text: K-theory and Homotopies of 2-cocycles on Group Bundles Elizabeth Gillaspy May 26, 2020 Abstract This paper continues the author’s program to investigate the question { } of when a homotopy of 2-cocycles Ω = ω ∈ on a locally compact G t t [0,1] Hausdorﬀ groupoid induces an isomorphism of the K-theory groups of ∗ the twisted groupoid C -algebras: ∗ G ∼ ∗ G K∗(C ( , ω0)) = K∗(C ( , ω1)). G → Building on our earlier work in [5, 4], we show that if π : M is a locally trivial bundle of amenable groups over a locally compact Hausdorﬀ { } G space M, a homotopy Ω = ω ∈ of 2-cocycles on gives rise to an t t [0,1] isomorphism ∗ G ∼ ∗ G K∗(C ( , ω0)) = K∗(C ( , ω1)). ∗ Keywords: Twisted groupoid C -algebra, K-theory, group bundle, 2- cocycle. MSC (2010): 46L80, 46L55 1 Introduction → Let π : V M be a real or complex vector bundle over a manifold M. A → R { } bilinear 2-form σ : V (2) induces a homotopy of 2-cocycles ω ∈ on V : t t [0,1] If π(v) = π(w), deﬁne ω (v, w) := e2πitσ(v,w). t Plymen proved in Theorem 1 of [11] that when V is an even-dimensional real ∗ vector bundle and σ is a symplectic 2-form on V , the twisted C -algebra of the ∗ ∗ vector bundle C (V, ω ) is a continuous-trace C -algebra over M, with trivial 1 Dixmier-Douady class, and hence is Morita equivalent to C (M). 0 Furthermore, applying a ﬁberwise Fourier transform to V , one sees immedi- ately that ∼ ∗ ∗ ∗ C (V, ω ) = C (V ) = C (V ), 0 0 ∗ where V is the dual bundle to V . Since V is even-dimensional, the Thom isomorphism in K-theory tells us that ∼ ∗ K∗(C (V )) = K∗(C (M)); 0 0 1

text: K-theory

text: and

text: Homotopies

text: of

text: on

text: Group

text: Bundles

text: Elizabeth

text: Gillaspy

text: May

label: date

text: Abstract

text: This

text: paper

text: continues

text: the

text: author’s

text: program

text: to

text: investigate

text: the

text: question

text: of

text: when

text: a

text: homotopy

text: of

text: Ω

text: ω

text: on

text: a

text: locally

text: compact

text: G

text: t

text: t

text: Hausdorﬀ

text: groupoid

text: induces

text: an

text: isomorphism

text: of

text: the

text: K-theory

text: groups

text: of

text: the

text: twisted

text: groupoid

text: C

text: -algebras:

text: G

text: G

text: K∗(C

text: K∗(C

text: G

text: Building

text: on

text: our

text: earlier

text: work

text: in

text: we

text: show

text: that

text: if

text: π

text: M

text: is

text: a

text: locally

text: trivial

text: bundle

text: of

text: amenable

text: groups

text: over

text: a

text: locally

text: compact

text: Hausdorﬀ

text: G

text: space

text: M,

text: a

text: homotopy

text: Ω

text: ω

text: of

text: on

text: gives

text: rise

text: to

text: an

text: t

text: t

text: isomorphism

text: G

text: G

text: K∗(C

text: K∗(C

text: Keywords:

text: Twisted

text: groupoid

text: C

text: -algebra,

text: K-theory,

text: group

text: bundle,

text: cocycle.

text: MSC

text: Introduction

text: Let

text: π

text: V

text: M

text: be

text: a

text: real

text: or

text: complex

text: vector

text: bundle

text: over

text: a

text: manifold

text: M.

text: A

text: R

text: bilinear

text: σ

text: V

text: induces

text: a

text: homotopy

text: of

text: ω

text: on

text: V

text: t

text: t

text: If

text: π(v)

text: π(w),

text: deﬁne

text: ω

text: (v,

text: w)

text: t

text: Plymen

text: proved

text: in

text: Theorem

text: of

text: that

text: when

text: V

text: is

text: an

text: even-dimensional

text: real

text: vector

text: bundle

text: and

text: σ

text: is

text: a

text: symplectic

text: on

text: V

text: the

text: twisted

text: C

text: -algebra

text: of

text: the

text: vector

text: bundle

text: C

text: (V,

text: ω

text: is

text: a

text: continuous-trace

text: C

text: -algebra

text: over

text: M,

text: with

text: trivial

text: Dixmier-Douady

text: class,

text: and

text: hence

text: is

text: Morita

text: equivalent

text: to

text: C

text: (M).

text: Furthermore,

text: applying

text: a

text: ﬁberwise

text: Fourier

text: transform

text: to

text: V

text: one

text: sees

text: immedi-

text: ately

text: that

text: C

text: (V,

text: ω

text: C

text: (V

text: C

text: (V

text: where

text: V

text: is

text: the

text: dual

text: bundle

text: to

text: V

text: Since

text: V

text: is

text: even-dimensional,

text: the

text: Thom

text: isomorphism

text: in

text: K-theory

text: tells

text: us

text: that

text: K∗(C

text: (V

text: K∗(C

text: (M));


text: induces a 6-term exact sequence of the K-groups K (J) K (A) K (B) 0 0 0 K (B) K (A) K (J) 1 1 1 Thus, the short exact sequence (7) gives rise to the following 6-term exact sequence in K-theory: ⊕ K (A ∪ ) K (A ) K (A ) K (A ∩ ) 0 F1 F2 0 F1 0 F2 0 F1 F2 ⊕ K (A ∩ ) K (A ) K (A ) K (A ∪ ) 1 F1 F2 1 F1 1 F2 1 F1 F2 (9) ∗ G G → Since C ( , ω) is a C (M)-algebra whenever π : M is a group bundle, 0 we propose to use this diagram to study the K-theory groups associated to a G homotopy of 2-cocycles on . The following deﬁnition is a special case of [5] Deﬁnition 2.11. G → Given a group bundle π : M, we can construct the associated group G × → × G × bundle π˜ : [0, 1] M [0, 1], which has total space [0, 1], and ﬁber − ∈ × ∈ π 1(m) over (m, t) M [0, 1] for any t [0, 1]. G → Deﬁnition 3.1. A homotopy of 2-cocycles on a group bundle π : M is a G × → × 2-cocycle Ω on the group bundle π˜ : [0, 1] M [0, 1]. { } Observe that a homotopy Ω of 2-cocycles gives rise to a family ω ∈ of G → t t [0,1] 2-cocycles on the original group bundle M, which varies continuously in t thanks to the continuity of Ω. → → R Example 3.2. Suppose that π : V M is a vector bundle and that σ : V (2) × → T is a 2-form on V . The function Ω : V (2) [0, 1] given by Ω((v, w, t)) = exp2πitσ(v,w) is a homotopy of 2-cocycles on V , with ω (v, w) = 1; ω (v, w) = exp2πiσ(v,w) . 0 1 { } G ∗ If Ω = ω ∈ is a homotopy of 2-cocycles on , we have a natural - t t [0,1] homomorphism ∗ G × → ∗ G Q : C ( [0, 1], Ω) C ( , ω ) t t ∈ G × for any t [0, 1], which is given on the dense subalgebra C ( [0, 1]) by ⊆ c evaluation at t. Observe that if F M is closed, then ◦ ◦ Q q × = q Q , t F [0,1] F t 14 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: induces

label: paragraph

text: a

text: exact

text: sequence

text: of

text: the

text: K-groups

text: K

text: (J)

text: K

text: (A)

text: K

text: (B)

text: K

text: (B)

text: K

text: (A)

text: K

text: (J)

text: Thus,

text: the

text: short

text: exact

text: sequence

text: gives

text: rise

text: to

text: the

text: following

text: exact

text: sequence

text: in

text: K-theory:

label: equation

text: K

text: (A

text: K

text: (A

text: K

text: (A

text: K

text: (A

text: K

text: (A

text: K

text: (A

text: K

text: (A

text: K

text: (A

text: G

text: G

text: Since

text: C

text: ω)

text: is

text: a

text: C

text: (M)-algebra

text: whenever

text: π

text: M

text: is

text: a

text: group

text: bundle,

text: we

text: propose

text: to

text: use

text: this

text: diagram

text: to

text: study

text: the

text: K-theory

text: groups

text: associated

text: to

text: a

text: G

text: homotopy

text: of

text: on

text: The

text: following

text: deﬁnition

text: is

text: a

text: special

text: case

text: of

text: Deﬁnition

text: G

text: Given

text: a

text: group

text: bundle

text: π

text: M,

text: we

text: can

text: construct

text: the

text: associated

text: group

text: G

text: G

text: bundle

text: π˜

text: M

text: which

text: has

text: total

text: space

text: and

text: ﬁber

text: π

text: over

text: (m,

text: t)

text: M

text: for

text: any

text: t

text: G

text: Deﬁnition

text: A

text: homotopy

text: of

text: on

text: a

text: group

text: bundle

text: π

text: M

text: is

text: a

text: G

text: Ω

text: on

text: the

text: group

text: bundle

text: π˜

text: M

text: Observe

text: that

text: a

text: homotopy

text: Ω

text: of

text: gives

text: rise

text: to

text: a

text: family

text: ω

text: of

text: G

text: t

text: t

text: on

text: the

text: original

text: group

text: bundle

text: M,

text: which

text: varies

text: continuously

text: in

text: t

text: thanks

text: to

text: the

text: continuity

text: of

text: Ω.

