text: ∗ Deﬁnition 1 (α-consistent strategy) Let N (T) be the num- p c N Action over channel N 0 0 ber of times that the user takes the optimal ofﬂine policy in a 0.50 0.15 4 sense frame over the ﬁrst T frames under an online strategy. Then, 0.50 0.17 3 sense ∀α ∈ (0, 1), if 0.50 0.21 2 sense E ∗ [T − N (T)] 0.50 0.23 1 guess = o(1), Tα 0.30 0.20 1 guess the strategy is α-consistent. 0.40 0.20 3 sense 0.60 0.20 2 sense Roughly speaking, an α-consistent strategy represents a cate- 0.65 0.20 1 sense gory of “good” online strategies under which the user selects the optimal ofﬂine policy in each frame with high probability. TABLE I: Structure of the optimal ofﬂine policy. Denote kl(θ , θ ) as the KL-distance between two Bernoulli 1 2 distributions with parameters θ , θ , respectively. Then, we 1 2 have the following lower bound for α-consistent strategies. would choose to transmit over the best channel directly without sensing. When p is large, the cost of a wrong guess outweighs 0 Theorem 3 Assume the optimal ofﬂine policy with θ ≥ θ ≥ that of sensing, thus the user should choose to sense instead of ∗ 1 2 . . . ≥ θK is to sense/guess the ﬁrst K channels sequentially guess. As p0 grows, the user will probe less channels because according to Theorem 1 and quit the remaining channels, the potential gain by sensing a channel (i.e., θ (b − p )) may ∗ i 0 0 where K < K. Then, under any α-consistent strategy, we not cover the sensing cost. have B. Online Algorithm Then, we resort to numerical simulations to verify the lim inf R(T) ≥ max min ∆π (cid:48) (5) effectiveness of Algorithm 1 in the online learning situation. T→∞ log T k=K∗+1,...,K π∈Πk kl(θk; max(θ1, uk)) Before we present the simulation results, we ﬁrst introduce two baseline learning algorithms for comparison, namely, the where Πk is the set of policies under which channel k might (cid:15)-greedy learning algorithm, and the Thompson Sampling be sampled or used wi(cid:48)th guess, ∆π is the per-frame regret (TS) based algorithm. The only difference between the (cid:15)- under policy π, and u is the upper threshold for channel greedy algorithm and Algorithm 1 is to replace the condition k k under the optimal ofﬂine policy, assuming k is the best for exploration with S = 1, where S is an independent channel. t t sample of a Bernoulli distribution with parameter (cid:15). Thus, Theorem 3 indicates that for most system settings where (cid:15)-greedy algorithm explores the system at a ﬁxed rate. TS is a randomized algorithm based on the Bayesian principle, at least one channel is left out under the optimal ofﬂine ∗ and has generated signiﬁcant interest due to its superior policy (i.e., K < K), the regret lower bound under any α- empirical performance [26]. We tailor the TS algorithm to consistent online strategy scales in log T. This scaling matches our setting, where the key idea of the algorithm is to sample the upper bound in Theorem 2, thus our online strategy is ∗ the channel statistics according to Beta distributions, whose order-optimal. We restrict to the situation K < K because, parameters are determined by past observations. Though we under this setting, we are able to identify some sub-optimal are not able to characterize its performance theoretically, we do policies adopted under an online strategy easily, i.e., any policy ∗ observe signiﬁcant performance improvement through simula- involves sense or guess over channel k, k > K , is strictly tions. Therefore, we include it in this section for comparison sub-optimal. By showing that the probability of choosing a purpose. Obtaining an upper bound on the regret under the policy that involves channel k cannot be too small, we obtain TS algorithm is one of our future directions. a lower bound on the regret. The proof of Theorem 3 can be We now compare the performances under Algorithm 1, the found in Appendix A. (cid:15)-greedy algorithm, and the TS algorithm through simulations. We set θ = {0.6, 0.5, 0.4, 0.3, 0.2, 0.1}, b = 1, p = 0.5, V. SIMULATION RESULTS 0 0 c = 0.2, and let the sensing cost, transmission cost and reward 0 In this section, we evaluate the ofﬂine and online spectrum be uniform random variables with ∆ = ∆ = ∆ = 0.1. b p c access policies through numerical results. According to Theorem 1, the optimal ofﬂine policy is to A. Ofﬂine Policy sense the ﬁrst three channels sequentially until it ﬁnds the We ﬁrst study how of system parameters would affect the ﬁrst available channel, and quit the current frame if all of optimal ofﬂine policy. We set θ = {0.6, 0.5, 0.4, 0.3, 0.2, 0.1}, them are unavailable. The expected per-frame net reward b = 1, and change the values of p and c separately. under the optimal policy is 0.12. We ﬁrst set L = 20, 0 0 0 As shown in Table I, for ﬁxed p , the maximum number D = 24.85 for Algorithm 1, and (cid:15) = 0.001 for the (cid:15)-greedy 0 of channels to sense or access (i.e., N) is monotonically algorithm. We run each algorithm 100 times and plot the decreasing as c increases. However, when c is ﬁxed, we do sample-path average in Fig. 1(a) and Fig. 1(b). As we can see 0 0 not observe such monotonicity in p . This can be explained in Fig. 1(a), Algorithm 1 and TS algorithm outperform the (cid:15)- 0 as follows: When p is small, the cost of a wrong guess is greedy algorithm signiﬁcantly as time T becomes sufﬁciently 0 small compared with the cost of sensing, thus the system large. In addition, the (cid:15)-greedy algorithm achieves smaller ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Deﬁnition

text: (α-consistent

text: strategy)

text: Let

text: N

text: (T)

text: be

text: the

text: num-

text: p

label: table

text: c

text: N

text: Action

text: over

text: channel

text: N

text: ber

text: of

text: times

text: that

text: the

text: user

text: takes

text: the

text: optimal

text: ofﬂine

text: policy

text: in

text: a

text: sense

text: frame

text: over

text: the

text: ﬁrst

text: T

text: frames

text: under

text: an

text: online

text: strategy.

text: Then,

text: sense

text: ∀α

text: if

text: sense

text: E

text: [T

text: N

text: (T)]

text: guess

text: Tα

text: guess

text: the

text: strategy

text: is

text: α-consistent.

text: sense

text: sense

text: Roughly

text: speaking,

text: an

text: α-consistent

text: strategy

text: represents

text: a

text: cate-

text: sense

text: gory

text: of

text: “good”

text: online

text: strategies

text: under

text: which

text: the

text: user

text: selects

text: the

text: optimal

text: ofﬂine

text: policy

text: in

text: each

text: frame

text: with

text: high

text: probability.

text: TABLE

text: I:

text: Structure

text: of

text: the

text: optimal

text: ofﬂine

text: policy.

text: Denote

text: kl(θ

text: θ

text: as

text: the

text: KL-distance

text: between

text: two

text: Bernoulli

text: distributions

text: with

text: parameters

text: θ

text: θ

text: respectively.

text: Then,

text: we

text: have

text: the

text: following

text: lower

text: bound

text: for

text: α-consistent

text: strategies.

text: would

text: choose

text: to

text: transmit

text: over

text: the

text: best

text: channel

text: directly

text: without

text: sensing.

text: When

text: p

text: is

text: large,

text: the

text: cost

text: of

text: a

text: wrong

text: guess

text: outweighs

text: Theorem

text: Assume

text: the

text: optimal

text: ofﬂine

text: policy

text: with

text: θ

text: θ

text: that

text: of

text: sensing,

text: thus

text: the

text: user

text: should

text: choose

text: to

text: sense

text: instead

text: of

text: θK

text: is

text: to

text: sense/guess

text: the

text: ﬁrst

text: K

text: channels

text: sequentially

text: guess.

text: As

text: grows,

text: the

text: user

text: will

text: probe

text: less

text: channels

text: because

text: according

text: to

text: Theorem

text: and

text: quit

text: the

text: remaining

text: channels,

text: the

text: potential

text: gain

text: by

text: sensing

text: a

text: channel

text: (i.e.,

text: θ

text: (b

text: p

text: may

text: i

text: where

text: K

text: K.

text: Then,

text: under

text: any

text: α-consistent

text: strategy,

text: we

text: not

text: cover

text: the

text: sensing

text: cost.

text: have

text: B.

text: Online

text: Algorithm

text: Then,

text: we

text: resort

text: to

text: numerical

text: simulations

text: to

text: verify

text: the

text: lim

text: inf

text: R(T)

text: max

text: min

text: ∆π

text: effectiveness

text: of

text: Algorithm

text: in

text: the

text: online

text: learning

text: situation.

text: T→∞

text: log

text: T

text: π∈Πk

text: kl(θk;

text: uk))

text: Before

text: we

text: present

text: the

text: simulation

text: results,

text: we

text: ﬁrst

text: introduce

text: two

text: baseline

text: learning

text: algorithms

text: for

text: comparison,

text: namely,

text: the

text: where

text: Πk

text: is

text: the

text: set

text: of

text: policies

text: under

text: which

text: channel

text: k

text: might

text: learning

text: algorithm,

text: and

text: the

text: Thompson

text: Sampling

text: be

text: sampled

text: or

text: used

text: guess,

text: ∆π

text: is

text: the

text: per-frame

text: regret

text: (TS)

text: based

text: algorithm.

text: The

text: only

text: difference

text: between

text: the

text: under

text: policy

text: π,

text: and

text: u

text: is

text: the

text: upper

text: threshold

text: for

text: channel

text: greedy

text: algorithm

text: and

text: Algorithm

text: is

text: to

text: replace

text: the

text: condition

text: k

text: k

text: under

text: the

text: optimal

text: ofﬂine

text: policy,

text: assuming

text: k

text: is

text: the

text: best

text: for

text: exploration

text: with

text: S

text: where

text: S

text: is

text: an

text: independent

text: channel.

text: t

text: t

text: sample

text: of

text: a

text: Bernoulli

text: distribution

text: with

text: parameter

text: Thus,

text: Theorem

text: indicates

text: that

text: for

text: most

text: system

text: settings

text: where

text: algorithm

text: explores

text: the

text: system

text: at

text: a

text: ﬁxed

text: rate.

text: TS

text: is

text: a

text: randomized

text: algorithm

text: based

text: on

text: the

text: Bayesian

text: principle,

text: at

text: least

text: one

text: channel

text: is

text: left

text: out

text: under

text: the

text: optimal

text: ofﬂine

text: and

text: has

text: generated

text: signiﬁcant

text: interest

text: due

text: to

text: its

text: superior

text: policy

text: (i.e.,

text: K

text: K),

text: the

text: regret

text: lower

text: bound

text: under

text: any

text: α-

text: empirical

text: performance

text: We

text: tailor

text: the

text: TS

text: algorithm

text: to

text: consistent

text: online

text: strategy

text: scales

text: in

text: log

text: T.

text: This

text: scaling

text: matches

text: our

text: setting,

text: where

text: the

text: key

text: idea

text: of

text: the

text: algorithm

text: is

text: to

text: sample

text: the

text: upper

text: bound

text: in

text: Theorem

text: thus

text: our

text: online

text: strategy

text: is

text: the

text: channel

text: statistics

text: according

text: to

text: Beta

text: distributions,

text: whose

text: order-optimal.

text: We

text: restrict

text: to

text: the

text: situation

text: K

text: K

text: because,

text: parameters

text: are

text: determined

text: by

text: past

text: observations.

text: Though

text: we

text: under

text: this

text: setting,

text: we

text: are

text: able

text: to

text: identify

text: some

text: sub-optimal

text: are

text: not

text: able

text: to

text: characterize

text: its

text: performance

text: theoretically,

text: we

text: do

text: policies

text: adopted

text: under

text: an

text: online

text: strategy

text: easily,

text: i.e.,

text: any

text: policy

text: observe

text: signiﬁcant

text: performance

text: improvement

text: through

text: simula-

text: involves

text: sense

text: or

text: guess

text: over

text: channel

text: k,

text: k

text: K

text: is

text: strictly

text: tions.

