text: 8 W. SUN, V. SIMON, S. MONNET, PH. ROBERT, AND P. SENS Figure 2. Distribution function of load distribution of a random node after 729 days. As a conclusion, the simulations show that, with a limited cost in terms of complexity, the power of choice policy has remarkable properties. The load of each node is bounded by 300. It may be remarked that each possible load between 0 and 300 is represented by the same amount of nodes on average. Figure 2 shows that there is approximately the same number of nodes having 0 data blocks, than nodes having 150 data block or nodes having 300 data blocks. Note that this is a stationary state. Additionally, the variation is low, we can observe in Table 1 that among the 132,090 samples, the most loaded node was never above 328. From a practical point of view, it means that a slightly oversized storage device at each node (a bit more than twice the average) is enough to guarantee the durability of the system. In the following sections we investigate simpliﬁed mathematical models of two placement policies: Random and Power of Choice. The goal is to explain these striking qualitative properties of these policies. 4. Mathematical Models The main goal of the paper is to investigate the performance of duplication algorithms in terms of the overhead for the loads of the nodes of the network. Without loss of generality, we will assume that the breakdown of each server occurs according to a Poisson process with rate 1. After a breakdown, a server restarts empty (in fact a new server replaces it). The replication degree of data blocks is ≥ d 2, each data block has at most d copies. A complete Markovian description of ∗ such a system is quite complex. Indeed, if there are N servers and F initial data ≤ ≤ ∗ N blocks, for 1 i F , the locations of the ith data block are given by the indices N ≤ of k distinct servers if there are k d copies of this data block. Consequently the ∗ size of the state space of the Markov process is at least of the order of (Nd)FN ∗ which is huge if it is remembered that F is of the order of N. For this reason, we N shall simplify the mathematical model. ##LTFigure##

text: W.

text: SUN,

text: V.

text: SIMON,

text: S.

text: MONNET,

text: PH.

text: ROBERT,

text: AND

text: P.

text: SENS

text: Figure

text: Distribution

text: function

text: of

text: load

text: distribution

text: of

text: a

text: random

text: node

text: after

text: days.

text: As

text: a

text: conclusion,

text: the

text: simulations

text: show

text: that,

text: with

text: a

text: limited

text: cost

text: in

text: terms

text: of

text: complexity,

text: the

text: power

text: of

text: choice

text: policy

text: has

text: remarkable

text: properties.

text: The

text: load

text: of

text: each

text: node

text: is

text: bounded

text: by

text: It

text: may

text: be

text: remarked

text: that

text: each

text: possible

text: load

text: between

text: and

text: is

text: represented

text: by

text: the

text: same

text: amount

text: of

text: nodes

text: on

text: average.

text: Figure

text: shows

text: that

text: there

text: is

text: approximately

text: the

text: same

text: number

text: of

text: nodes

text: having

text: data

text: blocks,

text: than

text: nodes

text: having

text: data

text: block

text: or

text: nodes

text: having

text: data

text: blocks.

text: Note

text: that

text: this

text: is

text: a

text: stationary

text: state.

text: Additionally,

text: the

text: variation

text: is

text: low,

text: we

text: can

text: observe

text: in

text: Table

text: that

text: among

text: the

text: samples,

text: the

text: most

text: loaded

text: node

text: was

text: never

text: above

text: From

text: a

text: practical

text: point

text: of

text: view,

text: it

text: means

text: that

text: a

text: slightly

text: oversized

text: storage

text: device

text: at

text: each

text: node

text: (a

text: bit

text: more

text: than

text: twice

text: the

text: average)

text: is

text: enough

text: to

text: guarantee

text: the

text: durability

text: of

text: the

text: system.

text: In

text: the

text: following

text: sections

text: we

text: investigate

text: simpliﬁed

text: mathematical

text: models

text: of

text: two

text: placement

text: policies:

text: Random

text: and

text: Power

text: of

text: Choice.

text: The

text: goal

text: is

text: to

text: explain

text: these

text: striking

text: qualitative

text: properties

text: of

text: these

text: policies.

text: Mathematical

text: Models

text: The

text: main

text: goal

text: of

text: the

text: paper

text: is

text: to

text: investigate

text: the

text: performance

text: of

text: duplication

text: algorithms

text: in

text: terms

text: of

text: the

text: overhead

text: for

text: the

text: loads

text: of

text: the

text: nodes

text: of

text: the

text: network.

text: Without

text: loss

text: of

text: generality,

text: we

text: will

text: assume

text: that

text: the

text: breakdown

text: of

text: each

text: server

text: occurs

text: according

text: to

text: a

text: Poisson

text: process

text: with

text: rate

text: After

text: a

text: breakdown,

text: a

text: server

text: restarts

text: empty

text: (in

text: fact

text: a

text: new

text: server

text: replaces

text: it).

text: The

text: replication

text: degree

text: of

text: data

text: blocks

text: is

text: d

text: each

text: data

text: block

text: has

text: at

text: most

text: d

text: copies.

text: A

text: complete

text: Markovian

text: description

text: of

text: such

text: a

text: system

text: is

text: quite

text: complex.

text: Indeed,

text: if

text: there

text: are

text: N

text: servers

text: and

text: F

text: initial

text: data

text: N

text: blocks,

text: for

text: i

text: F

text: the

text: locations

text: of

text: the

text: ith

text: data

text: block

text: are

text: given

text: by

text: the

text: indices

text: N

text: of

text: k

text: distinct

text: servers

text: if

text: there

text: are

text: k

text: d

text: copies

text: of

text: this

text: data

text: block.

text: Consequently

text: the

text: size

text: of

text: the

text: state

text: space

text: of

text: the

text: Markov

text: process

text: is

text: at

text: least

text: of

text: the

text: order

text: of

text: (Nd)FN

text: which

text: is

text: huge

text: if

text: it

text: is

text: remembered

text: that

text: F

text: is

text: of

text: the

text: order

text: of

text: N.

text: For

text: this

text: reason,

text: we

text: N

text: shall

text: simplify

text: the

text: mathematical

text: model.

text: ##LTFigure##


text: ASSESSING UNCERTAINTIES IN X-RAY SINGLE-PARTICLE THREE-DIMENSIONAL RECONSTRUCTIONS STEFAN ENGBLOM, CARL NETTELBLAD, AND JING LIU Abstract. Modern technology for producing extremely bright and coher- ent X-ray laser pulses provides the possibility to acquire a large number of diﬀraction patterns from individual biological nanoparticles, including pro- teins, viruses, and DNA. These two-dimensional diﬀraction patterns can be practically reconstructed and retrieved down to a resolution of a few ˚Angstr¨om. In principle, a suﬃciently large collection of diﬀraction patterns will contain the required information for a full three-dimensional reconstruction of the biomolecule. The computational methodology for this reconstruction task is still under development and highly resolved reconstructions have not yet been produced. We analyze the Expansion-Maximization-Compression scheme, the current state of the art approach for this very challenging application, by isolating diﬀerent sources of uncertainty. Through numerical experiments on synthetic data we evaluate their respective impact. We reach conclusions of relevance for handling actual experimental data, as well as pointing out certain improve- ments to the underlying estimation algorithm. We also introduce a practically applicable computational methodology in the form of bootstrap procedures for assessing reconstruction uncertainty in the real data case. We evaluate the sharpness of this approach and argue that this type of procedure will be critical in the near future when handling the increasing amount of data. 1. Introduction Determing the structure of a very small biological object, such as a protein or a virus, is both fascinating and hard. The most classical and common way to deter- mine the atomic and molecular structures of small biological objects is to crystallize them and use X-rays to investigate the resulting macroscopic crystals. This method, X-ray crystallography, has succeeded in determining more than 97,200 structures [24]. With X-ray crystallography, high-quality structures can be obtained from crystals whose atoms are formed in a near perfect periodic arrangement. However, due to conformational ﬂexibility not all biological samples can form crystals. Modern X-ray Free Electron Laser (X-FEL) technology potentially provides the ability to determine biological structure without crystals. X-FEL pulses are intense and short enough to create an observable diﬀraction signal from one single parti- cle, outrunning the radiation damage. Digital detectors are used to capture the diﬀracted wave, depicting the sample before it explodes and turns into a plasma. Date: May 29, 2020. 2010 Mathematics Subject Classiﬁcation. Primary: 62F40, 68U10; Secondary: 68W10, 82D99. Key words and phrases. Maximum-Likelihood; Expectation-Maximization; Bootstrap; Single- particle imaging; X-ray lasers; Diﬀraction patterns. Corresponding author: S. Engblom, telephone +46-18-471 27 54, fax +46-18-51 19 25. 1 ##LTLine##

text: ASSESSING

label: paragraph

text: UNCERTAINTIES

text: IN

text: X-RAY

text: SINGLE-PARTICLE

text: THREE-DIMENSIONAL

text: RECONSTRUCTIONS

text: STEFAN

text: ENGBLOM,

text: CARL

text: NETTELBLAD,

text: AND

text: JING

text: LIU

text: Abstract.

text: Modern

label: abstract

text: technology

text: for

text: producing

text: extremely

text: bright

text: and

text: coher-

text: ent

text: X-ray

text: laser

text: pulses

text: provides

text: the

text: possibility

text: to

text: acquire

text: a

text: large

text: number

text: of

text: diﬀraction

text: patterns

text: from

text: individual

text: biological

text: nanoparticles,

text: including

text: pro-

text: teins,

text: viruses,

text: and

text: DNA.

text: These

text: two-dimensional

text: diﬀraction

text: patterns

text: can

text: be

text: practically

text: reconstructed

text: and

text: retrieved

text: down

text: to

text: a

text: resolution

text: of

text: a

text: few

text: ˚Angstr¨om.

text: In

text: principle,

text: a

text: suﬃciently

text: large

text: collection

text: of

text: diﬀraction

text: patterns

text: will

text: contain

text: the

text: required

text: information

text: for

text: a

text: full

text: three-dimensional

text: reconstruction

text: of

text: the

text: biomolecule.

text: The

text: computational

text: methodology

text: for

text: this

text: reconstruction

text: task

text: is

text: still

text: under

text: development

text: and

text: highly

text: resolved

text: reconstructions

text: have

text: not

text: yet

text: been

text: produced.

text: We

text: analyze

text: the

text: Expansion-Maximization-Compression

text: scheme,

text: the

text: current

text: state

text: of

text: the

text: art

text: approach

text: for

text: this

text: very

text: challenging

text: application,

text: by

text: isolating

text: diﬀerent

text: sources

text: of

text: uncertainty.

text: Through

text: numerical

text: experiments

text: on

text: synthetic

text: data

text: we

text: evaluate

text: their

text: respective

text: impact.

text: We

text: reach

text: conclusions

text: of

text: relevance

text: for

text: handling

text: actual

text: experimental

text: data,

text: as

text: well

text: as

text: pointing

text: out

text: certain

text: improve-

text: ments

text: to

text: the

text: underlying

text: estimation

text: algorithm.

text: We

text: also

text: introduce

text: a

text: practically

text: applicable

text: computational

text: methodology

text: in

text: the

text: form

text: of

text: bootstrap

text: procedures

text: for

text: assessing

text: reconstruction

text: uncertainty

text: in

text: the

text: real

text: data

text: case.

