text: SPECTRAL SETS AND OPERATOR RADII 3 explicitly in terms of the usual norm. In fact, our methods work for homomorphisms from any unital Banach algebra satisfying the von Neumann inequality. The paper is organized as follows. A proof that a set is K-spectral for T if and only if it is a K˜ -numerical radius set (Theorem 2.1) is given in the next section. Although a more general result for the operator radii w will be proved later on, we hope that the proof of ρ the numerical radius case will highlight the simple ideas used in the proofs. In Section 3 we give the deﬁnition of operator radii of a unital homomorphism from an operator algebra (or a B Banach algebra which satisﬁes the von Neumann inequality) into (H). In the same section the operator radii w are revisited and the main result (Theorem 3.2) is stated. Section 4 ρ is devoted to the proof of the main result while in Section 5 we give several applications of Theorem 3.2 to (complete) K-spectral sets, including an extension of Drury’s result (1.3) for the w radii. The last section contains an application of the main result to the joint numerical ρ radius of non-commuting n-tuples of operators (a notion introduced in [17]). A version of Drury’s result is obtained in this non-commutative context. 2. Spectral sets as numerical radius sets The following result, showing the equivalence of K-spectral sets and K˜ -numerical radius sets, is a counterpart of the result from [5]. (cid:0) (cid:1) ∈ B ≥ Theorem 2.1. Let T (H), let Ω be an open domain in the complex plane and let K 1 − be a real number. Denote K˜ = 1 K + K 1 . The following assertions are equivalent: 2 (1) Ω is a K-spectral set for T ; (2) Ω is a K˜ -numerical radius set for T . (cid:112) Note ﬁrst that inverting the formula deﬁning K˜ we obtain − K = K˜ + K˜ 2 1. → → Since a spectral set is also (1 + ε)-spectral set and K˜ 1 if and only if K 1, it is possible to assume, without loss of any generality, that K > 1. Denote 1 + z φ(z) = − 1 z and D − D (cid:16) (cid:17):= φ(K 1 ) K D the open disc with center K2+−1 and radius 2K− . An equivalent description of is that K2 1 K2 1 K − D the endpoints of the interval K 1, K−+1 are diametrically opposed in . K+1 K 1 K The following lemma reformulates the deﬁnition of a K-spectral set. Recall ﬁrst that the ∗ real part of an operator A is deﬁned as Re(A) = (A + A )/2. ≥ Lemma 2.2. The set Ω is K-spectral for T if and only if Re(f(T )) 0 for all analytic → D functions f : Ω . K ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: SPECTRAL

text: SETS

text: AND

text: OPERATOR

text: RADII

text: explicitly

text: in

text: terms

text: of

text: the

text: usual

text: norm.

text: In

text: fact,

text: our

text: methods

text: work

text: for

text: homomorphisms

text: from

text: any

text: unital

text: Banach

text: algebra

text: satisfying

text: the

text: von

text: Neumann

text: inequality.

text: The

text: paper

text: is

text: organized

text: as

text: follows.

text: A

text: proof

text: that

text: a

text: set

text: is

text: K-spectral

text: for

text: T

text: if

text: and

text: only

text: if

text: it

text: is

text: a

text: K˜

text: -numerical

text: radius

text: set

text: (Theorem

text: is

text: given

text: in

text: the

text: next

text: section.

text: Although

text: a

text: more

text: general

text: result

text: for

text: the

text: operator

text: radii

text: w

text: will

text: be

text: proved

text: later

text: on,

text: we

text: hope

text: that

text: the

text: proof

text: of

text: ρ

text: the

text: numerical

text: radius

text: case

text: will

text: highlight

text: the

text: simple

text: ideas

text: used

text: in

text: the

text: proofs.

text: In

text: Section

text: we

text: give

text: the

text: deﬁnition

text: of

text: operator

text: radii

text: of

text: a

text: unital

text: homomorphism

text: from

text: an

text: operator

text: algebra

text: (or

text: a

text: B

text: Banach

text: algebra

text: which

text: satisﬁes

text: the

text: von

text: Neumann

text: inequality)

text: into

text: (H).

text: In

text: the

text: same

text: section

text: the

text: operator

text: radii

text: w

text: are

text: revisited

text: and

text: the

text: main

text: result

text: (Theorem

text: is

text: stated.

text: Section

text: ρ

text: is

text: devoted

text: to

text: the

text: proof

text: of

text: the

text: main

text: result

text: while

text: in

text: Section

text: we

text: give

text: several

text: applications

text: of

text: Theorem

text: to

text: (complete)

text: K-spectral

text: sets,

text: including

text: an

text: extension

text: of

text: Drury’s

text: result

text: for

text: the

text: w

text: radii.

text: The

text: last

text: section

text: contains

text: an

text: application

text: of

text: the

text: main

text: result

text: to

text: the

text: joint

text: numerical

text: ρ

text: radius

text: of

text: non-commuting

text: n-tuples

text: of

text: operators

text: (a

text: notion

text: introduced

text: in

text: A

text: version

text: of

text: Drury’s

text: result

text: is

text: obtained

text: in

text: this

text: non-commutative

text: context.

text: Spectral

text: sets

text: as

text: numerical

text: radius

text: sets

text: The

text: following

text: result,

text: showing

text: the

text: equivalence

text: of

text: K-spectral

text: sets

text: and

text: K˜

text: -numerical

text: radius

text: sets,

text: is

text: a

text: counterpart

text: of

text: the

text: result

text: from

text: B

text: Theorem

text: Let

text: T

text: (H),

text: let

text: Ω

text: be

text: an

text: open

text: domain

text: in

text: the

text: complex

text: plane

text: and

text: let

text: K

text: be

text: a

text: real

text: number.

text: Denote

text: K˜

text: K

text: K

text: The

text: following

text: assertions

text: are

text: equivalent:

text: Ω

text: is

text: a

text: K-spectral

text: set

text: for

text: T

text: Ω

text: is

text: a

text: K˜

text: -numerical

text: radius

text: set

text: for

text: T

text: Note

text: ﬁrst

text: that

text: inverting

text: the

text: formula

text: deﬁning

text: K˜

text: we

text: obtain

text: K

text: K˜

text: K˜

text: Since

text: a

text: spectral

text: set

text: is

text: also

text: ε)-spectral

text: set

text: and

text: K˜

text: if

text: and

text: only

text: if

text: K

text: it

text: is

text: possible

text: to

text: assume,

text: without

text: loss

text: of

text: any

text: generality,

text: that

text: K

text: Denote

text: z

text: φ(z)

text: z

text: and

text: D

text: D

text: φ(K

text: K

text: D

text: the

text: open

text: disc

text: with

text: center

text: and

text: radius

text: An

text: equivalent

text: description

text: of

text: is

text: that

text: K

text: D

text: the

text: endpoints

text: of

text: the

text: interval

text: K

text: are

text: diametrically

text: opposed

text: in

text: K

text: K

text: The

text: following

text: lemma

text: reformulates

text: the

text: deﬁnition

text: of

text: a

text: K-spectral

text: set.

text: Recall

text: ﬁrst

text: that

text: the

text: real

text: part

text: of

text: an

text: operator

text: A

text: is

text: deﬁned

text: as

text: Re(A)

text: (A

text: A

text: Lemma

text: The

text: set

text: Ω

text: is

text: K-spectral

text: for

text: T

text: if

text: and

text: only

text: if

text: Re(f(T

text: for

text: all

text: analytic

text: D

text: functions

text: f

text: Ω

text: K

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 2 YUNPING JIANG extendable. Since any simply connected one-dimensional hyperbolic complex manifold is isomorphic to the unit dsk due to the uniformiza- tion theorem, any holomorphic motion of a subset in the Riemann sphere over a simply connected one-dimensional hyperbolic complex manifold is fully extendable. There is a counter-example of a holomor- phic motion of a subset in the Riemann sphere over a simply connected ≥ n-dimensional complex manifold for n 2, which is not fully extend- able (see, for example, [7]). In the one-dimensional case, there is a counter-example of a holo- morphic motion of a subset in the Riemann sphere over a non-simply connected one-dimensional hyperbolic complex manifold, which is not fully extendable (see, for examples, [3, 4, 1]). This leads to consider the following zero winding number condition in holomorphic motions. Deﬁnition 1. Suppose Y is a topological space and suppose γ is a → closed curve in Y , that is, γ(θ) : [0, 1] Y is a continuous map satisfying γ(0) = γ(1). We say γ trivial (or homotopic to a point in Y ) if there exists a continuous map (called a homotopy), × → H(s, θ) : [0, 1] [0, 1] Y ∈ such that H(0, θ) = p, a constant point in Y , for all θ [0, 1], H(s, 0) = ∈ ∈ H(s, 1) for all s [0, 1], and, moreover, H(1, θ) = γ(θ) for all θ [0, 1]. Otherwise, we say that γ non-trivial (or not homotopoic to a point in Y ) . Suppose X is a connected one-dimensional hyperbolic complex man- ifold with a basepoint t . Gi(cid:98)ven a holomorphic motion φ of E over X 0 (cid:54) ∈ and a non-trivial simple closed curve α in X, for every pair z = z E, 1 2 C consider the closed curve in , − (cid:73) δ = δ(α, z , z ) = φ(α, z ) φ(α, z ). 1 2 1 2 Let 1 · η = η(δ(α, z , z )) = d arg δ( , z , z ) 1 2 1 2 2π α denote the winding number of δ about 0. It is a non-negative integer and 2πη is just the variation of argument of δ over α. Deﬁnition 2. The holomorphic motion φ is said to satisfy the zero winding number condition if (1) η(δ(α, z , z )) = 0 1 2 ##LTLine##

label: paragraph

text: YUNPING

text: JIANG

text: extendable.

text: Since

text: any

text: simply

text: connected

text: one-dimensional

text: hyperbolic

text: complex

text: manifold

text: is

text: isomorphic

text: to

text: the

text: unit

text: dsk

text: due

text: to

text: the

text: uniformiza-

text: tion

text: theorem,

text: any

text: holomorphic

text: motion

text: of

text: a

text: subset

text: in

text: the

text: Riemann

text: sphere

text: over

text: a

text: simply

text: connected

text: one-dimensional

text: hyperbolic

text: complex

text: manifold

text: is

text: fully

text: extendable.

text: There

text: is

text: a

text: counter-example

text: of

text: a

text: holomor-

text: phic

text: motion

text: of

text: a

text: subset

text: in

text: the

text: Riemann

text: sphere

text: over

text: a

text: simply

text: connected

text: n-dimensional

text: complex

text: manifold

text: for

text: n

text: which

text: is

text: not

text: fully

text: extend-

text: able

text: (see,

text: for

text: example,

text: In

text: the

text: one-dimensional

text: case,

text: there

text: is

text: a

text: counter-example

text: of

text: a

text: holo-

text: morphic

text: motion

text: of

text: a

text: subset

text: in

text: the

text: Riemann

text: sphere

text: over

text: a

text: non-simply

text: connected

text: one-dimensional

text: hyperbolic

text: complex

text: manifold,

text: which

text: is

text: not

text: fully

text: extendable

text: (see,

text: for

text: examples,

text: This

text: leads

text: to

text: consider

text: the

text: following

text: zero

text: winding

text: number

text: condition

text: in

text: holomorphic

text: motions.

