text: (a) Stitched constraints. (b) Sliding constraints. Figure 4: Multishape Experiment. 6 Conclusion and open problems The purpose of this paper was to develop a very general framework for the analysis of shape deformations, along with practical methods to ﬁnd an optimal deformation in that framework. The point of view of control theory gives powerful tools to attain this goal. This allows in particular for the treatment of constrained deformation, which had not been done before. Now that a concrete setting has been ﬁxed, many new developments can be expected. First of all, the minimization algorithms in the case of constrained shapes are quite slow for a very high number of constraints. Moreover, we did not study any of the geometric aspects of shape defor- mation spaces. For example, we only brieﬂy mentioned the inﬁnite-dimensional sub-Riemannian structure that the RKHS induces on both groups of diﬀeomorphism and shape spaces. Sub- Riemannian geometry in inﬁnite dimension and codimension is still a very open subject with very few results. The sub-Riemannian geometry in this paper is particularly diﬃcult and interesting to study because the horizontal spaces may not be closed in the ambient space. A key diﬀerence with ﬁnite dimension is that some geodesics might exist that are neither normal nor abnormal. More general control problems can be designed. One can, for example, use a second-order approach, with a control system taking the form − q˙ = ∂ H(q, p), p˙ = ∂ H(q, p) + f(q, u) p q → in which the original state is lifted to the cotangent space (q (q, p)), and the new control is u. These models have been introduced for shapes in [30], with f(q, u) = u, providing a way to interpolate smoothly between multiple shapes. We are currently exploring applications of this approach to model muscle-like motions, with external forces constrained to being collinear to the ﬁbers. Another glaring issue comes from the assumption of the surjectivity of the constraints in The- orem 2. Indeed, in most practical cases, such as multishapes in Section , this assumption fails. In 36 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: (a)

text: Stitched

text: constraints.

text: (b)

text: Sliding

text: constraints.

text: Figure

text: Multishape

text: Experiment.

text: Conclusion

text: and

text: open

text: problems

text: The

text: purpose

text: of

text: this

text: paper

text: was

text: to

text: develop

text: a

text: very

text: general

text: framework

text: for

text: the

text: analysis

text: of

text: shape

text: deformations,

text: along

text: with

text: practical

text: methods

text: to

text: ﬁnd

text: an

text: optimal

text: deformation

text: in

text: that

text: framework.

text: The

text: point

text: of

text: view

text: of

text: control

text: theory

text: gives

text: powerful

text: tools

text: to

text: attain

text: this

text: goal.

text: This

text: allows

text: in

text: particular

text: for

text: the

text: treatment

text: of

text: constrained

text: deformation,

text: which

text: had

text: not

text: been

text: done

text: before.

text: Now

text: that

text: a

text: concrete

text: setting

text: has

text: been

text: ﬁxed,

text: many

text: new

text: developments

text: can

text: be

text: expected.

text: First

text: of

text: all,

text: the

text: minimization

text: algorithms

text: in

text: the

text: case

text: of

text: constrained

text: shapes

text: are

text: quite

text: slow

text: for

text: a

text: very

text: high

text: number

text: of

text: constraints.

text: Moreover,

text: we

text: did

text: not

text: study

text: any

text: of

text: the

text: geometric

text: aspects

text: of

text: shape

text: defor-

text: mation

text: spaces.

text: For

text: example,

text: we

text: only

text: brieﬂy

text: mentioned

text: the

text: inﬁnite-dimensional

text: sub-Riemannian

text: structure

text: that

text: the

text: RKHS

text: induces

text: on

text: both

text: groups

text: of

text: diﬀeomorphism

text: and

text: shape

text: spaces.

text: Sub-

text: Riemannian

text: geometry

text: in

text: inﬁnite

text: dimension

text: and

text: codimension

text: is

text: still

text: a

text: very

text: open

text: subject

text: with

text: very

text: few

text: results.

text: The

text: sub-Riemannian

text: geometry

text: in

text: this

text: paper

text: is

text: particularly

text: diﬃcult

text: and

text: interesting

text: to

text: study

text: because

text: the

text: horizontal

text: spaces

text: may

text: not

text: be

text: closed

text: in

text: the

text: ambient

text: space.

text: A

text: key

text: diﬀerence

text: with

text: ﬁnite

text: dimension

text: is

text: that

text: some

text: geodesics

text: might

text: exist

text: that

text: are

text: neither

text: normal

text: nor

text: abnormal.

text: More

text: general

text: control

text: problems

text: can

text: be

text: designed.

text: One

text: can,

text: for

text: example,

text: use

text: a

text: second-order

text: approach,

text: with

text: a

text: control

text: system

text: taking

text: the

text: form

text: q˙

text: H(q,

text: p),

text: p˙

text: H(q,

text: p)

text: f(q,

text: u)

text: p

text: q

text: in

text: which

text: the

text: original

text: state

text: is

text: lifted

text: to

text: the

text: cotangent

text: space

text: (q

text: (q,

text: p)),

text: and

text: the

text: new

text: control

text: is

text: u.

text: These

text: models

text: have

text: been

text: introduced

text: for

text: shapes

text: in

text: with

text: f(q,

text: u)

text: u,

text: providing

text: a

text: way

text: to

text: interpolate

text: smoothly

text: between

text: multiple

text: shapes.

text: We

text: are

text: currently

text: exploring

text: applications

text: of

text: this

text: approach

text: to

text: model

text: muscle-like

text: motions,

text: with

text: external

text: forces

text: constrained

text: to

text: being

text: collinear

text: to

text: the

text: ﬁbers.

text: Another

text: glaring

text: issue

text: comes

text: from

text: the

text: assumption

text: of

text: the

text: surjectivity

text: of

text: the

text: constraints

text: in

text: The-

text: orem

text: Indeed,

text: in

text: most

text: practical

text: cases,

text: such

text: as

text: multishapes

text: in

text: Section

text: this

text: assumption

text: fails.

text: In

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##


text: 2 Modelling shape deformation problems with optimal con- trol ∈ ∈ The following notations will be used throughout the paper. Let d IN ﬁxed. A vector a IRd can be as well viewed as a column matrix of length d. The Euclidean norm of a is denoted by | | · ∈ a . The inner product a b between two vectors a, b IRd can as well be written, with matrix | | · notations, as aT b, where aT is the transpose of a. In particular one has a 2 = a a = aT a. (cid:107)·(cid:107) · · Let X be a Banach space. The norm on X is denoted by , and the inner product by ( , ) ∗ X X whenever X is a Hilbert space. The topological dual X of X is deﬁned as the set of all linear → (cid:107) (cid:107) { | ∈ continuous mappings p : X IR. Endowed with the usual dual norm p ∗ = sup p(x) x (cid:107) (cid:107) } ∈ ∗ X ∈ X, x = 1 , it is a Banach space. For p X , the natural pairing between p and w X is X(cid:104) (cid:105) p(w) = p, w ∗ , with the duality bracket. If X = IRn then p can be identiﬁed with a column X ,X vector through the equality p(w) = pT w. Let M be an open subset of X, and let Y be another Banach space. The Fr´echet derivative → ∈ of a map f : M Y at a point q M is written as df . When it is applied to a vector w, it is q (cid:104) (cid:105) denoted by df .w or df (w). When Y = IR, we may also write df , w ∗ . q q q X ,X We denote by W1,p(0, 1; M) (resp. H1(0, 1; M)) the usual Sobolev space of elements of Lp(0, 1; M), (cid:54) (cid:54) ∞ ∈ with 1 p + (resp., with p = 2) having a weak derivative in Lp(0, 1; X). For q M we denote ∈ 0∈ by W1,p(0, 1; M) (resp., by H1 (0, 1; M)) the space of all q W1,p(0, 1; M) (resp., q H1 (0, 1; M)) q0 q0 q0 such that q(0) = q . ∈ 0 → C For every (cid:96) IN, a mapping ϕ : M M is called a (cid:96) diﬀeomorphism if it is a bijective C C mapping of class (cid:96) with an inverse of class (cid:96). The space of all such diﬀeomorphisms is denoted by Diﬀ (cid:96)(M). Note(cid:40)t(cid:13)hat Diﬀ 0→(M) is(cid:13)the spCace of all homeomorphisms of M. (cid:41) For every mappin(cid:13)g f : IRd X o(cid:13)f clas(cid:12)s (cid:96) with compact support, we deﬁne the usual semi-norm (cid:13) (cid:13) (cid:12) (cid:13) (cid:13) ··· (cid:107) (cid:107) ∂(cid:96)1+ +(cid:96)df(x) ∈ ∈ · · · (cid:54) f = sup x IRd, ((cid:96) , . . . , (cid:96) ) INd, (cid:96) + + (cid:96) (cid:96) . (cid:96) ∂x(cid:96)1 . . . ∂x(cid:96)d 1 d 1 d 1 d X C (cid:107) · (cid:107) We deﬁne the Banach space (cid:96)(IRd, IRd) (endowed with the norm ) as the completion of C0 (cid:96) (cid:107) · (cid:107) the space of vector ﬁelds of class (cid:96) with compact support on IRd with respect to the norm . C C (cid:96) In other words, (cid:96)(IRd, IRd) is the space of vector ﬁelds of class (cid:96) on IRd whose derivatives of order 0 less than or equal to (cid:96) converge to zero at inﬁnity. C We deﬁne Diﬀ (cid:96)(IRd) as the set of all diﬀeomorphisms of class (cid:96) that converge to identity at 0 ∈ − ∈ C inﬁnity. Clearly, Diﬀ (cid:96)(IRd) is the set of all ϕ Diﬀ (cid:96)(IRd) such that ϕ Id (cid:96)(IRd, IRd). It is 0 (cid:55)→ ◦ IRd 0 a group for the composition law (ϕ, ψ) ϕ ψ. (cid:62) Note that, if (cid:96) 1, then Diﬀ (cid:96)(IRd) is an open subset of the aﬃne Banach space Id + C 0 IRd (cid:96)(IRd, IRd). This allows one to develop a diﬀerential calculus on Diﬀ (cid:96)(IRd). 0 0 2.1 Preliminaries: deformations and RKHS of vector ﬁelds Our approach to shape analysis is based on optimizing evolving deformations. A deformation is a one-parameter family of ﬂows in IRd generated by time-dependent vector ﬁelds on IRd. Let us deﬁne this concept more rigorously. ∈ ∗ Diﬀeomorphic deformations. Let (cid:96) IN . Let −→ C v : [0, 1] (cid:96)(IRd, IRd) (cid:55)−→ 0 t v(t) (cid:55)→ (cid:107) (cid:107) be a time-dependent vector ﬁeld such that the real-valued function t v(t) is integrable. In C (cid:96) other words, we consider an element v of the space L1(0, 1; (cid:96)(IRd, IRd)). 0 5 ##LTLine##

label: section

text: Modelling

text: shape

text: deformation

text: problems

text: with

text: optimal

text: con-

text: trol

label: paragraph

text: The

text: following

text: notations

text: will

text: be

text: used

text: throughout

text: the

text: paper.

