text: 5 II X N→N+1 N*→N+1 N-1→N* N*←N+1 S D S N-1←N D S D S D ΓSD ΓS ΓD ΓS N→N* N→N+1 N+1→N* N*→N+1 ex. N N+1 N* N+1 N* N relax. ΓD ΓS ΓD N+1→N N*→N+1 N+1→N* ΓD N+1→N relax. ex. N←N+1 N←N+1 N*←N+1 N*→N+1 S D S D S D S D FIG. 4. (Color online) Quantum state graph with associated FIG. 5. (Color online) Quantum state graph with associated electrochemical-potential diagrams, for device A, region II. electrochemical-potential diagrams, for device B, region X. | ∗(cid:105) The gray rectangles in tunnel barriers stand for weaker tun- N is a blocking state if the main escape process, namely | (cid:105) ∗ → neling via the neighboring transition. N is a blocking state, tunneling from S via transition N N + 1, is hindered from which the quantum dot can escape only by low-rate pro- (marked with a gray rectangle in the tunnel barrier). Cotun- cesses (inelastic cotunneling in the second diagram; excita- neling transitions are not displayed. tion). Elastic cotunneling is not displayed. In the bottom- right chemical-potential diagram, tunneling into S is equally possible. below as a blocking state. First, we match the large-Fano-factor data of the two (i) one electron tunnels from S into the QD (N→N +1, devices onto labeled regions of Fig. 3(d). For device A, we ∗ |N +1(cid:105) is a virtual state) and farther into D (N +1→N ), expect inside diamond N the existence of regions II, II’, or (ii) an electron tunnels into D (N→N −1, |N −1(cid:105) is III. For device B, we argue that the band below diamond a virtual state) and a second electron tunnels from S N is region X. The involved processes are summarized in ∗ (N −1→N ). Both paths contribute to the IEC rate. Figs. 4 and 5. We gather all such processes under the concise notation Inside the Coulomb diamond N of device A, IEC ± ∗ N N−→1 N . The IEC energy condition, |eV | ≥ ∆ , triangles are indicated by current and by diﬀerential- SD 0 deﬁnes two triangles inside the Coulomb diamond. conductance lines [Fig. 2(b),(a)]. The triangles are en- Of particular interest inside the IEC triangles are tirely covered by the COSET sidebands, composed by the sidebands delimited by the excited lines (e.g. re- regions II, II’, III. Hence, both IEC and sequential tun- gions II+III and II’+III in negative bias). As pointed neling are present. Fig. 4 speciﬁcally illustrates that ± ∗ out above, these regions also allow sequential tunneling IEC breaks the blockade (N N−→1 N , red arrow), being ∗ through state |N± (cid:105). ∗This combination of IEC eﬀective followed by sequential tunneling (N∗ →S N +1 →D N∗ →S excitation, N N−→1 N , with sequential transport through etc., orange and blue arrows). The transport sequence the excited state is often labeled as cotunneling-assisted is interrupted by spontaneous relaxation (wavy gray ar- sequential tunneling (COSET) [19]. In the absence of row) or by the N + 1 → N transition (green arrow), IEC, other excitation mechanisms are outweighed by both resulting in a return to the blocking state, |N(cid:105). relaxation. Therefore, at a bias below the excitation en- Because cotunneling is a second-order process, the IEC ergy, relaxation annuls the probability of the excited state rate, ΓSD→ ∗, is relatively weak, hence the blocking time N N and hence the sequential tunneling through it; this is why is relatively long. Thus, electronic-ﬂow periods alternate excited lines do not appear at low bias, |eV | < ∆ . with zero-current periods, implying telegraphic trans- SD 0 INTERPRETATION port, i.e. augmented noise. This is indeed what the Fano factor map [Fig. 2(e)] reveals. We stress here that the re- We further investigate our noise data within the model, laxation rate (wavy gray arrow) should be lower than the by looking for transitions through which tunneling is rel- sequential-tunneling rates (orange and blue arrows) in atively weak. These may indicate a delayed-escape path order allow alternation of current and blocking periods, from a reachable state. Such a particular state is referred i.e. super-Poissonian noise. state blocking gnikcolb etats ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: II

text: X

text: S

text: D

text: S

text: D

text: S

text: D

text: S

text: D

text: ΓSD

text: ΓS

text: ΓD

text: ΓS

text: N→N*

text: ex.

text: N

text: N*

text: N*

text: N

text: relax.

text: ΓD

text: ΓS

text: ΓD

text: ΓD

text: relax.

text: ex.

text: S

text: D

text: S

text: D

text: S

text: D

text: S

text: D

text: FIG.

text: (Color

text: online)

text: Quantum

text: state

text: graph

text: with

text: associated

text: FIG.

text: (Color

text: online)

text: Quantum

text: state

text: graph

text: with

text: associated

text: electrochemical-potential

text: diagrams,

text: for

text: device

text: A,

text: region

text: II.

text: electrochemical-potential

text: diagrams,

text: for

text: device

text: B,

text: region

text: X.

text: The

text: gray

text: rectangles

text: in

text: tunnel

text: barriers

text: stand

text: for

text: weaker

text: tun-

text: N

text: is

text: a

text: blocking

text: state

text: if

text: the

text: main

text: escape

text: process,

text: namely

text: neling

text: via

text: the

text: neighboring

text: transition.

text: N

text: is

text: a

text: blocking

text: state,

text: tunneling

text: from

text: S

text: via

text: transition

text: N

text: N

text: is

text: hindered

text: from

text: which

text: the

text: quantum

text: dot

text: can

text: escape

text: only

text: by

text: low-rate

text: pro-

text: (marked

text: with

text: a

text: gray

text: rectangle

text: in

text: the

text: tunnel

text: barrier).

text: Cotun-

text: cesses

text: (inelastic

text: cotunneling

text: in

text: the

text: second

text: diagram;

text: excita-

text: neling

text: transitions

text: are

text: not

text: displayed.

text: tion).

text: Elastic

text: cotunneling

text: is

text: not

text: displayed.

text: In

text: the

text: bottom-

text: right

text: chemical-potential

text: diagram,

text: tunneling

text: into

text: S

text: is

text: equally

text: possible.

text: below

text: as

text: a

text: blocking

text: state.

text: First,

text: we

text: match

text: the

text: large-Fano-factor

text: data

text: of

text: the

text: two

text: (i)

text: one

text: electron

text: tunnels

text: from

text: S

text: into

text: the

text: QD

text: (N→N

text: devices

text: onto

text: labeled

text: regions

text: of

text: Fig.

text: For

text: device

text: A,

text: we

text: |N

text: is

text: a

text: virtual

text: state)

text: and

text: farther

text: into

text: D

text: (N

text: expect

text: inside

text: diamond

text: N

text: the

text: existence

text: of

text: regions

text: II,

text: II’,

text: or

text: (ii)

text: an

text: electron

text: tunnels

text: into

text: D

text: (N→N

text: |N

text: is

text: III.

text: For

text: device

text: B,

text: we

text: argue

text: that

text: the

text: band

text: below

text: diamond

text: a

text: virtual

text: state)

text: and

text: a

text: second

text: electron

text: tunnels

text: from

text: S

text: N

text: is

text: region

text: X.

text: The

text: involved

text: processes

text: are

text: summarized

text: in

text: (N

text: −1→N

text: Both

text: paths

text: contribute

text: to

text: the

text: IEC

text: rate.

text: Figs.

text: and

text: We

text: gather

text: all

text: such

text: processes

text: under

text: the

text: concise

text: notation

text: Inside

text: the

text: Coulomb

text: diamond

text: N

text: of

text: device

text: A,

text: IEC

text: N

text: N−→1

text: N

text: The

text: IEC

text: energy

text: condition,

text: |eV

text: triangles

text: are

text: indicated

text: by

text: current

text: and

text: by

text: diﬀerential-

text: SD

text: deﬁnes

text: two

text: triangles

text: inside

text: the

text: Coulomb

text: diamond.

text: conductance

text: lines

text: [Fig.

text: The

text: triangles

text: are

text: en-

text: Of

text: particular

text: interest

text: inside

text: the

text: IEC

text: triangles

text: are

text: tirely

text: covered

text: by

text: the

text: COSET

text: sidebands,

text: composed

text: by

text: the

text: sidebands

text: delimited

text: by

text: the

text: excited

text: lines

text: (e.g.

text: re-

text: regions

text: II,

text: II’,

text: III.

text: Hence,

text: both

text: IEC

text: and

text: sequential

text: tun-

text: gions

text: II+III

text: and

text: II’+III

text: in

text: negative

text: bias).

text: As

text: pointed

text: neling

text: are

text: present.

