text: 11 produced values larger than the critical one at 5 %, namely, 3.841459. This was repeated 5000 times, and the results are shown in Fig. 7. h h GD h TD 200 400 600 800 1000 Data size Fig. 7. Empirical p-values for different sample sizes. Both tests tend to reject less than expected, since the empirical p-values are larger than the asymptotic one. The test based on the GD, SGD converges faster to η = 0.05 than the one that relies on the TD. It is also consistently closer to the asymptotic value. C. Results with actual data In the following, we show the results of applying the proposed technique to an E-SAR image [61] from the surroundings of Munich, Germany. We employed L band, HH polarization in single-look complex format to produce intensity data. The × × original resolution is 2.2 m 3.0 m (range azimuth). The intensity of each pixel is henceforth referred to as “full resolution” data; the subsampled average of four contiguous pixels is “reduced resolution”. Theoretically, the former has one look while the latter has four looks. Figure 8(a) shows the SAR image, and presents the area where the edge detection was performed. Figure 8(b) is the corresponding optical image from Google Maps. Figure 8(c) shows the result of applying the edge detector with both GD and TD to equally-spaced horizontal strips of side 3. It is remarkable that both led to the same edge points, differing only in the time it took to calculate them, except when the integral of the TD does not converge and therefore it returns no result. The computation time of TD is 70 times larger the computation time of GD, and it fails to converge with small samples. Figure 8(d) shows the edge detection using only GD, to the reduced resolution image. It can be seen that two points are shifted from the right place, but the result is acceptable. In order to show the advantage of our method, the ROA edge detector proposed by Touzi et al. [62] was applied to the same data of Figs. 8(c) and 8(d). The results are shown in Figs 9(a) and 9(b). The parameters used are: (i) increasing window size: × × × 3 3, 5 5, and 7 7, (ii) lower and higher thresholds: 0.14 and 0.22, resp. When compared to the edge points obtained using distances, ROA produces noisier results with higher false alarm results. ROA improves when applied to the reduced resolution data set, in agreement with the fact that this method is especially designed to segment smooth targets. In the following we discuss the effect of reducing the resolution on the distances using two actual images. Fig. 10 presents the samples used to estimate the parameters, and to then measure GD and TD between them. We identiﬁed ﬁ(cid:98)ve sam(cid:98)ples: A1 and A5 correspond to pasture regions, A2 and A4 areas are forest regions, and A3 corresponds to an urban G region. We estimated the texture parameter α of the 0 distribution in both full and reduced resolution; denote the estimates I α and αRR (reduced resolution), respectively. These estimates are shown in Table I, and help in the identiﬁcation of the types of land cover because they are related to the area roughness. The table also shows the size of each sample in full resolution (the reduced resolution samples are one fourth of the original size). These estimates are consistent with the land cover types visually identiﬁed. The estimates reduce with the resolution in most eulav−p 8.0 6.0 4.0 2.0 0.0 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: produced

text: values

text: larger

text: than

text: the

text: critical

text: one

text: at

text: namely,

text: This

text: was

text: repeated

text: times,

text: and

text: the

text: results

text: are

text: shown

text: in

text: Fig.

text: h

text: h

text: GD

text: h

text: TD

text: Data

text: size

text: Fig.

text: Empirical

text: p-values

text: for

text: different

text: sample

text: sizes.

text: Both

text: tests

text: tend

text: to

text: reject

text: less

text: than

text: expected,

text: since

text: the

text: empirical

text: p-values

text: are

text: larger

text: than

text: the

text: asymptotic

text: one.

text: The

text: test

text: based

text: on

text: the

text: GD,

text: SGD

text: converges

text: faster

text: to

text: η

text: than

text: the

text: one

text: that

text: relies

text: on

text: the

text: TD.

text: It

text: is

text: also

text: consistently

text: closer

text: to

text: the

text: asymptotic

text: value.

text: C.

text: Results

text: with

text: actual

text: data

text: In

text: the

text: following,

text: we

text: show

text: the

text: results

text: of

text: applying

text: the

text: proposed

text: technique

text: to

text: an

text: E-SAR

text: image

text: from

text: the

text: surroundings

text: of

text: Munich,

text: Germany.

text: We

text: employed

text: L

text: band,

text: HH

text: polarization

text: in

text: single-look

text: complex

text: format

text: to

text: produce

text: intensity

text: data.

text: The

text: original

text: resolution

text: is

text: m

text: m

text: (range

text: azimuth).

text: The

text: intensity

text: of

text: each

text: pixel

text: is

text: henceforth

text: referred

text: to

text: as

text: “full

text: resolution”

text: data;

text: the

text: subsampled

text: average

text: of

text: four

text: contiguous

text: pixels

text: is

text: “reduced

text: resolution”.

text: Theoretically,

text: the

text: former

text: has

text: one

text: look

text: while

text: the

text: latter

text: has

text: four

text: looks.

text: Figure

text: shows

text: the

text: SAR

text: image,

text: and

text: presents

text: the

text: area

text: where

text: the

text: edge

text: detection

text: was

text: performed.

text: Figure

text: is

text: the

text: corresponding

text: optical

text: image

text: from

text: Google

text: Maps.

text: Figure

text: shows

text: the

text: result

text: of

text: applying

text: the

text: edge

text: detector

text: with

text: both

text: GD

text: and

text: TD

text: to

text: equally-spaced

text: horizontal

text: strips

text: of

text: side

text: It

text: is

text: remarkable

text: that

text: both

text: led

text: to

text: the

text: same

text: edge

text: points,

text: differing

text: only

text: in

text: the

text: time

text: it

text: took

text: to

text: calculate

text: them,

text: except

text: when

text: the

text: integral

text: of

text: the

text: TD

text: does

text: not

text: converge

text: and

text: therefore

text: it

text: returns

text: no

text: result.

text: The

text: computation

text: time

text: of

text: TD

text: is

text: times

text: larger

text: the

text: computation

text: time

text: of

text: GD,

text: and

text: it

text: fails

text: to

text: converge

text: with

text: small

text: samples.

text: Figure

text: shows

text: the

text: edge

text: detection

text: using

text: only

text: GD,

text: to

text: the

text: reduced

text: resolution

text: image.

text: It

text: can

text: be

text: seen

text: that

text: two

text: points

text: are

text: shifted

text: from

text: the

text: right

text: place,

text: but

text: the

text: result

text: is

text: acceptable.

text: In

text: order

text: to

text: show

text: the

text: advantage

text: of

text: our

text: method,

text: the

text: ROA

text: edge

text: detector

text: proposed

text: by

text: Touzi

text: et

text: al.

text: was

text: applied

text: to

text: the

text: same

text: data

text: of

text: Figs.

text: and

text: The

text: results

text: are

text: shown

text: in

text: Figs

text: and

text: The

text: parameters

text: used

text: are:

text: (i)

text: increasing

text: window

text: size:

text: and

text: (ii)

text: lower

text: and

text: higher

text: thresholds:

text: and

text: resp.

text: When

text: compared

text: to

text: the

text: edge

text: points

text: obtained

text: using

text: distances,

text: ROA

text: produces

text: noisier

text: results

text: with

text: higher

text: false

text: alarm

text: results.

text: ROA

text: improves

text: when

text: applied

text: to

text: the

text: reduced

text: resolution

text: data

text: set,

text: in

text: agreement

text: with

text: the

text: fact

text: that

text: this

text: method

text: is

text: especially

text: designed

text: to

text: segment

text: smooth

text: targets.

text: In

text: the

text: following

text: we

text: discuss

text: the

text: effect

text: of

text: reducing

text: the

text: resolution

text: on

text: the

text: distances

text: using

text: two

text: actual

text: images.

text: Fig.

text: presents

text: the

text: samples

text: used

text: to

text: estimate

text: the

text: parameters,

text: and

text: to

text: then

text: measure

text: GD

text: and

text: TD

text: between

text: them.

text: We

text: identiﬁed

text: and

text: correspond

text: to

text: pasture

text: regions,

text: and

text: areas

text: are

text: forest

text: regions,

text: and

text: corresponds

text: to

text: an

text: urban

text: G

text: region.

text: We

text: estimated

text: the

text: texture

text: parameter

text: α

text: of

text: the

text: distribution

text: in

text: both

text: full

text: and

text: reduced

text: resolution;

text: denote

text: the

text: estimates

text: I

text: α

text: and

text: αRR

text: (reduced

text: resolution),

text: respectively.

text: These

text: estimates

text: are

text: shown

text: in

text: Table

text: I,

text: and

text: help

text: in

text: the

text: identiﬁcation

text: of

text: the

text: types

text: of

text: land

text: cover

text: because

text: they

text: are

text: related

text: to

text: the

text: area

text: roughness.

text: The

text: table

text: also

text: shows

text: the

text: size

text: of

text: each

text: sample

text: in

text: full

text: resolution

text: (the

text: reduced

text: resolution

text: samples

text: are

text: one

text: fourth

text: of

text: the

text: original

text: size).

text: These

text: estimates

text: are

text: consistent

text: with

text: the

text: land

text: cover

text: types

text: visually

text: identiﬁed.

text: The

text: estimates

text: reduce

text: with

text: the

text: resolution

text: in

text: most

text: eulav−p

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 13 (a) Result of the Touzi edge detector, full resolution data. (b) Result of the Touzi edge detector, reduced resolution data. Fig. 9. Results of applying the Touzi edge detectors to actual data. TABLE I ESTIMATES OF α IN AREAS FROM THE IMAGE IN FIG. 10. (cid:98) (cid:98) Sample Sample Size α αRR − − A1 1360 6.09 11.53 − − A2 1225 9.72 20.00 − − A3 2088 1.01 1.00 − − A4 1824 2.75 4.27 − − A5 1152 11.51 20.00 between pasture and forest areas. The TD correctly discriminates regions having very different texture, such as A3 (extreme textured area) and A5 (textureless area) but in the case of A1 and A2 which correspond to a textureless area and a moderate textured area, respectively, TD looses the ability to separate: its value is almost zero. This implicates an advantage of GD over TD. Figure 11 illustrates the distances between A1 and the other regions in both full and reduced resolution. GD and TD are represented with orange and green colors, respectively. The line thickness is proportional to the distances between region models(cid:98). Figure 12 shows an AIRSAR image over San Francisco, USA, HH polarization in intensity, with the equivalent number of looks L = 2. We identiﬁed three samples: A1 corresponding to water, A2 an urban region, and A3 a forest area. The same procedure was performed as before for obtaining an image with reduced resolution. Table III shows the estimates of the α parameter for each region and the sample size, for both resolutions. It can be seen that the estimates agree with the visual interpretation of the classes. Table IV presents the GD and the TD between regions. This example highlights the difﬁculty TD has to distinguish regions with a similar roughness. It can be seen that the TD between A1 and A3 and between A2 and A3 are almost the same in both resolutions. These experiments show that the methodology is not affected if the image has reduced resolution. Furthermore, as the ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: (a)

text: Result

text: of

text: the

text: Touzi

text: edge

text: detector,

text: full

text: resolution

text: data.