text: Let

text: d

text: IN

text: ﬁxed.

text: A

text: vector

text: a

text: IRd

text: can

text: be

text: as

text: well

text: viewed

text: as

text: a

text: column

text: matrix

text: of

text: length

text: d.

text: The

text: Euclidean

text: norm

text: of

text: a

text: is

text: denoted

text: by

text: a

text: The

text: inner

text: product

text: a

text: b

text: between

text: two

text: vectors

text: a,

text: b

text: IRd

text: can

text: as

text: well

text: be

text: written,

text: with

text: matrix

text: notations,

text: as

text: aT

text: b,

text: where

text: aT

text: is

text: the

text: transpose

text: of

text: a.

text: In

text: particular

text: one

text: has

text: a

text: a

text: a

text: aT

text: a.

text: Let

text: X

text: be

text: a

text: Banach

text: space.

text: The

text: norm

text: on

text: X

text: is

text: denoted

text: by

text: and

text: the

text: inner

text: product

text: by

text: X

text: X

text: whenever

text: X

text: is

text: a

text: Hilbert

text: space.

text: The

text: topological

text: dual

text: X

text: of

text: X

text: is

text: deﬁned

text: as

text: the

text: set

text: of

text: all

text: linear

text: continuous

text: mappings

text: p

text: X

text: IR.

text: Endowed

text: with

text: the

text: usual

text: dual

text: norm

text: p

text: sup

text: p(x)

text: x

text: X

text: X,

text: x

text: it

text: is

text: a

text: Banach

text: space.

text: For

text: p

text: X

text: the

text: natural

text: pairing

text: between

text: p

text: and

text: w

text: X

text: is

text: p(w)

text: p,

text: w

text: with

text: the

text: duality

text: bracket.

text: If

text: X

text: IRn

text: then

text: p

text: can

text: be

text: identiﬁed

text: with

text: a

text: column

text: X

text: ,X

text: vector

text: through

text: the

text: equality

text: p(w)

text: pT

text: w.

text: Let

text: M

text: be

text: an

text: open

text: subset

text: of

text: X,

text: and

text: let

text: Y

text: be

text: another

text: Banach

text: space.

text: The

text: Fr´echet

text: derivative

text: of

text: a

text: map

text: f

text: M

text: Y

text: at

text: a

text: point

text: q

text: M

text: is

text: written

text: as

text: df

text: When

text: it

text: is

text: applied

text: to

text: a

text: vector

text: w,

text: it

text: is

text: q

text: denoted

text: by

text: df

text: .w

text: or

text: df

text: (w).

text: When

text: Y

text: IR,

text: we

text: may

text: also

text: write

text: df

text: w

text: q

text: q

text: q

text: X

text: ,X

text: We

text: denote

text: by

text: M)

text: (resp.

text: M))

text: the

text: usual

text: Sobolev

text: space

text: of

text: elements

text: of

text: M),

text: with

text: p

text: (resp.,

text: with

text: p

text: having

text: a

text: weak

text: derivative

text: in

text: X).

text: For

text: q

text: M

text: we

text: denote

text: 0∈

text: by

text: M)

text: (resp.,

text: by

text: M))

text: the

text: space

text: of

text: all

text: q

text: M)

text: (resp.,

text: q

text: M))

text: such

text: that

text: q

text: C

text: For

text: every

text: IN,

text: a

text: mapping

text: ϕ

text: M

text: M

text: is

text: called

text: a

text: diﬀeomorphism

text: if

text: it

text: is

text: a

text: bijective

text: C

text: C

text: mapping

text: of

text: class

text: with

text: an

text: inverse

text: of

text: class

text: The

text: space

text: of

text: all

text: such

text: diﬀeomorphisms

text: is

text: denoted

text: by

text: Diﬀ

text: Diﬀ

text: spCace

text: of

text: all

text: homeomorphisms

text: of

text: M.

text: For

text: every

text: f

text: IRd

text: X

text: with

text: compact

text: support,

text: we

text: deﬁne

text: the

text: usual

text: semi-norm

text: f

text: sup

text: x

text: IRd,

text: INd,

text: d

text: d

text: d

text: X

text: C

text: We

text: deﬁne

text: the

text: Banach

text: space

text: IRd)

text: (endowed

text: with

text: the

text: norm

text: as

text: the

text: completion

text: of

text: the

text: space

text: of

text: vector

text: ﬁelds

text: of

text: class

text: with

text: compact

text: support

text: on

text: IRd

text: with

text: respect

text: to

text: the

text: norm

text: C

text: C

text: In

text: other

text: words,

text: IRd)

text: is

text: the

text: space

text: of

text: vector

text: ﬁelds

text: of

text: class

text: on

text: IRd

text: whose

text: derivatives

text: of

text: order

text: less

text: than

text: or

text: equal

text: to

text: converge

text: to

text: zero

text: at

text: inﬁnity.

text: C

text: We

text: deﬁne

text: Diﬀ

text: as

text: the

text: set

text: of

text: all

text: diﬀeomorphisms

text: of

text: class

text: that

text: converge

text: to

text: identity

text: at

text: C

text: inﬁnity.

text: Clearly,

text: Diﬀ

text: is

text: the

text: set

text: of

text: all

text: ϕ

text: Diﬀ

text: such

text: that

text: ϕ

text: Id

text: IRd).

text: It

text: is

text: IRd

text: a

text: group

text: for

text: the

text: composition

text: law

text: (ϕ,

text: ψ)

text: ϕ

text: ψ.

text: Note

text: that,

text: if

text: then

text: Diﬀ

text: is

text: an

text: open

text: subset

text: of

text: the

text: aﬃne

text: Banach

text: space

text: Id

text: C

text: IRd

text: IRd).

text: This

text: allows

text: one

text: to

text: develop

text: a

text: diﬀerential

text: calculus

text: on

text: Diﬀ

text: Preliminaries:

text: deformations

text: and

text: RKHS

text: of

text: vector

text: ﬁelds

text: Our

text: approach

text: to

text: shape

text: analysis

text: is

text: based

text: on

text: optimizing

text: evolving

text: deformations.

text: A

text: deformation

text: is

text: a

text: one-parameter

text: family

text: of

text: ﬂows

text: in

text: IRd

text: generated

text: by

text: time-dependent

text: vector

text: ﬁelds

text: on

text: IRd.

text: Let

text: us

text: deﬁne

text: this

text: concept

text: more

text: rigorously.

text: Diﬀeomorphic

text: deformations.

text: Let

text: IN

text: Let

text: C

text: v

text: IRd)

text: t

text: v(t)

text: be

text: a

text: time-dependent

text: vector

text: ﬁeld

text: such

text: that

text: the

text: real-valued

text: function

text: t

text: v(t)

text: is

text: integrable.

text: In

text: C

text: other

text: words,

text: we

text: consider

text: an

text: element

text: v

text: of

text: the

text: space

text: IRd)).