text: Fig.

text: speciﬁcally

text: illustrates

text: that

text: out

text: above,

text: these

text: regions

text: also

text: allow

text: sequential

text: tunneling

text: IEC

text: breaks

text: the

text: blockade

text: (N

text: N−→1

text: N

text: red

text: arrow),

text: being

text: through

text: state

text: |N±

text: ∗This

text: combination

text: of

text: IEC

text: eﬀective

text: followed

text: by

text: sequential

text: tunneling

text: (N∗

text: →S

text: N

text: →D

text: N∗

text: →S

text: excitation,

text: N

text: N−→1

text: N

text: with

text: sequential

text: transport

text: through

text: etc.,

text: orange

text: and

text: blue

text: arrows).

text: The

text: transport

text: sequence

text: the

text: excited

text: state

text: is

text: often

text: labeled

text: as

text: cotunneling-assisted

text: is

text: interrupted

text: by

text: spontaneous

text: relaxation

text: (wavy

text: gray

text: ar-

text: sequential

text: tunneling

text: (COSET)

text: In

text: the

text: absence

text: of

text: row)

text: or

text: by

text: the

text: N

text: N

text: transition

text: (green

text: arrow),

text: IEC,

text: other

text: excitation

text: mechanisms

text: are

text: outweighed

text: by

text: both

text: resulting

text: in

text: a

text: return

text: to

text: the

text: blocking

text: state,

text: relaxation.

text: Therefore,

text: at

text: a

text: bias

text: below

text: the

text: excitation

text: en-

text: Because

text: cotunneling

text: is

text: a

text: second-order

text: process,

text: the

text: IEC

text: ergy,

text: relaxation

text: annuls

text: the

text: probability

text: of

text: the

text: excited

text: state

text: rate,

text: ΓSD→

text: is

text: relatively

text: weak,

text: hence

text: the

text: blocking

text: time

text: N

text: N

text: and

text: hence

text: the

text: sequential

text: tunneling

text: through

text: it;

text: this

text: is

text: why

text: is

text: relatively

text: long.

text: Thus,

text: electronic-ﬂow

text: periods

text: alternate

text: excited

text: lines

text: do

text: not

text: appear

text: at

text: low

text: bias,

text: |eV

text: with

text: zero-current

text: periods,

text: implying

text: telegraphic

text: trans-

text: SD

text: INTERPRETATION

text: port,

text: i.e.

text: augmented

text: noise.

text: This

text: is

text: indeed

text: what

text: the

text: Fano

text: factor

text: map

text: [Fig.

text: reveals.

text: We

text: stress

text: here

text: that

text: the

text: re-

text: We

text: further

text: investigate

text: our

text: noise

text: data

text: within

text: the

text: model,

text: laxation

text: rate

text: (wavy

text: gray

text: arrow)

text: should

text: be

text: lower

text: than

text: the

text: by

text: looking

text: for

text: transitions

text: through

text: which

text: tunneling

text: is

text: rel-

text: sequential-tunneling

text: rates

text: (orange

text: and

text: blue

text: arrows)

text: in

text: atively

text: weak.

text: These

text: may

text: indicate

text: a

text: delayed-escape

text: path

text: order

text: allow

text: alternation

text: of

text: current

text: and

text: blocking

text: periods,

text: from

text: a

text: reachable

text: state.

text: Such

text: a

text: particular

text: state

text: is

text: referred

text: i.e.

text: super-Poissonian

text: noise.

text: state

text: blocking

text: gnikcolb

text: etats

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text: 16 SAMUEL BOISSIE`RE, CHIARA CAMERE, AND ALESSANDRA SARTI (cid:102) ⊥ ∩ ⊂ 9-dimensional complex ball Ωρδ,ξ = δ Ωρ0,ξ H . By [7, Theorem 9.1], the U(3) S n period map considered in the proof of Lemm(cid:102)a 4.5 induces a bijection (cid:102) (cid:102) \ (cid:83) Pρδ,ξ : Kρδ,ξ −→ ΩρUδ(,3ξ)(cid:102)HU(3) U(3) U(3) (cid:102) ΓρUδ(,3ξ) ⊥ where H := ε . The group Γρδ,ξ is the set of isometries of the K3 U(3) ∈ (cid:101) U(3) ε S−δ Z ⊥ ε2= 2 lattice ( e ) which commute with ρ . We thus clearly have an equality of orbit n δ (cid:102) spaces Ω(cid:102)ρδ,ξ δ⊥ ∩ Ωρ0,ξ U(3) = S . Γρδ,ξ Γρ0,ξ U(3) S P C → B \ ∪ The extension of the period map ρ0,ξ : sm 10 (HP n Hc) to a general nodal S 3 Γ cubic C is deﬁned in [4] by using the per(cid:102)iod of the associated K3 surface Σ : n P P ∈ ρ0,ξ(C) := ρδ,ξ (Σ ) H . S U(3) n n By studying the limit Hodge structure of the nodal degeneration of a cubic threefold, Allcock–Carlson–Toledo [4, Chapter 6] show that this deﬁnition extends holomor- P C B \ phically ρ0,ξ to an isomorphism between s and 10P Hc mapping bijectively ∆s S P 3 Γ 3 to the divisor H / Γ. n Turning back to the automorphism τ[2] on Σ[2], which gives a point in the mo- duli space Nρδ,ξ , the period map constnructed (cid:0)inn [14, Theo(cid:1)rem 5.0.17] produces a K(Tδ) bijection \ ∪ (cid:48) P N −→ Ωρδ,ξ H H ρδ,ξ : ρδ,ξ Tδ Tδ Tδ . Tδ K(Tδ) Γρδ,ξ (cid:102) Tδ The notation is sim(cid:102)ilar to those of Section , it is easy to check that Ωρδ,ξ = Ωρδ,ξ (cid:91) Tδ U(3) and that Γρδ,ξ = Γρδ,ξ . The hyperplane arrangements are Tδ U(3) ⊥ H = ε = H Tδ ∈ U(3) ε S−δ ε2= 2 (cid:48) ⊥ and H is the union of the hyperplanes ε which meet K(T ), for those MBM Tδ ∈ δ ∈ ⊗ Q classes ε L which admit a decomposition ε = ε + ε with ε T Z , ∈ ⊗ Q 1 2 1 δ ε S Z(cid:102), ε2 < 0, and ε2 < 0. (cid:102) 2 δ 1 2 Since Pρδ,ξ and Pρδ,ξ are bijections and ΩρUδ(,3ξ)(cid:102)\HU(3) is clearly birational to U(3) Tδ Γρδ,ξ U(3) ΩρTδδ,ξ\(HTδ∪HT(cid:48)δ), it follows that the moduli spaces ∆s and Kρδ,ξ, which are bi- Γρδ,ξ 3 Tδ Tδ N (cid:3) rationally equivalent by [17, Proposition 5.3], are also birational to ρδ,ξ . K(Tδ) Remark 4.7. Let us denote by M the moduli space of genus 4 curves. From the 4 proof of Proposition 4.6, we get bir(cid:102)ational isomorphisms ∼ K ∼ N ∼ ∆s ρδ,ξ ρδ,ξ M 3 U(3) K(Tδ) 4 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: SAMUEL

text: BOISSIE`RE,

text: CHIARA

text: CAMERE,

text: AND

text: ALESSANDRA

text: SARTI

text: complex

text: ball

text: Ωρδ,ξ

text: δ

text: Ωρ0,ξ

text: H

text: By

text: Theorem

text: the

text: S

text: n

text: period

text: map

text: considered

text: in

text: the

text: proof

text: of

text: induces

text: a

text: bijection

text: Pρδ,ξ

text: Kρδ,ξ

text: where

text: H

text: ε

text: The

text: group

text: Γρδ,ξ

text: is

text: the

text: set

text: of

text: isometries

text: of

text: the

text: ε

text: S−δ

text: Z

text: lattice

text: e

text: which

text: commute

text: with

text: ρ

text: We

text: thus

text: clearly

text: have

text: an

text: equality

text: of

text: orbit

text: n

text: δ

text: spaces

text: δ⊥

text: Ωρ0,ξ

text: S

text: Γρδ,ξ

text: Γρ0,ξ

text: S

text: P

text: C

text: B

text: The

text: extension

text: of

text: the

text: period

text: map

text: ρ0,ξ

text: sm

text: (HP

text: n

text: Hc)

text: to

text: a

text: general

text: nodal

text: S

text: Γ

text: cubic

text: C

text: is

text: deﬁned

text: in

text: by

text: using

text: the

text: of

text: the

text: associated

text: surface

text: Σ

text: n

text: P

text: P

text: ρδ,ξ

text: (Σ

text: H

text: S

text: n

text: n

text: By

text: studying

text: the

text: limit

text: Hodge

text: structure

text: of

text: the

text: nodal

text: degeneration

text: of

text: a

text: cubic

text: threefold,

text: Allcock–Carlson–Toledo

text: Chapter

text: show

text: that

text: this

text: deﬁnition

text: extends

text: holomor-

text: P

text: C

text: B

text: phically

text: ρ0,ξ

text: to

text: an

text: isomorphism

text: between

text: s

text: and

text: Hc

text: mapping

text: bijectively

text: ∆s

text: S

text: P

text: Γ

text: to

text: the

text: divisor

text: H

text: Γ.