text: We

text: evaluate

text: the

text: sharpness

text: of

text: this

text: approach

text: and

text: argue

text: that

text: this

text: type

text: of

text: procedure

text: will

text: be

text: critical

text: in

text: the

text: near

text: future

text: when

text: handling

text: the

text: increasing

text: amount

text: of

text: data.

text: Introduction

text: Determing

text: the

text: structure

text: of

text: a

text: very

text: small

text: biological

text: object,

text: such

text: as

text: a

text: protein

text: or

text: a

text: virus,

text: is

text: both

text: fascinating

text: and

text: hard.

text: The

text: most

text: classical

text: and

text: common

text: way

text: to

text: deter-

text: mine

text: the

text: atomic

text: and

text: molecular

text: structures

text: of

text: small

text: biological

text: objects

text: is

text: to

text: crystallize

text: them

text: and

text: use

text: X-rays

text: to

text: investigate

text: the

text: resulting

text: macroscopic

text: crystals.

text: This

text: method,

text: X-ray

text: crystallography,

text: has

text: succeeded

text: in

text: determining

text: more

text: than

text: structures

text: With

text: X-ray

text: crystallography,

text: high-quality

text: structures

text: can

text: be

text: obtained

text: from

text: crystals

text: whose

text: atoms

text: are

text: formed

text: in

text: a

text: near

text: perfect

text: periodic

text: arrangement.

text: However,

text: due

text: to

text: conformational

text: ﬂexibility

text: not

text: all

text: biological

text: samples

text: can

text: form

text: crystals.

text: Modern

text: X-ray

text: Free

text: Electron

text: Laser

text: (X-FEL)

text: technology

text: potentially

text: provides

text: the

text: ability

text: to

text: determine

text: biological

text: structure

text: without

text: crystals.

text: X-FEL

text: pulses

text: are

text: intense

text: and

text: short

text: enough

text: to

text: create

text: an

text: observable

text: diﬀraction

text: signal

text: from

text: one

text: single

text: parti-

text: cle,

text: outrunning

text: the

text: radiation

text: damage.

text: Digital

text: detectors

text: are

text: used

text: to

text: capture

text: the

text: diﬀracted

text: wave,

text: depicting

text: the

text: sample

text: before

text: it

text: explodes

text: and

text: turns

text: into

text: a

text: plasma.

text: Date:

text: May

text: Mathematics

text: Subject

text: Classiﬁcation.

text: Primary:

text: Secondary:

text: Key

text: words

text: and

text: phrases.

text: Maximum-Likelihood;

text: Expectation-Maximization;

text: Bootstrap;

text: Single-

text: particle

text: imaging;

text: X-ray

text: lasers;

text: Diﬀraction

text: patterns.

text: Corresponding

text: author:

text: S.

text: Engblom,

text: telephone

text: fax

text: ##LTLine##


text: UNCERTAINTIES IN X-RAY 3D RECONSTRUCTIONS 3 once inside the object (i.e. only the ﬁrst order Born approximation Ψ is required); 1 iii) small-angle scattering is assumed, i.e. that the object-detector distance is much longer than the wavelength; iv) the X-FEL sources generate plane, coherent, and homogeneous waves. The scattered wav(cid:112)e on the Ewald sphere can then be written as follows: ≈ 2π { } ∝ { } (2.2) Ψ¯ I Ω F δn⊥(r⊥) ∆x3 F δn⊥(r⊥) , 0 p λ2 2 2 where F is a 2D Fourier transformation, ∆x is the sampling distance, δn⊥(r⊥) 2 is the refractive component of the refractive index for the Ewald sphere, I is the X- 0 ray pulse intensity at the object-beam interaction point, and λ is the wave length. Further, Ω equals to P2/D2, where P is the physical pitch of a single detector p pixel, and D is the object-detector distance. The noiseless diﬀraction pattern detected by the detector is the square of this scattered wave, i.e. | | ∝ | { }| (2.3) I = Ψ¯ 2 F δn⊥(r⊥) 2. 2 ∗ ∗ We denote a collection of noiseless diﬀraction patterns by K = (K )Mdata, ∗ k k=1 where each frame K is obtained from (2.3) by specifying eﬀectively a rotation of k the object. Since the X-ray pulse intensity at the object-beam interaction point I will vary in practice, we denote this variation by φ - the (photon) ﬂuence, such 0 ∗ that diﬀraction pattern with varying ﬂuence is obtained by scaling φK . Moreover, k since digital detectors are pixelized, we also discretize each diﬀraction pattern and write K∗ = (K∗ )Mpix, where M is the number of pixels. k ik i=1 pix In (2.2), the refractive component of the refractive index for the Ewald sphere δn⊥(r⊥) is dependent on the rotation of particle. This is directly observable from FXI experiments. Several methods [2, 6, 20] can be used to estimate the unknown particle rotations from FXI diﬀraction patterns, but the most successful approach so far is the EMC algorithm [11, 20, 21]. Besides calculating maximum-likelihood (ML) estimates, the EMC algorithm interpolates between 2D diﬀraction patterns and a 3D model. The EMC algorithm consists of 4 steps per iteration: i) the expansion step (e step) slices the 3D model through the model center according to the sampled rotation, i.e. expands the 3D model into a set of 2D slices; ii) the expectation step (E step) estimates the probability of each pattern to be in any given rotation; iii) the maximization step (M step) updates the 2D slices and their ﬂuences using the estimated rotational probability; iv) the compression step (C step) inserts the updated 2D slices back into the 3D model. We ﬁrst introduce the e and the C step, which interpolates between a 3D model and 2D slices. Let W = {W }Mgrid be a 3D discrete model, an estimation of the 3D l l=1 Fourier intensity of a b(cid:80)iomolecule, where M = M3/2. The rotational space R is grid pix discretized by (R )Mrot, and the corresponding prior weight for rotation R is w , j j=1 j j normalized such that w = 1. Similarly, the intensity space is discretized by a j j ##LTLine## ##LTLine##

text: UNCERTAINTIES

text: IN

text: X-RAY

text: RECONSTRUCTIONS

text: once

text: inside

text: the

text: object

text: (i.e.

text: only

text: the

text: ﬁrst

text: order

text: Born

text: approximation

text: Ψ

text: is

text: required);

text: iii)

text: small-angle

text: scattering

text: is

text: assumed,

text: i.e.

text: that

text: the

text: object-detector

text: distance

text: is

text: much

text: longer

text: than

text: the

text: wavelength;

text: iv)

text: the

text: X-FEL

text: sources

text: generate

text: plane,

text: coherent,

text: and

text: homogeneous

text: waves.

text: The

text: scattered

label: equation

text: on

text: the

text: Ewald

text: sphere

text: can

text: then

text: be

text: written

text: as

text: follows:

text: 2π

text: Ψ¯

text: I

text: Ω

text: F

text: δn⊥(r⊥)

text: F

text: δn⊥(r⊥)

text: p

text: λ2

text: where

text: F

text: is

text: a

text: Fourier

text: transformation,

text: ∆x

text: is

text: the

text: sampling

text: distance,

text: δn⊥(r⊥)

text: is

text: the

text: refractive

text: component

text: of

text: the

text: refractive

text: index

text: for

text: the

text: Ewald

text: sphere,

text: I

text: is

text: the

text: X-

text: ray

text: pulse

text: intensity

text: at

text: the

text: object-beam

text: interaction

text: point,

text: and

text: λ

text: is

text: the

text: wave

text: length.

text: Further,

text: Ω

text: equals

text: to

text: where

text: P

text: is

text: the

text: physical

text: pitch

text: of

text: a

text: single

text: detector

text: p

text: pixel,

text: and

text: D

text: is

text: the

text: object-detector

text: distance.

text: The

text: noiseless

text: diﬀraction

text: pattern

text: detected

text: by

text: the

text: detector

text: is

text: the

text: square

text: of

text: this

text: scattered

text: wave,

text: i.e.

text: I

text: Ψ¯

text: F

text: δn⊥(r⊥)

text: We

text: denote

text: a

text: collection

text: of

text: noiseless

text: diﬀraction

text: patterns

text: by

text: K

text: (K

text: )Mdata,

text: k

text: where

text: each

text: frame

text: K

text: is

text: obtained

text: from

text: by

text: specifying

text: eﬀectively

text: a

text: rotation

text: of

text: k

text: the

text: object.

text: Since

text: the

text: X-ray

text: pulse

text: intensity

text: at

text: the

text: object-beam

text: interaction

text: point

text: I

text: will

text: vary

text: in

text: practice,

text: we

text: denote

text: this

text: variation

text: by

text: φ

text: the

text: (photon)

text: ﬂuence,

text: such

text: that

text: diﬀraction

text: pattern

text: with

text: varying

text: ﬂuence

text: is

text: obtained

text: by

text: scaling

text: φK

text: Moreover,

text: k

text: since

text: digital

text: detectors

text: are

text: pixelized,

text: we

text: also

text: discretize

text: each

text: diﬀraction

text: pattern

text: and

text: write

text: K∗

text: (K∗

text: )Mpix,

text: where

text: M

text: is

text: the

text: number

text: of

text: pixels.

text: k

text: ik

text: pix

text: In

text: the

text: refractive

text: component

text: of

text: the

text: refractive

text: index

text: for

text: the

text: Ewald

text: sphere

text: δn⊥(r⊥)

text: is

text: dependent

text: on

text: the

text: rotation

text: of

text: particle.

text: This

text: is

text: directly

text: observable

text: from

text: FXI

text: experiments.

text: Several

text: methods

text: can

text: be

text: used

text: to

text: estimate

text: the

text: unknown

text: particle

text: rotations

text: from

text: FXI

text: diﬀraction

text: patterns,

text: but

text: the

text: most

text: successful

text: approach

text: so

text: far

text: is

text: the

text: EMC

text: algorithm

text: Besides

text: calculating

text: maximum-likelihood

text: (ML)

text: estimates,

text: the

text: EMC

text: algorithm

text: interpolates

text: between

text: diﬀraction

text: patterns

text: and

text: a

text: model.

text: The

text: EMC

text: algorithm

text: consists

text: of

text: steps

text: per

text: iteration:

text: i)

text: the

text: expansion

text: step

text: (e

text: step)

text: slices

text: the

text: model

text: through

text: the

text: model

text: center

text: according

text: to

text: the

text: sampled

text: rotation,

text: i.e.

text: expands

text: the

text: model

text: into

text: a

text: set

text: of

text: slices;

text: ii)

text: the

text: expectation

text: step

text: (E

text: step)

text: estimates

text: the

text: probability

text: of

text: each

text: pattern

text: to

text: be

text: in

text: any

text: given

text: rotation;

text: iii)

text: the

text: maximization

text: step

text: (M

text: step)

text: updates

text: the

text: slices

text: and

text: their

text: ﬂuences

text: using

text: the

text: estimated

text: rotational

text: probability;

text: iv)

text: the

text: compression

text: step

text: (C

text: step)

text: inserts

text: the

text: updated

text: slices

text: back

text: into

text: the

text: model.

text: We

text: ﬁrst

text: introduce

text: the

text: e

text: and

text: the

text: C

text: step,

text: which

text: interpolates

text: between

text: a

text: model

text: and

text: slices.