text: R

text: Example

text: Suppose

text: that

text: π

text: V

text: M

text: is

text: a

text: vector

text: bundle

text: and

text: that

text: σ

text: V

text: T

text: is

text: a

text: on

text: V

text: The

text: function

text: Ω

text: V

text: given

text: by

text: Ω((v,

text: w,

text: t))

text: is

text: a

text: homotopy

text: of

text: on

text: V

text: with

text: ω

text: (v,

text: w)

text: ω

text: (v,

text: w)

text: G

text: If

text: Ω

text: ω

text: is

text: a

text: homotopy

text: of

text: on

text: we

text: have

text: a

text: natural

text: t

text: t

text: homomorphism

text: G

text: G

text: Q

text: C

text: Ω)

text: C

text: ω

text: t

text: t

text: G

text: for

text: any

text: t

text: which

text: is

text: given

text: on

text: the

text: dense

text: subalgebra

text: C

text: by

text: c

text: evaluation

text: at

text: t.

text: Observe

text: that

text: if

text: F

text: M

text: is

text: closed,

text: then

text: Q

text: q

text: q

text: Q

text: t

text: F

text: F

text: t

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: → When we consider the particular case when V M is a vector bundle, we obtain the following generalization of Theorem 1 from [11]: → → R Corollary 3.5. Let V M be a vector bundle, and let σ : V (2) be a bilinear 2-form on V . Setting ω(v, w) = e2πiσ(v,w), we have ∼ ∗ ∗ ∗ K∗(C (V, ω)) = K∗(C (V )) = K∗(C (V )). 0 In particular, if V is even dimensional, then ∼ ∗ K∗(C (V, ω)) = K∗(C (M)). 0 References [1] B. Blackadar, K-theory for operator algebras, Cambridge University Press, 1998. ∗ [2] J.H. Brown and A. an Huef, Decomposing the C -algebras of groupoid ex- tensions, Proceedings of the American Mathematical Society 142 (2014), 1261–1274. ∗ [3] L.O. Clark and A. an Huef, The representation theory of C -algebras as- sociated to groupoids, Mathematical Proceedings of the Cambridge Philo- sophical Society 153 (2012), 167–191. [4] E. Gillaspy, K-theory and homotopies of 2-cocycles on higher-rank graphs, arXiv:1403.3799 (2014). [5] , K-theory and homotopies of 2-cocycles on transformation groups, Journal of Operator Theory (to appear). [6] G. Goehle, The Mackey machine for crossed products by regular groupoids. II, Rocky Mountain Journal of Mathematics 42 (2012), 873–900. [7] A. an Huef, A. Kumjian, and A. Sims, A Dixmier-Douady theorem for Fell algebras, Journal of Functional Analysis 260 (2011), 1543–1581. ∗ [8] A. Kumjian, On C -diagonals, Canadian Journal of Mathematics 38 (1986), 969–1008. [9] P.S. Muhly, J.N. Renault, and D.P. Williams, Continuous-trace groupoid ∗ C -algebras. III, Transactions of the American Mathematical Society 348 (1996), 3621–3641. ∗ [10] P.S. Muhly and D.P. Williams, Continuous trace groupoid C -algebras II, Mathematica Scandinavica 70 (1992), 127–145. [11] R.J. Plymen, The Weyl bundle, Journal of Functional Analysis 49 (1982), 186–197. 17 ##LTLine##

text: When

text: we

text: consider

text: the

text: particular

text: case

text: when

text: V

text: M

text: is

text: a

text: vector

text: bundle,

text: we

text: obtain

text: the

text: following

text: generalization

text: of

text: Theorem

text: from

text: R

text: Corollary

text: Let

text: V

text: M

text: be

text: a

text: vector

text: bundle,

text: and

text: let

text: σ

text: V

text: be

text: a

text: bilinear

text: on

text: V

text: Setting

text: ω(v,

text: w)

text: we

text: have

text: K∗(C

text: (V,

text: ω))

text: K∗(C

text: (V

text: K∗(C

text: (V

text: In

text: particular,

text: if

text: V

text: is

text: even

text: dimensional,

text: then

text: K∗(C

text: (V,

text: ω))

text: K∗(C

text: (M)).

text: References

label: reference

text: B.

text: Blackadar,

text: K-theory

text: for

text: operator

text: algebras,

text: Cambridge

text: University

text: Press,

text: J.H.

text: Brown

text: and

text: A.

text: an

text: Huef,

text: Decomposing

text: the

text: C

text: -algebras

text: of

text: groupoid

text: ex-

text: tensions,

text: Proceedings

text: of

text: the

text: American

text: Mathematical

text: Society

text: L.O.

text: Clark

text: and

text: A.

text: an

text: Huef,

text: The

text: representation

text: theory

text: of

text: C

text: -algebras

text: as-

text: sociated

text: to

text: groupoids,

text: Mathematical

text: Proceedings

text: of

text: the

text: Cambridge

text: Philo-

text: sophical

text: Society

text: E.

text: Gillaspy,

text: K-theory

text: and

text: homotopies

text: of

text: on

text: higher-rank

text: graphs,

text: K-theory

text: and

text: homotopies

text: of

text: on

text: transformation

text: groups,

text: Journal

text: of

text: Operator

text: Theory

text: (to

text: appear).

text: G.

text: Goehle,

text: The

text: Mackey

text: machine

text: for

text: crossed

text: products

text: by

text: regular

text: groupoids.

text: II,

text: Rocky

text: Mountain

text: Journal

text: of

text: Mathematics

text: A.

text: an

text: Huef,

text: A.

text: Kumjian,

text: and

text: A.

text: Sims,

text: A

text: Dixmier-Douady

text: theorem

text: for

text: Fell

text: algebras,

text: Journal

text: of

text: Functional

text: Analysis

text: A.

text: Kumjian,

text: On

text: C

text: -diagonals,

text: Canadian

text: Journal

text: of

text: Mathematics

text: P.S.

text: Muhly,

text: J.N.

text: Renault,

text: and

text: D.P.

text: Williams,

text: Continuous-trace

text: groupoid

text: C

text: -algebras.

text: III,

text: Transactions

text: of

text: the

text: American

text: Mathematical

text: Society

text: P.S.

text: Muhly

text: and

text: D.P.

text: Williams,

text: Continuous

text: trace

text: groupoid

text: C

text: -algebras

text: II,

text: Mathematica

text: Scandinavica

text: R.J.

text: Plymen,

text: The

text: Weyl

text: bundle,

text: Journal

text: of

text: Functional

text: Analysis

text: ##LTLine##


text: [12] M. Rørdam, F. Larsen, and N.J. Laustsen, An introduction to K-theory for ∗ C -algebras, London Mathematical Society Student Texts, no. 49, Cam- bridge University Press, 2000. ∗ [13] J. Renault, A groupoid approach to C -algebras, Lecture Notes in Mathe- matics, vol. 793, Springer-Verlag, 1980. [14] , Topological amenability is a Borel property, arXiv:1302:0636 (2013). [15] J.-L. Tu, P. Xu, and C. Laurent-Gengoux, Twisted K-theory of diﬀer- entiable stacks, Annales Scientiﬁques de l’E´cole Normale Sup´erieure 37 (2004), 841–910. ∗ [16] N.E. Wegge-Olsen, K-theory and C -algebras: A friendly approach, Oxford University Press, 1993. ∗ [17] D.P. Williams, Crossed products of C -algebras, Mathematical Surveys & Monographs, vol. 134, AMS, 2007. 18 ##LTLine##

text: M.

text: Rørdam,

text: F.

text: Larsen,

text: and

text: N.J.

text: Laustsen,

text: An

text: introduction

text: to

text: K-theory

text: for

text: C

text: -algebras,

text: London

text: Mathematical

text: Society

text: Student

text: Texts,

text: no.

text: Cam-

text: bridge

text: University

text: Press,

text: J.

text: Renault,

text: A

text: groupoid

text: approach

text: to

text: C

text: -algebras,

text: Lecture

text: Notes

text: in

text: Mathe-

text: matics,

text: vol.

text: Springer-Verlag,

text: Topological

text: amenability

text: is

text: a

text: Borel

text: property,

text: J.-L.

