text: Prepared for submission to JINST Instrumentation for Colliding Beam Physics 2017 (INSTR17) 27 February - 3 March, 2017 Novosibirsk, Russia Highly Granular Calorimeters: Technologies and Results Yong Liu,a,b,1 aInsitut für Physik, Johannes Gutenberg-Universität Mainz, Staudinger Weg 7, 55128 Mainz, Germany bCluster of Excellence PRISMA Detector Lab, Johannes Gutenberg-Universität Mainz, Staudinger Weg 9, 55128 Mainz, Germany E-mail: yong.liu@uni-mainz.de Abstract: The CALICE collaboration is developing highly granular calorimeters for experiments at a future lepton collider primarily to establish technologies for particle ﬂow event reconstruction. These technologies also ﬁnd applications elsewhere, such as detector upgrades for the LHC. Mean- while, the large data sets collected in an extensive series of beam tests have enabled detailed studies of the properties of hadronic showers in calorimeter systems, resulting in improved simulation mod- els and development of sophisticated reconstruction techniques. In this proceeding, highlights are included from studies of the structure of hadronic showers and results on reconstruction techniques for imaging calorimetry. In addition, current R&D activities within CALICE are summarized, fo- cusing on technological prototypes that address challenges from full detector system integration and production techniques amenable to mass production for electromagnetic and hadronic calorimeters based on silicon, scintillator, and gas techniques. Keywords: Calorimeters, Calorimeter methods, Detector modelling and simulations I on behalf of the CALICE collaboration 1 ##LTLine##

text: Prepared

font: BQTWWF+TeXGyreTermes-Regular

text: for

text: submission

text: to

text: JINST

text: Instrumentation

font: ODVMXY+TeXGyreHeros-Regular

text: for

text: Colliding

text: Beam

text: Physics

text: February

text: March,

text: Novosibirsk,

text: Russia

text: Highly

label: title

font: XLAYMU+TeXGyreHeros-Bold

text: Granular

text: Calorimeters:

text: Technologies

text: and

text: Results

text: Yong

label: author

text: aInsitut

font: EVDLQE+TeXGyreTermes-Italic

text: für

text: Physik,

text: Johannes

text: Gutenberg-Universität

text: Mainz,

text: Staudinger

text: Weg

text: Mainz,

text: Germany

text: bCluster

text: of

text: Excellence

text: PRISMA

text: Detector

text: Lab,

text: Johannes

text: Gutenberg-Universität

text: Mainz,

text: Staudinger

text: Weg

text: Mainz,

text: Germany

text: E-mail:

text: yong.liu@uni-mainz.de

text: Abstract:

text: The

text: CALICE

text: collaboration

text: is

text: developing

text: highly

text: granular

text: calorimeters

text: for

text: experiments

text: at

text: a

text: future

text: lepton

text: collider

text: primarily

text: to

text: establish

text: technologies

text: for

text: particle

text: ﬂow

text: event

text: reconstruction.

text: These

text: technologies

text: also

text: ﬁnd

text: applications

text: elsewhere,

text: such

text: as

text: detector

text: upgrades

text: for

text: the

text: LHC.

text: Mean-

text: while,

text: the

text: large

text: data

text: sets

text: collected

text: in

text: an

text: extensive

text: series

text: of

text: beam

text: tests

text: have

text: enabled

text: detailed

text: studies

text: of

text: the

text: properties

text: of

text: hadronic

text: showers

text: in

text: calorimeter

text: systems,

text: resulting

text: in

text: improved

text: simulation

text: mod-

text: els

text: and

text: development

text: of

text: sophisticated

text: reconstruction

text: techniques.

text: In

text: this

text: proceeding,

text: highlights

text: are

text: included

text: from

text: studies

text: of

text: the

text: structure

text: of

text: hadronic

text: showers

text: and

text: results

text: on

text: reconstruction

text: techniques

text: for

text: imaging

text: calorimetry.

text: In

text: addition,

text: current

text: R&D

text: activities

text: within

text: CALICE

text: are

text: summarized,

text: fo-

text: cusing

text: on

text: technological

text: prototypes

text: that

text: address

text: challenges

text: from

text: full

text: detector

text: system

text: integration

text: and

text: production

text: techniques

text: amenable

text: to

text: mass

text: production

text: for

text: electromagnetic

text: and

text: hadronic

text: calorimeters

text: based

text: on

text: silicon,

text: scintillator,

text: and

text: gas

text: techniques.

text: Keywords:

text: Calorimeters,

text: Calorimeter

text: methods,

text: Detector

text: modelling

text: and

text: simulations

text: I

text: on

text: behalf

text: of

text: the

text: CALICE

text: collaboration

text: ##LTLine##


text: ON COURANT’S NODAL DOMAIN PROPERTY FOR LINEAR COMBINATIONS OF EIGENFUNCTIONS PIERRE BÉRARD AND BERNARD HELFFER Abstract. We revisit Courant’s nodal domain property for linear combinations of eigenfunctions, and propose new, simple and ex- plicit counterexamples for domains in R2, S2, T2, and R3, including convex domains. 1. Introduction Let Ω ⊂ R be a bounded open domain or, more generally, a compact d Riemannian manifold with boundary. Consider the eigenvalue problem ( − ∆ = in Ω u λ u , (1.1) ( ) = 0 on Ω B u ∂ , where ( ) is some homogeneous boundary condition on Ω, so that B u ∂ we have a self-adjoint boundary value problem (including the empty condition if Ω is a closed manifold). For example, we can choose | | ( ) = for the Dirichlet boundary condition, or ( ) = for D u u Ω N u ∂u Ω the Neuma∂nn boundary condition. ∂ν ∂ − Call (Ω ) the associated self-adjoint extension of ∆, and list its H , B eigenvalues in nondecreasing order, counting multiplicities, ≤ ≤ ≤ · · · (1.2) 0 (Ω ) (Ω ) (Ω ) λ1 , B < λ2 , B λ3 , B ≥ For any 1, deﬁne the number n { | } (1.3) (Ω ) = min (Ω ) = (Ω ) τ , B, n k λ , B λ , B . k n Date: October 19, 2017 (berard-helﬀer-courant-herrmann-V3-171019.tex). 2010 Mathematics Subject Classiﬁcation. 35P99, 35Q99, 58J50. Key words and phrases. Eigenfunction, Nodal domain, Courant nodal domain theorem. 1 ##LTLine## ##LTLine##

text: ON

label: paragraph

text: COURANT’S

text: NODAL

text: DOMAIN

text: PROPERTY

text: FOR

text: LINEAR

text: COMBINATIONS

text: OF

text: EIGENFUNCTIONS

text: PIERRE

text: BÉRARD

text: AND

text: BERNARD

text: HELFFER

text: Abstract.

text: We

label: abstract

text: revisit

text: Courant’s

text: nodal

text: domain

text: property

text: for

text: linear

text: combinations

text: of

text: eigenfunctions,

text: and

text: propose

text: new,

text: simple

text: and

text: ex-

text: plicit

text: counterexamples

text: for

text: domains

text: in

text: and

text: including

text: convex

text: domains.

label: section

text: Introduction

text: Let

text: Ω

text: R

text: be

text: a

text: bounded

text: open

text: domain

text: or,

text: more

text: generally,

text: a

text: compact

text: d

text: Riemannian

text: manifold

text: with

text: boundary.

text: Consider

text: the

text: eigenvalue

text: problem

label: equation

text: in

text: Ω

text: u

text: λ

text: u

text: on

text: Ω

text: B

text: u

text: where

text: is

text: some

text: homogeneous

text: boundary

text: condition

text: on

text: Ω,

text: so

text: that

text: B

text: u

text: we

text: have

text: a

text: self-adjoint

text: boundary

text: value

text: problem

text: (including

text: the

text: empty

text: condition

text: if

text: Ω

text: is

text: a

text: closed

text: manifold).

text: For

text: example,

text: we

text: can

text: choose

text: for

text: the

text: Dirichlet

text: boundary

text: condition,

text: or

text: for

text: D

text: u

text: u

text: Ω

text: N

text: u

text: ∂u

text: Ω

text: the

text: Neuma∂nn

text: boundary

text: condition.

text: ∂ν

text: Call

text: (Ω

text: the

text: associated

text: self-adjoint

text: extension

text: of

text: and

text: list

text: its

text: H

text: B

text: eigenvalues

text: in

text: nondecreasing

text: order,

text: counting

text: multiplicities,

text: (Ω

text: (Ω

text: (Ω

text: λ1

text: B

text: λ2

text: B

text: λ3

text: B

text: For

text: any

text: deﬁne

text: the

text: number

text: n

text: (Ω

text: min

text: (Ω

text: (Ω

text: τ

text: B,

text: n

text: k

text: λ

text: B

text: λ

text: B

text: k

text: n

text: Date:

text: October

text: Mathematics

text: Subject

text: Classiﬁcation.

text: Key

text: words

text: and

text: phrases.

text: Eigenfunction,

text: Nodal

text: domain,

text: Courant

text: nodal

text: domain

text: theorem.

text: ##LTLine##

text: ##LTLine##


text: 2 P. BÉRARD AND B. HELFFER ∈ Shorthand. We use (Ω ) as a shorthand for is an eigen- u λ , B u n function of (Ω ) associated with the eigenvalue (Ω ) i.e., H , B λ , B n (Ω )( ) = (Ω ) H , B u λ , B u . n Given a real continuous function on Ω, deﬁne its nodal set v Z { ∈ | } (1.4) ( ) = Ω ( ) = 0 v x v x , \ Z and call ( ) the number of connected components of Ω ( ) i.e., β0 v v the number of nodal domains of . v Theorem 1.1. [Courant (1923)] ∈ For any nonzero (Ω ), u λ , B n ≤ ≤ (1.5) ( ) (Ω ) β0 u τ , B, n n . Courant’s nodal domain theorem can be found in [11, Chap. V.6]. A footnote in [11, p. 454] (see also the second footnote in [10, p. 394]) indicates that this theorem also holds for any linear combination of the ﬁrst eigenfunctions, and refers to the PhD thesis of Horst Herrmann n (Göttingen, 1932) [19]. For later reference, we write a precise statement. Given 0, de- r > L note by (Ω ) the space of linear combinations of eigenfunctions , B, r of (Ω ) associated with eigenvalues less than or equal to , H , B  r  X    (1.6) L(Ω ) = | ∈ R ∈ (Ω ) , B, r c u c , u λ , B . j j j j j (Ω )≤ λj ,B r Statement 1.2. [Extended Courant Property] ∈ L Let (Ω (Ω )) be any linear combination of eigenfunctions v , B, λ , B n associated with the ﬁrst eigenvalues of the eigenvalue problem (1.1). n Then, ≤ ≤ (1.7) ( ) (Ω ) β0 v τ , B, n n . The footnote in [11, p. 454] claims that Statement 1.2 is true. Remarks on the Extended Courant Property. 1. Statement 1.2 is true for Sturm-Liouville equations. This was ﬁrst announced by C. Sturm in 1833, [29] and proved in [30]. Other proofs were later on given by J. Liouville and Lord Rayleigh who both cite Sturm explicitly, see [7] for more details. ##LTLine##

text: P.

text: BÉRARD

text: AND

text: B.

text: HELFFER

text: Shorthand.

text: We

text: use

text: (Ω

text: as

text: a

text: shorthand

text: for

text: is

text: an

text: eigen-

text: u

text: λ

text: B

text: u

text: n

text: function

text: of

text: (Ω

text: associated

text: with

text: the

text: eigenvalue

text: (Ω

text: i.e.,

text: H

text: B

text: λ

text: B

text: n

text: (Ω

text: (Ω

text: H

text: B

text: u

text: λ

text: B

text: u

text: n

text: Given

text: a

text: real

text: continuous

text: function

text: on

text: Ω,

text: deﬁne

text: its

text: nodal

text: set

text: v

text: Z

text: Ω

text: v

text: x

text: v

text: x

text: Z

text: and

text: call

text: the

text: number

text: of

text: connected

text: components

text: of

text: Ω

text: i.e.,

text: β0

text: v

text: v

text: the

text: number

text: of

text: nodal

text: domains

text: of

text: v

text: Theorem

text: [Courant

text: For

text: any

text: nonzero

text: (Ω

text: u

text: λ

text: B

text: n

text: (Ω

text: β0

text: u

text: τ

text: B,

text: n

text: n

text: Courant’s

text: nodal

text: domain

text: theorem

text: can

text: be

text: found

text: in

text: Chap.

text: A

text: footnote

text: in

text: p.

text: (see

text: also

text: the

text: second

text: footnote

text: in

text: p.

