υ at inﬁnity implies that the function (cid:55)→ a α (a) is well deﬁned for all (cid:96) > 0 large enough. Moreover, we have the (cid:96) (cid:18) (cid:19) expansion | | ln a O 1 (4.23) α (a) = + , (cid:96) (cid:96) (cid:96)2 −{ } which holds for all a in a given compact of R 0 and for all (cid:96) > 0 large enough. Finally, diﬀerentiating (4.22) yields ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: WEIWEI

text: AO,

text: MONICA

text: MUSSO,

text: FRANK

text: PACARD,

text: AND

text: JUNCHENG

text: WEI

text: In

text: particular,

text: there

text: exists

text: balanced

text: networks

text: which

text: are

text: ﬂexible

text: and

text: which

text: have

text: no

text: symmetry.

text: In

text: fact,

text: more

text: is

text: true

text: and,

text: in

text: the

text: spirit

text: of

text: the

text: result

text: of

text: Proposition

text: we

text: have

text: the

text: Proposition

text: The

text: set

text: of

text: ﬂexible

text: balanced

text: networks

text: in

text: a

text: given

text: homotopy

text: class

text: is

text: Zariski

text: open.

text: We

text: will

text: see

text: in

text: section

text: some

text: explicit

text: networks

text: which

text: are

text: balanced

text: and

text: which

text: have

text: no

text: symmetry.

text: Applications

text: We

text: now

text: explain

text: how

text: the

text: previous

text: framework

text: can

text: be

text: used

text: to

text: construct

text: approxi-

text: mate

text: solutions

text: to

label: equation

text: We

text: deﬁne

text: the

text: interaction

text: function

text: Υ

text: by

text: Υ(s)

text: u

text: (z

text: se)

text: div

text: e

text: dx

text: dy,

text: C

text: where

text: e

text: C

text: is

text: any

text: unit

text: vector

text: (and

text: z

text: x

text: iy).

text: Since

text: the

text: function

text: u

text: is

text: radial,

text: it

text: is

text: easy

text: to

text: check

text: that

text: Υ(s)

text: does

text: not

text: depend

text: on

text: the

text: choice

text: of

text: e.

text: If

text: the

text: exact

text: formula

text: for

text: Υ

text: is

text: not

text: known,

text: its

text: asymptotic

text: behavior

text: as

text: s

text: tends

text: to

text: inﬁnity

text: is

text: well

text: understood

text: and,

text: for

text: example,

text: we

text: know

text: that

text: there

text: exists

text: a

text: constant

text: C

text: R

text: such

text: that

text: O

text: ln

text: Υ(t)

text: t

text: ln

text: t

text: C

text: t

text: at

text: inﬁnity

text: and

text: also

text: that

text: Υ(t)

text: O

text: Υ

text: (t)

text: at

text: inﬁnity.

text: For

text: all

text: and

text: a

text: R

text: we

text: deﬁne,

text: if

text: it

text: exists,

text: α

text: α

text: (a)

text: R

text: by

text: the

text: identity

text: Υ

text: α))

text: a

text: The

text: asymptotic

text: behavior

text: of

text: the

text: function

text: Υ

text: at

text: inﬁnity

text: implies

text: that

text: the

text: function

text: a

text: α

text: (a)

text: is

text: well

text: deﬁned

text: for

text: all

text: large

text: enough.

text: Moreover,

text: we

text: have

text: the

text: expansion

text: ln

text: a

text: O

text: α

text: (a)

text: which

text: holds

text: for

text: all

text: a

text: in

text: a

text: given

text: compact

text: of

text: R

text: and

text: for

text: all

text: large

text: enough.

text: Finally,

text: diﬀerentiating

text: yields

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 34 WEIWEI AO, MONICA MUSSO, FRANK PACARD, AND JUNCHENG WEI ∈ Fig 8 : Example of a vertex p V with the edges of E ending at p, a sub-network N p and the same sub-network where the rays are drawn. ∈ Fig 9 : Example of a vertex p V with the edges of E ending at p, a sub-network N p and the same subnetwork where the rays are drawn. We require that the following properties hold : (cid:88) (i) The barycenter of the vertices of N p is equal to 0, namely r = 0. ∈ r V p ∈ (ii) (Embeddedness of the sub-netkorks with rays) For each p V , any two distinct elements of E p (which might be edges or rays) are either disjoint ext or intersect at their end points. ∈ (iii) (Internal vertices are balanced) If r V p is an internal vertex, i.e. is not (cid:88) equal to any of the rp, then q (cid:48) − r r ap (cid:48) | (cid:48) − | = 0, [r r] r r r(cid:48)∈Vrp (cid:48) ∈ (cid:48) ∈ where V p is the set of vertices r V p such that [r , r] E p. r ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine##

text: WEIWEI

text: AO,

text: MONICA

text: MUSSO,

text: FRANK

text: PACARD,

text: AND

text: JUNCHENG

text: WEI

text: Fig

text: Example

text: of

text: a

text: vertex

text: p

text: V

text: with

text: the

text: edges

text: of

text: E

text: ending

text: at

text: p,

text: a

text: sub-network

text: N

text: p

text: and

text: the

text: same

text: sub-network

text: where

text: the

text: rays

text: are

text: drawn.

text: Fig

text: Example

text: of

text: a

text: vertex

text: p

text: V

text: with

text: the

text: edges

text: of

text: E

text: ending

text: at

text: p,

text: a

text: sub-network

text: N

text: p

text: and

text: the

text: same

text: subnetwork

text: where

text: the

text: rays

text: are

text: drawn.