text: ##LTLine##


text: → According to the Cauchy-Lipshitz theorem, v generates a (unique) ﬂow ϕ : [0, 1] Diﬀ 1(IRd) 0 (see, e.g., [1] or [27, Chapter 11]), that is a one-parameter family of diﬀeomorphisms such that ∂ϕ (t, x) = v(t, ϕ(t, x)), ∂t ϕ(0, x) = x, ∈ ∈ for almost every t [0, 1] and every x IRd. In other words, considering ϕ as a curve in the space Diﬀ 1(IRd), the ﬂow ϕ is the unique solution of 0 ◦ ϕ˙(t) = v(t) ϕ(t), (1) ϕ(0) = Id . IRd C Such a ﬂow ϕ is called a deformation of IRd of class (cid:96). (cid:110) (cid:111) C Proposition 1. The set of deformations of IRd of class (cid:96) coincides with the set ∈ | ϕ W1,1(0, 1; Diﬀ (cid:96)(IRd)) ϕ(0) = Id . 0 IRd C (cid:55)→ In other words, the deformations of IRd of class (cid:96) are exactly the curves t ϕ(t) on Diﬀ (cid:96)(IRd) 0 that are integrable on (0, 1) as well as their derivative, such that ϕ(0) = Id . IRd Proof. Let us ﬁrst prove that there exists a sequence of positive real numbers (D ) ∈ such that C ∈ ∗ n n IN for every deformation ϕ of IRd of class (cid:96), with (cid:96) IN , induced by the time-dependent vector ﬁeld ∈ C (cid:16) (cid:90) (cid:17) v L1(0, 1; (cid:96)(IRd, IRd)), one has 0 (cid:107) − (cid:107) (cid:54) 1 (cid:107) (cid:107) sup ϕ(t) Id D exp D v(t) , (2) ∈ IRd i i i i t [0,1] 0 ∈ { } for every i 0, . . . , (cid:96) . The case i = 0 is an immediate consequence of the integral formulation of (1). Combining the formula for computing derivatives of a composition of mappings with an induction argument shows ◦ that the derivatives of order i of v ϕ are polynomials in the derivatives of v and ϕ of order less than or equ(cid:12)al to i. Mo(cid:12)reover, these polynomials are of degree one with respect to the derivatives (cid:12) (cid:12) of v, and a(cid:12)lso of degree(cid:12)one with respect to the derivatives of ϕ of order i. Therefore we can write (cid:12) (cid:12) d (cid:54) (cid:107) (cid:107) | | (cid:107) (cid:107) | | | − | ∂i ϕ(t, x) v(t) ∂i ϕ(t, x) + v(t) − P ( ∂1ϕ(t, x) , . . . , ∂i 1ϕ(t, x) ), (3) dt x i x i 1 i x x where P is a polynomial independent of v and ϕ, and the norms of the derivatives of the i ∂jϕ(t, x) are computed in the space of IRd-valued multilinear maps. The result then follows from x Gronwall estimates and from an induction argument on i. C That any deformation of IRd of class (cid:96) is a curve of class W1,1 in Diﬀ (cid:96)(IRd) is then a direct 0 consequence of (2) and (3). ◦ − Conversely, for every curve ϕ on Diﬀ (cid:96)(IRd) of class W1,1, we set v(t) = ϕ˙(t) ϕ 1(t), for every ∈ ◦ 0 ∈ t [0, 1]. We have ϕ˙(t) = v(t) ϕ(t) for almost every t [0, 1], and hence it suﬃces to prove (cid:55)→ (cid:107) (cid:107) that t v(t) is integrable. The curve ϕ is continuous on [0, 1] and therefore is bounded. This (cid:55)→(cid:96) − implies that t ϕ(t) 1 is bounded as well. The formula for computing derivatives of compositions (cid:107) (cid:107) (cid:55)→ (cid:107) (cid:107) of maps then shows that v(t) is integrable whenever t ϕ˙(t) is integrable, which completes (cid:96) (cid:96) the proof since ϕ is of class W1,1. 6 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: According

text: to

text: the

text: Cauchy-Lipshitz

text: theorem,

text: v

text: generates

text: a

text: (unique)

text: ﬂow

text: ϕ

text: Diﬀ

text: (see,

text: e.g.,

text: or

text: Chapter

text: that

text: is

text: a

text: one-parameter

text: family

text: of

text: diﬀeomorphisms

text: such

text: that

text: ∂ϕ

text: (t,

text: x)

text: v(t,

text: ϕ(t,

text: x)),

text: ∂t

text: x)

text: x,

text: for

text: almost

text: every

text: t

text: and

text: every

text: x

text: IRd.

text: In

text: other

text: words,

text: considering

text: ϕ

text: as

text: a

text: curve

text: in

text: the

text: space

text: Diﬀ

text: the

text: ﬂow

text: ϕ

text: is

text: the

text: unique

text: solution

text: of

label: equation

text: ϕ˙(t)

text: v(t)

text: ϕ(t),

text: Id

text: IRd

text: C

text: Such

text: a

text: ﬂow

text: ϕ

text: is

text: called

text: a

text: deformation

text: of

text: IRd

text: of

text: class

text: C

text: Proposition

text: The

text: set

text: of

text: deformations

text: of

text: IRd

text: of

text: class

text: coincides

text: with

text: the

text: set

text: ϕ

text: Diﬀ

text: Id

text: IRd

text: C

text: In

text: other

text: words,

text: the

text: deformations

text: of

text: IRd

text: of

text: class

text: are

text: exactly

text: the

text: curves

text: t

text: ϕ(t)

text: on

text: Diﬀ

text: that

text: are

text: integrable

text: on

text: as

text: well

text: as

text: their

text: derivative,

text: such

text: that

text: Id

text: IRd

text: Proof.

text: Let

text: us

text: ﬁrst

text: prove

text: that

text: there

text: exists

text: a

text: sequence

text: of

text: positive

text: real

text: numbers

text: (D

text: such

text: that

text: C

text: n

text: n

text: IN

text: for

text: every

text: deformation

text: ϕ

text: of

text: IRd

text: of

text: class

text: with

text: IN

text: induced

text: by

text: the

text: time-dependent

text: vector

text: ﬁeld

text: C

text: v

text: IRd)),

text: one

text: has

text: sup

text: ϕ(t)

text: Id

text: D

text: exp

text: D

text: v(t)

text: IRd

text: i

text: i

text: i

text: i

text: t

text: for

text: every

text: i

text: The

text: case

text: i

text: is

text: an

text: immediate

text: consequence

text: of

text: the

text: integral

text: formulation

text: of

text: Combining

text: the

text: formula

text: for

text: computing

text: derivatives

text: of

text: a

text: composition

text: of

text: mappings

text: with

text: an

text: induction

text: argument

text: shows

text: that

text: the

text: derivatives

text: of

text: order

text: i

text: of

text: v

text: ϕ

text: are

text: polynomials

text: in

text: the

text: derivatives

text: of

text: v

text: and

text: ϕ

text: of

text: order

text: less

text: than

text: or

text: to

text: i.

text: these

text: polynomials

text: are

text: of

text: degree

text: one

text: with

text: respect

text: to

text: the

text: derivatives

text: of

text: v,

text: and

text: of

text: with

text: respect

text: to

text: the

text: derivatives

text: of

text: ϕ

text: of

text: order

text: i.

text: Therefore

text: we

text: can

text: write

text: d

text: ∂i

text: ϕ(t,

text: x)

text: v(t)

text: ∂i

text: ϕ(t,

text: x)

text: v(t)

text: P

text: x)

text: ∂i

text: x)

text: dt

text: x

text: i

text: x

text: i

text: i

text: x

text: x

text: where

text: P

text: is

text: a

text: polynomial

text: independent

text: of

text: v

text: and

text: ϕ,

text: and

text: the

text: norms

text: of

text: the

text: derivatives

text: of

text: the

text: i

text: ∂jϕ(t,

text: x)

text: are

text: computed

text: in

text: the

text: space

text: of

text: IRd-valued

text: multilinear

text: maps.

text: The

text: result

text: then

text: follows

text: from

text: x

text: Gronwall

text: estimates

text: and

text: from

text: an

text: induction

text: argument

text: on

text: i.

text: C

text: That

text: any

text: deformation

text: of

text: IRd

text: of

text: class

text: is

text: a

text: curve

text: of

text: class

text: in

text: Diﬀ

text: is

text: then

text: a

text: direct

text: consequence

text: of

text: and

text: Conversely,

text: for

text: every

text: curve

text: ϕ

text: on

text: Diﬀ

text: of

text: class

text: we

text: set

text: v(t)

text: ϕ˙(t)

text: ϕ

text: for

text: every

text: t

text: We

text: have

text: ϕ˙(t)

text: v(t)

text: ϕ(t)

text: for

text: almost

text: every

text: t

text: and

text: hence

text: it

text: suﬃces

text: to

text: prove

text: that

text: t

text: v(t)

text: is

text: integrable.

text: The

text: curve

text: ϕ

text: is

text: continuous

text: on

text: and

text: therefore

text: is

text: bounded.

text: This

text: implies

text: that

text: t

text: ϕ(t)

text: is

text: bounded

text: as

text: well.

text: The

text: formula

text: for

text: computing

text: derivatives

text: of

text: compositions

text: of

text: maps

text: then

text: shows

text: that

text: v(t)

text: is

text: integrable

text: whenever

text: t

text: ϕ˙(t)