text: indicates

text: that

text: this

text: theorem

text: also

text: holds

text: for

text: any

text: linear

text: combination

text: of

text: the

text: ﬁrst

text: eigenfunctions,

text: and

text: refers

text: to

text: the

text: PhD

text: thesis

text: of

text: Horst

text: Herrmann

text: n

text: (Göttingen,

text: For

text: later

text: reference,

text: we

text: write

text: a

text: precise

text: statement.

text: Given

text: de-

text: r

text: L

text: note

text: by

text: (Ω

text: the

text: space

text: of

text: linear

text: combinations

text: of

text: eigenfunctions

text: B,

text: r

text: of

text: (Ω

text: associated

text: with

text: eigenvalues

text: less

text: than

text: or

text: equal

text: to

text: H

text: B

text: r

text: X

text: L(Ω

text: R

text: (Ω

text: B,

text: r

text: c

text: u

text: c

text: u

text: λ

text: B

text: j

text: j

text: j

text: j

text: j

text: (Ω

text: λj

text: ,B

text: r

text: Statement

text: [Extended

text: Courant

text: Property]

text: L

text: Let

text: (Ω

text: (Ω

text: be

text: any

text: linear

text: combination

text: of

text: eigenfunctions

text: v

text: B,

text: λ

text: B

text: n

text: associated

text: with

text: the

text: ﬁrst

text: eigenvalues

text: of

text: the

text: eigenvalue

text: problem

text: n

text: Then,

text: (Ω

text: β0

text: v

text: τ

text: B,

text: n

text: n

text: The

text: footnote

text: in

text: p.

text: claims

text: that

text: Statement

text: is

text: true.

text: Remarks

text: on

text: the

text: Extended

text: Courant

text: Property.

text: Statement

text: is

text: true

text: for

text: Sturm-Liouville

text: equations.

text: This

text: was

text: ﬁrst

text: announced

text: by

text: C.

text: Sturm

text: in

text: and

text: proved

text: in

text: Other

text: proofs

text: were

text: later

text: on

text: given

text: by

text: J.

text: Liouville

text: and

text: Lord

text: Rayleigh

text: who

text: both

text: cite

text: Sturm

text: explicitly,

text: see

text: for

text: more

text: details.

text: ##LTLine##


text: 12 P. BÉRARD AND B. HELFFER Figure 6.1. Equilateral triangle T = [ ] OAB e ( √ − ( ) = cos(4 ) + cos(2 ( + 3 )) (6.2) ϕN2 x, y 3πx √ 3π x y + cos(2 ( + 3 )) π x y , 3 or, more simply, (cid:18) (cid:19) (cid:18) (cid:19) !! 2 2 2 πx πx √πy − (6.3) ( ) = 2 cos cos + cos 1 ϕN2 x, y 3 3 3 . { } { } ∩ T The set + 1 = 0 consists of the two line segments = 3 ϕ√N2 √ x 4 e { } ∩ T T and + 3 = 3 , which meet at the point (3 3) on . x y , ∂ 2 e 4 4 e { } ∈ { − } The sets + = 0 , with 0 1 1 1 + , and small positive ϕ2 a a , ε, , ε − , are shown in Figure 6.2. When varies from 1 to 1 + , the ε a ε ε T number of nodal domains of + in jumps from 2 to 3, with the ϕ2 a e jump occurring for = 1. a ≤ ≤ It follows that + = 0, for 1 1 1, provides a counter- ϕN2 a a . example to the Extended Courant Property for the equilateral triangle with Neumann boundary condition. Remark. The eigenfunction restricted to the hemiequilateral tri- ϕN2 T angle is the second Neumann eigenfunction of = [ ]. The re- OAM h striction of to the hemiequilateral triangle is an eigenfunction of T ϕN3 with mixed boundary condition (Dirichlet on and Neumann on OM h the other sides). T The sequence of Dirichlet eigenvalues of the equilateral triangle be- e gins as follows, ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: P.

text: BÉRARD

text: AND

text: B.

text: HELFFER

text: Figure

text: Equilateral

label: caption

text: triangle

text: T

text: OAB

text: e

text: x,

text: y

text: 3π

text: x

text: y

text: π

text: x

text: y

text: or,

text: more

text: simply,

text: πx

text: πx

text: √πy

text: cos

text: cos

text: cos

text: x,

text: y

text: T

text: The

text: set

text: consists

text: of

text: the

text: two

text: line

text: segments

text: ϕ√N2

text: x

text: e

text: T

text: T

text: and

text: which

text: meet

text: at

text: the

text: point

text: on

text: x

text: y

text: e

text: e

text: The

text: sets

text: with

text: and

text: small

text: positive

text: ϕ2

text: a

text: a

text: ε,

text: ε

text: are

text: shown

text: in

text: Figure

text: When

text: varies

text: from

text: to

text: the

text: ε

text: a

text: ε

text: ε

text: T

text: number

text: of

text: nodal

text: domains

text: of

text: in

text: jumps

text: from

text: to

text: with

text: the

text: ϕ2

text: a

text: e

text: jump

text: occurring

text: for

text: a

text: It

text: follows

text: that

text: for

text: provides

text: a

text: counter-

text: a

text: a

text: example

text: to

text: the

text: Extended

text: Courant

text: Property

text: for

text: the

text: equilateral

text: triangle

text: with

text: Neumann

text: boundary

text: condition.

text: Remark.

text: The

text: eigenfunction

text: restricted

text: to

text: the

text: hemiequilateral

text: tri-

text: T

text: angle

text: is

text: the

text: second

text: Neumann

text: eigenfunction

text: of

text: The

text: re-

text: OAM

text: h

text: striction

text: of

text: to

text: the

text: hemiequilateral

text: triangle

text: is

text: an

text: eigenfunction

text: of

text: T

text: with

text: mixed

text: boundary

text: condition

text: (Dirichlet

text: on

text: and

text: Neumann

text: on

text: OM

text: h

text: the

text: other

text: sides).

text: T

text: The

text: sequence

text: of

text: Dirichlet

text: eigenvalues

text: of

text: the

text: equilateral

text: triangle

text: be-

text: e

text: gins

text: as

text: follows,

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 16 P. BÉRARD AND B. HELFFER evidence that (ECP) is not true for the equilateral triangle with rounded corners. 7. The regular hexagon We are looking for another counterexample for the Extended Courant Property in a convex domain of R2. We have already mentioned that this quest was unsuccessful for the square. It is natural to think of H other polygons and, among them, the regular hexagon . H T Call = [ ] the regular hexagon with sides of length 1, = ABCDEF T e [ ] the equilateral triangle, and = [ ] the hemiequilateral OAB OAM h triangle. See Figure 7.1. In this section, we consider both the Dirichlet, and the Neumann H boundary conditions on . ∂ Only a small portion (asymptotically one-sixth) of the eigenvalues, and H of the eigenfunctions, of the regular hexagon are known explicitly, namely those which arise from the equilateral triangle, or from the hemiequilateral triangle (with Dirichlet or Neumann boundary condi- tion). Numerical computations of the Dirichlet eigenvalues, and of the nodal patterns of Dirichlet eigenfunctions, are available in the literature, see for example [4, 12]. They strongly rely on the symmetries of the hexagon. We did not ﬁnd similar computations for the Neumann eigenvalues H and eigenfunctions of in the literature. We performed numerical computations for this case with matlab, making use of the symmetries as in [12]. Assuming that the computed eigenvalues are close enough to the true eigenvalues, using symmetries and Courant’s nodal domain theorem, it is possible to identify the ﬁrst six Dirichlet or Neumann eigenfunctions H of , and their nodal patterns. As a consequence, we obtain numerical evidence that the regular hexa- gon provides counterexamples for the Extended Courant Property, for either Dirichlet or Neumann boundary condition. Figure 7.2 displays the lines of mirror symmetry of the regular hexagon. For simplicity, we use the same notation for a line, and for the mirror symmetry across that line. ∈ { −} Given + , we consider the following sets of functions, σ, τ ,

text: P.

text: BÉRARD

text: AND

text: B.

text: HELFFER

text: evidence

text: that

text: (ECP)

text: is

text: not

text: true

text: for

text: the

text: equilateral

text: triangle

text: with

text: rounded

text: corners.

text: The

text: regular

text: hexagon

text: We

text: are

text: looking

text: for

text: another

text: counterexample

text: for

text: the

text: Extended

text: Courant

text: Property

text: in

text: a

text: convex

text: domain

text: of

text: We

text: have

text: already

text: mentioned

text: that

text: this

text: quest

text: was

text: unsuccessful

text: for

text: the

text: square.

text: It

text: is

text: natural

text: to

text: think

text: of

text: H

text: other

text: polygons

text: and,

text: among

text: them,

text: the

text: regular

text: hexagon

text: H

text: T

text: Call

text: the

text: regular

text: hexagon

text: with

text: sides

text: of

text: length

text: ABCDEF

text: T

text: e

text: the

text: equilateral

text: triangle,

text: and

text: the

text: hemiequilateral

text: OAB

text: OAM

text: h

text: triangle.

text: See

text: Figure

text: In

text: this

text: section,

text: we

text: consider

text: both

text: the

text: Dirichlet,

text: and

text: the

text: Neumann

text: H

text: boundary

text: conditions

text: on

text: Only

text: a

text: small

text: portion

text: (asymptotically

text: one-sixth)

text: of

text: the

text: eigenvalues,

text: and

text: H

text: of

text: the

text: eigenfunctions,

text: of

text: the

text: regular

text: hexagon

text: are

text: known

text: explicitly,

text: namely

text: those

text: which

text: arise

text: from

text: the

text: equilateral

text: triangle,

text: or

text: from

text: the

text: hemiequilateral

text: triangle

text: (with

text: Dirichlet

text: or

text: Neumann

text: boundary

text: condi-

text: tion).

text: Numerical

text: computations

text: of

text: the

text: Dirichlet

text: eigenvalues,

text: and

text: of

text: the

text: nodal

text: patterns

text: of

text: Dirichlet

text: eigenfunctions,

text: are

text: available

text: in

text: the

text: literature,

text: see

text: for

text: example

text: They

text: strongly

text: rely

text: on

text: the

text: symmetries

text: of

text: the

text: hexagon.

text: We

text: did

text: not

text: ﬁnd

text: similar

text: computations

text: for

text: the

text: Neumann

text: eigenvalues

text: H

text: and

text: eigenfunctions

text: of

text: in

text: the

text: literature.

text: We

text: performed

text: numerical

text: computations

text: for

text: this

text: case

text: with

text: matlab,

text: making

text: use

text: of

text: the

text: symmetries

text: as

text: in

text: Assuming

text: that

text: the

text: computed

text: eigenvalues

text: are

text: close

text: enough

text: to

text: the

text: true

text: eigenvalues,

text: using

text: symmetries

text: and

text: Courant’s

text: nodal

text: domain

text: theorem,

text: it

text: is

text: possible

text: to

text: identify

text: the

text: ﬁrst

text: six

text: Dirichlet

text: or

text: Neumann

text: eigenfunctions

text: H

text: of

text: and

text: their

text: nodal

text: patterns.

text: As

text: a

text: consequence,

text: we

text: obtain

text: numerical

text: evidence

text: that

text: the

text: regular

text: hexa-

text: gon

text: provides

text: counterexamples

text: for

text: the

text: Extended

text: Courant

text: Property,

text: for

text: either

text: Dirichlet

text: or

text: Neumann

text: boundary

text: condition.

text: Figure

text: displays

text: the

text: lines

text: of

text: mirror

text: symmetry

text: of

text: the

text: regular

text: hexagon.

text: For

text: simplicity,

text: we

text: use

text: the

text: same

text: notation

text: for

text: a

text: line,

text: and

text: for

text: the

text: mirror

text: symmetry

text: across

text: that

text: line.

text: Given

text: we

text: consider

text: the

text: following

text: sets

text: of

text: functions,

text: σ,

text: τ


text: 18 P. BÉRARD AND B. HELFFER Figure 7.3. The domain R or [6] and Appendix for a summary. The eigenvalues and eigenfunc- T tions of with mixed boundary conditions are, generally speaking, h not known explicitly. H For example, the ﬁrst Dirichlet eigenfunction of arises from the u1 T ,D ﬁrst eigenfunction for , with mixed boundary condition NND (sides h listed in decreasing order of length), i.e., Dirichlet condition on the T smaller side, Neumann on the other sides of . h H S S Note that the eigenfunctions of which belong to 0 or to 0 + − − + have at least six nodal domains. According to Courant’s n, odal domai,n theorem, they correspond to eigenvalues with index at least 6. Simi- H S larly, the eigenfunctions of which belong to 0 have at least twelve − − nodal domains, and hence correspond to eigenvalu, es with index at least twelve. Letting denote the rotation with center the origin, and angle 2 , R π 3 deﬁne the map ◦ − ◦ (7.3) ( ) = 2 T ϕ ϕ R ϕ R . Denote by ∗ the action of on functions, ∗ = ◦ . It is easy to R R R ϕ ϕ R see that (7.4) S0 = S ∩ ker( ∗ − ) R I . σ,τ σ,τ Introduce the subspace (7.5) S1 = S ∩ ker( ∗2 + ∗ + ) R R I . σ,τ σ,τ Then, ##LTFigure## ##LTLine##

text: P.