text: We

text: require

text: that

text: the

text: following

text: properties

text: hold

text: (i)

text: The

text: barycenter

text: of

text: the

text: vertices

text: of

text: N

text: p

text: is

text: equal

text: to

text: namely

text: r

text: r

text: V

text: p

text: (ii)

text: (Embeddedness

text: of

text: the

text: sub-netkorks

text: with

text: rays)

text: For

text: each

text: p

text: V

text: any

text: two

text: distinct

text: elements

text: of

text: E

text: p

text: (which

text: might

text: be

text: edges

text: or

text: rays)

text: are

text: either

text: disjoint

text: ext

text: or

text: intersect

text: at

text: their

text: end

text: points.

text: (iii)

text: (Internal

text: vertices

text: are

text: balanced)

text: If

text: r

text: V

text: p

text: is

text: an

text: internal

text: vertex,

text: i.e.

text: is

text: not

text: equal

text: to

text: any

text: of

text: the

text: rp,

text: then

text: q

text: r

text: r

text: ap

text: [r

text: r]

text: r

text: r

text: where

text: V

text: p

text: is

text: the

text: set

text: of

text: vertices

text: r

text: V

text: p

text: such

text: that

text: [r

text: r]

text: E

text: p.

text: r

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##


text: SOLUTIONS WITHOUT ANY SYMMETRY FOR SEMILINEAR ELLIPTIC PROBLEMS 43 which belongs to H. In other words, if we deﬁne the operator − (cid:104) ∇ (cid:105) L(cid:91)(v, c) := ∆v v + c, u , 0 0 C we have obtained the existence and uniqueness of a solution of L(cid:91)(v, c) = h, 0 ∈ ∈ with v H and c C. The solvability of − (cid:104) ∇ (cid:105) ∆v v + 3 u2 v + c, u = h, 0 0 C × in H C can then by rephrased in the invertibility of the operator I + K, where by deﬁnition − (6.32) K(v, c) := (L(cid:91)) 1(3 u2v). 0 0 Using the fact that u decays exponentially at inﬁnity, it is easy to check that 0 the operator K is compact, hence the invertibility of (6.32) follows from the appli- cation of Fredholm theory. Since injectivity follows from the results of Lemma 6.1. Fredholm alternative implies that I + K is therefore an isomorphism. (cid:104) ∇ (cid:105) So far, we have obtained a function v solution of L v + c, u = h which ∈0 ∞ 0 C belongs to H1(C) but elliptic regularity implies that v L (C) and that (cid:107) (cid:107) ≤ (cid:107) (cid:107) v ∞ C f ∞ , L (C) L (C) δ for some constant C > 0. We need to check that the solution v has the correct behavior at inﬁnity. To this aim, just remark that if we deﬁne − v (r) := eδr + (cid:15) e δr, (cid:15) then − ≥ −(1 δ2) (cid:16) L v(cid:17) v 0 (cid:15) 2 0 on the complement of the ball of radius r , provided is ﬁxed large enough. Hence, 0 (cid:107) (cid:107) (cid:107)(cid:107) the function v ∞ + −2 ∞ v is certainly a super-solution for our problem L 1 δ2 Lδ (cid:15) on the complement of the disc of radius r > 0 and, passing to the limit as (cid:15) tends 0 to 0, this proves that there exists a constant C > 0 such that (cid:107) (cid:107) ≤ (cid:107) (cid:107) v ∞ C f ∞ , L (C) L (C) δ δ This completes the proof of the existence of the solution. The uniqueness and the (cid:3) corresponding estimate follow at once from the result of Lemma 6.1. Building on the previous result, we prove a similar result for the operator L˜ (see also [16] for more details). First(cid:32)we need to deﬁne(cid:33)weighted spaces adapted to L˜. (cid:88) Given δ < 0, we deﬁne the weighted space √ L∞ |·− | ∞ (C) := eδ 1+ z 2 L (C), δ (cid:13) ∈ (cid:13) (cid:13)(cid:32) z Z (cid:33) (cid:13) (cid:13) (cid:88) (cid:13) with the natural associated nor(cid:13)m which is deﬁned to be(cid:13) (cid:13) √ − (cid:13) (cid:13) 1 (cid:13) (cid:107) (cid:107) |·− | v L∞ := eδ 1+ z 2 v . (C) δ ∈ z Z ∞ L (C) ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: SOLUTIONS

text: WITHOUT

text: ANY

text: SYMMETRY

text: FOR

text: SEMILINEAR

text: ELLIPTIC

text: PROBLEMS

text: which

text: belongs

text: to

text: H.

text: In

text: other

text: words,

text: if

text: we

text: deﬁne

text: the

text: operator

text: c)

text: ∆v

text: v

text: c,

text: u

text: C

text: we

text: have

text: obtained

text: the

text: existence

text: and

text: uniqueness

text: of

text: a

text: solution

text: of

text: c)

text: h,

text: with

text: v

text: H

text: and

text: c

text: C.

text: The

text: solvability

text: of

text: ∆v

text: v

text: v

text: c,

text: u

text: h,

text: C

text: in

text: H

text: C

text: can

text: then

text: by

text: rephrased

text: in

text: the

text: invertibility

text: of

text: the

text: operator

text: I

text: K,

text: where

text: by

text: deﬁnition

text: K(v,

text: c)

text: Using

text: the

text: fact

text: that

text: u

text: decays

text: exponentially

text: at

text: inﬁnity,

text: it

text: is

text: easy

text: to

text: check

text: that

text: the

text: operator

text: K

text: is

text: compact,

text: hence

text: the

text: invertibility

text: of

text: follows

text: from

text: the

text: appli-

text: cation

text: of

text: Fredholm

text: theory.

text: Since

text: injectivity

text: follows

text: from

text: the

text: results

text: of

text: Lemma

text: Fredholm

text: alternative

text: implies

text: that

text: I

text: K

text: is

text: therefore

text: an

text: isomorphism.