text: Let

text: W

text: {W

text: }Mgrid

text: be

text: a

text: discrete

text: model,

text: an

text: estimation

text: of

text: the

text: l

text: Fourier

text: intensity

text: of

text: a

text: where

text: M

text: The

text: rotational

text: space

text: R

text: is

text: grid

text: pix

text: discretized

text: by

text: (R

text: )Mrot,

text: and

text: the

text: corresponding

text: prior

text: weight

text: for

text: rotation

text: R

text: is

text: w

text: j

text: j

text: j

text: normalized

text: such

text: that

text: w

text: Similarly,

text: the

text: intensity

text: space

text: is

text: discretized

text: by

text: a

text: j

text: j

text: ##LTLine##

text: ##LTLine##


text: UNCERTAINTIES IN X-RAY 3D RECONSTRUCTIONS 5 The original EMC algorithm [20] assumed that the ith pixel of the kth measured diﬀraction pattern K is Poissonian around the unknown Fourier intensity W , ik (cid:89) ij (2.10) P(K = κ|W , R ) = Mrot (Wij)κe−Wij . ik ij j κ! j=1 Some attempts [11, 21] have been made to better take the photon ﬂuence into account by approximating the Poisson distribution by a Gaussian distribution for high-intensity FXI diﬀraction patterns, which unfortunately makes (2.9) nonlinear. In this paper, instead of solving a scaled Poissonian probability model directly, we propose a solution within the EMC framework using ideas borrowed from non- negative matrix factorization (NNMF). More precisely, we assume that the mea- sured intensity of the ith pixel in the kth diﬀraction pattern is Poissonian around the scaled unknown Fourier intensity φ W , i.e. , jk ij | ∝ − (2.11) log P(K W , R , φ ) K log W + log φ φ W := Q . ik ij j jk ik ij jk jk ij ijk Summing over i, we obtain the joint log-likelihood function, (cid:88) Mpix − (2.12) Q := (K log W + log φ φ W ) . jk ik ij jk jk ij i=1 In the E step, we assume that the 2D slices W and their ﬂuences φ are known, so that the rotational probability is explicitly available by integrating the joint log-likelihood function (2.12) over the rotational space at the (n + 1)th iteration. | Pn+1 =P(cid:80)n+1(Wn, φn) =: P(R K , φn, Wn) jk jk j k w exp(Q (Wn)) (2.13) = j jk . Mj(cid:48)=ro1t wj(cid:48) exp(Qj(cid:48)k(Wn)) The M step freezes the rotational probability P at the (n + 1)th iteration, so jk (cid:88) that φ and W may be obtained as solutions to the following optimization problem: − (2.14) arg max (P K log(φ W ) P φ W ) . jk ik jk ij jk jk ij φ,W ijk We propose to solve for φ and W jointly by directly translating the optimization (cid:18) (cid:19) problem into an NNMF prob(cid:88)lem in the form of minimizing the Klein divergence, min D(PK||PφW) = min  P K log PjkKik − P K + P φ W φ,W φ,W ijk (cid:88)jk ik PjkφjkWij jk ik jk jk ij − − (2.15) = min (P K log(φ W ) P φ W ) + C , jk ik jk ij jk jk ij φ,W ijk ##LTLine## ##LTLine## ##LTLine##

text: UNCERTAINTIES

text: IN

text: X-RAY

text: RECONSTRUCTIONS

text: The

text: original

text: EMC

text: algorithm

text: assumed

text: that

text: the

text: ith

text: pixel

text: of

text: the

text: kth

text: measured

text: diﬀraction

text: pattern

text: K

text: is

text: Poissonian

text: around

text: the

text: unknown

text: Fourier

text: intensity

text: W

text: ik

text: ij

text: P(K

text: κ|W

text: R

text: Mrot

text: (Wij)κe−Wij

text: ik

text: ij

text: j

text: κ!

text: Some

text: attempts

text: have

text: been

text: made

text: to

text: better

text: take

text: the

text: photon

text: ﬂuence

text: into

text: account

text: by

text: approximating

text: the

text: Poisson

text: distribution

text: by

text: a

text: Gaussian

text: distribution

text: for

text: high-intensity

text: FXI

text: diﬀraction

text: patterns,

text: which

text: unfortunately

text: makes

text: nonlinear.

text: In

text: this

text: paper,

text: instead

text: of

text: solving

text: a

text: scaled

text: Poissonian

text: probability

text: model

text: directly,

text: we

text: propose

text: a

text: solution

text: within

text: the

text: EMC

text: framework

text: using

text: ideas

text: borrowed

text: from

text: non-

text: negative

text: matrix

text: factorization

text: (NNMF).

text: More

text: precisely,

text: we

text: assume

text: that

text: the

text: mea-

text: sured

text: intensity

text: of

text: the

text: ith

text: pixel

text: in

text: the

text: kth

text: diﬀraction

text: pattern

text: is

text: Poissonian

text: around

text: the

text: scaled

text: unknown

text: Fourier

text: intensity

text: φ

text: W

text: i.e.

text: jk

text: ij

text: log

text: P(K

text: W

text: R

text: φ

text: K

text: log

text: W

text: log

text: φ

text: φ

text: W

text: Q

text: ik

text: ij

text: j

text: jk

text: ik

text: ij

text: jk

text: jk

text: ij

text: ijk

text: Summing

text: over

text: i,

text: we

text: obtain

text: the

text: joint

text: log-likelihood

text: function,

text: Mpix

text: Q

text: (K

text: log

text: W

text: log

text: φ

text: φ

text: W

text: jk

text: ik

text: ij

text: jk

text: jk

text: ij

text: In

text: the

text: E

text: step,

text: we

text: assume

text: that

text: the

text: slices

text: W

text: and

text: their

text: ﬂuences

text: φ

text: are

text: known,

text: so

text: that

text: the

text: rotational

text: probability

text: is

text: explicitly

text: available

text: by

text: integrating

text: the

text: joint

text: log-likelihood

text: function

text: over

text: the

text: rotational

text: space

text: at

text: the

text: (n

text: iteration.

text: φn)

text: P(R

text: K

text: φn,

text: Wn)

text: jk

text: jk

text: j

text: k

text: w

text: exp(Q

text: (Wn))

text: j

text: jk

text: The

text: M

text: step

text: freezes

text: the

text: rotational

text: probability

text: P

text: at

text: the

text: (n

text: iteration,

text: so

text: jk

text: that

text: φ

text: and

text: W

text: may

text: be

text: obtained

text: as

text: solutions

text: to

text: the

text: following

text: optimization

text: problem:

text: arg

text: max

text: (P

text: K

text: log(φ

text: W

text: P

text: φ

text: W

text: jk

text: ik

text: jk

text: ij

text: jk

text: jk

text: ij

text: φ,W

text: ijk

text: We

text: propose

text: to

text: solve

text: for

text: φ

text: and

text: W

text: jointly

text: by

text: directly

text: translating

text: the

text: optimization

text: problem

text: into

text: an

text: NNMF

text: in

text: the

text: form

text: of

text: minimizing

text: the

text: Klein

text: divergence,

text: min

text: D(PK||PφW)

text: min

text: P

text: K

text: log

text: PjkKik

text: P

text: K

text: P

text: φ

text: W

text: φ,W

text: φ,W

text: ijk

text: ik

text: PjkφjkWij

text: jk

text: ik

text: jk

text: jk

text: ij

text: min

text: (P

text: K

text: log(φ

text: W

text: P

text: φ

text: W

text: C

text: jk

text: ik

text: jk

text: ij

text: jk

text: jk

text: ij

text: φ,W

text: ijk

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ∞ , but since we are performing a numerical calculation we have to choose a suitable cutoﬀ. Since our integrand involves highly oscillatory function we use the Levin method for doing the numerical integration, which is tailor-made for functions that have oscillatory behavior. The Levin method gives us converging results as compared to conventional numerical methods like Monte Carlo (MC). In our previous work [41] on Wigner distributions we had relied on MC integration and thus the results were cutoﬀ dependent. However, for lower values of ∆ both the MC and Levin methods are in good agreement with each other (see Figure. 1). max In Figure. 1 we show the behavior of all the Wigner distributions with ∆ using two max methods for numerical integration, i.e., the Levin and MC methods. In Figures. 1 (a) and 1(c) we show the distributions ρx , ρx , ρx , and ρx for the Levin and MC methods, TL TU LT UT respectively. Similarly in Figures. 1 (b) and 1(d) we show the distributions for ρ , ρ , ρ , UU LL TT − − and ρ . The 2D plots are for a ﬁxed value of b = 0.4 GeV 1 , b = 0.5 GeV 1, k = LU x y x 0.0 GeV, and k = 0.4 GeV. We study the ∆ dependence up to ∆ = 1000 GeV y max max and the results clearly show that the Levin method is ideal since it shows convergence, whereas MC fails to converge. As ∆ increases, the results from MC begin to diverge max more and more. On the contrary, the Levin method starts to give constant results from around ∆ = 20 GeV and it stays constant thereafter. Based on these results we set max ∆ = 20 GeV for all the 3D plots. In all the plots we have taken m = 0.33 GeV, and max divided by a normalization constant. IV. RESULTS AND DISCUSSION In this section, we start by discussing ρ (k⊥, b⊥), which is the Wigner distribution for an UU unpolarized quark in an unpolarized dressed quark state. Figure. 2(a) shows the distribution in b⊥ space with a ﬁxed transverse momentum k⊥ = 0.4 GeV eˆ . We observe a positive j peak centered around b = b = 0 as observed in Refs.[44, 47]. In k⊥ space, we obtain x y a sharp negative peak shown in Figure. 2(b). Figure. 2(c) shows the Wigner distribution ρ (k⊥, b⊥) in mixed space where we have integrated out k and b dependence, thus, we UU x y get the probability densities in the b - k plane. The Wigner distribution ρ can be related x y UU to unpolarized GPD and the unpolarized TMDs by taking the appropriate limit. In Figure. 2(d), we present ρ (k⊥, b⊥), which is the Wigner distribution for a longi- UL tudinally polarized quark in an unpolarized dressed quark state in k⊥ space with b⊥ = 11

text: but

text: since

text: we

text: are

text: performing

text: a

text: numerical

text: calculation

text: we

text: have

text: to

text: choose

text: a

text: suitable

text: cutoﬀ.

text: Since

text: our

text: integrand

text: involves

text: highly

text: oscillatory

text: function

text: we

text: use

text: the

text: Levin

text: method

text: for

text: doing

text: the

text: numerical

text: integration,

text: which

text: is

text: tailor-made

text: for

text: functions

text: that

text: have

text: oscillatory

text: behavior.

text: The

text: Levin

text: method

text: gives

text: us

text: converging

text: results

text: as

text: compared

text: to

text: conventional

text: numerical

text: methods

text: like

text: Monte

text: Carlo

text: (MC).

text: In

text: our

text: previous

text: work

text: on

text: Wigner

text: distributions

text: we

text: had

text: relied

text: on

text: MC

text: integration

text: and

text: thus

text: the

text: results

text: were

text: cutoﬀ

text: dependent.

text: However,

text: for

text: lower

text: values

text: of

text: both

text: the

text: MC

text: and

text: Levin

text: methods

text: are

text: in

text: good

text: agreement

text: with

text: each

text: other

text: (see

text: Figure.

text: max

text: In

text: Figure.