text: is

text: integrable,

text: which

text: completes

text: the

text: proof

text: since

text: ϕ

text: is

text: of

text: class

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Reproducing Kernel Hilbert Spaces of vector ﬁelds. Let us brieﬂy recall the deﬁnition ∈ ∗ and a few properties of RKHS’s (see [5, 32] for more details). Let k IN be ﬁxed. · · Given a Hilbert space (V, ( , ) ), according to the Riesz representation theorem, the mapping (cid:55)→ · V ∗ v (v, ) is a bijective isometry between V and V , whose inverse is denoted by K . Then V ∈ ∗ ∈ (cid:104) (cid:105) (cid:107) (cid:107) (cid:107) V (cid:107) f(cid:104)or ever(cid:105)y p V and every v V one has p, v V ∗,V = (KV p, v)V and p 2V ∗ = KV p 2V = p, K p ∗ . V V ,V C Deﬁnition 1. A Reproducing Kernel Vector Space (RKHS) of vector ﬁelds of class (cid:96) is a Hilbert · · ⊂ C 0 space (V, ( , ) ) of vector ﬁelds on IRd, such that V (cid:96)(IRd, IRd) with continuous inclusion. V 0 C ∈ Let V be an RKHS of vector ﬁelds of class (cid:96). Then, for all (b, y) (IRd)2, by deﬁnition the ⊗ ⊗ 0 ∈ linear form b δ on V , deﬁned by b δ (v) = bT v(y) for every v V , is continuous (actually y y this continuity property holds as well for every compactly supported vector-valued distribution of (cid:104) ⊗ (cid:105) ⊗ order at most (cid:96) on IRd). By deﬁnition of K , there holds b δ , v ∗ = (K (b δ ), v) . The V ×y V ,V V y V reproducing kernel K of V is then the mapping deﬁned on IRd IRd, with values in the set of real square matrices of size d, deﬁned by ⊗ K(x, y)b = K (b δ )(x), (4) V y ∈ · ∈ for all (b, x, y) (IRd)3. In other words, there holds (K( , y)b, v) = bT v(y), for all (b, y) ∈ · ⊗ V C (IRd)2 and every v V , and K( , y)b = K (b δ ) is a vector ﬁeld of class (cid:96) in IRd, element of V . · · V y ∈ It is easy to see that (K( , x)a, K( , y)b) = aT K(x, y)b, for all (a, b, x, y) (IRd)4, and hence V (cid:82) that K(x, y)T = K(y, x) and that K(x, x) is positive semi-deﬁnite under the assumption that no · · · no∈ntrivial linear combination a⊗T1 v(x1)+ +a(cid:90)Tnv(xn), with given distinct xj’s can vanish for every v V . Finally, writing K (a δ )(x) = K(x, y)a = K(x, s)a dδ (s), we have V y IRd y K p(x) = K(x, y) dp(y), (5) V IRd for every compactly supported vector-valued distribution p on IRd of order less than or equal to (cid:96).1 As explained in [5, 34], one of the interests of such a structure is that we can deﬁne the kernel itself instead of deﬁning the space V . Indeed a given kernel K yields a unique associated RKHS. It | − | ∈ C∞ is usual to consider kernels of the form K(x, y) = γ( x y )Id with γ (IR). Such a kernel IRd yields a metric that is invariant under rotation and translation. The most common model is when γ is a Gaussian function but other families of kernels can be used as well [31, 34]. 2.2 From shape space problems to optimal control We deﬁne a shape space in IRd as an open subset M of a Banach space X on which the group of diﬀeomorphisms of IRd acts in a certain way. The elements of M, called states of the shape, are denoted by q. They are usually subsets or immersed submanifolds of IRd, with a typical deﬁnition C of the shape space as the set M = Emb1(S, IRd) of all embeddings of class 1 of a given Riemannian C manifold S into IRd. For example, if S is the unit circle then M is the set of all parametrized 1 simple closed curves in IRd. In practical applications or in numerical implementations, one has to (cid:90) (cid:90) 1Indeed, it suﬃces to note that ⊗ ⊗ bTKV p(x) = (b δx,KV p)V ∗,V = (p,KV b δx) = (K(y,x)b)Tdp(y) = bT K(x,y)dp(y). IRd IRd 7 ##LTLine##

text: Reproducing

text: Kernel

text: Hilbert

text: Spaces

text: of

text: vector

text: ﬁelds.

text: Let

text: us

text: brieﬂy

text: recall

text: the

text: deﬁnition

text: and

text: a

text: few

text: properties

text: of

text: RKHS’s

text: (see

text: for

text: more

text: details).

text: Let

text: k

text: IN

text: be

text: ﬁxed.

text: Given

text: a

text: Hilbert

text: space

text: (V,

text: according

text: to

text: the

text: Riesz

text: representation

text: theorem,

text: the

text: mapping

text: V

text: v

text: (v,

text: is

text: a

text: bijective

text: isometry

text: between

text: V

text: and

text: V

text: whose

text: inverse

text: is

text: denoted

text: by

text: K

text: Then

text: V

text: V

text: p

text: V

text: and

text: every

text: v

text: V

text: one

text: has

text: p,

text: v

text: V

text: ∗,V

text: (KV

text: p,

text: v)V

text: and

text: p

text: KV

text: p

text: p,

text: K

text: p

text: V

text: V

text: ,V

text: C

text: Deﬁnition

text: A

text: Reproducing

text: Kernel

text: Vector

text: Space

text: (RKHS)

text: of

text: vector

text: ﬁelds

text: of

text: class

text: is

text: a

text: Hilbert

text: C

text: space

text: (V,

text: of

text: vector

text: ﬁelds

text: on

text: IRd,

text: such

text: that

text: V

text: IRd)

text: with

text: continuous

text: inclusion.

text: V

text: C

text: Let

text: V

text: be

text: an

text: RKHS

text: of

text: vector

text: ﬁelds

text: of

text: class

text: Then,

text: for

text: all

text: (b,

text: y)

text: by

text: deﬁnition

text: the

text: linear

text: form

text: b

text: δ

text: on

text: V

text: deﬁned

text: by

text: b

text: δ

text: (v)

text: bT

text: v(y)

text: for

text: every

text: v

text: V

text: is

text: continuous

text: (actually

text: y

text: y

text: this

text: continuity

text: property

text: holds

text: as

text: well

text: for

text: every

text: compactly

text: supported

text: vector-valued

text: distribution

text: of

text: order

text: at

text: most

text: on

text: IRd).

text: By

text: deﬁnition

text: of

text: K

text: there

text: holds

text: b

text: δ

text: v

text: (K

text: (b

text: δ

text: v)

text: The

text: V

text: ×y

text: V

text: ,V

text: V

text: y

text: V

text: reproducing

text: kernel

text: K

text: of

text: V

text: is

text: then

text: the

text: mapping

text: deﬁned

text: on

text: IRd

text: IRd,

text: with

text: values

text: in

text: the

text: set

text: of

text: real

text: square

text: matrices

text: of

text: size

text: d,

text: deﬁned

text: by

text: K(x,

text: y)b

text: K

text: (b

text: δ

text: )(x),

text: V

text: y

text: for

text: all

text: (b,

text: x,

text: y)

text: In

text: other

text: words,

text: there

text: holds

text: (K(

text: y)b,

text: v)

text: bT

text: v(y),

text: for

text: all

text: (b,

text: y)

text: V

text: C

text: and

text: every

text: v

text: V

text: and

text: K(

text: y)b

text: K

text: (b

text: δ

text: is

text: a

text: vector

text: ﬁeld

text: of

text: class

text: in

text: IRd,

text: element

text: of

text: V

text: V

text: y

text: It

text: is

text: easy

text: to

text: see

text: that

text: (K(

text: x)a,

text: K(

text: y)b)

text: aT

text: K(x,

text: y)b,

text: for

text: all

text: (a,

text: b,

text: x,

text: y)

text: and

text: hence

text: V

text: that

text: K(x,

text: y)T

text: K(y,

text: x)

text: and

text: that

text: K(x,

text: x)

text: is

text: positive

text: semi-deﬁnite

text: under

text: the

text: assumption

text: that

text: no

text: no∈ntrivial

text: linear

text: combination

text: with

text: given

text: distinct

text: xj’s

text: can

text: vanish

text: for

text: every

text: v

text: V

text: Finally,

text: writing

text: K

text: (a

text: δ

text: )(x)

text: K(x,

text: y)a

text: K(x,

text: s)a

text: dδ

text: (s),

text: we

text: have

text: V

text: y

text: IRd

text: y

text: K

text: p(x)

text: K(x,

text: y)

text: dp(y),

text: V

text: IRd

text: for

text: every

text: compactly

text: supported

text: vector-valued

text: distribution

text: p

text: on

text: IRd

text: of

text: order

text: less

text: than

text: or

text: equal

text: to

text: As

text: explained

text: in

text: one

text: of

text: the

text: interests

text: of

text: such

text: a

text: structure

text: is

text: that

text: we

text: can

text: deﬁne

text: the

text: kernel

text: itself

text: instead

text: of

text: deﬁning

text: the

text: space

text: V

text: Indeed

text: a

text: given

text: kernel

text: K

text: yields

text: a

text: unique

text: associated

text: RKHS.

text: It

text: C∞

text: is

text: usual

text: to

text: consider

text: kernels

text: of

text: the

text: form

text: K(x,

text: y)

text: γ(

text: x

text: y

text: )Id

text: with

text: γ

text: (IR).

text: Such

text: a

text: kernel

text: IRd

text: yields

text: a

text: metric

text: that

text: is

text: invariant

text: under

text: rotation

text: and

text: translation.

text: The

text: most

text: common

text: model

text: is

text: when

text: γ

text: is

text: a

text: Gaussian

text: function

text: but

text: other

text: families

text: of

text: kernels

text: can

text: be

text: used

text: as

text: well

text: From

text: shape

text: space

text: problems

text: to

text: optimal

text: control

text: We

text: deﬁne

text: a

text: shape

text: space

text: in

text: IRd

text: as

text: an

text: open

text: subset

text: M

text: of

text: a

text: Banach

text: space

text: X

text: on

text: which

text: the

text: group

text: of

text: diﬀeomorphisms

text: of

text: IRd

text: acts

text: in

text: a

text: certain

text: way.

text: The

text: elements

text: of

text: M,

text: called

text: states

text: of

text: the

text: shape,

text: are

text: denoted

text: by

text: q.

text: They

text: are

text: usually

text: subsets

text: or

text: immersed

text: submanifolds

text: of

text: IRd,

text: with

text: a

text: typical

text: deﬁnition

text: C

text: of

text: the

text: shape

text: space

text: as

text: the

text: set

text: M

text: IRd)

text: of

text: all

text: embeddings

text: of

text: class

text: of

text: a

text: given

text: Riemannian

text: C

text: manifold

text: S

text: into

text: IRd.

text: For

text: example,

text: if

text: S

text: is

text: the

text: unit

text: circle

text: then

text: M

text: is

text: the

text: set

text: of

text: all

text: parametrized

text: simple

text: closed

text: curves

text: in

text: IRd.

text: In

text: practical

text: applications

text: or

text: in

text: numerical

text: implementations,

text: one

text: has

text: to

text: it

text: suﬃces

text: to

text: note

text: that

text: bTKV

text: p(x)

text: (b

text: δx,KV

text: p)V

text: ∗,V

text: (p,KV

text: b

text: δx)

text: (K(y,x)b)Tdp(y)

text: bT

text: K(x,y)dp(y).