text: So

text: far,

text: we

text: have

text: obtained

text: a

text: function

text: v

text: solution

text: of

text: L

text: v

text: c,

text: u

text: h

text: which

text: ∈0

text: C

text: belongs

text: to

text: but

text: elliptic

text: regularity

text: implies

text: that

text: v

text: L

text: (C)

text: and

text: that

text: v

text: C

text: f

text: L

text: (C)

text: L

text: (C)

text: δ

text: for

text: some

text: constant

text: C

text: We

text: need

text: to

text: check

text: that

text: the

text: solution

text: v

text: has

text: the

text: correct

text: behavior

text: at

text: inﬁnity.

text: To

text: this

text: aim,

text: just

text: remark

text: that

text: if

text: we

text: deﬁne

text: v

text: (r)

text: eδr

text: e

text: δr,

text: then

text: L

text: v

text: on

text: the

text: complement

text: of

text: the

text: ball

text: of

text: radius

text: r

text: provided

text: is

text: ﬁxed

text: large

text: enough.

text: Hence,

text: the

text: function

text: v

text: −2

text: v

text: is

text: certainly

text: a

text: super-solution

text: for

text: our

text: problem

text: L

text: δ2

text: Lδ

text: on

text: the

text: complement

text: of

text: the

text: disc

text: of

text: radius

text: r

text: and,

text: passing

text: to

text: the

text: limit

text: as

text: tends

text: to

text: this

text: proves

text: that

text: there

text: exists

text: a

text: constant

text: C

text: such

text: that

text: v

text: C

text: f

text: L

text: (C)

text: L

text: (C)

text: δ

text: δ

text: This

text: completes

text: the

text: proof

text: of

text: the

text: existence

text: of

text: the

text: solution.

text: The

text: uniqueness

text: and

text: the

text: corresponding

text: estimate

text: follow

text: at

text: once

text: from

text: the

text: result

text: of

text: Lemma

text: Building

text: on

text: the

text: previous

text: result,

text: we

text: prove

text: a

text: similar

text: result

text: for

text: the

text: operator

text: L˜

text: (see

text: also

text: for

text: more

text: details).

text: need

text: to

text: spaces

text: adapted

text: to

text: L˜.

text: Given

text: δ

text: we

text: deﬁne

text: the

text: weighted

text: space

text: L∞

text: (C)

text: eδ

text: z

text: L

text: (C),

text: δ

text: z

text: Z

text: with

text: the

text: natural

text: associated

text: which

text: is

text: deﬁned

text: to

text: v

text: L∞

text: eδ

text: z

text: v

text: (C)

text: δ

text: z

text: Z

text: L

text: (C)

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 44 WEIWEI AO, MONICA MUSSO, FRANK PACARD, AND JUNCHENG WEI Observe that we could have used the points of Z˜ instead of the points of Z to deﬁne these spaces and this would not have changed anything since the respective norms (cid:29) (cid:40) would have been uniformly equivalent independently of (cid:96) 1. We deﬁne a cutoﬀ function ≤ − 1 if s 1 χ(s) := ≥ 0 if s 1, and, for all s¯ > 0 we deﬁne − χ (s) := χ (s s¯) . s¯ ∈ We also deﬁne for all z Z˜, the vector ﬁeld | · − | ∇ · − (6.33) Ξ := χ ( z ) u ( z), z (cid:96)/4 0 being understood that we identify vectors in R2 with complex numbers. The main result of this section reads : ∈ − Proposition 6.2. Assume that δ ( 1, 0). Then, there exists (cid:96)∗ > 0 (larger than ≥ or equal to the one deﬁned in Proposition 5.1) and, for all (cid:96) (cid:96)∗, there exists a linear operator G˜ : L∞(C) → L∞(C) × Cn, where n is the cardinal of Z, suδch(cid:88)that, foδr all f ∈ L∞(C), G˜ f =: (v, (c ) ∈ ) δ z z Z satisﬁes (cid:104) (cid:105) L˜ v + c , Ξ = f, z z C ∈ z Z˜ in C. Moreover, (cid:107) (cid:107) | | ≤ (cid:107) (cid:107) v L∞ + sup c C f L∞ , (C) z (C) δ z∈Z˜ δ for some constant C > 0 which does not depend on f and, if, for i = 1, 2, G˜(i) is (cid:12) (cid:12) the right inverse corresponding to z˜[p,q],(i), we have (cid:12) (cid:12) j (cid:12) (cid:12) (cid:57) (cid:57) − ≤ − (6.34) G˜(2) G˜(1) C sup sup z˜[p,q],(2) z˜[p,q],(1) , ∈ − j j [p,q] E j=1,...,2m[p,q] 1 for some constant C > 0. (cid:32) (cid:33) (cid:88) (cid:88) Proof. We decompose f as − | · − | | · − | f = 1 χ ( z ) f + χ ( z ) f (cid:96)/4 (cid:96)/4 ∈ ∈ z Z z Z ∈ For each z Z, we use the result of Proposition 6.1 to solve (cid:104) ∇ (cid:105) · L v + c , u = χ f( + z). 0 z z 0 C (cid:96)/4 We know that we have (cid:107) (cid:107) | | ≤ (cid:107) (cid:107) v ∞ + c C f ∞ , z L (C) z L (C;Z) δ δ with similar estimates for the ﬁrst partial derivatives of v . z

text: WEIWEI

text: AO,

text: MONICA

text: MUSSO,

text: FRANK

text: PACARD,

text: AND

text: JUNCHENG

text: WEI

text: Observe

text: that

text: we

text: could

text: have

text: used

text: the

text: points

text: of

text: Z˜

text: instead

text: of

text: the

text: points

text: of

text: Z

text: to

text: deﬁne

text: these

text: spaces

text: and

text: this

text: would

text: not

text: have

text: changed

text: anything

text: since

text: the

text: respective

text: norms

text: would

text: have

text: been

text: uniformly

text: equivalent

text: independently

text: of

text: We

text: deﬁne

text: a

text: cutoﬀ

text: function

text: if

text: s

text: χ(s)

text: if

text: s

text: and,

text: for

text: all

text: s¯

text: we

text: deﬁne

text: χ

text: (s)

text: χ

text: (s

text: s¯)

text: s¯

text: We

text: also

text: deﬁne

text: for

text: all

text: z

text: Z˜,

text: the

text: vector

text: ﬁeld

text: Ξ

text: χ

text: z

text: u

text: z),

text: z

text: being

text: understood

text: that

text: we

text: identify

text: vectors

text: in

text: with

text: complex

text: numbers.