text: Therefore,

text: we

text: include

text: it

text: in

text: this

text: section

text: for

text: comparison

text: sub-optimal.

text: By

text: showing

text: that

text: the

text: probability

text: of

text: choosing

text: a

text: purpose.

text: Obtaining

text: an

text: upper

text: bound

text: on

text: the

text: regret

text: under

text: the

text: policy

text: that

text: involves

text: channel

text: k

text: cannot

text: be

text: too

text: small,

text: we

text: obtain

text: TS

text: algorithm

text: is

text: one

text: of

text: our

text: future

text: directions.

text: a

text: lower

text: bound

text: on

text: the

text: regret.

text: The

text: proof

text: of

text: Theorem

text: can

text: be

text: We

text: now

text: compare

text: the

text: performances

text: under

text: Algorithm

text: the

text: found

text: in

text: Appendix

text: A.

text: algorithm,

text: and

text: the

text: TS

text: algorithm

text: through

text: simulations.

text: We

text: set

text: θ

text: b

text: p

text: V.

text: SIMULATION

text: RESULTS

text: c

text: and

text: let

text: the

text: sensing

text: cost,

text: transmission

text: cost

text: and

text: reward

text: In

text: this

text: section,

text: we

text: evaluate

text: the

text: ofﬂine

text: and

text: online

text: spectrum

text: be

text: uniform

text: random

text: variables

text: with

text: b

text: p

text: c

text: access

text: policies

text: through

text: numerical

text: results.

text: According

text: to

text: Theorem

text: the

text: optimal

text: ofﬂine

text: policy

text: is

text: to

text: A.

text: Ofﬂine

text: Policy

text: sense

text: the

text: ﬁrst

text: three

text: channels

text: sequentially

text: until

text: it

text: ﬁnds

text: the

text: We

text: ﬁrst

text: study

text: how

text: of

text: system

text: parameters

text: would

text: affect

text: the

text: ﬁrst

text: available

text: channel,

text: and

text: quit

text: the

text: current

text: frame

text: if

text: all

text: of

text: optimal

text: ofﬂine

text: policy.

text: We

text: set

text: θ

text: them

text: are

text: unavailable.

text: The

text: expected

text: per-frame

text: net

text: reward

text: b

text: and

text: change

text: the

text: values

text: of

text: p

text: and

text: c

text: separately.

text: under

text: the

text: optimal

text: policy

text: is

text: We

text: ﬁrst

text: set

text: L

text: As

text: shown

text: in

text: Table

text: I,

text: for

text: ﬁxed

text: p

text: the

text: maximum

text: number

text: D

text: for

text: Algorithm

text: and

text: for

text: the

text: of

text: channels

text: to

text: sense

text: or

text: access

text: (i.e.,

text: N)

text: is

text: monotonically

text: algorithm.

text: We

text: run

text: each

text: algorithm

text: times

text: and

text: plot

text: the

text: decreasing

text: as

text: c

text: increases.

text: However,

text: when

text: c

text: is

text: ﬁxed,

text: we

text: do

text: sample-path

text: average

text: in

text: Fig.

text: and

text: Fig.

text: As

text: we

text: can

text: see

text: not

text: observe

text: such

text: monotonicity

text: in

text: p

text: This

text: can

text: be

text: explained

text: in

text: Fig.

text: Algorithm

text: and

text: TS

text: algorithm

text: outperform

text: the

text: as

text: follows:

text: When

text: p

text: is

text: small,

text: the

text: cost

text: of

text: a

text: wrong

text: guess

text: is

text: greedy

text: algorithm

text: signiﬁcantly

text: as

text: time

text: T

text: becomes

text: sufﬁciently

text: small

text: compared

text: with

text: the

text: cost

text: of

text: sensing,

text: thus

text: the

text: system

text: large.

text: In

text: addition,

text: the

text: algorithm

text: achieves

text: smaller

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Measurement of incandescent microparticle acceleration using stereoscopic imaginga) Pinghan Chu ,1, b) Bradley T. Wolfe,1 and Zhehui Wang1, c) (Dated: 31 May 2020) Microparticles ranging from sub-microns to millimeter in size are a common form of matter in magnetic fusion environment, and they are highly mobile due to their small mass. Diﬀerent forces in addition to gravity can aﬀect their motion both inside and outside the plasmas. Several recent advances open up new diagnostic possibilities to characterize the particle motion and their forces: high-speed imaging camera technology, microparticle injection techniques developed for fusion, and image processing software. Extending our earlier work on high-temperature 4D microparticle tracking using exploding wires1, we report latest results on time- resolved microparticle acceleration measurement. New particle tracking algorithm is found to be eﬀective in particle tracking even when there are a large number of particles close to each other. Epipolar constraint is used for track-pairing from two-camera views. Error ﬁeld based on epi-geometry model is characterized based on a large set of 2D track data and 3D track reconstructions. Accelerations based on individual reconstructed 3D tracks are obtained. Force sensitivity on the order of ten gravitational acceleration has achieved. High- speed imaging is a useful diagnostic tool for microparticle physics, computer model validation and mass injection technology development for magnetic fusion. PACS numbers: Keywords: Particle Tracking, High-speed Imaging Camera I. INTRODUCTION rate above 106 frames per second (‘mega-frame cameras’) and with a relatively limited ﬁeld of view. In compari- son, mega-frame cameras can now capture movies con- Production of microparticles or dust of diﬀerent sizes tinueously for seconds at a time1. A growing number of is not avoidable in burning plasmas like ITER2,3. Due mass injection techniques are used in magnetic fusion7, to their high mobility, one of the main concerns is that allowing controlled mass injection and therefore quanti- the microparticles could migrate inside the closed mag- tative analysis of microparticle dynamics. Additionally, netic ﬂux surfaces and contaminate the plasma core. For the broad uses of imaging techniques, including applica- low-Z atoms like hydrogen to carbon, the undesired eﬀect tions in machine vision, have led to rather sophisticated is fuel density dilution. For high-Z atoms like tungsten, tool kits that are availability for three-dimensional (3D) radiative cooling could be detrimental4. In ITER-like particle tracking from multiple camera views8,9. plasmas, beryllium as an impurity cannot exceed 15 mg in the plasma core. Tungsten as an impurity cannot ex- In 2016, an exploding-wire apparatus was built to ceed 0.6 mg in the core. The dynamics of microparti- generate high-temperature molten metallic microparti- cles can be complex due to the interplay of many forces cles, emulating the microparticles anticipated near the associated with plasma ﬂow, edge pressure and density plasma-facing surfaces in magnetic fusion1. The mo- gradient, edge turbulence, magnetic ﬁeld strength, heat- tion of these microparticles was then recorded by two ing and ablation. Gravity in comparison was shown to fast cameras in-sync. We showed that time-resolved be relatively small in fusion-edge-like plasma conditions5. 3D motion, or ‘four-dimensional’ (4D) tracking of high- Additional concerns with the microparticles include the temperature microparticles was feasible. These cameras spread of tritium and neutron-activated radioactive ma- took images at a frame rate of 105 Hz, i.e., the inter- − terials throughout the vacuum chamber, exacerbating the frame time of 10 5 sec and an exposure time of a few µs. tritium retention and removal problems. Experimental Metallic microparticles with sizes ranging from 10 to 50 techniques such as high-speed imaging sometimes is the µm were generated along the wire after an initial high- only experimental option to examine the microparticle voltage (up to 2 keV) high-current (estimated to be a dynamics. Furthermore, imaging of high-temperature few kA) pulse. High-temperature molten microparticles particles or dust can contribute to the understanding of were created within 10 to 20 µs based on particle incan- plasma-dust interactions and plasma-surface interactions descence and camera images. Two algorithms showed and development of ﬁrst wall materials6. similar results for 4D reconstruction of microparticle motion without statistical analysis. However, it showed Several recent advances open up new diagnostic pos- the diﬃculty of successfully pairing a large number of sibilities to characterize the particle motion and their tracks in the left and right cameras. forces: high-speed imaging camera technology, micropar- ticle injection techniques developed for fusion, and image In this work, the combination of 2D track identiﬁca- processing software. About 10 years ago5, commercial tion using nearest velocity with track pairing using both fast cameras can only take a few frames of images at a epipolar constraint and temporal brightness are used in

text: Measurement

label: title

text: of

text: incandescent

text: microparticle

text: acceleration

text: using

text: stereoscopic

text: imaginga)

text: Pinghan

label: author

text: Chu

label: paragraph

text: b)

text: Bradley

text: T.

text: and

text: Zhehui

text: c)

text: (Dated:

label: date

text: May

text: Microparticles

label: abstract

text: ranging

text: from

text: sub-microns

text: to

text: millimeter

text: in

text: size

text: are

text: a

text: common

text: form

text: of

text: matter

text: in

text: magnetic

text: fusion

text: environment,

text: and

text: they

text: are

text: highly

text: mobile

text: due

text: to

text: their

text: small

text: mass.

text: Diﬀerent

text: forces

text: in

text: addition

text: to

text: gravity

text: can

text: aﬀect

text: their

text: motion

text: both

text: inside

text: and

text: outside

text: the

text: plasmas.

text: Several

text: recent

text: advances

text: open

text: up

text: new

text: diagnostic

text: possibilities

text: to

text: characterize

text: the

text: particle

text: motion

text: and

text: their

text: forces:

text: high-speed

text: imaging

text: camera

text: technology,

text: microparticle

text: injection

text: techniques

text: developed

text: for

text: fusion,

text: and

text: image

text: processing

text: software.

text: Extending

text: our

text: earlier

text: work

text: on

text: high-temperature

text: microparticle

text: tracking

text: using

text: exploding

text: we

text: report

text: latest

text: results

text: on

text: time-

text: resolved

text: microparticle

text: acceleration

text: measurement.

text: New

text: particle

text: tracking

text: algorithm

text: is

text: found

text: to

text: be

text: eﬀective

text: in

text: particle

text: tracking

text: even

text: when

text: there

text: are

text: a

text: large

text: number

text: of

text: particles

text: close

text: to

text: each

text: other.

text: Epipolar

text: constraint

text: is

text: used

text: for

text: track-pairing

text: from

text: two-camera

text: views.

text: Error

text: ﬁeld

text: based

text: on

text: epi-geometry

text: model

text: is

text: characterized

text: based

text: on

text: a

text: large

text: set

text: of

text: track

text: data

text: and

text: track

text: reconstructions.

text: Accelerations

text: based

text: on

text: individual

text: reconstructed

text: tracks

text: are

text: obtained.

text: Force

text: sensitivity

text: on

text: the

text: order

text: of

text: ten

text: gravitational

text: acceleration

text: has

text: achieved.

text: High-

text: speed

text: imaging

text: is

text: a

text: useful

text: diagnostic

text: tool

text: for

text: microparticle

text: physics,

text: computer

text: model

text: validation

text: and

text: mass

text: injection

text: technology

text: development

text: for

text: magnetic

text: fusion.

text: PACS

text: numbers:

text: Keywords:

text: Particle

text: Tracking,

text: High-speed

text: Imaging

text: Camera

text: I.

text: INTRODUCTION

text: rate

text: above

text: frames

text: per

text: second

text: (‘mega-frame

text: cameras’)

text: and

text: with

text: a

text: relatively

text: limited

text: ﬁeld

text: of

text: view.

text: In

text: compari-

text: son,

text: mega-frame

text: cameras

text: can

text: now

text: capture

text: movies

text: con-

text: Production

text: of

text: microparticles

text: or

text: dust

text: of

text: diﬀerent

text: sizes

text: tinueously

text: for

text: seconds

text: at

text: a

text: A

text: growing

text: number

text: of

text: is

text: not

text: avoidable

text: in

text: burning

text: plasmas

text: like

text: Due

text: mass

text: injection

text: techniques

text: are

text: used

text: in

text: magnetic

text: to

text: their

text: high

text: mobility,

text: one

text: of

text: the

text: main

text: concerns

text: is

text: that

text: allowing

text: controlled

text: mass

text: injection

text: and

text: therefore

text: quanti-

text: the

text: microparticles

text: could

text: migrate

text: inside

text: the

text: closed

text: mag-

text: tative

text: analysis

text: of

text: microparticle

text: dynamics.