text: BÉRARD

text: AND

text: B.

text: HELFFER

text: Figure

text: The

text: domain

text: R

text: or

text: and

text: Appendix

text: for

text: a

text: summary.

text: The

text: eigenvalues

text: and

text: eigenfunc-

text: T

text: tions

text: of

text: with

text: mixed

text: boundary

text: conditions

text: are,

text: generally

text: speaking,

text: h

text: not

text: known

text: explicitly.

text: H

text: For

text: example,

text: the

text: ﬁrst

text: Dirichlet

text: eigenfunction

text: of

text: arises

text: from

text: the

text: T

text: ,D

text: ﬁrst

text: eigenfunction

text: for

text: with

text: mixed

text: boundary

text: condition

text: NND

text: (sides

text: h

text: listed

text: in

text: decreasing

text: order

text: of

text: length),

text: i.e.,

text: Dirichlet

text: condition

text: on

text: the

text: T

text: smaller

text: side,

text: Neumann

text: on

text: the

text: other

text: sides

text: of

text: h

text: H

text: S

text: S

text: Note

text: that

text: the

text: eigenfunctions

text: of

text: which

text: belong

text: to

text: or

text: to

text: have

text: at

text: least

text: six

text: nodal

text: domains.

text: According

text: to

text: Courant’s

text: n,

text: odal

text: domai,n

text: theorem,

text: they

text: correspond

text: to

text: eigenvalues

text: with

text: index

text: at

text: least

text: Simi-

text: H

text: S

text: larly,

text: the

text: eigenfunctions

text: of

text: which

text: belong

text: to

text: have

text: at

text: least

text: twelve

text: nodal

text: domains,

text: and

text: hence

text: correspond

text: to

text: eigenvalu,

text: es

text: with

text: index

text: at

text: least

text: twelve.

text: Letting

text: denote

text: the

text: rotation

text: with

text: center

text: the

text: origin,

text: and

text: angle

text: R

text: π

text: deﬁne

text: the

text: map

text: T

text: ϕ

text: ϕ

text: R

text: ϕ

text: R

text: Denote

text: by

text: the

text: action

text: of

text: on

text: functions,

text: It

text: is

text: easy

text: to

text: R

text: R

text: R

text: ϕ

text: ϕ

text: R

text: see

text: that

text: S

text: ker(

text: R

text: I

text: σ,τ

text: σ,τ

text: Introduce

text: the

text: subspace

text: S

text: ker(

text: ∗2

text: R

text: R

text: I

text: σ,τ

text: σ,τ

text: Then,

text: ##LTFigure##

text: ##LTLine##


text: COURANT NODAL DOMAIN PROPERTY 3 2. Å. Pleijel mentions Statement 1.2 in his well-known paper [28] on the asymptotic behaviour of the number of nodal domains of a Dirichlet eigenfunction associated with the -th eigenvalue for a plane domain. n At the end of the paper, he also considers the Neumann boundary condition. “In order to treat, for instance the case of the free three-dimensional membrane [0 ]3, it would be necessary to use, in a special case, the , π theorem quoted in [10], p. 3941. This theorem which generalizes part of the Liouville-Rayleigh theorem for the string asserts that a linear combination, with constant coeﬃcients, of the ﬁrst eigenfunctions n can have at most nodal domains. However, as far as I have been able n to ﬁnd there is no proof of this assertion in the literature.” 3. Arnold [2], see also [22, 23] and [21], mentions that he actually discussed the footnote in [11, p. 454] with R. Courant, that the Ex- tended Courant Property cannot be true in general, and that O. Viro produced counterexamples for the sphere. More precisely, as early as 1973, V. Arnold [1] pointed out that, when Ω is the round sphere S , N the Extended Courant Property is related to Hilbert’s 16th problem. Indeed, the eigenfunctions of the Laplace-Beltrami operator on S are N the spherical harmonics i.e., the restrictions to the sphere of the har- monic homogeneous polynomials, so that the linear combinations of spherical harmonics of degree less than or equal to are the restric- n tions to the sphere of the homogeneous polynomials of degree in n ( + 1) variables. The following citation is taken from [3]. N Eigen oscillations of the sphere with the standard metric are described by spherical functions, i.e., polynomials. Therefore the Courant state- ment cited above implies the following estimate dim (R − R) ≤ + 1 (1) RH0 P N Vn, CNN+n−2 for the number of connected components of the complement to an alge- braic hypersurface of degree in the -dimensional projective space. n N For planar curves ( = 2), the estimation (1) is exact (it turns into N equality on a conﬁguration of lines in general position) and can be n proven independently of the Courant statement. For smooth surfaces of degree 4 in R 3 the estimation is also exact and proved (by V.M. Khar- P lamov). In the general case, the Courant statement is false (a counterexample can be constructed by a small perturbation of the standard metric on the sphere). Nonetheless the estimation (1) seems to be plausible: for proving it one has to verify the Courant statement only for oscillation of the sphere (or the projective space) with the standard metric.1 1 Translator’s remark: the inequality (1) does not hold true for surfaces of even degree ≥ 6 in R 3. Counterexamples to (1) were constructed P 1Pleijel refers to the German edition, this is p. 454 in the English edition [11]. ##LTLine##

text: COURANT

text: NODAL

text: DOMAIN

text: PROPERTY

text: Å.

text: Pleijel

text: mentions

text: Statement

text: in

text: his

text: well-known

text: paper

text: on

text: the

text: asymptotic

text: behaviour

text: of

text: the

text: number

text: of

text: nodal

text: domains

text: of

text: a

text: Dirichlet

text: eigenfunction

text: associated

text: with

text: the

text: -th

text: eigenvalue

text: for

text: a

text: plane

text: domain.

text: n

text: At

text: the

text: end

text: of

text: the

text: paper,

text: he

text: also

text: considers

text: the

text: Neumann

text: boundary

text: condition.

text: “In

text: order

text: to

text: treat,

text: for

text: instance

text: the

text: case

text: of

text: the

text: free

text: three-dimensional

text: membrane

text: it

text: would

text: be

text: necessary

text: to

text: use,

text: in

text: a

text: special

text: case,

text: the

text: π

text: theorem

text: quoted

text: in

text: p.

text: This

text: theorem

text: which

text: generalizes

text: part

text: of

text: the

text: Liouville-Rayleigh

text: theorem

text: for

text: the

text: string

text: asserts

text: that

text: a

text: linear

text: combination,

text: with

text: constant

text: coeﬃcients,

text: of

text: the

text: ﬁrst

text: eigenfunctions

text: n

text: can

text: have

text: at

text: most

text: nodal

text: domains.

text: However,

text: as

text: far

text: as

text: I

text: have

text: been

text: able

text: n

text: to

text: ﬁnd

text: there

text: is

text: no

text: proof

text: of

text: this

text: assertion

text: in

text: the

text: literature.”

text: Arnold

text: see

text: also

text: and

text: mentions

text: that

text: he

text: actually

text: discussed

text: the

text: footnote

text: in

text: p.

text: with

text: R.

text: Courant,

text: that

text: the

text: Ex-

text: tended

text: Courant

text: Property

text: cannot

text: be

text: true

text: in

text: general,

text: and

text: that

text: O.

text: Viro

text: produced

text: counterexamples

text: for

text: the

text: sphere.

text: More

text: precisely,

text: as

text: early

text: as

text: V.

text: Arnold

text: pointed

text: out

text: that,

text: when

text: Ω

text: is

text: the

text: round

text: sphere

text: S

text: N

text: the

text: Extended

text: Courant

text: Property

text: is

text: related

text: to

text: Hilbert’s

text: problem.

text: Indeed,

text: the

text: eigenfunctions

text: of

text: the

text: Laplace-Beltrami

text: operator

text: on

text: S

text: are

text: N

text: the

text: spherical

text: harmonics

text: i.e.,

text: the

text: restrictions

text: to

text: the

text: sphere

text: of

text: the

text: har-

text: monic

text: homogeneous

text: polynomials,

text: so

text: that

text: the

text: linear

text: combinations

text: of

text: spherical

text: harmonics

text: of

text: degree

text: less

text: than

text: or

text: equal

text: to

text: are

text: the

text: restric-

text: n

text: tions

text: to

text: the

text: sphere

text: of

text: the

text: homogeneous

text: polynomials

text: of

text: degree

text: in

text: n

text: variables.

text: The

text: following

text: citation

text: is

text: taken

text: from

text: N

text: Eigen

text: oscillations

text: of

text: the

text: sphere

text: with

text: the

text: standard

text: metric

text: are

text: described

text: by

text: spherical

text: functions,

text: i.e.,

text: polynomials.

text: Therefore

text: the

text: Courant

text: state-

text: ment

text: cited

text: above

text: implies

text: the

text: following

text: estimate

text: dim

text: (R

text: R)

text: P

text: N

text: Vn,

text: for

text: the

text: number

text: of

text: connected

text: components

text: of

text: the

text: complement

text: to

text: an

text: alge-

text: braic

text: hypersurface

text: of

text: degree

text: in

text: the

text: -dimensional

text: projective

text: space.

text: n

text: N

text: For

text: planar

text: curves

text: the

text: estimation

text: is

text: exact

text: (it

text: turns

text: into

text: N

text: equality

text: on

text: a

text: conﬁguration

text: of

text: lines

text: in

text: general

text: position)

text: and

text: can

text: be

text: n

text: proven

text: independently

text: of

text: the

text: Courant

text: statement.

text: For

text: smooth

text: surfaces

text: of

text: degree

text: in

text: R

text: the

text: estimation

text: is

text: also

text: exact

text: and

text: proved

text: (by

text: V.M.

text: Khar-

text: P

text: lamov).

text: In

text: the

text: general

text: case,

text: the

text: Courant

text: statement

text: is

text: false

text: (a

text: counterexample

text: can

text: be

text: constructed

text: by

text: a

text: small

text: perturbation

text: of

text: the

text: standard

text: metric

text: on

text: the

text: sphere).

text: Nonetheless

text: the

text: estimation

text: seems

text: to

text: be

text: plausible:

text: for

text: proving

text: it

text: one

text: has

text: to

text: verify

text: the

text: Courant

text: statement

text: only

text: for

text: oscillation

text: of

text: the

text: sphere

text: (or

text: the

text: projective

text: space)

text: with

text: the

text: standard

text: Translator’s

text: remark:

text: the

text: inequality

text: does

text: not

text: hold

text: true

text: for

text: surfaces

text: of

text: even

text: degree

text: in

text: R

text: Counterexamples

text: to

text: were

text: constructed

text: P

text: refers

text: to

text: the

text: German

text: edition,

text: this

text: is

text: p.