text: IRd

text: IRd

text: ##LTLine##


text: • The mapping × C −→ ξ : M (cid:96)(IRd, IRd) X 0 (cid:55)−→ (8) (q, v) ξ v q × C C is continuous, and its restriction to M (cid:96)+1(IRd, IRd) is of class 1. In particular the (cid:55)→ C 0 C mapping q ξ v is of class 1, for every bounded vector ﬁeld v of class (cid:96)+1. q (cid:62) Example 1. For (cid:96) 1, the action of Diﬀ (cid:96)(IRd) on itself by left composition makes it a shape 0 space of order (cid:96) in IRd. ∈ C Example 2. Let (cid:96) IN and let S be a (cid:96) smooth compact Riemannian manifold. Consider the C space M = X = (cid:96)(S, IRd) equipped with its usual Banach norm. Then M is a shape space of · ◦ order (cid:96), where the action of Diﬀ max(1,(cid:96))(IRd) is given by the composition ϕ q = ϕ q. Indeed, it 0 is continuous thanks to the rule for computing derivatives of a composition, and we also have (cid:107) · − · (cid:107) (cid:54) (cid:107) − (cid:107) ϕ q ϕ q γ ϕ ϕ . 1 2 X 1 2 (cid:96) ∈ ∈ C ◦ ∈ Moreover, given q M and v (cid:96)(IRd, IRd), ξ v is the vector ﬁeld along q given by ξ (v) = v q C 0 q q (cid:96)(M, IRd). Finally, the formula for computing derivatives of a composition yields (cid:107) ◦ − ◦ − (cid:107) (cid:107) (cid:107) v (q + δq) v q dv .δq = o( δq ), q X X ∈ for every δq M, and the last part of the deﬁnition follows. This framework describes most of shape spaces. An interesting particular case of this general example is when S = (s , . . . , s ) is a ﬁnite set 1 n (zero-dimensional manifold), X = (IRd)n and { ∈ | (cid:54) (cid:54) } M = Lmk (n) = (x , . . . , x ) (IRd)n x = x if i = j d 1 n i j is a (so-called) space of n landmarks in IRd. For q = (x , . . . , x ), the smooth action of order 0 · ∈ C 1 n is ϕ q = (ϕ(x ), . . . , ϕ(x )). For v 0(IR2, IR2), the inﬁnitesimal action of v at q is given by 1 n 0 ξ (v) = (v(x ), . . . , v(x )). q 1 n Remark 1. In most cases, and in all examples given throughout this paper, the mapping ξ × C C ∈ restricted to M (cid:96)+k(IRd, IRd) is of class k, for every k IN. 0 ∈ (cid:55)→ · C Proposition 2. For every q M, the mapping ϕ ϕ q is of class 1, and its diﬀerential at ∈ ϕ is given by ξ · dR − . In particular, given q M and given ϕ a deformation of IRd of class C ϕ q ϕ 1 0 (cid:55)→ · max1,(cid:96), which is the ﬂow of the time-dependent vector ﬁeld v, the curve t q(t) = ϕ(t) q is of 0 0 class W1,1 and one has ◦ − q˙(t) = ξ · ϕ˙(t) ϕ(t) 1 = ξ v(t), (9) ϕ(t) q0 q(t) ∈ for almost every t [0, 1]. ∈ ∈ ∈ C ∈ Proof. Let q M, ﬁx ϕ Diﬀ (cid:96)(IRd) and take δϕ T Diﬀ (cid:96)(IRd) = (cid:96)(IRd, IRd). Then ϕ + δϕ 0 (cid:107) (cid:107) 0 ϕ 0 ◦ 0− Diﬀ (cid:96)(IRd) for δϕ small enough. We deﬁne v = (dR − ) δϕ = δϕ ϕ 1. We have 0 (cid:96) ϕ 1 ϕ · · · · · ◦ − (ϕ + δϕ) q = (Id + v) (ϕ q) = ϕ q + ξ · v + o(v) = ϕ q + ξ · ϕ˙ ϕ 1 + o(δϕ), IRd ϕ q ϕ q (cid:55)→ · and therefore the mapping ϕ ϕ q is diﬀerentiable at ϕ, with continuous diﬀerential ξ · dR − . ϕ q ϕ 1 The result follows. 9 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: list

text: The

text: mapping

text: C

text: ξ

text: M

text: IRd)

text: X

text: (q,

text: v)

text: ξ

text: v

text: q

text: C

text: C

text: is

text: continuous,

text: and

text: its

text: restriction

text: to

text: M

text: IRd)

text: is

text: of

text: class

text: In

text: particular

text: the

text: C

text: C

text: mapping

text: q

text: ξ

text: v

text: is

text: of

text: class

text: for

text: every

text: bounded

text: vector

text: ﬁeld

text: v

text: of

text: class

text: q

text: Example

text: For

text: the

text: action

text: of

text: Diﬀ

text: on

text: itself

text: by

text: left

text: composition

text: makes

text: it

text: a

text: shape

text: space

text: of

text: order

text: in

text: IRd.

text: C

text: Example

text: Let

text: IN

text: and

text: let

text: S

text: be

text: a

text: smooth

text: compact

text: Riemannian

text: manifold.

text: Consider

text: the

text: C

text: space

text: M

text: X

text: IRd)

text: equipped

text: with

text: its

text: usual

text: Banach

text: norm.

text: Then

text: M

text: is

text: a

text: shape

text: space

text: of

text: order

text: where

text: the

text: action

text: of

text: Diﬀ

text: is

text: given

text: by

text: the

text: composition

text: ϕ

text: q

text: ϕ

text: q.

text: Indeed,

text: it

text: is

text: continuous

text: thanks

text: to

text: the

text: rule

text: for

text: computing

text: derivatives

text: of

text: a

text: composition,

text: and

text: we

text: also

text: have

text: ϕ

text: q

text: ϕ

text: q

text: γ

text: ϕ

text: ϕ

text: X

text: C

text: Moreover,

text: given

text: q

text: M

text: and

text: v

text: IRd),

text: ξ

text: v

text: is

text: the

text: vector

text: ﬁeld

text: along

text: q

text: given

text: by

text: ξ

text: (v)

text: v

text: q

text: C

text: q

text: q

text: IRd).

text: Finally,

text: the

text: formula

text: for

text: computing

text: derivatives

text: of

text: a

text: composition

text: yields

text: v

text: (q

text: δq)

text: v

text: q

text: dv

text: .δq

text: o(

text: δq

text: q

text: X

text: X

text: for

text: every

text: δq

text: M,

text: and

text: the

text: last

text: part

text: of

text: the

text: deﬁnition

text: follows.

text: This

text: framework

text: describes

text: most

text: of

text: shape

text: spaces.

text: An

text: interesting

text: particular

text: case

text: of

text: this

text: general

text: example

text: is

text: when

text: S

text: (s

text: s

text: is

text: a

text: ﬁnite

text: set

text: n

text: (zero-dimensional

text: manifold),

text: X

text: (IRd)n

text: and

text: M

text: Lmk

text: (n)

text: (x

text: x

text: (IRd)n

text: x

text: x

text: if

text: i

text: j

text: d

text: n

text: i

text: j

text: is

text: a

text: (so-called)

text: space

text: of

text: n

text: landmarks

text: in

text: IRd.

text: For

text: q

text: (x

text: x

text: the

text: smooth

text: action

text: of

text: order

text: C

text: n

text: is

text: ϕ

text: q

text: (ϕ(x

text: ϕ(x

text: For

text: v

text: the

text: inﬁnitesimal

text: action

text: of

text: v

text: at

text: q

text: is

text: given

text: by

text: n

text: ξ

text: (v)

text: (v(x

text: v(x

text: q

text: n

text: Remark

text: In

text: most

text: cases,

text: and

text: in

text: all

text: examples

text: given

text: throughout

text: this

text: paper,

text: the

text: mapping

text: ξ

text: C

text: C

text: restricted

text: to

text: M

text: IRd)

text: is

text: of

text: class

text: k,

text: for

text: every

text: k

text: IN.

text: C

text: Proposition

text: For

text: every

text: q

text: M,

text: the

text: mapping

text: ϕ

text: ϕ

text: q

text: is

text: of

text: class

text: and

text: its

text: diﬀerential

text: at

text: ϕ

text: is

text: given

text: by

text: ξ

text: dR

text: In

text: particular,

text: given

text: q

text: M

text: and

text: given

text: ϕ

text: a

text: deformation

text: of

text: IRd

text: of

text: class

text: C

text: ϕ

text: q

text: ϕ

text: which

text: is

text: the

text: ﬂow

text: of

text: the

text: time-dependent

text: vector

text: ﬁeld

text: v,

text: the

text: curve

text: t

text: q(t)

text: ϕ(t)

text: q

text: is

text: of

text: class

text: and

text: one

text: has

text: q˙(t)

text: ξ

text: ϕ˙(t)

text: ϕ(t)

text: ξ

text: v(t),

text: ϕ(t)

text: q(t)

text: for

text: almost

text: every

text: t

text: C

text: Proof.

text: Let

text: q

text: M,

text: ﬁx

text: ϕ

text: Diﬀ

text: and

text: take

text: δϕ

text: T

text: Diﬀ

text: IRd).

text: Then

text: ϕ

text: δϕ

text: ϕ

text: 0−

text: Diﬀ

text: for

text: δϕ

text: small

text: enough.