text: Deﬁnition

text: Suppose

text: Y

text: is

text: a

text: topological

text: space

text: and

text: suppose

text: γ

text: is

text: a

text: closed

text: curve

text: in

text: Y

text: that

text: is,

text: γ(θ)

text: Y

text: is

text: a

text: continuous

text: map

text: satisfying

text: We

text: say

text: γ

text: trivial

text: (or

text: homotopic

text: to

text: a

text: point

text: in

text: Y

text: if

text: there

text: exists

text: a

text: continuous

text: map

text: (called

text: a

text: homotopy),

text: H(s,

text: θ)

text: Y

text: such

text: that

text: θ)

text: p,

text: a

text: constant

text: point

text: in

text: Y

text: for

text: all

text: θ

text: H(s,

text: H(s,

text: for

text: all

text: s

text: and,

text: moreover,

text: θ)

text: γ(θ)

text: for

text: all

text: θ

text: Otherwise,

text: we

text: say

text: that

text: γ

text: non-trivial

text: (or

text: not

text: homotopoic

text: to

text: a

text: point

text: in

text: Y

text: Suppose

text: X

text: is

text: a

text: connected

text: one-dimensional

text: hyperbolic

text: complex

text: man-

text: ifold

text: with

text: a

text: basepoint

text: t

text: a

text: holomorphic

text: motion

text: φ

text: of

text: E

text: over

text: X

text: and

text: a

text: non-trivial

text: simple

text: closed

text: curve

text: α

text: in

text: X,

text: for

text: every

text: pair

text: z

text: z

text: E,

text: C

text: consider

text: the

text: closed

text: curve

text: in

text: δ

text: δ(α,

text: z

text: z

text: φ(α,

text: z

text: φ(α,

text: z

text: Let

text: η

text: η(δ(α,

text: z

text: z

text: d

text: arg

text: δ(

text: z

text: z

text: 2π

text: α

text: denote

text: the

text: winding

text: number

text: of

text: δ

text: about

text: It

text: is

text: a

text: non-negative

text: integer

text: and

text: 2πη

text: is

text: just

text: the

text: variation

text: of

text: argument

text: of

text: δ

text: over

text: α.

text: Deﬁnition

text: The

text: holomorphic

text: motion

text: φ

text: is

text: said

text: to

text: satisfy

text: the

text: zero

text: winding

text: number

text: condition

text: if

text: η(δ(α,

text: z

text: z

text: ##LTLine##


text: 8 YUNPING JIANG (cid:26) We deﬁne a normalized holomorphic motion of E over X as follows: ∞ ∈ z if z = 0, 1, and t X; (11) φ(t, z) = ∈ ξg(z t) if z = ξ and t X. 0 We check that it is a holomorphic motion: ∈ i) since g(1) = 1, we have φ(t , z) = z for al(cid:98)l z E; ∞ (cid:54)∈ 0 ∞ (cid:54)∈ ∈ ii) since 0, z , g(A), we have that 0, 1, ξg(A) but φ(t, ξ) 0 ∈ · → C ξg(A) for all t X, thus φ(t, ) : E is injective for any ∈ ﬁxed t X; (cid:98) ∞ iii) for z = 0, 1, , φ(t, z) = z and φ(t, ξ) = ξg(z t), therefore, for ∈ · → C 0 any ﬁxed z E, φ( , z) : X is holomorphic. Thus φ(t, z) is indeed a normalized holomorphic motion of E over X. The main statement of this paper is Theorem 1 (Main Theorem). The holomorphic motion φ of E over X deﬁned in (11) satisﬁes the zero winding number condition (1) but is not fully extendable. We prove this theorem in the next section. 4. Proof of the Main Result (Theorem 1) We ﬁrst check (1) for φ in (11). The set E contains only four points ∞ 0, 1, ξ = 1/z , and . The parameter space X has only one non-trivial 0 simply closed curve (cid:98) → C (12) α(θ) = ξe2πθi : [0, 1] in the sense of homotopy. So we only need to calculate the winding (cid:54) ∈ number η(δ(α, z , z )) of δ(α, z(cid:98), z ) for every pair z = z E. ∈ { 1 2∞} 1 2 1 2 If z , z 0, 1, , then it is clear that η(δ(α, z , z )) = 0 since 1 2 (cid:54) C 1 2 δ(α, z , z ) is a point (= 0) in . 1 2 Let γ be the curve in (8). Then we have that γ = z α. Let us 0 calculate the winding number of δ(α, ξ, 0) = φ(α, ξ) = ξg(γ). (cid:73) From the argument principle in complex analysis, we have that the winding number (cid:48) 1 g (z) η(δ(α, ξ, 0)) = dz 2πi g(z) γ ##LTLine## ##LTLine##

text: YUNPING

text: JIANG

label: equation

text: We

text: deﬁne

text: a

text: normalized

text: holomorphic

text: motion

text: of

text: E

text: over

text: X

text: as

text: follows:

text: z

text: if

text: z

text: and

text: t

text: X;

text: φ(t,

text: z)

text: ξg(z

text: t)

text: if

text: z

text: ξ

text: and

text: t

text: X.

text: We

text: check

text: that

text: it

text: is

text: a

text: holomorphic

text: motion:

label: list

text: i)

text: since

text: we

text: have

text: φ(t

text: z)

text: z

text: for

text: z

text: E;

text: ii)

text: since

text: z

text: g(A),

text: we

text: have

text: that

text: ξg(A)

text: but

text: φ(t,

text: ξ)

text: C

text: ξg(A)

text: for

text: all

text: t

text: X,

text: thus

text: φ(t,

text: E

text: is

text: injective

text: for

text: any

text: ﬁxed

text: t

text: X;

text: iii)

text: for

text: z

text: φ(t,

text: z)

text: z

text: and

text: φ(t,

text: ξ)

text: ξg(z

text: t),

text: therefore,

text: for

text: C

text: any

text: ﬁxed

text: z

text: E,

text: φ(

text: z)

text: X

text: is

text: holomorphic.

text: Thus

text: φ(t,

text: z)

text: is

text: indeed

text: a

text: normalized

text: holomorphic

text: motion

text: of

text: E

text: over

text: X.

text: The

text: main

text: statement

text: of

text: this

text: paper

text: is

text: Theorem

text: (Main

text: Theorem).

text: The

text: holomorphic

text: motion

text: φ

text: of

text: E

text: over

text: X

text: deﬁned

text: in

text: satisﬁes

text: the

text: zero

text: winding

text: number

text: condition

text: but

text: is

text: not

text: fully

text: extendable.

text: We

text: prove

text: this

text: theorem

text: in

text: the

text: next

text: section.

text: Proof

text: of

text: the

text: Main

text: Result

text: (Theorem

text: We

text: ﬁrst

text: check

text: for

text: φ

text: in

text: The

text: set

text: E

text: contains

text: only

text: four

text: points

text: ξ

text: and

text: The

text: parameter

text: space

text: X

text: has

text: only

text: one

text: non-trivial

text: simply

text: closed

text: curve

text: C

text: α(θ)

text: ξe2πθi

text: in

text: the

text: sense

text: of

text: homotopy.

text: So

text: we

text: only

text: need

text: to

text: calculate

text: the

text: winding

text: number

text: η(δ(α,

text: z

text: z

text: of

text: δ(α,

text: z

text: for

text: every

text: pair

text: z

text: z

text: E.

text: If

text: z

text: z

text: then

text: it

text: is

text: clear

text: that

text: η(δ(α,

text: z

text: z

text: since

text: C

text: δ(α,

text: z

text: z

text: is

text: a

text: point

text: in

text: Let

text: γ

text: be

text: the

text: curve

text: in

text: Then

text: we

text: have

text: that

text: γ

text: z

text: α.

text: Let

text: us

text: calculate

text: the

text: winding

text: number

text: of

text: δ(α,

text: ξ,

text: φ(α,

text: ξ)

text: ξg(γ).

text: From

text: the

text: argument

text: principle

text: in

text: complex

text: analysis,

text: we

text: have

text: that

text: the

text: winding

text: number

text: g

text: (z)

text: η(δ(α,

text: ξ,

text: dz

text: g(z)

text: γ

text: ##LTLine##

text: ##LTLine##


text: HOLOMORPHIC MOTIONS 9 which equals the number of zeros minus the number of poles of g inside γ (counted by multiplicity). Since g has only one zero at 0 of order n and only one pole at 1/a of order n insider γ, this implies that the n − winding number η(δ(α, ξ, 0)) = n n = 0. For the closed curve − δ(α, ξ, 1) = φ(α, ξ) φ(cid:73)(α, 1) = ξh(γ), the winding number (cid:48) 1 h (z) η(δ(α, ξ, 1)) = dz. 2πi h(z) γ Lemma 1 says that h(z) has exactly n zeros counted by multiplicity − and only one pole of order n insider γ, thus η(δ(α, ξ, 1)) = n n = 0. To calculate the winding number of ∞ − ∞ − ∞ δ(α, ξ, ) = φ(α, ξ) φ(α, ) = φ(α, ξ) , ∞ we consider the local coordinate z = 1/w at . By the symmetry, we ∞ have g(w) = 1/g(1/w). Under this coordinate, δ(α, ξ, ) has the form, 1 1 − = = z g( γ). 0 φ(α, ξ) (cid:73) ξg(γ) Thus the winding number (cid:48) ∞ 1 g (z) − η(δ(α, ξ, )) = dz = η(δ(α, ξ, 0)) = 0. 2πi − g(z) γ Therefore, we have proved that the holomorphic motion φ deﬁned in (11) satisﬁes the zero winding number condition (1). Next we will prove that φ is not fully extendable. Before to give our proof, we need to prepare some background in Teichmu¨ller theory. We use [5] and [8] as two references, in which the reader can ﬁnd other references. C ∞ C Let M( ) be the space of all L complex functions µ on with (cid:107) (cid:107) C norms µ ∞ < 1. Every element µ in M( ) is also called a Beltrami C coeﬃcient. The space M( ) is a simply connected complex Banach ∞ manifold since it is the open unit ball in the space of L complex C functions on which is a complex Banach space. ∈ C For every µ M( ), the partial diﬀeren(cid:98)tial equation C (13) w = µw on z z (cid:98) is called a Beltrami equation and its solution w is called a quasicon- C formal mapping of . We say a solution w is normalized if w(0) = 0, ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: HOLOMORPHIC

text: MOTIONS

text: which

text: equals

text: the

text: number

text: of

text: zeros

text: minus

text: the

text: number

text: of

text: poles

text: of

text: g

text: inside

text: γ

text: (counted

text: by

text: multiplicity).

text: Since

text: g

text: has

text: only

text: one

text: zero

text: at

text: of

text: order

text: n

text: and

text: only

text: one

text: pole

text: at

text: of

text: order

text: n

text: insider

text: γ,

text: this

text: implies

text: that

text: the

text: n

text: winding

text: number

text: η(δ(α,

text: ξ,

text: n

text: n

text: For

text: the

text: closed

text: curve

text: δ(α,

text: ξ,

text: φ(α,

text: ξ)

text: ξh(γ),

text: the

text: winding

text: number

text: h

text: (z)

text: η(δ(α,

text: ξ,

text: dz.