text: in

text: the

text: English

text: edition

text: ##LTLine##


text: (cid:102) (cid:101) (cid:103) (cid:102) (cid:102) (cid:102) CYCLIC COCYCLE AND RELATIVE-PARTITIONED INDEX THEOREM 15 (cid:101) (cid:102) R × ⊗ W(cid:101) i = ZRi =RWi and φi = 1 φNi. Then (Mi, Wi, Di) is→anReven relative index data over ( 2, ), here a continuous coarse map f : M 2 is deﬁned to be i i f (x, y) = (x, dist(y, Z )). The counterpart of Lemma 3.5 is as follows. i i (cid:102) (cid:102) (cid:102) ∗ ⊂ → ∗ ⊂ Lemma 4.3. By using the map Γ : K (C (W M )) K (C (W M )) → 1 1 (cid:102) (cid:102)1 (cid:102) (cid:102) 1 1 1 deﬁned by the identity map N (W ) N (W ), we have r 1 r 1 Γ(c-ind(φ , D , φ , D )) = c-ind(φ , D , φ , D ). 1 1 2 2 1 1 2 2 (cid:101) (cid:101) (cid:101) Proof. Take unitaries U and U such that U = U on L2(N (W ), S ) and propaga- r 1 1 tion of U and U is less than r/8 as in the proof of Lemma 3.5. Take a chopping ⊂ − − ∗ fu−nct−ion ∗χ −such that Supp(χˆ) ( r/8, r/8). Then op(cid:101)erator(cid:101)s uφ1 U∗uφ−2U a−nd u 1 U u 1U are supported in N (W ), so that an operator u U u 1U 1 φ1 (cid:102) φ2 r/4 1 (cid:102) ∗ −(cid:102) − (cid:101)φ1 φ(cid:101)2 is supported in N (W ). Similarly, an operator u U u 1U 1 is supported in r/4 1 ∗φ1 − φ2 (cid:102) ∗ −(cid:102) N (W ) = N (W ). Since propagation of u U u 1U and u U u 1U is less r/4 1 r/4 1 φ1(cid:102) (cid:102)φ2 (cid:102) (cid:102) φ1 φ2 than 3r/4, we have Γ(c-ind(φ , D , φ , D )) = c-ind(φ , D , φ , D ). 1 1 2 2 1 1 2 2 (cid:3) Finally, we comlete the proof of Theorem 4.1 for the product case. By Lemma (cid:101) 2.10, we have (cid:101) (cid:102) (cid:102) (cid:102) (cid:102) (cid:102) (cid:102) ζ∗(c-ind(φ , D , φ , D ) = ζ∗(Γ(c-ind(φ , D , φ , D ))). 1 1 2 (cid:101)2 (cid:101) (cid:102) (cid:102)1 (cid:101)1 2 2 By Lemma 4.3, the value equals ζ∗(c-ind(φ , D , φ , D )). Since N and N are (cid:101) (cid:102) (cid:102) (cid:102) (cid:102) 1 1 2 2 1 2 closed and we can take a unitary U satisﬁes UΠ = Π U, we have (cid:16) 1(cid:17) 2 (cid:102) (cid:101) (cid:101) (cid:102) ζ∗(c-ind(φ , D , φ , D )) 1 1 2 2 − 1 (cid:16) (cid:102) ∗(cid:17)−(cid:102) (cid:16) (cid:17) = index Π(cid:102)u U(cid:102) u 1UΠ (cid:102) (cid:102) 8πi 1 φ(cid:102)1 φ2 1 (cid:102) − 1 1 = index Π u Π + index Π u Π . 8πi 1 φ1 1 8πi 2 φ2 2 By an index theorem for To(cid:16)eplitz o(cid:16)perato(cid:17)rs on part(cid:16)itioned(cid:17)(cid:17)mannifolds [8, Theorem 2.6], the value equals (cid:103) (cid:103) − 1 − index T index T . 8πi φN1 φN2 This is nothing but the relative topological Toeplitz index, so that the proof is completed. References 1. A. Connes, Noncommutative diﬀerential geometry, Inst. Hautes E´tudes Sci. Publ. Math. (1985), no. 62, 257–360. MR 823176 (87i:58162) 2. M. Gromov and H. Blaine Lawson, Jr., Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Inst. Hautes E´tudes Sci. Publ. Math. (1983), no. 58, 83–196 (1984). MR 720933 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: CYCLIC

text: COCYCLE

text: AND

text: RELATIVE-PARTITIONED

text: INDEX

text: THEOREM

text: R

text: i

text: ZRi

text: =RWi

text: and

text: φi

text: φNi.

text: Then

text: (Mi,

text: Wi,

text: Di)

text: is→anReven

text: relative

text: index

text: data

text: over

text: here

text: a

text: continuous

text: coarse

text: map

text: f

text: M

text: is

text: deﬁned

text: to

text: be

text: i

text: i

text: f

text: (x,

text: y)

text: (x,

text: dist(y,

text: Z

text: The

text: counterpart

text: of

text: Lemma

text: is

text: as

text: follows.

text: i

text: i

text: Lemma

text: By

text: using

text: the

text: map

text: Γ

text: K

text: (C

text: (W

text: M

text: K

text: (C

text: (W

text: M

text: deﬁned

text: by

text: the

text: identity

text: map

text: N

text: (W

text: N

text: (W

text: we

text: have

text: r

text: r

text: Γ(c-ind(φ

text: D

text: φ

text: D

text: c-ind(φ

text: D

text: φ

text: D

text: Proof.

text: Take

text: unitaries

text: U

text: and

text: U

text: such

text: that

text: U

text: U

text: on

text: (W

text: S

text: and

text: propaga-

text: r

text: tion

text: of

text: U

text: and

text: U

text: is

text: less

text: than

text: as

text: in

text: the

text: proof

text: of

text: Lemma

text: Take

text: a

text: chopping

text: fu−nct−ion

text: ∗χ

text: −such

text: that

text: Supp(χˆ)

text: Then

text: a−nd

text: u

text: U

text: u

text: are

text: supported

text: in

text: N

text: (W

text: so

text: that

text: an

text: operator

text: u

text: U

text: u

text: φ1

text: φ2

text: is

text: supported

text: in

text: N

text: (W

text: Similarly,

text: an

text: operator

text: u

text: U

text: u

text: is

text: supported

text: in

text: ∗φ1

text: φ2

text: N

text: (W

text: N

text: (W

text: Since

text: propagation

text: of

text: u

text: U

text: u

text: and

text: u

text: U

text: u

text: is

text: less

text: φ1

text: φ2

text: than

text: we

text: have

text: Γ(c-ind(φ

text: D

text: φ

text: D

text: c-ind(φ

text: D

text: φ

text: D

text: Finally,

text: we

text: comlete

text: the

text: proof

text: of

text: Theorem

text: for

text: the

text: product

text: case.

text: By

text: Lemma

text: we

text: have

text: ζ∗(c-ind(φ

text: D

text: φ

text: D

text: ζ∗(Γ(c-ind(φ

text: D

text: φ

text: D

text: By

text: Lemma

text: the

text: value

text: equals

text: ζ∗(c-ind(φ

text: D

text: φ

text: D

text: Since

text: N

text: and

text: N

text: are

text: closed

text: and

text: we

text: can

text: take

text: a

text: unitary

text: U

text: satisﬁes

text: UΠ

text: Π

text: U,

text: we

text: have

text: ζ∗(c-ind(φ

text: D

text: φ

text: D

text: index

text: u

text: φ2

text: index

text: Π

text: u

text: Π

text: index

text: Π

text: u

text: Π

text: φ1

text: φ2

text: By

text: an

text: index

text: theorem

text: for

text: on

text: Theorem

text: the

text: value

text: equals

text: index

text: T

text: index

text: T

text: This

text: is

text: nothing

text: but

text: the

text: relative

text: topological

text: Toeplitz

text: index,

text: so

text: that

text: the

text: proof

text: is

text: completed.

text: References

text: A.

text: Connes,

text: Noncommutative

text: diﬀerential

text: geometry,

text: Inst.

text: Hautes

text: E´tudes

text: Sci.

text: Publ.

text: Math.

text: no.

text: MR

text: M.

text: Gromov

text: and

text: H.

text: Blaine

text: Lawson,

text: Jr.,

text: Positive

text: scalar

text: curvature

text: and

text: the

text: Dirac

text: operator

text: on

text: complete

text: Riemannian

text: manifolds,

text: Inst.

text: Hautes

text: E´tudes

text: Sci.

text: Publ.

text: Math.

text: no.

text: MR

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 16 TATSUKI SETO 3. N. Higson, A note on the cobordism invariance of the index, Topology 30 (1991), no. 3, 439–443. MR 1113688 (92f:58171) 4. M. Karami, A.H.S. Sadegh, and M.E. Zadeh, Relative-partitioned index theorem, 2014, arXiv:1411.6090. 5. J. Roe, Partitioning noncompact manifolds and the dual Toeplitz problem, Operator algebras and applications, Vol. 1, London Math. Soc. Lecture Note Ser., vol. 135, Cambridge Univ. Press, Cambridge, 1988, pp. 187–228. MR 996446 6. , Index theory, coarse geometry, and topology of manifolds, Providence, RI: American Mathematical Society, 1996. MR 1399087 7. , Positive curvature, partial vanishing theorems and coarse indices, Proc. Edinb. Math. Soc. (2) 59 (2016), no. 1, 223–233. MR 3439130 8. T. Seto, Toeplitz operators and the Roe-Higson type index theorem, 2014, arXiv:1405.4852 (to appear in J. of Noncommut. Geom.). 9. , Toeplitz operators and the Roe-Higson type index theorem in Riemannian surfaces, Tokyo J. Math. 39 (2016), no. 2, 423–439. MR 3599501 10. P. Siegel, Homological calculations with the analytic structure group, ProQuest LLC, Ann Arbor, MI, 2012, Thesis (Ph.D.)–The Pennsylvania State University. MR 3153339 Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya, Japan Email address: m11034y@math.nagoya-u.ac.jp ##LTLine## ##LTLine## ##LTLine##

text: TATSUKI

text: SETO

label: reference

text: N.

text: Higson,

text: A

text: note

text: on

text: the

text: cobordism

text: invariance

text: of

text: the

text: index,

text: Topology

text: no.

text: MR

text: M.

text: Karami,

text: A.H.S.

text: Sadegh,

text: and

text: M.E.

text: Zadeh,

text: Relative-partitioned

text: index

text: theorem,

text: J.

text: Roe,

text: Partitioning

text: noncompact

text: manifolds

text: and

text: the

text: dual

text: Toeplitz

text: problem,

text: Operator

text: algebras

text: and

text: applications,

text: Vol.

text: London

text: Math.

text: Soc.

text: Lecture

text: Note

text: Ser.,

text: vol.

text: Cambridge

text: Univ.

text: Press,

text: Cambridge,

text: pp.

text: MR

text: Index

text: theory,

text: coarse

text: geometry,

text: and

text: topology

text: of

text: manifolds,

text: Providence,

text: RI:

text: American

text: Mathematical

text: Society,

text: MR

text: Positive

text: curvature,

text: partial

text: vanishing

text: theorems

text: and

text: coarse

text: indices,

text: Proc.

text: Edinb.

text: Math.

text: Soc.

text: no.

text: MR

text: T.

text: Seto,

text: Toeplitz

text: operators

text: and

text: the

text: Roe-Higson

text: type

text: index

text: theorem,

text: (to

text: appear

text: in

text: J.

text: of

text: Noncommut.

text: Geom.).

text: Toeplitz

text: operators

text: and

text: the

text: Roe-Higson

text: type

text: index

text: theorem

text: in

text: Riemannian

text: surfaces,

text: Tokyo

text: J.

text: Math.

text: no.

text: MR

text: P.

text: Siegel,

text: Homological

text: calculations

text: with

text: the

text: analytic

text: structure

text: group,

text: ProQuest

text: LLC,

text: Ann

text: Arbor,

text: MI,

text: Thesis

text: (Ph.D.)–The

text: Pennsylvania

text: State

text: University.

text: MR

text: Graduate

text: School

text: of

text: Mathematics,

text: Nagoya

text: University,

text: Furocho,

text: Chikusaku,

text: Nagoya,

text: Japan

text: Email

text: address:

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 8 TATSUKI SETO Deﬁnition 2.5.(cid:80)[1, p.109] Deﬁne the map ∗ ⊂ → C ζ∗ : K (C (W M)) 1 − by ζ∗([u]) = 1 ζ((u 1) , u ), where we assume [u] is represented by an el- ∈ 8πi i,j ji ij ement u GL (A ) and u is the (i, j)-component of u. We note that this is l W ij Connes’ pairing of cyclic cohomology with K-theory, and 1/8πi is a constant mul- tiple in Connes’ pairing. By standard calculation implies the following; see, for instance, [8, Proposition 1.13]. ∈ ∗ ⊂ Proposition 2.6. For any u GL (C (W M)), one has l − 1 → ζ∗([u]) = index(ΠuΠ : Π(L2(M, S))l Π(L2(M, S))l). 8πi Remark 2.7. By an isomorphism [10, Proposition 4.3.12] ∗ ∼ ∗ ⊂ K∗(C (W)) = K∗(C (W M)) → ∈ ∗ ⊂ induced by the inclusion W M, any x K (C (W M)) can be represented ∈ ∗ ⊂ 1 − by u GL (C (W M)) such that an operator u 1 is supported in N (W) for l r arbitrary small r > 0. Denote by p the characteristic function of N (W). Under r r the notations, we have − 1 → ζ∗(x) = index(Πp up Π : p Π(L2(M, S))l p Π(L2(M, S))l). 8πi r r r r Let (M, W) be a pair of a complete Riemannian manifold M and a closed subset ⊂ → W M and S M a Hermitian vector bundle. In this subsection, we prove properties of the map ζ∗ which we use. Firstly, we prove “cobordism” invariance of ζ∗. (cid:48) (cid:48) Lemma 2.8. Let N and N be two partitions of (M, W). Denote by ζ and ζ (cid:48) the cyclic cocycle introduced in Deﬁnition 2.3 by using the partitions N and N , (cid:48) ⊂ respectively. Assume that N N (N) for some r > 0. Then we have r (cid:48) ∗ ⊂ → C ζ∗ = ζ∗ : K (C (W M)) . 1 (cid:48) (cid:48) − (cid:48) Proof. Denote by Π the characteristic function of M+ and set φ = Π Π . Take ∈ ∗ ⊂ ∈ ∗ ⊂ any x K (C (W M)). x can be represented by u = v + 1 GL (C (W M)) 1 ∈ ∗ ⊂ l such that v M (C (W M)) is supported near W as in Remark 2.7. Then we l have − 1 − 1 ζ∗(x) = index(ΠuΠ on Π(L2(M, S))l) = index(Πv + 1) 8πi 8πi and (cid:48) − 1 (cid:48) (cid:48) (cid:48) − 1 (cid:48) ζ∗(x) = index(Π uΠ on Π (L2(M, S))l) = index(Π v + 1) 8πi 8πi (cid:48) ⊂ since operators [Π, v] and [Π , v] are compact. By the way, properties Supp(φ) − N (N), v is locally compact and v is supported near W imply an operator (Πv+1) r ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: TATSUKI

text: SETO

text: Deﬁnition

text: Deﬁne

text: the

text: map

text: C

text: ζ∗

text: K

text: (C

text: (W

text: M))

text: by

text: ζ∗([u])

text: ζ((u

text: u

text: where

text: we

text: assume

text: [u]

text: is

text: represented

text: by

text: an

text: el-

text: i,j

text: ji

text: ij

text: ement

text: u

text: GL

text: (A

text: and

text: u

text: is

text: the

text: (i,

text: j)-component

text: of

text: u.