text: Additionally,

text: netic

text: ﬂux

text: surfaces

text: and

text: contaminate

text: the

text: plasma

text: core.

text: For

text: the

text: broad

text: uses

text: of

text: imaging

text: techniques,

text: including

text: applica-

text: low-Z

text: atoms

text: like

text: hydrogen

text: to

text: carbon,

text: the

text: undesired

text: eﬀect

text: tions

text: in

text: machine

text: vision,

text: have

text: led

text: to

text: rather

text: sophisticated

text: is

text: fuel

text: density

text: dilution.

text: For

text: high-Z

text: atoms

text: like

text: tungsten,

text: tool

text: kits

text: that

text: are

text: availability

text: for

text: three-dimensional

text: radiative

text: cooling

text: could

text: be

text: In

text: ITER-like

text: particle

text: tracking

text: from

text: multiple

text: camera

text: plasmas,

text: beryllium

text: as

text: an

text: impurity

text: cannot

text: exceed

text: mg

text: in

text: the

text: plasma

text: core.

text: Tungsten

text: as

text: an

text: impurity

text: cannot

text: ex-

text: In

text: an

text: exploding-wire

text: apparatus

text: was

text: built

text: to

text: ceed

text: mg

text: in

text: the

text: core.

text: The

text: dynamics

text: of

text: microparti-

text: generate

text: high-temperature

text: molten

text: metallic

text: microparti-

text: cles

text: can

text: be

text: complex

text: due

text: to

text: the

text: interplay

text: of

text: many

text: forces

text: cles,

text: emulating

text: the

text: microparticles

text: anticipated

text: near

text: the

text: associated

text: with

text: plasma

text: ﬂow,

text: edge

text: pressure

text: and

text: density

text: plasma-facing

text: surfaces

text: in

text: magnetic

text: The

text: mo-

text: gradient,

text: edge

text: turbulence,

text: magnetic

text: ﬁeld

text: strength,

text: heat-

text: tion

text: of

text: these

text: microparticles

text: was

text: then

text: recorded

text: by

text: two

text: ing

text: and

text: ablation.

text: Gravity

text: in

text: comparison

text: was

text: shown

text: to

text: fast

text: cameras

text: in-sync.

text: We

text: showed

text: that

text: time-resolved

text: be

text: relatively

text: small

text: in

text: fusion-edge-like

text: plasma

text: motion,

text: or

text: ‘four-dimensional’

text: tracking

text: of

text: high-

text: Additional

text: concerns

text: with

text: the

text: microparticles

text: include

text: the

text: temperature

text: microparticles

text: was

text: feasible.

text: These

text: cameras

text: spread

text: of

text: tritium

text: and

text: neutron-activated

text: radioactive

text: ma-

text: took

text: images

text: at

text: a

text: frame

text: rate

text: of

text: Hz,

text: i.e.,

text: the

text: inter-

text: terials

text: throughout

text: the

text: vacuum

text: chamber,

text: exacerbating

text: the

text: frame

text: time

text: of

text: sec

text: and

text: an

text: exposure

text: time

text: of

text: a

text: few

text: µs.

text: tritium

text: retention

text: and

text: removal

text: problems.

text: Experimental

text: Metallic

text: microparticles

text: with

text: sizes

text: ranging

text: from

text: to

text: techniques

text: such

text: as

text: high-speed

text: imaging

text: sometimes

text: is

text: the

text: µm

text: were

text: generated

text: along

text: the

text: wire

text: after

text: an

text: initial

text: high-

text: only

text: experimental

text: option

text: to

text: examine

text: the

text: microparticle

text: voltage

text: (up

text: to

text: keV)

text: high-current

text: (estimated

text: to

text: be

text: a

text: dynamics.

text: Furthermore,

text: imaging

text: of

text: high-temperature

text: few

text: kA)

text: pulse.

text: High-temperature

text: molten

text: microparticles

text: particles

text: or

text: dust

text: can

text: contribute

text: to

text: the

text: understanding

text: of

text: were

text: created

text: within

text: to

text: µs

text: based

text: on

text: particle

text: incan-

text: plasma-dust

text: interactions

text: and

text: plasma-surface

text: interactions

text: descence

text: and

text: camera

text: images.

text: Two

text: algorithms

text: showed

text: and

text: development

text: of

text: ﬁrst

text: wall

text: similar

text: results

text: for

text: reconstruction

text: of

text: microparticle

text: motion

text: without

text: statistical

text: analysis.

text: However,

text: it

text: showed

text: Several

text: recent

text: advances

text: open

text: up

text: new

text: diagnostic

text: pos-

text: the

text: diﬃculty

text: of

text: successfully

text: pairing

text: a

text: large

text: number

text: of

text: sibilities

text: to

text: characterize

text: the

text: particle

text: motion

text: and

text: their

text: tracks

text: in

text: the

text: left

text: and

text: right

text: cameras.

text: forces:

text: high-speed

text: imaging

text: camera

text: technology,

text: micropar-

text: ticle

text: injection

text: techniques

text: developed

text: for

text: fusion,

text: and

text: image

text: In

text: this

text: work,

text: the

text: combination

text: of

text: track

text: identiﬁca-

text: processing

text: software.

text: About

text: years

text: commercial

text: tion

text: using

text: nearest

text: velocity

text: with

text: track

text: pairing

text: using

text: both

text: fast

text: cameras

text: can

text: only

text: take

text: a

text: few

text: frames

text: of

text: images

text: at

text: a

text: epipolar

text: constraint

text: and

text: temporal

text: brightness

text: are

text: used

text: in


text: On Geodesically Convex Formulations for the Brascamp-Lieb Constant Nisheeth K. Vishnoi and Ozan Yıldız ´ Ecole Polytechnique F´ed´erale de Lausanne (EPFL), Switzerland May 31, 2020 Abstract We consider two non-convex formulations for computing the optimal constant in the Brascamp- Lieb inequality corresponding to a given datum, and show that they are geodesically log-concave on the manifold of positive deﬁnite matrices endowed with the Riemannian metric corresponding to the Hessian of the log-determinant function. The ﬁrst formulation is present in the work of Lieb [Lie90] and the second is inspired by the work of Bennett et al. [BCCT08]. Recent works of Garg et al. [GGdOW17] and Allen-Zhu et al. [AZGL+18] also imply a geodesically log-concave formulation of the Brascamp-Lieb constant through a reduction to the operator scaling problem. However, the dimension of the arising optimization problem in their reduction depends expo- nentially on the number of bits needed to describe the Brascamp-Lieb datum. The formulations presented here have dimensions that are polynomial in the bit complexity of the input datum. 1 Introduction The Brascamp-Lieb Inequality. Brascamp and Lieb [BL76] presented a class of inequalities that generalize many well-known inequalities and, as a consequence, have played an important role in various mathematical disciplines. Formally, they presented the following class of inequalities where each inequality is described by a “datum”, referred to as the Brascamp-Lieb datum. Deﬁnition 1.1 (The Brascamp-Lieb Inequality, Datum, Constant). Let n, m, and (n ) ∈ be j j [m] positive integers and p := (p ) ∈ be non-negative real numbers. Let B := (B ) ∈ be an j j [m] j j [m] R R m-tuple of linear transformations where Bj is a surjective linear transformation from n to nj. The corresponding Brascamp-Lieb datum denoted by (B, p). The Brascamp-Lieb inequality states that for each Brascamp-Lieb datum (B, p) there exists a constant C(B, p) not necessarily ﬁnite, such that for any selection of real-valued, non-negative, Lebesgue measurable functions f where fj : Rnj → R, (cid:90)  (cid:89)  (cid:89) (cid:18)(cid:90) (cid:19) j   ≤ pj f (B x)pj dx C(B, p) f (x)dx . (1) x∈Rn ∈ j j ∈ x∈Rnj j j [m] j [m] The smallest constant that satisﬁes (1) for any choice of f := (f ) ∈ satisfying the properties j j [m] mentioned above is called the Brascamp-Lieb constant and denoted by BL(B, p). A Brascamp-Lieb datum (B, p) is called feasible if BL(B, p) is ﬁnite, otherwise, it is called infeasible. For a given m-tuple B, the set of real vectors p such that (B, p) is feasible is denoted by P . B

text: On

text: Geodesically

text: Convex

text: Formulations

text: for

text: the

text: Brascamp-Lieb

text: Constant

text: Nisheeth

text: K.

text: Vishnoi

text: and

text: Ozan

text: Yıldız

text: Ecole

text: Polytechnique

text: F´ed´erale

text: de

text: Lausanne

text: (EPFL),

text: Switzerland

text: May

text: Abstract

text: We

text: consider

text: two

text: non-convex

text: formulations

text: for

text: computing

text: the

text: optimal

text: constant

text: in

text: the

text: Brascamp-

text: Lieb

text: inequality

text: corresponding

text: to

text: a

text: given

text: datum,

text: and

text: show

text: that

text: they

text: are

text: geodesically

text: log-concave

text: on

text: the

text: manifold

text: of

text: positive

text: deﬁnite

text: matrices

text: endowed

text: with

text: the

text: Riemannian

text: metric

text: corresponding

text: to

text: the

text: Hessian

text: of

text: the

text: log-determinant

text: function.

text: The

text: ﬁrst

text: formulation

text: is

text: present

text: in

text: the

text: work

text: of

text: Lieb

text: and

text: the

text: second

text: is

text: inspired

text: by

text: the

text: work

text: of

text: Bennett

text: et

text: al.

text: Recent

text: works

text: of

text: Garg

text: et

text: al.

text: and

text: Allen-Zhu

text: et

text: al.

text: also

text: imply

text: a

text: geodesically

text: log-concave

text: formulation

text: of

text: the

text: Brascamp-Lieb

text: constant

text: through

text: a

text: reduction

text: to

text: the

text: operator

text: scaling

text: problem.

text: However,

text: the

text: dimension

text: of

text: the

text: arising

text: optimization

text: problem

text: in

text: their

text: reduction

text: depends

text: expo-

text: nentially

text: on

text: the

text: number

text: of

text: bits

text: needed

text: to

text: describe

text: the

text: Brascamp-Lieb

text: datum.

text: The

text: formulations

text: presented

text: here

text: have

text: dimensions

text: that

text: are

text: polynomial

text: in

text: the

text: bit

text: complexity

text: of

text: the

text: input

text: datum.

text: Introduction

text: The

text: Brascamp-Lieb

text: Inequality.

text: Brascamp

text: and

text: Lieb

text: presented

text: a

text: class

text: of

text: inequalities

text: that

text: generalize

text: many

text: well-known

text: inequalities

text: and,

text: as

text: a

text: consequence,

text: have

text: played

text: an

text: important

text: role

text: in

text: various

text: mathematical

text: disciplines.

text: Formally,

text: they

text: presented

text: the

text: following

text: class

text: of

text: inequalities

text: where

text: each

text: inequality

text: is

text: described

text: by

text: a

text: “datum”,

text: referred

text: to

text: as

text: the

text: Brascamp-Lieb

text: datum.

text: Deﬁnition

text: (The

text: Brascamp-Lieb

text: Inequality,

text: Datum,

text: Constant).

text: Let

text: n,

text: m,

text: and

text: (n

text: be

text: j

text: j

text: [m]

text: positive

text: integers

text: and

text: p

text: (p

text: be

text: non-negative

text: real

text: numbers.

text: Let

text: B

text: (B

text: be

text: an

text: j

text: j

text: [m]

text: j

text: j

text: [m]

text: R

text: R

text: m-tuple

text: of

text: linear

text: transformations

text: where

text: Bj

text: is

text: a

text: surjective

text: linear

text: transformation

text: from

text: n

text: to

text: nj.

text: The

text: corresponding

text: Brascamp-Lieb

text: datum

text: denoted

text: by

text: (B,

text: p).