text: Tu,

text: P.

text: Xu,

text: and

text: C.

text: Laurent-Gengoux,

text: Twisted

text: K-theory

text: of

text: diﬀer-

text: entiable

text: stacks,

text: Annales

text: Scientiﬁques

text: de

text: l’E´cole

text: Normale

text: Sup´erieure

text: N.E.

text: Wegge-Olsen,

text: K-theory

text: and

text: C

text: -algebras:

text: A

text: friendly

text: approach,

text: Oxford

text: University

text: Press,

text: D.P.

text: Williams,

text: Crossed

text: products

text: of

text: C

text: -algebras,

text: Mathematical

text: Surveys

text: Monographs,

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text: References

text: Elkon

text: R,

text: Linhart

text: C,

text: Sharan

text: R,

text: Shamir

text: R,

text: Shiloh

text: Y:

text: Genome-wide

text: in

text: silico

text: identiﬁcation

text: of

text: transcriptional

text: regulators

text: controlling

text: the

text: cell

text: cycle

text: in

text: human

text: cells.

text: Genome

text: Research

text: Srihari

text: S,

text: Leong

text: HW:

text: Temporal

text: dynamics

text: of

text: protein

text: complexes

text: in

text: PPI

text: networks:

text: a

text: case

text: study

text: using

text: yeast

text: cell

text: cycle

text: dynamics.

text: BMC

text: Bioinformatics

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text: Siegfried

text: Z,

text: Brandeis

text: M

text: et

text: al.:

text: Genome-wide

text: transcriptional

text: analysis

text: of

text: the

text: human

text: cell

text: cycle

text: identiﬁes

text: genes

text: diﬀerentially

text: regulated

text: in

text: normal

text: and

text: cancer

text: cells.

text: Proceedings

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text: National

text: Academy

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text: Sciences

text: USA

text: Spriggs

text: KA,

text: Bushell

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text: Translational

text: regulation

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text: Molecular

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text: Luscombe

text: NM,

text: Babu

text: MM,

text: Yu

text: H,

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text: M,

text: Teichmann

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text: Gerstein

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text: Genomic

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text: regulatory

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text: large

text: topological

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text: Nature

text: Balaji

text: S,

text: Babu

text: MD,

text: Lakshminarayan

text: MI,

text: Luscombe

text: NM,

text: Aravind

text: L:

text: Comprehensive

text: analysis

text: of

text: combinato-

text: rial

text: regulation

text: using

text: the

text: transcriptional

text: regulatory

text: network

text: of

text: yeast.

text: Journal

text: of

text: Molecular

text: Biology

text: Nebert

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text: Transcription

text: factors

text: and

text: cancer:

text: an

text: overview.

text: Toxicology

text: Darnell

text: JE:

text: Transcription

text: factors

text: as

text: targets

text: for

text: cancer

text: therapy.

text: Nature

text: Reviews

text: Cancer

text: Karamouzis

text: MV,

text: Papavassiliou

text: AG:

text: Transcription

text: factor

text: networks

text: as

text: targets

text: for

text: therapeutic

text: intervention

text: of

text: cancer:

text: the

text: breast

text: cancer

text: paradigm.

text: Molecular

text: Medicine

text: Vanunu

text: O,

text: Magger

text: P,

text: Ruppin

text: E,

text: Shlomi

text: T,

text: Sharan

text: R:

text: Associating

text: genes

text: and

text: protein

text: complexes

text: with

text: disease

text: via

text: network

text: propagation.

text: PLoS

text: Computational

text: Biology

text: Zhao

text: J,

text: Lee

text: SH,

text: Huss

text: M,

text: Holme

text: P:

text: The

text: network

text: organization

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text: cancer-associated

text: protein

text: complexes

text: in

text: human

text: tissues.

text: Scientiﬁc

text: Reports

text: Jansen

text: R.,

text: Greenbaum

text: D.,

text: Gernstein

text: M.:

text: Relating

text: whole-genome

text: expression

text: data

text: with

text: protein-protein

text: interactions.

text: Genome

text: Research

text: Grigoriev

text: A.:

text: A

text: relationship

text: between

text: gene

text: expression

text: and

text: protein

text: interactions

text: on

text: the

text: proteome

text: scale:

text: analysis

text: of

text: the

text: bacteriophage

text: and

text: the

text: yeast

text: Saccharomyces

text: cerevisiae.

text: Nucleic

text: Acids

text: Research

text: Safran

text: M,

text: Solomon

text: I,

text: Shmueli

text: O,

text: Lapidot

text: M

text: et

text: al.:

text: GeneCards

text: towards

text: a

text: complete,

text: object-oriented,

text: human

text: gene

text: compendium.

text: Bioinformatics

text: Fujioka

text: S,

text: Sclabas

text: GM,

text: Schmidt

text: C,

text: Frederick

text: WA

text: et

text: al.:

text: Function

text: of

text: nuclear

text: factor

text: kappaB

text: in

text: pancreatic

text: cancer

text: metastasis.

text: Clinical

text: Cancer

text: Research

text: Jones

text: S,

text: Zhang

text: X,

text: Parsons

text: WD

text: et

text: al.:

text: Core

text: signalling

text: pathways

text: in

text: human

text: pancreatic

text: cancers

text: revealed

text: by

text: global

text: genomic

text: analysis.

text: Science

text: Wei

text: D,

text: Le

text: X,

text: Zheng

text: L,

text: Wang

text: L

text: et

text: al.:

text: activation

text: regulates

text: the

text: expression

text: of

text: vascular

text: endothelial

text: growth

text: factor

text: and

text: human

text: pancreatic

text: cancer

text: angiogenesis

text: and

text: metastasis.

text: Oncogene

text: Perou

text: CM,

text: Sorlie

text: T,

text: Eisen

text: MB,

text: van

text: de

text: Rijn

text: M,

text: Jeﬀrey

text: SS

text: et

text: al.:

text: Molecular

text: portraits

text: of

text: human

text: breast

text: tumours.

text: Nature

text: Sorlie

text: T,

text: Perou

text: CM,

text: Tibshirani

text: R,

text: Aas

text: T

text: et

text: al.:

text: Gene

text: expression

text: patterns

text: of

text: breast

text: carcinomas

text: distin-

text: guish

text: tumor

text: subclasses

text: with

text: clinical

text: implications.

text: Proceedings

text: of

text: the

text: National

text: Academy

text: of

text: Sciences

text: USA

text: Taherian-Fard

text: A,

text: Srihari

text: S,

text: Ragan

text: MA:

text: Breast

text: cancer

text: classiﬁcation:

text: linking

text: molecular

text: mechanisms

text: to

text: disease

text: prognosis.

text: Brieﬁngs

text: in

text: Bioinformatics

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text: Waddell

text: N,

text: Arnold

text: J,

text: Cocciardi

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text: Silva

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text: et

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text: familial

text: breast

text: tumours

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text: expression

text: and

text: copy

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text: Breast

text: Cancer

text: Research

text: and

text: Treatment

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text: SR,

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text: ferences

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text: Journal

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text: National

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text: Table 3 - Changes in co-expression of complexes in pancreatic (normal-PDAC) and breast (BRCA1- BRCA2) conditions (The co-expression values are Fisher-transformed) • ∆C represents the (absolute) change in co-expression of complexes between conditions, that is, increase or decrease in co-expression of complexes from normal to PDAC or BRCA1 to BRCA2 tumours. • M(cid:48) – complexes with co-expression for normal > PDAC tumour or BRCA1 > BRCA2 tumour M(cid:48)(cid:48) – complexes with co-expression for PDAC tumour > normal or BRCA2 > BRCA1 tumour. Pancreatic Breast Complex Size of ∆C Size of ∆C type Subset subset Max Avg subset Max Avg M(cid:48) 159 0.969 0.336 225 0.761 0.281 M(cid:48)(cid:48) Our 96 0.421 0.192 52 0.543 0.197 M(cid:48) 138 0.602 0.219 386 0.642 0.178 M(cid:48)(cid:48) CORUM 51 0.512 0.209 55 0.347 0.212 (cid:80) Table 4 - Relationship between changes in co-expression of complexes (∆C) and changes in transcrip- tional regulation by TFs ( ∆R) (cid:80) All coeﬃcient values are positive, and Spearman coeﬃcients were higher than Pearson, indicating a non-linear relationship between ∆C and ∆R. Pancreatic Breast Complex #Complex-TF #Complex-TF type pairs Pearson Spearman pairs Pearson Spearman Our 226 0.273 0.436 241 0.206 0.413 CORUM 71 0.298 0.434 32 0.212 0.393 21 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Table

label: section

text: Changes

text: in

text: co-expression

text: of

text: complexes

text: in

text: pancreatic

text: (normal-PDAC)

text: and

text: breast

text: conditions

text: (The

text: co-expression

text: values

text: are

text: Fisher-transformed)

label: list

text: ∆C

text: represents

text: the

text: (absolute)

text: change

text: in

text: co-expression

text: of

text: complexes

text: between

text: conditions,

text: that

text: is,

text: increase

text: or

text: decrease

text: in

text: co-expression

text: of

text: complexes

text: from

text: normal

text: to

text: PDAC

text: or

text: to

text: tumours.