text: n

text: Turning

text: back

text: to

text: the

text: automorphism

text: on

text: which

text: gives

text: a

text: point

text: in

text: the

text: mo-

text: duli

text: space

text: Nρδ,ξ

text: the

text: period

text: map

text: constnructed

text: produces

text: a

text: K(Tδ)

text: bijection

text: P

text: N

text: Ωρδ,ξ

text: H

text: H

text: ρδ,ξ

text: ρδ,ξ

text: Tδ

text: Tδ

text: Tδ

text: Tδ

text: K(Tδ)

text: Γρδ,ξ

text: Tδ

text: The

text: notation

text: is

text: to

text: those

text: of

text: Section

text: it

text: is

text: easy

text: to

text: check

text: that

text: Ωρδ,ξ

text: Ωρδ,ξ

text: Tδ

text: and

text: that

text: Γρδ,ξ

text: Γρδ,ξ

text: The

text: hyperplane

text: arrangements

text: are

text: Tδ

text: H

text: ε

text: H

text: Tδ

text: ε

text: S−δ

text: and

text: H

text: is

text: the

text: union

text: of

text: the

text: hyperplanes

text: ε

text: which

text: meet

text: K(T

text: for

text: those

text: MBM

text: Tδ

text: δ

text: Q

text: classes

text: ε

text: L

text: which

text: admit

text: a

text: decomposition

text: ε

text: ε

text: ε

text: with

text: ε

text: T

text: Z

text: Q

text: δ

text: ε

text: S

text: ε2

text: and

text: ε2

text: δ

text: Since

text: Pρδ,ξ

text: and

text: Pρδ,ξ

text: are

text: bijections

text: and

text: is

text: clearly

text: birational

text: to

text: Tδ

text: Γρδ,ξ

text: it

text: follows

text: that

text: the

text: moduli

text: spaces

text: ∆s

text: and

text: Kρδ,ξ,

text: which

text: are

text: bi-

text: Γρδ,ξ

text: Tδ

text: Tδ

text: N

text: rationally

text: equivalent

text: by

text: Proposition

text: are

text: also

text: birational

text: to

text: ρδ,ξ

text: K(Tδ)

text: Remark

text: Let

text: us

text: denote

text: by

text: M

text: the

text: moduli

text: space

text: of

text: genus

text: curves.

text: From

text: the

text: proof

text: of

text: Proposition

text: we

text: get

text: isomorphisms

text: K

text: N

text: ∆s

text: ρδ,ξ

text: ρδ,ξ

text: M

text: K(Tδ)

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 10 SAMUEL BOISSIE`RE, CHIARA CAMERE, AND ALESSANDRA SARTI The term “occult” in the section title is a reference to a paper of Kudla– Rapoport [29]. As suggested to us by Michael Rapoport and thanks to enlightening discussions with Gregory Sankaran, we can draw direct paths between all the in- N volved period maps, producing in particular a period map from (cid:104)ρ,(cid:105)ξ to a moduli 6 space of principally polarized abelian varieties. The classical period map for cubic ∈ C threefolds of Clemens–Griﬃths [18] maps a cubic C sm to its intermediate Ja- 3 cobian IJ(C) := H1,2(CZ) , which gives a point in the 15-dimensional moduli space A H3(C, ) of principally polarized abelian variety of genus 5. 5 N → A Corollary 3.7. There exists an embedding (cid:104)ρ,(cid:105)ξ (cid:44) mapping a (ρ, j)-polarized 6 5 ⊂ tuple (X, η, σ, ι) to the Albanese variety Alb(Xσ) of the ﬁxed locus Xσ X of the automorphism σ. Proof. By [18, (0.8)] the intermediate Jacobian of C is∼isomorphic to the Albanese variety of the Fano variety of lines on C, i.e. IJ(C) = Alb(F(C)). As observed in Section , the variety F(C) is the ﬁxed locus in F(Y ) of the automorphism σ → P of the ramiﬁed covering map Y 4 branched along C. Since the morphism C → N F: sm (cid:104)ρ,(cid:105)ξ mapping C to (F(V ), σ) (with the appropriate markings and po- 3 6 (cid:3) larizations as above) is an isomorphism, we get the result. 4. Degenerations § Following an idea used by Dolgachev–Kond¯o [21, 11] to study the degenera- tion of K3 surfaces with a non-symplectic automorphism of prime order, we study the degenerations of those irreducible holomorphic symplectic manifolds (X, η, σ, ι) M M parametrized by ρ(cid:104),ξ(cid:105), inside the moduli space (cid:104) (cid:105) of j-polarized marked triples. ◦ 6 { ∈ P6 | } Denote by Ω(cid:104) (cid:105) the connected component of ω (SC) h − (ω, ω) > 0 which 6 S( 1) contains the ball Ωρ(cid:104),ξ(cid:105), it is easy to deduce from the theorem of surjectivity of the pe- 6 P M → ◦ riod map of Huybrechts [26, Theorem 8.1] that the period map (cid:104) (cid:105) : (cid:104) (cid:105) Ω(cid:104) (cid:105) 6 6◦ 6◦ is surjective. Note that, comparing with the notation in Section , we have Ω(cid:104) (cid:105) = Ω 6 S but we use both notations depending on the context. We thus have a commutative diagram P Mρ(cid:104),ξ(cid:105) (cid:104)ρ6,(cid:105)ξ(cid:47)(cid:47) Ωρ(cid:104),ξ(cid:105) \ H (cid:127)6(cid:95) 6 (cid:127) (cid:95) (cid:15)(cid:15) P (cid:15)(cid:15) M(cid:104) (cid:105) (cid:104)6(cid:105) (cid:47)(cid:47) (cid:47)(cid:47) Ω◦(cid:104) (cid:105) 6 6 ∈ ∈ P− Let ω H and (X, η, ι) (cid:104) 1(cid:105)(ω). By [33, Theorem 1.2], any two points in this P 6 ﬁber of (cid:104) (cid:105) correspond to birational manifolds. 6 ∈ We ﬁrst observe that the isometry ρ O(L) is not represented by any auto- morphism of X. As explained in the proof of Theorem 3.4, this is a special case ∈ ∈ of [15, Theorem 4.0.8]: if ω H , then ω is orthogonal to some class δ S such − that η 1(δ) is a MBM class of X. If ρ was represented by an automorphism of ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: SAMUEL

text: BOISSIE`RE,

text: CHIARA

text: CAMERE,

text: AND

text: ALESSANDRA

text: SARTI

text: The

text: term

text: “occult”

text: in

text: the

text: section

text: title

text: is

text: a

text: reference

text: to

text: a

text: paper

text: of

text: Kudla–

text: Rapoport

text: As

text: suggested

text: to

text: us

text: by

text: Michael

text: Rapoport

text: and

text: thanks

text: to

text: enlightening

text: discussions

text: with

text: Gregory

text: Sankaran,

text: we

text: can

text: draw

text: direct

text: paths

text: between

text: all

text: the

text: in-

text: N

text: volved

text: period

text: maps,

text: producing

text: in

text: particular

text: a

text: period

text: map

text: from

text: to

text: a

text: moduli

text: space

text: of

text: principally

text: polarized

text: abelian

text: varieties.

text: The

text: classical

text: period

text: map

text: for

text: cubic

text: C

text: threefolds

text: of

text: Clemens–Griﬃths

text: maps

text: a

text: cubic

text: C

text: sm

text: to

text: its

text: intermediate

text: Ja-

text: cobian

text: IJ(C)

text: which

text: gives

text: a

text: point

text: in

text: the

text: moduli

text: space

text: A

text: of

text: principally

text: polarized

text: abelian

text: variety

text: of

text: genus

text: N

text: A

text: Corollary

text: There

text: exists

text: an

text: embedding

text: mapping

text: a

text: (ρ,

text: j)-polarized

text: tuple

text: (X,

text: η,

text: σ,

text: ι)

text: to

text: the

text: Albanese

text: variety

text: Alb(Xσ)

text: of

text: the

text: ﬁxed

text: locus

text: Xσ

text: X

text: of

text: the

text: automorphism

text: σ.