text: The

text: Brascamp-Lieb

text: inequality

text: states

text: that

text: for

text: each

text: Brascamp-Lieb

text: datum

text: (B,

text: p)

text: there

text: exists

text: a

text: constant

text: C(B,

text: p)

text: not

text: necessarily

text: ﬁnite,

text: such

text: that

text: for

text: any

text: selection

text: of

text: real-valued,

text: non-negative,

text: Lebesgue

text: measurable

text: functions

text: f

text: where

text: fj

text: Rnj

text: R,

text: j

text: pj

text: f

text: (B

text: x)pj

text: dx

text: C(B,

text: p)

text: f

text: (x)dx

text: x∈Rn

text: j

text: j

text: x∈Rnj

text: j

text: j

text: [m]

text: j

text: [m]

text: The

text: smallest

text: constant

text: that

text: satisﬁes

text: for

text: any

text: choice

text: of

text: f

text: (f

text: satisfying

text: the

text: properties

text: j

text: j

text: [m]

text: mentioned

text: above

text: is

text: called

text: the

text: Brascamp-Lieb

text: constant

text: and

text: denoted

text: by

text: BL(B,

text: p).

text: A

text: Brascamp-Lieb

text: datum

text: (B,

text: p)

text: is

text: called

text: feasible

text: if

text: BL(B,

text: p)

text: is

text: ﬁnite,

text: otherwise,

text: it

text: is

text: called

text: infeasible.

text: For

text: a

text: given

text: m-tuple

text: B,

text: the

text: set

text: of

text: real

text: vectors

text: p

text: such

text: that

text: (B,

text: p)

text: is

text: feasible

text: is

text: denoted

text: by

text: P

text: B


text: Geodesic Convexity. One deﬁnition of convexity of a function f in a Euclidean space is that the average of the function at the end points of each line in the domain is at-least the value of the function at the average point on the line. Geodesic convexity is a natural extension of this notion of convexity from Euclidean spaces to Riemannian manifolds that are geodesically convex. A set in the manifold is said to be geodesically convex if for every pair of points in the set, the geodesic combining these points lies entirely in the set. ⊆ S Deﬁnition 2.2 (Geodesically Convex Sets). A set S d is called geodesically convex if for any ∈ ∈ ∈ ++ X, Y S and t [0, 1], X# Y S. t A function deﬁned on a geodesically convex set is said to be geodesically convex if the average of the function at the end points of any geodesic in the domain is at-least the value of the function at the average point on the geodesic. ⊆ S Deﬁnition 2.3 (Geodesically Convex Functions). Let S d be a geodesically convex set. A → R ++∈ S ∈ function f : S is called geodesically convex if for any X, Y d and t [0, 1], ++ ≤ − f(X# Y ) (1 t)f(X) + tf(Y ). t − f is called geodesically concave if f is geodesically convex. An important point regarding geodesic convexity is that a non-convex function might be geodesically convex or vice-versa. In general, one cannot convert a geodesically convex function to a convex function by a change of variables. A well-known example for this is the log det(X) function whose concavity a classical result from the matrix calculus. On the other hand, a folklore result is that log det(X) is both geodesically convex and geodesically concave on the space of positive deﬁnite matrices with the metric eq. (5). Proposition 2.4 (Geodesic Linearity of log det). The log det(X) function is geodesically linear, S i.e, it is both geodesically convex and geodesically concave over n . ++ ∈ S ∈ Proof. Let X, Y n and t [0, 1]. Then, ++ − − − log det(X# Y ) Theor=em 2.1 log det(X1/2(X 1/2Y X 1/2)tX1/2) = (1 t) log det(X) + t log det(Y ). t Therefore, log det(X) is a geodesically linear function over positive deﬁnite cone with respect to the metric in eq. (5). Henceforth, when we mention geodesic convexity it is with respect to the metric in eq. (5). Geodesi- cally convex functions share some properties with usual convex functions. One such property is the relation between local and global minimizers. ⊆ Theorem 2.5 (Minimizers of Geodesically Convex Functions [Rap97], Theorem 6.1.1). Let S S → R d be a geodesically convex set and f : S be a geodesically convex function. Then, any local ++ minimum point of f is also a global minimum of f. More precisely, if x(cid:63) := argmin ∈ f(x) for x O some open geodesically convex subset O of S, then f(x(cid:63)) = inf ∈ f(x). x S (cid:80) Geometric Mean of Matrices and Linear Maps. While the function log det(P ) is geodesi- (cid:62) cally linear, our proof of Theorem 1.6 relies on the geodesic convexity of log det( ∈ p B A B ). (cid:62) j [m] j j j j A simple but important observation is that, if (B, p) is feasible, then p B A B is a strictly positive j j j j linear map for each j as proved below. 6 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Geodesic

text: Convexity.

text: One

text: deﬁnition

text: of

text: convexity

text: of

text: a

text: function

text: f

text: in

text: a

text: Euclidean

text: space

text: is

text: that

text: the

text: average

text: of

text: the

text: function

text: at

text: the

text: end

text: points

text: of

text: each

text: line

text: in

text: the

text: domain

text: is

text: at-least

text: the

text: value

text: of

text: the

text: function

text: at

text: the

text: average

text: point

text: on

text: the

text: line.

text: Geodesic

text: convexity

text: is

text: a

text: natural

text: extension

text: of

text: this

text: notion

text: of

text: convexity

text: from

text: Euclidean

text: spaces

text: to

text: Riemannian

text: manifolds

text: that

text: are

text: geodesically

text: convex.

text: A

text: set

text: in

text: the

text: manifold

text: is

text: said

text: to

text: be

text: geodesically

text: convex

text: if

text: for

text: every

text: pair

text: of

text: points

text: in

text: the

text: set,

text: the

text: geodesic

text: combining

text: these

text: points

text: lies

text: entirely

text: in

text: the

text: set.

text: S

text: Deﬁnition

text: (Geodesically

text: Convex

text: Sets).

text: A

text: set

text: S

text: d

text: is

text: called

text: geodesically

text: convex

text: if

text: for

text: any

text: X,

text: Y

text: S

text: and

text: t

text: X#

text: Y

text: S.

text: t

text: A

text: function

text: deﬁned

text: on

text: a

text: geodesically

text: convex

text: set

text: is

text: said

text: to

text: be

text: geodesically

text: convex

text: if

text: the

text: average

text: of

text: the

text: function

text: at

text: the

text: end

text: points

text: of

text: any

text: geodesic

text: in

text: the

text: domain

text: is

text: at-least

text: the

text: value

text: of

text: the

text: function

text: at

text: the

text: average

text: point

text: on

text: the

text: geodesic.

text: S

text: Deﬁnition

text: (Geodesically

text: Convex

text: Functions).

text: Let

text: S

text: d

text: be

text: a

text: geodesically

text: convex

text: set.

text: A

text: R

text: S

text: function

text: f

text: S

text: is

text: called

text: geodesically

text: convex

text: if

text: for

text: any

text: X,

text: Y

text: d

text: and

text: t

text: f(X#

text: Y

text: t)f(X)

text: tf(Y

text: t

text: f

text: is

text: called

text: geodesically

text: concave

text: if

text: f

text: is

text: geodesically

text: convex.

text: An

text: important

text: point

text: regarding

text: geodesic

text: convexity

text: is

text: that

text: a

text: non-convex

text: function

text: might

text: be

text: geodesically

text: convex

text: or

text: vice-versa.

text: In

text: general,

text: one

text: cannot

text: convert

text: a

text: geodesically

text: convex

text: function

text: to

text: a

text: convex

text: function

text: by

text: a

text: change

text: of

text: variables.

text: A

text: well-known

text: example

text: for

text: this

text: is

text: the

text: log

text: det(X)

text: function

text: whose

text: concavity

text: a

text: classical

text: result

text: from

text: the

text: matrix

text: calculus.

text: On

text: the

text: other

text: hand,

text: a

text: folklore

text: result

text: is

text: that

text: log

text: det(X)

text: is

text: both

text: geodesically

text: convex

text: and

text: geodesically

text: concave

text: on

text: the

text: space

text: of

text: positive

text: deﬁnite

text: matrices

text: with

text: the

text: metric

text: eq.

text: Proposition

text: (Geodesic

text: Linearity

text: of

text: log

text: det).

text: The

text: log

text: det(X)

text: function

text: is

text: geodesically

text: linear,

text: S

text: i.e,

text: it

text: is

text: both

text: geodesically

text: convex

text: and

text: geodesically

text: concave

text: over

text: n

text: S

text: Proof.

text: Let

text: X,

text: Y

text: n

text: and

text: t

text: Then,

text: log

text: det(X#

text: Y

text: Theor=em

text: log

text: X

text: t)

text: log

text: det(X)

text: t

text: log

text: det(Y

text: t

text: Therefore,

text: log

text: det(X)

text: is

text: a

text: geodesically

text: linear

text: function

text: over

text: positive

text: deﬁnite

text: cone

text: with

text: respect

text: to

text: the

text: metric

text: in

text: eq.

text: Henceforth,

text: when

text: we

text: mention

text: geodesic

text: convexity

text: it

text: is

text: with

text: respect

text: to

text: the

text: metric

text: in

text: eq.

text: Geodesi-

text: cally

text: convex

text: functions

text: share

text: some

text: properties

text: with

text: usual

text: convex

text: functions.

text: One

text: such

text: property

text: is

text: the

text: relation

text: between

text: local

text: and

text: global

text: minimizers.

text: Theorem

text: (Minimizers

text: of

text: Geodesically

text: Convex

text: Functions

text: Theorem

text: Let

text: S

text: S

text: R

text: d

text: be

text: a

text: geodesically

text: convex

text: set

text: and

text: f

text: S

text: be

text: a

text: geodesically

text: convex

text: function.

text: Then,

text: any

text: local

text: minimum

text: point

text: of

text: f

text: is

text: also

text: a

text: global

text: minimum

text: of

text: f.

text: More

text: precisely,

text: if

text: argmin

text: f(x)

text: for

text: x

text: O

text: some

text: open

text: geodesically

text: convex

text: subset

text: O

text: of

text: S,

text: then

text: inf

text: f(x).

text: x

text: S

text: Geometric

text: Mean

text: of

text: Matrices

text: and

text: Linear

text: Maps.