text: We

text: note

text: that

text: this

text: is

text: l

text: W

text: ij

text: Connes’

text: pairing

text: of

text: cyclic

text: cohomology

text: with

text: K-theory,

text: and

text: is

text: a

text: constant

text: mul-

text: tiple

text: in

text: Connes’

text: pairing.

text: By

text: standard

text: calculation

text: implies

text: the

text: following;

text: see,

text: for

text: instance,

text: Proposition

text: Proposition

text: For

text: any

text: u

text: GL

text: (C

text: (W

text: M)),

text: one

text: has

text: l

text: ζ∗([u])

text: index(ΠuΠ

text: S))l

text: S))l).

text: Remark

text: By

text: an

text: isomorphism

text: Proposition

text: K∗(C

text: (W))

text: K∗(C

text: (W

text: M))

text: induced

text: by

text: the

text: inclusion

text: W

text: M,

text: any

text: x

text: K

text: (C

text: (W

text: M))

text: can

text: be

text: represented

text: by

text: u

text: GL

text: (C

text: (W

text: M))

text: such

text: that

text: an

text: operator

text: u

text: is

text: supported

text: in

text: N

text: (W)

text: for

text: l

text: r

text: arbitrary

text: small

text: r

text: Denote

text: by

text: p

text: the

text: characteristic

text: function

text: of

text: N

text: (W).

text: Under

text: r

text: r

text: the

text: notations,

text: we

text: have

text: ζ∗(x)

text: index(Πp

text: up

text: Π

text: p

text: S))l

text: p

text: S))l).

text: r

text: r

text: r

text: r

text: Let

text: (M,

text: W)

text: be

text: a

text: pair

text: of

text: a

text: complete

text: Riemannian

text: manifold

text: M

text: and

text: a

text: closed

text: subset

text: W

text: M

text: and

text: S

text: M

text: a

text: Hermitian

text: vector

text: bundle.

text: In

text: this

text: subsection,

text: we

text: prove

text: properties

text: of

text: the

text: map

text: ζ∗

text: which

text: we

text: use.

text: Firstly,

text: we

text: prove

text: “cobordism”

text: invariance

text: of

text: ζ∗.

text: Lemma

text: Let

text: N

text: and

text: N

text: be

text: two

text: partitions

text: of

text: (M,

text: W).

text: Denote

text: by

text: ζ

text: and

text: ζ

text: the

text: cyclic

text: cocycle

text: introduced

text: in

text: Deﬁnition

text: by

text: using

text: the

text: partitions

text: N

text: and

text: N

text: respectively.

text: Assume

text: that

text: N

text: N

text: (N)

text: for

text: some

text: r

text: Then

text: we

text: have

text: r

text: C

text: ζ∗

text: ζ∗

text: K

text: (C

text: (W

text: M))

text: Proof.

text: Denote

text: by

text: Π

text: the

text: characteristic

text: function

text: of

text: M+

text: and

text: set

text: φ

text: Π

text: Π

text: Take

text: any

text: x

text: K

text: (C

text: (W

text: M)).

text: x

text: can

text: be

text: represented

text: by

text: u

text: v

text: GL

text: (C

text: (W

text: M))

text: l

text: such

text: that

text: v

text: M

text: (C

text: (W

text: M))

text: is

text: supported

text: near

text: W

text: as

text: in

text: Remark

text: Then

text: we

text: l

text: have

text: ζ∗(x)

text: index(ΠuΠ

text: on

text: S))l)

text: index(Πv

text: and

text: ζ∗(x)

text: index(Π

text: uΠ

text: on

text: Π

text: S))l)

text: index(Π

text: v

text: since

text: operators

text: [Π,

text: v]

text: and

text: [Π

text: v]

text: are

text: compact.

text: By

text: the

text: way,

text: properties

text: Supp(φ)

text: N

text: (N),

text: v

text: is

text: locally

text: compact

text: and

text: v

text: is

text: supported

text: near

text: W

text: imply

text: an

text: operator

text: r

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: CYCLIC COCYCLE AND RELATIVE-PARTITIONED INDEX THEOREM 9 (cid:48) (cid:48) ∈ ∗ ⊂ (Π v + 1) = φv is compact. Thus we have ζ∗(x) = ζ∗(x) for any x K (C (W 1 (cid:3) M)). Secondly, we shall prove an analogue of Higson’s Lemma [3, Lemma 3.1]. (cid:48) (cid:48) Lemma 2.9. Let (M, W) and (M , W ) be two pairs which is partitioned by N (cid:48) → (cid:48) → (cid:48) and N , respectively, and S M and S M two Hermitian vector bundles. (cid:48) (cid:48) Let Π and Π be the characteristic function of M+ and M +, respectively. We (cid:48) → assume that there exists an isometry γ : M + M+ which lifts an isomorphism ∗ | → (cid:48)| ∗ γ : S S (cid:48) . We denote the Hilbert space isometry deﬁned by γ by the M+ ∗ M + → (cid:48) (cid:48) (cid:48) ∈ ∗ ⊂ same letter γ : Π(L2(M, S)) Π (L2(M , S )). Take u GL (C (W M)) and (cid:48) ∈ ∗ (cid:48) ⊂ (cid:48) ∗ ∼ (cid:48) (cid:48) ∗ l (cid:48) (cid:48) u GL (C (W M )) such that γ uΠ Π u γ . Then one has ζ∗([u]) = ζ∗([u ]). l (cid:48)− → − Similarly, if there exists an isometry γ : M M which lifts an isomorphism ∗ (cid:48)| → | ∗ ∼ (cid:48) (cid:48) ∗ (cid:48) (cid:48) γ : S (cid:48)− S − and γ uΠ Π u γ , then one has ζ∗([u]) = ζ∗([u ]). M M − → Proof. It suﬃces to show the case when l = 1. Let v : (1 Π)(L2(M, S)) − (cid:48) (cid:48) (cid:48) ∗ − (1 Π )(L2(M , S )) be any invertible operator. Then V = γ Π + v(1 Π) : → (cid:48) (cid:48) L2(M, S) L2(M , S ) is also invertible. Hence we obtain − − − (cid:48) (cid:48) (cid:48) (cid:48) V ((1 Π) + ΠuΠ) ((1 Π ) + Π u Π )V ∗ − (cid:48) (cid:48) (cid:48) ∗ ∼ ∗ − (cid:48) ∗ ∼ = γ ΠuΠ Π u Π γ γ uΠ Π uγ 0. (cid:48) (cid:48) Therefore, we obtain ζ∗([u]) = ζ∗([u ]) since V is an invertible operator and one has − − − (cid:48) (cid:48) − 8πiζ∗([u]) = index(ΠuΠ) = index((1 Π) + ΠuΠ) and 8πiζ∗([u ]) = index((1 (cid:48) (cid:48) (cid:48) (cid:48) (cid:3) Π ) + Π u Π ). (cid:48) (cid:48) (cid:48) Let (M, W) and (M , W ) be two pairs which is partitioned by N and N , re- → (cid:48) → (cid:48) spectively, and S M and S M two Hermitian vector bundles. Let Π (cid:48) (cid:48) and Π be the characteristic function of M+ and M +, respectively. We assume (cid:48) → that there exists an isometry γ : N (W ) N (W) which lifts an isomorphism ∗ | → | r r ∗ ⊂ → γ : S S (cid:48) . We deﬁne an additive map Γ : K (C (W M)) ∗1 Nr(cid:48)(W⊂) (cid:48) 2 Nr(W ) ∈ ∗ ⊂ 1 K (C (W M )) as follows. Take any x K (C (W M)). x can be rep- 1 ∈ ∗ ⊂ 1 ∈ ∗ ⊂ resented by u = v + 1 GL (C (W M)) such that v M (C (W M)) is l ∗ ∗ − l ∗ ∗ − ∈ supported in N (W); see Remark 2.7. Then we have γ u(γ ) 1 = γ v(γ ) 1 + 1 ∗ (cid:48) ⊂ r (cid:48) ∗ ∗ − ∈ ∗ (cid:48) ⊂ (cid:48) (cid:48) GL (C (W M )), here γ v(γ ) 1 M (C (W M )) is supported in N (W ). l ∗ ∗ − l r The K -class of γ u(γ ) 1 does not depend on the choice of such an above u. Set 1 ∗ ∗ − ∈ ∗ (cid:48) ⊂ (cid:48) Γ(x) = [γ u(γ ) 1] K (C (W M )). 1 (cid:48) → Lemma 2.10. Moreover, we assume that an isometry γ : N (W ) N (W) (cid:48) ◦ (cid:48)r r preserves partitions, that is, γ satisﬁes Π = Π γ on N (W ). Then we have (cid:48) ◦ ∗ ⊂ r ζ∗ Γ = ζ∗ on K (C (W M)). 1 (cid:48) (cid:48) Proof. Let p and p be the characteristic function of N (W) and N (W ), re- r r ∈ ∗ ⊂ r r ∈ spectively. Take any x K (C (W M)). We represent x by u = v + 1 ∗ ⊂ 1 (cid:0)∈ ∗ ⊂ (cid:1) GL (C (W M)) such that v M (C (W M)) is supported in N (W). We l l r have (cid:48) ◦ − (cid:48) (cid:48) (cid:48) (cid:48) ∗ ∗ − (cid:48) (cid:48) ζ∗ Γ(x) = index (1 p Π ) + Π p γ u(γ ) 1p Π . r r r

text: CYCLIC

text: COCYCLE

text: AND

text: RELATIVE-PARTITIONED

text: INDEX

text: THEOREM

text: (Π

text: v

text: φv

text: is

text: compact.

text: Thus

text: we

text: have

text: ζ∗(x)

text: ζ∗(x)

text: for

text: any

text: x

text: K

text: (C

text: (W

text: M)).

text: Secondly,

text: we

text: shall

text: prove

text: an

text: analogue

text: of

text: Higson’s

text: Lemma

text: Lemma

text: Lemma

text: Let

text: (M,

text: W)

text: and

text: (M

text: W

text: be

text: two

text: pairs

text: which

text: is

text: partitioned

text: by

text: N

text: and

text: N

text: respectively,

text: and

text: S

text: M

text: and

text: S

text: M

text: two

text: Hermitian

text: vector

text: bundles.

text: Let

text: Π

text: and

text: Π

text: be

text: the

text: characteristic

text: function

text: of

text: M+

text: and

text: M

text: respectively.

text: We

text: assume

text: that

text: there

text: exists

text: an

text: isometry

text: γ

text: M

text: M+

text: which

text: lifts

text: an

text: isomorphism

text: γ

text: S

text: S

text: We

text: denote

text: the

text: Hilbert

text: space

text: isometry

text: deﬁned

text: by

text: γ

text: by

text: the

text: M+

text: M

text: same

text: letter

text: γ

text: S))

text: Π

text: S

text: Take

text: u

text: GL

text: (C

text: (W

text: M))

text: and

text: l

text: u

text: GL

text: (C

text: (W

text: M

text: such

text: that

text: γ

text: uΠ

text: Π

text: u

text: γ

text: Then

text: one

text: has

text: ζ∗([u])

text: ζ∗([u

text: l

text: Similarly,

text: if

text: there

text: exists

text: an

text: isometry

text: γ

text: M

text: M

text: which

text: lifts

text: an

text: isomorphism

text: γ

text: S

text: S

text: and

text: γ

text: uΠ

text: Π

text: u

text: γ

text: then

text: one

text: has

text: ζ∗([u])

text: ζ∗([u

text: M

text: M

text: Proof.