text: complexes

text: with

text: co-expression

text: for

text: normal

text: PDAC

text: tumour

text: or

text: tumour

text: complexes

text: with

text: co-expression

text: for

text: PDAC

text: tumour

text: normal

text: or

text: tumour.

text: Pancreatic

label: table

text: Breast

text: Complex

text: Size

text: of

text: ∆C

text: Size

text: of

text: ∆C

text: type

text: Subset

text: subset

text: Max

text: Avg

text: subset

text: Max

text: Avg

text: Our

text: CORUM

text: Table

text: Relationship

text: between

text: changes

text: in

text: co-expression

text: of

text: complexes

text: (∆C)

text: and

text: changes

text: in

text: transcrip-

text: tional

text: regulation

text: by

text: TFs

text: ∆R)

text: All

text: coeﬃcient

text: values

text: are

text: positive,

text: and

text: Spearman

text: coeﬃcients

text: were

text: higher

text: than

text: Pearson,

text: indicating

text: a

text: non-linear

text: relationship

text: between

text: ∆C

text: and

text: ∆R.

text: Pancreatic

text: Breast

text: Complex

text: #Complex-TF

text: #Complex-TF

text: type

text: pairs

text: Pearson

text: Spearman

text: pairs

text: Pearson

text: Spearman

text: Our

text: CORUM

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text:             (cid:48) × −→ (cid:48) × f| (cid:48)× : I J I J (25) 1 c 0 x u v 1 0 e I J 0 d 0 u y w c d f (32) (cid:48) × (cid:48) × e f 1 v w z 0 0 1 of f on I J is a bijection from I J to itself. x u v Combine the formula (13), (17) and (23), in fact we got a ≡ { } u y w mod p (33) family f ∈ (cid:48) (cid:45) of bijections λ,c λ,c I ,p (λc+1) v w z (cid:48) × −→ (cid:48) × fλ,c : I J (cid:55)→ I J Then we know that the systems of linear equations   (i, j) ((λc + 1)i mod p, (ci + j) mod p) (26)        0 c2 2c 0 0     −   x  satisfying the condition (1), (2), (3), where the invariant of  0 cd d 1 0 0    −    y  f is (λj i) mod p.  e cf ce + f 0 c  λ,c − u = 0 (34) On the basis of above discussion, in fact we get: 0 d2 1 0 0 0 − v 0 df de 0 d 1 w (cid:48) e2 f2 2ef 2e 2f Proposition III.1. Let p be a odd prime number, I := { · · · − } { · · · − } ∈ (cid:48) 1, 2, , p 1 , J := 0, 1, , p 1 . Let λ, c I such F (cid:54)≡ on has non trivial solutions. Because (c, p) = 1, left that λc + 1 0 mod p. Deﬁne the map p multiply theinvertible matrix  (cid:48) × −→ (cid:48) × f : I J I J (cid:55)→   (i, j) ((λc + 1)i mod p, (ci + j) mod p).    1 0 0 0 0 0  −  −  Then f satisﬁes the condition (i), (ii), (iii), where (λj i)  1 d 2 0 0 0 0  mod p is a hopping invariant.  c02 0c 1 0 0 0  − − (35) (d 1)2 2(1 d) 0 1 0 0 c2 c We suppose that the hopping invariant in (3) has degree 2, 0 0 0 0 1 0 i.e. it has the form 0 0 0 0 0 1     tothe both side of the equation, we get the equations × −→ Z Z   I J /p       x u v i     (cid:55)→  0 c2 2c 0 0    (i, j) (i, j, 1) u y w j mod p    x   0 d + 1 0 0 0    v w z 1    y  (27)  e cf ce + f 0 c  u = 0 (36) 0 0 0 0 0 − v Hence      0 df de 0 d 1 w e2 f2 2ef 2e 2f       a c 0 x u v a b e F b d 0 u y w c d f (28) on p. It is equivalent to the equations  e f 1 v w z 0 0 1         ≡ x u v  0 c2 2c 0 0   x  u y w mod p (29)  0 d + 1 0 0 0   y  v w z e cf ce + f 0 −c u = 0 (37) 0 df de 0 d 1 v From (2), we know that the equation about (i, j) e2 f2 2ef 2e 2f w − It easy to see that the determined of the coefﬁcient matrix (a 1)i + bj + e = 0 (30) − is 4ce2(d2 1). Because (c, p) = 1, (e, p) = 1 and p is a odd prime, we know that the equation has non-trivial solution if has not solve F . Hence and only if p − ≡ ≡ (cid:54)≡ − ≡ a 1 0 mod p, b 0 mod p, e 0 mod p d2 1 0 mod p. (38) (31) ≡ Hence When d 1 mod p, we get the solution ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: v

text: a

text: b

text: e

text: F

text: b

text: d

text: u

text: y

text: w

text: c

text: d

text: f

text: on

text: p.

text: It

text: is

text: equivalent

text: to

text: the

text: equations

text: e

text: f

text: v

text: w

text: z

text: 0

text: x

text: u

text: v

text: x

text: u

text: y

text: w

text: mod

text: p

text: d

text: y

text: v

text: w

text: z

text: e

text: cf

text: ce

text: f

text: −c

text: u

text: df

text: de

text: d

text: v

text: From

text: we

text: know

text: that

text: the

text: equation

text: about

text: (i,

text: j)

text: w

text: It

text: easy

text: to

text: see

text: that

text: the

text: determined

text: of

text: the

text: coefﬁcient

text: matrix

text: (a

text: bj

text: e

text: is

text: Because

text: (c,

text: p)

text: (e,

text: p)

text: and

text: p

text: is

text: a

text: odd

text: prime,

text: we

text: know

text: that

text: the

text: equation

text: has

text: non-trivial

text: solution

text: if

text: has

text: not

text: solve

text: F

text: Hence

text: and

text: only

text: if

text: p

text: a

text: mod

text: p,

text: b

text: mod

text: p,

text: e

text: mod

text: p

text: mod

text: p.

text: Hence

text: When

text: d

text: mod

text: p,

text: we

text: get

text: the

text: solution

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: formulation is different than proposed by Peng [17] because the latter uses an expansion of the driving ﬁeld gradient. In addition to SHG from centrosymmetric structures such as Si, some work has been performed to investigate SHG in GaAs [18-22] which has a zincblende structure. Such a system is - in contrary to Si - not centrosymmetric due to the different atomic potentials between Ga and As. Therefore SHG produced by dipole inside the bulk is not forbidden and can be studied using the bond model which so far has been sucessful in predicting dipole radiation from centrosymmetric surfaces. Although the 4 fold symmetry has been obtained from various experiments, the exact origin is unknown [18]. Further investigation on vicinal GaAs structure also exist where the SHG contribution seems to arise from surface and bulk dipoles [19]. An experiment to measure fourth harmonic generation from GaAs has been performed and found a strong 4 fold anisotropic surface speciﬁc signal that is not present in SHG [20]. In addition SHG analysis from induced electric currents due to assymetric distribution of carriers in GaAs has been investigated using quantum mechanical arguments involving the creation of an exciton and band transition analysis [21]. To the best of our knowledge however, the classical bond model has never been applied by other researchers to analyze SHG from a diatomic unit cell, in this case GaAs. In our previous work [22] we have brieﬂy discussed the relation between SBHM and group theory for diamond monocrystal and diatomic zincblende crystal. In this work we elaborate our points more clearly and go further by explaining how SBHM can really reproduce experimental results using only one ﬁtting parameter and describe in length the mechanism and physical assumption that lead to the ﬁnal SHG intensity formula and how it should be intepreted. For clarity we organize this work as follows: we start by describing the instrumental setup of our experiment in Section II. We then proceed into Section III by introducing the force equation followed by deﬁning the corresponding bond direction for the unit cell and how it is inserted into the nonlinear SHG polarization. In Section IV we compare the model with the point group symme- try of diatomic zinc blende and check whether the two apporaches yields the same susceptibility tensor. Section V gives a complete expression for the 4 polarization SHG intensity of GaAs that is obtained using SBHM. The experimental result is then ﬁtted with the additional requirement of Fresnel equation as an important incorporation to the model. Finally, a brief summary describing the major points is presented. II. ROTATIONAL ANISOTROPY SHG EXPERIMENTAL SETUP 4

text: formulation

text: is

text: different

text: than

text: proposed

text: by

text: Peng

text: because

text: the

text: latter

text: uses

text: an

text: expansion

text: of

text: the

text: driving

text: ﬁeld

text: gradient.