text: While

text: the

text: function

text: log

text: det(P

text: is

text: geodesi-

text: cally

text: linear,

text: our

text: proof

text: of

text: Theorem

text: relies

text: on

text: the

text: geodesic

text: convexity

text: of

text: log

text: det(

text: p

text: B

text: A

text: B

text: j

text: [m]

text: j

text: j

text: j

text: j

text: A

text: simple

text: but

text: important

text: observation

text: is

text: that,

text: if

text: (B,

text: p)

text: is

text: feasible,

text: then

text: p

text: B

text: A

text: B

text: is

text: a

text: strictly

text: positive

text: j

text: j

text: j

text: j

text: linear

text: map

text: for

text: each

text: j

text: as

text: proved

text: below.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ∈ R × ∈ Lemma 2.6. Let (cid:62)(B, p) be a feasible Brascamp-Lieb datum with ∈Bj nj n for each j [m]. Then, Φ (X) := B XB is a strictly positive linear map for each j [m]. j j j ∈ Proof. Let us assume that for some j [m], Φ (X) is not strictly positive linear map. Then, ∈ S 0 j0 ∈ R there e(cid:62)xists X0 ≤n++ such that Φj0(X0) is (cid:62)not positive≤deﬁnite. Thus, there exists v nj0 such that v Φ (X )v 0. Equivalently, (B v) X (B v) 0. Since X is positive deﬁnite, we get j0 0 (cid:62) (cid:62) (cid:88)j0 0 j0 (cid:88) 0 − R B v = 0. Hence, v B B v = 0. Consequently, rank of B is at most n 1 and dim(B n) < n . j0 j0 j0 j0 j j j Therefore, R ≤ R n = dim( n) p dim(B n) < p n = n, j j j j ∈ ∈ j [m] j [m] ∈ (cid:62) by Theorem 1.3, a contradiction. Con(cid:80)sequently, for any j [m], Φj(X) := Bj XBj is strictly positive linear whenever (B, p) is feasible. (cid:62) (cid:80) The joint geodesic convexity of log det( ∈ p B A B ) follows from a more general observation j [m] j j j j (that we prove) that asserts that if Φ s are strictly positive linear maps from Snj to Sn , then j + + log det( ∈ Φ (A )) is geodescially convex. Sra and Hosseini [SH15] observed this when m = 1. j [m] j j Their result follows from a result of Ando [And79] about “geometric means” that is also important for us and we explain it next. The geometric mean of two matrices was introduced by Pusz and Woronowicz [PW75]. If ∈ S P, Q d , then the geometric mean of P and Q is deﬁned as ++ − − P # Q = P 1/2(P 1/2QP 1/2)1/2P 1/2. (7) 1/2 By abuse of notation, we drop 1/2 and denote geometric mean by P #Q. Recall that, the geodesic S → R ∈ S ∈ convexity of a function f : d is equivalent to for any P, Q d and t [0, 1] ++ ++ ≤ − f(P # Q) (1 t)f(P ) + tf(Q). t If f is continuous, then the geodesic convexity of f can be deduced from the following: ∀ ∈ S ≤ 1 1 P, Q d , f(P #Q) f(P ) + f(Q). ++ 2 2 Ando proved the following result about the eﬀect of a strictly positive linear map on the geometric mean of two matrices. S → S (cid:48) Theorem 2.7 (Eﬀect of a Linear Map over Geometric Mean [And79], Theorem 3). Let Φ : d d ∈ S + + be a strictly positive linear map. If P, Q d , then ++ (cid:22) Φ(P #Q) Φ(P )#Φ(Q). (8) (cid:22) ≤ The monotonicity of logdet (P Q implies log det(P ) log det(Q)) and the multiplicativity of the determinant, combined with Theorem 2.7, imply the following result. Corollary 2.8 (Geodesic Convexity of the Logarithm of Linear Maps [SH15], Corollary 12). If S → S (cid:48) Φ : d d is a strictly positive linear map, then log det(Φ(P )) is geodesically con(cid:80)vex. + + While the proof of Theorem 1.7 uses Corollary 2.8, it is not enough for the proof of Theorem 1.6. Instead of geodesic convexity of log det(Φ(P )), the joint geodesic convexity of log det( ∈ Φ (P )) is needed where Φ : Snj → Sn is a linear map for each j ∈ [m]. We conclude this sectjio[nm]witjh tjhe j + + following two results on a maximal characterization of geometric mean and the eﬀect of positive linear maps on positive deﬁniteness of block diagonal matrices. 7 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: R

text: Lemma

text: Let

text: p)

text: be

text: a

text: feasible

text: Brascamp-Lieb

text: datum

text: with

text: ∈Bj

text: nj

text: n

text: for

text: each

text: j

text: [m].

text: Then,

text: Φ

text: (X)

text: B

text: XB

text: is

text: a

text: strictly

text: positive

text: linear

text: map

text: for

text: each

text: j

text: [m].

text: j

text: j

text: j

text: Proof.

text: Let

text: us

text: assume

text: that

text: for

text: some

text: j

text: [m],

text: Φ

text: (X)

text: is

text: not

text: strictly

text: positive

text: linear

text: map.

text: Then,

text: S

text: R

text: there

text: ≤n++

text: such

text: that

text: is

text: positive≤deﬁnite.

text: Thus,

text: there

text: exists

text: v

text: such

text: that

text: v

text: Φ

text: (X

text: )v

text: Equivalently,

text: (B

text: v)

text: X

text: (B

text: v)

text: Since

text: X

text: is

text: positive

text: deﬁnite,

text: we

text: get

text: R

text: B

text: v

text: Hence,

text: v

text: B

text: B

text: v

text: Consequently,

text: rank

text: of

text: B

text: is

text: at

text: most

text: n

text: and

text: dim(B

text: n)

text: n

text: j

text: j

text: j

text: Therefore,

text: R

text: R

text: n

text: dim(

text: n)

text: p

text: dim(B

text: n)

text: p

text: n

text: n,

text: j

text: j

text: j

text: j

text: j

text: [m]

text: j

text: [m]

text: by

text: Theorem

text: a

text: contradiction.

text: for

text: any

text: j

text: [m],

text: Φj(X)

text: Bj

text: XBj

text: is

text: strictly

text: positive

text: linear

text: whenever

text: (B,

text: p)

text: is

text: feasible.

text: The

text: joint

text: geodesic

text: convexity

text: of

text: log

text: det(

text: p

text: B

text: A

text: B

text: follows

text: from

text: a

text: more

text: general

text: observation

text: j

text: [m]

text: j

text: j

text: j

text: j

text: (that

text: we

text: prove)

text: that

text: asserts

text: that

text: if

text: Φ

text: s

text: are

text: strictly

text: positive

text: linear

text: maps

text: from

text: Snj

text: to

text: Sn

text: then

text: j

text: log

text: det(

text: Φ

text: (A

text: is

text: geodescially

text: convex.

text: Sra

text: and

text: Hosseini

text: observed

text: this

text: when

text: m

text: j

text: [m]

text: j

text: j

text: Their

text: result

text: follows

text: from

text: a

text: result

text: of

text: Ando

text: about

text: “geometric

text: means”

text: that

text: is

text: also

text: important

text: for

text: us

text: and

text: we

text: explain

text: it

text: next.

text: The

text: geometric

text: mean

text: of

text: two

text: matrices

text: was

text: introduced

text: by

text: Pusz

text: and

text: Woronowicz

text: If

text: S

text: P,

text: Q

text: d

text: then

text: the

text: geometric

text: mean

text: of

text: P

text: and

text: Q

text: is

text: deﬁned

text: as

text: P

text: Q

text: P

text: By

text: abuse

text: of

text: notation,

text: we

text: drop

text: and

text: denote

text: geometric

text: mean

text: by

text: P

text: #Q.

text: Recall

text: that,

text: the

text: geodesic

text: S

text: R

text: S

text: convexity

text: of

text: a

text: function

text: f

text: d

text: is

text: equivalent

text: to

text: for

text: any

text: P,

text: Q

text: d

text: and

text: t

text: f(P

text: Q)

text: t)f(P

text: tf(Q).

text: t

text: If

text: f

text: is

text: continuous,

text: then

text: the

text: geodesic

text: convexity

text: of

text: f

text: can

text: be

text: deduced

text: from

text: the

text: following:

text: S

text: P,

text: Q

text: d

text: f(P

text: #Q)

text: f(P

text: f(Q).

text: Ando

text: proved

text: the

text: following

text: result

text: about

text: the

text: eﬀect

text: of

text: a

text: strictly

text: positive

text: linear

text: map

text: on

text: the

text: geometric

text: mean

text: of

text: two

text: matrices.

text: S

text: S

text: Theorem

text: (Eﬀect

text: of

text: a

text: Linear

text: Map

text: over

text: Geometric

text: Mean

text: Theorem

text: Let

text: Φ

text: d

text: d

text: S

text: be

text: a

text: strictly

text: positive

text: linear

text: map.

text: If

text: P,

text: Q

text: d

text: then

text: Φ(P

text: #Q)

text: Φ(P

text: )#Φ(Q).

text: The

text: monotonicity

text: of

text: logdet

text: (P

text: Q

text: implies

text: log

text: det(P

text: log

text: det(Q))

text: and

text: the

text: multiplicativity

text: of

text: the

text: determinant,

text: combined

text: with

text: Theorem

text: imply

text: the

text: following

text: result.

text: Corollary

text: (Geodesic

text: Convexity

text: of

text: the

text: Logarithm

text: of

text: Linear

text: Maps

text: Corollary

text: If

text: S

text: S

text: Φ

text: d

text: d

text: is

text: a

text: strictly

text: positive

text: linear

text: map,

text: then

text: log

text: det(Φ(P

text: is

text: geodesically

text: While

text: the

text: proof

text: of

text: Theorem

text: uses

text: Corollary

text: it

text: is

text: not

text: enough

text: for

text: the

text: proof

text: of

text: Theorem

text: Instead

text: of

text: geodesic

text: convexity

text: of

text: log

text: det(Φ(P

text: the

text: joint

text: geodesic

text: convexity

text: of

text: log

text: det(

text: Φ

text: (P

text: is

text: needed

text: where

text: Φ

text: Snj

text: Sn

text: is

text: a

text: linear

text: map

text: for

text: each

text: j

text: [m].

text: We

text: conclude

text: this

text: sectjio[nm]witjh

text: tjhe

text: j

text: following

text: two

text: results

text: on

text: a

text: maximal

text: characterization

text: of

text: geometric

text: mean

text: and

text: the

text: eﬀect

text: of

text: positive

text: linear

text: maps

text: on

text: positive

text: deﬁniteness

text: of

text: block

text: diagonal

text: matrices.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: S ×· · ·×S → S Theorem 4.2 (Eﬀect of Jointly Linear Maps over Geometric Means). Let Φ : n1 nm n + + + be a jointly linear map. Then, (cid:22) Φ(G) Φ(P )#Φ(Q) where P := (P ) ∈ , Q := (Q ) ∈ , and G := (G ) ∈ with P , Q ∈ Snj and G := P #Q . j j [m] j j [m] j j [m] j j ++ j j j The following is a corollary of the theorem above and a generalization of Corollary 2.8. S × · · · × Corollary 4.3 (Joint Geodesic Convexity of Logarithm of Jointly Linear Maps). If Φ : n1 S → S + nm n is a jointly linear map, then + + g(P , . . . , P ) := log det(Φ(P , . . . , P )) (13) 1 m 1 m S × · · · × S is jointly geodesically convex in n1 nm . ++ ++ Proof of Corollary 4.3. We show that g is jointly geodesically mid-point concave. Theorem 4.2 implies that (cid:22) Φ(G) Φ(P )#Φ(Q) (14) for any P := (P ) ∈ and Q := (Q ) ∈ with P , Q ∈ Snj. Therefore, j j [m] j j [m] j j + (13) g(G) = log det(Φ(G)) ≤ log det(Φ(P )#Φ(Q)) (monotonicity of log det and (14)) ≤1 1 log det(Φ(P )) + log det(Φ(Q)) (multiplicativity of det) 2 2 (13)1 = (g(P ) + g(Q)) . 2 Thus, g satisﬁes mid-point geodesic convexity. Consequently, we establish the geodesic convexity of g using the continuity of g. The proof of Theorem 1.6 is a simple application of Corollary 4.3 and Proposition 2.4. Proof of Theorem 1.6. We show that BL(B, p; A) is jointly geodesically mid-point log-concave with respect to A. In other words, we show that for arbitrary P = (P ) ∈ , Q = (Q ) ∈ j j [m] j j [m] − ≤ −1 log BL(B, p; G) (log BL(B, p; P ) + log BL(B, p; Q)) 2 where G = (G ) ∈ with G := P #Q , being the midpoint of geodesic combining P to Q . This j j [m] j j j j j implies that BL(B, p; A) is jointly geodesically log-concave with respect to A due to the continuity (cid:80) (cid:80) of BL(B, p; A) with respect to A. (cid:62) Let Φ (P ) := p B P B . Φ is strictly positive linear map by Lemma 2.6. Then, Φ(P ) := (cid:62)j j j j j j j p B P B is jointly linear, as Φ(P ) = Φ (P ). Hence, log det(Φ(P )) is jointly geodesically ∈ j j j j ∈ j j j [m] j [m] convex by Corollary 4.3. Also, log det(X) is geodesically linear (Proposition 2.4). Thus, for any 10 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: S

text: ·×S

text: S

text: Theorem

text: (Eﬀect

text: of

text: Jointly

text: Linear

text: Maps

text: over

text: Geometric

text: Means).

text: Let

text: Φ

text: nm

text: n

text: be

text: a

text: jointly

text: linear

text: map.