text: It

text: suﬃces

text: to

text: show

text: the

text: case

text: when

text: l

text: Let

text: v

text: S))

text: Π

text: S

text: be

text: any

text: invertible

text: operator.

text: Then

text: V

text: γ

text: Π

text: Π)

text: S)

text: S

text: is

text: also

text: invertible.

text: Hence

text: we

text: obtain

text: V

text: Π)

text: ΠuΠ)

text: Π

text: Π

text: u

text: Π

text: )V

text: γ

text: ΠuΠ

text: Π

text: u

text: Π

text: γ

text: γ

text: uΠ

text: Π

text: uγ

text: Therefore,

text: we

text: obtain

text: ζ∗([u])

text: ζ∗([u

text: since

text: V

text: is

text: an

text: invertible

text: operator

text: and

text: one

text: has

text: index(ΠuΠ)

text: Π)

text: ΠuΠ)

text: and

text: Π

text: Π

text: u

text: Π

text: Let

text: (M,

text: W)

text: and

text: (M

text: W

text: be

text: two

text: pairs

text: which

text: is

text: partitioned

text: by

text: N

text: and

text: N

text: re-

text: spectively,

text: and

text: S

text: M

text: and

text: S

text: M

text: two

text: Hermitian

text: vector

text: bundles.

text: Let

text: Π

text: and

text: Π

text: be

text: the

text: characteristic

text: function

text: of

text: M+

text: and

text: M

text: respectively.

text: We

text: assume

text: that

text: there

text: exists

text: an

text: isometry

text: γ

text: N

text: (W

text: N

text: (W)

text: which

text: lifts

text: an

text: isomorphism

text: r

text: r

text: γ

text: S

text: S

text: We

text: deﬁne

text: an

text: additive

text: map

text: Γ

text: K

text: (C

text: (W

text: M))

text: ∗1

text: Nr(W

text: K

text: (C

text: (W

text: M

text: as

text: follows.

text: Take

text: any

text: x

text: K

text: (C

text: (W

text: M)).

text: x

text: can

text: be

text: rep-

text: resented

text: by

text: u

text: v

text: GL

text: (C

text: (W

text: M))

text: such

text: that

text: v

text: M

text: (C

text: (W

text: M))

text: is

text: l

text: l

text: supported

text: in

text: N

text: (W);

text: see

text: Remark

text: Then

text: we

text: have

text: γ

text: u(γ

text: γ

text: v(γ

text: r

text: GL

text: (C

text: (W

text: M

text: here

text: γ

text: v(γ

text: M

text: (C

text: (W

text: M

text: is

text: supported

text: in

text: N

text: (W

text: l

text: l

text: r

text: The

text: K

text: -class

text: of

text: γ

text: u(γ

text: does

text: not

text: depend

text: on

text: the

text: choice

text: of

text: such

text: an

text: above

text: u.

text: Set

text: Γ(x)

text: [γ

text: u(γ

text: K

text: (C

text: (W

text: M

text: Lemma

text: Moreover,

text: we

text: assume

text: that

text: an

text: isometry

text: γ

text: N

text: (W

text: N

text: (W)

text: r

text: preserves

text: partitions,

text: that

text: is,

text: γ

text: satisﬁes

text: Π

text: Π

text: γ

text: on

text: N

text: (W

text: Then

text: we

text: have

text: r

text: ζ∗

text: Γ

text: ζ∗

text: on

text: K

text: (C

text: (W

text: M)).

text: Proof.

text: Let

text: p

text: and

text: p

text: be

text: the

text: characteristic

text: function

text: of

text: N

text: (W)

text: and

text: N

text: (W

text: re-

text: r

text: r

text: r

text: r

text: spectively.

text: Take

text: any

text: x

text: K

text: (C

text: (W

text: M)).

text: We

text: represent

text: x

text: by

text: u

text: v

text: GL

text: (C

text: (W

text: M))

text: such

text: that

text: v

text: M

text: (C

text: (W

text: M))

text: is

text: supported

text: in

text: N

text: (W).

text: We

text: l

text: l

text: r

text: have

text: ζ∗

text: Γ(x)

text: index

text: p

text: Π

text: Π

text: p

text: γ

text: u(γ

text: Π

text: r

text: r

text: r


text: Nuclear Physics A Nuclear Physics A 00 (2020) 1–4 www.elsevier.com/locate/procedia Proton structure ﬂuctuations: constraints from HERA and applications to p + A collisions Heikki Ma¨ntysaari, Bjo¨rn Schenke, Chun Shen, Prithwish Tribedy Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA Abstract We constrain proton structure ﬂuctuations at small-x by comparing with the HERA diﬀractive vector meson production data, and ﬁnd that large geometric ﬂuctuations in the proton wave function are needed. Hydrodynamical simulations of proton-nucleus collisions including the ﬂuctuating proton structure are found to be compatible with the momentum anisotropies measured at the LHC. Keywords: diﬀraction, ﬂuctuations, CGC, Color Glass Condensate, proton-nucleus collisions 1. Introduction One of the striking ﬁndings of the proton-nucleus collisions at the LHC has been the presence of strong collective phenomena in high multiplicity events. These signals, particularly long range rapidity correlations with distinct anisotropies in the azimuthal direction, are very similar to those that have been traditionally interpreted as evidence for the creation of an almost perfect ﬂuid in ultrarelativistic heavy ion collisions (for a review of recent collectivity measurements, see Ref. [1]). This has raised a natural question whether the high-multiplicity proton-nucleus, and even proton-proton collisions, can be described using the same relativistic hydrodynamic framework that has been successful in describing a vast variety of LHC and RHIC heavy ion data (see e.g. Ref [2]). In hydrodynamical simulations the initial state geometric anisotropies are converted into momentum space correlations and are observed e.g. as an elliptic (v ) and triangular ﬂow (v ). Thus, in order to 2 3 model proton-nucleus collisions using relativistic hydrodynamics, a detailed knowledge of the initial state geometry is crucial. In particular, the event-by-event ﬂuctuations of the (on average round) proton are needed to model the initial state. Without large eccentricities in the initial state it is diﬃcult to explain the large ﬂow harmonics observed at the LHC [3]. One possibility to constrain the proton structure ﬂuctuations is given by diﬀractive deep inelastic scat- tering measured at the HERA electron-proton collider. Unlike other DIS observables, diﬀractive scattering processes are sensitive to the transverse geometry of the target. The actual process we have in mind here is exclusive vector meson production (in this work J/Ψ). These processes can be divided into two cate- gories. In coherent diﬀraction, where the target hadron remains in the same quantum state, the transverse momentum spectra of the produced vector mesons are directly related to the average density proﬁle of the ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Nuclear

font: MEUDAV+NimbusSanL-Regu

text: Physics

text: A

text: Nuclear

text: Physics

text: A

text: Proton

text: structure

text: ﬂuctuations:

text: constraints

text: from

text: HERA

text: and

text: applications

text: to

text: p

font: JMJXTF+rtxr

text: A

text: collisions

text: Heikki

text: Ma¨ntysaari,

text: Bjo¨rn

text: Schenke,

text: Chun

text: Shen,

text: Prithwish

text: Tribedy

text: Physics

text: Department,

text: Brookhaven

text: National

text: Laboratory,

text: Upton,

text: NY

text: USA

text: Abstract

text: We

text: constrain

text: proton

text: structure

text: ﬂuctuations

text: at

text: small-x

text: by

text: comparing

text: with

text: the

text: HERA

text: diﬀractive

text: vector

text: meson

text: production

text: data,

text: and

text: ﬁnd

text: that

text: large

text: geometric

text: ﬂuctuations

text: in

text: the

text: proton

text: wave

text: function

text: are

text: needed.

text: Hydrodynamical

text: simulations

text: of

text: proton-nucleus

text: collisions

text: including

text: the

text: ﬂuctuating

text: proton

text: structure

text: are

text: found

text: to

text: be

text: compatible

text: with

text: the

text: momentum

text: anisotropies

text: measured

text: at

text: the

text: LHC.

text: Keywords:

text: diﬀraction,

text: ﬂuctuations,

text: CGC,

text: Color

text: Glass

text: Condensate,

text: proton-nucleus

text: collisions

text: Introduction

text: One

text: of

text: the

text: striking

text: ﬁndings

text: of

text: the

text: proton-nucleus

text: collisions

text: at

text: the

text: LHC

text: has

text: been

text: the

text: presence

text: of

text: strong

text: collective

text: phenomena

text: in

text: high

text: multiplicity

text: events.

text: These

text: signals,

text: particularly

text: long

text: range

text: rapidity

text: correlations

text: with

text: distinct

text: anisotropies

text: in

text: the

text: azimuthal

text: direction,

text: are

text: very

text: similar

text: to

text: those

text: that

text: have

text: been

text: traditionally

text: interpreted

text: as

text: evidence

text: for

text: the

text: creation

text: of

text: an

text: almost

text: perfect

text: ﬂuid

text: in

text: ultrarelativistic

text: heavy

text: ion

text: collisions

text: (for

text: a

text: review

text: of

text: recent

text: collectivity

text: measurements,

text: see

text: Ref.

text: This

text: has

text: raised

text: a

text: natural

text: question

text: whether

text: the

text: high-multiplicity

text: proton-nucleus,

text: and

text: even

text: proton-proton

text: collisions,

text: can

text: be

text: described

text: using

text: the

text: same

text: relativistic

text: hydrodynamic

text: framework

text: that

text: has

text: been

text: successful

text: in

text: describing

text: a

text: vast

text: variety

text: of

text: LHC

text: and

text: RHIC

text: heavy

text: ion

text: data

text: (see

text: e.g.

text: Ref

text: In

text: hydrodynamical

text: simulations

text: the

text: initial

text: state

text: geometric

text: anisotropies

text: are

text: converted

text: into

text: momentum

text: space

text: correlations

text: and

text: are

text: observed

text: e.g.

text: as

text: an

text: elliptic

text: (v

text: and

text: triangular

text: ﬂow

text: (v

text: Thus,

text: in

text: order

text: to

text: model

text: proton-nucleus

text: collisions

text: using

text: relativistic

text: hydrodynamics,

text: a

text: detailed

text: knowledge

text: of

text: the

text: initial

text: state

text: geometry

text: is

text: crucial.

text: In

text: particular,

text: the

text: event-by-event

text: ﬂuctuations

text: of

text: the

text: (on

text: average

text: round)

text: proton

text: are

text: needed

text: to

text: model

text: the

text: initial

text: state.

text: Without

text: large

text: eccentricities

text: in

text: the

text: initial

text: state

text: it

text: is

text: diﬃcult

text: to

text: explain

text: the

text: large

text: ﬂow

text: harmonics

text: observed

text: at

text: the

text: LHC

text: One

text: possibility

text: to

text: constrain

text: the

text: proton

text: structure

text: ﬂuctuations

text: is

text: given

text: by

text: diﬀractive

text: deep

text: inelastic

text: scat-

text: tering

text: measured

text: at

text: the

text: HERA

text: electron-proton

text: collider.

text: Unlike

text: other

text: DIS

text: observables,

text: diﬀractive

text: scattering

text: processes

text: are

text: sensitive

text: to

text: the

text: transverse

text: geometry

text: of

text: the

text: target.

text: The

text: actual

text: process

text: we

text: have

text: in

text: mind

text: here

text: is

text: exclusive

text: vector

text: meson

text: production

text: (in

text: this

text: work

text: J/Ψ).

text: These

text: processes

text: can

text: be

text: divided

text: into

text: two

text: cate-

text: gories.