text: The

text: main

text: result

text: of

text: this

text: section

text: reads

text: Proposition

text: Assume

text: that

text: δ

text: Then,

text: there

text: exists

text: (larger

text: than

text: or

text: equal

text: to

text: the

text: one

text: deﬁned

text: in

text: Proposition

text: and,

text: for

text: all

text: there

text: exists

text: a

text: linear

text: operator

text: G˜

text: L∞(C)

text: L∞(C)

text: Cn,

text: where

text: n

text: is

text: the

text: cardinal

text: of

text: Z,

text: foδr

text: all

text: f

text: L∞(C),

text: G˜

text: f

text: (v,

text: (c

text: δ

text: z

text: z

text: Z

text: satisﬁes

text: L˜

text: v

text: c

text: Ξ

text: f,

text: z

text: z

text: C

text: z

text: Z˜

text: in

text: C.

text: Moreover,

text: v

text: L∞

text: sup

text: c

text: C

text: f

text: L∞

text: (C)

text: z

text: (C)

text: δ

text: z∈Z˜

text: δ

text: for

text: some

text: constant

text: C

text: which

text: does

text: not

text: depend

text: on

text: f

text: and,

text: if,

text: for

text: i

text: G˜(i)

text: is

text: the

text: right

text: inverse

text: corresponding

text: to

text: z˜[p,q],(i),

text: we

text: have

text: j

text: C

text: sup

text: sup

text: j

text: j

text: [p,q]

text: E

text: for

text: some

text: constant

text: C

text: Proof.

text: We

text: decompose

text: f

text: as

text: f

text: χ

text: z

text: f

text: χ

text: z

text: f

text: z

text: Z

text: z

text: Z

text: For

text: each

text: z

text: Z,

text: we

text: use

text: the

text: result

text: of

text: Proposition

text: to

text: solve

text: L

text: v

text: c

text: u

text: χ

text: f(

text: z).

text: z

text: z

text: C

text: We

text: know

text: that

text: we

text: have

text: v

text: c

text: C

text: f

text: z

text: L

text: (C)

text: z

text: L

text: (C;Z)

text: δ

text: δ

text: with

text: similar

text: estimates

text: for

text: the

text: ﬁrst

text: partial

text: derivatives

text: of

text: v

text: z


text: SOLUTIONS WITHOUT ANY SYMMETRY FOR SEMILINEAR ELLIPTIC PROBLEMS 47 ∈ − Lemma 7.1. Assume that δ ( 1, 0) is ﬁxed. Then, there exists a constant ≥ C > 0, independent of (cid:96) (cid:96)∗ and all parameters of the construction, such that 0 (cid:107) (cid:107) ≤ E˜ L∞ C Υ((cid:96)). (C) 0 δ   (cid:88) (cid:88) Proof. We start from the fact that   3 · − (cid:48) − · − (cid:48) E˜ = η (cid:48)u ( z ) (η (cid:48)u ( z ))3 . z 0 z 0 z(cid:48)∈Z˜ z(cid:48)∈Z˜ ∈ Wethen estimate E˜ near a given point z Z˜. In a ball of radius (cid:96)/2 centered (cid:88) (cid:88) at z, we can write   3 · − · − (cid:48) − · − − · − (cid:48) E˜ = η u ( z) + η (cid:48)u ( z ) (η u ( z))3 (η (cid:48)u ( z ))3 , z 0 z 0 z 0 z 0 z(cid:48)(cid:54)=z (cid:88) z(cid:48)(cid:54)=z and hence, we get | | ≤ |·− | ≤ |·− (cid:48)| E˜ C Υ((cid:96)) eδ z C Υ((cid:96)) eδ z , z(cid:48)∈Z˜ for some constant C > 0. While, away from the balls of radius (cid:96)/2 centered at the points of Z˜, we take the advantage that u decays exponentially fast to 0 at 0 (cid:88) (cid:88) inﬁnity, to prove that | | ≤ − − |·− | ≤ |·− | E˜ C (cid:96) 3/2e 3 z CΥ((cid:96)) eδ z , ∈ ∈ z Z˜ z Z˜ for some constant C > 0. The estimate for E˜ then follows at once. Observe that (cid:3) the estimate is achieved near the points of Z˜. We will also need the ∈ − Lemma 7.2. Assume that δ ( 1, 0) is ﬁxed. Then, there exists a constant ≥ C > 0, independent of (cid:96) (cid:96)∗ and all parameters of the construction, such that 1 (cid:107) (cid:48) − (cid:107) ≤ (cid:107) (cid:48) − (cid:107) Q˜(v ) Q˜(v) L∞ C Υ((cid:96)) v v L∞ , (C) 1 (C) (cid:107) (cid:48)(cid:107) ≤ δ δ provided v L∞ 2 C Υ((cid:96)), (C) 0 δ Proof. The estimate follows from the expression Q˜(v) = v3 + 3u˜ v2, (cid:3) we leave the details to the reader. The result of Proposition 7.1 then follows from these two results, the result of Proposition 6.2 and a simple application of a ﬁxed point theorem for contraction L∞ mappings in the closed ball of radius 2 C Υ((cid:96)) in (C), provided (cid:96) is chosen large 0 δ enough. Proofs with all details are given in [16]. The estimate (7.35) follows from taking the diﬀerence between the equations satisﬁed by the two solutions and using (6.34).