text: Then,

text: Φ(G)

text: Φ(P

text: )#Φ(Q)

text: where

text: P

text: (P

text: Q

text: (Q

text: and

text: G

text: (G

text: with

text: P

text: Q

text: Snj

text: and

text: G

text: P

text: #Q

text: j

text: j

text: [m]

text: j

text: j

text: [m]

text: j

text: j

text: [m]

text: j

text: j

text: j

text: j

text: j

text: The

text: following

text: is

text: a

text: corollary

text: of

text: the

text: theorem

text: above

text: and

text: a

text: generalization

text: of

text: Corollary

text: S

text: Corollary

text: (Joint

text: Geodesic

text: Convexity

text: of

text: Logarithm

text: of

text: Jointly

text: Linear

text: Maps).

text: If

text: Φ

text: S

text: S

text: nm

text: n

text: is

text: a

text: jointly

text: linear

text: map,

text: then

text: g(P

text: P

text: log

text: det(Φ(P

text: P

text: m

text: m

text: S

text: S

text: is

text: jointly

text: geodesically

text: convex

text: in

text: nm

text: Proof

text: of

text: Corollary

text: We

text: show

text: that

text: g

text: is

text: jointly

text: geodesically

text: mid-point

text: concave.

text: Theorem

text: implies

text: that

text: Φ(G)

text: Φ(P

text: )#Φ(Q)

text: for

text: any

text: P

text: (P

text: and

text: Q

text: (Q

text: with

text: P

text: Q

text: Snj.

text: Therefore,

text: j

text: j

text: [m]

text: j

text: j

text: [m]

text: j

text: j

text: g(G)

text: log

text: det(Φ(G))

text: log

text: det(Φ(P

text: )#Φ(Q))

text: (monotonicity

text: of

text: log

text: det

text: and

text: ≤1

text: log

text: det(Φ(P

text: log

text: det(Φ(Q))

text: (multiplicativity

text: of

text: det)

text: (g(P

text: g(Q))

text: Thus,

text: g

text: satisﬁes

text: mid-point

text: geodesic

text: convexity.

text: Consequently,

text: we

text: establish

text: the

text: geodesic

text: convexity

text: of

text: g

text: using

text: the

text: continuity

text: of

text: g.

text: The

text: proof

text: of

text: Theorem

text: is

text: a

text: simple

text: application

text: of

text: Corollary

text: and

text: Proposition

text: Proof

text: of

text: Theorem

text: We

text: show

text: that

text: BL(B,

text: p;

text: A)

text: is

text: jointly

text: geodesically

text: mid-point

text: log-concave

text: with

text: respect

text: to

text: A.

text: In

text: other

text: words,

text: we

text: show

text: that

text: for

text: arbitrary

text: P

text: (P

text: Q

text: (Q

text: j

text: j

text: [m]

text: j

text: j

text: [m]

text: −1

text: log

text: BL(B,

text: p;

text: G)

text: (log

text: BL(B,

text: p;

text: P

text: log

text: BL(B,

text: p;

text: Q))

text: where

text: G

text: (G

text: with

text: G

text: P

text: #Q

text: being

text: the

text: midpoint

text: of

text: geodesic

text: combining

text: P

text: to

text: Q

text: This

text: j

text: j

text: [m]

text: j

text: j

text: j

text: j

text: j

text: implies

text: that

text: BL(B,

text: p;

text: A)

text: is

text: jointly

text: geodesically

text: log-concave

text: with

text: respect

text: to

text: A

text: due

text: to

text: the

text: continuity

text: of

text: BL(B,

text: p;

text: A)

text: with

text: respect

text: to

text: A.

text: Let

text: Φ

text: (P

text: p

text: B

text: P

text: B

text: Φ

text: is

text: strictly

text: positive

text: linear

text: map

text: by

text: Lemma

text: Then,

text: Φ(P

text: j

text: j

text: j

text: j

text: j

text: j

text: p

text: B

text: P

text: B

text: is

text: jointly

text: linear,

text: as

text: Φ(P

text: Φ

text: (P

text: Hence,

text: log

text: det(Φ(P

text: is

text: jointly

text: geodesically

text: j

text: j

text: j

text: j

text: j

text: j

text: j

text: [m]

text: j

text: [m]

text: convex

text: by

text: Corollary

text: Also,

text: log

text: det(X)

text: is

text: geodesically

text: linear

text: (Proposition

text: Thus,

text: for

text: any

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 8.5 GR GR a=1 100 a=1 a=10 a=10 a=100 a=100 a=105 a=105 7.5 60 SLy SLy 6.5 20 1.5 2 2.5 1.5 2 2.5 M (M ) M (M ) O• O• Figure 3: (left) Frequency ω (in units of kHz) of the fundamental l = 2 curvature mode R versus the mass (in units of the solar mass M(cid:12)) of neutron stars with the SLy equation of state and diﬀerent values of the parameter a varying between GR (black) and a = 105 (red). (right) A similar ﬁgure showing the damping time τ (in µs). frequency is maximal in GR and decreases as the value of a is increased, becoming minimal for the massless Brans-Dicke theory. Note that since the maximum mass increases with the value of a, the lowest frequencies are reached by the massless Brans-Dicke neutron stars with masses close to the maximum. Similarly, in Fig. 3 (right panel) we show the damping time τ in µs as a function of the total mass. The damping time is largest in GR, and decreases with increasing value of a. Interestingly, for low mass conﬁgurations the values of the damping time become rather independent of a. The damping time increases monotonically with the mass in all cases. These features are present independently of the equation of state. In each panel of Fig. 4 we show the frequency ω as a function of the total mass for a diﬀerent EOS R and several values of a. Similarly in Fig. 5 we show the damping time τ. In Fig. 6 we show the frequency ω (left panel) and the damping time τ (right panel) as R a function of the parameter a for several EOS (SLy in brown, APR4 in green and BHZBM in purple) and for a ﬁxed value of the total mass, M = 2M(cid:12). Both the frequency and the damping time decrease monotonically as the value of a increases. This behaviour is observed for large values of the mass (i.e., larger compactness). However, for small values of the mass (around 1M(cid:12)), this behaviour can change for some equations of state, with the frequency slightly increasing for small values of a, as can be seen in Fig. 4 (e.g., for GNH3 and WCS1). → → ∞ The limit a 0 leads to the corresponding GR value of the mode, while the limit a gives the massless Brans-Dicke value. 3.2.2 Universal relations Let us now address (approximate) universal relations for the rescaled frequencies and the rescaled damping times [37, 24, 38, 35, 36]. In Fig. 7 (left panel) we exhibit the frequency ω rescaled with the radius R of the star versus the compactness M/R of the star. The R s s results are shown for GR, and for f(R) theory with a = 10 and a = 105. The three sets of 12 )zHk( Rω )sµ( τ ##LTFigure## ##LTFigure##

font: default

text: GR

text: GR

text: SLy

text: SLy

text: M

text: (M

text: M

text: (M

text: O•

text: O•

text: Figure

text: (left)

label: caption

text: Frequency

text: ω

text: (in

text: units

text: of

text: kHz)

text: of

text: the

text: fundamental

text: l

text: curvature

text: mode

text: R

text: versus

text: the

text: mass

text: (in

text: units

text: of

text: the

text: solar

text: mass

text: of

text: neutron

text: stars

text: with

text: the

text: SLy

text: equation

text: of

text: state

text: and

text: diﬀerent

text: values

text: of

text: the

text: parameter

text: a

text: varying

text: between

text: GR

text: (black)

text: and

text: a

text: (red).

text: (right)

text: A

text: similar

text: ﬁgure

text: showing

text: the

text: damping

text: time

text: τ

text: (in

text: µs).

text: frequency

text: is

text: maximal

text: in

text: GR

text: and

text: decreases

text: as

text: the

text: value

text: of

text: a

text: is

text: increased,

text: becoming

text: minimal

text: for

text: the

text: massless

text: Brans-Dicke

text: theory.

text: Note

text: that

text: since

text: the

text: maximum

text: mass

text: increases

text: with

text: the

text: value

text: of

text: a,

text: the

text: lowest

text: frequencies

text: are

text: reached

text: by

text: the

text: massless

text: Brans-Dicke

text: neutron

text: stars

text: with

text: masses

text: close

text: to

text: the

text: maximum.

text: Similarly,

text: in

text: Fig.

text: (right

text: panel)

text: we

text: show

text: the

text: damping

text: time

text: τ

text: in

text: µs

text: as

text: a

text: function

text: of

text: the

text: total

text: mass.

text: The

text: damping

text: time

text: is

text: largest

text: in

text: GR,

text: and

text: decreases

text: with

text: increasing

text: value

text: of

text: a.

text: Interestingly,

text: for

text: low

text: mass

text: conﬁgurations

text: the

text: values

text: of

text: the

text: damping

text: time

text: become

text: rather

text: independent

text: of

text: a.

text: The

text: damping

text: time

text: increases

text: monotonically

text: with

text: the

text: mass

text: in

text: all

text: cases.

text: These

text: features

text: are

text: present

text: independently

text: of

text: the

text: equation

text: of

text: state.

text: In

text: each

text: panel

text: of

text: Fig.

text: we

text: show

text: the

text: frequency

text: ω

text: as

text: a

text: function

text: of

text: the

text: total

text: mass

text: for

text: a

text: diﬀerent

text: EOS

text: R

text: and

text: several

text: values

text: of

text: a.

text: Similarly

text: in

text: Fig.

text: we

text: show

text: the

text: damping

text: time

text: τ.

text: In

text: Fig.

text: we

text: show

text: the

text: frequency

text: ω

text: (left

text: panel)

text: and

text: the

text: damping

text: time

text: τ

text: (right

text: panel)

text: as

text: R

text: a

text: function

text: of

text: the

text: parameter

text: a

text: for

text: several

text: EOS

text: (SLy

text: in

text: brown,

text: in

text: green

text: and

text: BHZBM

text: in

text: purple)

text: and

text: for

text: a

text: ﬁxed

text: value

text: of

text: the

text: total

text: mass,

text: M

text: Both

text: the

text: frequency

text: and

text: the

text: damping

text: time

text: decrease

text: monotonically

text: as

text: the

text: value

text: of

text: a

text: increases.

text: This

text: behaviour

text: is

text: observed

text: for

text: large

text: values

text: of

text: the

text: mass

text: (i.e.,

text: larger

text: compactness).

text: However,

text: for

text: small

text: values

text: of

text: the

text: mass

text: (around

text: this

text: behaviour

text: can

text: change

text: for

text: some

text: equations

text: of

text: state,

text: with

text: the

text: frequency

text: slightly

text: increasing

text: for

text: small

text: values

text: of

text: a,

text: as

text: can

text: be

text: seen

text: in

text: Fig.

text: (e.g.,

text: for

text: and

text: The

text: limit

text: a

text: leads

text: to

text: the

text: corresponding

text: GR

text: value

text: of

text: the

text: mode,

text: while

text: the

text: limit

text: a

text: gives

text: the

text: massless

text: Brans-Dicke

text: value.

text: Universal

text: relations

text: Let

text: us

text: now

text: address

text: (approximate)

text: universal

text: relations

text: for

text: the

text: rescaled

text: frequencies

text: and

text: the

text: rescaled

text: damping

text: times

text: In

text: Fig.

text: (left

text: panel)

text: we

text: exhibit

text: the

text: frequency

text: ω

text: rescaled

text: with

text: the

text: radius

text: R

text: of

text: the

text: star

text: versus

text: the

text: compactness

text: M/R

text: of

text: the

text: star.