text: h(z)

text: γ

text: Lemma

text: says

text: that

text: h(z)

text: has

text: exactly

text: n

text: zeros

text: counted

text: by

text: multiplicity

text: and

text: only

text: one

text: pole

text: of

text: order

text: n

text: insider

text: γ,

text: thus

text: η(δ(α,

text: ξ,

text: n

text: n

text: To

text: calculate

text: the

text: winding

text: number

text: of

text: δ(α,

text: ξ,

text: φ(α,

text: ξ)

text: φ(α,

text: φ(α,

text: ξ)

text: we

text: consider

text: the

text: local

text: coordinate

text: z

text: at

text: By

text: the

text: symmetry,

text: we

text: have

text: g(w)

text: Under

text: this

text: coordinate,

text: δ(α,

text: ξ,

text: has

text: the

text: form,

text: z

text: g(

text: γ).

text: φ(α,

text: ξ)

text: ξg(γ)

text: Thus

text: the

text: winding

text: number

text: g

text: (z)

text: η(δ(α,

text: ξ,

text: dz

text: η(δ(α,

text: ξ,

text: g(z)

text: γ

text: Therefore,

text: we

text: have

text: proved

text: that

text: the

text: holomorphic

text: motion

text: φ

text: deﬁned

text: in

text: satisﬁes

text: the

text: zero

text: winding

text: number

text: condition

text: Next

text: we

text: will

text: prove

text: that

text: φ

text: is

text: not

text: fully

text: extendable.

text: Before

text: to

text: give

text: our

text: proof,

text: we

text: need

text: to

text: prepare

text: some

text: background

text: in

text: Teichmu¨ller

text: theory.

text: We

text: use

text: and

text: as

text: two

text: references,

text: in

text: which

text: the

text: reader

text: can

text: ﬁnd

text: other

text: references.

text: C

text: C

text: Let

text: M(

text: be

text: the

text: space

text: of

text: all

text: L

text: complex

text: functions

text: µ

text: on

text: with

text: C

text: norms

text: µ

text: Every

text: element

text: µ

text: in

text: M(

text: is

text: also

text: called

text: a

text: Beltrami

text: C

text: coeﬃcient.

text: The

text: space

text: M(

text: is

text: a

text: simply

text: connected

text: complex

text: Banach

text: manifold

text: since

text: it

text: is

text: the

text: open

text: unit

text: ball

text: in

text: the

text: space

text: of

text: L

text: complex

text: C

text: functions

text: on

text: which

text: is

text: a

text: complex

text: Banach

text: space.

text: C

text: For

text: every

text: µ

text: M(

text: the

text: partial

text: equation

text: C

text: w

text: µw

text: on

text: z

text: z

text: is

text: called

text: a

text: Beltrami

text: equation

text: and

text: its

text: solution

text: w

text: is

text: called

text: a

text: quasicon-

text: C

text: formal

text: mapping

text: of

text: We

text: say

text: a

text: solution

text: w

text: is

text: normalized

text: if

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 4 Local density from J⊥ = 1 to J⊥ = 6. Interestingly, the results do not 10 show a clear linear signal at all. Instead, at short dis- 0.25 tances, a structure emerges, which then seems to freeze 8 and not to move out to further distances. In the course 0.20 of time, oscillations are seen, which appear more pro- nounced in the triplet pairing correlation function. 6 These results indicate that in both pairing correla- 0.15 tion functions the light-cone signal is strongly suppressed. 4 Similar behavior is found when quenching from a SC 0.10 phase to the LL phase at small J⊥. However, in this case, a faint linear signal can be obtained. Again, its slope is ∼ 2 0.05 4. This indicates that the light-cone in principle can also be realized in the pairing correlation functions, but 0 is substantially weaker than in the other correlation func- 10 20 30 40 50 60 70 tions, an issue we will address in more detail in Sec. . Sites Summarizing the results so far, the velocity of the quasiparticles causing the lightcone in SC phases is not Density FFT half the one of non-interacting particles. This is an in- 10 0.5 teresting ﬁnding and stands in contrast to the doubling of the frequency of the center-of-mass motion reported 8 in Ref. [27] for a similar one-dimensional system upon 0.4 entering the SC phase. It would be interesting to see if such a smaller velocity could be realized in systems with 6 0.3 true SC long-range order, like two-dimensional systems at zero temperature. The data presented in Fig. 3 (a) 4 0.2 allow also the observation of a pattern in the dynamics inside the correlated region. There, it is possible to see at least another ballistic signal identiﬁed by a local max- 2 0.1 imum spreading with a velocity slower than V . This is lc reminiscent of ﬁndings for exactly solvable models [18], 0 0.0 where internal patterns in the dynamics have been pre- /2 0 /2 k dicted and connected to phase velocities of the spectrum, in contrast to the lightcone given by the group velocity FIG. 2. tDMRG results for the time evolution of the local of excitations. density after a quench in the t-J⊥ model (3) from J⊥ = 5 IV. SUPPRESSION OF THE LIGHTCONE IN ,initial to J⊥ = 2. (a) Time evolution as a function of position. PAIRING CORRELATION FUNCTIONS IN BCS ,ﬁnal (b) Fourier transform of the results of (a) to k-space by only THEORY considering the bulk region of (a). The ﬁndings of the previous section may be speciﬁc to the t-J-model and to one spatial dimension. In or- ues J⊥ = 2, 3, 6 is comparable and lies between 3 and 4. der to test their validity beyond this set-up, we now turn to a simple, analytically tractable model, which contains Hence, the velocities of the quasiparticles, which one can superconductivity. A simple approach is to recall BCS associate to the light-cone, are comparable to the ones theory of superconductivity and to compute the time of free electrons, where the slope of the light-cone has evolution of correlation functions after a quench in this the value four [7]. This is also seen for quenches starting from J⊥ = 5 to the same ﬁnal values. Furthermore, the framework. T(cid:88)his (cid:16)leads us(cid:17)t(cid:16)o the simple toy(cid:17)Hamiltonian slope of the lightcones for charge correlation functions and for spin correlation functions has comparable values. H (cid:88) (cid:16)− † † (cid:17)− = (cid:15) µ c ↑c ↑ + c ↓c ↓ This indicates that the quasiparticles involved in the for- k k k k k mation of lightcones in these observables are not related k − † † ∗ to pairs of fermions, for which one could expect a slower ∆ c ↑c− ↓ + ∆ c− ↓c ↑ , (10) k k k k velocity, as the mass of such an object would be twice k the mass of a free particle. − Next, we ask the question if such a change of veloc- where (cid:15)k = 2 cos (k) is the dispersion of non-interact†ing ity might be solely realized in pairing correlation func- electrons on a one-dimensional lattice, the operators c( ) kσ tions. Figure 3 contrasts the time evolution of the singlet denote fermionic annihilation (creation) operators for a and triplet pairing correlation functions to the ones of particle with momentum k and spin σ (note that the re- the density and spin correlation functions for the quench striction of no double occupancy of the t-J-model is not t emiT t emiT ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##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##

text: Local

font: default

text: density

text: from

text: J⊥

text: to

text: J⊥

text: Interestingly,

text: the

text: results

text: do

text: not

text: show

text: a

text: clear

text: linear

text: signal

text: at

text: all.

text: Instead,

text: at

text: short

text: dis-

text: tances,

text: a

text: structure

text: emerges,

text: which

text: then

text: seems

text: to

text: freeze

text: and

text: not

text: to

text: move

text: out

text: to

text: further

text: distances.

text: In

text: the

text: course

text: of

text: time,

text: oscillations

text: are

text: seen,

text: which

text: appear

text: more

text: pro-

text: nounced

text: in

text: the

text: triplet

text: pairing

text: correlation

text: function.

text: These

text: results

text: indicate

text: that

text: in

text: both

text: pairing

text: correla-

text: tion

text: functions

text: the

text: light-cone

text: signal

text: is

text: strongly

text: suppressed.

text: Similar

text: behavior

text: is

text: found

text: when

text: quenching

text: from

text: a

text: SC

text: phase

text: to

text: the

text: LL

text: phase

text: at

text: small

text: J⊥.

text: However,

text: in

text: this

text: case,

text: a

text: faint

text: linear

text: signal

text: can

text: be

text: obtained.

text: Again,

text: its

text: slope

text: is

text: This

text: indicates

text: that

text: the

text: light-cone

text: in

text: principle

text: can

text: also

text: be

text: realized

text: in

text: the

text: pairing

text: correlation

text: functions,

text: but

text: is

text: substantially

text: weaker

text: than

text: in

text: the

text: other

text: correlation

text: func-

text: tions,

text: an

text: issue

text: we

text: will

text: address

text: in

text: more

text: detail

text: in

text: Sec.

text: Sites

text: Summarizing

text: the

text: results

text: so

text: far,

text: the

text: velocity

text: of

text: the

text: quasiparticles

text: causing

text: the

text: lightcone

text: in

text: SC

text: phases

text: is

text: not

text: Density

text: FFT

text: half

text: the

text: one

text: of

text: non-interacting

text: particles.

text: This

text: is

text: an

text: in-

text: teresting

text: ﬁnding

text: and

text: stands

text: in

text: contrast

text: to

text: the

text: doubling

text: of

text: the

text: frequency

text: of

text: the

text: center-of-mass

text: motion

text: reported

text: in

text: Ref.

text: for

text: a

text: similar

text: one-dimensional

text: system

text: upon

text: entering

text: the

text: SC

text: phase.

text: It

text: would

text: be

text: interesting

text: to

text: see

text: if

text: such

text: a

text: smaller

text: velocity

text: could

text: be

text: realized

text: in

text: systems

text: with

text: true

text: SC

text: long-range

text: order,

text: like

text: two-dimensional

text: systems

text: at

text: zero

text: temperature.

text: The

text: data

text: presented

text: in

text: Fig.

text: (a)

text: allow

text: also

text: the

text: observation

text: of

text: a

text: pattern

text: in

text: the

text: dynamics

text: inside

text: the

text: correlated

text: region.

text: There,

text: it

text: is

text: possible

text: to

text: see

text: at

text: least

text: another

text: ballistic

text: signal

text: identiﬁed

text: by

text: a

text: local

text: max-

text: imum

text: spreading

text: with

text: a

text: velocity

text: slower

text: than

text: V

text: This

text: is

text: lc

text: reminiscent

text: of

text: ﬁndings

text: for

text: exactly

text: solvable

text: models

text: where

text: internal

text: patterns

text: in

text: the

text: dynamics

text: have

text: been

text: pre-

text: k

text: dicted

text: and

text: connected

text: to

text: phase

text: velocities

text: of

text: the

text: spectrum,

text: in

text: contrast

text: to

text: the

text: lightcone

text: given

text: by

text: the

text: group

text: velocity

text: FIG.

text: tDMRG

text: results

text: for

text: the

text: time

text: evolution

text: of

text: the

text: local

text: of

text: excitations.