text: SOLUTIONS

text: WITHOUT

text: ANY

text: SYMMETRY

text: FOR

text: SEMILINEAR

text: ELLIPTIC

text: PROBLEMS

text: Lemma

text: Assume

text: that

text: δ

text: is

text: ﬁxed.

text: Then,

text: there

text: exists

text: a

text: constant

text: C

text: independent

text: of

text: and

text: all

text: parameters

text: of

text: the

text: construction,

text: such

text: that

text: E˜

text: L∞

text: C

text: (C)

text: δ

text: Proof.

text: We

text: start

text: from

text: the

text: fact

text: that

text: E˜

text: η

text: z

text: (η

text: z

text: z

text: z

text: Wethen

text: estimate

text: E˜

text: near

text: a

text: given

text: point

text: z

text: Z˜.

text: In

text: a

text: ball

text: of

text: radius

text: centered

text: at

text: z,

text: we

text: can

text: write

text: E˜

text: η

text: u

text: z)

text: η

text: z

text: (η

text: u

text: (η

text: z

text: z

text: z

text: z

text: z

text: and

text: hence,

text: we

text: get

text: E˜

text: C

text: eδ

text: z

text: C

text: eδ

text: z

text: for

text: some

text: constant

text: C

text: While,

text: away

text: from

text: the

text: balls

text: of

text: radius

text: centered

text: at

text: the

text: points

text: of

text: Z˜,

text: we

text: take

text: the

text: advantage

text: that

text: u

text: decays

text: exponentially

text: fast

text: to

text: at

text: inﬁnity,

text: to

text: prove

text: that

text: E˜

text: C

text: z

text: eδ

text: z

text: z

text: Z˜

text: z

text: Z˜

text: for

text: some

text: constant

text: C

text: The

text: estimate

text: for

text: E˜

text: then

text: follows

text: at

text: once.

text: Observe

text: that

text: the

text: estimate

text: is

text: achieved

text: near

text: the

text: points

text: of

text: Z˜.

text: We

text: will

text: also

text: need

text: the

text: Lemma

text: Assume

text: that

text: δ

text: is

text: ﬁxed.

text: Then,

text: there

text: exists

text: a

text: constant

text: C

text: independent

text: of

text: and

text: all

text: parameters

text: of

text: the

text: construction,

text: such

text: that

text: Q˜(v

text: Q˜(v)

text: L∞

text: C

text: v

text: v

text: L∞

text: (C)

text: (C)

text: δ

text: δ

text: provided

text: v

text: L∞

text: C

text: (C)

text: δ

text: Proof.

text: The

text: estimate

text: follows

text: from

text: the

text: expression

text: Q˜(v)

text: we

text: leave

text: the

text: details

text: to

text: the

text: reader.

text: The

text: result

text: of

text: Proposition

text: then

text: follows

text: from

text: these

text: two

text: results,

text: the

text: result

text: of

text: Proposition

text: and

text: a

text: simple

text: application

text: of

text: a

text: ﬁxed

text: point

text: theorem

text: for

text: contraction

text: L∞

text: mappings

text: in

text: the

text: closed

text: ball

text: of

text: radius

text: C

text: in

text: (C),

text: provided

text: is

text: chosen

text: large

text: δ

text: enough.

text: Proofs

text: with

text: all

text: details

text: are

text: given

text: in

text: The

text: estimate

text: follows

text: from

text: taking

text: the

text: diﬀerence

text: between

text: the

text: equations

text: satisﬁed

text: by

text: the

text: two

text: solutions

text: and

text: using


text: SOLUTIONS WITHOUT ANY SYMMETRY FOR SEMILINEAR ELLIPTIC PROBLEMS 53 (cid:110) while its set of edges is deﬁned to be (cid:111) − − − E := [0, tan θ], [0, i], [0, tan θ], [0, i], [tan θ, i], [i, tan θ], V − − − [ tan θ, i], [ i, tan θ] . Observe that the network is invariant under the symmetries with respect to the x-axis and the y-axis. Fig 13 : The network N . → − { } V We deﬁne the weight function a : E R 0 by V − − a = a − := 2 sin θ, a = a − := cos θ, [0,tanθ] [0, tanθ] [0,i] [0, i] and a = a − = a − − = a − := 1. [tanθ,i] [i, tanθ] [ tanθ, i] [ i,tanθ] It is easy to check that the network (N , a) is balanced. We also have the V Lemma 10.1. The balanced network (N , a) is ﬂexible in the sense of Deﬁni- V tion 3.8 and closable in the sense of Deﬁnition 4.1. − Proof. In this example m = 8 and n = 5 and hence 2m = n 2. Therefore, to prove that the network is ﬂexible, it is enough to show that D F has rank equal to a (Id,a) 7. Let us assume that D F (a˙) = 0 and also that a˙ = 0. Then looking at a (Id,a) [p1,p2] the component of D F at p , we ﬁnd that a (Id,a) 1 − − p p p p (10.39) a˙ | 0 − 1| + a˙ | 4 − 1| = 0. [p0,p1] p p [p1,p4] p p 0 1 4 1 − − Since p p and p p are not R-collinear, we conclude that a˙ = a˙ = 0 1 4 1 [p0,p1] [p1,p4] 0. Then one looks at the component of D F at p , we have a (Id,a) 2 − − p p p p (10.40) a˙ | 0 − 2| + a˙ | 3 − 2| = 0. [p0,p2] p p [p2,p3] p p 0 2 3 2 ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: SOLUTIONS

text: WITHOUT

text: ANY

text: SYMMETRY

text: FOR

text: SEMILINEAR

text: ELLIPTIC

text: PROBLEMS

text: while

text: its

text: set

text: of

text: edges

text: is

text: deﬁned

text: to

text: be

text: E

text: tan

text: θ],

text: i],

text: tan

text: θ],

text: i],

text: [tan

text: θ,

text: i],

text: [i,

text: tan

text: θ],

text: V

text: tan

text: θ,

text: i],

text: i,

text: tan

text: θ]

text: Observe

text: that

text: the

text: network

text: is

text: invariant

text: under

text: the

text: symmetries

text: with

text: respect

text: to

text: the

text: x-axis

text: and

text: the

text: y-axis.