text: Proof.

text: By

text: the

text: intermediate

text: Jacobian

text: of

text: C

text: is∼isomorphic

text: to

text: the

text: Albanese

text: variety

text: of

text: the

text: Fano

text: variety

text: of

text: lines

text: on

text: C,

text: i.e.

text: IJ(C)

text: Alb(F(C)).

text: As

text: observed

text: in

text: Section

text: the

text: variety

text: F(C)

text: is

text: the

text: ﬁxed

text: locus

text: in

text: F(Y

text: of

text: the

text: automorphism

text: σ

text: P

text: of

text: the

text: ramiﬁed

text: covering

text: map

text: Y

text: branched

text: along

text: C.

text: Since

text: the

text: morphism

text: C

text: N

text: F:

text: sm

text: mapping

text: C

text: to

text: (F(V

text: σ)

text: (with

text: the

text: appropriate

text: markings

text: and

text: po-

text: larizations

text: as

text: above)

text: is

text: an

text: isomorphism,

text: we

text: get

text: the

text: result.

text: Degenerations

text: Following

text: an

text: idea

text: used

text: by

text: Dolgachev–Kond¯o

text: to

text: study

text: the

text: degenera-

text: tion

text: of

text: surfaces

text: with

text: a

text: non-symplectic

text: automorphism

text: of

text: prime

text: order,

text: we

text: study

text: the

text: degenerations

text: of

text: those

text: irreducible

text: holomorphic

text: symplectic

text: manifolds

text: (X,

text: η,

text: σ,

text: ι)

text: M

text: M

text: parametrized

text: by

text: inside

text: the

text: moduli

text: space

text: of

text: j-polarized

text: marked

text: triples.

text: Denote

text: by

text: the

text: connected

text: component

text: of

text: ω

text: (SC)

text: h

text: (ω,

text: ω)

text: which

text: S(

text: contains

text: the

text: ball

text: it

text: is

text: easy

text: to

text: deduce

text: from

text: the

text: theorem

text: of

text: surjectivity

text: of

text: the

text: pe-

text: P

text: M

text: riod

text: map

text: of

text: Huybrechts

text: Theorem

text: that

text: the

text: period

text: map

text: 6◦

text: 6◦

text: is

text: surjective.

text: Note

text: that,

text: comparing

text: with

text: the

text: notation

text: in

text: Section

text: we

text: have

text: Ω

text: S

text: but

text: we

text: use

text: both

text: notations

text: depending

text: on

text: the

text: context.

text: We

text: thus

text: have

text: a

text: commutative

text: diagram

text: P

text: H

text: P

text: P−

text: Let

text: ω

text: H

text: and

text: (X,

text: η,

text: ι)

text: By

text: Theorem

text: any

text: two

text: points

text: in

text: this

text: P

text: ﬁber

text: of

text: correspond

text: to

text: birational

text: manifolds.

text: We

text: ﬁrst

text: observe

text: that

text: the

text: isometry

text: ρ

text: O(L)

text: is

text: not

text: represented

text: by

text: any

text: auto-

text: morphism

text: of

text: X.

text: As

text: explained

text: in

text: the

text: proof

text: of

text: Theorem

text: this

text: is

text: a

text: special

text: case

text: of

text: Theorem

text: if

text: ω

text: H

text: then

text: ω

text: is

text: orthogonal

text: to

text: some

text: class

text: δ

text: S

text: such

text: that

text: η

text: is

text: a

text: MBM

text: class

text: of

text: X.

text: If

text: ρ

text: was

text: represented

text: by

text: an

text: automorphism

text: of

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 2 DEMIR, UNAL: RESIDUAL POLISHING FOR IMAGE INPAINTING Coarse Painter Network Source Image Input + Fine Painter Network Output Figure 1: General structure of the Residual Polishing framework. measuring and comparing information on the general structure of the image, they tend to converge to the mean of pixel values that cause blurry outputs. Goodfellow et al. have taken image synthesis step forward by presenting Generative Adversarial Networks (GAN) [9]. An additional binary classiﬁer, which is called a dis- criminative network, is included in order to classify whether an image comes from a real distribution or a generator network output. The discriminative network is trained while the generative network tries to convince the former by producing more realistic patches. During the training, the generative network is scored by an adversarial loss that is calculated by the discriminator network. Another remarkable approach for image generation is given by a loss function that compares the features extracted from a pre-trained network instead of through direct pixel-wise measurements. It is known as the content loss in the literature [14, 18]. The idea of this approach lies in the recovery of high frequency details rather than blur. While using these loss functions, although plausible image patches are produced, they introduce undesired artifacts and noise due to complexity of the training procedure as can be seen in Figure 5. The contribution of our work can be summarized upfront as follows: we divide the prob- lem of image inpainting into two parts. First part, which we call the Coarse Painter Net (CPN), synthesizes the image texture by studying the uncorrupted parts. Second part of our method, which we call the Fine Painter Network (FPN) takes the reconstructed region and applies an enhancement to obtain an improved reconstruction. The latter is similar in spirit to the super-resolution problem, however, instead of enlarging the image, we aim to recover details without resizing. Furthermore, the second network aims to reduce noise if present. Overall, our FPN is designed to learn changes instead of the ﬁnal image patch itself. 2 Related Work As described above, we perform an inpainting process followed by an enhancement of the result by another process, overall of which we call the Residual Polishing framework. Thus, our work is related to a set of topics such as inpainting, denoising and super-resolution in the literature. Early studies on inpainting generally worked on a single damaged image [2, 6, 22]. Start- ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: DEMIR,

text: UNAL:

text: RESIDUAL

text: POLISHING

text: FOR

text: IMAGE

text: INPAINTING

text: Coarse

text: Painter

text: Network

text: Source

text: Image

text: Input

text: Fine

text: Painter

text: Network

text: Output

text: Figure

text: General

label: caption

text: structure

text: of

text: the

text: Residual

text: Polishing

text: framework.

text: measuring

text: and

text: comparing

text: information

text: on

text: the

text: general

text: structure

text: of

text: the

text: image,

text: they

text: tend

text: to

text: converge

text: to

text: the

text: mean

text: of

text: pixel

text: values

text: that

text: cause

text: blurry

text: outputs.

text: Goodfellow

text: et

text: al.

text: have

text: taken

text: image

text: synthesis

text: step

text: forward

text: by

text: presenting

text: Generative

text: Adversarial

text: Networks

text: (GAN)

text: An

text: additional

text: binary

text: classiﬁer,

text: which

text: is

text: called

text: a

text: dis-

text: criminative

text: network,

text: is

text: included

text: in

text: order

text: to

text: classify

text: whether

text: an

text: image

text: comes

text: from

text: a

text: real

text: distribution

text: or

text: a

text: generator

text: network

text: output.

text: The

text: discriminative

text: network

text: is

text: trained

text: while

text: the

text: generative

text: network

text: tries

text: to

text: convince

text: the

text: former

text: by

text: producing

text: more

text: realistic

text: patches.

text: During

text: the

text: training,

text: the

text: generative

text: network

text: is

text: scored

text: by

text: an

text: adversarial

text: loss

text: that

text: is

text: calculated

text: by

text: the

text: discriminator

text: network.

text: Another

text: remarkable

text: approach

text: for

text: image

text: generation

text: is

text: given

text: by

text: a

text: loss

text: function

text: that

text: compares

text: the

text: features

text: extracted

text: from

text: a

text: pre-trained

text: network

text: instead

text: of

text: through

text: direct

text: pixel-wise

text: measurements.

text: It

text: is

text: known

text: as

text: the

text: content

text: loss

text: in

text: the

text: literature

text: The

text: idea

text: of

text: this

text: approach

text: lies

text: in

text: the

text: recovery

text: of

text: high

text: frequency

text: details

text: rather

text: than

text: blur.

text: While

text: using

text: these

text: loss

text: functions,

text: although

text: plausible

text: image

text: patches

text: are

text: produced,

text: they

text: introduce

text: undesired

text: artifacts

text: and

text: noise

text: due

text: to

text: complexity

text: of

text: the

text: training

text: procedure

text: as

text: can

text: be

text: seen

text: in

text: Figure

text: The

text: contribution

text: of

text: our

text: work

text: can

text: be

text: summarized

text: upfront

text: as

text: follows:

text: we

text: divide

text: the

text: prob-

text: lem

text: of

text: image

text: inpainting

text: into

text: two

text: parts.