text: density

text: after

text: a

text: quench

text: in

text: the

text: t-J⊥

text: model

text: from

text: J⊥

text: IV.

text: SUPPRESSION

text: OF

text: THE

text: LIGHTCONE

text: IN

text: ,initial

text: to

text: J⊥

text: (a)

text: Time

text: evolution

text: as

text: a

text: function

text: of

text: position.

text: PAIRING

text: CORRELATION

text: FUNCTIONS

text: IN

text: BCS

text: ,ﬁnal

text: (b)

text: Fourier

text: transform

text: of

text: the

text: results

text: of

text: (a)

text: to

text: k-space

text: by

text: only

text: THEORY

text: considering

text: the

text: bulk

text: region

text: of

text: (a).

text: The

text: ﬁndings

text: of

text: the

text: previous

text: section

text: may

text: be

text: speciﬁc

text: to

text: the

text: t-J-model

text: and

text: to

text: one

text: spatial

text: dimension.

text: In

text: or-

text: ues

text: J⊥

text: is

text: comparable

text: and

text: lies

text: between

text: and

text: der

text: to

text: test

text: their

text: validity

text: beyond

text: this

text: set-up,

text: we

text: now

text: turn

text: to

text: a

text: simple,

text: analytically

text: tractable

text: model,

text: which

text: contains

text: Hence,

text: the

text: velocities

text: of

text: the

text: quasiparticles,

text: which

text: one

text: can

text: superconductivity.

text: A

text: simple

text: approach

text: is

text: to

text: recall

text: BCS

text: associate

text: to

text: the

text: light-cone,

text: are

text: comparable

text: to

text: the

text: ones

text: theory

text: of

text: superconductivity

text: and

text: to

text: compute

text: the

text: time

text: of

text: free

text: electrons,

text: where

text: the

text: slope

text: of

text: the

text: light-cone

text: has

text: evolution

text: of

text: correlation

text: functions

text: after

text: a

text: quench

text: in

text: this

text: the

text: value

text: four

text: This

text: is

text: also

text: seen

text: for

text: quenches

text: starting

text: from

text: J⊥

text: to

text: the

text: same

text: ﬁnal

text: values.

text: Furthermore,

text: the

text: framework.

text: the

text: simple

text: slope

text: of

text: the

text: lightcones

text: for

text: charge

text: correlation

text: functions

text: and

text: for

text: spin

text: correlation

text: functions

text: has

text: comparable

text: values.

text: H

text: µ

text: c

text: ↑c

text: c

text: ↓c

text: This

text: indicates

text: that

text: the

text: quasiparticles

text: involved

text: in

text: the

text: for-

text: k

text: k

text: k

text: k

text: k

text: mation

text: of

text: lightcones

text: in

text: these

text: observables

text: are

text: not

text: related

text: k

text: to

text: pairs

text: of

text: fermions,

text: for

text: which

text: one

text: could

text: expect

text: a

text: slower

text: c

text: ↑c−

text: c−

text: ↓c

text: k

text: k

text: k

text: k

text: velocity,

text: as

text: the

text: mass

text: of

text: such

text: an

text: object

text: would

text: be

text: twice

text: k

text: the

text: mass

text: of

text: a

text: free

text: particle.

text: Next,

text: we

text: ask

text: the

text: question

text: if

text: such

text: a

text: change

text: of

text: veloc-

text: where

text: cos

text: (k)

text: is

text: the

text: dispersion

text: of

text: non-interact†ing

text: ity

text: might

text: be

text: solely

text: realized

text: in

text: pairing

text: correlation

text: func-

text: electrons

text: on

text: a

text: one-dimensional

text: lattice,

text: the

text: operators

text: c(

text: kσ

text: tions.

text: Figure

text: contrasts

text: the

text: time

text: evolution

text: of

text: the

text: singlet

text: denote

text: fermionic

text: annihilation

text: (creation)

text: operators

text: for

text: a

text: and

text: triplet

text: pairing

text: correlation

text: functions

text: to

text: the

text: ones

text: of

text: particle

text: with

text: momentum

text: k

text: and

text: spin

text: σ

text: (note

text: that

text: the

text: re-

text: the

text: density

text: and

text: spin

text: correlation

text: functions

text: for

text: the

text: quench

text: striction

text: of

text: no

text: double

text: occupancy

text: of

text: the

text: t-J-model

text: is

text: not

text: t

text: emiT

text: t

text: emiT

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: 7 Singlet interesting to see if similar behavior is found in true su- 20.0 0.0030 perconducting phases, e.g., in two dimensional systems at zero temperature. 17.5 0.0025 ACKNOWLEDGEMENTS 15.0 0.0020 12.5 We thank Markus Schmitt for useful discussions, in particular for pointing out the possibility to do a 10.0 0.0015 BCS treatment. Financial support from the Deutsche 7.5 Forschungsgemeinschaft (DFG) through SFB/CRC1073 0.0010 (Project B03) is gratefully acknowledged. 5.0 Appendix A: Analytical expressions for the time 0.0005 evolution of the correlation functions in 2.5 time-dependent BCS theory 0.0 0.0000 20 40 60 80 Distance|i j| We study the BCS-Hamiltonian (10) using the Bogoljubov- transformation (11). In the set of equations (11), u and k v are complex parameters that can be chosen to diago- k − FIG. 6. Absolute value of the connected singlet correlation nalize the Hamiltonian(cid:32)(10) using ξ = (cid:15)k (cid:33) µ, function Eq. (A11) as function of distance and time after a → quench in the BCS Hamiltonian (10) from ∆ = 0 ∆ = (cid:112) i f 1/2. The red solid line has the slope of the maximal group ve- | | 1 (cid:32) ξ (cid:33) locity of the spectrum Eq. (12) multiplied by two and agrees u 2 = 1 + k| | (A1) very well with the border of the light-cone region. The col- k 2 (cid:112)ξk + ∆f 2 ormap has been set to enhance the contrast in the vicinity of ∼ the lightcone, where the maximal value is 0.001. Note that |v |2 = 1 1 − ξk| | . (A2) close to the lightcone, the strength of the signal is between k 2 ξ + ∆ 2 k f one and two orders of magnitude smaller than the one of the density-density correlation function in Fig. 4. To study the time evolution, we promote the ampli- tudes u and v to be time dependent complex variables. k k times, and then a coexistence of both wave vectors is Their explicit form is then determined solving the Heisen- seen. Instead, for quenches in the reverse direction, the berg equation of motion, see Ref. [77, 78] Friedel oscillations in the initial state are with wave vec- tor k /2, and then are suppressed in the course of the F time evolution. ni nj + ni nj 20.0 0.0200 In the correlation functions one clearly observes lightcone behavior in the density-density and spin correlation func- 17.5 0.0175 tions, but a strongly suppressed lightcone in the pairing 15.0 0.0150 correlation functions. The slope of the lightcone is ap- proximately the same in all cases, and similar to the one 12.5 0.0125 of non-interacting fermions on a lattice. This indicates 10.0 0.0100 that superconducting pairing does not modify the time scales, at which correlations move through the system. 7.5 0.0075 This is in contrast to the behavior of center-of-mass os- 5.0 0.0050 cillations in similar situations, for which the time scales are strongly aﬀected by the SC phase, as seen in the dou- 2.5 0.0025 bling of the frequency of this oscillation [27]. 0.0 0.0000 Similar behavior is obtained in quenches in a BCS 20 40 60 80 treatment. In one dimension, the values of the pairing Distance |i j| correlation functions in the vicinity of the lightcone are strongly suppressed compared to the ones of the density- FIG. 7. Absolute value of the dynamics of the observable density correlation function. This can be understood by Eq. (13) as function of distance and time after a quench in the lack of contribution of the term (13) in the pairing → the BCS Hamiltonian (10) from ∆ = 0 ∆ = 1/2. The red i f correlation functions, which carries most of the weight solid line has the slope of the maximal group velocity of the of the density correlation function in the vicinity of the spectrum Eq. (12) multiplied by two and agrees very well with lightcone. the border of the light-cone region. The colormap has been The superconducting phases in the one-dimensional set to enhance the contrast i∼n the vicinity of the lightcone, systems treated here are not true SC phases according where the maximal value is 0.01, which is comparable to to the Mermin-Wagner-Hohenberg theorem. It would be the one of the density-density correlation function in Fig. 4. t emiT t emiT ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##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##

text: Singlet

text: interesting

text: to

text: see

text: if

text: similar

text: behavior

text: is

text: found

text: in

text: true

text: su-

text: perconducting

text: phases,

text: e.g.,

text: in

text: two

text: dimensional

text: systems

text: at

text: zero

text: temperature.

text: ACKNOWLEDGEMENTS

label: section

text: We

text: thank

text: Markus

text: Schmitt

text: for

text: useful

text: discussions,

text: in

text: particular

text: for

text: pointing

text: out

text: the

text: possibility

text: to

text: do

text: a

text: BCS

text: treatment.

text: Financial

text: support

text: from

text: the

text: Deutsche

text: Forschungsgemeinschaft

text: (DFG)

text: through

text: (Project

text: is

text: gratefully

text: acknowledged.

text: Appendix

text: A:

text: Analytical

text: expressions

text: for

text: the

text: time

text: evolution

text: of

text: the

text: correlation

text: functions

text: in

text: time-dependent

text: BCS

text: theory

text: Distance|i

text: j|

text: We

text: study

text: the

text: BCS-Hamiltonian

text: using

text: the

text: Bogoljubov-

text: transformation

text: In

text: the

text: set

text: of

text: equations

text: u

text: and

text: k

text: v

text: are

text: complex

text: parameters

text: that

text: can

text: be

text: chosen

text: to

text: diago-

text: k

text: FIG.

text: Absolute

text: value

text: of

text: the

text: connected

text: singlet

text: correlation

text: nalize

text: the

text: using

text: ξ

text: µ,

text: function

text: Eq.

text: as

text: function

text: of

text: distance

text: and

text: time

text: after

text: a

text: quench

text: in

text: the

text: BCS

text: Hamiltonian

text: from

text: i

text: f

text: The

text: red

text: solid

text: line

text: has

text: the

text: slope

text: of

text: the

text: maximal

text: group

text: ve-

text: ξ

text: locity

text: of

text: the

text: spectrum

text: Eq.

text: multiplied

text: by

text: two

text: and

text: agrees

text: u

text: k|

text: very

text: well

text: with

text: the

text: border

text: of

text: the

text: light-cone

text: region.

text: The

text: col-

text: k

text: ∆f

text: ormap

text: has

text: been

text: set

text: to

text: enhance

text: the

text: contrast

text: in

text: the

text: vicinity

text: of

text: the

text: lightcone,

text: where

text: the

text: maximal

text: value

text: is

text: Note

text: that

text: |v

text: ξk|

text: close

text: to

text: the

text: lightcone,

text: the

text: strength

text: of

text: the

text: signal

text: is

text: between

text: k

text: ξ

text: k

text: f

text: one

text: and

text: two

text: orders

text: of

text: magnitude

text: smaller

text: than

text: the

text: one

text: of

text: the

text: density-density

text: correlation

text: function

text: in

text: Fig.