text: In

text: addition

text: to

text: SHG

text: from

text: centrosymmetric

text: structures

text: such

text: as

text: Si,

text: some

text: work

text: has

text: been

text: performed

text: to

text: investigate

text: SHG

text: in

text: GaAs

text: which

text: has

text: a

text: zincblende

text: structure.

text: Such

text: a

text: system

text: is

text: in

text: contrary

text: to

text: Si

text: not

text: centrosymmetric

text: due

text: to

text: the

text: different

text: atomic

text: potentials

text: between

text: Ga

text: and

text: As.

text: Therefore

text: SHG

text: produced

text: by

text: dipole

text: inside

text: the

text: bulk

text: is

text: not

text: forbidden

text: and

text: can

text: be

text: studied

text: using

text: the

text: bond

text: model

text: which

text: so

text: far

text: has

text: been

text: sucessful

text: in

text: predicting

text: dipole

text: radiation

text: from

text: centrosymmetric

text: surfaces.

text: Although

text: the

text: fold

text: symmetry

text: has

text: been

text: obtained

text: from

text: various

text: experiments,

text: the

text: exact

text: origin

text: is

text: unknown

text: Further

text: investigation

text: on

text: vicinal

text: GaAs

text: structure

text: also

text: exist

text: where

text: the

text: SHG

text: contribution

text: seems

text: to

text: arise

text: from

text: surface

text: and

text: bulk

text: dipoles

text: An

text: experiment

text: to

text: measure

text: fourth

text: harmonic

text: generation

text: from

text: GaAs

text: has

text: been

text: performed

text: and

text: found

text: a

text: strong

text: fold

text: anisotropic

text: surface

text: speciﬁc

text: signal

text: that

text: is

text: not

text: present

text: in

text: SHG

text: In

text: addition

text: SHG

text: analysis

text: from

text: induced

text: electric

text: currents

text: due

text: to

text: assymetric

text: distribution

text: of

text: carriers

text: in

text: GaAs

text: has

text: been

text: investigated

text: using

text: quantum

text: mechanical

text: arguments

text: involving

text: the

text: creation

text: of

text: an

text: exciton

text: and

text: band

text: transition

text: analysis

text: To

text: the

text: best

text: of

text: our

text: knowledge

text: however,

text: the

text: classical

text: bond

text: model

text: has

text: never

text: been

text: applied

text: by

text: other

text: researchers

text: to

text: analyze

text: SHG

text: from

text: a

text: diatomic

text: unit

text: cell,

text: in

text: this

text: case

text: GaAs.

text: In

text: our

text: previous

text: work

text: we

text: have

text: brieﬂy

text: discussed

text: the

text: relation

text: between

text: SBHM

text: and

text: group

text: theory

text: for

text: diamond

text: monocrystal

text: and

text: diatomic

text: zincblende

text: crystal.

text: In

text: this

text: work

text: we

text: elaborate

text: our

text: points

text: more

text: clearly

text: and

text: go

text: further

text: by

text: explaining

text: how

text: SBHM

text: can

text: really

text: reproduce

text: experimental

text: results

text: using

text: only

text: one

text: ﬁtting

text: parameter

text: and

text: describe

text: in

text: length

text: the

text: mechanism

text: and

text: physical

text: assumption

text: that

text: lead

text: to

text: the

text: ﬁnal

text: SHG

text: intensity

text: formula

text: and

text: how

text: it

text: should

text: be

text: intepreted.

text: For

text: clarity

text: we

text: organize

text: this

text: work

text: as

text: follows:

text: we

text: start

text: by

text: describing

text: the

text: instrumental

text: setup

text: of

text: our

text: experiment

text: in

text: Section

text: II.

text: We

text: then

text: proceed

text: into

text: Section

text: III

text: by

text: introducing

text: the

text: force

text: equation

text: followed

text: by

text: deﬁning

text: the

text: corresponding

text: bond

text: direction

text: for

text: the

text: unit

text: cell

text: and

text: how

text: it

text: is

text: inserted

text: into

text: the

text: nonlinear

text: SHG

text: polarization.

text: In

text: Section

text: IV

text: we

text: compare

text: the

text: model

text: with

text: the

text: point

text: group

text: symme-

text: try

text: of

text: diatomic

text: zinc

text: blende

text: and

text: check

text: whether

text: the

text: two

text: apporaches

text: yields

text: the

text: same

text: susceptibility

text: tensor.

text: Section

text: V

text: gives

text: a

text: complete

text: expression

text: for

text: the

text: polarization

text: SHG

text: intensity

text: of

text: GaAs

text: that

text: is

text: obtained

text: using

text: SBHM.

text: The

text: experimental

text: result

text: is

text: then

text: ﬁtted

text: with

text: the

text: additional

text: requirement

text: of

text: Fresnel

text: equation

text: as

text: an

text: important

text: incorporation

text: to

text: the

text: model.

text: Finally,

text: a

text: brief

text: summary

text: describing

text: the

text: major

text: points

text: is

text: presented.

text: II.

text: ROTATIONAL

text: ANISOTROPY

text: SHG

text: EXPERIMENTAL

text: SETUP


text: and are directed towards the Glan-Taylor calcite polarizer which is used as an analyzer of the polarization state of the SHG radiation. After that the radiation is directed to the Pellin-Broca prism near Brewster’s angle minimizing reﬂection losses in p-polarized light. Then, the radiation ◦ exits at 90 with respect to the input direction plus a small angle due to dispersion which is less for longer wavelengths. Due to this, SHG and residual fundamental radiations exit at different angles. The SHG ra- diation is passed to the photomultiplier tube (R7518, Hamamatsu) having a high sensitiveness at wavelength λ = 532 nm through the slit, which blocks the residual fundamental radiation. SHG At the output of PMT, electric signal is gained by lock-in-ampliﬁer SR830 (Stanford Research system). The gained electric signal is recorded in the memory of computer by an acquisition card PCI-6110 from "National Instrument". The process of measurement is performed in automatic mode under control of personal computer and software "Labview". The average power of the incident radiation in the focal plane was set 35 mW that is lower than the damage threshold. III. BOND MODEL FOR SHG IN ZINCBLENDE DIATOMIC CRYSTAL SBHM is a classical model that in its most simpliﬁed form takes the following assumptions: The nonlinear radiations are only produced by anharmonic dipoles that are oscillating along the bonds. Before proceeding further we would like to point out various sources that can generate SHG inside the material. For the case of a centrosymmetric structure such as Si, dipole radiation inside the bulk is forbidden due to the fact that a symmetric potential leads to zero even order hy- perpolarizability. However at the surface there is anisotropy and the symmetry is broken, therefore SHG can be generated from dipoles at the surface [23]. Also because there is a ﬁeld gradient due to absorption SHG can also be generated inside the bulk by quadrupolar effects. However, in a noncentrosymmetric crystal SHG dipole radiation is now allowed inside the bulk[24]. For GaAs this noncentrosymmetricity is due to the existence of two different atoms that sit in the center of each domain. It will be shown later that for this case the hyperpolarizability is not canceling out. Furthermore because the driving ﬁeld can of course experience signiﬁcant decay there can be bulk quadrupolar effects too whose strength depend on the ﬁeld decay gradient and penetration depth of the crystal. However if we assume that the ﬁeld decay is sufﬁciently low there is not much difference between the ﬁelds strength at the layers below the ﬁrst bulk layer and bulk dipoles then become the sole dominating SHG source. In this work we limit ourself to bulk 6

text: and

text: are

text: directed

text: towards

text: the

text: Glan-Taylor

text: calcite

text: polarizer

text: which

text: is

text: used

text: as

text: an

text: analyzer

text: of

text: the

text: polarization

text: state

text: of

text: the

text: SHG

text: radiation.

text: After

text: that

text: the

text: radiation

text: is

text: directed

text: to

text: the

text: Pellin-Broca

text: prism

text: near

text: Brewster’s

text: angle

text: minimizing

text: reﬂection

text: losses

text: in

text: p-polarized

text: light.

text: Then,

text: the

text: radiation

text: exits

text: at

text: with

text: respect

text: to

text: the

text: input

text: direction

text: plus

text: a

text: small

text: angle

text: due

text: to

text: dispersion

text: which

text: is

text: less

text: for

text: longer

text: wavelengths.

text: Due

text: to

text: this,

text: SHG

text: and

text: residual

text: fundamental

text: radiations

text: exit

text: at

text: different

text: angles.

text: The

text: SHG

text: ra-

text: diation

text: is

text: passed

text: to

text: the

text: photomultiplier

text: tube

text: Hamamatsu)

text: having

text: a

text: high

text: sensitiveness

text: at

text: wavelength

text: λ

text: nm

text: through

text: the

text: slit,

text: which

text: blocks

text: the

text: residual

text: fundamental

text: radiation.