text: Fig

text: The

text: network

text: N

text: V

text: We

text: deﬁne

text: the

text: weight

text: function

text: a

text: E

text: R

text: by

text: V

text: a

text: a

text: sin

text: θ,

text: a

text: a

text: cos

text: θ,

text: tanθ]

text: i]

text: and

text: a

text: a

text: a

text: a

text: [tanθ,i]

text: [i,

text: tanθ]

text: tanθ,

text: i]

text: i,tanθ]

text: It

text: is

text: easy

text: to

text: check

text: that

text: the

text: network

text: (N

text: a)

text: is

text: balanced.

text: We

text: also

text: have

text: the

text: V

text: Lemma

text: The

text: balanced

text: network

text: (N

text: a)

text: is

text: ﬂexible

text: in

text: the

text: sense

text: of

text: Deﬁni-

text: V

text: tion

text: and

text: closable

text: in

text: the

text: sense

text: of

text: Deﬁnition

text: Proof.

text: In

text: this

text: example

text: m

text: and

text: n

text: and

text: hence

text: n

text: Therefore,

text: to

text: prove

text: that

text: the

text: network

text: is

text: ﬂexible,

text: it

text: is

text: enough

text: to

text: show

text: that

text: D

text: F

text: has

text: rank

text: equal

text: to

text: a

text: (Id,a)

text: Let

text: us

text: assume

text: that

text: D

text: F

text: (a˙)

text: and

text: also

text: that

text: a˙

text: Then

text: looking

text: at

text: a

text: (Id,a)

text: the

text: component

text: of

text: D

text: F

text: at

text: p

text: we

text: ﬁnd

text: that

text: a

text: (Id,a)

text: p

text: p

text: p

text: p

text: a˙

text: a˙

text: p

text: p

text: p

text: p

text: Since

text: p

text: p

text: and

text: p

text: p

text: are

text: not

text: R-collinear,

text: we

text: conclude

text: that

text: a˙

text: a˙

text: Then

text: one

text: looks

text: at

text: the

text: component

text: of

text: D

text: F

text: at

text: p

text: we

text: have

text: a

text: (Id,a)

text: p

text: p

text: p

text: p

text: a˙

text: a˙

text: p

text: p

text: p

text: p

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 54 WEIWEI AO, MONICA MUSSO, FRANK PACARD, AND JUNCHENG WEI − − Since p p and p p are not R-collinear, we conclude that a˙ = a˙ = 0 2 2 3 [p0,p2] [p2,p3] 0. Arguing similarly at p and p , we conclude that a˙ = 0 and hence D F , 3 4 a (Id,a) restricted to the hyperplane a˙ = 0, is injective. Therefore, this map has rank − [p1,p2] at least equal to m 1 = 7. By Proposition 3.4, this shows that the network is ﬂexible. Now, it remains to check that the network is closable. This amounts to check that the image of DL does not contain the vector T. Namely, that the only Id solution of DL (Φ˙ ) = λ T, Id − − isλ = 0 and Φ˙ = 0. Writing p p = z and Φ˙ Φ˙ = z˙ , this amount to check that(cid:104)the o(cid:105)nly solution to i 0 i pi p0(cid:104) −i − (cid:105) 1, z˙ = λ tan θ ln(2 sin θ) z z , z˙ z˙ = 0 1 C 2 1 2 1 C  (cid:104)(cid:104)i, z˙2(cid:105)(cid:105)C = λ−ln(2 cos θ)  (cid:104)(cid:104)z3 −− z2, z˙3 −− z˙2(cid:105)(cid:105)C = 0 1, z˙ = λ tan θ ln(2 sin θ) z z , z˙ z˙ = 0 3 C 4 3 4 3 C (cid:104) (cid:105) − (cid:104) − − (cid:105) 1, z˙ = λ ln(2 cos θ) z z , z˙ z˙ = 0 4 C 1 4 1 4 C is given by z˙ = 0 for j = 1, . . . , 4 and λ = 0. Using the second system together j − − with the fact that z = z and z = z , we get 3 1 2 3 (cid:104) − − − (cid:105) (cid:104) − − − (cid:105) z z , z˙ + z˙ z˙ z˙ = z z , z˙ + z˙ z˙ z˙ = 0 2 1 1 3 2 4 C 3 2 1 3 2 4 C − − and, since z z and z z are R-independent, we conclude that 2 1 3 2 (10.41) z˙ + z˙ = z˙ + z˙ . 1 3 2 4 Using the ﬁrst system, we get (cid:104) (cid:105) (cid:104) (cid:105) 1, z˙ + z˙ = i, z˙ + z˙ = 0 1 3 C 2 4 C and, together with(10.41), this implies that z˙ + z˙ = z˙ + z˙ = 0. Using this 1 3 2 4 information back into the system(cid:104)1,yzi˙el(cid:105)ds = λ tan θ ln(2 sin θ) 1 C (cid:104) (cid:105)  (cid:104)z − z , z˙ −i, zz˙˙2(cid:105)C == λ0 ln(2 cos θ) 2 1 2 1 C (cid:104) − (cid:105) z z , z˙ + z˙ = 0 3 2 2 1 C − − − Since z z = tan θ i and z z = tan θ + i, the sum of the last two equations 1 2 2 3 implies that (cid:104) (cid:105) (cid:104) (cid:105) tan θ 1, z˙ + i, z˙ = 0, 1 C 2 C using the ﬁrst two equations, we conclude that T is not in the image of DL unless Id sin2 θ ln(2 sin θ) + cos2 θ ln(2 cos θ) = 0, (cid:3) which never happens. So, by Deﬁnition 4.1, the network is closable.