text: In

text: coherent

text: diﬀraction,

text: where

text: the

text: target

text: hadron

text: remains

text: in

text: the

text: same

text: quantum

text: state,

text: the

text: transverse

text: momentum

text: spectra

text: of

text: the

text: produced

text: vector

text: mesons

text: are

text: directly

text: related

text: to

text: the

text: average

text: density

text: proﬁle

text: of

text: the

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 3 Fig. 2. Simulated post-FEC BER as a function of (a) pre-FEC BER BERpre, (b) normalized AIR In, (c) SMI Is, and (d) ASI Ia. each dimension. To generate these distributions, a constant- to increase at lower code rates Rc. According to Fig. 2(a), composite distribution matcher was employed [11] with output BER works surprisingly well for these signal sets and will pre block length Ns = 1024.2 The probabilities used are the provide guidelines for real system performance. Fig. 2(b) same as in [13, Table I (a)–(c)], and some measures of their shows that the normalized AIR In is not a good metric nonuniformity are indicated in Table I. B1 is the sign bit, for nonuniform signaling. The required In for nonuniform carrying the uniform information bits of the PAM symbol signaling is clearly less than that for uniform signaling to · · · − or FEC parity bits, and B2, Bm are amplitude bits. The achieve the same BERpost value such as 10 4. Fig. 2(c) number of input bits of bit tributary Bi to the distribution indicates that the SMI Is is better correlated with BERpost for matcher per block is denoted by NBi. As benchmarks, uniform nonuniform signaling than In. This is a beneﬁt of changing the signaling using 16-QAM and 64-QAM was also simulated. As treatment from parallel bit channels into a single bit sequence, SD-FEC codes, we utilized the DVB-S2 binary low-density which is what the FEC encoder and decoder will see. Fig. 2(d) parity check codes [18], having a codeword length of 64800. shows that the ASI performs better than the other metrics by The bit mapper was (3,2,1) for 8-PAM, which is optimum using the asymmetric distribution of La. [9, Tab. V], and (2,1) for 4-PAM. The examined code rates Intuitively, the lower accuracy of In can be understood Rc were 2/3, 3/4, 5/6, and 9/10. The results were averaged because (7) does not properly take FEC coding in the PAS over 500 codewords in each case, and a couple of independent scheme into account. The maximum FEC code rate for error- − − simulations were performed to verify that 500 codewords is free transmission is Rc,max = 1 (H(B) RBMD)/−m, w−hich is indeed sufﬁcient for reliable statistics. The optical channel the code rate at which the information rate H(B) (1 Rc)m was assumed to be Gaussian with the signal-to-noise ratio equals the AIR RBMD. In tends (cid:80)to be lower than Rc,max and varied with 0.1 dB granularity. The FEC decoder’s internal the difference becomes larger for a strongly shaped signal. (cid:80) − − calculations used ﬂoating-point precision with 20 decoder The metric NGMI = 1 (H(B) GMI )/m [15], where − | − iterations. GMI can be deﬁned as H(B) m H(B Y ) = H(B) | i=1 i The metrics In, Is, and Ia were estimated from (7), (8), mi=1 H(Bi Li),3 is identical to Rc,max in the regime where and (11), resp. The corresponding differential entropies were RBMD > 0. On the other hand, one can show that Ia = NGMI calculated as in [19, Sec. 8.3] from discretized versions of the by considering [16, eqs. (3.67), (4.81)] and [9, Sec. VI-B]. LLRs Li, L, and La, whose distributions were estimated using Note that (4) represents the estimates of NGMI and Ia by histograms having 25 levels with optimized steps. The esti- Monte-Carlo integration for nonuniform signaling. − mated error introduced by this discretization is only < 10 3 F−ig. 3 summarizes the prediction accuracy at BERpost =− in each information quantity and hence negligible. 10 4. Fig. 3(a) compares the information quantities 1 The results are shown in Fig. 2, where it is desirable that BERpre, In, Is, and Ia. Relatively large deviations are mainly all curves for the same code rate should be close to each seen for nonuniform signaling (especially case (i), which is other, if the quantity on the horizontal axis is to be useful the most nonuniform) with the AIR metric. When considering to predict the post-FEC BER. For most of the metrics, the all signals at the same BER, the maximum metric variations difference due to modulation format and nonuniformity tends 3 H(B) minus the conditional entropy due to suboptimal reception is a 2A smaller Ns, such as Ns = 12 in [14], gives simple implementation but natural extension of the GMI [8, eq. (21)] to nonuniform signaling. Several reduces the rate. The rate loss is limited to < 0.55% in the cases tested here. kinds of rates in PAS are discussed in [10], too. ##LTFigure## ##LTFigure##

text: Fig.

text: Simulated

text: post-FEC

text: BER

text: as

text: a

text: function

text: of

text: (a)

text: pre-FEC

text: BER

text: BERpre,

text: (b)

text: normalized

text: AIR

text: In,

text: (c)

text: SMI

text: Is,

text: and

text: (d)

text: ASI

text: Ia.

text: each

text: dimension.

text: To

text: generate

text: these

text: distributions,

text: a

text: constant-

text: to

text: increase

text: at

text: lower

text: code

text: rates

text: Rc.

text: According

text: to

text: Fig.

text: composite

text: distribution

text: matcher

text: was

text: employed

text: with

text: output

text: BER

text: works

text: surprisingly

text: well

text: for

text: these

text: signal

text: sets

text: and

text: will

text: pre

text: block

text: length

text: Ns

text: The

text: probabilities

text: used

text: are

text: the

text: provide

text: guidelines

text: for

text: real

text: system

text: performance.

text: Fig.

text: same

text: as

text: in

text: Table

text: I

text: (a)–(c)],

text: and

text: some

text: measures

text: of

text: their

text: shows

text: that

text: the

text: normalized

text: AIR

text: In

text: is

text: not

text: a

text: good

text: metric

text: nonuniformity

text: are

text: indicated

text: in

text: Table

text: I.

text: is

text: the

text: sign

text: bit,

text: for

text: nonuniform

text: signaling.

text: The

text: required

text: In

text: for

text: nonuniform

text: carrying

text: the

text: uniform

text: information

text: bits

text: of

text: the

text: PAM

text: symbol

text: signaling

text: is

text: clearly

text: less

text: than

text: that

text: for

text: uniform

text: signaling

text: to

text: or

text: FEC

text: parity

text: bits,

text: and

text: Bm

text: are

text: amplitude

text: bits.

text: The

text: achieve

text: the

text: same

text: BERpost

text: value

text: such

text: as

text: Fig.

text: number

text: of

text: input

text: bits

text: of

text: bit

text: tributary

text: Bi

text: to

text: the

text: distribution

text: indicates

text: that

text: the

text: SMI

text: Is

text: is

text: better

text: correlated

text: with

text: BERpost

text: for

text: matcher

text: per

text: block

text: is

text: denoted

text: by

text: NBi.

text: As

text: benchmarks,

text: uniform

text: nonuniform

text: signaling

text: than

text: In.

text: This

text: is

text: a

text: beneﬁt

text: of

text: changing

text: the

text: signaling

text: using

text: and

text: was

text: also

text: simulated.

text: As

text: treatment

text: from

text: parallel

text: bit

text: channels

text: into

text: a

text: single

text: bit

text: sequence,

text: SD-FEC

text: codes,

text: we

text: utilized

text: the

text: binary

text: low-density

text: which

text: is

text: what

text: the

text: FEC

text: encoder

text: and

text: decoder

text: will

text: see.

text: Fig.

text: parity

text: check

text: codes

text: having

text: a

text: codeword

text: length

text: of

text: shows

text: that

text: the

text: ASI

text: performs

text: better

text: than

text: the

text: other

text: metrics

text: by

text: The

text: bit

text: mapper

text: was

text: for

text: which

text: is

text: optimum

text: using

text: the

text: asymmetric

text: distribution

text: of

text: La.

text: Tab.

text: V],

text: and

text: for

text: The

text: examined

text: code

text: rates

text: Intuitively,

text: the

text: lower

text: accuracy

text: of

text: In

text: can

text: be

text: understood

text: Rc

text: were

text: and

text: The

text: results

text: were

text: averaged

text: because

text: does

text: not

text: properly

text: take

text: FEC

text: coding

text: in

text: the

text: PAS

text: over

text: codewords

text: in

text: each

text: case,

text: and

text: a

text: couple

text: of

text: independent

text: scheme

text: into

text: account.

text: The

text: maximum

text: FEC

text: code

text: rate

text: for

text: error-

text: simulations

text: were

text: performed

text: to

text: verify

text: that

text: codewords

text: is

text: free

text: transmission

text: is

text: Rc,max

text: (H(B)

text: RBMD)/−m,

text: w−hich

text: is

text: indeed

text: sufﬁcient

text: for

text: reliable

text: statistics.

text: The

text: optical

text: channel

text: the

text: code

text: rate

text: at

text: which

text: the

text: information

text: rate

text: H(B)

text: Rc)m

text: was

text: assumed

text: to

text: be

text: Gaussian

text: with

text: the

text: signal-to-noise

text: ratio

text: equals

text: the

text: AIR

text: RBMD.

text: In

text: tends

text: be

text: lower

text: than

text: Rc,max

text: and

text: varied

text: with

text: dB

text: granularity.

text: The

text: FEC

text: decoder’s

text: internal

text: the

text: difference

text: becomes

text: larger

text: for

text: a

text: strongly

text: shaped

text: signal.

text: calculations

text: used

text: ﬂoating-point

text: precision

text: with

text: decoder

text: The

text: metric

text: NGMI

text: (H(B)

text: GMI

text: )/m

text: where

text: iterations.

text: GMI

text: can

text: be

text: deﬁned

text: as

text: H(B)

text: m

text: H(B

text: Y

text: H(B)

text: i

text: The

text: metrics

text: In,

text: Is,

text: and

text: Ia

text: were

text: estimated

text: from

text: H(Bi

text: is

text: identical

text: to

text: Rc,max

text: in

text: the

text: regime

text: where

text: and

text: resp.

text: The

text: corresponding

text: differential

text: entropies

text: were

text: RBMD

text: On

text: the

text: other

text: hand,

text: one

text: can

text: show

text: that

text: Ia

text: NGMI

text: calculated

text: as

text: in

text: Sec.

text: from

text: discretized

text: versions

text: of

text: the

text: by

text: considering

text: eqs.

text: and

text: Sec.

text: VI-B].

text: LLRs

text: Li,

text: L,

text: and

text: La,

text: whose

text: distributions

text: were

text: estimated

text: using

text: Note

text: that

text: represents

text: the

text: estimates

text: of

text: NGMI

text: and

text: Ia

text: by

text: histograms

text: having

text: levels

text: with

text: optimized

text: steps.

text: The

text: esti-

text: Monte-Carlo

text: integration

text: for

text: nonuniform

text: signaling.

text: mated

text: error

text: introduced

text: by

text: this

text: discretization

text: is

text: only

text: F−ig.

text: summarizes

text: the

text: prediction

text: accuracy

text: at

text: BERpost

text: in

text: each

text: information

text: quantity

text: and

text: hence

text: negligible.

text: Fig.

text: compares

text: the

text: information

text: quantities

text: The

text: results

text: are

text: shown

text: in

text: Fig.

text: where

text: it

text: is

text: desirable

text: that

text: BERpre,

text: In,

text: Is,

text: and

text: Ia.

text: Relatively

text: large

text: deviations

text: are

text: mainly

text: all

text: curves

text: for

text: the

text: same

text: code

text: rate

text: should

text: be

text: close

text: to

text: each

text: seen

text: for

text: nonuniform

text: signaling

text: (especially

text: case

text: (i),

text: which

text: is

text: other,

text: if

text: the

text: quantity

text: on

text: the

text: horizontal

text: axis

text: is

text: to

text: be

text: useful

text: the

text: most

text: nonuniform)

text: with

text: the

text: AIR

text: metric.

text: When

text: considering

text: to

text: predict

text: the

text: post-FEC

text: BER.

text: For

text: most

text: of

text: the

text: metrics,

text: the

text: all

text: signals

text: at

text: the

text: same

text: BER,

text: the

text: maximum

text: metric

text: variations

text: difference

text: due

text: to

text: modulation

text: format

text: and

text: nonuniformity

text: tends

text: H(B)

text: minus

text: the

text: conditional

text: entropy

text: due

text: to

text: suboptimal

text: reception

text: is

text: a

text: smaller

text: Ns,

text: such

text: as

text: Ns

text: in

text: gives

text: simple

text: implementation

text: but

text: natural

text: extension

text: of

text: the

text: GMI

text: eq.

text: to

text: nonuniform

text: signaling.

text: Several

text: reduces

text: the

text: rate.

text: The

text: rate

text: loss

text: is

text: limited

text: to

text: in

text: the

text: cases

text: tested

text: here.

text: kinds

text: of

text: rates

text: in

text: PAS

text: are

text: discussed

text: in

text: too.