text: To

text: study

text: the

text: time

text: evolution,

text: we

text: promote

text: the

text: ampli-

text: tudes

text: u

text: and

text: v

text: to

text: be

text: time

text: dependent

text: complex

text: variables.

text: k

text: k

text: times,

text: and

text: then

text: a

text: coexistence

text: of

text: both

text: wave

text: vectors

text: is

text: Their

text: explicit

text: form

text: is

text: then

text: determined

text: solving

text: the

text: Heisen-

text: seen.

text: Instead,

text: for

text: quenches

text: in

text: the

text: reverse

text: direction,

text: the

text: berg

text: equation

text: of

text: motion,

text: see

text: Ref.

text: Friedel

text: oscillations

text: in

text: the

text: initial

text: state

text: are

text: with

text: wave

text: vec-

text: tor

text: k

text: and

text: then

text: are

text: suppressed

text: in

text: the

text: course

text: of

text: the

text: F

text: time

text: evolution.

text: ni

text: nj

text: ni

text: nj

text: In

text: the

text: correlation

text: functions

text: one

text: clearly

text: observes

text: lightcone

text: behavior

text: in

text: the

text: density-density

text: and

text: spin

text: correlation

text: func-

text: tions,

text: but

text: a

text: strongly

text: suppressed

text: lightcone

text: in

text: the

text: pairing

text: correlation

text: functions.

text: The

text: slope

text: of

text: the

text: lightcone

text: is

text: ap-

text: proximately

text: the

text: same

text: in

text: all

text: cases,

text: and

text: similar

text: to

text: the

text: one

text: of

text: non-interacting

text: fermions

text: on

text: a

text: lattice.

text: This

text: indicates

text: that

text: superconducting

text: pairing

text: does

text: not

text: modify

text: the

text: time

text: scales,

text: at

text: which

text: correlations

text: move

text: through

text: the

text: system.

text: This

text: is

text: in

text: contrast

text: to

text: the

text: behavior

text: of

text: center-of-mass

text: os-

text: cillations

text: in

text: similar

text: situations,

text: for

text: which

text: the

text: time

text: scales

text: are

text: strongly

text: aﬀected

text: by

text: the

text: SC

text: phase,

text: as

text: seen

text: in

text: the

text: dou-

text: bling

text: of

text: the

text: frequency

text: of

text: this

text: oscillation

text: Similar

text: behavior

text: is

text: obtained

text: in

text: quenches

text: in

text: a

text: BCS

text: treatment.

text: In

text: one

text: dimension,

text: the

text: values

text: of

text: the

text: pairing

text: Distance

text: |i

text: j|

text: correlation

text: functions

text: in

text: the

text: vicinity

text: of

text: the

text: lightcone

text: are

text: strongly

text: suppressed

text: compared

text: to

text: the

text: ones

text: of

text: the

text: density-

text: FIG.

text: Absolute

text: value

text: of

text: the

text: dynamics

text: of

text: the

text: observable

text: density

text: correlation

text: function.

text: This

text: can

text: be

text: understood

text: by

text: Eq.

text: as

text: function

text: of

text: distance

text: and

text: time

text: after

text: a

text: quench

text: in

text: the

text: lack

text: of

text: contribution

text: of

text: the

text: term

text: in

text: the

text: pairing

text: the

text: BCS

text: Hamiltonian

text: from

text: The

text: red

text: i

text: f

text: correlation

text: functions,

text: which

text: carries

text: most

text: of

text: the

text: weight

text: solid

text: line

text: has

text: the

text: slope

text: of

text: the

text: maximal

text: group

text: velocity

text: of

text: the

text: of

text: the

text: density

text: correlation

text: function

text: in

text: the

text: vicinity

text: of

text: the

text: spectrum

text: Eq.

text: multiplied

text: by

text: two

text: and

text: agrees

text: very

text: well

text: with

text: lightcone.

text: the

text: border

text: of

text: the

text: light-cone

text: region.

text: The

text: colormap

text: has

text: been

text: The

text: superconducting

text: phases

text: in

text: the

text: one-dimensional

text: set

text: to

text: enhance

text: the

text: contrast

text: i∼n

text: the

text: vicinity

text: of

text: the

text: lightcone,

text: systems

text: treated

text: here

text: are

text: not

text: true

text: SC

text: phases

text: according

text: where

text: the

text: maximal

text: value

text: is

text: which

text: is

text: comparable

text: to

text: to

text: the

text: Mermin-Wagner-Hohenberg

text: theorem.

text: It

text: would

text: be

text: the

text: one

text: of

text: the

text: density-density

text: correlation

text: function

text: in

text: Fig.

text: t

text: emiT

text: t

text: emiT

text: ##LTFigure##

text: ##LTFigure##

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text: 1 titlecolorSeed-Point Detection of Clumped Convex Objects by Short-Range Attractive Long-Range Repulsive Particle Clustering authorcolorJames Kapaldo, Xu Han, and Domingo Mery, Member, IEEE Abstract—abstractcolor Locating the center of convex objects is important in both image processing and unsupervised machine learning/data clustering ﬁelds. The automated analysis of biological images uses both of these ﬁelds for locating cell nuclei and for discovering new biological effects or cell phenotypes. In this work, we develop a novel clustering method for locating the centers of overlapping convex objects by modeling particles that interact by a short-range attractive and long-range repulsive potential and are conﬁned to a potential well created from the data. We apply this method to locating the centers of clumped nuclei in cultured cells, where we show that it results in a signiﬁcant improvement over existing methods (8.2% in F score); and we apply it to unsupervised 1 learning on a difﬁcult data set that has rare classes without local density maxima, and show it is able to well locate cluster centers when other clustering techniques fail. Index Terms—seed-point detection; data mining; nuclei de-clumping; short-range attractive long-range repulsive sectioncolor (cid:70) 1 INTRODUCTION L OCATING object and distribution centers is a fundamen- at the nuclei centers. The fundamental problem with these tal step of both image processing and unsupervised ma- methods is that the extrema of the voting landscapes are chine learning or exploratory data mining. The automated not at the nuclei centers; further, there can be extra extrema, analysis and classiﬁcation of cells from microscopy images where false nuclei are detected, or missing extrema, where is an important ﬁeld that includes both of these disciplines true nuclei are not detected. Since the segmentation methods [1], [2], [3], [4], [5], [6]. In this ﬁeld, the ﬁrst step is often will return one region per seed-point, it is critical that each locating nuclei centers using image processing techniques, nuclei have only one seed-point as close as possible to the which allows for seed-point based segmentation algorithms nuclei center [10], [22]. For unsupervised learning, there are [7], [8], [9], [10] to determine the nuclei boundaries. Using many well known clustering techniques (K-means/fuzzy C- the segmentation results, features of each nucleus or cell means [25], [26], [27], expectation maximization/mixture of are measured and used to classify the cells into different Gaussians [28], hierarchical clustering [29], mean-shift [30]), groups (e.g. cancer/non-cancer, different phenotypes/cell- but a general drawback of most of these methods is that they phase [5], [11], [12], [13], [14], [15]). When searching for new produce cluster centers that are biased towards high density groups (new phenotypes, new effects, ...) or when it not pos- regions in the data, and many of them require the number sible for an expert to create a labeled data set, unsupervised of clusters (which is normally not known beforehand) as machine learning, where groups are determined by looking an input. Mean-shift clustering can be better in this regard; for similarities in the data, must be used. however, each cluster must have a local maximum in the Locating nuclei centers and using unsupervised learning point density. to locate cluster centers are, abstractly, very similar: they In this work, we develop a new clustering method (SALR both try to locate the centers of partially-overlapping convex clustering) that can be used for locating both nuclei centers objects or distributions. However, the techniques for solving and the cluster centers in scatter point data, while at the each of these problems are quite different and are not same time addressing the above issues. The premise of our ideal for several reasons. The current methods for locating method, which is graphically described in Figure 1 and nuclei centers (radial voting [10], [16], [17], [18], gradient which takes cues from the fundamental physics of modeling convergence and sliding band ﬁlters [19], [20], Laplacian classical Wigner crystals [31] and modeling the formation of Gaussian ﬁlters [21], [22], [23], [24]) try to compute a of clusters [32], [33], is that we can ﬁnd the centers of surface, named the voting landscape, that has local extrema overlapping convex regions by modeling the dynamics of particles trapped in an appropriate conﬁning well. • Consider a set of repulsive particles conﬁned to a region; This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may these particles will try to move as far apart from each other no longer be accessible. as possible. Figure 1a shows this for 10 particles conﬁned • J. Kapaldo and X. Han are with the Department of Physics, Univer- to a parabolic potential well. If the region the particles are sity of Notre Dame, Notre Dame, IN, 46545. (e-mail: jkapaldo@nd.edu, conﬁned to has several local minima, then the particles will xhan1@nd.edu) • D. Mery is with the Department of Computer Science, Pontiﬁcia Universi- preferentially locate as close as possible to these minima dad Cato´lica de Chile, Santiago 7820436, Chile. (email: dmery@ing.puc.cl) to lower the total energy. Assuming our goal is to have footercolorManuscript submitted Aug. 26, 2017. particles near each local minima and nowhere else, we ##LTLine## ##LTLine## ##LTLine##

font: UVSEEW+NimbusSanL-Regu

text: titlecolorSeed-Point

text: Detection

text: of

text: Clumped

text: Convex

text: Objects

text: by

text: Short-Range

text: Attractive

text: Long-Range

text: Repulsive

text: Particle

text: Clustering

text: authorcolorJames

text: Kapaldo,

text: Xu

text: Han,

text: and

text: Domingo

text: Mery,

text: Member,

font: MHTPBP+NimbusSanL-ReguItal

text: IEEE

text: Abstract—abstractcolor

text: Locating

text: the

text: center

text: of

text: convex

text: objects

text: is

text: important

text: in

text: both

text: image

text: processing

text: and

text: unsupervised

text: machine

text: learning/data

text: clustering

text: ﬁelds.

text: The

text: automated

text: analysis

text: of

text: biological

text: images

text: uses

text: both

text: of

text: these

text: ﬁelds

text: for

text: locating

text: cell

text: nuclei

text: and

text: for

text: discovering

text: new

text: biological

text: effects

text: or

text: cell

text: phenotypes.

text: In

text: this

text: work,

text: we

text: develop

text: a

text: novel

text: clustering

text: method

text: for

text: locating

text: the

text: centers

text: of

text: overlapping

text: convex

text: objects

text: by

text: modeling

text: particles

text: that

text: interact

text: by

text: a

text: short-range

text: attractive

text: and

text: long-range

text: repulsive

text: potential

text: and

text: are

text: conﬁned

text: to

text: a

text: potential

text: well

text: created

text: from

text: the

text: data.

text: We

text: apply

text: this

text: method

text: to

text: locating

text: the

text: centers

text: of

text: clumped

text: nuclei

text: in

text: cultured

text: cells,

text: where

text: we

text: show

text: that

text: it

text: results

text: in

text: a

text: signiﬁcant

text: improvement

text: over

text: existing

text: methods

text: in

text: F

text: score);

text: and

text: we

text: apply

text: it

text: to

text: unsupervised

text: learning

text: on

text: a

text: difﬁcult

text: data

text: set

text: that

text: has

text: rare

text: classes

text: without

text: local

text: density

text: maxima,

text: and

text: show

text: it

text: is

text: able

text: to

text: well

text: locate

text: cluster

text: centers

text: when

text: other

text: clustering

text: techniques

text: fail.