text: (b)

text: Result

text: of

text: the

text: Touzi

text: edge

text: detector,

text: reduced

text: resolution

text: data.

text: Fig.

text: Results

label: caption

text: of

text: applying

text: the

text: Touzi

text: edge

text: detectors

text: to

text: actual

text: data.

text: TABLE

text: I

text: ESTIMATES

text: OF

text: α

text: IN

text: AREAS

text: FROM

text: THE

text: IMAGE

text: IN

text: FIG.

label: table

text: Sample

text: Sample

text: Size

text: α

text: αRR

text: between

text: pasture

text: and

text: forest

text: areas.

text: The

text: TD

text: correctly

text: discriminates

text: regions

text: having

text: very

text: different

text: texture,

text: such

text: as

text: (extreme

text: textured

text: area)

text: and

text: (textureless

text: area)

text: but

text: in

text: the

text: case

text: of

text: and

text: which

text: correspond

text: to

text: a

text: textureless

text: area

text: and

text: a

text: moderate

text: textured

text: area,

text: respectively,

text: TD

text: looses

text: the

text: ability

text: to

text: separate:

text: its

text: value

text: is

text: almost

text: zero.

text: This

text: implicates

text: an

text: advantage

text: of

text: GD

text: over

text: TD.

text: Figure

text: illustrates

text: the

text: distances

text: between

text: and

text: the

text: other

text: regions

text: in

text: both

text: full

text: and

text: reduced

text: resolution.

text: GD

text: and

text: TD

text: are

text: represented

text: with

text: orange

text: and

text: green

text: colors,

text: respectively.

text: The

text: line

text: thickness

text: is

text: proportional

text: to

text: the

text: distances

text: between

text: region

text: Figure

text: shows

text: an

text: AIRSAR

text: image

text: over

text: San

text: Francisco,

text: USA,

text: HH

text: polarization

text: in

text: intensity,

text: with

text: the

text: equivalent

text: number

text: of

text: looks

text: L

text: We

text: identiﬁed

text: three

text: samples:

text: corresponding

text: to

text: water,

text: an

text: urban

text: region,

text: and

text: a

text: forest

text: area.

text: The

text: same

text: procedure

text: was

text: performed

text: as

text: before

text: for

text: obtaining

text: an

text: image

text: with

text: reduced

text: resolution.

text: Table

text: III

text: shows

text: the

text: estimates

text: of

text: the

text: α

text: parameter

text: for

text: each

text: region

text: and

text: the

text: sample

text: size,

text: for

text: both

text: resolutions.

text: It

text: can

text: be

text: seen

text: that

text: the

text: estimates

text: agree

text: with

text: the

text: visual

text: interpretation

text: of

text: the

text: classes.

text: Table

text: IV

text: presents

text: the

text: GD

text: and

text: the

text: TD

text: between

text: regions.

text: This

text: example

text: highlights

text: the

text: difﬁculty

text: TD

text: has

text: to

text: distinguish

text: regions

text: with

text: a

text: similar

text: roughness.

text: It

text: can

text: be

text: seen

text: that

text: the

text: TD

text: between

text: and

text: and

text: between

text: and

text: are

text: almost

text: the

text: same

text: in

text: both

text: resolutions.

text: These

text: experiments

text: show

text: that

text: the

text: methodology

text: is

text: not

text: affected

text: if

text: the

text: image

text: has

text: reduced

text: resolution.

text: Furthermore,

text: as

text: the

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


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text: on

text: May

label: reference

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text: et

text: al.

text: Collaboration],

text: “Results

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text: the

text: production

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text: with

text: —S—

text: and

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text: in

text: S

text: W

text: collisions

text: at

text: per

text: nucleon,”

text: Nucl.

text: Phys.

text: A

text: J.

text: B.

text: Kinson

text: et

text: al.

text: Collaboration],

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text: in

text: sulphur-sulphur

text: interactions

text: at

text: per

text: nucleon,”

text: Nucl.

text: Phys.

text: A

text: U.

text: W.

text: Heinz

text: and

text: M.

text: Jacob,

text: “Evidence

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text: a

text: new

text: state

text: of

text: matter:

text: An

text: Assess-

text: ment

text: of

text: the

text: results

text: from

text: the

text: CERN

text: lead

text: beam

text: program,”

text: J.

text: Rafelski

text: and

text: J.

text: Letessier,

text: “Hadrons

text: from

text: Pb

text: Pb

text: collisions

text: at

text: Rejected

text: by

text: PRL

text: on

text: Advice

text: of

text: Divisional

text: Associate

text: Editor

text: U.

text: Heinz

text: nucl-

text: I.

text: Arsene

text: et

text: al.

text: [BRAHMS

text: Collaboration],

text: “Quark

text: gluon

text: plasma

text: and

text: color

text: glass

text: condensate

text: at

text: RHIC?

text: The

text: Perspective

text: from

text: the

text: BRAHMS

text: experiment,”

text: Nucl.

text: Phys.

text: A

text: K.

text: Adcox

text: et

text: al.

text: [PHENIX

text: Collaboration],

text: “Formation

text: of

text: dense

text: partonic

text: mat-

text: ter

text: in

text: relativistic

text: nucleus-nucleus

text: collisions

text: at

text: RHIC:

text: Experimental

text: evalua-

text: tion

text: by

text: the

text: PHENIX

text: collaboration,”

text: Nucl.

text: Phys.

text: A

text: [nucl-

text: B.

text: B.

text: Back

text: et

text: al.

text: [PHOBOS

text: Collaboration],

text: “The

text: PHOBOS

text: perspective

text: on

text: discoveries

text: at

text: RHIC,”

text: Nucl.

text: Phys.

text: A

text: J.

text: Adams

text: et

text: al.

text: [STAR

text: Collaboration],

text: “Experimental

text: and

text: theoretical

text: chal-

text: lenges

text: in

text: the

text: search

text: for

text: the

text: quark

text: gluon

text: plasma:

text: The

text: STAR

text: Collaboration’s

text: critical

text: assessment

text: of

text: the

text: evidence

text: from

text: RHIC

text: collisions,”

text: Nucl.

text: Phys.

text: A

text: B.

text: B.

text: Abelev

text: et

text: al.

text: [ALICE

text: Collaborat√ion],

text: “Multi-strange

text: baryon

text: production

text: at

text: mid-rapidity

text: in

text: Pb-Pb

text: collisions

text: at

text: s

text: TeV,”

text: Phys.

text: Lett.

text: B

text: NN

text: [Corrigendum:

text: Phys.

text: Lett.

text: B

text: J.

text: Rafelski

text: and

text: J.

text: Letessier,

text: “Importance

text: of

text: reaction

text: volume

text: in

text: hadronic

text: colli-

text: sions:

text: Canonical

text: enhancement,”

text: J.

text: Phys.

text: G

text: M.

text: Petran

text: and

text: J.

text: Rafelski,

text: “Multistrange

text: Particle

text: Production

text: and

text: the

text: Statisti-

text: cal

text: Hadronization

text: Model,”

text: Phys.

text: Rev.

text: C

text: J.

text: Adam

text: et

text: al.

text: [ALIC√E

text: Collaboration],

text: “Multi-strange

text: baryon

text: production

text: in

text: p-Pb

text: collisions

text: at

text: s

text: TeV,”

text: Phys.

text: Lett.

text: B

text: NN

text: M.

text: Gazdzicki,

text: M.

text: Gorenstein

text: and

text: P.

text: Seyboth,

text: “Onset

text: of

text: deconﬁnement

text: in

text: nucleus-nucleus

text: collisions:

text: Review

text: for

text: pedestrians

text: and

text: experts,”

text: Acta

text: Phys.

text: Polon.

text: B

text: J.

text: Letessier

text: and

text: J.

text: Rafelski,

text: “Hadron

text: production

text: and

text: phase

text: changes

text: in

text: rela-

text: tivistic

text: heavy-ion

text: collisions,”

text: Eur.

text: Phys.

text: J.

text: A

text: J.

text: Rafelski

text: and

text: J.

text: Letessier,

text: “Particle

text: Production

text: and

text: Deconﬁnement

text: Thresh-

text: old,”

text: PoS

text: CONFINEMENT

text: G.

text: Torrieri,

text: S.

text: Steinke,

text: W.

text: Broniowski,

text: W.

text: Florkowski,

text: J.

text: Letessier

text: and

text: J.

text: Rafelski,

text: “SHARE:

text: Statistical

text: hadronization

text: with

text: resonances,”

text: Comput.

text: Phys.

text: Commun.

text: S.

text: Wheaton,

text: J.

text: Cleymans

text: and

text: M.