text: we

text: show

text: the

text: behavior

text: of

text: all

text: the

text: Wigner

text: distributions

text: with

text: using

text: two

text: max

text: methods

text: for

text: numerical

text: integration,

text: i.e.,

text: the

text: Levin

text: and

text: MC

text: methods.

text: In

text: Figures.

text: (a)

text: and

text: we

text: show

text: the

text: distributions

text: ρx

text: ρx

text: ρx

text: and

text: ρx

text: for

text: the

text: Levin

text: and

text: MC

text: methods,

text: TL

text: TU

text: LT

text: UT

text: respectively.

text: Similarly

text: in

text: Figures.

text: (b)

text: and

text: we

text: show

text: the

text: distributions

text: for

text: ρ

text: ρ

text: ρ

text: UU

text: LL

text: TT

text: and

text: ρ

text: The

text: plots

text: are

text: for

text: a

text: ﬁxed

text: value

text: of

text: b

text: GeV

text: b

text: GeV

text: k

text: LU

text: x

text: y

text: x

text: GeV,

text: and

text: k

text: GeV.

text: We

text: study

text: the

text: dependence

text: up

text: to

text: GeV

text: y

text: max

text: max

text: and

text: the

text: results

text: clearly

text: show

text: that

text: the

text: Levin

text: method

text: is

text: ideal

text: since

text: it

text: shows

text: convergence,

text: whereas

text: MC

text: fails

text: to

text: converge.

text: As

text: increases,

text: the

text: results

text: from

text: MC

text: begin

text: to

text: diverge

text: max

text: more

text: and

text: more.

text: On

text: the

text: contrary,

text: the

text: Levin

text: method

text: starts

text: to

text: give

text: constant

text: results

text: from

text: around

text: GeV

text: and

text: it

text: stays

text: constant

text: thereafter.

text: Based

text: on

text: these

text: results

text: we

text: set

text: max

text: GeV

text: for

text: all

text: the

text: plots.

text: In

text: all

text: the

text: plots

text: we

text: have

text: taken

text: m

text: GeV,

text: and

text: max

text: divided

text: by

text: a

text: normalization

text: constant.

text: IV.

text: RESULTS

text: AND

text: DISCUSSION

text: In

text: this

text: section,

text: we

text: start

text: by

text: discussing

text: ρ

text: (k⊥,

text: b⊥),

text: which

text: is

text: the

text: Wigner

text: distribution

text: for

text: an

text: UU

text: unpolarized

text: quark

text: in

text: an

text: unpolarized

text: dressed

text: quark

text: state.

text: Figure.

text: shows

text: the

text: distribution

text: in

text: b⊥

text: space

text: with

text: a

text: ﬁxed

text: transverse

text: momentum

text: k⊥

text: GeV

text: eˆ

text: We

text: observe

text: a

text: positive

text: j

text: peak

text: centered

text: around

text: b

text: b

text: as

text: observed

text: in

text: In

text: k⊥

text: space,

text: we

text: obtain

text: x

text: y

text: a

text: sharp

text: negative

text: peak

text: shown

text: in

text: Figure.

text: Figure.

text: shows

text: the

text: Wigner

text: distribution

text: ρ

text: (k⊥,

text: b⊥)

text: in

text: mixed

text: space

text: where

text: we

text: have

text: integrated

text: out

text: k

text: and

text: b

text: dependence,

text: thus,

text: we

text: UU

text: x

text: y

text: get

text: the

text: probability

text: densities

text: in

text: the

text: b

text: k

text: plane.

text: The

text: Wigner

text: distribution

text: ρ

text: can

text: be

text: related

text: x

text: y

text: UU

text: to

text: unpolarized

text: GPD

text: and

text: the

text: unpolarized

text: TMDs

text: by

text: taking

text: the

text: appropriate

text: limit.

text: In

text: Figure.

text: we

text: present

text: ρ

text: (k⊥,

text: b⊥),

text: which

text: is

text: the

text: Wigner

text: distribution

text: for

text: a

text: longi-

text: UL

text: tudinally

text: polarized

text: quark

text: in

text: an

text: unpolarized

text: dressed

text: quark

text: state

text: in

text: k⊥

text: space

text: with

text: b⊥


text: (a) (d) (b) (e) (c) (f) FIG. 5. 3D plot of Wigner distributions ρx (k⊥, b⊥) and ρx (k⊥, b⊥) at ∆ = 20 GeV. The LT TL max ﬁrst row displays the two distributions in b⊥ space with k⊥ = 0.4 GeV eˆ . The second row shows y − the two distributions in k⊥ space with b⊥ = 0.4 GeV 1 eˆ . The last row represents the two y distributions in mixed space. If one integrates the Wigner function ρ over k⊥ and b⊥, one would get the familiar plus UU distribution −1 [48] as expected in the parton distribution of a dressed quark. It is very (1 x)+ important to take into account the contribution from the normalization of the state to get the correct behavior at x = 1. In [55] the authors considered the two-particle contribution 15 ##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##

text: (a)

text: (d)

text: (b)

text: (e)

text: (c)

text: (f)

text: FIG.

label: caption

text: plot

text: of

text: Wigner

text: distributions

text: ρx

text: (k⊥,

text: b⊥)

text: and

text: ρx

text: (k⊥,

text: b⊥)

text: at

text: GeV.

text: The

text: LT

text: TL

text: max

text: ﬁrst

text: row

text: displays

text: the

text: two

text: distributions

text: in

text: b⊥

text: space

text: with

text: k⊥

text: GeV

text: eˆ

text: The

text: second

text: row

text: shows

text: y

text: the

text: two

text: distributions

text: in

text: k⊥

text: space

text: with

text: b⊥

text: GeV

text: eˆ

text: The

text: last

text: row

text: represents

text: the

text: two

text: y

text: distributions

text: in

text: mixed

text: space.

text: If

text: one

text: integrates

text: the

text: Wigner

text: function

text: ρ

text: over

text: k⊥

text: and

text: b⊥,

text: one

text: would

text: get

text: the

text: familiar

text: plus

text: UU

text: distribution

text: −1

text: as

text: expected

text: in

text: the

text: parton

text: distribution

text: of

text: a

text: dressed

text: quark.

text: It

text: is

text: very

text: x)+

text: important

text: to

text: take

text: into

text: account

text: the

text: contribution

text: from

text: the

text: normalization

text: of

text: the

text: state

text: to

text: get

text: the

text: correct

text: behavior

text: at

text: x

text: In

text: the

text: authors

text: considered

text: the

text: two-particle

text: contribution

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##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: phase space. So Wigner distributions do not have a probabilistic interpretation. Neverthe- less, they are very useful tools in understanding quark/gluon spin and angular momentum correlations inside the target nucleon, and certain model-based relations, and under certain conditions, it is possible to have a semiclassical interpretation [29]. Moreover, Wigner distri- butions have previously been studied in various ﬁelds such as quantum molecular dynamics, quantum information, image processing etc. [30–32] and in fact, there are experiments in which it is measured [33–36]. In QCD, the Wigner distributions were ﬁrst looked into using the nonrelativistic approximation in Refs.[37, 38] where the distribution was studied as a six-dimensional function. The six-dimensional space consists of three position and three momentum coordinates. Then in Ref. [29], the authors deﬁned a ﬁve-dimensional Wigner distribution consistent with relativity using the light-cone framework. The ﬁve-dimensional space consists of two transverse position and three momentum coordinates. Various phe- nomenological models like the light-cone constituent quark model [39], chiral quark soliton model [40], light-front dressed quark model [41, 42], light-cone spectator model [43, 44] and diquark model [45, 46] have been used to study Wigner distributions. A complete multipole analysis of the quark Wigner distributions including transverse polarization was recently studied [47]. In this work, we study the Wigner distribution of quarks using the light-front Hamiltonian gauge-ﬁxed formulation [48]. The Hamiltonian approach in front form is advantageous compared to the conventional equal-time form mainly because of the absence of the square- root operator in the bound state eigenvalue equation and due to the triviality of the QCD vacuum structure. Instead of a proton state, we take a simple composite spin-1/2 state, namely a quark dressed at one loop with a gluon. Like the proton state, the dressed quark state can also be expanded in multiparticle occupation number Fock states and because of the trivial vacuum such an expansion gives a complete basis for diagonalizing the full theory [49]. The advantage is that unlike the proton light-front wave functions (LFWFs), the two- particle LFWFs of the dressed quark state can be calculated analytically in perturbation theory, and thus this can be thought of as a ﬁeld theory based perturbative model having a gluonic degree of freedom. In our previous work [41] we studied the three independent quark Wigner distributions for unpolarized and longitudinal polarization of quark and the target state. Now, we present the complete study involving unpolarized, longitudinal, and transverse polarization combi- 3

text: phase

text: space.

text: So

text: Wigner

text: distributions

text: do

text: not

text: have

text: a

text: probabilistic

text: interpretation.

text: Neverthe-

text: less,

text: they

text: are

text: very

text: useful

text: tools

text: in

text: understanding

text: quark/gluon

text: spin

text: and

text: angular

text: momentum

text: correlations

text: inside

text: the

text: target

text: nucleon,

text: and

text: certain

text: model-based

text: relations,

text: and

text: under

text: certain

text: conditions,

text: it

text: is

text: possible

text: to

text: have

text: a

text: semiclassical

text: interpretation

text: Moreover,

text: Wigner

text: distri-

text: butions

text: have

text: previously

text: been

text: studied

text: in

text: various

text: ﬁelds

text: such

text: as

text: quantum

text: molecular

text: dynamics,

text: quantum

text: information,

text: image

text: processing

text: etc.

text: and

text: in

text: fact,

text: there

text: are

text: experiments

text: in

text: which

text: it

text: is

text: measured

text: In

text: QCD,

text: the

text: Wigner

text: distributions

text: were

text: ﬁrst

text: looked

text: into

text: using

text: the

text: nonrelativistic

text: approximation

text: in

text: where

text: the

text: distribution

text: was

text: studied

text: as

text: a

text: six-dimensional

text: function.

text: The

text: six-dimensional

text: space

text: consists

text: of

text: three

text: position

text: and

text: three

text: momentum

text: coordinates.

text: Then

text: in

text: Ref.

text: the

text: authors

text: deﬁned

text: a

text: ﬁve-dimensional

text: Wigner

text: distribution

text: consistent

text: with

text: relativity

text: using

text: the

text: light-cone

text: framework.

text: The

text: ﬁve-dimensional

text: space

text: consists

text: of

text: two

text: transverse

text: position

text: and

text: three

text: momentum

text: coordinates.

text: Various

text: phe-

text: nomenological

text: models

text: like

text: the

text: light-cone

text: constituent

text: quark

text: model

text: chiral

text: quark

text: soliton

text: model

text: light-front

text: dressed

text: quark

text: model

text: light-cone

text: spectator

text: model

text: and

text: diquark

text: model

text: have

text: been

text: used

text: to

text: study

text: Wigner

text: distributions.