text: The

text: R

text: s

text: s

text: results

text: are

text: shown

text: for

text: GR,

text: and

text: for

text: f(R)

text: theory

text: with

text: a

text: and

text: a

text: The

text: three

text: sets

text: of

text: )zHk(

text: Rω

text: )sµ(

text: τ

text: ##LTFigure##

text: ##LTFigure##


text: Let us comment on how do we calculate the physical distance from the center of the compact object to a ﬁxed point, since this (cid:90)will be used later. It would be diﬀerent from the Einstein frame distance, and it is computed in the following way r r = A(ϕ)eλdr, (8) phys 0 which follows from the fact that the coupling function A(ϕ) is actually the conformal factor connecting the metrics of the Jordan frame and the Einstein frame. Another quantity that should be evaluated in the physical Jordan frame is the physical radius of the star, which is determined by the requirement that the pressure should vanish at the stellar surface r = r(p¯ = 0). It is easily seen that the physical radius is given by R = A(ϕ )r , where S S S S ϕ is the value of the scalar ﬁeld at the stellar surface. The neutron star mass on the other S hand is the same in both frames [29]. Likewise, the frequencies and damping times of the QNMs are the same in the Jordan frame and in the Einstein frame. 2.2 Axial perturbations and exterior complex scaling Following the standard procedure [16, 17, 18] the axial perturbations of the metric can be written as       0 0 h (t, r)Slm(θ, φ) h (t, r)Slm(θ, φ)  0 θ 0 φ 0 0 h (t, r)Slm(θ, φ) h (t, r)Slm(θ, φ) Haxial = 1 θ 1 φ , (9) µν h (t, r)Slm(θ, φ) h (t, r)Slm(θ, φ) 0 0 0 θ 1 θ h (t, r)(r)Slm(θ, φ) h (t, r)Slm(θ, φ) 0 0 0 φ 1 φ − where (Slm(θ, φ), Slm(θ, φ)) = ( ∂ Y (θ, φ)/ sin θ, sin θ∂ Y (θ, φ)) with Y (θ, φ) being θ φ φ lm θ lm lm the spherical harmonics. Since the scalar ﬁeld ϕ, the pressure p and the energy density ρ transform as true scalars under reﬂections of the angular coordinates their axial perturbations vanish. The relevant perturbed ﬁeld equat(cid:18)ions are(cid:19)then given by − − − (cid:0)− (cid:1)− 2 − (l 1)(l + 2) e 2ν∂2h + e 2ν ∂ ∂ h h = 0, (10) t 1 r r t 0 r2 1 − − − ∂ h eν λ∂ eν λh = 0. (11) t 0 r 1 Extracting ∂ h from the second equation and substituting it into the ﬁrst, we ﬁnd the t 0 equation for h . But instead of using h it is convenient to use the function 1 1 − eν λ X = h . (12) 1 r 5 ##LTLine## ##LTLine## ##LTLine##

text: Let

text: us

text: comment

text: on

text: how

text: do

text: we

text: calculate

text: the

text: physical

text: distance

text: from

text: the

text: center

text: of

text: the

text: compact

text: object

text: to

text: a

text: ﬁxed

text: point,

text: since

text: this

label: equation

text: be

text: used

text: later.

text: It

text: would

text: be

text: diﬀerent

text: from

text: the

text: Einstein

text: frame

text: distance,

text: and

text: it

text: is

text: computed

text: in

text: the

text: following

text: way

text: r

text: r

text: A(ϕ)eλdr,

text: phys

text: which

text: follows

text: from

text: the

text: fact

text: that

text: the

text: coupling

text: function

text: A(ϕ)

text: is

text: actually

text: the

text: conformal

text: factor

text: connecting

text: the

text: metrics

text: of

text: the

text: Jordan

text: frame

text: and

text: the

text: Einstein

text: frame.

text: Another

text: quantity

text: that

text: should

text: be

text: evaluated

text: in

text: the

text: physical

text: Jordan

text: frame

text: is

text: the

text: physical

text: radius

text: of

text: the

text: star,

text: which

text: is

text: determined

text: by

text: the

text: requirement

text: that

text: the

text: pressure

text: should

text: vanish

text: at

text: the

text: stellar

text: surface

text: r

text: r(p¯

text: It

text: is

text: easily

text: seen

text: that

text: the

text: physical

text: radius

text: is

text: given

text: by

text: R

text: A(ϕ

text: )r

text: where

text: S

text: S

text: S

text: S

text: ϕ

text: is

text: the

text: value

text: of

text: the

text: scalar

text: ﬁeld

text: at

text: the

text: stellar

text: surface.

text: The

text: neutron

text: star

text: mass

text: on

text: the

text: other

text: S

text: hand

text: is

text: the

text: same

text: in

text: both

text: frames

text: Likewise,

text: the

text: frequencies

text: and

text: damping

text: times

text: of

text: the

text: QNMs

text: are

text: the

text: same

text: in

text: the

text: Jordan

text: frame

text: and

text: in

text: the

text: Einstein

text: frame.

text: Axial

text: perturbations

text: and

text: exterior

text: complex

text: scaling

text: Following

text: the

text: standard

text: procedure

text: the

text: axial

text: perturbations

text: of

text: the

text: metric

text: can

text: be

text: written

text: as

text: h

text: (t,

text: r)Slm(θ,

text: φ)

text: h

text: (t,

text: r)Slm(θ,

text: φ)

text: θ

text: φ

text: h

text: (t,

text: r)Slm(θ,

text: φ)

text: h

text: (t,

text: r)Slm(θ,

text: φ)

text: Haxial

text: θ

text: φ

text: µν

text: h

text: (t,

text: r)Slm(θ,

text: φ)

text: h

text: (t,

text: r)Slm(θ,

text: φ)

text: θ

text: θ

text: h

text: (t,

text: r)(r)Slm(θ,

text: φ)

text: h

text: (t,

text: r)Slm(θ,

text: φ)

text: φ

text: φ

text: where

text: (Slm(θ,

text: φ),

text: Slm(θ,

text: φ))

text: Y

text: (θ,

text: φ)/

text: sin

text: θ,

text: sin

text: θ∂

text: Y

text: (θ,

text: φ))

text: with

text: Y

text: (θ,

text: φ)

text: being

text: θ

text: φ

text: φ

text: lm

text: θ

text: lm

text: lm

text: the

text: spherical

text: harmonics.

text: Since

text: the

text: scalar

text: ﬁeld

text: ϕ,

text: the

text: pressure

text: p

text: and

text: the

text: energy

text: density

text: ρ

text: transform

text: as

text: true

text: scalars

text: under

text: reﬂections

text: of

text: the

text: angular

text: coordinates

text: their

text: axial

text: perturbations

text: vanish.

text: The

text: relevant

text: perturbed

text: ﬁeld

text: given

text: by

text: (l

text: e

text: e

text: 2ν

text: h

text: h

text: t

text: r

text: r

text: t

text: h

text: eν

text: λ∂

text: eν

text: λh

text: t

text: r

text: Extracting

text: h

text: from

text: the

text: second

text: equation

text: and

text: substituting

text: it

text: into

text: the

text: ﬁrst,

text: we

text: ﬁnd

text: the

text: t

text: equation

text: for

text: h

text: But

text: instead

text: of

text: using

text: h

text: it

text: is

text: convenient

text: to

text: use

text: the

text: function

text: eν

text: λ

text: X

text: h

text: r

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: equation (13), the outgoing wave condition does not have to be imposed explicitly. Instead, we should just locate the rightmost boundary of the integration domain far enough from the center of the star, so that any ingoing contamination of the signal would have to travel too long to the point of extraction of the gravitational wave signal and, thus, would not inﬂuence the ﬁnal results. In order to control the error coming from the numerical contamination with an ingoing wave when calculating the resonant frequencies using the time-independent perturbation equation (15), we employ the method based on exterior complex scaling (see e.g., [42, 39, 40, 24]). Instead of integrating directly Eq. (15) for the perturbation X, we consider the phase function of the perturbation. Thus we transform Eq. (15) to the form of a Riccati equation [43] by deﬁning the phase function o(cid:20)f the perturbation via dX = gX. Then we(cid:21)obtain dr (cid:0) (cid:1) (cid:20) (cid:21) dg − − − − r − 1(cid:0) − − (cid:1) = g2 e2λ 2νω2 + ge2λ 4πrA4(ϕ)(ρ¯ p¯) + V (ϕ) 1 e 2λ dr 2 r − 1 l(l + 1) − 3 − − +e2λ 4πA4(ϕ)(ρ¯ p¯) + V (ϕ) + 1 e 2λ . (17) 2 r2 r2 Regular perturbations at the center of the neutron star will possess a phase function ∼ behaving like g l+1 close to r = 0. At inﬁnity, however, a general solution of the equation r will look like − − ∼ A eiωr A e iωr in out g iω − , (18) A eiωr + A e iωr in out | − which tends to g ∞ = iω asymptotically, for both purely outgoing waves and mixed (ingoing plus outgoing) waves. In order to guarantee that the solutions correspond to purely outgoing waves, we make an analytical continuation of the equation into the complex plane. This can be done by deﬁning a complex coordinate r [44] − r = r + ye iζ, (19) j ∈ ∞ where the real numbers r and ζ are auxiliary constants, and y [0, ). The asymptotic j → ∞ behaviour of the phase function g then becomes for y − − ∼ A eiξy A e iξy in out g iω − , (20) A eiξy + A e iξy in out 7 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: equation

text: the

text: outgoing

text: wave

text: condition

text: does

text: not

text: have

text: to

text: be

text: imposed

text: explicitly.

text: Instead,

text: we

text: should

text: just

text: locate

text: the

text: rightmost

text: boundary

text: of

text: the

text: integration

text: domain

text: far

text: enough

text: from

text: the

text: center

text: of

text: the

text: star,

text: so

text: that

text: any

text: ingoing

text: contamination

text: of

text: the

text: signal

text: would

text: have

text: to

text: travel

text: too

text: long

text: to

text: the

text: point

text: of

text: extraction

text: of

text: the

text: gravitational

text: wave

text: signal

text: and,

text: thus,

text: would

text: not

text: inﬂuence

text: the

text: ﬁnal

text: results.

text: In

text: order

text: to

text: control

text: the

text: error

text: coming

text: from

text: the

text: numerical

text: contamination

text: with

text: an

text: ingoing

text: wave

text: when

text: calculating

text: the

text: resonant

text: frequencies

text: using

text: the

text: time-independent

text: perturbation

text: equation

text: we

text: employ

text: the

text: method

text: based

text: on

text: exterior

text: complex

text: scaling

text: (see

text: e.g.,

text: Instead

text: of

text: integrating

text: directly

text: Eq.

text: for

text: the

text: perturbation

text: X,

text: we

text: consider

text: the

text: phase

text: function

text: of

text: the

text: perturbation.

text: Thus

text: we

text: transform

text: Eq.

text: to

text: the

text: form

text: of

text: a

text: Riccati

text: equation

text: by

text: deﬁning

text: the

text: phase

text: function

text: the

text: perturbation

text: via

text: dX

text: gX.

text: Then

text: dr

text: dg

text: r

text: 2νω2

text: p¯)

text: V

text: (ϕ)

text: e

text: 2λ

text: dr

text: r

text: l(l

text: p¯)

text: V

text: (ϕ)

text: e

text: 2λ

text: Regular

text: perturbations

text: at

text: the

text: center

text: of

text: the

text: neutron

text: star

text: will

text: possess

text: a

text: phase

text: function

text: behaving

text: like

text: g

text: close

text: to

text: r

text: At

text: inﬁnity,

text: however,

text: a

text: general

text: solution

text: of

text: the

text: equation

text: r

text: will

text: look

text: like

text: A

text: eiωr

text: A

text: e

text: iωr

text: in

text: out

text: g

text: iω

text: A

text: eiωr

text: A

text: e

text: iωr

text: in

text: out

text: which

text: tends

text: to

text: g

text: iω

text: asymptotically,

text: for

text: both

text: purely

text: outgoing

text: waves

text: and

text: mixed

text: (ingoing

text: plus

text: outgoing)

text: waves.

text: In

text: order

text: to

text: guarantee

text: that

text: the

text: solutions

text: correspond

text: to

text: purely

text: outgoing

text: waves,

text: we

text: make

text: an

text: analytical

text: continuation

text: of

text: the

text: equation

text: into

text: the

text: complex

text: plane.