text: Hauer,

text: “Thermus,”

text: ##LTLine##

text: ##LTLine##


text: 4 1609CPODWroclaw˙Marek˙shortJR printed on May 29, 2020 The work of Marek that followed supports the conjecture that he spear- headed the Dubna group interest and insight into strangeness. Following his Dubna debut he was able to join the CERN-NA35 collaboration where he developed the strangeness signature of QGP. NA35 was a scion of the LBNL-GSI BEVELAC experimental eﬀort with LBNL’s Howell Pugh being the main administrative and intellectual force for strangeness, as my per- sonal correspondence from this period shows. Howell was a key member of both the NA35 and the NA36 experiments; however, NA36 had instrumen- tal diﬃculties. While initially the objective of NA35 was the exploration of equations of state of dense nuclear matter, which was a direct contin- uation of the eﬀort carried out at LBNL-BEVALAC, Marek’s arrival, and the fade-out of NA36 presented the NA35 experimental program with the opportunity to enter forcefully into a novel domain of heavy-ion physics, the strangeness signature of QGP. But as the following events show, not every- one in the collaboration was ready to move in this direction immediately. Writing in year 2000 [10], Graz˙yna Odyniec of LBNL largely corrobo- rates my memory and the contents of my correspondence: “From the very beginning Howell (Pugh), with ﬁrmness and clarity, advocated the study of strange baryon and antibaryon production. He played a leading role in launching two of the major CERN heavy-ion experiments: NA35 and NA36, the latter being exclusively dedicated to measurements of hyperons. Strangeness enhancement predicted by theorists was discovered by NA35 and reported at the Quark Matter Conference in 1988.” I have reviewed the proceedings of this meeting held in September 1988 and while Marek [11] presents NA35 strangeness Λ and K0 enhancement results in S-S collisions, everything shown in the proceedings of the meeting published July 1989 is overprinted with the warning, ‘preliminary’. These results are presented in their extended and ﬁnal format two years later [12] (submitted 17 April 1990; in revised form 2 July 1990). As Marek remarked in a personal conversation, the title of this paper should have been “Strangeness Enhancement as the Evidence of QGP,” but both the collabo- ration dynamics (I count 67 authors on this manuscript, a large number as measured by the norm of the time period) and the review process prevented such a claim. Thus instead the manuscript has the non-telling title, “Neutral strange particle production in sulphur-sulphur and proton-sulphur collisions at 200 GeV/nucleon.” The abstract states, “Signiﬁcant enhancement of the multiplicities of all observed strange particles relative to negative hadrons was observed in central S-S collisions, as compared to p + p and p+S col- lisions.” In the concluding section, buried in a lot of ink, one ﬁnds, “Thus our observation . . . appears to be consistent with a dynamical evolution that passes through a deconﬁnement stage.” Had the NA35 collaboration ended the concluding section here one could rightly see them agreeing with Marek and claiming discovery of QGP as well. Reading on in Ref. [12] we see: “However (according to) [11, 14]

text: printed

text: on

text: May

text: The

text: work

text: of

text: Marek

text: that

text: followed

text: supports

text: the

text: conjecture

text: that

text: he

text: spear-

text: headed

text: the

text: Dubna

text: group

text: interest

text: and

text: insight

text: into

text: strangeness.

text: Following

text: his

text: Dubna

text: debut

text: he

text: was

text: able

text: to

text: join

text: the

text: collaboration

text: where

text: he

text: developed

text: the

text: strangeness

text: signature

text: of

text: QGP.

text: was

text: a

text: scion

text: of

text: the

text: LBNL-GSI

text: BEVELAC

text: experimental

text: eﬀort

text: with

text: LBNL’s

text: Howell

text: Pugh

text: being

text: the

text: main

text: administrative

text: and

text: intellectual

text: force

text: for

text: strangeness,

text: as

text: my

text: per-

text: sonal

text: correspondence

text: from

text: this

text: period

text: shows.

text: Howell

text: was

text: a

text: key

text: member

text: of

text: both

text: the

text: and

text: the

text: experiments;

text: however,

text: had

text: instrumen-

text: tal

text: diﬃculties.

text: While

text: initially

text: the

text: objective

text: of

text: was

text: the

text: exploration

text: of

text: equations

text: of

text: state

text: of

text: dense

text: nuclear

text: matter,

text: which

text: was

text: a

text: direct

text: contin-

text: uation

text: of

text: the

text: eﬀort

text: carried

text: out

text: at

text: LBNL-BEVALAC,

text: Marek’s

text: arrival,

text: and

text: the

text: fade-out

text: of

text: presented

text: the

text: experimental

text: program

text: with

text: the

text: opportunity

text: to

text: enter

text: forcefully

text: into

text: a

text: novel

text: domain

text: of

text: heavy-ion

text: physics,

text: the

text: strangeness

text: signature

text: of

text: QGP.

text: But

text: as

text: the

text: following

text: events

text: show,

text: not

text: every-

text: one

text: in

text: the

text: collaboration

text: was

text: ready

text: to

text: move

text: in

text: this

text: direction

text: immediately.

text: Writing

text: in

text: year

text: Graz˙yna

text: Odyniec

text: of

text: LBNL

text: largely

text: corrobo-

text: rates

text: my

text: memory

text: and

text: the

text: contents

text: of

text: my

text: correspondence:

text: “From

text: the

text: very

text: beginning

text: Howell

text: (Pugh),

text: with

text: ﬁrmness

text: and

text: clarity,

text: advocated

text: the

text: study

text: of

text: strange

text: baryon

text: and

text: antibaryon

text: production.

text: He

text: played

text: a

text: leading

text: role

text: in

text: launching

text: two

text: of

text: the

text: major

text: CERN

text: heavy-ion

text: experiments:

text: and

text: the

text: latter

text: being

text: exclusively

text: dedicated

text: to

text: measurements

text: of

text: hyperons.

text: Strangeness

text: enhancement

text: predicted

text: by

text: theorists

text: was

text: discovered

text: by

text: and

text: reported

text: at

text: the

text: Quark

text: Matter

text: Conference

text: in

text: I

text: have

text: reviewed

text: the

text: proceedings

text: of

text: this

text: meeting

text: held

text: in

text: September

text: and

text: while

text: Marek

text: presents

text: strangeness

text: Λ

text: and

text: enhancement

text: results

text: in

text: S-S

text: collisions,

text: everything

text: shown

text: in

text: the

text: proceedings

text: of

text: the

text: meeting

text: published

text: July

text: is

text: overprinted

text: with

text: the

text: warning,

text: ‘preliminary’.

text: These

text: results

text: are

text: presented

text: in

text: their

text: extended

text: and

text: ﬁnal

text: format

text: two

text: years

text: later

text: (submitted

text: April

text: in

text: revised

text: form

text: July

text: As

text: Marek

text: remarked

text: in

text: a

text: personal

text: conversation,

text: the

text: title

text: of

text: this

text: paper

text: should

text: have

text: been

text: “Strangeness

text: Enhancement

text: as

text: the

text: Evidence

text: of

text: QGP,”

text: but

text: both

text: the

text: collabo-

text: ration

text: dynamics

text: (I

text: count

text: authors

text: on

text: this

text: manuscript,

text: a

text: large

text: number

text: as

text: measured

text: by

text: the

text: norm

text: of

text: the

text: time

text: period)

text: and

text: the

text: review

text: process

text: prevented

text: such

text: a

text: claim.

text: Thus

text: instead

text: the

text: manuscript

text: has

text: the

text: non-telling

text: title,

text: “Neutral

text: strange

text: particle

text: production

text: in

text: sulphur-sulphur

text: and

text: proton-sulphur

text: collisions

text: at

text: GeV/nucleon.”

text: The

text: abstract

text: states,

text: “Signiﬁcant

text: enhancement

text: of

text: the

text: multiplicities

text: of

text: all

text: observed

text: strange

text: particles

text: relative

text: to

text: negative

text: hadrons

text: was

text: observed

text: in

text: central

text: S-S

text: collisions,

text: as

text: compared

text: to

text: p

text: p

text: and

text: p+S

text: col-

text: lisions.”

text: In

text: the

text: concluding

text: section,

text: buried

text: in

text: a

text: lot

text: of

text: ink,

text: one

text: ﬁnds,

text: “Thus

text: our

text: observation

text: appears

text: to

text: be

text: consistent

text: with

text: a

text: dynamical

text: evolution

text: that

text: passes

text: through

text: a

text: deconﬁnement

text: stage.”

text: Had

text: the

text: collaboration

text: ended

text: the

text: concluding

text: section

text: here

text: one

text: could

text: rightly

text: see

text: them

text: agreeing

text: with

text: Marek

text: and

text: claiming

text: discovery

text: of

text: QGP

text: as

text: well.

text: Reading

text: on

text: in

text: Ref.

text: we

text: see:

text: “However

text: (according

text: to)


text: In [1, 4, 11] this type of control was applied to stabilize small amplitude cycles born via a Hopf bifurcation, while in [17] analysis of the stability of large amplitude cycles was done by exploiting the additional S1 symmetry of Stewart-Landau oscillators. In this paper, we extend the odd-number limitation type results considered in [6] to treat the case when control of the form (5) is applied to a system with a ﬁnite group of symmetries (Section 2); and, the case when the standard Pyragas control such as in (2) is applied to a target cycle, which is not hyperbolic, because the system is S1-symmetric (Section 3). Analytical results are illustrated with examples. 2 Modiﬁed Pyragas control of systems with ﬁ- nite symmetry group 2.1 Necessary condition for stabilization ∗ Suppose that system (11) has a periodic solution x with period T. Assume that this system respects some group of symmetries, and for one particular g relation (4) holds. We denote by Φ(t) the fundamental matrix of the lineariza- tion ∗ y˙ = B(t)y, B(t) := Df (x (t)), (6) x ∗ of system (11) near x (t), where Df denotes the Jacobi matrix of f. Con- x dition (4) implies that − ∗ A 1Φ(T )ψ(0) = ψ(0), ψ(t) := x˙ (t), (7) g g − i.e. the matrix A 1Φ(T ) has an eigenvalue 1. We assume that g g − (H ) 1 is a simple eigenvalue for the matrix A 1Φ(T ). 1 g g Following [14], we introduce a modiﬁed Pyragas control as in (5), where we assume that KA = A K. (8) g g This commutativity property can be a natural restriction on feasible con- trols. For example, it is typical of laser systems. On the other hand, gain matrices, which are simple enough to allow for eﬃcient analysis of stability of the controlled equation (5), also usually satisfy condition (8) (cf. [11,16]). (cid:48) † Let D denote the transpose of a matrix D. Using (H ), denote by ψ the 1 − (cid:48) normalized adjoint eigenvector with the eigenvalue 1 for the matrix [A 1Φ(T )] : g g − (cid:48) † † † · [A 1Φ(T )] ψ = ψ ; ψ ψ(0) = 1, g g 0 0 0 3

text: In

text: this

text: type

text: of

text: control

text: was

text: applied

text: to

text: stabilize

text: small

text: amplitude

text: cycles

text: born

text: via

text: a

text: Hopf

text: bifurcation,

text: while

text: in

text: analysis

text: of

text: the

text: stability

text: of

text: large

text: amplitude

text: cycles

text: was

text: done

text: by

text: exploiting

text: the

text: additional

text: symmetry

text: of

text: Stewart-Landau

text: oscillators.