text: A

text: complete

text: multipole

text: analysis

text: of

text: the

text: quark

text: Wigner

text: distributions

text: including

text: transverse

text: polarization

text: was

text: recently

text: studied

text: In

text: this

text: work,

text: we

text: study

text: the

text: Wigner

text: distribution

text: of

text: quarks

text: using

text: the

text: light-front

text: Hamiltonian

text: gauge-ﬁxed

text: formulation

text: The

text: Hamiltonian

text: approach

text: in

text: front

text: form

text: is

text: advantageous

text: compared

text: to

text: the

text: conventional

text: equal-time

text: form

text: mainly

text: because

text: of

text: the

text: absence

text: of

text: the

text: square-

text: root

text: operator

text: in

text: the

text: bound

text: state

text: eigenvalue

text: equation

text: and

text: due

text: to

text: the

text: triviality

text: of

text: the

text: QCD

text: vacuum

text: structure.

text: Instead

text: of

text: a

text: proton

text: state,

text: we

text: take

text: a

text: simple

text: composite

text: state,

text: namely

text: a

text: quark

text: dressed

text: at

text: one

text: loop

text: with

text: a

text: gluon.

text: Like

text: the

text: proton

text: state,

text: the

text: dressed

text: quark

text: state

text: can

text: also

text: be

text: expanded

text: in

text: multiparticle

text: occupation

text: number

text: Fock

text: states

text: and

text: because

text: of

text: the

text: trivial

text: vacuum

text: such

text: an

text: expansion

text: gives

text: a

text: complete

text: basis

text: for

text: diagonalizing

text: the

text: full

text: theory

text: The

text: advantage

text: is

text: that

text: unlike

text: the

text: proton

text: light-front

text: wave

text: functions

text: (LFWFs),

text: the

text: two-

text: particle

text: LFWFs

text: of

text: the

text: dressed

text: quark

text: state

text: can

text: be

text: calculated

text: analytically

text: in

text: perturbation

text: theory,

text: and

text: thus

text: this

text: can

text: be

text: thought

text: of

text: as

text: a

text: ﬁeld

text: theory

text: based

text: perturbative

text: model

text: having

text: a

text: gluonic

text: degree

text: of

text: freedom.

text: In

text: our

text: previous

text: work

text: we

text: studied

text: the

text: three

text: independent

text: quark

text: Wigner

text: distributions

text: for

text: unpolarized

text: and

text: longitudinal

text: polarization

text: of

text: quark

text: and

text: the

text: target

text: state.

text: Now,

text: we

text: present

text: the

text: complete

text: study

text: involving

text: unpolarized,

text: longitudinal,

text: and

text: transverse

text: polarization

text: combi-


text: (cid:104) (cid:105) The transverse-unpolarized Wigner distribution 1 − − ρi (b⊥, k⊥, x) = ρ[γ+](b⊥, k⊥, x, eˆ ) ρ[γ+](b⊥, k⊥, x, eˆ ) (17) TU 2 i i (cid:104) (cid:105) The transverse-longitudinally polarized Wigner distribution 1 − − ρi (b⊥, k⊥, x) = ρ[γ+γ5](b⊥, k⊥, x, eˆ ) ρ[γ+γ5](b⊥, k⊥, x, eˆ ) (18) TL 2 i i (cid:104) (cid:105) The transversely polarized Wigner distribution 1 − − ρ (b⊥, k⊥, x) = δ ρ[iσ+jγ5](b⊥, k⊥, x, eˆ ) ρ[iσ+jγ5](b⊥, k⊥, x, eˆ ) (19) TT ij i i 2 For i = j = 1 and i = j = 2, the result is the same. (cid:104) (cid:105) The pretzelous Wigner distribution ⊥ 1 − − ρ (b⊥, k⊥, x) = (cid:15) ρ[iσ+jγ5](b⊥, k⊥, x, eˆ ) ρ[iσ+jγ5](b⊥, k⊥, x, eˆ ) (20) TT 2 ij i i − | (cid:105) | − (cid:105) where +eˆ ( eˆ ) corresponds to helicity up (down), i.e, 1 ( 1 ) of the target state. eˆ z z 2 2 i corresponds to the transversity state and can be expressed in terms of the helicity state. For | ± (cid:105) | (cid:105) ± | − (cid:105) instance, eˆ = √1 ( 1 1 ). x 2 2 2 Here Equation(20) corresponds to the transversely polarized dressed quark and internal quark along the two orthogonal directions. For example, consider the case, i = 1, j = 2, − that refers to the dressed quark polarized along the x direction and the internal quark − polarized along the y direction. There are two terms corresponding to this case as seen on the rhs of Equation (20). We obtain an equal contribution from both terms. Thus, for case ⊥ i = 1, j = 2 the overall contribution to ρ (b⊥, k⊥, x) is zero. Similarly, for the case i = 2, TT ⊥ j = 1, we obtain that the distribution ρ (b⊥, k⊥, x) vanishes. Thus the pretzelous Wigner TT distribution vanishes in our model. We also would like to point out that the pretzelous distribution in Ref. [43] vanishes in the scalar spectator case but not in the axial-vector spectator case. In this model, we obtained ρ (b⊥, k⊥, x) equal to ρ (b⊥, k⊥, x). So, UL LU we have ten independent Wigner distributions, out of which ρj (b⊥, k⊥, x) with j = 1, 2 TT vanishes as discussed above. Finally, we study the eight independent Wigner distributions, and their analytical ex- (cid:90) pressions are as follows: · (cid:16) d2∆⊥ cos(∆⊥ b(cid:17)⊥) ρUU(b⊥, k⊥, x) = N(cid:104) 2(2π)2 D(q⊥)D(q(cid:48)⊥) (cid:105) − − − × 4k⊥2 ∆2⊥(1 x)2 (1 + x2) 4m2(1 x) − + (21) x2(1 x)3 x2 7 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: The

text: transverse-unpolarized

text: Wigner

text: distribution

text: ρi

text: (b⊥,

text: k⊥,

text: x)

text: ρ[γ+](b⊥,

text: k⊥,

text: x,

text: eˆ

text: ρ[γ+](b⊥,

text: k⊥,

text: x,

text: eˆ

text: TU

text: i

text: i

text: The

text: transverse-longitudinally

text: polarized

text: Wigner

text: distribution

text: ρi

text: (b⊥,

text: k⊥,

text: x)

text: k⊥,

text: x,

text: eˆ

text: k⊥,

text: x,

text: eˆ

text: TL

text: i

text: i

text: The

text: transversely

text: polarized

text: Wigner

text: distribution

text: ρ

text: (b⊥,

text: k⊥,

text: x)

text: δ

text: k⊥,

text: x,

text: eˆ

text: k⊥,

text: x,

text: eˆ

text: TT

text: ij

text: i

text: i

text: For

text: i

text: j

text: and

text: i

text: j

text: the

text: result

text: is

text: the

text: same.

text: The

text: pretzelous

text: Wigner

text: distribution

text: ρ

text: (b⊥,

text: k⊥,

text: x)

text: k⊥,

text: x,

text: eˆ

text: k⊥,

text: x,

text: eˆ

text: TT

text: ij

text: i

text: i

text: where

text: +eˆ

text: eˆ

text: corresponds

text: to

text: helicity

text: up

text: (down),

text: i.e,

text: of

text: the

text: target

text: state.

text: eˆ

text: z

text: z

text: i

text: corresponds

text: to

text: the

text: transversity

text: state

text: and

text: can

text: be

text: expressed

text: in

text: terms

text: of

text: the

text: helicity

text: state.

text: For

text: instance,

text: eˆ

text: √1

text: x

text: Here

text: corresponds

text: to

text: the

text: transversely

text: polarized

text: dressed

text: quark

text: and

text: internal

text: quark

text: along

text: the

text: two

text: orthogonal

text: directions.

text: For

text: example,

text: consider

text: the

text: case,

text: i

text: j

text: that

text: refers

text: to

text: the

text: dressed

text: quark

text: polarized

text: along

text: the

text: x

text: direction

text: and

text: the

text: internal

text: quark

text: polarized

text: along

text: the

text: y

text: direction.

text: There

text: are

text: two

text: terms

text: corresponding

text: to

text: this

text: case

text: as

text: seen

text: on

text: the

text: rhs

text: of

text: Equation

text: We

text: obtain

text: an

text: equal

text: contribution

text: from

text: both

text: terms.

text: Thus,

text: for

text: case

text: i

text: j

text: the

text: overall

text: contribution

text: to

text: ρ

text: (b⊥,

text: k⊥,

text: x)

text: is

text: zero.

text: Similarly,

text: for

text: the

text: case

text: i

text: TT

text: j

text: we

text: obtain

text: that

text: the

text: distribution

text: ρ

text: (b⊥,

text: k⊥,

text: x)

text: vanishes.

text: Thus

text: the

text: pretzelous

text: Wigner

text: TT

text: distribution

text: vanishes

text: in

text: our

text: model.

text: We

text: also

text: would

text: like

text: to

text: point

text: out

text: that

text: the

text: pretzelous

text: distribution

text: in

text: Ref.

text: vanishes

text: in

text: the

text: scalar

text: spectator

text: case

text: but

text: not

text: in

text: the

text: axial-vector

text: spectator

text: case.

text: In

text: this

text: model,

text: we

text: obtained

text: ρ

text: (b⊥,

text: k⊥,

text: x)

text: equal

text: to

text: ρ

text: (b⊥,

text: k⊥,

text: x).

text: So,

text: UL

text: LU

text: we

text: have

text: ten

text: independent

text: Wigner

text: distributions,

text: out

text: of

text: which

text: ρj

text: (b⊥,

text: k⊥,

text: x)

text: with

text: j

text: TT

text: vanishes

text: as

text: discussed

text: above.

text: Finally,

text: we

text: study

text: the

text: eight

text: independent

text: Wigner

text: distributions,

text: and

text: their

text: analytical

text: ex-

text: pressions

text: are

text: as

text: follows:

text: d2∆⊥

text: cos(∆⊥

text: ρUU(b⊥,

text: k⊥,

text: x)

text: x)