text: We

text: deﬁne

text: v

text: (dR

text: δϕ

text: δϕ

text: ϕ

text: We

text: have

text: ϕ

text: ϕ

text: (ϕ

text: δϕ)

text: q

text: (Id

text: v)

text: (ϕ

text: q)

text: ϕ

text: q

text: ξ

text: v

text: o(v)

text: ϕ

text: q

text: ξ

text: ϕ˙

text: ϕ

text: o(δϕ),

text: IRd

text: ϕ

text: q

text: ϕ

text: q

text: and

text: therefore

text: the

text: mapping

text: ϕ

text: ϕ

text: q

text: is

text: diﬀerentiable

text: at

text: ϕ,

text: with

text: continuous

text: diﬀerential

text: ξ

text: dR

text: ϕ

text: q

text: ϕ

text: The

text: result

text: follows.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: How well does the FoF algorithm recover group properties? 7 (a) Projected space galaxy mock catalog (b) Redshift space galaxy mock catalog Figure 2. Views of our initial, ﬂux-limited, 2.2 sr SDSS mock galaxy catalog, projection on the celestial sphere (left) and 3D cone (right, not sliced). The colors provide the absolute r-band magnitude (green for low luminosity). TG to the EG that contains the TG’s MMG. With this deﬁnition for We deﬁned reliability as the fraction of galaxies in the EG that linking, we could easily associate FoF groups to real groups. were members of the parent TG (i.e., within the sphere of radius 4.2 Global tests r ). Here, we also limited our tests to the primary EGs. 200 Mathematically speaking, these deﬁnitions of galaxy com- Our deﬁnition of the link between EGs and TGs allowed us pleteness, C, and reliability, R, can respectively be written as to search for cases where there is no one-to-one correspondence between the groups in real and redshift space: a TG can suffer from ∩ fragmentation into several EGs, while an EG can be built from the TG EG C = , merging of several TGs. T∩G Figure 3 illustrates different cases (following an analogous ﬁg- TG EG R = . ure in Knobel et al. 2009). The top panel shows a one-to-one cor- EG respondence between the true and extracted groups. We deﬁned a fragmented TG as one that contains the MMGs Looking at Figure 3, the completeness is the fraction of galax- of several EGs. Multiple situations can cause fragmentation of TGs. ies in the TG (left, green circles) recovered in the EG (right, red In some cases, the FoF algorithm fails to recover entire TGs, se- circles), while the reliability is the fraction of galaxies in the EG lecting instead its primary and secondary substructures (see panel that belong to the TG. Fig. 3b). In other cases, an EG is mostly composed of galaxies from These four quantities allow one to deﬁne the capacity of the one TG, but the MMG of another TG is ‘accidentally’ linked to the FoF grouping algorithm (or any other grouping algorithm) to re- ﬁrst TG. In consequence, the EG could be linked to a TG providing cover groups in real space from galaxy catalogs in redshift space. only a single member galaxy to the EG, in comparison with more Note that EGs that are fragments can have high reliability, members arising from another TG. When fragmentation occurred, while fragmentation causes primary EGs to have reduced complete- we distinguished the primary EG, as that whose MMG corresponds ness. When EGs are mergers of TGs, the secondary TGs lead to a to the MMG of the parent TG, from the other EGs, which we called decrease in the reliability, but can have high completeness. fragments. 4.4 Mass accuracy The dual of fragmentation is merging. In this situation, an EG There are many properties of groups that one wishes to recover contains the MMGs of several TGs. Proceeding similarly as for the with optimal accuracy (see Sect. ). We focused here on one single case of fragmentation, we denoted primary TG of a given EG the TG whose MMG corresponds to the MMG of that EG, denoting property that appeared to us as the(cid:80)most relevant: the group total mass. We measured the masses of our EGs using the virial theorem the other TGs as secondary. An example of merging is shown in (cid:80) formula of Heisler, Tremaine, & Bahcall (1985) Figure 3c. Note that a true group can be fragmented and its primary extracted group can be the result of a merger of the true group with M = 3π (cid:104)R(cid:105) σ2 = 3π N vi2 , (17) another one, as illustrated in Figure 3d. EG G h v 2 G i<j 1/Rij 4.3 Local tests (cid:104) (cid:105) (cid:104) (cid:105)− where R = 1/R 1 is the harmonic mean projected h ij Our local tests check the membership of the EGs. We deﬁned separation, while σ is the unbiased measure of the standard devia- v completeness as the fraction of galaxies in the TG (i.e. within the tion of the group velocities, given as solutions of equation (16) for sphere of radius r ) that were members of the primary EG. Given vLOS, replacing z by the redshift of the MMG of the EG. 200 p cos this deﬁnition, it did not make sense to consider the completeness More precisely, we computed the accuracy of the log masses, for secondary fragments, hence we limited our tests to the primary respectively deﬁning the bias and inefﬁciency as the median EGs. and equivalent standard deviation (half 16–84 interpercentile) of (cid:13) c 2014 RAS, MNRAS 000, 000–000 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: How

text: well

text: does

text: the

text: FoF

text: algorithm

text: recover

text: group

text: properties?

text: (a)

text: Projected

text: space

text: galaxy

text: mock

text: catalog

text: (b)

text: Redshift

text: space

text: galaxy

text: mock

text: catalog

text: Figure

text: Views

label: caption

text: of

text: our

text: initial,

text: ﬂux-limited,

text: sr

text: SDSS

text: mock

text: galaxy

text: catalog,

text: projection

text: on

text: the

text: celestial

text: sphere

text: (left)

text: and

text: cone

text: (right,

text: not

text: sliced).

text: The

text: colors

text: provide

text: the

text: absolute

text: r-band

text: magnitude

text: (green

text: for

text: low

text: luminosity).

text: TG

text: to

text: the

text: EG

text: that

text: contains

text: the

text: TG’s

text: MMG.

text: With

text: this

text: deﬁnition

text: for

text: We

text: deﬁned

text: reliability

text: as

text: the

text: fraction

text: of

text: galaxies

text: in

text: the

text: EG

text: that

text: linking,

text: we

text: could

text: easily

text: associate

text: FoF

text: groups

text: to

text: real

text: groups.

text: were

text: members

text: of

text: the

text: parent

text: TG

text: (i.e.,

text: within

text: the

text: sphere

text: of

text: radius

text: Global

text: tests

text: r

text: Here,

text: we

text: also

text: limited

text: our

text: tests

text: to

text: the

text: primary

text: EGs.

text: Mathematically

text: speaking,

text: these

text: deﬁnitions

text: of

text: galaxy

text: com-

text: Our

text: deﬁnition

text: of

text: the

text: link

text: between

text: EGs

text: and

text: TGs

text: allowed

text: us

text: pleteness,

text: C,

text: and

text: reliability,

text: R,

text: can

text: respectively

text: be

text: written

text: as

text: to

text: search

text: for

text: cases

text: where

text: there

text: is

text: no

text: one-to-one

text: correspondence

text: between

text: the

text: groups

text: in

text: real

text: and

text: redshift

text: space:

text: a

text: TG

text: can

text: suffer

text: from

text: fragmentation

text: into

text: several

text: EGs,

text: while

text: an

text: EG

text: can

text: be

text: built

text: from

text: the

text: TG

text: EG

text: C

text: merging

text: of

text: several

text: TGs.

text: T∩G

text: Figure

text: illustrates

text: different

text: cases

text: (following

text: an

text: analogous

text: ﬁg-

text: TG

text: EG

text: R

text: ure

text: in

text: Knobel

text: et

text: al.

text: The

text: top

text: panel

text: shows

text: a

text: one-to-one

text: cor-

text: EG

text: respondence

text: between

text: the

text: true

text: and

text: extracted

text: groups.

text: We

text: deﬁned

text: a

text: fragmented

text: TG

text: as

text: one

text: that

text: contains

text: the

text: MMGs

text: Looking

text: at

text: Figure

text: the

text: completeness

text: is

text: the

text: fraction

text: of

text: galax-

text: of

text: several

text: EGs.

text: Multiple

text: situations

text: can

text: cause

text: fragmentation

text: of

text: TGs.

text: ies

text: in

text: the

text: TG

text: (left,

text: green

text: circles)

text: recovered

text: in

text: the

text: EG

text: (right,

text: red

text: In

text: some

text: cases,

text: the

text: FoF

text: algorithm

text: fails

text: to

text: recover

text: entire

text: TGs,

text: se-

text: circles),

text: while

text: the

text: reliability

text: is

text: the

text: fraction

text: of

text: galaxies

text: in

text: the

text: EG

text: lecting

text: instead

text: its

text: primary

text: and

text: secondary

text: substructures

text: (see

text: panel

text: that

text: belong

text: to

text: the

text: TG.

text: Fig.

text: In

text: other

text: cases,

text: an

text: EG

text: is

text: mostly

text: composed

text: of

text: galaxies

text: from

text: These

text: four

text: quantities

text: allow

text: one

text: to

text: deﬁne

text: the

text: capacity

text: of

text: the

text: one

text: TG,

text: but

text: the

text: MMG

text: of

text: another

text: TG

text: is

text: ‘accidentally’

text: linked

text: to

text: the

text: FoF

text: grouping

text: algorithm

text: (or

text: any

text: other

text: grouping

text: algorithm)

text: to

text: re-

text: ﬁrst

text: TG.

text: In

text: consequence,

text: the

text: EG

text: could

text: be

text: linked

text: to

text: a

text: TG

text: providing

text: cover

text: groups

text: in

text: real

text: space

text: from

text: galaxy

text: catalogs

text: in

text: redshift

text: space.

text: only

text: a

text: single

text: member

text: galaxy

text: to

text: the

text: EG,

text: in

text: comparison

text: with

text: more

text: Note

text: that

text: EGs

text: that

text: are

text: fragments

text: can

text: have

text: high

text: reliability,

text: members

text: arising

text: from

text: another

text: TG.

text: When

text: fragmentation

text: occurred,

text: while

text: fragmentation

text: causes

text: primary

text: EGs

text: to

text: have

text: reduced

text: complete-

text: we

text: distinguished

text: the

text: primary

text: EG,

text: as

text: that

text: whose

text: MMG

text: corresponds

text: ness.