text: Index

text: Terms—seed-point

text: detection;

text: data

text: mining;

text: nuclei

text: de-clumping;

text: short-range

text: attractive

text: long-range

text: repulsive

text: sectioncolor

text: INTRODUCTION

text: L

text: OCATING

text: object

text: and

text: distribution

text: centers

text: is

text: a

text: fundamen-

text: at

text: the

text: nuclei

text: centers.

text: The

text: fundamental

text: problem

text: with

text: these

text: tal

text: step

text: of

text: both

text: image

text: processing

text: and

text: unsupervised

text: ma-

text: methods

text: is

text: that

text: the

text: extrema

text: of

text: the

text: voting

text: landscapes

text: are

text: chine

text: learning

text: or

text: exploratory

text: data

text: mining.

text: The

text: automated

text: not

text: at

text: the

text: nuclei

text: centers;

text: further,

text: there

text: can

text: be

text: extra

text: extrema,

text: analysis

text: and

text: classiﬁcation

text: of

text: cells

text: from

text: microscopy

text: images

text: where

text: false

text: nuclei

text: are

text: detected,

text: or

text: missing

text: extrema,

text: where

text: is

text: an

text: important

text: ﬁeld

text: that

text: includes

text: both

text: of

text: these

text: disciplines

text: true

text: nuclei

text: are

text: not

text: detected.

text: Since

text: the

text: segmentation

text: methods

text: In

text: this

text: ﬁeld,

text: the

text: ﬁrst

text: step

text: is

text: often

text: will

text: return

text: one

text: region

text: per

text: seed-point,

text: it

text: is

text: critical

text: that

text: each

text: locating

text: nuclei

text: centers

text: using

text: image

text: processing

text: techniques,

text: nuclei

text: have

text: only

text: one

text: seed-point

text: as

text: close

text: as

text: possible

text: to

text: the

text: which

text: allows

text: for

text: seed-point

text: based

text: segmentation

text: algorithms

text: nuclei

text: center

text: For

text: unsupervised

text: learning,

text: there

text: are

text: to

text: determine

text: the

text: nuclei

text: boundaries.

text: Using

text: many

text: well

text: known

text: clustering

text: techniques

text: (K-means/fuzzy

text: C-

text: the

text: segmentation

text: results,

text: features

text: of

text: each

text: nucleus

text: or

text: cell

text: means

text: expectation

text: maximization/mixture

text: of

text: are

text: measured

text: and

text: used

text: to

text: classify

text: the

text: cells

text: into

text: different

text: Gaussians

text: hierarchical

text: clustering

text: mean-shift

text: groups

text: (e.g.

text: cancer/non-cancer,

text: different

text: phenotypes/cell-

text: but

text: a

text: general

text: drawback

text: of

text: most

text: of

text: these

text: methods

text: is

text: that

text: they

text: phase

text: When

text: searching

text: for

text: new

text: produce

text: cluster

text: centers

text: that

text: are

text: biased

text: towards

text: high

text: density

text: groups

text: (new

text: phenotypes,

text: new

text: effects,

text: or

text: when

text: it

text: not

text: pos-

text: regions

text: in

text: the

text: data,

text: and

text: many

text: of

text: them

text: require

text: the

text: number

text: sible

text: for

text: an

text: expert

text: to

text: create

text: a

text: labeled

text: data

text: set,

text: unsupervised

text: of

text: clusters

text: (which

text: is

text: normally

text: not

text: known

text: beforehand)

text: as

text: machine

text: learning,

text: where

text: groups

text: are

text: determined

text: by

text: looking

text: an

text: input.

text: Mean-shift

text: clustering

text: can

text: be

text: better

text: in

text: this

text: regard;

text: for

text: similarities

text: in

text: the

text: data,

text: must

text: be

text: used.

text: however,

text: each

text: cluster

text: must

text: have

text: a

text: local

text: maximum

text: in

text: the

text: Locating

text: nuclei

text: centers

text: and

text: using

text: unsupervised

text: learning

text: point

text: density.

text: to

text: locate

text: cluster

text: centers

text: are,

text: abstractly,

text: very

text: similar:

text: they

text: In

text: this

text: work,

text: we

text: develop

text: a

text: new

text: clustering

text: method

text: (SALR

text: both

text: try

text: to

text: locate

text: the

text: centers

text: of

text: partially-overlapping

text: convex

text: clustering)

text: that

text: can

text: be

text: used

text: for

text: locating

text: both

text: nuclei

text: centers

text: objects

text: or

text: distributions.

text: However,

text: the

text: techniques

text: for

text: solving

text: and

text: the

text: cluster

text: centers

text: in

text: scatter

text: point

text: data,

text: while

text: at

text: the

text: each

text: of

text: these

text: problems

text: are

text: quite

text: different

text: and

text: are

text: not

text: same

text: time

text: addressing

text: the

text: above

text: issues.

text: The

text: premise

text: of

text: our

text: ideal

text: for

text: several

text: reasons.

text: The

text: current

text: methods

text: for

text: locating

text: method,

text: which

text: is

text: graphically

text: described

text: in

text: Figure

text: and

text: nuclei

text: centers

text: (radial

text: voting

text: gradient

text: which

text: takes

text: cues

text: from

text: the

text: fundamental

text: physics

text: of

text: modeling

text: convergence

text: and

text: sliding

text: band

text: ﬁlters

text: Laplacian

text: classical

text: Wigner

text: crystals

text: and

text: modeling

text: the

text: formation

text: of

text: Gaussian

text: ﬁlters

text: try

text: to

text: compute

text: a

text: of

text: clusters

text: is

text: that

text: we

text: can

text: ﬁnd

text: the

text: centers

text: of

text: surface,

text: named

text: the

text: voting

text: landscape,

text: that

text: has

text: local

text: extrema

text: overlapping

text: convex

text: regions

text: by

text: modeling

text: the

text: dynamics

text: of

text: particles

text: trapped

text: in

text: an

text: appropriate

text: conﬁning

text: well.

text: Consider

text: a

text: set

text: of

text: repulsive

text: particles

text: conﬁned

text: to

text: a

text: region;

text: This

text: work

text: has

text: been

text: submitted

text: to

text: the

text: IEEE

text: for

text: possible

text: publication.

text: Copyright

text: may

text: be

text: transferred

text: without

text: notice,

text: after

text: which

text: this

text: version

text: may

text: these

text: particles

text: will

text: try

text: to

text: move

text: as

text: far

text: apart

text: from

text: each

text: other

text: no

text: longer

text: be

text: accessible.

text: as

text: possible.

text: Figure

text: shows

text: this

text: for

text: particles

text: conﬁned

text: J.

text: Kapaldo

text: and

text: X.

text: Han

text: are

text: with

text: the

text: Department

text: of

text: Physics,

text: Univer-

text: to

text: a

text: parabolic

text: potential

text: well.

text: If

text: the

text: region

text: the

text: particles

text: are

text: sity

text: of

text: Notre

text: Dame,

text: Notre

text: Dame,

text: IN,

text: (e-mail:

text: jkapaldo@nd.edu,

text: conﬁned

text: to

text: has

text: several

text: local

text: minima,

text: then

text: the

text: particles

text: will

text: D.

text: Mery

text: is

text: with

text: the

text: Department

text: of

text: Computer

text: Science,

text: Pontiﬁcia

text: Universi-

text: preferentially

text: locate

text: as

text: close

text: as

text: possible

text: to

text: these

text: minima

text: dad

text: Cato´lica

text: de

text: Chile,

text: Santiago

text: Chile.

text: (email:

text: dmery@ing.puc.cl)

text: to

text: lower

text: the

text: total

text: energy.

text: Assuming

text: our

text: goal

text: is

text: to

text: have

text: footercolorManuscript

text: submitted

text: Aug.

text: particles

text: near

text: each

text: local

text: minima

text: and

text: nowhere

text: else,

text: we

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 2 Fig. 1. captioncolorPremise of SALR clustering. a–d, Intuition: schematic of the lowest energy conﬁguration for a, 10 repulsive particles in a parabolic well (this is one of the two isomeric ground states [31]); b, three repulsive particles with three potential minima; c, seven repulsive particles with three potential minima; and d, seven SALR interacting particles with three potential minima. Contour levels represent potential energy. e, SALR interaction potential formed with a Gaussian attractive term and a 1/r repulsive term. The distance ra, named the attractive extent, is the distance where particles switch from being attractive to repulsive. f, Starting from a region of overlapping convex objects, a conﬁning potential is created (third dimension corresponds to potential value); particles are randomly placed and the dynamics modeled; at the end the particles will form clusters located approximately at the center of each convex object. (cid:88)(cid:16) (cid:17) (cid:88) (cid:0) (cid:1) must have the same number of particles as the number of equationcolor potential minima, since two particles cannot be near each H 1 kq q | − | other, see Figure 1b,c. This is a problem as the number of = p2 + V (r ) + i j V r r , (1) 2m i i 2 int i j potential minima is normally not known before hand, but it i i (cid:54)i j=i can be solved by modifying how the particles interact with each other such that particles near each other are attracted to where m, p, r, and q represent the particle’s mass, mo- each other instead of repulsed—this short-range attractive mentum, position, and charge, respectfully, k is a coupling | − | long-range repulsive (SALR) particle interaction potential constant, and r r is the distance between r and r . · i j i j can be seen in Figure 1e. Now, we can use (many) more V ( ) is the SALR particle interaction potential created with int particles than we expect there to be potential minima, and a Gaussian attractive term and a 1/(cid:32)r repulsive t(cid:33)erm we can have a cluster of particles located at each potential equationcolor minima, see Figure 1d. − 1 − −(r µ)2 Using this premise, we create a conﬁning potential from V (r) = A exp , (2) int r + (cid:15) 2σ2 the region of overlapping convex objects that has a higher energy at the region’s edges and a lower energy (valley) near where r is the distance between two of the particles the object centers, we randomly place many SALR particles and (cid:15) is a small constant value ((cid:15) = 0.2) to prevent the in this region, and then we model the particle dynamics divergence at r = 0. An example of this potential is shown while slowly decreasing their speed; the ﬁnal position of in Figure 1e. The values A, µ, and σ in (2) are set by solving each cluster of particles will approximately give the center of a least-squares problem so that the depth d , the location each locally convex region, see Figure 1f and Supplemental 0 of the potential minimum r , and the distance at which the Video 1. 0 potential goes from being attractive to repulsive, referred to We validate SALR clustering by applying it to the prob- as the attractive extent, r may be directly speciﬁed, see Fig- a lem of locating nuclei centers, and show that it can signif- ure 1e. For example, the correspondence between two sets of → − → icantly improve the location performance compared with these parameters (d , r , r ) (A, µ, σ) are ( 1, 2, 10) 0 −0 a → − previous methods and discuss how/why it is able to im- (1.58, 0.87, 2.82) and ( 1, 2, 15) (1.84, 1.32, 4.84). prove the performance. We also demonstrate the application The particles’ equations of motion can be found from the of SALR clustering to unsupervised machine learning, and Hamiltonian (1) to be (cid:88) show that it can determine the correct number and position equationcolor of clusters and better locate rare clusters that do not have dp −∇ − ∇ | − | local density maxima. i = V (r ) kq q V ( r r ) (3) dt i i (cid:54) i j i int i j sectioncolor j=i 2 MODELING PARTICLE DYNAMICS dri = 1 p , (4) dt m i i ∇ The equations governing the particle dynamics are de- where is the gradient operator acting on vector r . i i rived from the Hamiltonian of a set of N interacting charged The system described so far conserves energy, which particles, means that the particles/clusters will never stop moving. ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Fig.