text: First

text: part,

text: which

text: we

text: call

text: the

text: Coarse

text: Painter

text: Net

text: (CPN),

text: synthesizes

text: the

text: image

text: texture

text: by

text: studying

text: the

text: uncorrupted

text: parts.

text: Second

text: part

text: of

text: our

text: method,

text: which

text: we

text: call

text: the

text: Fine

text: Painter

text: Network

text: (FPN)

text: takes

text: the

text: reconstructed

text: region

text: and

text: applies

text: an

text: enhancement

text: to

text: obtain

text: an

text: improved

text: reconstruction.

text: The

text: latter

text: is

text: similar

text: in

text: spirit

text: to

text: the

text: super-resolution

text: problem,

text: however,

text: instead

text: of

text: enlarging

text: the

text: image,

text: we

text: aim

text: to

text: recover

text: details

text: without

text: resizing.

text: Furthermore,

text: the

text: second

text: network

text: aims

text: to

text: reduce

text: noise

text: if

text: present.

text: Overall,

text: our

text: FPN

text: is

text: designed

text: to

text: learn

text: changes

text: instead

text: of

text: the

text: ﬁnal

text: image

text: patch

text: itself.

text: Related

text: Work

text: As

text: described

text: above,

text: we

text: perform

text: an

text: inpainting

text: process

text: followed

text: by

text: an

text: enhancement

text: of

text: the

text: result

text: by

text: another

text: process,

text: overall

text: of

text: which

text: we

text: call

text: the

text: Residual

text: Polishing

text: framework.

text: Thus,

text: our

text: work

text: is

text: related

text: to

text: a

text: set

text: of

text: topics

text: such

text: as

text: inpainting,

text: denoising

text: and

text: super-resolution

text: in

text: the

text: literature.

text: Early

text: studies

text: on

text: inpainting

text: generally

text: worked

text: on

text: a

text: single

text: damaged

text: image

text: Start-

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: DEMIR, UNAL: RESIDUAL POLISHING FOR IMAGE INPAINTING 3 ing from the border of the missing region, similar patches are searched inside the uncorrupted image segments. The closest discovered patches are placed into the missing region, and the process is repeated until the blank area is completely ﬁlled. Most of those earlier works paid attention to improving the speed of similar patch ﬁnding procedure. Although those methods have shown superior performance on the images which contain similar texture details, they suffered from the lack of global structural information, which led to undesirable outputs. CNN has shown great success in both classiﬁcation [12, 26, 28] and regression tasks [3, 9, 25]. Especially AE architecture has been used widely for image generation and recon- struction [8, 19, 24]. Adversarial training scheme has shown striking success for realistic image generation. Lately, a wide range of GAN type architectures, which offer novel objec- tive functions by changing the structure of discriminator [23, 27, 35], have been proposed. Pathak et al. use the GAN for an inpainting CNN which is called Context-Encoder [24]. Their inpainting network takes the center cropped images and regress the missing part. In- deed, our CPN architecture is inspired from the Context-Encoder. On the other hand, most GANs have been focusing solely on the realistic image generation instead of generation of an image patch well-matched to the global image, and that property of GANs is incompatible with the original goal of the inpainting. Using features extracted from pre-trained networks for comparing images has become popular recently in art transfer [14, 21], AE training [18] and inpainting [30]. In the liter- ature, this idea is known as either mainly content loss, perceptual loss, style loss or feature loss, as we call here. Generally, features are extracted from a classiﬁcation network, which is trained on huge data collections due to their representation strength. During the opti- mization, features of the generated image and the ground truth are forced to be close. The learned feature loss measures produce more robust results compared to those of the pixel- based distance measures [18]. Notable inpainting results are obtained recently through using a combination of the feature loss and the adversarial loss [31]. Also, in [30], Yang et al. reported state-of-the-art results by applying a local texture constraint that was inspired from [21] along with the content loss and the adversarial loss. Residual Networks have been developed to improve gradient ﬂow in the deep networks through addition of skip connections to the architecture [11, 12, 28, 29, 32]. They stack the residual blocks which consist of several convolutional layers and a residual connection. This operation reduces the training time dramatically while classiﬁcation accuracies on the public datasets are improved signiﬁcantly. In another area of regression problems, residual connections are used differently. Kim et al. proposed a super-resolution model that learns the difference between the high resolution image and its low resolution counterparts [15]. The ﬁnal result is obtained by adding the difference to the input image. It was shown that regressing the difference between input and the desired output can produce considerable performance improvement [10, 20, 34]. In this paper, we present a novel method that combines a coarse painter and a ﬁne painter network, which is described in Section 3. Our ﬁne painter idea is inspired from the super- resolution problem where a high resolution image is produced based on a given low resolu- tion input. To our knowledge, our proposed residual polishing method is the ﬁrst application that uses the residual connections to improve the results in image inpainting problem, which we will demonstrate in Experiments (Section ), followed by Conclusions (Section ). 3 Residual Polishing In Residual Polishing framework, the intention is completion of the inpainting process at ##LTLine##

text: DEMIR,

text: UNAL:

text: RESIDUAL

text: POLISHING

text: FOR

text: IMAGE

text: INPAINTING

text: ing

text: from

text: the

text: border

text: of

text: the

text: missing

text: region,

text: similar

text: patches

text: are

text: searched

text: inside

text: the

text: uncorrupted

text: image

text: segments.

text: The

text: closest

text: discovered

text: patches

text: are

text: placed

text: into

text: the

text: missing

text: region,

text: and

text: the

text: process

text: is

text: repeated

text: until

text: the

text: blank

text: area

text: is

text: completely

text: ﬁlled.

text: Most

text: of

text: those

text: earlier

text: works

text: paid

text: attention

text: to

text: improving

text: the

text: speed

text: of

text: similar

text: patch

text: ﬁnding

text: procedure.

text: Although

text: those

text: methods

text: have

text: shown

text: superior

text: performance

text: on

text: the

text: images

text: which

text: contain

text: similar

text: texture

text: details,

text: they

text: suffered

text: from

text: the

text: lack

text: of

text: global

text: structural

text: information,

text: which

text: led

text: to

text: undesirable

text: outputs.

text: CNN

text: has

text: shown

text: great

text: success

text: in

text: both

text: classiﬁcation

text: and

text: regression

text: tasks

text: Especially

text: AE

text: architecture

text: has

text: been

text: used

text: widely

text: for

text: image

text: generation

text: and

text: recon-

text: struction

text: Adversarial

text: training

text: scheme

text: has

text: shown

text: striking

text: success

text: for

text: realistic

text: image

text: generation.

text: Lately,

text: a

text: wide

text: range

text: of

text: GAN

text: type

text: architectures,

text: which

text: offer

text: novel

text: objec-

text: tive

text: functions

text: by

text: changing

text: the

text: structure

text: of

text: discriminator

text: have

text: been

text: proposed.

text: Pathak

text: et

text: al.

text: use

text: the

text: GAN

text: for

text: an

text: inpainting

text: CNN

text: which

text: is

text: called

text: Context-Encoder

text: Their

text: inpainting

text: network

text: takes

text: the

text: center

text: cropped

text: images

text: and

text: regress

text: the

text: missing

text: part.

text: In-

text: deed,

text: our

text: CPN

text: architecture

text: is

text: inspired

text: from

text: the

text: Context-Encoder.

text: On

text: the

text: other

text: hand,

text: most

text: GANs

text: have

text: been

text: focusing

text: solely

text: on

text: the

text: realistic

text: image

text: generation

text: instead

text: of

text: generation

text: of

text: an

text: image

text: patch

text: well-matched

text: to

text: the

text: global

text: image,

text: and

text: that

text: property

text: of

text: GANs

text: is

text: incompatible

text: with

text: the

text: original

text: goal

text: of

text: the

text: inpainting.

text: Using

text: features

text: extracted

text: from

text: pre-trained

text: networks

text: for

text: comparing

text: images

text: has

text: become

text: popular

text: recently

text: in

text: art

text: transfer

text: AE

text: training

text: and

text: inpainting

text: In

text: the

text: liter-

text: ature,

text: this

text: idea

text: is

text: known

text: as

text: either

text: mainly

text: content

text: loss,

text: perceptual

text: loss,

text: style

text: loss

text: or

text: feature

text: loss,

text: as

text: we

text: call

text: here.

text: Generally,

text: features

text: are

text: extracted

text: from

text: a

text: classiﬁcation

text: network,

text: which

text: is

text: trained

text: on

text: huge

text: data

text: collections

text: due

text: to

text: their

text: representation

text: strength.

text: During

text: the

text: opti-

text: mization,

text: features

text: of

text: the

text: generated

text: image

text: and

text: the

text: ground

text: truth

text: are

text: forced

text: to

text: be

text: close.