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text: (cid:16) (cid:17) (cid:90) (cid:104) (cid:105) − · 4 k ∆ k ∆ (1 + x) d2∆⊥ sin(∆⊥ b⊥) y x x y ρUL(b⊥, k⊥, x) = N 2(2π)2(cid:90)D(q⊥)D(q(cid:48)⊥) x(cid:104)2(1 − x(cid:105)) (22) · d2∆⊥ sin(∆⊥ b⊥) 4m∆ ρx (b⊥, k⊥, x) = N (cid:48) x (23) UT (cid:90) 2(2π)2 D(q⊥)D(q ⊥) x2 · (cid:16) d2∆⊥ cos(∆⊥ (cid:17)b⊥) ρLL(b⊥, k⊥, x) = N(cid:104) 2(2π)2 D(q⊥)D(q(cid:48)⊥) (cid:105) − − − × 4k⊥2 ∆2⊥(1 x)2 (1 + x2) − 4m2(1 x) − (24) (cid:90) x2(1 x)3 (cid:104) x2 (cid:105) · d2∆⊥ cos(∆⊥ b⊥) 8mk ρx (b⊥, k⊥, x) = N (cid:48) −x (25) LT (cid:90)2(2π)2 D(q⊥)D(q ⊥) x(cid:104)2(1 x(cid:105)) · d2∆⊥ sin(∆⊥ b⊥) 4m∆ ρx (b⊥, k⊥, x) = N (cid:48) x (26) TU (cid:90) 2(2π)2 D(q⊥)D(q ⊥(cid:104)) x (cid:105) · − d2∆⊥ cos(∆⊥ b⊥) 8mk ρx (b⊥, k⊥, x) = N (cid:48) − x (27) TL 2(2π)2 D(q⊥)D(q(cid:16) ⊥) x(1 x) (cid:17) (cid:90) (cid:104) (cid:105) − − · d2∆⊥ cos(∆⊥ b⊥) 2 4k⊥2 ∆2⊥(1 x)2 ρTT (b⊥, k⊥, x) = N 2(2π)2 D(q⊥)D(q(cid:48)⊥) x(1 − x)3 (28) where, g2C F N = , C is the color factor (cid:104) (2π)3 F (cid:105) − − (cid:104) − m2 + (k⊥ + ∆⊥(1 x))2 − (k⊥ + ∆⊥(1 x))2 (cid:105) D(q⊥) = m2 2 − 2 x 1 x − − − − (cid:48) − m2 + (k⊥ ∆⊥(1 x))2 − (k⊥ ∆⊥(1 x))2 D(q⊥) = m2 2 − 2 (29) x 1 x III. NUMERICAL STRATEGY The eight independent Wigner distributions obtained in the previous section are function of ﬁve continuous variables, two transverse position b⊥, two transverse momentum k⊥ and one longitudinal momentum fraction x. We are interested in studying the transverse phase space, so we integrate over the x dependence from all the distributions and purely study them 8 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: k

text: k

text: x)

text: d2∆⊥

text: sin(∆⊥

text: b⊥)

text: y

text: x

text: x

text: y

text: ρUL(b⊥,

text: k⊥,

text: x)

text: N

text: d2∆⊥

text: sin(∆⊥

text: b⊥)

text: ρx

text: (b⊥,

text: k⊥,

text: x)

text: N

text: x

text: UT

text: D(q⊥)D(q

text: d2∆⊥

text: cos(∆⊥

text: ρLL(b⊥,

text: k⊥,

text: x)

text: x)

text: d2∆⊥

text: cos(∆⊥

text: b⊥)

text: ρx

text: (b⊥,

text: k⊥,

text: x)

text: N

text: −x

text: LT

text: D(q⊥)D(q

text: d2∆⊥

text: sin(∆⊥

text: b⊥)

text: ρx

text: (b⊥,

text: k⊥,

text: x)

text: N

text: x

text: TU

text: D(q⊥)D(q

text: x

text: d2∆⊥

text: cos(∆⊥

text: b⊥)

text: ρx

text: (b⊥,

text: k⊥,

text: x)

text: N

text: x

text: TL

text: x)

text: d2∆⊥

text: cos(∆⊥

text: b⊥)

text: ρTT

text: (b⊥,

text: k⊥,

text: x)

text: N

text: where,

text: F

text: N

text: C

text: is

text: the

text: color

text: factor

text: F

text: (k⊥

text: ∆⊥(1

text: (k⊥

text: ∆⊥(1

text: D(q⊥)

text: x

text: x

text: (k⊥

text: ∆⊥(1

text: (k⊥

text: ∆⊥(1

text: D(q⊥)

text: x

text: x

text: III.

text: NUMERICAL

text: STRATEGY

text: The

text: eight

text: independent

text: Wigner

text: distributions

text: obtained

text: in

text: the

text: previous

text: section

text: are

text: function

text: of

text: ﬁve

text: continuous

text: variables,

text: two

text: transverse

text: position

text: b⊥,

text: two

text: transverse

text: momentum

text: k⊥

text: and

text: one

text: longitudinal

text: momentum

text: fraction

text: x.

text: We

text: are

text: interested

text: in

text: studying

text: the

text: transverse

text: phase

text: space,

text: so

text: we

text: integrate

text: over

text: the

text: x

text: dependence

text: from

text: all

text: the

text: distributions

text: and

text: purely

text: study

text: them

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text: ##LTLine##

text: ##LTLine##


text: 4 This way, we are able to map out the regions of existence of the diﬀerent dynamical patterns. III. SPOTS AND RING PATTERNS In this section, we focus on chimera states of spot and ring type in both FHN and LIF models. A. FitzHugh-Nagumo: Spots and ring patterns × In the 100 100 FHN system described by Eqs. (2) and (3), we observe various chimera regions in the pa- ∈ rameter space spanned by the coupling radius r [1, 49] ∈ and coupling phase ϕ [0, 2π). A complete scan of the parameter space and several enlargements can be found in Appendix . Here, we will focus on the range ∈ FIG. 2: (Color online) FHN system: Chimera state of (a) ϕ [1.3, 1.65]. incoherent-spot and (b) coherent-spot type. The left panels An example for a single-headed incoherent spot sur- depict a snapshot of the activator variable x at t = 2000. ij rounded by coherent oscillators is shown in the top panel The center and right panels show the mean phase velocity ω ij of Fig. 2(a). The left, center, and right panels depict a and a horizontal cut of ω . Coupling parameters: (a) r = 33, − ij snapshot of xij, the mean phase velocity ωij, where the ϕ = π/2 0.2 and (b) r = 42, ϕ = π/2. Other parameters: average is computed over 1000 time units, and a horizon- σ = 0.1, a = 0.5, and (cid:15) = 0.05. tal cut of ω through the center of the spot, respectively. ij The latter exhibits the typical arc-shape proﬁle of a chimera state. When the spots deform to a square shape, the maximum of the ω ’s is less pronounced and be- ij comes a plateau (not shown). The spots and the squares grow when the coupling radius r increases, as shown in Fig. 3(a) for all observed spots. The color code refers to diﬀerent choices of ϕ. We observe two diﬀerent domains of large and small coupling radii. These correspond to diﬀerent areas in parameter space, which support spot ≤ ≤ chimeras, see Appendix . In the range 20 r 30 the spot patterns lose their stability in space, wander around the grid in an unpredictable way, change their shape over time or even split into pieces and reemerge after some time. Another chimera spot pattern – again found as a single- headed and multi-headed version – is a spot with a coher- ent center. An example is shown in the bottom panel of Fig. 2(b). As in the previous example, the ω -values are ij lower in the coherent area than in the incoherent region (middle panel). Furthermore, the section of the mean phase velocities (right panel) has a shape similar to in- coherent spot type and many other chimera patterns in one dimension. This state can also be found with two coherent heads (not shown). The size of the coherent region is depicted in Fig. 3(b). We observe only a weak dependence on the coupling radius r, while the coupling phase ϕ has a stronger eﬀect. Compared to the incoher- ent spot chimeras, the size increases with decreasing ϕ. In some cases, we observe that at the opposite side of the coherent region of the torus, a spiral can form, see Ap- FIG. 3: (Color online) FHN system: Radius ρ of the spot-like pendix for an example. The radius of the spiral increases chimera states, (a) incoherent spot and (b) coherent spot, with the coupling radius (not shown). in dependence on the coupling radius r for diﬀerent coupling Between regions of spots (see Appendix ) and fully syn- phases ϕ given as color code. The error bars indicate an uncer- chronous solutions, we observe that ring chimeras can be tainty of (a) ∆ρ = 0.3 and (b) ∆ρ = 0.5 in the measurements found for ϕ < 1.35, see Fig. 4. The label “ring chimera” of the radii. Parameters as in Fig. 2. refers to the dip in the mean phase velocity ω in the cen- ij ter of the incoherent ring that can reach the level of the ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: This

text: way,

text: we

text: are

text: able

text: to

text: map

text: out

text: the

text: regions

text: of

text: existence

text: of

text: the

text: diﬀerent

text: dynamical

text: patterns.

text: III.

label: section

text: SPOTS

text: AND

text: RING

text: PATTERNS

text: In

text: this

text: section,

text: we

text: focus

text: on

text: chimera

text: states

text: of

text: spot

text: and

text: ring

text: type

text: in

text: both

text: FHN

text: and

text: LIF

text: models.

text: A.

text: FitzHugh-Nagumo:

text: Spots

text: and

text: ring

text: patterns

text: In

text: the

text: FHN

text: system

text: described

text: by

text: Eqs.

text: and

text: we

text: observe

text: various

text: chimera

text: regions

text: in

text: the

text: pa-

text: rameter

text: space

text: spanned

text: by

text: the

text: coupling

text: radius

text: r

text: and

text: coupling

text: phase

text: ϕ

text: A

text: complete

text: scan

text: of

text: the

text: parameter

text: space

text: and

text: several

text: enlargements

text: can

text: be

text: found

text: in

text: Appendix

text: Here,

text: we

text: will

text: focus

text: on

text: the

text: range

text: FIG.

text: (Color

text: online)

text: FHN

text: system:

text: Chimera

text: state

text: of

text: (a)

text: ϕ

text: incoherent-spot

text: and

text: (b)

text: coherent-spot

text: type.

text: The

text: left

text: panels

text: An

text: example

text: for

text: a

text: single-headed

text: incoherent

text: spot

text: sur-

text: depict

text: a

text: snapshot

text: of

text: the

text: activator

text: variable

text: x

text: at

text: t

text: ij

text: rounded

text: by

text: coherent

text: oscillators

text: is

text: shown

text: in

text: the

text: top

text: panel

text: The

text: center

text: and

text: right

text: panels

text: show

text: the

text: mean

text: phase

text: velocity

text: ω

text: ij

text: of

text: Fig.

text: The

text: left,

text: center,

text: and

text: right

text: panels

text: depict

text: a

text: and

text: a

text: horizontal

text: cut

text: of

text: ω

text: Coupling

text: parameters:

text: (a)

text: r

text: ij

text: snapshot

text: of

text: xij,

text: the

text: mean

text: phase

text: velocity

text: ωij,

text: where

text: the

text: ϕ

text: and

text: (b)

text: r

text: ϕ

text: Other

text: parameters:

text: average

text: is

text: computed

text: over

text: time

text: units,

text: and

text: a

text: horizon-

text: σ

text: a

text: and

text: tal

text: cut

text: of

text: ω

text: through

text: the

text: center

text: of

text: the

text: spot,

text: respectively.

text: ij

text: The

text: latter

text: exhibits

text: the

text: typical

text: arc-shape

text: proﬁle

text: of

text: a

text: chimera

text: state.

text: When

text: the

text: spots

text: deform

text: to

text: a

text: square

text: shape,

text: the

text: maximum

text: of

text: the

text: ω

text: ’s

text: is

text: less

text: pronounced

text: and

text: be-

text: ij

text: comes

text: a

text: plateau

text: (not

text: shown).

text: The

text: spots

text: and

text: the

text: squares

text: grow

text: when

text: the

text: coupling

text: radius

text: r

text: increases,

text: as

text: shown

text: in

text: Fig.

text: for

text: all

text: observed

text: spots.

text: The

text: color

text: code

text: refers

text: to

text: diﬀerent

text: choices

text: of

text: ϕ.

text: We

text: observe

text: two

text: diﬀerent

text: domains

text: of

text: large

text: and

text: small

text: coupling

text: radii.

text: These

text: correspond

text: to

text: diﬀerent

text: areas

text: in

text: parameter

text: space,

text: which

text: support

text: spot

text: chimeras,

text: see

text: Appendix

text: In

text: the

text: range

text: r

text: the

text: spot

text: patterns

text: lose

text: their

text: stability

text: in

text: space,

text: wander

text: around

text: the

text: grid

text: in

text: an

text: unpredictable

text: way,

text: change

text: their

text: shape

text: over

text: time

text: or

text: even

text: split

text: into

text: pieces

text: and

text: reemerge

text: after

text: some

text: time.