text: In

text: this

text: paper,

text: we

text: extend

text: the

text: odd-number

text: limitation

text: type

text: results

text: considered

text: in

text: to

text: treat

text: the

text: case

text: when

text: control

text: of

text: the

text: form

text: is

text: applied

text: to

text: a

text: system

text: with

text: a

text: ﬁnite

text: group

text: of

text: symmetries

text: (Section

text: and,

text: the

text: case

text: when

text: the

text: standard

text: Pyragas

text: control

text: such

text: as

text: in

text: is

text: applied

text: to

text: a

text: target

text: cycle,

text: which

text: is

text: not

text: hyperbolic,

text: because

text: the

text: system

text: is

text: (Section

text: Analytical

text: results

text: are

text: illustrated

text: with

text: examples.

text: Modiﬁed

text: Pyragas

text: control

text: of

text: systems

text: with

text: ﬁ-

text: nite

text: symmetry

text: group

text: Necessary

text: condition

text: for

text: stabilization

text: Suppose

text: that

text: system

text: has

text: a

text: periodic

text: solution

text: x

text: with

text: period

text: T.

text: Assume

text: that

text: this

text: system

text: respects

text: some

text: group

text: of

text: symmetries,

text: and

text: for

text: one

text: particular

text: g

text: relation

text: holds.

text: We

text: denote

text: by

text: Φ(t)

text: the

text: fundamental

text: matrix

text: of

text: the

text: lineariza-

text: tion

text: y˙

text: B(t)y,

text: B(t)

text: Df

text: (x

text: (t)),

text: x

text: of

text: system

text: near

text: x

text: (t),

text: where

text: Df

text: denotes

text: the

text: Jacobi

text: matrix

text: of

text: f.

text: Con-

text: x

text: dition

text: implies

text: that

text: A

text: ψ(t)

text: x˙

text: (t),

text: g

text: g

text: i.e.

text: the

text: matrix

text: A

text: has

text: an

text: eigenvalue

text: We

text: assume

text: that

text: g

text: g

text: (H

text: is

text: a

text: simple

text: eigenvalue

text: for

text: the

text: matrix

text: A

text: g

text: g

text: Following

text: we

text: introduce

text: a

text: modiﬁed

text: Pyragas

text: control

text: as

text: in

text: where

text: we

text: assume

text: that

text: KA

text: A

text: K.

text: g

text: g

text: This

text: commutativity

text: property

text: can

text: be

text: a

text: natural

text: restriction

text: on

text: feasible

text: con-

text: trols.

text: For

text: example,

text: it

text: is

text: typical

text: of

text: laser

text: systems.

text: On

text: the

text: other

text: hand,

text: gain

text: matrices,

text: which

text: are

text: simple

text: enough

text: to

text: allow

text: for

text: eﬃcient

text: analysis

text: of

text: stability

text: of

text: the

text: controlled

text: equation

text: also

text: usually

text: satisfy

text: condition

text: (cf.

text: Let

text: D

text: denote

text: the

text: transpose

text: of

text: a

text: matrix

text: D.

text: Using

text: (H

text: denote

text: by

text: ψ

text: the

text: normalized

text: adjoint

text: eigenvector

text: with

text: the

text: eigenvalue

text: for

text: the

text: matrix

text: [A

text: g

text: g

text: [A

text: ψ

text: ψ

text: ψ

text: g

text: g


text: | | − − − z˙ = (α + i + γ z 2)z + a(z z ) + b(z (t π/ω) z )), 1 1 1 2 1 2 1 | | − − − (13) z˙ = (α + i + γ z 2)z + a(z z ) + b(z (t π/ω) z )) 2 2 2 1 2 1 2 with a complex gain parameter b. To study stability of the anti-phase branch close to the bifurcation point in system (13), linear stability analysis of the origin combined with explicit knowledge of the branch made it possible to ﬁnd suﬃcient conditions under which for some interval of α suﬃciently close to 2a the branch is stable. Due to the simple nature of the Landau oscillator, explicit computation of the fundamental matrix of the linearization of system (11) near the solution (12) allows us to compute expression (10). This gives us the following necessary condition for stabilization of any of the anti-phase cycles (12): Re γ Re b + Im γ Im b 1 + π < 0. (14) ωRe γ At the point α = 2a, ω = 1, this coincides with formula (6.10) from [4] which, together with other conditions, deﬁnes the stability domain for small periodic orbits. In particular, for a ﬁxed gain parameter b, condition (14) provides an upper bound for the interval of α where the anti-phase cycle is stable. 2.3 Proof of Theorem 2.1 ∗ Linearizing system (5) near x gives − − y˙(t) = B(t)y(t) + K(A y(t T ) y(t)). (15) g g ∗ To prove that x is an unstable periodic solution of (5) we will show that system (15) has a solution ∗ − y (t) = µt/Tgp(t), A p(t T ) = p(t) (16) µ g g − with µ > 1, where the relation A p(t T ) = p(t) ensures that p is periodic. (cid:0) g g (cid:1) It is easy to see that if the ordinary diﬀerential system − − y˙ = B(t) + (µ 1 1)K y (17) has a solution y of type (16), then y is also a solution of (15). Denote by µ µ Ψ (t) the fundamental matrix of (17). µ − Lemma 2.2 If for some µ > 1 the matrix A 1Ψ (T ) has the eigenvalue µ, g µ g ∗ then system (17) has a solution of type (16) and hence the periodic solution x of (5) is unstable. 5 ##LTLine##

label: equation

text: z˙

text: (α

text: i

text: γ

text: z

text: a(z

text: z

text: b(z

text: (t

text: π/ω)

text: z

text: z˙

text: (α

text: i

text: γ

text: z

text: a(z

text: z

text: b(z

text: (t

text: π/ω)

text: z

text: with

text: a

text: complex

text: gain

text: parameter

text: b.

text: To

text: study

text: stability

text: of

text: the

text: anti-phase

text: branch

text: close

text: to

text: the

text: bifurcation

text: point

text: in

text: system

text: linear

text: stability

text: analysis

text: of

text: the

text: origin

text: combined

text: with

text: explicit

text: knowledge

text: of

text: the

text: branch

text: made

text: it

text: possible

text: to

text: ﬁnd

text: suﬃcient

text: conditions

text: under

text: which

text: for

text: some

text: interval

text: of

text: α

text: suﬃciently

text: close

text: to

text: the

text: branch

text: is

text: stable.

text: Due

text: to

text: the

text: simple

text: nature

text: of

text: the

text: Landau

text: oscillator,

text: explicit

text: computation

text: of

text: the

text: fundamental

text: matrix

text: of

text: the

text: linearization

text: of

text: system

text: near

text: the

text: solution

text: allows

text: us

text: to

text: compute

text: expression

text: This

text: gives

text: us

text: the

text: following

text: necessary

text: condition

text: for

text: stabilization

text: of

text: any

text: of

text: the

text: anti-phase

text: cycles

text: Re

text: γ

text: Re

text: b

text: Im

text: γ

text: Im

text: b

text: π

text: ωRe

text: γ

text: At

text: the

text: point

text: α

text: ω

text: this

text: coincides

text: with

text: formula

text: from

text: which,

text: together

text: with

text: other

text: conditions,

text: deﬁnes

text: the

text: stability

text: domain

text: for

text: small

text: periodic

text: orbits.

text: In

text: particular,

text: for

text: a

text: ﬁxed

text: gain

text: parameter

text: b,

text: condition

text: provides

text: an

text: upper

text: bound

text: for

text: the

text: interval

text: of

text: α

text: where

text: the

text: anti-phase

text: cycle

text: is

text: stable.

text: Proof

text: of

text: Theorem

text: Linearizing

text: system

text: near

text: x

text: gives

text: y˙(t)

text: B(t)y(t)

text: K(A

text: y(t

text: T

text: y(t)).

text: g

text: g

text: To

text: prove

text: that

text: x

text: is

text: an

text: unstable

text: periodic

text: solution

text: of

text: we

text: will

text: show

text: that

text: system

text: has

text: a

text: solution

text: y

text: (t)

text: µt/Tgp(t),

text: A

text: p(t

text: T

text: p(t)

text: µ

text: g

text: g

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text: Dynamic Deep Neural Networks: Optimizing Accuracy-Efﬁciency Trade-offs by Selective Execution Lanlan Liu Jia Deng llanlan@umich.edu jiadeng@umich.edu University of Michigan 2260 Hayward St, Ann Arbor, MI, 48105, USA Abstract network whose output is a decision that controls whether other modules can execute. Fig. 1 (left) illustrates a simple We introduce Dynamic Deep Neural Networks (D2NN), D2NN with one control module (Q) and two regular mod- a new type of feed-forward deep neural network that allows ules (N1, N2), where the controller Q outputs a binary de- selective execution. Given an input, only a subset of D2NN cision on whether module N2 executes. For certain inputs, neurons are executed, and the particular subset is deter- the controller may decide that N2 is unnecessary and in- mined by the D2NN itself. By pruning unnecessary com- stead execute a dummy node D to save on computation. As putation depending on input, D2NNs provide a way to im- an example application, this D2NN can be used for binary prove computational efﬁciency. To achieve dynamic selec- classiﬁcation of images, where some images can be rapidly tive execution, a D2NN augments a feed-forward deep neu- classiﬁed as negative after only a small amount of compu- ral network (directed acyclic graph of differentiable mod- tation. ules) with controller modules. Each controller module is D2NNs are motivated by the need for computational ef- a sub-network whose output is a decision that controls ﬁciency, in particular, by the need to deploy deep networks whether other modules can execute. A D2NN is trained end on mobile devices and data centers. Mobile devices are con- to end. Both regular and controller modules in a D2NN strained by energy and power, limiting the amount of com- are learnable and are jointly trained to optimize both ac- putation that can be executed. Data centers need energy curacy and efﬁciency. Such training is achieved by inte- efﬁciency to scale to higher throughput and to save operat- grating backpropagation with reinforcement learning. With ing cost. D2NNs provide a way to improve computational extensive experiments of various D2NN architectures on im- efﬁciency by selective execution, pruning unnecessary com- age classiﬁcation tasks, we demonstrate that D2NNs are putation depending on input. D2NNs also make it possible general and ﬂexible, and can effectively optimize accuracy- to use a bigger network under a computation budget by ex- efﬁciency trade-offs. ecuting only a subset of the neurons each time. A D2NN is trained end to end. That is, regular modules and control modules are jointly trained to optimize both ac- 1. Introduction curacy and efﬁciency. We achieve such training by integrat- ing backpropagation with reinforcement learning, necessi- This paper introduces Dynamic Deep Neural Networks tated by the non-differentiability of control modules. (D2NN), a new type of feed-forward deep neural network Compared to prior work that optimizes computational ef- (DNN) that allows selective execution. That is, given an ﬁciency in computer vision and machine learning, our work input, only a subset of neurons are executed, and the partic- is distinctive in four aspects: (1) the decisions on selective ular subset is determined by the network itself based on the execution are part of the network inference and are learned particular input. In other words, the amount of computa- end to end together with the rest of the network, as op- tion and computation sequence are dynamic based on input. posed to hand-designed or separately learned [23, 29, 2]; This is different from standard feed-forward networks that (2) D2NNs allow more ﬂexible network architectures and always execute the same computation sequence regardless execution sequences including parallel paths, as opposed to of input. architectures with less variance [12, 27]; (3) our D2NNs di- A D2NN is a feed-forward deep neural network (directed rectly optimize arbitrary efﬁciency metric that is deﬁned by acyclic graph of differentiable modules) augmented with the user, while previous work has no such ﬂexibility be- one or more control modules. A control module is a sub- cause they improve efﬁciency indirectly through sparsity 1