text: ##LTFigure##

text: ##LTFigure##


text: To optimize the network, we use Adam solver with β = Method without RN withRN − 1 0.9, β2 = 0.9999, (cid:15) = 10 8 [21]. We set the training- fps 52.19 51.18 batch-size to 64. To initialize the weights in convolution #Param 1.40M 1.40M layers, we followed the method of [12]. We trained our MSE 218.04 213.04 networks with a total of 30 epochs with the ﬁxed learn- MAE 5.77 5.54 − ing rate of 10 4. It takes one day for training in a single Nvidia GTX Titan X GPU with 12GB memory. The afore- Table 1. Performance of DeepView with/without the rendering ren mentioned training conﬁgurations are identically used in all network (RN). experiments unless otherwise mentioned. \ Since there are no publicly available SIVG datasets, we Test Train Scale Scale Scale Scale 4 5 6 4,5,6 use our dataset introduced in Section 4. Our dataset con- MSE 216.18 218.79 220.48 218.79 Scale sists of 27 non-animated stereoscopic movies. We divided 4 MAE 5.67 5.70 5.79 5.70 them into 18 training and 9 testing movies, such that there MSE 217.49 213.04 219.49 217.86 Scale is no overlap between the training and testing datasets. To 5 MAE 5.74 5.54 5.75 5.66 reduce computational complexity for training, we selected MSE 197.20 194.12 193.58 195.13 Scale training/testing frames every 2 seconds among a total of 2M 6 MAE 5.68 5.44 5.34 5.47 frames, resulting in 58K training frames and 22K testing frames. For all the training/testing frames, we performed Table 2. Prediction performance of DeepView for different rec a downscaling process by preserving the frame aspect ratio scale training/testing datasets. and slightly cropped the upper and lower pixel boundaries, such that the spatial resolution of all the training/testing × as described in Section 5.1. However, here we downscaled frames becomes 384 160. the data with different factors of (4, 5, 6). This generates To measure computational efﬁciency, we use memory 3 datasets of the same content but at different scales. We consumption (#Param), i.e., the number of weight and bias use this data to train our DeepView . We performed four values used in convolution and batch-normalization layers. rec trainings. The ﬁrst three train each scale separately while Those parameters should be kept in the memory during the the last trains all scales together at once. The performance entire process. Also, the average running speed in frames of each model is calculated. An architecture is scalable if per second (fps) is measured for all the testing images. We the testing performance of all the all scales model is similar use MSE and mean absolute error (MAE) to measure pre- | − | to the performance of the scale-speciﬁc model. diction accuracy. MAE is calculated as E[ Rˆ R ], where · Table 2 shows the prediction performance of E[ ] is expectation operator over pixels in an image. Note DeepView for different scale training/testing datasets. that, higher fps indicates higher performance, while higher rec As shown in Table 2, the dedicated model for each scale #Param, MSE and MAE mean lower performance. shows the lowest error in correspondingly-sized testing 5.2. Effectiveness of rendering network data. While the performance of the trained model with We verify the effectiveness of the rendering network in all the scales also approximates all the testing data. This DeepView by performing experiments on DeepView indicates that our approach can support spatial scalability in ren ren with and without the rendering network. Table 1 shows the a single network. Fig. 4 shows the qualitative performance performance of DeepView with and without the render- of DeepView for three different spatial resolutions ren × re×c × ing network. As shown in Table 1, the rendering network (320 128, 384 160, and 480 192). In Fig. 4, the upper improves prediction accuracy in terms of MSE and MAE. and bottom rows illustrate the estimated right-view images It also does not introduce noticeable computational com- and their ground-truth, respectively. As shown in Fig. 4, plexity in both fps and #Param. This is because the ren- DeepViewrec is able to estimate right-view images for three dering network explicitly performs pixel-translations of the different spatial resolutions consistently in a single net- left-image and selects the best translation based on the gen- work. Therefore, contrary to the existing method requiring erated disparity maps. dedicated networks depending on different spatial scales, 5.3. Spatial scalability our architectures can effectively be used for practical applications in a single network. Our architectures have spatial scalability, i.e., they can 6. Experiments support multiple input/output sizes in a single network. To verify the spatial scalability, we train DeepView To verify the effectiveness of our architectures, we com- rec for various-sized images. In order to generate the train- pare them against the state of the art Deep3D [32]. Note that ing/testing with different spatial resolutions, we use the we do not compare our approach with the existing depth- same 18 and 9 movies for training and testing, respectively, estimation methods, since they require an additional DIBR 5 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: To

text: optimize

text: the

text: network,

text: we

text: use

text: Adam

text: solver

text: with

text: β

text: Method

label: table

text: without

text: RN

text: withRN

text: β2

text: We

text: set

text: the

text: training-

text: fps

text: batch-size

text: to

text: To

text: initialize

text: the

text: weights

text: in

text: convolution

text: #Param

text: layers,

text: we

text: followed

text: the

text: method

text: of

text: We

text: trained

text: our

text: MSE

text: networks

text: with

text: a

text: total

text: of

text: epochs

text: with

text: the

text: ﬁxed

text: learn-

text: MAE

text: ing

text: rate

text: of

text: It

text: takes

text: one

text: day

text: for

text: training

text: in

text: a

text: single

text: Nvidia

text: GTX

text: Titan

text: X

text: GPU

text: with

text: memory.

text: The

text: afore-

text: Table

text: Performance

text: of

text: DeepView

text: with/without

text: the

text: rendering

text: ren

text: mentioned

text: training

text: conﬁgurations

text: are

text: identically

text: used

text: in

text: all

text: network

text: (RN).

text: experiments

text: unless

text: otherwise

text: mentioned.

text: Since

text: there

text: are

text: no

text: publicly

text: available

text: SIVG

text: datasets,

text: we

text: Test

text: Train

text: Scale

text: Scale

text: Scale

text: Scale

text: use

text: our

text: dataset

text: introduced

text: in

text: Section

text: Our

text: dataset

text: con-

text: MSE

text: Scale

text: sists

text: of

text: non-animated

text: stereoscopic

text: movies.

text: We

text: divided

text: MAE

text: them

text: into

text: training

text: and

text: testing

text: movies,

text: such

text: that

text: there

text: MSE

text: Scale

text: is

text: no

text: overlap

text: between

text: the

text: training

text: and

text: testing

text: datasets.

text: To

text: MAE

text: reduce

text: computational

text: complexity

text: for

text: training,

text: we

text: selected

text: MSE

text: Scale

text: training/testing

text: frames

text: every

text: seconds

text: among

text: a

text: total

text: of

text: MAE

text: frames,

text: resulting

text: in

text: training

text: frames

text: and

text: testing

text: frames.

text: For

text: all

text: the

text: training/testing

text: frames,

text: we

text: performed

text: Table

text: Prediction

text: performance

text: of

text: DeepView

text: for

text: different

text: rec

text: a

text: downscaling

text: process

text: by

text: preserving

text: the

text: frame

text: aspect

text: ratio

text: scale

text: training/testing

text: datasets.

text: and

text: slightly

text: cropped

text: the

text: upper

text: and

text: lower

text: pixel

text: boundaries,

text: such

text: that

text: the

text: spatial

text: resolution

text: of

text: all

text: the

text: training/testing

text: as

text: described

text: in

text: Section

text: However,

text: here

text: we

text: downscaled

text: frames

text: becomes

text: the

text: data

text: with

text: different

text: factors

text: of

text: This

text: generates

text: To

text: measure

text: computational

text: efﬁciency,

text: we

text: use

text: memory

text: datasets

text: of

text: the

text: same

text: content

text: but

text: at

text: different

text: scales.

text: We

text: consumption

text: (#Param),

text: i.e.,

text: the

text: number

text: of

text: weight

text: and

text: bias

text: use

text: this

text: data

text: to

text: train

text: our

text: DeepView

text: We

text: performed

text: four

text: values

text: used

text: in

text: convolution

text: and

text: batch-normalization

text: layers.

text: rec

text: trainings.

text: The

text: ﬁrst

text: three

text: train

text: each

text: scale

text: separately

text: while

text: Those

text: parameters

text: should

text: be

text: kept

text: in

text: the

text: memory

text: during

text: the

text: the

text: last

text: trains

text: all

text: scales

text: together

text: at

text: once.

text: The

text: performance

text: entire

text: process.

text: Also,

text: the

text: average

text: running

text: speed

text: in

text: frames

text: of

text: each

text: model

text: is

text: calculated.

text: An

text: architecture

text: is

text: scalable

text: if

text: per

text: second

text: (fps)

text: is

text: measured

text: for

text: all

text: the

text: testing

text: images.

text: We

text: the

text: testing

text: performance

text: of

text: all

text: the

text: all

text: scales

text: model

text: is

text: similar

text: use

text: MSE

text: and

text: mean

text: absolute

text: error

text: (MAE)

text: to

text: measure

text: pre-

text: to

text: the

text: performance

text: of

text: the

text: scale-speciﬁc

text: model.

text: diction

text: accuracy.

text: MAE

text: is

text: calculated

text: as

text: E[

text: Rˆ

text: R

text: where

text: Table

text: shows

text: the

text: prediction

text: performance

text: of

text: E[

text: is

text: expectation

text: operator

text: over

text: pixels

text: in

text: an

text: image.

text: Note

text: DeepView

text: for

text: different

text: scale

text: training/testing

text: datasets.

text: that,

text: higher

text: fps

text: indicates

text: higher

text: performance,

text: while

text: higher

text: rec

text: As

text: shown

text: in

text: Table

text: the

text: dedicated

text: model

text: for

text: each

text: scale

text: #Param,

text: MSE

text: and

text: MAE

text: mean

text: lower

text: performance.

text: shows

text: the

text: lowest

text: error

text: in

text: correspondingly-sized

text: testing

text: Effectiveness

text: of

text: rendering

text: network

text: data.

text: While

text: the

text: performance

text: of

text: the

text: trained

text: model

text: with

text: We

text: verify

text: the

text: effectiveness

text: of

text: the

text: rendering

text: network

text: in

text: all

text: the

text: scales

text: also

text: approximates

text: all

text: the

text: testing

text: data.

text: This

text: DeepView

text: by

text: performing

text: experiments

text: on

text: DeepView

text: indicates

text: that

text: our

text: approach

text: can

text: support

text: spatial

text: scalability

text: in

text: ren

text: ren

text: with

text: and

text: without

text: the

text: rendering

text: network.

text: Table

text: shows

text: the

text: a

text: single

text: network.

text: Fig.

text: shows

text: the

text: qualitative

text: performance

text: performance

text: of

text: DeepView

text: with

text: and

text: without

text: the

text: render-

text: of

text: DeepView

text: for

text: three

text: different

text: spatial

text: resolutions

text: ren

text: re×c

text: ing

text: network.

text: As

text: shown

text: in

text: Table

text: the

text: rendering

text: network

text: and

text: In

text: Fig.

text: the

text: upper

text: improves

text: prediction

text: accuracy

text: in

text: terms

text: of

text: MSE

text: and

text: MAE.

text: and

text: bottom

text: rows

text: illustrate

text: the

text: estimated

text: right-view

text: images

text: It

text: also

text: does

text: not

text: introduce

text: noticeable

text: computational

text: com-

text: and

text: their

text: ground-truth,

text: respectively.

text: As

text: shown

text: in

text: Fig.

text: plexity

text: in

text: both

text: fps

text: and

text: #Param.

text: This

text: is

text: because

text: the

text: ren-

text: DeepViewrec

text: is

text: able

text: to

text: estimate

text: right-view

text: images

text: for

text: three

text: dering

text: network

text: explicitly

text: performs

text: pixel-translations

text: of

text: the

text: different

text: spatial

text: resolutions

text: consistently

text: in

text: a

text: single

text: net-

text: left-image

text: and

text: selects

text: the

text: best

text: translation

text: based

text: on

text: the

text: gen-

text: work.

text: Therefore,

text: contrary

text: to

text: the

text: existing

text: method

text: requiring

text: erated

text: disparity

text: maps.

text: dedicated

text: networks

text: depending

text: on

text: different

text: spatial

text: scales,

text: Spatial

text: scalability

text: our

text: architectures

text: can

text: effectively

text: be

text: used

text: for

text: practical

text: applications

text: in

text: a

text: single

text: network.

text: Our

text: architectures

text: have

text: spatial

text: scalability,

text: i.e.,

text: they

text: can

text: Experiments

text: support

text: multiple

text: input/output

text: sizes

text: in

text: a

text: single

text: network.

text: To

text: verify

text: the

text: spatial

text: scalability,

text: we

text: train

text: DeepView

text: To

text: verify

text: the

text: effectiveness

text: of

text: our

text: architectures,

text: we

text: com-

text: rec

text: for

text: various-sized

text: images.

text: In

text: order

text: to

text: generate

text: the

text: train-

text: pare

text: them

text: against

text: the

text: state

text: of

text: the

text: art

text: Note

text: that

text: ing/testing

text: with

text: different

text: spatial

text: resolutions,

text: we

text: use

text: the

text: we

text: do

text: not

text: compare

text: our

text: approach

text: with

text: the

text: existing

text: depth-

text: same

text: and

text: movies

text: for

text: training

text: and

text: testing,

text: respectively,

text: estimation

text: methods,

text: since

text: they

text: require

text: an

text: additional

text: DIBR

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##