text: Another

text: chimera

text: spot

text: pattern

text: again

text: found

text: as

text: a

text: single-

text: headed

text: and

text: multi-headed

text: version

text: is

text: a

text: spot

text: with

text: a

text: coher-

text: ent

text: center.

text: An

text: example

text: is

text: shown

text: in

text: the

text: bottom

text: panel

text: of

text: Fig.

text: As

text: in

text: the

text: previous

text: example,

text: the

text: ω

text: -values

text: are

text: ij

text: lower

text: in

text: the

text: coherent

text: area

text: than

text: in

text: the

text: incoherent

text: region

text: (middle

text: panel).

text: Furthermore,

text: the

text: section

text: of

text: the

text: mean

text: phase

text: velocities

text: (right

text: panel)

text: has

text: a

text: shape

text: similar

text: to

text: in-

text: coherent

text: spot

text: type

text: and

text: many

text: other

text: chimera

text: patterns

text: in

text: one

text: dimension.

text: This

text: state

text: can

text: also

text: be

text: found

text: with

text: two

text: coherent

text: heads

text: (not

text: shown).

text: The

text: size

text: of

text: the

text: coherent

text: region

text: is

text: depicted

text: in

text: Fig.

text: We

text: observe

text: only

text: a

text: weak

text: dependence

text: on

text: the

text: coupling

text: radius

text: r,

text: while

text: the

text: coupling

text: phase

text: ϕ

text: has

text: a

text: stronger

text: eﬀect.

text: Compared

text: to

text: the

text: incoher-

text: ent

text: spot

text: chimeras,

text: the

text: size

text: increases

text: with

text: decreasing

text: ϕ.

text: In

text: some

text: cases,

text: we

text: observe

text: that

text: at

text: the

text: opposite

text: side

text: of

text: the

text: coherent

text: region

text: of

text: the

text: torus,

text: a

text: spiral

text: can

text: form,

text: see

text: Ap-

text: FIG.

text: (Color

text: online)

text: FHN

text: system:

text: Radius

text: ρ

text: of

text: the

text: spot-like

text: pendix

text: for

text: an

text: example.

text: The

text: radius

text: of

text: the

text: spiral

text: increases

text: chimera

text: states,

text: (a)

text: incoherent

text: spot

text: and

text: (b)

text: coherent

text: spot,

text: with

text: the

text: coupling

text: radius

text: (not

text: shown).

text: in

text: dependence

text: on

text: the

text: coupling

text: radius

text: r

text: for

text: diﬀerent

text: coupling

text: Between

text: regions

text: of

text: spots

text: (see

text: Appendix

text: and

text: fully

text: syn-

text: phases

text: ϕ

text: given

text: as

text: color

text: code.

text: The

text: error

text: bars

text: indicate

text: an

text: uncer-

text: chronous

text: solutions,

text: we

text: observe

text: that

text: ring

text: chimeras

text: can

text: be

text: tainty

text: of

text: (a)

text: ∆ρ

text: and

text: (b)

text: ∆ρ

text: in

text: the

text: measurements

text: found

text: for

text: ϕ

text: see

text: Fig.

text: The

text: label

text: “ring

text: chimera”

text: of

text: the

text: radii.

text: Parameters

text: as

text: in

text: Fig.

text: refers

text: to

text: the

text: dip

text: in

text: the

text: mean

text: phase

text: velocity

text: ω

text: in

text: the

text: cen-

text: ij

text: ter

text: of

text: the

text: incoherent

text: ring

text: that

text: can

text: reach

text: the

text: level

text: of

text: the

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##


text: 5 coherent oscillators or have a diﬀerent, larger ω -value many other patterns, like square- or cross-shaped inco- ij as in the displayed example. We also ﬁnd that rings can herent spot, stripe (cf. Sec. ), alternating and more com- occur as a single-headed state and a multi-headed state plex chimeras, many of them in a multistable conﬁgura- with 2 rings existing simultaneously at diﬀerent positions tion. Examples will be discussed in Appendix . More- → (not shown). Similar to the incoherent spots, the rings over, in the limit r 0 which corresponds to the contin- can either be round or have a square shape. Figure 5 uous reaction-diﬀusion FHN model, the well-known Tur- shows the dependence of the inner and outer diameter of ing patterns can be found. In addition, we ﬁnd grid-like the ring chimeras on the coupling radius r using diﬀerent structures and lines of incoherent spots in a diﬀerent area levels of brightness to indicate the number of occurrences of the parameter space (cf. Sec. ). on the considered ϕ-range. We ﬁnd that the outer diam- eter increases linear with r, while the size of the inner ring does not show such a strong dependence. B. Leaky Integrate-and-Fire: Spots and ring patterns Spots and ring patterns are also observed in the LIF model given by Eqs. (9) and (10), in particular for small values of the coupling strength σ. The spot pattern exists for p = 0, while for ﬁnite refractory periods the interior r of the spot synchronizes and ring patterns are obtained. × As in the case of the FHN model we also use a N N = × 100 100 lattice with toroidal boundary conditions, while FIG. 4: (Color online) FHN system: Ring chimera, found for − r = 33 and ϕ = π/2 0.24. A snapshot at t = 1900 of the here the control parameters are the coupling radius R and (color coded) activator variable xij is shown on the left side the refractory period pr. and the mean phase velocity and a section through the ring Figure 6 depicts an incoherent-spot chimera state for in the center and right panels, respectively. Other parameters small values of the coupling constant and zero refractory as in Fig. 2. period. A typical snapshot of the LIF spot chimera is shown in the left panel, the corresponding mean phase velocity ω is depicted in the middle and a horizontal cut ij crossing the spot is depicted on the right. Single spots develop in the LIF system for small values of σ (here σ = 0.1) when the coupling range R is small. As the coupling range increases while keeping the refractory period at zero the spot breaks into several unequal secondary spots whose size and distance increase with the coupling range (not shown). FIG. 6: (Color online) LIF system: Chimera state of incoherent-spot type. Snapshot of u (left) and the corre- ij sponding mean phase velocity (middle and right) for small FIG. 5: (Color online) FHN system: Inner (red triangles) values of the coupling constant σ = 0.1 and for a coupling and outer (black squares) diameter D of the incoherent ring range R = 10. Other parameters: µ = 1.0, p = 0 and r in dependence on the coupling radius r. The error bars in- N = 100. dicate an uncertainty of ∆D = 0.9 in the measurements of the diameters. The lines are linear ﬁts as a guide to the eye. ∈ − − Ring patterns are produced, when the refractory pe- All ring patterns found for ϕ [π/2 0.24, π/2 0.2] are riod takes ﬁnite values p > 0 and the interior of the shown. The inset shows the data with rescaled axes: Frac- r tion of incoherent elements Ninc/N2 vs. number of coupled spots synchronizes. The snapshot of uij, ωij and its cor- neighbors N /N2. The brightness of the dots corresponds to responding horizontal cut of a ring chimera is shown in r the number of observed ring patterns for every (D, r)-pair in Fig. 7. The outer border of the ring has a square shape the considered ϕ-range. Other parameters as in Fig. 2. due to the square conﬁguration of the connectivity ma- trix given by Eq. (10). This pattern diﬀers from the ones The region in parameter space discussed above includes appearing in the one-dimensional conﬁguration: From ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: coherent

text: oscillators

text: or

text: have

text: a

text: diﬀerent,

text: larger

text: ω

text: -value

text: many

text: other

text: patterns,

text: like

text: square-

text: or

text: cross-shaped

text: inco-

text: ij

text: as

text: in

text: the

text: displayed

text: example.

text: We

text: also

text: ﬁnd

text: that

text: rings

text: can

text: herent

text: spot,

text: stripe

text: (cf.

text: Sec.

text: alternating

text: and

text: more

text: com-

text: occur

text: as

text: a

text: single-headed

text: state

text: and

text: a

text: multi-headed

text: state

text: plex

text: chimeras,

text: many

text: of

text: them

text: in

text: a

text: multistable

text: conﬁgura-

text: with

text: rings

text: existing

text: simultaneously

text: at

text: diﬀerent

text: positions

text: tion.

text: Examples

text: will

text: be

text: discussed

text: in

text: Appendix

text: More-

text: (not

text: shown).

text: Similar

text: to

text: the

text: incoherent

text: spots,

text: the

text: rings

text: over,

text: in

text: the

text: limit

text: r

text: which

text: corresponds

text: to

text: the

text: contin-

text: can

text: either

text: be

text: round

text: or

text: have

text: a

text: square

text: shape.

text: Figure

text: uous

text: reaction-diﬀusion

text: FHN

text: model,

text: the

text: well-known

text: Tur-

text: shows

text: the

text: dependence

text: of

text: the

text: inner

text: and

text: outer

text: diameter

text: of

text: ing

text: patterns

text: can

text: be

text: found.

text: In

text: addition,

text: we

text: ﬁnd

text: grid-like

text: the

text: ring

text: chimeras

text: on

text: the

text: coupling

text: radius

text: r

text: using

text: diﬀerent

text: structures

text: and

text: lines

text: of

text: incoherent

text: spots

text: in

text: a

text: diﬀerent

text: area

text: levels

text: of

text: brightness

text: to

text: indicate

text: the

text: number

text: of

text: occurrences

text: of

text: the

text: parameter

text: space

text: (cf.

text: Sec.

text: on

text: the

text: considered

text: ϕ-range.

text: We

text: ﬁnd

text: that

text: the

text: outer

text: diam-

text: eter

text: increases

text: linear

text: with

text: r,

text: while

text: the

text: size

text: of

text: the

text: inner

text: ring

text: does

text: not

text: show

text: such

text: a

text: strong

text: dependence.

text: B.

text: Leaky

text: Integrate-and-Fire:

text: Spots

text: and

text: ring

text: patterns

text: Spots

text: and

text: ring

text: patterns

text: are

text: also

text: observed

text: in

text: the

text: LIF

text: model

text: given

text: by

text: Eqs.

text: and

text: in

text: particular

text: for

text: small

text: values

text: of

text: the

text: coupling

text: strength

text: σ.

text: The

text: spot

text: pattern

text: exists

text: for

text: p

text: while

text: for

text: ﬁnite

text: refractory

text: periods

text: the

text: interior

text: r

text: of

text: the

text: spot

text: synchronizes

text: and

text: ring

text: patterns

text: are

text: obtained.

text: As

text: in

text: the

text: case

text: of

text: the

text: FHN

text: model

text: we

text: also

text: use

text: a

text: N

text: N

text: lattice

text: with

text: toroidal

text: boundary

text: conditions,

text: while

text: FIG.

text: (Color

text: online)

text: FHN

text: system:

text: Ring

text: chimera,

text: found

text: for

text: r

text: and

text: ϕ

text: A

text: snapshot

text: at

text: t

text: of

text: the

text: here

text: the

text: control

text: parameters

text: are

text: the

text: coupling

text: radius

text: R

text: and

text: (color

text: coded)

text: activator

text: variable

text: xij

text: is

text: shown

text: on

text: the

text: left

text: side

text: the

text: refractory

text: period

text: pr.