text: When

text: EGs

text: are

text: mergers

text: of

text: TGs,

text: the

text: secondary

text: TGs

text: lead

text: to

text: a

text: to

text: the

text: MMG

text: of

text: the

text: parent

text: TG,

text: from

text: the

text: other

text: EGs,

text: which

text: we

text: called

text: decrease

text: in

text: the

text: reliability,

text: but

text: can

text: have

text: high

text: completeness.

text: fragments.

text: Mass

text: accuracy

text: The

text: dual

text: of

text: fragmentation

text: is

text: merging.

text: In

text: this

text: situation,

text: an

text: EG

text: There

text: are

text: many

text: properties

text: of

text: groups

text: that

text: one

text: wishes

text: to

text: recover

text: contains

text: the

text: MMGs

text: of

text: several

text: TGs.

text: Proceeding

text: similarly

text: as

text: for

text: the

text: with

text: optimal

text: accuracy

text: (see

text: Sect.

text: We

text: focused

text: here

text: on

text: one

text: single

text: case

text: of

text: fragmentation,

text: we

text: denoted

text: primary

text: TG

text: of

text: a

text: given

text: EG

text: the

text: TG

text: whose

text: MMG

text: corresponds

text: to

text: the

text: MMG

text: of

text: that

text: EG,

text: denoting

text: property

text: that

text: appeared

text: to

text: us

text: as

text: relevant:

text: the

text: group

text: total

text: mass.

text: We

text: measured

text: the

text: masses

text: of

text: our

text: EGs

text: using

text: the

text: virial

text: theorem

text: the

text: other

text: TGs

text: as

text: secondary.

text: An

text: example

text: of

text: merging

text: is

text: shown

text: in

text: formula

text: of

text: Heisler,

text: Tremaine,

text: Bahcall

text: Figure

text: Note

text: that

text: a

text: true

text: group

text: can

text: be

text: fragmented

text: and

text: its

text: primary

text: extracted

text: group

text: can

text: be

text: the

text: result

text: of

text: a

text: merger

text: of

text: the

text: true

text: group

text: with

text: M

text: 3π

text: σ2

text: 3π

text: N

text: another

text: one,

text: as

text: illustrated

text: in

text: Figure

text: EG

text: G

text: h

text: v

text: G

text: i<j

text: Local

text: tests

text: where

text: R

text: is

text: the

text: harmonic

text: mean

text: projected

text: h

text: ij

text: Our

text: local

text: tests

text: check

text: the

text: membership

text: of

text: the

text: EGs.

text: We

text: deﬁned

text: separation,

text: while

text: σ

text: is

text: the

text: unbiased

text: measure

text: of

text: the

text: standard

text: devia-

text: v

text: completeness

text: as

text: the

text: fraction

text: of

text: galaxies

text: in

text: the

text: TG

text: (i.e.

text: within

text: the

text: tion

text: of

text: the

text: group

text: velocities,

text: given

text: as

text: solutions

text: of

text: equation

text: for

text: sphere

text: of

text: radius

text: r

text: that

text: were

text: members

text: of

text: the

text: primary

text: EG.

text: Given

text: vLOS,

text: replacing

text: z

text: by

text: the

text: redshift

text: of

text: the

text: MMG

text: of

text: the

text: EG.

text: p

text: cos

text: this

text: deﬁnition,

text: it

text: did

text: not

text: make

text: sense

text: to

text: consider

text: the

text: completeness

text: More

text: precisely,

text: we

text: computed

text: the

text: accuracy

text: of

text: the

text: log

text: masses,

text: for

text: secondary

text: fragments,

text: hence

text: we

text: limited

text: our

text: tests

text: to

text: the

text: primary

text: respectively

text: deﬁning

text: the

text: bias

text: and

text: inefﬁciency

text: as

text: the

text: median

text: EGs.

text: and

text: equivalent

text: standard

text: deviation

text: (half

text: interpercentile)

text: of

text: c

text: RAS,

text: MNRAS

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: − band for the range of values of η previously quoted (10 20 eV or less) that correspond to 0 the allowed values of f . a It may be interesting to think what would happen if one attempts to produce a ‘forbidden’ photon, i.e. one whose momentum falls in one of the forbidden bands. A photon with such a wave number is ‘oﬀ-shell’ and as such it will always decay. For instance, it could decay into three other photons with appropiately lower energies. However, because the oﬀ-shellness − is so small (typically 10 20 eV or less) it could live for a long time as a metastable state, travelling distances commensurable with the solar system. For more technical details see e.g. Ref. [? ]. We realize that the small bandwidth of the forbidden momentum bands make them unobservable in practice. However their mere existence is of theoretical interest. Conclusions might be diﬀerent for other axion-like backgrounds. IV. PROPER MODES IN A MAGNETIC FIELD AND AXION BACKGROUND (cid:54) In the presence of a magnetic ﬁeld, but no CAB (b = 0, η = 0) there is no longer a time 0 dependence in the coeﬃcients of the equations, so we can Fourier transform with respect to time as well. We ﬁnd the following dispersion relations: (cid:18) (cid:19) (cid:112) (cid:18) (cid:19) m2 + b2 1(cid:112) ≈ b2 ω2 = k2 + a + (m2 + b2)2 + 4b2k2 (k2 + m2) 1 + a 2 2 a a m2 a m2 + b2 − 1 ≈ − b2 ω2 = k2 + a (m2 + b2)2 + 4b2k2 k2 1 1 2 2 a m2 a ω2 = k2, (15) 2 ≈ (cid:28) where the symbol indicates the limit bk 1. These results are well known [? ]. We m2 a have identiﬁed as corresponding to ‘photons’ the two modes that if b = 0 reduce to the two usual polarization modes. The third frequency corresponds predominantly to the axion (or (cid:107) axion-like particle), but of course it has also a small photon component as the polarized photon mixes with the axion. If laser light of frequency ω is injected into a cavity, the diﬀerent components will develop diﬀerent wave-numbers resulting in the appeareance of changes in the plane of polarization (ellipticity and rotation) unless the photon polarization is initially exactly parallel or exactly 11 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: band

text: for

text: the

text: range

text: of

text: values

text: of

text: η

text: previously

text: quoted

text: eV

text: or

text: less)

text: that

text: correspond

text: to

text: the

text: allowed

text: values

text: of

text: f

text: a

text: It

text: may

text: be

text: interesting

text: to

text: think

text: what

text: would

text: happen

text: if

text: one

text: attempts

text: to

text: produce

text: a

text: ‘forbidden’

text: photon,

text: i.e.

text: one

text: whose

text: momentum

text: falls

text: in

text: one

text: of

text: the

text: forbidden

text: bands.

text: A

text: photon

text: with

text: such

text: a

text: wave

text: number

text: is

text: ‘oﬀ-shell’

text: and

text: as

text: such

text: it

text: will

text: always

text: decay.

text: For

text: instance,

text: it

text: could

text: decay

text: into

text: three

text: other

text: photons

text: with

text: appropiately

text: lower

text: energies.

text: However,

text: because

text: the

text: oﬀ-shellness

text: is

text: so

text: small

text: (typically

text: eV

text: or

text: less)

text: it

text: could

text: live

text: for

text: a

text: long

text: time

text: as

text: a

text: metastable

text: state,

text: travelling

text: distances

text: commensurable

text: with

text: the

text: solar

text: system.

text: For

text: more

text: technical

text: details

text: see

text: e.g.

text: Ref.

text: We

text: realize

text: that

text: the

text: small

text: bandwidth

text: of

text: the

text: forbidden

text: momentum

text: bands

text: make

text: them

text: unobservable

text: in

text: practice.

text: However

text: their

text: mere

text: existence

text: is

text: of

text: theoretical

text: interest.

text: Conclusions

text: might

text: be

text: diﬀerent

text: for

text: other

text: axion-like

text: backgrounds.

text: IV.

text: PROPER

text: MODES

text: IN

text: A

text: MAGNETIC

text: FIELD

text: AND

text: AXION

text: BACKGROUND

text: In

text: the

text: presence

text: of

text: a

text: magnetic

text: ﬁeld,

text: but

text: no

text: CAB

text: (b

text: η

text: there

text: is

text: no

text: longer

text: a

text: time

text: dependence

text: in

text: the

text: coeﬃcients

text: of

text: the

text: equations,

text: so

text: we

text: can

text: Fourier

text: transform

text: with

text: respect

text: to

text: time

text: as

text: well.

text: We

text: ﬁnd

text: the

text: following

text: dispersion

text: relations:

text: ω2

text: a

text: a

text: a

text: a

text: a

text: ω2

text: a

text: a

text: a

text: ω2

text: where

text: the

text: symbol

text: indicates

text: the

text: limit

text: bk

text: These

text: results

text: are

text: well

text: known

text: We

text: a

text: have

text: identiﬁed

text: as

text: corresponding

text: to

text: ‘photons’

text: the

text: two

text: modes

text: that

text: if

text: b

text: reduce

text: to

text: the

text: two

text: usual

text: polarization

text: modes.

text: The

text: third

text: frequency

text: corresponds

text: predominantly

text: to

text: the

text: axion

text: (or

text: axion-like

text: particle),

text: but

text: of

text: course

text: it

text: has

text: also

text: a

text: small

text: photon

text: component

text: as

text: the

text: polarized

text: photon

text: mixes

text: with

text: the

text: axion.