text: The

text: learned

text: feature

text: loss

text: measures

text: produce

text: more

text: robust

text: results

text: compared

text: to

text: those

text: of

text: the

text: pixel-

text: based

text: distance

text: measures

text: Notable

text: inpainting

text: results

text: are

text: obtained

text: recently

text: through

text: using

text: a

text: combination

text: of

text: the

text: feature

text: loss

text: and

text: the

text: adversarial

text: loss

text: Also,

text: in

text: Yang

text: et

text: al.

text: reported

text: state-of-the-art

text: results

text: by

text: applying

text: a

text: local

text: texture

text: constraint

text: that

text: was

text: inspired

text: from

text: along

text: with

text: the

text: content

text: loss

text: and

text: the

text: adversarial

text: loss.

text: Residual

text: Networks

text: have

text: been

text: developed

text: to

text: improve

text: gradient

text: ﬂow

text: in

text: the

text: deep

text: networks

text: through

text: addition

text: of

text: skip

text: connections

text: to

text: the

text: architecture

text: They

text: stack

text: the

text: residual

text: blocks

text: which

text: consist

text: of

text: several

text: convolutional

text: layers

text: and

text: a

text: residual

text: connection.

text: This

text: operation

text: reduces

text: the

text: training

text: time

text: dramatically

text: while

text: classiﬁcation

text: accuracies

text: on

text: the

text: public

text: datasets

text: are

text: improved

text: signiﬁcantly.

text: In

text: another

text: area

text: of

text: regression

text: problems,

text: residual

text: connections

text: are

text: used

text: differently.

text: Kim

text: et

text: al.

text: proposed

text: a

text: super-resolution

text: model

text: that

text: learns

text: the

text: difference

text: between

text: the

text: high

text: resolution

text: image

text: and

text: its

text: low

text: resolution

text: counterparts

text: The

text: ﬁnal

text: result

text: is

text: obtained

text: by

text: adding

text: the

text: difference

text: to

text: the

text: input

text: image.

text: It

text: was

text: shown

text: that

text: regressing

text: the

text: difference

text: between

text: input

text: and

text: the

text: desired

text: output

text: can

text: produce

text: considerable

text: performance

text: improvement

text: In

text: this

text: paper,

text: we

text: present

text: a

text: novel

text: method

text: that

text: combines

text: a

text: coarse

text: painter

text: and

text: a

text: ﬁne

text: painter

text: network,

text: which

text: is

text: described

text: in

text: Section

text: Our

text: ﬁne

text: painter

text: idea

text: is

text: inspired

text: from

text: the

text: super-

text: resolution

text: problem

text: where

text: a

text: high

text: resolution

text: image

text: is

text: produced

text: based

text: on

text: a

text: given

text: low

text: resolu-

text: tion

text: input.

text: To

text: our

text: knowledge,

text: our

text: proposed

text: residual

text: polishing

text: method

text: is

text: the

text: ﬁrst

text: application

text: that

text: uses

text: the

text: residual

text: connections

text: to

text: improve

text: the

text: results

text: in

text: image

text: inpainting

text: problem,

text: which

text: we

text: will

text: demonstrate

text: in

text: Experiments

text: (Section

text: followed

text: by

text: Conclusions

text: (Section

text: Residual

text: Polishing

text: In

text: Residual

text: Polishing

text: framework,

text: the

text: intention

text: is

text: completion

text: of

text: the

text: inpainting

text: process

text: at

text: ##LTLine##


text: ∈ We then generalise this for removing any number of points. For all n { − } − P 1, 2, ..., r 2 , we can remove n 1 points from , using abuse of notation to − ∈ { − } label the r n points left as t , ..., t − . Then there exists j 1, ..., r n 1 r n n such that || || ≤ 2 || || γ p − γ p . (10) p,[tjn−1,tjn+1] r n p,[0,1] − P P\{ } Removing r 2 points from , we are left with the path γ tj1,tj2,...,tjr−2 . As the partition only has three points left, and P\{ } P\{ } γ tj1,tj2,...,tjr−2 (t0) = γ tj1,tj2,...,tjr−2 (tr), the path only has two points left. Therefore, P\{ } η(γ tj1,tj2,...,tjr−2 , (x, y))) = 0 (11) P\{ } for all (x, y) outside γ tj1,tj2,...,tjr−2 . It is then clear that for all (x, y) P outside γ , we have P P − P\{ } η(γ , (x, y)) = (η(γ , (x, y)) η(γ tj1 ), (x, y)) P\{ } − P\{ } +(η(γ tj1 , (x, y)) η(γ tj1,tj2 , (x, y))) (12) +... P\{ } − P\{ } +(η(γ tj1,tj2,...,tjr−3 , (x, y)) η(γ tj1,tj2,...,tjr−2 , (x, y))). − (cid:90)T(cid:90)herefore, by an extension of the Minkowski inequality (4) to r 2 func- tions, | P | ( (cid:90)(cid:90)η(γ , (x, y)) q dx dy)1/q R 2 ≤ | P − P\{ } | ((cid:90)(cid:90) η(γ , (x, y)) η(γ tj1 , (x, y)) q dx dy)1/q R 2 | P\{ } − P\{ } | + ( η(γ tj1 , (x, y)) η(γ tj1,tj2 , (x, y)) q dx dy)1/q (cid:90)(cid:90) R 2 + ... | P\{ } − P\{ } | + ( η(γ tj1,tj2,...,tjr−3 , (x, y)) η(γ tj1,tj2,...,tjr−2 , (x, y)) q dx dy)1/q. R 2 (13) We can now create the inequality which forms the base of the ﬁnal proof. 7 ##LTLine##

text: We

text: then

text: generalise

text: this

text: for

text: removing

text: any

text: number

text: of

text: points.

text: For

text: all

text: n

text: P

text: r

text: we

text: can

text: remove

text: n

text: points

text: from

text: using

text: abuse

text: of

text: notation

text: to

text: label

text: the

text: r

text: n

text: points

text: left

text: as

text: t

text: t

text: Then

text: there

text: exists

text: j

text: r

text: n

text: r

text: n

text: n

text: such

text: that

label: equation

text: γ

text: p

text: γ

text: p

text: r

text: n

text: P

text: P\{

text: Removing

text: r

text: points

text: from

text: we

text: are

text: left

text: with

text: the

text: path

text: γ

text: As

text: the

text: partition

text: only

text: has

text: three

text: points

text: left,

text: and

text: P\{

text: P\{

text: γ

text: γ

text: (tr),

text: the

text: path

text: only

text: has

text: two

text: points

text: left.

text: Therefore,

text: P\{

text: η(γ

text: (x,

text: y)))

text: P\{

text: for

text: all

text: (x,

text: y)

text: outside

text: γ

text: It

text: is

text: then

text: clear

text: that

text: for

text: all

text: (x,

text: y)

text: P

text: outside

text: γ

text: we

text: have

text: P

text: P

text: P\{

text: η(γ

text: (x,

text: y))

text: (η(γ

text: (x,

text: y))

text: η(γ

text: (x,

text: y))

text: P\{

text: P\{

text: +(η(γ

text: (x,

text: y))

text: η(γ

text: (x,

text: y)))

text: P\{

text: P\{

text: +(η(γ

text: (x,

text: y))

text: η(γ

text: (x,

text: y))).

text: by

text: an

text: extension

text: of

text: the

text: Minkowski

text: inequality

text: to

text: r

text: func-

text: tions,

text: P

text: (x,

text: y))

text: q

text: dx

text: R

text: P

text: P\{

text: η(γ

text: (x,

text: y))

text: η(γ

text: (x,

text: y))

text: q

text: dx

text: R

text: P\{

text: P\{

text: η(γ

text: (x,

text: y))

text: η(γ

text: (x,

text: y))

text: q

text: dx

text: R

text: P\{

text: P\{

text: η(γ

text: (x,

text: y))

text: η(γ

text: (x,

text: y))

text: q

text: dx

text: R

text: We

text: can

text: now

text: create

text: the

text: inequality

text: which

text: forms

text: the

text: base

text: of

text: the

text: ﬁnal

text: proof.

text: ##LTLine##


text: 4 Acknowledgements I would like to acknowledge the support of the EPSRC Vacation Bursary. I would like to thank my supervisor Horatio Boedihardjo for his support and insights. I also thank the referee for their useful suggestions and comments. 10

label: section

text: Acknowledgements

text: I

text: would

text: like

text: to

text: acknowledge

text: the

text: support

text: of

text: the

text: EPSRC

text: Vacation

text: Bursary.