text: and

text: the

text: mean

text: phase

text: velocity

text: and

text: a

text: section

text: through

text: the

text: ring

text: Figure

text: depicts

text: an

text: incoherent-spot

text: chimera

text: state

text: for

text: in

text: the

text: center

text: and

text: right

text: panels,

text: respectively.

text: Other

text: parameters

text: small

text: values

text: of

text: the

text: coupling

text: constant

text: and

text: zero

text: refractory

text: as

text: in

text: Fig.

text: period.

text: A

text: typical

text: snapshot

text: of

text: the

text: LIF

text: spot

text: chimera

text: is

text: shown

text: in

text: the

text: left

text: panel,

text: the

text: corresponding

text: mean

text: phase

text: velocity

text: ω

text: is

text: depicted

text: in

text: the

text: middle

text: and

text: a

text: horizontal

text: cut

text: ij

text: crossing

text: the

text: spot

text: is

text: depicted

text: on

text: the

text: right.

text: Single

text: spots

text: develop

text: in

text: the

text: LIF

text: system

text: for

text: small

text: values

text: of

text: σ

text: (here

text: σ

text: when

text: the

text: coupling

text: range

text: R

text: is

text: small.

text: As

text: the

text: coupling

text: range

text: increases

text: while

text: keeping

text: the

text: refractory

text: period

text: at

text: zero

text: the

text: spot

text: breaks

text: into

text: several

text: unequal

text: secondary

text: spots

text: whose

text: size

text: and

text: distance

text: increase

text: with

text: the

text: coupling

text: range

text: (not

text: shown).

text: FIG.

text: (Color

text: online)

text: LIF

text: system:

text: Chimera

text: state

text: of

text: incoherent-spot

text: type.

text: Snapshot

text: of

text: u

text: (left)

text: and

text: the

text: corre-

text: ij

text: sponding

text: mean

text: phase

text: velocity

text: (middle

text: and

text: right)

text: for

text: small

text: FIG.

text: (Color

text: online)

text: FHN

text: system:

text: Inner

text: (red

text: triangles)

text: values

text: of

text: the

text: coupling

text: constant

text: σ

text: and

text: for

text: a

text: coupling

text: and

text: outer

text: (black

text: squares)

text: diameter

text: D

text: of

text: the

text: incoherent

text: ring

text: range

text: R

text: Other

text: parameters:

text: µ

text: p

text: and

text: r

text: in

text: dependence

text: on

text: the

text: coupling

text: radius

text: r.

text: The

text: error

text: bars

text: in-

text: N

text: dicate

text: an

text: uncertainty

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text: of

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text: The

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text: as

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text: The

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text: tion

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text: elements

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text: neighbors

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text: brightness

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text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##


text: INDUCTION FOR LOCALLY COMPACT QUANTUM GROUPS REVISITED MEHRDAD KALANTAR, PAWEL(cid:32) KASPRZAK, ADAM SKALSKI, AND PIOTR SOL(cid:32) TAN Abstract. In this paper we revisit the theory of induced representations in the setting of locally compact quantum groups. In the case of induction from open quantum subgroups, we show that constructions of Kustermans and Vaes are equivalent to the classical, and much simpler, construction of Rieﬀel. We also prove in general setting the continuity of induction in the sense of Vaes with respect to weak containment. 1. Introduction The theory of induced representations, allowing one to manufacture in a canonical way a representation of a group out of that of a subgroup, dates back to the work of Frobenius for ﬁnite groups and was later developed by Mackey for locally compact groups. It has played a very important role in developing the general theory of group representations and has seen many applications in various areas of abstract harmonic analysis (we refer to the recent book [7] for a precise description of the construction, the historical background and an exhaustive list of references). From the modern point of view a very important point in the history of the subject was the paper of Rieﬀel ([12]), which phrased the construction in the language ∗ of Hilbert C -modules. This led to later generalizations of similar induction procedures to several operator algebraic contexts. Thus it was natural that in the beginning of the 21st century, with the widely accepted notion of locally compact quantum groups emerging in the work of Kustermans and Vaes ([9]), an interest has developed in the study of the theory of induced representations in the quantum context. In fact the two relevant papers were written respectively by Kustermans ([8]) and Vaes ([15]). In the ﬁrst of these the construction was presented under the assump- tion of integrability of the natural action of the quantum subgroup on the larger quantum group (automatically satisﬁed for classical groups – and later shown to be always true also in the quantum context in [5]), in the second an alternative approach was proposed, explicitly disposing of the integrability constraint and also allowing for a more general framework of ∗ representations on Hilbert C -modules, as opposed to Hilbert spaces. Both constructions of Kustermans and Vaes are technically complicated; indeed even the classical procedure of Mackey in general requires overcoming certain measure-theoretic and topological diﬃculties. The starting point of our paper is a classical observation that in the classical case if the subgroup from which we induce the representation is open, most of the technical problems disappear (see again [7]). More importantly, one can apply then a very straightforward version of the Rieﬀel procedure, simply observing that if H is an open subgroup of a locally compact ∗ ∗ ⊂ ∗ group G, then one has a natural embedding of the group C -algebras C (H) C (G), which ∗ → ∗ admits also a canonical conditional expectation E :: C (G) C (H). In a recent work [6] 2010 Mathematics Subject Classiﬁcation. Primary: 22D30, 46L89, Secondary: 22D25, 20G42. Key words and phrases. Locally compact quantum group, induction of representations. 1 ##LTLine##

text: INDUCTION

text: FOR

text: LOCALLY

text: COMPACT

text: QUANTUM

text: GROUPS

text: REVISITED

text: MEHRDAD

text: KALANTAR,

text: KASPRZAK,

text: ADAM

text: SKALSKI,

text: AND

text: PIOTR

text: TAN

text: Abstract.

text: In

text: this

text: paper

text: we

text: revisit

text: the

text: theory

text: of

text: induced

text: representations

text: in

text: the

text: setting

text: of

text: locally

text: compact

text: quantum

text: groups.

text: In

text: the

text: case

text: of

text: induction

text: from

text: open

text: quantum

text: subgroups,

text: we

text: show

text: that

text: constructions

text: of

text: Kustermans

text: and

text: Vaes

text: are

text: equivalent

text: to

text: the

text: classical,

text: and

text: much

text: simpler,

text: construction

text: of

text: Rieﬀel.

text: We

text: also

text: prove

text: in

text: general

text: setting

text: the

text: continuity

text: of

text: induction

text: in

text: the

text: sense

text: of

text: Vaes

text: with

text: respect

text: to

text: weak

text: containment.

text: Introduction

text: The

text: theory

text: of

text: induced

text: representations,

text: allowing

text: one

text: to

text: manufacture

text: in

text: a

text: canonical

text: way

text: a

text: representation

text: of

text: a

text: group

text: out

text: of

text: that

text: of

text: a

text: subgroup,

text: dates

text: back

text: to

text: the

text: work

text: of

text: Frobenius

text: for

text: ﬁnite

text: groups

text: and

text: was

text: later

text: developed

text: by

text: Mackey

text: for

text: locally

text: compact

text: groups.

text: It

text: has

text: played

text: a

text: very

text: important

text: role

text: in

text: developing

text: the

text: general

text: theory

text: of

text: group

text: representations

text: and

text: has

text: seen

text: many

text: applications

text: in

text: various

text: areas

text: of

text: abstract

text: harmonic

text: analysis

text: (we

text: refer

text: to

text: the

text: recent

text: book

text: for

text: a

text: precise

text: description

text: of

text: the

text: construction,

text: the

text: historical

text: background

text: and

text: an

text: exhaustive

text: list

text: of

text: references).

text: From

text: the

text: modern

text: point

text: of

text: view

text: a

text: very

text: important

text: point

text: in

text: the

text: history

text: of

text: the

text: subject

text: was

text: the

text: paper

text: of

text: Rieﬀel

text: which

text: phrased

text: the

text: construction

text: in

text: the

text: language

text: of

text: Hilbert

text: C

text: -modules.

text: This

text: led

text: to

text: later

text: generalizations

text: of

text: similar

text: induction

text: procedures

text: to

text: several

text: operator

text: algebraic

text: contexts.

text: Thus

text: it

text: was

text: natural

text: that

text: in

text: the

text: beginning

text: of

text: the

text: century,

text: with

text: the

text: widely

text: accepted

text: notion

text: of

text: locally

text: compact

text: quantum

text: groups

text: emerging

text: in

text: the

text: work

text: of

text: Kustermans

text: and

text: Vaes

text: an

text: interest

text: has

text: developed

text: in

text: the

text: study

text: of

text: the

text: theory

text: of

text: induced

text: representations

text: in

text: the

text: quantum

text: context.

text: In

text: fact

text: the

text: two

text: relevant

text: papers

text: were

text: written

text: respectively

text: by

text: Kustermans

text: and

text: Vaes

text: In

text: the

text: ﬁrst

text: of

text: these

text: the

text: construction

text: was

text: presented

text: under

text: the

text: assump-

text: tion

text: of

text: integrability

text: of

text: the

text: natural

text: action

text: of

text: the

text: quantum

text: subgroup

text: on

text: the

text: larger

text: quantum

text: group

text: (automatically

text: satisﬁed

text: for

text: classical

text: groups

text: and

text: later

text: shown

text: to

text: be

text: always

text: true

text: also

text: in

text: the

text: quantum

text: context

text: in

text: in

text: the

text: second

text: an

text: alternative

text: approach

text: was

text: proposed,

text: explicitly

text: disposing

text: of

text: the

text: integrability

text: constraint

text: and

text: also

text: allowing

text: for

text: a

text: more

text: general

text: framework

text: of

text: representations

text: on

text: Hilbert

text: C

text: -modules,

text: as

text: opposed

text: to

text: Hilbert

text: spaces.

text: Both

text: constructions

text: of

text: Kustermans

text: and

text: Vaes

text: are

text: technically

text: complicated;

text: indeed

text: even

text: the

text: classical

text: procedure

text: of

text: Mackey

text: in

text: general

text: requires

text: overcoming

text: certain

text: measure-theoretic

text: and

text: topological

text: diﬃculties.

text: The

text: starting

text: point

text: of

text: our

text: paper

text: is

text: a

text: classical

text: observation

text: that

text: in

text: the

text: classical

text: case

text: if

text: the

text: subgroup

text: from

text: which

text: we

text: induce

text: the

text: representation

text: is

text: open,

text: most

text: of

text: the

text: technical

text: problems

text: disappear

text: (see

text: again

text: More

text: importantly,

text: one

text: can

text: apply

text: then

text: a

text: very

text: straightforward

text: version

text: of

text: the

text: Rieﬀel

text: procedure,

text: simply

text: observing

text: that

text: if

text: H

text: is

text: an

text: open

text: subgroup

text: of

text: a

text: locally

text: compact

text: group

text: G,

text: then

text: one

text: has

text: a

text: natural

text: embedding

text: of

text: the

text: group

text: C

text: -algebras

text: C

text: (H)

text: C

text: (G),

text: which

text: admits

text: also

text: a

text: canonical

text: conditional

text: expectation

text: E

text: C

text: (G)

text: C

text: (H).

text: In

text: a

text: recent

text: work

text: Mathematics

text: Subject

text: Classiﬁcation.

text: Primary:

text: Secondary:

text: Key

text: words

text: and

text: phrases.

text: Locally

text: compact

text: quantum

text: group,

text: induction

text: of

text: representations.

text: ##LTLine##