text: SHG

text: At

text: the

text: output

text: of

text: PMT,

text: electric

text: signal

text: is

text: gained

text: by

text: lock-in-ampliﬁer

text: (Stanford

text: Research

text: system).

text: The

text: gained

text: electric

text: signal

text: is

text: recorded

text: in

text: the

text: memory

text: of

text: computer

text: by

text: an

text: acquisition

text: card

text: from

text: "National

text: Instrument".

text: The

text: process

text: of

text: measurement

text: is

text: performed

text: in

text: automatic

text: mode

text: under

text: control

text: of

text: personal

text: computer

text: and

text: software

text: "Labview".

text: The

text: average

text: power

text: of

text: the

text: incident

text: radiation

text: in

text: the

text: focal

text: plane

text: was

text: set

text: mW

text: that

text: is

text: lower

text: than

text: the

text: damage

text: threshold.

text: III.

text: BOND

text: MODEL

text: FOR

text: SHG

text: IN

text: ZINCBLENDE

text: DIATOMIC

text: CRYSTAL

text: SBHM

text: is

text: a

text: classical

text: model

text: that

text: in

text: its

text: most

text: simpliﬁed

text: form

text: takes

text: the

text: following

text: assumptions:

text: The

text: nonlinear

text: radiations

text: are

text: only

text: produced

text: by

text: anharmonic

text: dipoles

text: that

text: are

text: oscillating

text: along

text: the

text: bonds.

text: Before

text: proceeding

text: further

text: we

text: would

text: like

text: to

text: point

text: out

text: various

text: sources

text: that

text: can

text: generate

text: SHG

text: inside

text: the

text: material.

text: For

text: the

text: case

text: of

text: a

text: centrosymmetric

text: structure

text: such

text: as

text: Si,

text: dipole

text: radiation

text: inside

text: the

text: bulk

text: is

text: forbidden

text: due

text: to

text: the

text: fact

text: that

text: a

text: symmetric

text: potential

text: leads

text: to

text: zero

text: even

text: order

text: hy-

text: perpolarizability.

text: However

text: at

text: the

text: surface

text: there

text: is

text: anisotropy

text: and

text: the

text: symmetry

text: is

text: broken,

text: therefore

text: SHG

text: can

text: be

text: generated

text: from

text: dipoles

text: at

text: the

text: surface

text: Also

text: because

text: there

text: is

text: a

text: ﬁeld

text: gradient

text: due

text: to

text: absorption

text: SHG

text: can

text: also

text: be

text: generated

text: inside

text: the

text: bulk

text: by

text: quadrupolar

text: effects.

text: However,

text: in

text: a

text: noncentrosymmetric

text: crystal

text: SHG

text: dipole

text: radiation

text: is

text: now

text: allowed

text: inside

text: the

text: For

text: GaAs

text: this

text: noncentrosymmetricity

text: is

text: due

text: to

text: the

text: existence

text: of

text: two

text: different

text: atoms

text: that

text: sit

text: in

text: the

text: center

text: of

text: each

text: domain.

text: It

text: will

text: be

text: shown

text: later

text: that

text: for

text: this

text: case

text: the

text: hyperpolarizability

text: is

text: not

text: canceling

text: out.

text: Furthermore

text: because

text: the

text: driving

text: ﬁeld

text: can

text: of

text: course

text: experience

text: signiﬁcant

text: decay

text: there

text: can

text: be

text: bulk

text: quadrupolar

text: effects

text: too

text: whose

text: strength

text: depend

text: on

text: the

text: ﬁeld

text: decay

text: gradient

text: and

text: penetration

text: depth

text: of

text: the

text: crystal.

text: However

text: if

text: we

text: assume

text: that

text: the

text: ﬁeld

text: decay

text: is

text: sufﬁciently

text: low

text: there

text: is

text: not

text: much

text: difference

text: between

text: the

text: ﬁelds

text: strength

text: at

text: the

text: layers

text: below

text: the

text: ﬁrst

text: bulk

text: layer

text: and

text: bulk

text: dipoles

text: then

text: become

text: the

text: sole

text: dominating

text: SHG

text: source.

text: In

text: this

text: work

text: we

text: limit

text: ourself

text: to

text: bulk


text: 5   ( a) ( b ) ρ1 = I2 + (cid:88)j=31 q1jSj ⊗ (V Π1V †), (34) Concurrence000...123 000...000001692 h=2.0 D = 0 hh.==500..05 Concurrence000...123 0000....00002345 h=3.0 D = 1hh==. 015..00 0.003 0.01 where (cid:20) (cid:21) 0 0 0.1 0.2 0.3 0 0 0.25 0.5 0.75 1 0.1 0.2 T0.3 0.4 0.5 0.2 0.4 T 0.6 0.8 1 qk1 = (−1)kc(cid:20)1 1 +s(i−n1θ)ckocs3φcos θ(cid:21) , 0.3 ( c )h=0.0 D = 0 . 5 0.3 ( d ) 0.09 D = 1 . 5 h=0.5 h=3.0 qk2 = tan φqk1, 0.06 q = (−1)k c2 cos−θ + (−1)kc3 . (35) QD0.2 00..0034 h=2.0 QD0.2 0.03 k3 1 + ( 1)kc3 cos θ 0.1 00..0012 0.1 00 0.5 1 1.5 2 00 0.25 0.5 0.75 1 h=0.0 h=1.0 Then, by evaluating the von Neumann entropy from † 0 0 Eqs. (33) and (34) and using S(V Π V ) = 0, we obtain 0.5 1 1.5 2 0.5 1 1.5 2 0 T T − − FIG. 1: (Color online.) The concurrence ((a) and (b)) and − 1 + θ 1 + θ 1 θ 1 θ S(ρ ) = ( k ) log( k ) + ( k ) log( k ), QD ((c) and (d)) between NN spins as a function of the tem- k 2 2 2 2 perature for values of the LF less than quantum critical point (36) and diﬀerent values of the DM interaction. The inset shows the same ﬁgure for a value of the LF larger than h . c with (cid:118) (cid:117) (cid:117)(cid:88) principle, at the mentioned critical temperature a part of (cid:116) quantum correlations which contributes in the entangle- 3 ment of formation will be destroyed by classical thermal θ = q2 . (37) ﬂuctuations. So at T = T the concurrence is zero and k kj c j=1 one can derive the e(cid:114)quatio(cid:113)n Therefore, the classical correlation for the spin pair is − − − given by γ = 1 1 γ2 + γ4. (39) 2 1 1 (cid:18) (cid:19) C (ρ) = By solving this equation, the critical temperature, T , − c = max S(ρi) − S(ρ0) + S(ρ1) − c cos θS(ρ0) S(ρ1()38).will be obtained in whole range of parameters. Note in {ΠBi } 2 3 2 the absence of the LF γ1 = 1/2, at any value of the temperature and DM interaction, thus the critical tem- C In general, (ρ) has to be numerically evaluated by perature can be recast as optimizing over the angles θ and φ. Once the classical √ correlation is obtained, insertion of Eq. (30) and Eq. (38) (cid:90) − 1 2 into Eq. (10) can be used to determine the quantum dis- γ (T , h = 0, D) = (40) cord. 2 c 2 IV. NUMERICAL RESULTS 1 π a(k) 1 − = cos(k)( )( 1/2)dk. 2π −π ε(k) 1 + eβcε(k) Here we present our numerical results based on the the- oretical formulation introduced in the previous sections. Moreover, for the LF more than the quantum critical . In Fig. 1, we have presented thermal behavior of the point, NN spins are unentangled at zero temperature (see concurrence (Fig. 1(a), (b)) and the QD (Fig. 1(c), (d)). the inset of Fig. 1). By increasing the temperature from It can be clearly seen that at zero temperature, NN spins zero, NN spins remain unentangled up to a critical tem- are entangled in the quantum LL phase (h < h ). By perature, T . The existence of this critical temperature c c1 increasing temperature both concurrence and QD start can be related to the fact that the system is gapped in to decrease. There is a ﬁeld-independent critical temper- the saturated ferromagnetic phase. It shows that in the ature (T ) point in which concurrence vanishes, whereas gapped saturated ferromagnetic phase, the quantum cor- c QD shows an asymptotic behavior which is in agreement relations appear only at a ﬁnite value of the temperature with results obtained by Bethe ansatz approach33,34. In proportional to the spin gap. As soon as the temperature ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: Mode hunting and clustering in infinite dimensions based on a second kernel, G. The kernel G, which is often referred to as the shadow of K, has a proﬁle g which is related to the proﬁle k of K in a particular way. We further discuss the notion of shadow of a kernel and the link between the proﬁle k and g later in this section. We show that the gradient ascent direction can be characterized as the element of X that has unit norm and maximizes the Gateˆaux derivative of p˜. Before proceeding, let us brieﬂy recall the deﬁnition of Gateˆaux diﬀerential and deriva- tive. B Deﬁnition 3 (Gateˆaux diﬀerentiability, Ambrosetti and Prodi, 1995) Let be a B (cid:55)→ a Banach space and let U be an open subset of . The map F : U V is Gateaˆux ∈ B diﬀerentiable at u U if there exists a continuous linear map F from to V , called the ∈ u Gateˆaux diﬀerential of F at u, such that for all w U − F (u + αw) F (u) lim = F (w), → u α 0 α ∈ R ∈ where α . If F is Gateˆaux diﬀerentiable at all u U, then F is said to be Gateˆaux (cid:48) (cid:55)→ ∈ diﬀerentiable in U. The map F : u F , which associates to each u U the continuous G u linear operator F , is called the Gateˆaux derivative of F . u The Gateˆaux derivative of the estimated surrogate density p˜ is obtained in the following Lemma. Lemma 4 (Gateˆaux diﬀerential of the estimated surrogate density) Let p˜ (y) de- ∈ X ∈ X x note the Gateˆaux diﬀerential of p˜ at x evaluated at y . Let h(X) be an adaptive ∈ S bandwidth possibly varying across the data points X which possibly depends on the S (cid:42) (cid:43) entire sample , but not on x. Then, (cid:88) S 1 − p˜ (y) = y, Cw ( ) K (X, x) (X x) , (12) x G ∈S h2(X) h X where the proﬁles g and k of the kernel G used in p˜ and the kernel K satisfy k(t) = − (cid:48) g (t)/(Ct) for a constant C > 0. (cid:0)(cid:83) (cid:1) Remark 5 If the proﬁle k of K is not continuous at 1 (as it is the case for the truncated Gaussian proﬁle for instance), the Gateˆaux derivative of p˜ is still almost surely continuous ∈ X { } at any ﬁxed x as long as P ∈S d(X, x) = h(X) = 0. X In the second last line of the proof of the above Lemma (see Appendix), we set (cid:48) −g (t) k(t) = (13) (cid:90) Ct with (cid:48) −g (t) C = dt > 0. R t One can verify that equation (13) (which is a diﬀerential version of equation (15) of Cheng, 1995) implies, among others, the correspondences of Table 1. For our purposes, we give the following deﬁnition of the shadow of a kernel. 11 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Mode