text: I

text: would

text: like

text: to

text: thank

text: my

text: supervisor

text: Horatio

text: Boedihardjo

text: for

text: his

text: support

text: and

text: insights.

text: I

text: also

text: thank

text: the

text: referee

text: for

text: their

text: useful

text: suggestions

text: and

text: comments.


text: 5.2 Numerical solution Figure 11: The simulated electronics, radiator and cooling pipe (in cm) (a) (b) Figure 12: Simulation (dimension is in cm) with COMSOL [8] : (a) Temperature of the model after 100 min of heating (b) Temperature of the model 140 min after the cooling is applied. The cooling is done at 10oC and the coolant is ﬂowing at 2 g/sec. 11 ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine##

text: Numerical

text: solution

text: Figure

text: The

text: simulated

text: electronics,

text: radiator

text: and

text: cooling

text: pipe

text: (in

text: cm)

text: (a)

text: (b)

text: Figure

text: Simulation

text: (dimension

text: is

text: in

text: cm)

text: with

text: COMSOL

text: (a)

text: Temperature

text: of

text: the

text: model

text: after

text: min

text: of

text: heating

text: (b)

text: Temperature

text: of

text: the

text: model

text: min

text: after

text: the

text: cooling

text: is

text: applied.

text: The

text: cooling

text: is

text: done

text: at

text: and

text: the

text: coolant

text: is

text: ﬂowing

text: at

text: g/sec.

text: ##LTFigure##

label: figure

font: default

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##


text: Figure 5: Isotherms of CO showing the two-phase region and critical point. 2 ◦ ◦ the LC-TPC may be even higher by 5 C-10 C. It depends upon the climate and also on the fact that the TPC will be at the center of ILD and will share the heat generated by the other devices as well. The sampling frequency of the electronics for the experiments at LP-TPC is presently set at 25 MHz, that is, time binning of 40 ns. At LC-TPC, where the event generation would be in a much higher rate, the sampling frequency could be even higher to ensure better time reconstruction over a large drift length. Though the exact value is not yet decided, a higher sampling rate will very likely lead to an increase in the power consumption of the electronics. 3 Two-phase CO cooling for a single MM module 2 In this section, the usefulness of CO cooling and the results are discussed. 2 3.1 Experimental setup In order to estimate the heating and the eﬀect of cooling, one temperature sensor is plugged in each FEC and the FEM to read temperature at regular time intervals. To apply cooling, the coolant is allowed to ﬂow through a small pipe. The ﬁve-fold pipe is equipped with six radiators (ﬁgure 1) and the radiators are coupled with each FEC to ensure good thermal contact. The material of the cooling pipe is chosen to be stainless steel, which are essentially non-ferromagnetic to avoid any magnetic induction (so that the setup is ready to be used for the LP-TPC [5]). The pipe has standard inner and outer diameters of 0.079 cm and 0.159 cm respectively. Finally, the inlet and outlet of the entire cooling pipe are placed at the two diagonally opposite corners of the module. 3.2 Why two-phase CO cooling 2 In the beginning, air cooling was tried. However, during the beam test, it was found that the cooling provided by air is not suﬃcient. This is because, air is not an eﬃcient coolant as it has low thermal conductivity and high viscosity. Two-phase CO is considered as an eﬃcient coolant. The principal 2 advantage of using two-phase cooling is that, the heat transfer happens during phase change and so the temperature remains constant. This property allows to maintain a very stable temperature during experiment. The temperature range at which the detector is desired to be cooled is quite below the critical temperature and coinside with the two-phase zone of CO (see ﬁg 5). 2 Steeper slope of the vapor pressure curve (see ﬁgure 6a) helps to change the temperature in smaller steps. It also means a smaller temperature variations against pressure ﬂuctuations. Low ratio of liquid to gas densities results in a more homogeneous two-phase ﬂow (see ﬁgure 6b). The ratio plays an important role for an evaporator as it determines the ﬂow pattern and hence the heat transfer coeﬃcients [11]. The liquid CO is well compressible in the range of temperature where the experiment is carried out. Besides 2 5 ##LTFigure## ##LTLine## ##LTLine##

text: Figure

text: Isotherms

text: of

text: CO

text: showing

text: the

text: two-phase

text: region

text: and

text: critical

text: point.

text: the

text: LC-TPC

text: may

text: be

text: even

text: higher

text: by

text: C.

text: It

text: depends

text: upon

text: the

text: climate

text: and

text: also

text: on

text: the

text: fact

text: that

text: the

text: TPC

text: will

text: be

text: at

text: the

text: center

text: of

text: ILD

text: and

text: will

text: share

text: the

text: heat

text: generated

text: by

text: the

text: other

text: devices

text: as

text: well.

text: The

text: sampling

text: frequency

text: of

text: the

text: electronics

text: for

text: the

text: experiments

text: at

text: LP-TPC

text: is

text: presently

text: set

text: at

text: MHz,

text: that

text: is,

text: time

text: binning

text: of

text: ns.

text: At

text: LC-TPC,

text: where

text: the

text: event

text: generation

text: would

text: be

text: in

text: a

text: much

text: higher

text: rate,

text: the

text: sampling

text: frequency

text: could

text: be

text: even

text: higher

text: to

text: ensure

text: better

text: time

text: reconstruction

text: over

text: a

text: large

text: drift

text: length.

text: Though

text: the

text: exact

text: value

text: is

text: not

text: yet

text: decided,

text: a

text: higher

text: sampling

text: rate

text: will

text: very

text: likely

text: lead

text: to

text: an

text: increase

text: in

text: the

text: power

text: consumption

text: of

text: the

text: electronics.

text: Two-phase

text: CO

text: cooling

text: for

text: a

text: single

text: MM

text: module

text: In

text: this

text: section,

text: the

text: usefulness

text: of

text: CO

text: cooling

text: and

text: the

text: results

text: are

text: discussed.

text: Experimental

text: setup

text: In

text: order

text: to

text: estimate

text: the

text: heating

text: and

text: the

text: eﬀect

text: of

text: cooling,

text: one

text: temperature

text: sensor

text: is

text: plugged

text: in

text: each

text: FEC

text: and

text: the

text: FEM

text: to

text: read

text: temperature

text: at

text: regular

text: time

text: intervals.

text: To

text: apply

text: cooling,

text: the

text: coolant

text: is

text: allowed

text: to

text: ﬂow

text: through

text: a

text: small

text: pipe.

text: The

text: ﬁve-fold

text: pipe

text: is

text: equipped

text: with

text: six

text: radiators

text: (ﬁgure

text: and

text: the

text: radiators

text: are

text: coupled

text: with

text: each

text: FEC

text: to

text: ensure

text: good

text: thermal

text: contact.

text: The

text: material

text: of

text: the

text: cooling

text: pipe

text: is

text: chosen

text: to

text: be

text: stainless

text: steel,

text: which

text: are

text: essentially

text: non-ferromagnetic

text: to

text: avoid

text: any

text: magnetic

text: induction

text: (so

text: that

text: the

text: setup

text: is

text: ready

text: to

text: be

text: used

text: for

text: the

text: LP-TPC

text: The

text: pipe

text: has

text: standard

text: inner

text: and

text: outer

text: diameters

text: of

text: cm

text: and

text: cm

text: respectively.

text: Finally,

text: the

text: inlet

text: and

text: outlet

text: of

text: the

text: entire

text: cooling

text: pipe

text: are

text: placed

text: at

text: the

text: two

text: diagonally

text: opposite

text: corners

text: of

text: the

text: module.

text: Why

text: two-phase

text: CO

text: cooling

text: In

text: the

text: beginning,

text: air

text: cooling

text: was

text: tried.

text: However,

text: during

text: the

text: beam

text: test,

text: it

text: was

text: found

text: that

text: the

text: cooling

text: provided

text: by

text: air

text: is

text: not

text: suﬃcient.

text: This

text: is

text: because,

text: air

text: is

text: not

text: an

text: eﬃcient

text: coolant

text: as

text: it

text: has

text: low

text: thermal

text: conductivity

text: and

text: high

text: viscosity.

text: Two-phase

text: CO

text: is

text: considered

text: as

text: an

text: eﬃcient

text: coolant.

text: The

text: principal

text: advantage

text: of

text: using

text: two-phase

text: cooling

text: is

text: that,

text: the

text: heat

text: transfer

text: happens

text: during

text: phase

text: change

text: and

text: so

text: the

text: temperature

text: remains

text: constant.

text: This

text: property

text: allows

text: to

text: maintain

text: a

text: very

text: stable

text: temperature

text: during

text: experiment.