text: captioncolorPremise

text: of

text: SALR

text: clustering.

text: a–d,

font: WXYCWA+URWPalladioL-Bold

text: Intuition:

text: schematic

text: of

text: the

text: lowest

text: energy

text: conﬁguration

text: for

text: a,

text: repulsive

text: particles

text: in

text: a

text: parabolic

text: well

text: (this

text: is

text: one

text: of

text: the

text: two

text: isomeric

text: ground

text: states

text: b,

text: three

text: repulsive

text: particles

text: with

text: three

text: potential

text: minima;

text: c,

text: seven

text: repulsive

text: particles

text: with

text: three

text: potential

text: minima;

text: and

text: d,

text: seven

text: SALR

text: interacting

text: particles

text: with

text: three

text: potential

text: minima.

text: Contour

text: levels

text: represent

text: potential

text: energy.

text: e,

text: SALR

text: interaction

text: potential

text: formed

text: with

text: a

text: Gaussian

text: attractive

text: term

text: and

text: a

text: repulsive

text: term.

text: The

text: distance

text: ra,

text: named

text: the

text: attractive

text: extent,

text: is

text: the

text: distance

text: where

text: particles

text: switch

text: from

text: being

text: attractive

text: to

text: repulsive.

text: f,

text: Starting

text: from

text: a

text: region

text: of

text: overlapping

text: convex

text: objects,

text: a

text: conﬁning

text: potential

text: is

text: created

text: (third

text: dimension

text: corresponds

text: to

text: potential

text: value);

text: particles

text: are

text: randomly

text: placed

text: and

text: the

text: dynamics

text: modeled;

text: at

text: the

text: end

text: the

text: particles

text: will

text: form

text: clusters

text: located

text: approximately

text: at

text: the

text: center

text: of

text: each

text: convex

text: object.

text: must

text: have

text: the

text: same

text: number

text: of

text: particles

text: as

text: the

text: number

text: of

text: equationcolor

text: potential

text: minima,

text: since

text: two

text: particles

text: cannot

text: be

text: near

text: each

text: H

text: kq

text: q

text: other,

text: see

text: Figure

text: This

text: is

text: a

text: problem

text: as

text: the

text: number

text: of

text: V

text: (r

text: i

text: j

text: V

text: r

text: r

text: i

text: i

text: int

text: i

text: j

text: potential

text: minima

text: is

text: normally

text: not

text: known

text: before

text: hand,

text: but

text: it

text: i

text: i

text: j=i

text: can

text: be

text: solved

text: by

text: modifying

text: how

text: the

text: particles

text: interact

text: with

text: each

text: other

text: such

text: that

text: particles

text: near

text: each

text: other

text: are

text: attracted

text: to

text: where

text: m,

text: p,

text: r,

text: and

text: q

text: represent

text: the

text: particle’s

text: mass,

text: mo-

text: each

text: other

text: instead

text: of

text: repulsed—this

text: short-range

text: attractive

text: mentum,

text: position,

text: and

text: charge,

text: respectfully,

text: k

text: is

text: a

text: coupling

text: long-range

text: repulsive

text: (SALR)

text: particle

text: interaction

text: potential

text: constant,

text: and

text: r

text: r

text: is

text: the

text: distance

text: between

text: r

text: and

text: r

text: i

text: j

text: i

text: j

text: can

text: be

text: seen

text: in

text: Figure

text: Now,

text: we

text: can

text: use

text: (many)

text: more

text: V

text: is

text: the

text: SALR

text: particle

text: interaction

text: potential

text: created

text: with

text: int

text: particles

text: than

text: we

text: expect

text: there

text: to

text: be

text: potential

text: minima,

text: and

text: a

text: Gaussian

text: attractive

text: term

text: and

text: a

text: repulsive

text: we

text: can

text: have

text: a

text: cluster

text: of

text: particles

text: located

text: at

text: each

text: potential

text: equationcolor

text: minima,

text: see

text: Figure

text: −(r

text: Using

text: this

text: premise,

text: we

text: create

text: a

text: conﬁning

text: potential

text: from

text: V

text: (r)

text: A

text: exp

text: int

text: r

text: the

text: region

text: of

text: overlapping

text: convex

text: objects

text: that

text: has

text: a

text: higher

text: energy

text: at

text: the

text: region’s

text: edges

text: and

text: a

text: lower

text: energy

text: (valley)

text: near

text: where

text: r

text: is

text: the

text: distance

text: between

text: two

text: of

text: the

text: particles

text: the

text: object

text: centers,

text: we

text: randomly

text: place

text: many

text: SALR

text: particles

text: and

text: is

text: a

text: small

text: constant

text: value

text: to

text: prevent

text: the

text: in

text: this

text: region,

text: and

text: then

text: we

text: model

text: the

text: particle

text: dynamics

text: divergence

text: at

text: r

text: An

text: example

text: of

text: this

text: potential

text: is

text: shown

text: while

text: slowly

text: decreasing

text: their

text: speed;

text: the

text: ﬁnal

text: position

text: of

text: in

text: Figure

text: The

text: values

text: A,

text: µ,

text: and

text: σ

text: in

text: are

text: set

text: by

text: solving

text: each

text: cluster

text: of

text: particles

text: will

text: approximately

text: give

text: the

text: center

text: of

text: a

text: least-squares

text: problem

text: so

text: that

text: the

text: depth

text: d

text: the

text: location

text: each

text: locally

text: convex

text: region,

text: see

text: Figure

text: and

text: Supplemental

text: of

text: the

text: potential

text: minimum

text: r

text: and

text: the

text: distance

text: at

text: which

text: the

text: Video

text: potential

text: goes

text: from

text: being

text: attractive

text: to

text: repulsive,

text: referred

text: to

text: We

text: validate

text: SALR

text: clustering

text: by

text: applying

text: it

text: to

text: the

text: prob-

text: as

text: the

text: attractive

text: extent,

text: r

text: may

text: be

text: directly

text: speciﬁed,

text: see

text: Fig-

text: a

text: lem

text: of

text: locating

text: nuclei

text: centers,

text: and

text: show

text: that

text: it

text: can

text: signif-

text: ure

text: For

text: example,

text: the

text: correspondence

text: between

text: two

text: sets

text: of

text: icantly

text: improve

text: the

text: location

text: performance

text: compared

text: with

text: these

text: parameters

text: (d

text: r

text: r

text: (A,

text: µ,

text: σ)

text: are

text: −0

text: a

text: previous

text: methods

text: and

text: discuss

text: how/why

text: it

text: is

text: able

text: to

text: im-

text: and

text: prove

text: the

text: performance.

text: We

text: also

text: demonstrate

text: the

text: application

text: The

text: particles’

text: equations

text: of

text: motion

text: can

text: be

text: found

text: from

text: the

text: of

text: SALR

text: clustering

text: to

text: unsupervised

text: machine

text: learning,

text: and

text: Hamiltonian

text: to

text: be

text: show

text: that

text: it

text: can

text: determine

text: the

text: correct

text: number

text: and

text: position

text: equationcolor

text: of

text: clusters

text: and

text: better

text: locate

text: rare

text: clusters

text: that

text: do

text: not

text: have

text: dp

text: local

text: density

text: maxima.

text: i

text: V

text: (r

text: kq

text: q

text: V

text: r

text: r

text: dt

text: i

text: i

text: i

text: j

text: i

text: int

text: i

text: j

text: sectioncolor

text: j=i

text: MODELING

text: PARTICLE

text: DYNAMICS

text: dri

text: p

text: dt

text: m

text: i

text: i

text: The

text: equations

text: governing

text: the

text: particle

text: dynamics

text: are

text: de-

text: where

text: is

text: the

text: gradient

text: operator

text: acting

text: on

text: vector

text: r

text: i

text: i

text: rived

text: from

text: the

text: Hamiltonian

text: of

text: a

text: set

text: of

text: N

text: interacting

text: charged

text: The

text: system

text: described

text: so

text: far

text: conserves

text: energy,

text: which

text: particles,

text: means

text: that

text: the

text: particles/clusters

text: will

text: never

text: stop

text: moving.

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 8 Fig. 8. captioncolorSALR clustering for real data. a, projections of the 5D point count down to dimensions 1, 3, and 4 and dimensions 1, 2, and 5. Black points are the results from each of the 5 iterations, red points are the ﬁnal seed-points. Arrows label the three low density regions. b, results using k-means with 4, 5, and 6 clusters. The black arrow labels the one seed-point that approximately locates a low-density region; the circles show the two missing seed-points. achieved using the coupling constant, k, if one wanted). conﬁning potential gradient. (Note each repetition of k- In Supplemental Figure 8, we show how the interaction means and each iteration of our method can run in paral- potential and force in data space change as (cid:96) goes from lel. Computer speciﬁcations are shown below.) Thus, our 10 to 50. The interaction force is strongest at (cid:96) = 10 and method can lead to an important speed increase for some decreases as (cid:96) increases. data sets. We model particles ﬁve times (small black points in Fig- sectioncolor ure 8a) and cluster the results (red points in Figure 8a). You 5 DISCUSSION can see that our method located the three low density extru- sions (labeled by black arrows) very well, and our method It is interesting to consider how, and under what con- even ﬁnds the very small, but distinct, cluster marked by ditions, SALR clustering leads to these improvements. The the circle in right plot of Figure 8a. (Supplemental Video answer comes by analyzing the force on the particles in 2 shows an animation of the particle dynamics along with the simulation: the conﬁning potential exerts a force on the |∇ | the comparison to k-means with 5 clusters.) The parameter particles with a magnitude of V (which is the slope of values used in these simulations were as follows; uniform the conﬁning potential in Figure 1f), and the particles repel ≈ − random distribution, r 1.5, [V , V ] = [1/4, 1/6], each other with a force of about r 2. If the force from the s − min max a ra = 2, r0 = 0.15 ra, d0 = 1, (cid:96) = 12, p = 4, and the c|∇onﬁ|ni≫ng po−tential is much larger than the repulsive force, potential force was scaled so that the 99% value is 0.4. (The V r 2, then the particles will only ﬁnd the local a potential bound requirements helps to reduce the number minimum of the conﬁning potential—this would lead to the of particles, which is 147, and to position the particles near results labeled Dist. Trans. in Figure 5, and is the same thing the outskirts of the region, which is better, in general.) that the previous methods do in ﬁnding the local extrema of As means of comparison, we used k-means clustering the voting landscape or ﬁning the local maxima of the point with two repetitions for 4, 5, and 6 clusters; we repeated density (mean-shift). Alternatively, if the repulsive force is − ≫ |∇ | this 20 times and show the results in Figure 8b. The results much larger than the conﬁning force, r 2 V , then a using 4 or 5 clusters are stable (the clusters are in the same the particles will ignore the conﬁning potential minima and place in all 20 iterations), but are only able to approximately spread out—this would result in particle clusters separated locate one of the three low density regions, while missing by at least a distance r that uniformly cover the region. a two of them (labeled by the circles). The reason k-means In between these two regimes, when the two forces are |∇ | ∼ − does not locate the low density regions well is that the approximately the same order of magnitude, V r 2, a cluster centers it ﬁnds are the locations which minimize the there is an interaction between the conﬁning potential and variance in each cluster, and the variance can be minimized the particle repulsion, and this is the regime we operate in by moving the cluster centers closer to the high density and that leads to SALR clustering’s improved performance. regions. (The same/similar reasoning also applies for fuzzy (When using the distance transform to create the conﬁning ∝ − c-means or mixture of Gaussians.) This demonstrates an potential in our images, our conﬁning force is r 2, where important feature of our method: our method can locate the r is the distance to the nearest region boundary, and the ∝ − center of low density regions (also called rare classes) even repulsive force between the particles is also r 2, where r when the low density regions do not have local maximum. is the distance between particles.) This need for the forces Looking at the computation time, the k-means with 5 to be approximately equal is why it is necessary to scale the ± clusters, which was the fastest, takes 15 5 sec per repetition; magnitude of the potential gradient when the inverse point ± ∼ our method takes 1.15 0.08 sec per iteration with 5.5 density is used as the conﬁning potential. sec overhead for binning, smoothing, and computing the Our method can be applied to even higher dimensional ##LTFigure##