text: Dynamic

label: title

text: Deep

text: Neural

text: Networks:

text: Optimizing

text: Accuracy-Efﬁciency

text: Trade-offs

text: by

text: Selective

text: Execution

text: Lanlan

label: author

text: Liu

text: Jia

text: Deng

text: llanlan@umich.edu

font: IDHMQJ+NimbusMonL-Regu

text: jiadeng@umich.edu

text: University

text: of

text: Michigan

text: Hayward

text: St,

text: Ann

text: Arbor,

text: MI,

text: USA

text: Abstract

text: network

text: whose

text: output

text: is

text: a

text: decision

text: that

text: controls

text: whether

text: other

text: modules

text: can

text: execute.

text: Fig.

text: (left)

text: illustrates

text: a

text: simple

text: We

label: abstract

text: introduce

text: Dynamic

text: Deep

text: Neural

text: Networks

text: with

text: one

text: control

text: module

text: (Q)

text: and

text: two

text: regular

text: mod-

text: a

text: new

text: type

text: of

text: feed-forward

text: deep

text: neural

text: network

text: that

text: allows

text: ules

text: where

text: the

text: controller

text: Q

text: outputs

text: a

text: binary

text: de-

text: selective

text: execution.

text: Given

text: an

text: input,

text: only

text: a

text: subset

text: of

text: cision

text: on

text: whether

text: module

text: executes.

text: For

text: certain

text: inputs,

text: neurons

text: are

text: executed,

text: and

text: the

text: particular

text: subset

text: is

text: deter-

text: the

text: controller

text: may

text: decide

text: that

text: is

text: unnecessary

text: and

text: in-

text: mined

text: by

text: the

text: itself.

text: By

text: pruning

text: unnecessary

text: com-

text: stead

text: execute

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text: dummy

text: node

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text: to

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text: computation.

text: As

text: putation

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text: provide

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text: computational

text: efﬁciency.

text: To

text: achieve

text: dynamic

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text: classiﬁcation

text: of

text: images,

text: where

text: some

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text: be

text: rapidly

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text: execution,

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text: a

text: feed-forward

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text: neu-

text: classiﬁed

text: as

text: negative

text: after

text: only

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text: small

text: amount

text: of

text: compu-

text: ral

text: network

text: (directed

text: acyclic

text: graph

text: of

text: differentiable

text: mod-

text: tation.

text: ules)

text: with

text: controller

text: modules.

text: Each

text: controller

text: module

text: is

text: are

text: motivated

text: by

text: the

text: need

text: for

text: computational

text: ef-

text: a

text: sub-network

text: whose

text: output

text: is

text: a

text: decision

text: that

text: controls

text: ﬁciency,

text: in

text: particular,

text: by

text: the

text: need

text: to

text: deploy

text: deep

text: networks

text: whether

text: other

text: modules

text: can

text: execute.

text: A

text: is

text: trained

text: end

text: on

text: mobile

text: devices

text: and

text: data

text: centers.

text: Mobile

text: devices

text: are

text: con-

text: to

text: end.

text: Both

text: regular

text: and

text: controller

text: modules

text: in

text: a

text: strained

text: by

text: energy

text: and

text: power,

text: limiting

text: the

text: amount

text: of

text: com-

text: are

text: learnable

text: and

text: are

text: jointly

text: trained

text: to

text: optimize

text: both

text: ac-

text: putation

text: that

text: can

text: be

text: executed.

text: Data

text: centers

text: need

text: energy

text: curacy

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text: efﬁciency.

text: Such

text: training

text: is

text: achieved

text: by

text: inte-

text: efﬁciency

text: to

text: scale

text: to

text: higher

text: throughput

text: and

text: to

text: save

text: operat-

text: grating

text: backpropagation

text: with

text: reinforcement

text: learning.

text: With

text: ing

text: cost.

text: provide

text: a

text: way

text: to

text: improve

text: computational

text: extensive

text: experiments

text: of

text: various

text: architectures

text: on

text: im-

text: efﬁciency

text: by

text: selective

text: execution,

text: pruning

text: unnecessary

text: com-

text: age

text: classiﬁcation

text: tasks,

text: we

text: demonstrate

text: that

text: are

text: putation

text: depending

text: on

text: input.

text: also

text: make

text: it

text: possible

text: general

text: and

text: ﬂexible,

text: and

text: can

text: effectively

text: optimize

text: accuracy-

text: to

text: use

text: a

text: bigger

text: network

text: under

text: a

text: computation

text: budget

text: by

text: ex-

text: efﬁciency

text: trade-offs.

text: ecuting

text: only

text: a

text: subset

text: of

text: the

text: neurons

text: each

text: time.

text: A

text: is

text: trained

text: end

text: to

text: end.

text: That

text: is,

text: regular

text: modules

text: and

text: control

text: modules

text: are

text: jointly

text: trained

text: to

text: optimize

text: both

text: ac-

text: Introduction

text: curacy

text: and

text: efﬁciency.

text: We

text: achieve

text: such

text: training

text: by

text: integrat-

text: ing

text: backpropagation

text: with

text: reinforcement

text: learning,

text: necessi-

text: This

text: paper

text: introduces

text: Dynamic

text: Deep

text: Neural

text: Networks

text: tated

text: by

text: the

text: non-differentiability

text: of

text: control

text: modules.

text: a

text: new

text: type

text: of

text: feed-forward

text: deep

text: neural

text: network

text: Compared

text: to

text: prior

text: work

text: that

text: optimizes

text: computational

text: ef-

text: (DNN)

text: that

text: allows

text: selective

text: execution.

text: That

text: is,

text: given

text: an

text: ﬁciency

text: in

text: computer

text: vision

text: and

text: machine

text: learning,

text: our

text: work

text: input,

text: only

text: a

text: subset

text: of

text: neurons

text: are

text: executed,

text: and

text: the

text: partic-

text: is

text: distinctive

text: in

text: four

text: aspects:

text: the

text: decisions

text: on

text: selective

text: ular

text: subset

text: is

text: determined

text: by

text: the

text: network

text: itself

text: based

text: on

text: the

text: execution

text: are

text: part

text: of

text: the

text: network

text: inference

text: and

text: are

text: learned

text: particular

text: input.

text: In

text: other

text: words,

text: the

text: amount

text: of

text: computa-

text: end

text: to

text: end

text: together

text: with

text: the

text: rest

text: of

text: the

text: network,

text: as

text: op-

text: tion

text: and

text: computation

text: sequence

text: are

text: dynamic

text: based

text: on

text: input.

text: posed

text: to

text: hand-designed

text: or

text: separately

text: learned

text: This

text: is

text: different

text: from

text: standard

text: feed-forward

text: networks

text: that

text: allow

text: more

text: ﬂexible

text: network

text: architectures

text: and

text: always

text: execute

text: the

text: same

text: computation

text: sequence

text: regardless

text: execution

text: sequences

text: including

text: parallel

text: paths,

text: as

text: opposed

text: to

text: of

text: input.

text: architectures

text: with

text: less

text: variance

text: our

text: di-

text: A

text: is

text: a

text: feed-forward

text: deep

text: neural

text: network

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text: optimize

text: arbitrary

text: efﬁciency

text: metric

text: that

text: is

text: deﬁned

text: by

text: acyclic

text: graph

text: of

text: differentiable

text: modules)

text: augmented

text: with

text: the

text: user,

text: while

text: previous

text: work

text: has

text: no

text: such

text: ﬂexibility

text: be-

text: one

text: or

text: more

text: control

text: modules.