text: hunting

text: and

text: clustering

text: in

text: infinite

text: dimensions

text: based

text: on

text: a

text: second

text: kernel,

text: G.

text: The

text: kernel

text: G,

text: which

text: is

text: often

text: referred

text: to

text: as

text: the

text: shadow

text: of

text: K,

text: has

text: a

text: proﬁle

text: g

text: which

text: is

text: related

text: to

text: the

text: proﬁle

text: k

text: of

text: K

text: in

text: a

text: particular

text: way.

text: We

text: further

text: discuss

text: the

text: notion

text: of

text: shadow

text: of

text: a

text: kernel

text: and

text: the

text: link

text: between

text: the

text: proﬁle

text: k

text: and

text: g

text: later

text: in

text: this

text: section.

text: We

text: show

text: that

text: the

text: gradient

text: ascent

text: direction

text: can

text: be

text: characterized

text: as

text: the

text: element

text: of

text: X

text: that

text: has

text: unit

text: norm

text: and

text: maximizes

text: the

text: Gateˆaux

text: derivative

text: of

text: p˜.

text: Before

text: proceeding,

text: let

text: us

text: brieﬂy

text: recall

text: the

text: deﬁnition

text: of

text: Gateˆaux

text: diﬀerential

text: and

text: deriva-

text: tive.

text: B

text: Deﬁnition

text: (Gateˆaux

text: diﬀerentiability,

text: Ambrosetti

text: and

text: Prodi,

text: Let

text: be

text: a

text: B

text: a

text: Banach

text: space

text: and

text: let

text: U

text: be

text: an

text: open

text: subset

text: of

text: The

text: map

text: F

text: U

text: V

text: is

text: Gateaˆux

text: B

text: diﬀerentiable

text: at

text: u

text: U

text: if

text: there

text: exists

text: a

text: continuous

text: linear

text: map

text: F

text: from

text: to

text: V

text: called

text: the

text: u

text: Gateˆaux

text: diﬀerential

text: of

text: F

text: at

text: u,

text: such

text: that

text: for

text: all

text: w

text: U

text: F

text: (u

text: αw)

text: F

text: (u)

text: lim

text: F

text: (w),

text: u

text: α

text: α

text: R

text: where

text: α

text: If

text: F

text: is

text: Gateˆaux

text: diﬀerentiable

text: at

text: all

text: u

text: U,

text: then

text: F

text: is

text: said

text: to

text: be

text: Gateˆaux

text: diﬀerentiable

text: in

text: U.

text: The

text: map

text: F

text: u

text: F

text: which

text: associates

text: to

text: each

text: u

text: U

text: the

text: continuous

text: G

text: u

text: linear

text: operator

text: F

text: is

text: called

text: the

text: Gateˆaux

text: derivative

text: of

text: F

text: u

text: The

text: Gateˆaux

text: derivative

text: of

text: the

text: estimated

text: surrogate

text: density

text: p˜

text: is

text: obtained

text: in

text: the

text: following

text: Lemma.

text: Lemma

text: (Gateˆaux

text: diﬀerential

text: of

text: the

text: estimated

text: surrogate

text: density)

text: Let

text: p˜

text: (y)

text: de-

text: X

text: X

text: x

text: note

text: the

text: Gateˆaux

text: diﬀerential

text: of

text: p˜

text: at

text: x

text: evaluated

text: at

text: y

text: Let

text: h(X)

text: be

text: an

text: adaptive

text: S

text: bandwidth

text: possibly

text: varying

text: across

text: the

text: data

text: points

text: X

text: which

text: possibly

text: depends

text: on

text: the

text: S

text: entire

text: sample

text: but

text: not

text: on

text: x.

text: Then,

text: S

text: p˜

text: (y)

text: y,

text: Cw

text: K

text: (X,

text: x)

text: (X

text: x)

text: x

text: G

text: ∈S

text: h

text: X

text: where

text: the

text: proﬁles

text: g

text: and

text: k

text: of

text: the

text: kernel

text: G

text: used

text: in

text: p˜

text: and

text: the

text: kernel

text: K

text: satisfy

text: k(t)

text: g

text: (t)/(Ct)

text: for

text: a

text: constant

text: C

text: Remark

text: If

text: the

text: proﬁle

text: k

text: of

text: K

text: is

text: not

text: continuous

text: at

text: (as

text: it

text: is

text: the

text: case

text: for

text: the

text: truncated

text: Gaussian

text: proﬁle

text: for

text: instance),

text: the

text: Gateˆaux

text: derivative

text: of

text: p˜

text: is

text: still

text: almost

text: surely

text: continuous

text: X

text: at

text: any

text: ﬁxed

text: x

text: as

text: long

text: as

text: P

text: ∈S

text: d(X,

text: x)

text: h(X)

text: X

text: In

text: the

text: second

text: last

text: line

text: of

text: the

text: proof

text: of

text: the

text: above

text: Lemma

text: (see

text: Appendix),

text: we

text: set

text: −g

text: (t)

text: k(t)

text: Ct

text: with

text: −g

text: (t)

text: C

text: dt

text: R

text: t

text: One

text: can

text: verify

text: that

text: equation

text: (which

text: is

text: a

text: diﬀerential

text: version

text: of

text: equation

text: of

text: Cheng,

text: implies,

text: among

text: others,

text: the

text: correspondences

text: of

text: Table

text: For

text: our

text: purposes,

text: we

text: give

text: the

text: following

text: deﬁnition

text: of

text: the

text: shadow

text: of

text: a

text: kernel.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Mode hunting and clustering in infinite dimensions Figure 6: Clusters obtained with the functional mean-shift algorithm. The modal curves are represented by the red dashed lines. that is H : µ˜ is not a mode of p 0 j vs. H : µ˜ is a mode of p. 1 j The pointwise nature of the alternative hypothesis on the functional gradient may constitute a problem in the speciﬁcation of the above hypothesis test. For instance, in the Euclidean ∇ setting, the condition p = 0 would correspond to a zero (Lebesgue) measure set. At µ˜j least in the Euclidean setting, one possibility is to recast the hypothesis test as follows (Genovese et al., 2013): ¬ H : H 0 1 vs. (22) (cid:107)∇ (cid:107) H : λ < 0 and p < δ 1 µ˜j µ˜j for some δ > 0, so that we are instead testing the hypothesis that µ˜ is an approximate j mode of p. The sample splitting procedure introduced in the test of Genovese et al. (2013) is such that constraint on the gradient has essentially no eﬀect in (22) (we will later return to this point in Remark 14). In the following, we consider a similar test that is adapted to the functional case. We focus on ≥ H : λ 0 0 µ˜j vs. (23) H : λ < 0. 1 µ˜j 19 ##LTFigure##

text: Mode

text: hunting

text: and

text: clustering

text: in

text: infinite

text: dimensions

text: Figure

text: Clusters

label: caption

text: obtained

text: with

text: the

text: functional

text: mean-shift

text: algorithm.