text: WEIWEI

text: AO,

text: MONICA

text: MUSSO,

text: FRANK

text: PACARD,

text: AND

text: JUNCHENG

text: WEI

text: Since

text: p

text: p

text: and

text: p

text: p

text: are

text: not

text: R-collinear,

text: we

text: conclude

text: that

text: a˙

text: a˙

text: Arguing

text: similarly

text: at

text: p

text: and

text: p

text: we

text: conclude

text: that

text: a˙

text: and

text: hence

text: D

text: F

text: a

text: (Id,a)

text: restricted

text: to

text: the

text: hyperplane

text: a˙

text: is

text: injective.

text: Therefore,

text: this

text: map

text: has

text: rank

text: at

text: least

text: equal

text: to

text: m

text: By

text: Proposition

text: this

text: shows

text: that

text: the

text: network

text: is

text: ﬂexible.

text: Now,

text: it

text: remains

text: to

text: check

text: that

text: the

text: network

text: is

text: closable.

text: This

text: amounts

text: to

text: check

text: that

text: the

text: image

text: of

text: DL

text: does

text: not

text: contain

text: the

text: vector

text: T.

text: Namely,

text: that

text: the

text: only

text: Id

text: solution

text: of

text: DL

text: (Φ˙

text: λ

text: T,

text: Id

text: isλ

text: and

text: Φ˙

text: Writing

text: p

text: p

text: z

text: and

text: Φ˙

text: Φ˙

text: z˙

text: this

text: amount

text: to

text: check

text: solution

text: to

text: i

text: i

text: pi

text: −i

text: z˙

text: λ

text: tan

text: θ

text: sin

text: θ)

text: z

text: z

text: z˙

text: z˙

text: C

text: C

text: cos

text: θ)

text: z˙

text: λ

text: tan

text: θ

text: sin

text: θ)

text: z

text: z

text: z˙

text: z˙

text: C

text: C

text: z˙

text: λ

text: cos

text: θ)

text: z

text: z

text: z˙

text: z˙

text: C

text: C

text: is

text: given

text: by

text: z˙

text: for

text: j

text: and

text: λ

text: Using

text: the

text: second

text: system

text: together

text: j

text: with

text: the

text: fact

text: that

text: z

text: z

text: and

text: z

text: z

text: we

text: get

text: z

text: z

text: z˙

text: z˙

text: z˙

text: z˙

text: z

text: z

text: z˙

text: z˙

text: z˙

text: z˙

text: C

text: C

text: and,

text: since

text: z

text: z

text: and

text: z

text: z

text: are

text: R-independent,

text: we

text: conclude

text: that

text: z˙

text: z˙

text: z˙

text: z˙

text: Using

text: the

text: ﬁrst

text: system,

text: we

text: get

text: z˙

text: z˙

text: i,

text: z˙

text: z˙

text: C

text: C

text: and,

text: together

text: this

text: implies

text: that

text: z˙

text: z˙

text: z˙

text: z˙

text: Using

text: this

text: information

text: back

text: into

text: the

text: λ

text: tan

text: θ

text: sin

text: θ)

text: C

text: z

text: z˙

text: −i,

text: λ0

text: cos

text: θ)

text: C

text: z

text: z

text: z˙

text: z˙

text: C

text: Since

text: z

text: z

text: tan

text: θ

text: i

text: and

text: z

text: z

text: tan

text: θ

text: i,

text: the

text: sum

text: of

text: the

text: last

text: two

text: equations

text: implies

text: that

text: tan

text: θ

text: z˙

text: i,

text: z˙

text: C

text: C

text: using

text: the

text: ﬁrst

text: two

text: equations,

text: we

text: conclude

text: that

text: T

text: is

text: not

text: in

text: the

text: image

text: of

text: DL

text: unless

text: Id

text: θ

text: sin

text: θ)

text: θ

text: cos

text: θ)

text: which

text: never

text: happens.

text: So,

text: by

text: Deﬁnition

text: the

text: network

text: is

text: closable.