text: A

text: control

text: module

text: is

text: a

text: sub-

text: cause

text: they

text: improve

text: efﬁciency

text: indirectly

text: through

text: sparsity


text: • × After the preprocessing, the training step is similar to Node N1: a convolutional layer with a 3 3 ﬁlter size, × a regular DNN: a forward pass and a backward pass. All 2 ﬁlters and a stride of 2, followed by a 3 3 max- nodes are visited according to a topological ordering in a pooling layer with a stride of 2. forward pass and the reverse ordering in a backward pass. • × For each function node, the forward pass has three steps: Node N2: a convolutional layer with a 3 3 ﬁlter size × fetch inputs, forward inside the node, and send data or con- and 8 ﬁlters, followed by a 3 3 max-pooling layer trol signals to children nodes. When dealing with multiple with a stride of 2. data inputs and multiple control signals, the D2NN will ﬁl- • × Node N3: a convolutional layer with a 3 3 ﬁlter size ter examples with more than one null inputs or all negative and 32 ﬁlters. control signals. When a default value has been set for a node, all examples have to send out data. If the node is not • × Node N4: two convolutional layers with both 3 3 ﬁl- activated for a particular example, the output will take the ter sizes and 32, 64 ﬁlters repectively, augmented by a default value. A backward pass has similar logic: fetch gra- × 3 3 max-pooling layer with a stride of 2. The output dients from children, perform the backward pass inside and is reshaped and fed into a fully connected layer with send out gradients to parents. It is worth noting that when a 512 neurons followed by another fully connected layer default value is used in a node, the gradients can be blocked with the 2-class output. by this node because it is not actually executed. B. ILSVRC-10 Semantic Hierarchy • × Node Q1: three 3 3 max-pooling layers with strides of 2. The output is reshaped and fed into a fully con- The ILSVRC-10 dataset is a subset of the ILSVRC-65 nected layer with the 2-action output. dataset [9]. In our ILSVRC-10, there are 10 classes or- ganized into a 3-layer hierarchy: 2 superclasses, 5 coarse • × Node Q2: two 3 3 max-pooling layers with strides of classes and 10 leaf classes as in Fig 7. Each class has 500 × 2, followed by a convolutional layer with a 1 1 ﬁlter training images, 50 validation images, and 150 test images. size and 2 ﬁlters. The output is reshaped and fed into a C. Conﬁgurations fully connected layer with the 2-action output. • × High-Low Capacity D2NN The high-low capacity D2NN Node Q3: a 3 3 max-pooling layer with a stride of 2. consists of a single control node (Q) and three regular nodes The output is reshaped and fed into a fully connected (N1,N2,N3) as illustrated in Fig. 3a). layer with the 2-action output. • × Node N1: a convolutional layer with a 3 3 ﬁlter size, × 8 ﬁlters and a stride of 2, followed by a 3 3 max- Chain D2NN The Chain D2NN is shaped as a chain, where pooling layer with a stride of 2. each link consists of a control node selecting between two regular nodes. In the experiments of LFW-B dataset, we use • × Node N2: a convolutional layer with a 3 3 ﬁlter size a 3-stage Chain D2NN as in Fig. 3c). × and 16 ﬁlters, followed by a 3 3 max-pooling layer with a stride of 2. The output is reshaped and fed into • Node N1: a convolutional layer with a 3×3 ﬁlter size, a fully connected layer with 512 neurons followed by 2 ﬁlters and a stride of 2, followed by a 3×3 max- another fully connected layer with the 2-class output. pooling layer with a stride of 2. • × • × Node N3: three 3 3 max-pooling layers, each with a Node N2: a convolutional layer with a 1 1 ﬁlter size stride of 2, followed by two fully connected layers with and 16 ﬁlters. 32 neurons and the 2-class output. • × Node N3: a convolutional layer with a 3 3 ﬁlter size • × Node Q1: a convolutional layer with a 3 3 ﬁlter size and 16 ﬁlters. × and 2 ﬁlters, followed by a 3 3 max-pooling layer • × with a stride of 2. The output is reshaped and fed into Node N4: a 3 3 max-pooling layer with a stride of 2. a fully connected layer with 128 neurons followed by • × Node N5: a convolutional layer with a 1 1 ﬁlter size another fully connected layer with the 2-action output. and 32 ﬁlters. • × Node N6: two convolutional layers with both 3 3 ﬁl- Cascade D2NN The cascade D2NN consists of a sequence ter sizes and 32, 32 ﬁlters repectively. of four regular nodes (N1 to N4) and three control nodes • × (Q1-Q3) as in Fig. 3b). Node N7: a 3 3 max-pooling layer with a stride of 2.

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text: node

text: (Q)

text: and

text: three

text: regular

text: nodes

text: The

text: output

text: is

text: reshaped

text: and

text: fed

text: into

text: a

text: fully

text: connected

text: as

text: illustrated

text: in

text: Fig.

text: layer

text: with

text: the

text: output.

text: Node

text: a

text: convolutional

text: layer

text: with

text: a

text: ﬁlter

text: size,

text: ﬁlters

text: and

text: a

text: stride

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text: followed

text: by

text: a

text: max-

text: Chain

text: The

text: Chain

text: is

text: shaped

text: as

text: a

text: chain,

text: where

text: pooling

text: layer

text: with

text: a

text: stride

text: of

text: each

text: link

text: consists

text: of

text: a

text: control

text: node

text: selecting

text: between

text: two

text: regular

text: nodes.

text: In

text: the

text: experiments

text: of

text: LFW-B

text: dataset,

text: we

text: use

text: Node

text: a

text: convolutional

text: layer

text: with

text: a

text: ﬁlter

text: size

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text: Chain

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text: in

text: Fig.

text: and

text: ﬁlters,

text: followed

text: by

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text: max-pooling

text: layer

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text: stride

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text: The

text: output

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text: reshaped

text: and

text: fed

text: into

text: Node

text: a

text: convolutional

text: layer

text: with

text: a

text: ﬁlter

text: size,

text: a

text: fully

text: connected

text: layer

text: with

text: neurons

text: followed

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text: ﬁlters

text: and

text: a

text: stride

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text: followed

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text: a

text: max-

text: another

text: fully

text: connected

text: layer

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text: the

text: output.

text: pooling

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text: with

text: a

text: stride

text: of

text: Node

text: three

text: max-pooling

text: layers,

text: each

text: with

text: a

text: Node

text: a

text: convolutional

text: layer

text: with

text: a

text: ﬁlter

text: size

text: stride

text: of

text: followed

text: by

text: two

text: fully

text: connected

text: layers

text: with

text: and

text: ﬁlters.

text: neurons

text: and

text: the

text: output.

text: Node

text: a

text: convolutional

text: layer

text: with

text: a

text: ﬁlter

text: size

text: Node

text: a

text: convolutional

text: layer

text: with

text: a

text: ﬁlter

text: size

text: and

text: ﬁlters.

text: and

text: ﬁlters,

text: followed

text: by

text: a

text: max-pooling

text: layer

text: with

text: a

text: stride

text: of

text: The

text: output

text: is

text: reshaped

text: and

text: fed

text: into

text: Node

text: a

text: max-pooling

text: layer

text: with

text: a

text: stride

text: of

text: a

text: fully

text: connected

text: layer

text: with

text: neurons

text: followed

text: by

text: Node

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text: convolutional

text: layer

text: with

text: a

text: ﬁlter

text: size

text: another

text: fully

text: connected

text: layer

text: with

text: the

text: output.

text: and

text: ﬁlters.

text: Node

text: two

text: convolutional

text: layers

text: with

text: both

text: ﬁl-

text: Cascade

text: The

text: cascade

text: consists

text: of

text: a

text: sequence

text: ter

text: sizes

text: and

text: ﬁlters

text: repectively.

text: of

text: four

text: regular

text: nodes

text: to

text: and

text: three

text: control

text: nodes

text: as

text: in

text: Fig.

text: Node

text: a

text: max-pooling

text: layer

text: with

text: a

text: stride

text: of


text: 3 FIG. 5. Circuit schematic of the diode assembly showing the 16 kΩ resistor that reduces current ﬂow during arcing in order to protect the machine (cid:13) FIG. 4. COMSOL MultiphysicsR electron path simulation of weakened magnetic ﬁeld, the electric ﬁeld dominates and the electrons accelerate to the vacuum chamber wall. Experimental tests conﬁrmed the electrons were colliding with the vacuum chamber wall exactly in the location shown by the simulation. Further analyzing the simulation shows that when the electrons spiral back toward the target, they get close to the surface but do not collide. The magnetic force a particle feels is proportional to its velocity. The electron slows down as it spirals back toward the surface due to (a) (b) the electric ﬁeld, causing the magnetic force to weaken. FIG. 6. Electrostatic shroud: (a) assembled, (b) exploded Ultimately, this is a consequence of the fact that as mag- view netic ﬁelds can only change the path of the electron and do not add energy, it is diﬃcult for the electrons to over- come the electric ﬁeld force and recombine with the tar- zener diode chain was made as high as 2400 V. get. A. Electric Field Simulation Results IV. ELECTROSTATIC SHROUD Figure 7 shows the simulation of the electric ﬁeld with An electrostatic shroud is a device that suppresses the installation of the shroud design shown in Figure 6. back-streaming electrons by introducing an opposing In the simulation, the shroud voltage is -102 kV, while electric ﬁeld near the target surface. This is done by the target is at -100 kV. Installation of the shroud re- installing a shroud around the target that is kept at a duced the distance between the ground and high voltage, more negative potential. Locally, the resulting electric but by designing the shroud to have a large radius of ﬁeld is directed away from the target surface, causing curvature everywhere, and installing insulating spacers electrons to be repelled from the shroud and return to to move the ion sources back, it was possible to limit the the target. electric ﬁeld below the 80 kV/cm electrical breakdown The shroud design used for the HFNG is shown in Fig- limit determined earlier. The maximum electric ﬁeld is ure 6. The clamshell design allows for easy removal. approximately 47 kV/cm near the top of the shroud. Exchangeable face-plates allow for the ability to easily The minimum voltage diﬀerential between the target change the beam entrance window geometry. The volt- and shroud required to suppress electrons was deter- age diﬀerence between the shroud and target is created mined by analyzing the electric ﬁeld within the shroud. through the use of multiple zener diodes in series, as Figure 8 shows the magnitude of the electric ﬁeld along shown in Figure 5. The total breakdown voltage of the the direction of the deuteron beam on the central plane ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: FIG.

text: Circuit

text: schematic

text: of

text: the

text: diode

text: assembly

text: showing

text: the

text: kΩ

text: resistor

text: that

text: reduces

text: current

text: ﬂow

text: during

text: arcing

text: in

text: order

text: to

text: protect

text: the

text: machine

text: FIG.

text: COMSOL

text: MultiphysicsR

text: electron

text: path

text: simulation

text: of

text: weakened

text: magnetic

text: ﬁeld,

text: the

text: electric

text: ﬁeld

text: dominates

text: and

text: the

text: electrons

text: accelerate

text: to

text: the

text: vacuum

text: chamber

text: wall.

text: Experimental

text: tests

text: conﬁrmed

text: the

text: electrons

text: were

text: colliding

text: with

text: the

text: vacuum

text: chamber

text: wall

text: exactly

text: in

text: the

text: location

text: shown

text: by

text: the

text: simulation.

text: Further

text: analyzing

text: the

text: simulation

text: shows

text: that

text: when

text: the

text: electrons

text: spiral

text: back

text: toward

text: the

text: target,

text: they

text: get

text: close

text: to

text: the

text: surface

text: but

text: do

text: not

text: collide.

text: The

text: magnetic

text: force

text: a

text: particle

text: feels

text: is

text: proportional

text: to

text: its

text: velocity.

text: The

text: electron

text: slows

text: down

text: as

text: it

text: spirals

text: back

text: toward

text: the

text: surface

text: due

text: to

text: (a)

text: (b)

text: the

text: electric

text: ﬁeld,

text: causing

text: the

text: magnetic

text: force

text: to

text: weaken.