text: This

text: can

text: be

text: done

text: by

text: deﬁning

text: a

text: complex

text: coordinate

text: r

text: r

text: r

text: ye

text: iζ,

text: j

text: where

text: the

text: real

text: numbers

text: r

text: and

text: ζ

text: are

text: auxiliary

text: constants,

text: and

text: y

text: The

text: asymptotic

text: j

text: behaviour

text: of

text: the

text: phase

text: function

text: g

text: then

text: becomes

text: for

text: y

text: A

text: eiξy

text: A

text: e

text: iξy

text: in

text: out

text: g

text: iω

text: A

text: eiξy

text: A

text: e

text: iξy

text: in

text: out

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: − with ξ = ξ + iξ and ξ = ω cos ζ + ω sin ζ, ξ = ω cos ζ ω sin ζ. If we now choose R I −R R I I | I − R ζ such that ξ = ω cos ζ ω sin ζ < 0, the condition g ∞ = iω enforces A = 0, and the I I R in solution will describe purely outgoing waves. With this setting, the resulting numerical procedure is the following. First we generate a static background solution, where we have to ﬁx the parameters of the theory (i.e., the constant a), the equation of state, and the central pressure of the conﬁguration. In the next step, we calculate the coeﬃcients of Eq. (17) using the background conﬁguration. Then we generate an interior solution of this equation extending from the center of the star at r = 0, up to a point at r = r , where the point r is chosen to be outside the star (r > R ), and j j · j s we demand regularity of the phase function by imposing lim → r g = l + 1. r 0 Next, a second solution of Eq. (17) is generated, but this time after performing the − transformation (18) with ω cos ζ ω sin ζ < 0. This solution extends from r = r to I R | − j inﬁnity, and we demand the exterior phase to satisfy g ∞ = iω, which imposes the outgoing wave behaviour. QNMs are obtained when the interior phase solution and the exterior phase solution are continuous at the point r = r . In our case, this only happens for a discrete j set of values of ω. We implement a search algorithm that minimizes the diﬀerence between both solutions at r = r using the gradient descent. Once we obtain a QNM, we test the j numerical stability by changing the auxiliary parameters r and ζ. Typically the QNMs can − j be obtained with a relative accuracy of 10 3 or better. 3 Numerical results We used two independent numerical techniques for solving the problem in order to verify the results and gain deeper insight. First we performed a time evolution of Eq. (13) perturb- ing the star with some initial pulse. This is the most general way of constructing a solution since we just impose some initial perturbation, and then let the system evolve. The resulting signal will contain an admixture of all the excited modes. The drawback of this method is that it normally yields only the ﬁrst axial QNM because of the short damping times of the higher axial modes. In addition, only a few oscillations can be observed before the signal dies out due to the fast damping, which leads to large errors in the extracted frequencies and damping times. Solving the time-independent perturbation equation (15) via the complex scaling method is of course more involved, but it is a much more robust procedure, that can be used to explore the parameter space with high accuracy. Therefore, we used this method to obtain the QNMs of a large variety of EOS and extract a set of universal relations. The background solutions were obtained using the ODE solver Colsys [45]. Typically, the mesh sizes of − the solutions include 104-105 points, with a relative accuracy of 10 10 for both functions and derivatives. To implement the realistic equations of state, we used a monotonic cubic Hermite interpolation of the data points for the tabulated EOS, or alternatively, a piece- wise polytropic interpolation as described in [46]. The results obtained with the two methods proved to be in very good agreement, pointing to the correctness of the calculations. 3.1 Time evolution of the axial perturbation equation We start our discussion of the numerical results by examining the axial oscillations of 8

text: with

text: ξ

text: ξ

text: iξ

text: and

text: ξ

text: ω

text: cos

text: ζ

text: ω

text: sin

text: ζ,

text: ξ

text: ω

text: cos

text: ζ

text: ω

text: sin

text: ζ.

text: If

text: we

text: now

text: choose

text: R

text: I

text: −R

text: R

text: I

text: I

text: I

text: R

text: ζ

text: such

text: that

text: ξ

text: ω

text: cos

text: ζ

text: ω

text: sin

text: ζ

text: the

text: condition

text: g

text: iω

text: enforces

text: A

text: and

text: the

text: I

text: I

text: R

text: in

text: solution

text: will

text: describe

text: purely

text: outgoing

text: waves.

text: With

text: this

text: setting,

text: the

text: resulting

text: numerical

text: procedure

text: is

text: the

text: following.

text: First

text: we

text: generate

text: a

text: static

text: background

text: solution,

text: where

text: we

text: have

text: to

text: ﬁx

text: the

text: parameters

text: of

text: the

text: theory

text: (i.e.,

text: the

text: constant

text: a),

text: the

text: equation

text: of

text: state,

text: and

text: the

text: central

text: pressure

text: of

text: the

text: conﬁguration.

text: In

text: the

text: next

text: step,

text: we

text: calculate

text: the

text: coeﬃcients

text: of

text: Eq.

text: using

text: the

text: background

text: conﬁguration.

text: Then

text: we

text: generate

text: an

text: interior

text: solution

text: of

text: this

text: equation

text: extending

text: from

text: the

text: center

text: of

text: the

text: star

text: at

text: r

text: up

text: to

text: a

text: point

text: at

text: r

text: r

text: where

text: the

text: point

text: r

text: is

text: chosen

text: to

text: be

text: outside

text: the

text: star

text: (r

text: R

text: and

text: j

text: j

text: j

text: s

text: we

text: demand

text: regularity

text: of

text: the

text: phase

text: function

text: by

text: imposing

text: lim

text: r

text: g

text: l

text: r

text: Next,

text: a

text: second

text: solution

text: of

text: Eq.

text: is

text: generated,

text: but

text: this

text: time

text: after

text: performing

text: the

text: transformation

text: with

text: ω

text: cos

text: ζ

text: ω

text: sin

text: ζ

text: This

text: solution

text: extends

text: from

text: r

text: r

text: to

text: I

text: R

text: j

text: inﬁnity,

text: and

text: we

text: demand

text: the

text: exterior

text: phase

text: to

text: satisfy

text: g

text: iω,

text: which

text: imposes

text: the

text: outgoing

text: wave

text: behaviour.

text: QNMs

text: are

text: obtained

text: when

text: the

text: interior

text: phase

text: solution

text: and

text: the

text: exterior

text: phase

text: solution

text: are

text: continuous

text: at

text: the

text: point

text: r

text: r

text: In

text: our

text: case,

text: this

text: only

text: happens

text: for

text: a

text: discrete

text: j

text: set

text: of

text: values

text: of

text: ω.

text: We

text: implement

text: a

text: search

text: algorithm

text: that

text: minimizes

text: the

text: diﬀerence

text: between

text: both

text: solutions

text: at

text: r

text: r

text: using

text: the

text: gradient

text: descent.

text: Once

text: we

text: obtain

text: a

text: QNM,

text: we

text: test

text: the

text: j

text: numerical

text: stability

text: by

text: changing

text: the

text: auxiliary

text: parameters

text: r

text: and

text: ζ.

text: Typically

text: the

text: QNMs

text: can

text: j

text: be

text: obtained

text: with

text: a

text: relative

text: accuracy

text: of

text: or

text: better.

text: Numerical

text: results

text: We

text: used

text: two

text: independent

text: numerical

text: techniques

text: for

text: solving

text: the

text: problem

text: in

text: order

text: to

text: verify

text: the

text: results

text: and

text: gain

text: deeper

text: insight.

text: First

text: we

text: performed

text: a

text: time

text: evolution

text: of

text: Eq.

text: perturb-

text: ing

text: the

text: star

text: with

text: some

text: initial

text: pulse.

text: This

text: is

text: the

text: most

text: general

text: way

text: of

text: constructing

text: a

text: solution

text: since

text: we

text: just

text: impose

text: some

text: initial

text: perturbation,

text: and

text: then

text: let

text: the

text: system

text: evolve.

text: The

text: resulting

text: signal

text: will

text: contain

text: an

text: admixture

text: of

text: all

text: the

text: excited

text: modes.

text: The

text: drawback

text: of

text: this

text: method

text: is

text: that

text: it

text: normally

text: yields

text: only

text: the

text: ﬁrst

text: axial

text: QNM

text: because

text: of

text: the

text: short

text: damping

text: times

text: of

text: the

text: higher

text: axial

text: modes.

text: In

text: addition,

text: only

text: a

text: few

text: oscillations

text: can

text: be

text: observed

text: before

text: the

text: signal

text: dies

text: out

text: due

text: to

text: the

text: fast

text: damping,

text: which

text: leads

text: to

text: large

text: errors

text: in

text: the

text: extracted

text: frequencies

text: and

text: damping

text: times.

text: Solving

text: the

text: time-independent

text: perturbation

text: equation

text: via

text: the

text: complex

text: scaling

text: method

text: is

text: of

text: course

text: more

text: involved,

text: but

text: it

text: is

text: a

text: much

text: more

text: robust

text: procedure,

text: that

text: can

text: be

text: used

text: to

text: explore

text: the

text: parameter

text: space

text: with

text: high

text: accuracy.

text: Therefore,

text: we

text: used

text: this

text: method

text: to

text: obtain

text: the

text: QNMs

text: of

text: a

text: large

text: variety

text: of

text: EOS

text: and

text: extract

text: a

text: set

text: of

text: universal

text: relations.

text: The

text: background

text: solutions

text: were

text: obtained

text: using

text: the

text: ODE

text: solver

text: Colsys

text: Typically,

text: the

text: mesh

text: sizes

text: of

text: the

text: solutions

text: include

text: points,

text: with

text: a

text: relative

text: accuracy

text: of

text: for

text: both

text: functions

text: and

text: derivatives.

text: To

text: implement

text: the

text: realistic

text: equations

text: of

text: state,

text: we

text: used

text: a

text: monotonic

text: cubic

text: Hermite

text: interpolation

text: of

text: the

text: data

text: points

text: for

text: the

text: tabulated

text: EOS,

text: or

text: alternatively,

text: a

text: piece-

text: wise

text: polytropic

text: interpolation

text: as

text: described

text: in

text: The

text: results

text: obtained

text: with

text: the

text: two

text: methods

text: proved

text: to

text: be

text: in

text: very

text: good

text: agreement,

text: pointing

text: to

text: the

text: correctness

text: of

text: the

text: calculations.

text: Time

text: evolution

text: of

text: the

text: axial

text: perturbation

text: equation

text: We

text: start

text: our

text: discussion

text: of

text: the

text: numerical

text: results

text: by

text: examining

text: the

text: axial

text: oscillations

text: of


text: 14 C. BADEA, M. CROUZEIX, AND H. KLAJA (C. Badea) Univ. Lille, CNRS, Laboratoire Paul Painleve´ - UMR 8524, Lille, France Email address: catalin.badea@univ-lille.fr (M. Crouzeix) Univ. Rennes, CNRS, IRMAR - UMR 6625, Rennes, France Email address: michel.crouzeix@univ-rennes1.fr (H. Klaja) E´cole Centrale de Lille, CNRS, Laboratoire Paul Painleve´ - UMR 8524, Lille, France Email address: hubert.klaja@gmail.com

text: C.

text: BADEA,

text: M.

text: CROUZEIX,

text: AND

text: H.

text: KLAJA

text: (C.

text: Badea)

text: Univ.

text: Lille,

text: CNRS,

text: Laboratoire

text: Paul

text: Painleve´

text: UMR

text: Lille,

text: France

text: Email

text: address:

text: catalin.badea@univ-lille.fr

label: footer

text: (M.

text: Crouzeix)

text: Univ.

text: Rennes,

text: CNRS,

text: IRMAR

text: UMR

text: Rennes,

text: France

text: Email

text: address:

text: (H.

text: Klaja)

text: E´cole

text: Centrale

text: de

text: Lille,

text: CNRS,

text: Laboratoire

text: Paul

text: Painleve´

text: UMR

text: Lille,

text: France

text: Email

text: address:

text: hubert.klaja@gmail.com