text: 58 WEIWEI AO, MONICA MUSSO, FRANK PACARD, AND JUNCHENG WEI The proof now proceeds as in the proof of Lemma 10.1. For example, looking at the component of DF corresponding to z , we get (Id,a) 0 − − z z z z a˙ | 1 − 0| + a˙ | 4 − 0| = 0 [z1,z0] z z [z4,z0] z z 1 0 4 0 − − and, since z z and z z are R-independent, we conclude that a˙ = 1 0 4 0 [z1,z0] a˙ = 0. Proceeding similarly for the other components of DF , we prove [z4,z0] (Id,a) that a˙ = 0. This completes the proof of the fact that the network is ﬂexible. It remains to check that the network is closable. This amounts to check that the only solution to DL (Φ˙ ) = λ T is Φ˙ = 0 and λ = 0. Now observe that the Id equations in this system are of the form (cid:104) − (cid:48) − (cid:48) (cid:105) z z , Φ˙ Φ˙ | (cid:48) − | | | |z(cid:48) −z z| z C = λ z z ln a[z(cid:48),z] , or equivalently (cid:104) − (cid:48) − (cid:48) (cid:105) | (cid:48) − | | | z z , Φ˙ Φ˙ = λ z z 2 ln a (cid:48) . z z C [z ,z] This is this last system we will consider. We write Φ˙ = x + ix , Φ˙ = x + ix , Φ = x + ix , Φ = x + ix , z4 1 2 z1 3 4 z2 5 6 z3 7 8 and Φ˙ = x + ix , Φ˙ = x + ix zˆ0 9 10 z0 11 12 and we identify the image of DL with R9 starting by labeling the edges in the fol- Id lowing order [zˆ , z ], [zˆ , z ], [z , z ], [z , z ], [zˆ , z ] and next [z , z ], [z , z ], [z , z ], [z , z ] 0 0 0 4 0 1 0 2 0 3 1 2 1 3 3 4 2 4 to give a vector in R9. We re(cid:0)call that cos θ and sin θ have been deﬁned in (10.42). We need(cid:1)to check that the vector of R9 whose coordinates are given by ν2 ln(2 cos θ), 0, 0, 0, 0, ln(sin θ), µ2 ln(cos θ), ln(sin θ), µ2 ln(cos θ)   is not in the image of    −   0 0 0 0 0 0 0 0 2ν 0 2ν 0   − − −   ν µ 1 0 0 0 0 0 0 µ ν 1 0 0   − − −   0 0 µ ν 1 0 0 0 0 0 0 ν µ 1   − − −   0 0 0 0 µ ν 1 0 0 0 0 ν µ 1   − − −   0 0 0 0 0 0 ν µ 1 µ ν 1 0 0   −   0 0 0 2 0 2 0 0 0 0 0 0  − 0 0 2µ 0 0 0 2µ 0 0 0 0 0 − 0 2 0 0 0 0 0 2 0 0 0 0 − 2µ 0 0 0 2µ 0 0 0 0 0 0 0 (cid:3) This can be checked using Mathematica. Example 10.3 : Finally, we give an explicit example of ﬂexible closable network whose symmetry group reduces to the identity. The set of vertices of the network N is given by C { − − − − } V := a + ib, 1 + i, 1 + i, 1 i, 1 i , C ##LTLine## ##LTLine## ##LTLine##

text: WEIWEI

text: AO,

text: MONICA

text: MUSSO,

text: FRANK

text: PACARD,

text: AND

text: JUNCHENG

text: WEI

text: The

text: proof

text: now

text: proceeds

text: as

text: in

text: the

text: proof

text: of

text: Lemma

text: For

text: example,

text: looking

text: at

text: the

text: component

text: of

text: DF

text: corresponding

text: to

text: z

text: we

text: get

text: (Id,a)

text: z

text: z

text: z

text: z

text: a˙

text: a˙

text: z

text: z

text: z

text: z

text: and,

text: since

text: z

text: z

text: and

text: z

text: z

text: are

text: R-independent,

text: we

text: conclude

text: that

text: a˙

text: a˙

text: Proceeding

text: similarly

text: for

text: the

text: other

text: components

text: of

text: DF

text: we

text: prove

text: (Id,a)

text: that

text: a˙

text: This

text: completes

text: the

text: proof

text: of

text: the

text: fact

text: that

text: the

text: network

text: is

text: ﬂexible.

text: It

text: remains

text: to

text: check

text: that

text: the

text: network

text: is

text: closable.

text: This

text: amounts

text: to

text: check

text: that

text: the

text: only

text: solution

text: to

text: DL

text: (Φ˙

text: λ

text: T

text: is

text: Φ˙

text: and

text: λ

text: Now

text: observe

text: that

text: the

text: Id

text: equations

text: in

text: this

text: system

text: are

text: of

text: the

text: form

text: z

text: z

text: Φ˙

text: Φ˙

text: −z

text: z|

text: z

text: C

text: λ

text: z

text: z

text: ln

text: or

text: equivalently

text: z

text: z

text: Φ˙

text: Φ˙

text: λ

text: z

text: z

text: ln

text: a

text: z

text: z

text: C

text: [z

text: ,z]

text: This

text: is

text: this

text: last

text: system

text: we

text: will

text: consider.

text: We

text: write

text: Φ˙

text: x

text: ix

text: Φ˙

text: x

text: ix

text: Φ

text: x

text: ix

text: Φ

text: x

text: ix

text: and

text: Φ˙

text: x

text: ix

text: Φ˙

text: x

text: ix

text: and

text: we

text: identify

text: the

text: image

text: of

text: DL

text: with

text: starting

text: by

text: labeling

text: the

text: edges

text: in

text: the

text: fol-

text: Id

text: lowing

text: order

text: [zˆ

text: z

text: [zˆ

text: z

text: [z

text: z

text: [z

text: z

text: [zˆ

text: z

text: and

text: next

text: [z

text: z

text: [z

text: z

text: [z

text: z

text: [z

text: z

text: to

text: give

text: a

text: vector

text: in

text: We

text: that

text: cos

text: θ

text: and

text: sin

text: θ

text: have

text: been

text: deﬁned

text: in

text: We

text: check

text: that

text: the

text: vector

text: of

text: whose

text: coordinates

text: are

text: given

text: by

text: ν2

text: cos

text: θ),

text: ln(sin

text: θ),

text: µ2

text: ln(cos

text: θ),

text: ln(sin

text: θ),

text: µ2

text: ln(cos

text: θ)

text: is

text: not

text: in

text: the

text: image

text: of

text: 2ν

text: 2ν

text: ν

text: µ

text: µ

text: ν

text: µ

text: ν

text: ν

text: µ

text: µ

text: ν

text: ν

text: µ

text: ν

text: µ

text: µ

text: ν

text: 2µ

text: 2µ

text: 2µ

text: 2µ

text: This

text: can

text: be

text: checked

text: using

text: Mathematica.

text: Example

text: Finally,

text: we

text: give

text: an

text: explicit

text: example

text: of

text: ﬂexible

text: closable

text: network

text: whose

text: symmetry

text: group

text: reduces

text: to

text: the

text: identity.

text: The

text: set

text: of

text: vertices

text: of

text: the

text: network

text: N

text: is

text: given

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