text: FIG.

text: Electrostatic

text: shroud:

text: (a)

text: assembled,

text: (b)

text: exploded

text: Ultimately,

text: this

text: is

text: a

text: consequence

text: of

text: the

text: fact

text: that

text: as

text: mag-

text: view

text: netic

text: ﬁelds

text: can

text: only

text: change

text: the

text: path

text: of

text: the

text: electron

text: and

text: do

text: not

text: add

text: energy,

text: it

text: is

text: diﬃcult

text: for

text: the

text: electrons

text: to

text: over-

text: come

text: the

text: electric

text: ﬁeld

text: force

text: and

text: recombine

text: with

text: the

text: tar-

text: zener

text: diode

text: chain

text: was

text: made

text: as

text: high

text: as

text: V.

text: get.

text: A.

text: Electric

text: Field

text: Simulation

text: Results

text: IV.

text: ELECTROSTATIC

text: SHROUD

text: Figure

text: shows

text: the

text: simulation

text: of

text: the

text: electric

text: ﬁeld

text: with

text: An

text: electrostatic

text: shroud

text: is

text: a

text: device

text: that

text: suppresses

text: the

text: installation

text: of

text: the

text: shroud

text: design

text: shown

text: in

text: Figure

text: back-streaming

text: electrons

text: by

text: introducing

text: an

text: opposing

text: In

text: the

text: simulation,

text: the

text: shroud

text: voltage

text: is

text: kV,

text: while

text: electric

text: ﬁeld

text: near

text: the

text: target

text: surface.

text: This

text: is

text: done

text: by

text: the

text: target

text: is

text: at

text: kV.

text: Installation

text: of

text: the

text: shroud

text: re-

text: installing

text: a

text: shroud

text: around

text: the

text: target

text: that

text: is

text: kept

text: at

text: a

text: duced

text: the

text: distance

text: between

text: the

text: ground

text: and

text: high

text: voltage,

text: more

text: negative

text: potential.

text: Locally,

text: the

text: resulting

text: electric

text: but

text: by

text: designing

text: the

text: shroud

text: to

text: have

text: a

text: large

text: radius

text: of

text: ﬁeld

text: is

text: directed

text: away

text: from

text: the

text: target

text: surface,

text: causing

text: curvature

text: everywhere,

text: and

text: installing

text: insulating

text: spacers

text: electrons

text: to

text: be

text: repelled

text: from

text: the

text: shroud

text: and

text: return

text: to

text: to

text: move

text: the

text: ion

text: sources

text: back,

text: it

text: was

text: possible

text: to

text: limit

text: the

text: the

text: target.

text: electric

text: ﬁeld

text: below

text: the

text: kV/cm

text: electrical

text: breakdown

text: The

text: shroud

text: design

text: used

text: for

text: the

text: HFNG

text: is

text: shown

text: in

text: Fig-

text: limit

text: determined

text: earlier.

text: The

text: maximum

text: electric

text: ﬁeld

text: is

text: ure

text: The

text: clamshell

text: design

text: allows

text: for

text: easy

text: removal.

text: approximately

text: kV/cm

text: near

text: the

text: top

text: of

text: the

text: shroud.

text: Exchangeable

text: face-plates

text: allow

text: for

text: the

text: ability

text: to

text: easily

text: The

text: minimum

text: voltage

text: diﬀerential

text: between

text: the

text: target

text: change

text: the

text: beam

text: entrance

text: window

text: geometry.

text: The

text: volt-

text: and

text: shroud

text: required

text: to

text: suppress

text: electrons

text: was

text: deter-

text: age

text: diﬀerence

text: between

text: the

text: shroud

text: and

text: target

text: is

text: created

text: mined

text: by

text: analyzing

text: the

text: electric

text: ﬁeld

text: within

text: the

text: shroud.

text: through

text: the

text: use

text: of

text: multiple

text: zener

text: diodes

text: in

text: series,

text: as

text: Figure

text: shows

text: the

text: magnitude

text: of

text: the

text: electric

text: ﬁeld

text: along

text: shown

text: in

text: Figure

text: The

text: total

text: breakdown

text: voltage

text: of

text: the

text: the

text: direction

text: of

text: the

text: deuteron

text: beam

text: on

text: the

text: central

text: plane

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##


text: 4 (a) FIG. 7. Electric ﬁeld simulation outside shroud with target potential at -100 kV and shroud potential at -102 kV within the shroud (same cross section plane shown in Figure 1). Red areas indicate regions where E > 0 and z blue areas indicate regions where E < 0 (z is the direc- z tion along the deuteron beam). For a deuteron extracted from the top of Figure 7, the electric ﬁeld will accelerate (b) the deuteron downwards toward the target. When the deuteron enters the shroud and reaches the intersection of the red and blue region, it will have an energy of ap- proximately 102 keV. The red region will then decelerate the deuteron to 100 keV. When the deuteron strikes the target surface and ejects electrons, the electrons will feel the electric force pushing it back toward the target. When designing the shroud geometry, one must ensure that the red region in the top of Figure 8 does not touch the target (as it does in Figure 8a), in order to prevent an electron from being ejected into an electric ﬁeld that would accelerate the particle toward the vacuum cham- (c) ber wall. Factors that aﬀect the electric ﬁeld within the shroud include the voltage diﬀerential and the distance FIG. 8. Electric ﬁeld direction near shroud window at (a) 400 between the shroud and target, as well as the window V, (b) 800 V, (c) and 1400 V voltage diﬀerential. Red indi- size. cates ﬁeld along the direction of the deuteron beam (shown by B. Simulation of Electron Trajectories Using arrow), blue indicates direction opposite the deuteron beam. Electrostatic Shroud T and S denote the target and shroud, respectively. A simulation of the secondary electrons released into ber. After implementing the shroud, no visual sign of 2π from a 1.3 mA deuterium beam striking the ﬂat tar- chamber heating could be seen during machine opera- get surface is shown in Figure 9. Assuming 1.2 electrons tion due to back-streaming electrons. Also, the dose of released per deuteron ion striking the target, this results bremsstrahlung X-rays detected by a Geiger counter in in 1.56 mA of electrons. As expected and described pre- the vault decreased by a factor of more than 20 compared viously due to the electric ﬁeld shown in Figure 8a, for to the magnetic suppression technique. a 400 V voltage diﬀerential between the shroud and the target, secondary electrons are not suppressed and leave through the shroud window in a beam that eventually V. CONCLUSIONS strikes the chamber with an energy of 100 keV. A shroud voltage diﬀerential of 800 V results in the electrons re- Two main methods of electron suppression were tested: turning to the surface of the shroud, as seen in Figure using permanent magnets to bend electrons back to the 9b. The maximum energy the electron receives is equal target, and implementing an electrostatic shroud creating to the energy at release. a suppressing electric ﬁeld. Simulations of the permanent Testing of the shroud at 800 V voltage diﬀerential or magnet design showed that even with magnetic ﬁelds sur- greater on the HFNG has shown a large reduction in the passing 1000 gauss, electrons would migrate along the × amount of secondary electrons hitting the vacuum cham- target due to the E B force without coming into con- ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: (a)

text: FIG.

text: Electric

text: ﬁeld

text: simulation

text: outside

text: shroud

text: with

text: target

text: potential

text: at

text: kV

text: and

text: shroud

text: potential

text: at

text: kV

text: within

text: the

text: shroud

text: (same

text: cross

text: section

text: plane

text: shown

text: in

text: Figure

text: Red

text: areas

text: indicate

text: regions

text: where

text: E

text: and

text: z

text: blue

text: areas

text: indicate

text: regions

text: where

text: E

text: (z

text: is

text: the

text: direc-

text: z

text: tion

text: along

text: the

text: deuteron

text: beam).

text: For

text: a

text: deuteron

text: extracted

text: from

text: the

text: top

text: of

text: Figure

text: the

text: electric

text: ﬁeld

text: will

text: accelerate

text: (b)

text: the

text: deuteron

text: downwards

text: toward

text: the

text: target.

text: When

text: the

text: deuteron

text: enters

text: the

text: shroud

text: and

text: reaches

text: the

text: intersection

text: of

text: the

text: red

text: and

text: blue

text: region,

text: it

text: will

text: have

text: an

text: energy

text: of

text: ap-

text: proximately

text: keV.

text: The

text: red

text: region

text: will

text: then

text: decelerate

text: the

text: deuteron

text: to

text: keV.

text: When

text: the

text: deuteron

text: strikes

text: the

text: target

text: surface

text: and

text: ejects

text: electrons,

text: the

text: electrons

text: will

text: feel

text: the

text: electric

text: force

text: pushing

text: it

text: back

text: toward

text: the

text: target.

text: When

text: designing

text: the

text: shroud

text: geometry,

text: one

text: must

text: ensure

text: that

text: the

text: red

text: region

text: in

text: the

text: top

text: of

text: Figure

text: does

text: not

text: touch

text: the

text: target

text: (as

text: it

text: does

text: in

text: Figure

text: in

text: order

text: to

text: prevent

text: an

text: electron

text: from

text: being

text: ejected

text: into

text: an

text: electric

text: ﬁeld

text: that

text: would

text: accelerate

text: the

text: particle

text: toward

text: the

text: vacuum

text: cham-

text: (c)

text: ber

text: wall.

text: Factors

text: that

text: aﬀect

text: the

text: electric

text: ﬁeld

text: within

text: the

text: shroud

text: include

text: the

text: voltage

text: diﬀerential

text: and

text: the

text: distance

text: FIG.

text: Electric

text: ﬁeld

text: direction

text: near

text: shroud

text: window

text: at

text: (a)

text: between

text: the

text: shroud

text: and

text: target,

text: as

text: well

text: as

text: the

text: window

text: V,

text: (b)

text: V,

text: (c)

text: and

text: V

text: voltage

text: diﬀerential.

text: Red

text: indi-

text: size.

text: cates

text: ﬁeld

text: along

text: the

text: direction

text: of

text: the

text: deuteron

text: beam

text: (shown

text: by

text: B.

text: Simulation

text: of

text: Electron

text: Trajectories

text: Using

text: arrow),

text: blue

text: indicates

text: direction

text: opposite

text: the

text: deuteron

text: beam.