text: If

text: laser

text: light

text: of

text: frequency

text: ω

text: is

text: injected

text: into

text: a

text: cavity,

text: the

text: diﬀerent

text: components

text: will

text: develop

text: diﬀerent

text: wave-numbers

text: resulting

text: in

text: the

text: appeareance

text: of

text: changes

text: in

text: the

text: plane

text: of

text: polarization

text: (ellipticity

text: and

text: rotation)

text: unless

text: the

text: photon

text: polarization

text: is

text: initially

text: exactly

text: parallel

text: or

text: exactly

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: CSD2007-00042 (CPAN). A. Renau acknowledges the ﬁnancial support of a FPU pre- doctoral grant. It is a pleasure to thank several of the participants in the 9th Patras Workshop for discussions. The authors are grateful to G. Cantatore for discussions on polarimetric experiments. Appendix A: Propagator Considering only the spatial components, eq. (52) of Ref. [? ] becomes: (cid:40) D bibj (cid:41) ij(ω, k) = Dij + iω2 − − − (k4 η2(cid:126)k2)(k2 m2) ω2k2b2 0 a − iη k2(biqj qibj) 0 + − − − − , (A1) (k4 η2(cid:126)k2)[(k4 η2(cid:126)k2)(k2 m2) ω2k2b2] 0 0 a (cid:32) (cid:33) where − P ij P−ij × Dij = i −+ | | + | | , (cid:126)q = ((cid:126)b (cid:126)k) (A2) k2 η (cid:126)k k2 + η (cid:126)k 0 0 and the projectors P± have been deﬁned in Ref. [? ]. Terms proportional to kikj have been dropped, since we are interested in contracting the propagator with a photon | | (cid:126) polarization vector. The roots of the denominators are k = F , with j (cid:113) η2 ∓ η ≈ ∓ F 2 = ω2 + 0 0 4ω2 + η2 ω2 ωη , 1,2 2 − 2 √ 0 √ 0 − m2 η2 ∓ F 2 = ω2 a 0 + W (cos χ 3 sin χ), 3,4 3− √ − m2 η2 − F 2 = ω2 a 0 2 W cos χ. (A3) 5 3 (cid:18) (cid:19) (cid:18) (cid:19) ≈ m2 2 3ω2b2 W a 1 + , 3√ m4 (cid:115)a ≈ (cid:18)1 (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) χ 3ωξ, m2 a − ≈ 9ω2b2 1 ωb2 2 ω2b3 2 ξ 1 + η2 + + . (A4) 2m4 0 2m2 m4 a 21 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: (CPAN).

text: A.

text: Renau

text: acknowledges

text: the

text: ﬁnancial

text: support

text: of

text: a

text: FPU

text: pre-

text: doctoral

text: grant.

text: It

text: is

text: a

text: pleasure

text: to

text: thank

text: several

text: of

text: the

text: participants

text: in

text: the

text: Patras

text: Workshop

text: for

text: discussions.

text: The

text: authors

text: are

text: grateful

text: to

text: G.

text: Cantatore

text: for

text: discussions

text: on

text: polarimetric

text: experiments.

text: Appendix

text: A:

text: Propagator

text: Considering

text: only

text: the

text: spatial

text: components,

text: eq.

text: of

text: Ref.

text: becomes:

text: D

text: bibj

text: ij(ω,

text: k)

text: Dij

text: a

text: iη

text: qibj)

text: a

text: where

text: P

text: ij

text: P−ij

text: Dij

text: i

text: η

text: η

text: and

text: the

text: projectors

text: P±

text: have

text: been

text: deﬁned

text: in

text: Ref.

text: Terms

text: proportional

text: to

text: kikj

text: have

text: been

text: dropped,

text: since

text: we

text: are

text: interested

text: in

text: contracting

text: the

text: propagator

text: with

text: a

text: photon

text: polarization

text: vector.

text: The

text: roots

text: of

text: the

text: denominators

text: are

text: k

text: F

text: with

text: j

text: η2

text: η

text: F

text: ω2

text: η2

text: ω2

text: ωη

text: η2

text: F

text: ω2

text: a

text: W

text: (cos

text: χ

text: sin

text: χ),

text: 3−

text: η2

text: F

text: ω2

text: a

text: W

text: cos

text: χ.

text: W

text: a

text: 3√

text: χ

text: 3ωξ,

text: a

text: ξ

text: η2

text: a

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: The propagator in position space is, after dropping an overall factor, (cid:16) (cid:17) (cid:16) (cid:17) ≈ (cid:88) η x −(cid:88) η x dij(ω, x) (P ij + P−ij) cos 0 + i(P ij P−ij) sin 0 + 2 + 2 5 − − 5 +ˆbiˆbj a eiαlx i(ˆbiqˆj qˆiˆbj) a˜ eiαlx, (A11) l l l=3 l=1 where ω3b2A − a = l , a˜ = ω3b2η A˜ , α = F ω. (A12) l l 0 l l l F l ≈ − All the α are proportional to η or b2, except for α m2a . Restricting ourselves only l 0 5 2ω −   to y z components, we can write d(ω, x) in matrix form.   1 0 P i + P−i = , (A13) +j j 0 1     − 0 1 i(P i P−i ) = − , (A14) +j j 1 0     − 1 0 ˆbiˆb = , (A15) j 0 0     − − − 0 i i(ˆbiqˆ qˆiˆb ) = . (A16) j j i 0 If we write (cid:88) (cid:88) − − a eiαlx = ((cid:15) + iϕ), i a˜ eiαlx = ((cid:15)˜+ iϕ˜), (A17) l l l l we have 23 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: The

text: propagator

text: in

text: position

text: space

text: is,

text: after

text: dropping

text: an

text: overall

text: factor,

text: η

text: x

text: η

text: x

text: dij(ω,

text: x)

text: (P

text: ij

text: P−ij)

text: cos

text: i(P

text: ij

text: P−ij)

text: sin

text: +ˆbiˆbj

text: a

text: eiαlx

text: i(ˆbiqˆj

text: qˆiˆbj)

text: a˜

text: eiαlx,

text: l

text: l

text: where

text: a

text: l

text: a˜

text: A˜

text: α

text: F

text: ω.

text: l

text: l

text: l

text: l

text: l

text: F

text: l

text: All

text: the

text: α

text: are

text: proportional

text: to

text: η

text: or

text: except

text: for

text: α

text: Restricting

text: ourselves

text: only

text: l

text: 2ω

text: to

text: y

text: z

text: components,

text: we

text: can

text: write

text: d(ω,

text: x)

text: in

text: matrix

text: form.

text: P

text: i

text: P−i

text: +j

text: j

text: i(P

text: i

text: P−i

text: +j

text: j

text: ˆbiˆb

text: j

text: i

text: i(ˆbiqˆ

text: qˆiˆb

text: j

text: j

text: i

text: If

text: we

text: write

text: a

text: eiαlx

text: iϕ),

text: i

text: a˜

text: eiαlx

text: iϕ˜),

text: l

text: l

text: l

text: l

text: we

text: have

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 12 (cid:104) (cid:105) (cid:104) (cid:105) 1 (cid:104) − | | −(cid:105) | | σˆ1zσˆ2z = 2 1 Ln(16 α 2)e 8 α 2 (B4) As is the case with the Bloch vector length calculation, Eq. B5 agrees with our earlier result using coherent states (cid:104)(cid:104) (cid:105)(cid:105) 1 − − | | ∼ − σˆzσˆz = 1 e 8 α 2(2n¯+1) . (B5) (Eq. 30) in the limit n¯ β 1. 1 2 th 2 [1] B. C. Sawyer et al., Phys. Rev. Lett. 108, 213003 (2012). (1981). [2] T. Rosenband et al., Science 319, 1808 (2008). [32] E. L. Hahn, Phys. Rev. 80, 580 (1950). [3] N. Shiga, W. M. Itano, and J. J. Bollinger, Phys. Rev. A [33] M. J. Biercuk, H. Uys, A. P. VanDevender, N. Shiga, 84, 012510 (2011). W. M. Itano, and J. J. Bollinger, Nature 458, 996 (2009). [4] R. Blatt and D. J. Wineland, Nature 453, 1008 (2008). [34] M. J. Biercuk, H. Uys, A. P. VanDevender, N. Shiga, [5] J. P. Home, Advances in Atomic, Molecular, and Optical W. M. Itano, and J. J. Bollinger, Quantum Info. and Physics, vol. 62 (Academic Press, Oxford, 2013). Comp. 9, 920 (2009). [6] R. Blatt and C. F. Roos, Nature Phys. 8, 277 (2012). [35] D. M. Meekhoﬀ, C. Monroe, B. E. King, W. M. Itano, [7] R. Islam et al., Science 340, 583 (2013). and D. J. Wineland, Phys. Rev. Lett. 76, 1796 (1996). [8] J. W. Britton et al., Nature 484, 489 (2012). [36] H. Uys, M. J. Biercuk, A. VanDevender, C. Ospelkaus, [9] H. M. V. Horn, Science 252, 384 (1991). D. Meiser, R. Ozeri, and J. J. Bollinger, Phys. Rev. Lett. [10] S. Ichimaru, H. Iyetomi, and S. Tanaka, Phys. Rep. 149, 105, 200401 (2010). 92 (1987). [37] M. J. Biercuk, J. W. Britton, H. Uys, A. VanDevender, [11] D. H. E. Dubin and H. Dewitt, Phys. Rev. B 49, 3043 and J. J. Bollinger, Nat. Nanotechnol. 5, 646 (2010). (1994). [38] L. Deslauriers, S. Olmschenk, D. Stick, W. K. Hensinger, [12] D. A. Baiko, Phys. Rev. E 80, 046405 (2009). J. Sterk, and C. Monroe, Phys. Rev. Lett. 97, 103007 [13] F. Anderegg, D. H. E. Dubin, T. M. O’Neil, and C. F. (2006). Driscoll, Phys. Rev. Lett. 102, 185001 (2009). 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