text: The

text: modal

text: curves

text: are

text: represented

text: by

text: the

text: red

text: dashed

text: lines.

text: that

text: is

text: H

text: µ˜

text: is

text: not

text: a

text: mode

text: of

text: p

text: j

text: vs.

text: H

text: µ˜

text: is

text: a

text: mode

text: of

text: p.

text: j

text: The

text: pointwise

text: nature

text: of

text: the

text: alternative

text: hypothesis

text: on

text: the

text: functional

text: gradient

text: may

text: constitute

text: a

text: problem

text: in

text: the

text: speciﬁcation

text: of

text: the

text: above

text: hypothesis

text: test.

text: For

text: instance,

text: in

text: the

text: Euclidean

text: setting,

text: the

text: condition

text: p

text: would

text: correspond

text: to

text: a

text: zero

text: (Lebesgue)

text: measure

text: set.

text: At

text: µ˜j

text: least

text: in

text: the

text: Euclidean

text: setting,

text: one

text: possibility

text: is

text: to

text: recast

text: the

text: hypothesis

text: test

text: as

text: follows

text: (Genovese

text: et

text: al.,

text: H

text: H

text: vs.

text: H

text: λ

text: and

text: p

text: δ

text: µ˜j

text: µ˜j

text: for

text: some

text: δ

text: so

text: that

text: we

text: are

text: instead

text: testing

text: the

text: hypothesis

text: that

text: µ˜

text: is

text: an

text: approximate

text: j

text: mode

text: of

text: p.

text: The

text: sample

text: splitting

text: procedure

text: introduced

text: in

text: the

text: test

text: of

text: Genovese

text: et

text: al.

text: is

text: such

text: that

text: constraint

text: on

text: the

text: gradient

text: has

text: essentially

text: no

text: eﬀect

text: in

text: (we

text: will

text: later

text: return

text: to

text: this

text: point

text: in

text: Remark

text: In

text: the

text: following,

text: we

text: consider

text: a

text: similar

text: test

text: that

text: is

text: adapted

text: to

text: the

text: functional

text: case.

text: We

text: focus

text: on

text: H

text: λ

text: µ˜j

text: vs.

text: H

text: λ

text: µ˜j

text: ##LTFigure##


text: Mode hunting and clustering in infinite dimensions 7 6 Number of non-atomic clusters 2345 1 (35 out of 40) (39 out of 40) 0 0.1 0.2 0.3 0.4 0.5 0.6 Bandwidth / max. distance Figure 15: Number of non-atomic clusters as a function of the bandwidth parameter, h. The number of non-atomic clusters stabilizes in the 2 highlighted ranges. The candidate bandwidth parameters are then set to be the midpoint of these 2 relevant ranges. The number in parentheses above the candidate bandwidths denote the number of sample curves that are clustered in non-atomic clusters using the corresponding candidate bandwidth. and the ﬁfth clusters (red, blue and yellow). These clusters contain 6 original and 1 fake signature (red), 2 original and 1 fake signature (blue), and 2 original signatures (yellow), respectively. The middle cluster (green) contains 15 of the 20 fake signatures. Overall, it seems that the clustering may be too ﬁne. Figure 17 displays the output of the functional mean-shift algorithm when h = h . This time, the algorithm produces two internally 2 homogeneous clusters which appear to meaningfully summarize the structure of the data: the top cluster (black) contains 19 of the 20 original signatures and the second cluster (green) contains all the 20 fake signatures. 10. Discussion The mean-shift algorithm is an iterative algorithm that corresponds to a form of adaptive gradient ascent. The mean-shift algorithm has been extensively studied for scalar and vector data, but a counterpart of this algorithm for inﬁnite-dimensional data has not been devised. In this paper, we introduce the functional mean-shift algorithm, a functional version of the mean-shift algorithm of Fukunaga and Hostetler (1975) and Cheng (1995) which is designed to deal with data that are intrinsically inﬁnite-dimensional. The functional mean-shift algorithm expands the applicability of the mean-shift algo- rithm well beyond the familiar Euclidean case. In Section , we show that the functional mean-shift algorithm corresponds to a form of adaptive gradient ascent on the estimated surrogate density of random variables that are valued in an inﬁnite-dimensional Hilbert space: for practitioners, this fact establishes a reassuring formal analogy with the standard mean-shift algorithm for scalar and vector data. In Section , we provide a bootstrap-based 29 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Mode

text: hunting

text: and

text: clustering

text: in

text: infinite

text: dimensions

text: Number

text: of

text: non-atomic

text: clusters

text: out

text: of

text: out

text: of

text: Bandwidth

text: max.

text: distance

text: Figure

text: Number

text: of

text: non-atomic

text: clusters

text: as

text: a

text: function

text: of

text: the

text: bandwidth

text: parameter,

text: h.

text: The

text: number

text: of

text: non-atomic

text: clusters

text: stabilizes

text: in

text: the

text: highlighted

text: ranges.

text: The

text: candidate

text: bandwidth

text: parameters

text: are

text: then

text: set

text: to

text: be

text: the

text: midpoint

text: of

text: these

text: relevant

text: ranges.

text: The

text: number

text: in

text: parentheses

text: above

text: the

text: candidate

text: bandwidths

text: denote

text: the

text: number

text: of

text: sample

text: curves

text: that

text: are

text: clustered

text: in

text: non-atomic

text: clusters

text: using

text: the

text: corresponding

text: candidate

text: bandwidth.

text: and

text: the

text: ﬁfth

text: clusters

text: (red,

text: blue

text: and

text: yellow).

text: These

text: clusters

text: contain

text: original

text: and

text: fake

text: signature

text: (red),

text: original

text: and

text: fake

text: signature

text: (blue),

text: and

text: original

text: signatures

text: (yellow),

text: respectively.

text: The

text: middle

text: cluster

text: (green)

text: contains

text: of

text: the

text: fake

text: signatures.

text: Overall,

text: it

text: seems

text: that

text: the

text: clustering

text: may

text: be

text: too

text: ﬁne.

text: Figure

text: displays

text: the

text: output

text: of

text: the

text: functional

text: mean-shift

text: algorithm

text: when

text: h

text: h

text: This

text: time,

text: the

text: algorithm

text: produces

text: two

text: internally

text: homogeneous

text: clusters

text: which

text: appear

text: to

text: meaningfully

text: summarize

text: the

text: structure

text: of

text: the

text: data:

text: the

text: top

text: cluster

text: (black)

text: contains

text: of

text: the

text: original

text: signatures

text: and

text: the

text: second

text: cluster

text: (green)

text: contains

text: all

text: the

text: fake

text: signatures.

text: Discussion

text: The

text: mean-shift

text: algorithm

text: is

text: an

text: iterative

text: algorithm

text: that

text: corresponds

text: to

text: a

text: form

text: of

text: adaptive

text: gradient

text: ascent.

text: The

text: mean-shift

text: algorithm

text: has

text: been

text: extensively

text: studied

text: for

text: scalar

text: and

text: vector

text: data,

text: but

text: a

text: counterpart

text: of

text: this

text: algorithm

text: for

text: inﬁnite-dimensional

text: data

text: has

text: not

text: been

text: devised.

text: In

text: this

text: paper,

text: we

text: introduce

text: the

text: functional

text: mean-shift

text: algorithm,

text: a

text: functional

text: version

text: of

text: the

text: mean-shift

text: algorithm

text: of

text: Fukunaga

text: and

text: Hostetler

text: and

text: Cheng

text: which

text: is

text: designed

text: to

text: deal

text: with

text: data

text: that

text: are

text: intrinsically

text: inﬁnite-dimensional.

text: The

text: functional

text: mean-shift

text: algorithm

text: expands

text: the

text: applicability

text: of

text: the

text: mean-shift

text: algo-

text: rithm

text: well

text: beyond

text: the

text: familiar

text: Euclidean

text: case.

text: In

text: Section

text: we

text: show

text: that

text: the

text: functional

text: mean-shift

text: algorithm

text: corresponds

text: to

text: a

text: form

text: of

text: adaptive

text: gradient

text: ascent

text: on

text: the

text: estimated

text: surrogate

text: density

text: of

text: random

text: variables

text: that

text: are

text: valued

text: in

text: an

text: inﬁnite-dimensional

text: Hilbert

text: space:

text: for

text: practitioners,

text: this

text: fact

text: establishes

text: a

text: reassuring

text: formal

text: analogy

text: with

text: the

text: standard

text: mean-shift

text: algorithm

text: for

text: scalar

text: and

text: vector

text: data.

text: In

text: Section

text: we

text: provide

text: a

text: bootstrap-based

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: ##LTLine##

text: ##LTLine##