text: Fig.

text: captioncolorSALR

text: clustering

text: for

text: real

text: data.

text: a,

text: projections

text: of

text: the

text: point

text: count

text: down

text: to

text: dimensions

text: and

text: and

text: dimensions

text: and

text: Black

text: points

text: are

text: the

text: results

text: from

text: each

text: of

text: the

text: iterations,

text: red

text: points

text: are

text: the

text: ﬁnal

text: seed-points.

text: Arrows

text: label

text: the

text: three

text: low

text: density

text: regions.

text: b,

text: results

text: using

text: k-means

text: with

text: and

text: clusters.

text: The

text: black

text: arrow

text: labels

text: the

text: one

text: seed-point

text: that

text: approximately

text: locates

text: a

text: low-density

text: region;

text: the

text: circles

text: show

text: the

text: two

text: missing

text: seed-points.

text: achieved

text: using

text: the

text: coupling

text: constant,

text: k,

text: if

text: one

text: wanted).

text: conﬁning

text: potential

text: gradient.

text: (Note

text: each

text: repetition

text: of

text: k-

text: In

text: Supplemental

text: Figure

text: we

text: show

text: how

text: the

text: interaction

text: means

text: and

text: each

text: iteration

text: of

text: our

text: method

text: can

text: run

text: in

text: paral-

text: potential

text: and

text: force

text: in

text: data

text: space

text: change

text: as

text: goes

text: from

text: lel.

text: Computer

text: speciﬁcations

text: are

text: shown

text: below.)

text: Thus,

text: our

text: to

text: The

text: interaction

text: force

text: is

text: strongest

text: at

text: and

text: method

text: can

text: lead

text: to

text: an

text: important

text: speed

text: increase

text: for

text: some

text: decreases

text: as

text: increases.

text: data

text: sets.

text: We

text: model

text: particles

text: ﬁve

text: times

text: (small

text: black

text: points

text: in

text: Fig-

text: sectioncolor

text: ure

text: and

text: cluster

text: the

text: results

text: (red

text: points

text: in

text: Figure

text: You

text: DISCUSSION

text: can

text: see

text: that

text: our

text: method

text: located

text: the

text: three

text: low

text: density

text: extru-

text: sions

text: (labeled

text: by

text: black

text: arrows)

text: very

text: well,

text: and

text: our

text: method

text: It

text: is

text: interesting

text: to

text: consider

text: how,

text: and

text: under

text: what

text: con-

text: even

text: ﬁnds

text: the

text: very

text: small,

text: but

text: distinct,

text: cluster

text: marked

text: by

text: ditions,

text: SALR

text: clustering

text: leads

text: to

text: these

text: improvements.

text: The

text: the

text: circle

text: in

text: right

text: plot

text: of

text: Figure

text: (Supplemental

text: Video

text: answer

text: comes

text: by

text: analyzing

text: the

text: force

text: on

text: the

text: particles

text: in

text: shows

text: an

text: animation

text: of

text: the

text: particle

text: dynamics

text: along

text: with

text: the

text: simulation:

text: the

text: conﬁning

text: potential

text: exerts

text: a

text: force

text: on

text: the

text: the

text: comparison

text: to

text: k-means

text: with

text: clusters.)

text: The

text: parameter

text: particles

text: with

text: a

text: magnitude

text: of

text: V

text: (which

text: is

text: the

text: slope

text: of

text: values

text: used

text: in

text: these

text: simulations

text: were

text: as

text: follows;

text: uniform

text: the

text: conﬁning

text: potential

text: in

text: Figure

text: and

text: the

text: particles

text: repel

text: random

text: distribution,

text: r

text: [V

text: V

text: each

text: other

text: with

text: a

text: force

text: of

text: about

text: r

text: If

text: the

text: force

text: from

text: the

text: s

text: min

text: max

text: a

text: ra

text: ra,

text: p

text: and

text: the

text: c|∇onﬁ|ni≫ng

text: po−tential

text: is

text: much

text: larger

text: than

text: the

text: repulsive

text: force,

text: potential

text: force

text: was

text: scaled

text: so

text: that

text: the

text: value

text: is

text: (The

text: V

text: r

text: then

text: the

text: particles

text: will

text: only

text: ﬁnd

text: the

text: local

text: a

text: potential

text: bound

text: requirements

text: helps

text: to

text: reduce

text: the

text: number

text: minimum

text: of

text: the

text: conﬁning

text: potential—this

text: would

text: lead

text: to

text: the

text: of

text: particles,

text: which

text: is

text: and

text: to

text: position

text: the

text: particles

text: near

text: results

text: labeled

text: Dist.

text: Trans.

text: in

text: Figure

text: and

text: is

text: the

text: same

text: thing

text: the

text: outskirts

text: of

text: the

text: region,

text: which

text: is

text: better,

text: in

text: general.)

text: that

text: the

text: previous

text: methods

text: do

text: in

text: ﬁnding

text: the

text: local

text: extrema

text: of

text: As

text: means

text: of

text: comparison,

text: we

text: used

text: k-means

text: clustering

text: the

text: voting

text: landscape

text: or

text: ﬁning

text: the

text: local

text: maxima

text: of

text: the

text: point

text: with

text: two

text: repetitions

text: for

text: and

text: clusters;

text: we

text: repeated

text: density

text: (mean-shift).

text: Alternatively,

text: if

text: the

text: repulsive

text: force

text: is

text: this

text: times

text: and

text: show

text: the

text: results

text: in

text: Figure

text: The

text: results

text: much

text: larger

text: than

text: the

text: conﬁning

text: force,

text: r

text: V

text: then

text: a

text: using

text: or

text: clusters

text: are

text: stable

text: (the

text: clusters

text: are

text: in

text: the

text: same

text: the

text: particles

text: will

text: ignore

text: the

text: conﬁning

text: potential

text: minima

text: and

text: place

text: in

text: all

text: iterations),

text: but

text: are

text: only

text: able

text: to

text: approximately

text: spread

text: out—this

text: would

text: result

text: in

text: particle

text: clusters

text: separated

text: locate

text: one

text: of

text: the

text: three

text: low

text: density

text: regions,

text: while

text: missing

text: by

text: at

text: least

text: a

text: distance

text: r

text: that

text: uniformly

text: cover

text: the

text: region.

text: a

text: two

text: of

text: them

text: (labeled

text: by

text: the

text: circles).

text: The

text: reason

text: k-means

text: In

text: between

text: these

text: two

text: regimes,

text: when

text: the

text: two

text: forces

text: are

text: does

text: not

text: locate

text: the

text: low

text: density

text: regions

text: well

text: is

text: that

text: the

text: approximately

text: the

text: same

text: order

text: of

text: magnitude,

text: V

text: r

text: a

text: cluster

text: centers

text: it

text: ﬁnds

text: are

text: the

text: locations

text: which

text: minimize

text: the

text: there

text: is

text: an

text: interaction

text: between

text: the

text: conﬁning

text: potential

text: and

text: variance

text: in

text: each

text: cluster,

text: and

text: the

text: variance

text: can

text: be

text: minimized

text: the

text: particle

text: repulsion,

text: and

text: this

text: is

text: the

text: regime

text: we

text: operate

text: in

text: by

text: moving

text: the

text: cluster

text: centers

text: closer

text: to

text: the

text: high

text: density

text: and

text: that

text: leads

text: to

text: SALR

text: clustering’s

text: improved

text: performance.

text: regions.

text: (The

text: same/similar

text: reasoning

text: also

text: applies

text: for

text: fuzzy

text: (When

text: using

text: the

text: distance

text: transform

text: to

text: create

text: the

text: conﬁning

text: c-means

text: or

text: mixture

text: of

text: Gaussians.)

text: This

text: demonstrates

text: an

text: potential

text: in

text: our

text: images,

text: our

text: conﬁning

text: force

text: is

text: r

text: where

text: important

text: feature

text: of

text: our

text: method:

text: our

text: method

text: can

text: locate

text: the

text: r

text: is

text: the

text: distance

text: to

text: the

text: nearest

text: region

text: boundary,

text: and

text: the

text: center

text: of

text: low

text: density

text: regions

text: (also

text: called

text: rare

text: classes)

text: even

text: repulsive

text: force

text: between

text: the

text: particles

text: is

text: also

text: r

text: where

text: r

text: when

text: the

text: low

text: density

text: regions

text: do

text: not

text: have

text: local

text: maximum.

text: is

text: the

text: distance

text: between

text: particles.)

text: This

text: need

text: for

text: the

text: forces

text: Looking

text: at

text: the

text: computation

text: time,

text: the

text: k-means

text: with

text: to

text: be

text: approximately

text: equal

text: is

text: why

text: it

text: is

text: necessary

text: to

text: scale

text: the

text: clusters,

text: which

text: was

text: the

text: fastest,

text: takes

text: sec

text: per

text: repetition;

text: magnitude

text: of

text: the

text: potential

text: gradient

text: when

text: the

text: inverse

text: point

text: our

text: method

text: takes

text: sec

text: per

text: iteration

text: with

text: density

text: is

text: used

text: as

text: the

text: conﬁning

text: potential.

text: sec

text: overhead

text: for

text: binning,

text: smoothing,

text: and

text: computing

text: the

text: Our

text: method

text: can

text: be

text: applied

text: to

text: even

text: higher

text: dimensional

text: ##LTFigure##