text: Electrostatic

text: Shroud

text: T

text: and

text: S

text: denote

text: the

text: target

text: and

text: shroud,

text: respectively.

text: A

text: simulation

text: of

text: the

text: secondary

text: electrons

text: released

text: into

text: ber.

text: After

text: implementing

text: the

text: shroud,

text: no

text: visual

text: sign

text: of

text: 2π

text: from

text: a

text: mA

text: deuterium

text: beam

text: striking

text: the

text: ﬂat

text: tar-

text: chamber

text: heating

text: could

text: be

text: seen

text: during

text: machine

text: opera-

text: get

text: surface

text: is

text: shown

text: in

text: Figure

text: Assuming

text: electrons

text: tion

text: due

text: to

text: back-streaming

text: electrons.

text: Also,

text: the

text: dose

text: of

text: released

text: per

text: deuteron

text: ion

text: striking

text: the

text: target,

text: this

text: results

text: bremsstrahlung

text: X-rays

text: detected

text: by

text: a

text: Geiger

text: counter

text: in

text: in

text: mA

text: of

text: electrons.

text: As

text: expected

text: and

text: described

text: pre-

text: the

text: vault

text: decreased

text: by

text: a

text: factor

text: of

text: more

text: than

text: compared

text: viously

text: due

text: to

text: the

text: electric

text: ﬁeld

text: shown

text: in

text: Figure

text: for

text: to

text: the

text: magnetic

text: suppression

text: technique.

text: a

text: V

text: voltage

text: diﬀerential

text: between

text: the

text: shroud

text: and

text: the

text: target,

text: secondary

text: electrons

text: are

text: not

text: suppressed

text: and

text: leave

text: through

text: the

text: shroud

text: window

text: in

text: a

text: beam

text: that

text: eventually

text: V.

text: CONCLUSIONS

text: strikes

text: the

text: chamber

text: with

text: an

text: energy

text: of

text: keV.

text: A

text: shroud

text: voltage

text: diﬀerential

text: of

text: V

text: results

text: in

text: the

text: electrons

text: re-

text: Two

text: main

text: methods

text: of

text: electron

text: suppression

text: were

text: tested:

text: turning

text: to

text: the

text: surface

text: of

text: the

text: shroud,

text: as

text: seen

text: in

text: Figure

text: using

text: permanent

text: magnets

text: to

text: bend

text: electrons

text: back

text: to

text: the

text: The

text: maximum

text: energy

text: the

text: electron

text: receives

text: is

text: equal

text: target,

text: and

text: implementing

text: an

text: electrostatic

text: shroud

text: creating

text: to

text: the

text: energy

text: at

text: release.

text: a

text: suppressing

text: electric

text: ﬁeld.

text: Simulations

text: of

text: the

text: permanent

text: Testing

text: of

text: the

text: shroud

text: at

text: V

text: voltage

text: diﬀerential

text: or

text: magnet

text: design

text: showed

text: that

text: even

text: with

text: magnetic

text: ﬁelds

text: sur-

text: greater

text: on

text: the

text: HFNG

text: has

text: shown

text: a

text: large

text: reduction

text: in

text: the

text: passing

text: gauss,

text: electrons

text: would

text: migrate

text: along

text: the

text: amount

text: of

text: secondary

text: electrons

text: hitting

text: the

text: vacuum

text: cham-

text: target

text: due

text: to

text: the

text: E

text: B

text: force

text: without

text: coming

text: into

text: con-

text: ##LTFigure##

text: ##LTFigure##

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text: (cid:80) ∗ ∗∗ 2 YAN LI BIN ZHOU XIAOHUA ZHU ρ = 1 ∈ α and Ξ be the relative interior of the cone generated by Φ . Then 2 α Φ+ + Delcroix proved Theorem 1.1. Let M be a polarized compactiﬁcation of G with c (M) > 0. Then 1 M admits a Ka¨hler-Einstein metric if and only if ∈ (1.1) bar 4ρ + Ξ, (cid:82) (cid:82) (cid:81) yπ(y)dy where bar = 2P+ is the barycentre of 2P+ with respect to the weighted π(y)dy 2P+ (cid:104) (cid:105) measure π(y)dy and π(y) = ∈ α, y 2. α Φ+ It is pointed by Delcroix that (1.1) implies that the Futaki invariant vanishes for × holomorphic vector ﬁelds induced by G G, but the inverse is not true in general. Thus one may ask if (1.1) is related to the K-stability and is determined by a generalized Futaki invariant for some test-conﬁgurations. In the present paper, we will answer this question. In fact, motivated by the study on toric manifolds [14], we × investigate the K-energy on the space of K K-invariant K¨ahler potentials through K · the reduced K-energy ( ) via Legendre transformation. We show that condition K · (1.1) comes from our formula of ( ) naturally when c (M) > 0 (cf. Proposition 1 3.1, Proposition 3.4). Moreover, we give an alternative proof of Theorem 1.1 by showing the properness of the K-energy (cf. Section 4). The K¨ahler-Ricci solitons case can be discussed similarly (cf. Section 5). The main purpose of this paper is to give a criterion for the properness of the K-energy on a general polarized compactiﬁcation (M, L) of G as done on a toric ∩ { } manifold in [33]. We divide ∂(2P ) ∂(2P) into several pieces F d0 such that − + A A(cid:104)=1 (cid:105) for any A, FA lies on a∈n (r 1)-dimensionaZl hyperplane(cid:91)deﬁned by y, uA = λA for some primitive u N, where N is the -dual of M. Deﬁne a cone by E = A A { | ∈ ∈ } d0 ty t [0, 1], y F for any A. It is clear that 2P = E . Let A + A A=1 2 (cid:104) (cid:105) (1.2) Λ = (1 + 2ρ, u ) . A λ A A ∈ Then the average of scalar curvat(cid:80)ure S¯ of(cid:82)ω 2πc (L) is given by1 0 1 (cid:82) n Λ π dy (1.3) S¯ = A A EA . (cid:102) π dy 2P+ Deﬁne a weighted barycentre bar(cid:80)of 2P (cid:82)by + (cid:102) (cid:80) (cid:82) Λ yπ dy (1.4) bar = A A EA . Λ π dy A A EA 1(1.3) will be veriﬁed at the end of Section 2. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: YAN

text: LI

text: BIN

text: ZHOU

text: XIAOHUA

text: ZHU

text: ρ

text: α

text: and

text: Ξ

text: be

text: the

text: relative

text: interior

text: of

text: the

text: cone

text: generated

text: by

text: Φ

text: Then

text: α

text: Φ+

text: Delcroix

text: proved

text: Theorem

text: Let

text: M

text: be

text: a

text: polarized

text: compactiﬁcation

text: of

text: G

text: with

text: c

text: (M)

text: Then

text: M

text: admits

text: a

text: Ka¨hler-Einstein

text: metric

text: if

text: and

text: only

text: if

text: bar

text: 4ρ

text: Ξ,

text: yπ(y)dy

text: where

text: bar

text: is

text: the

text: barycentre

text: of

text: with

text: respect

text: to

text: the

text: weighted

text: π(y)dy

text: measure

text: π(y)dy

text: and

text: π(y)

text: α,

text: y

text: α

text: Φ+

text: It

text: is

text: pointed

text: by

text: Delcroix

text: that

text: implies

text: that

text: the

text: Futaki

text: invariant

text: vanishes

text: for

text: holomorphic

text: vector

text: ﬁelds

text: induced

text: by

text: G

text: G,

text: but

text: the

text: inverse

text: is

text: not

text: true

text: in

text: general.

text: Thus

text: one

text: may

text: ask

text: if

text: is

text: related

text: to

text: the

text: K-stability

text: and

text: is

text: determined

text: by

text: a

text: generalized

text: Futaki

text: invariant

text: for

text: some

text: test-conﬁgurations.

text: In

text: the

text: present

text: paper,

text: we

text: will

text: answer

text: this

text: question.

text: In

text: fact,

text: motivated

text: by

text: the

text: study

text: on

text: toric

text: manifolds

text: we

text: investigate

text: the

text: K-energy

text: on

text: the

text: space

text: of

text: K

text: K-invariant

text: K¨ahler

text: potentials

text: through

text: K

text: the

text: reduced

text: K-energy

text: via

text: Legendre

text: transformation.

text: We

text: show

text: that

text: condition

text: K

text: comes

text: from

text: our

text: formula

text: of

text: naturally

text: when

text: c

text: (M)

text: (cf.

text: Proposition

text: Proposition

text: Moreover,

text: we

text: give

text: an

text: alternative

text: proof

text: of

text: Theorem

text: by

text: showing

text: the

text: properness

text: of

text: the

text: K-energy

text: (cf.

text: Section

text: The

text: K¨ahler-Ricci

text: solitons

text: case

text: can

text: be

text: discussed

text: similarly

text: (cf.

text: Section

text: The

text: main

text: purpose

text: of

text: this

text: paper

text: is

text: to

text: give

text: a

text: criterion

text: for

text: the

text: properness

text: of

text: the

text: K-energy

text: on

text: a

text: general

text: polarized

text: compactiﬁcation

text: (M,

text: L)

text: of

text: G

text: as

text: done

text: on

text: a

text: toric

text: manifold

text: in

text: We

text: divide

text: into

text: several

text: pieces

text: F

text: such

text: that

text: A

text: for

text: any

text: A,

text: FA

text: lies

text: on

text: a∈n

text: (r

text: by

text: y,

text: uA

text: λA

text: for

text: some

text: primitive

text: u

text: N,

text: where

text: N

text: is

text: the

text: -dual

text: of

text: M.

text: Deﬁne

text: a

text: cone

text: by

text: E

text: A

text: A

text: ty

text: t

text: y

text: F

text: for

text: any

text: A.

text: It

text: is

text: clear

text: that

text: E

text: Let

text: A

text: A

text: Λ

text: u

text: A

text: λ

text: A

text: A

text: Then

text: the

text: average

text: of

text: scalar

text: S¯

text: (L)

text: is

text: given

text: n

text: Λ

text: π

text: dy

text: S¯

text: A

text: A

text: EA

text: π

text: dy

text: Deﬁne

text: a

text: weighted

text: barycentre

text: Λ

text: yπ

text: dy

text: bar

text: A

text: A

text: EA

text: Λ

text: π

text: dy

text: A

text: A

text: EA

text: will

text: be

text: veriﬁed

text: at

text: the

text: end

text: of

text: Section

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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