text: The

text: temperature

text: range

text: at

text: which

text: the

text: detector

text: is

text: desired

text: to

text: be

text: cooled

text: is

text: quite

text: below

text: the

text: critical

text: temperature

text: and

text: coinside

text: with

text: the

text: two-phase

text: zone

text: of

text: CO

text: (see

text: ﬁg

text: Steeper

text: slope

text: of

text: the

text: vapor

text: pressure

text: curve

text: (see

text: ﬁgure

text: helps

text: to

text: change

text: the

text: temperature

text: in

text: smaller

text: steps.

text: It

text: also

text: means

text: a

text: smaller

text: temperature

text: variations

text: against

text: pressure

text: ﬂuctuations.

text: Low

text: ratio

text: of

text: liquid

text: to

text: gas

text: densities

text: results

text: in

text: a

text: more

text: homogeneous

text: two-phase

text: ﬂow

text: (see

text: ﬁgure

text: The

text: ratio

text: plays

text: an

text: important

text: role

text: for

text: an

text: evaporator

text: as

text: it

text: determines

text: the

text: ﬂow

text: pattern

text: and

text: hence

text: the

text: heat

text: transfer

text: coeﬃcients

text: The

text: liquid

text: CO

text: is

text: well

text: compressible

text: in

text: the

text: range

text: of

text: temperature

text: where

text: the

text: experiment

text: is

text: carried

text: out.

text: Besides

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##


text: 3.3 The cooling system (b) (a) Figure 6: Two useful thermal properties of few commonly used refrigerants. Here R-744 implies CO . 2 (a) Change of vapor pressure with temperature. (b) Change of the ratio of liquid to vapor density with temperature. The results are taken from [11] that, liquid CO has large latent heat which suﬃces low mass ﬂow rate. Low viscosity provides uniform 2 ﬂow. High compressibility is another interesting property of liquid CO which provides high pressure 2 drop which in turn provides the advantage of small tubing. It should be mentioned in this context, that ◦ ◦ the cooling pipe is not supposed to be cooled down below 10 C. Because below 10 C, ambient water vapor may start to condense around the cooling pipe. If the condensation goes on, the water will ﬂow to the electronics causing damage to the readout of the detector. 3.3 The cooling system The cooling structure, which consists of the cooling pipe and six radiators, is tested with a cooling system [12] called TRACI (Transportable Refrigeration Apparatus for CO Investigation), at Nikhef, 2 Amsterdam in December 2013. The volume of the system is 1.7 l and the CO ﬁlling is 1 kg. The system 2 consists of chiller, CO pump, CO condenser, accumulator and a gas distribution box (described in the 2 2 schematic, ﬁgure 7) [13]. The distribution (or bypass) box is connected to system by a long concentric hose. The inlets and outlets to the cooling pipes are connected to the bypass box. Features of TRACI: 1. It circulates CO in close loop and does not throw out CO directly to the environment. 2 2 2. It can circulate CO in a stable pressure diﬀerence with a large range of operation. 2 3. Evaporation point of liquid CO can be varied over a long range by adjusting inlet pressure. A 2 calibration chart comes with the system. 4. It provides a long range of ﬂow rate control. 5. Adjusting the inlet and outlet knobs of the bypass box, pressure diﬀerence can be changed in desired way. 6. The interlock safety option takes care of pressure diﬀerence and automatically shuts down the system if it goes beyond the allowed range. Before ﬂowing CO , the cooling tubes are pumped out for around thirty minutes. In the meantime, 2 the electronic board of the MM module is powered on and it starts getting warm. Finally, CO ﬂow 2 is started. The embedded processor board starts taking signals from the temperature sensors and the temperature proﬁle is being written in a ﬁle. 6 ##LTFigure## ##LTFigure## ##LTLine## ##LTLine##

text: The

text: cooling

text: system

text: (b)

text: (a)

text: Figure

text: Two

text: useful

text: thermal

text: properties

text: of

text: few

text: commonly

text: used

text: refrigerants.

text: Here

text: implies

text: CO

text: (a)

text: Change

text: of

text: vapor

text: pressure

text: with

text: temperature.

text: (b)

text: Change

text: of

text: the

text: ratio

text: of

text: liquid

text: to

text: vapor

text: density

text: with

text: temperature.

text: The

text: results

text: are

text: taken

text: from

text: that,

text: liquid

text: CO

text: has

text: large

text: latent

text: heat

text: which

text: suﬃces

text: low

text: mass

text: ﬂow

text: rate.

text: Low

text: viscosity

text: provides

text: uniform

text: ﬂow.

text: High

text: compressibility

text: is

text: another

text: interesting

text: property

text: of

text: liquid

text: CO

text: which

text: provides

text: high

text: pressure

text: drop

text: which

text: in

text: turn

text: provides

text: the

text: advantage

text: of

text: small

text: tubing.

text: It

text: should

text: be

text: mentioned

text: in

text: this

text: context,

text: that

text: the

text: cooling

text: pipe

text: is

text: not

text: supposed

text: to

text: be

text: cooled

text: down

text: below

text: C.

text: Because

text: below

text: C,

text: ambient

text: water

text: vapor

text: may

text: start

text: to

text: condense

text: around

text: the

text: cooling

text: pipe.

text: If

text: the

text: condensation

text: goes

text: on,

text: the

text: water

text: will

text: ﬂow

text: to

text: the

text: electronics

text: causing

text: damage

text: to

text: the

text: readout

text: of

text: the

text: detector.

text: The

text: cooling

text: system

text: The

text: cooling

text: structure,

text: which

text: consists

text: of

text: the

text: cooling

text: pipe

text: and

text: six

text: radiators,

text: is

text: tested

text: with

text: a

text: cooling

text: system

text: called

text: TRACI

text: (Transportable

text: Refrigeration

text: Apparatus

text: for

text: CO

text: Investigation),

text: at

text: Nikhef,

text: Amsterdam

text: in

text: December

text: The

text: volume

text: of

text: the

text: system

text: is

text: l

text: and

text: the

text: CO

text: ﬁlling

text: is

text: kg.

text: The

text: system

text: consists

text: of

text: chiller,

text: CO

text: pump,

text: CO

text: condenser,

text: accumulator

text: and

text: a

text: gas

text: distribution

text: box

text: (described

text: in

text: the

text: schematic,

text: ﬁgure

text: The

text: distribution

text: (or

text: bypass)

text: box

text: is

text: connected

text: to

text: system

text: by

text: a

text: long

text: concentric

text: hose.

text: The

text: inlets

text: and

text: outlets

text: to

text: the

text: cooling

text: pipes

text: are

text: connected

text: to

text: the

text: bypass

text: box.

text: Features

text: of

text: TRACI:

text: It

text: circulates

text: CO

text: in

text: close

text: loop

text: and

text: does

text: not

text: throw

text: out

text: CO

text: directly

text: to

text: the

text: environment.

text: It

text: can

text: circulate

text: CO

text: in

text: a

text: stable

text: pressure

text: diﬀerence

text: with

text: a

text: large

text: range

text: of

text: operation.

text: Evaporation

text: point

text: of

text: liquid

text: CO

text: can

text: be

text: varied

text: over

text: a

text: long

text: range

text: by

text: adjusting

text: inlet

text: pressure.

text: A

text: calibration

text: chart

text: comes

text: with

text: the

text: system.

text: It

text: provides

text: a

text: long

text: range

text: of

text: ﬂow

text: rate

text: control.

text: Adjusting

text: the

text: inlet

text: and

text: outlet

text: knobs

text: of

text: the

text: bypass

text: box,

text: pressure

text: diﬀerence

text: can

text: be

text: changed

text: in

text: desired

text: way.

text: The

text: interlock

text: safety

text: option

text: takes

text: care

text: of

text: pressure

text: diﬀerence

text: and

text: automatically

text: shuts

text: down

text: the

text: system

text: if

text: it

text: goes

text: beyond

text: the

text: allowed

text: range.

text: Before

text: ﬂowing

text: CO

text: the

text: cooling

text: tubes

text: are

text: pumped

text: out

text: for

text: around

text: thirty

text: minutes.

text: In

text: the

text: meantime,

text: the

text: electronic

text: board

text: of

text: the

text: MM

text: module

text: is

text: powered

text: on

text: and

text: it

text: starts

text: getting

text: warm.

text: Finally,

text: CO

text: ﬂow

text: is

text: started.

text: The

text: embedded

text: processor

text: board

text: starts

text: taking

text: signals

text: from

text: the

text: temperature

text: sensors

text: and

text: the

text: temperature

text: proﬁle

text: is

text: being

text: written

text: in

text: a

text: ﬁle.

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##


text: Figure 7: The layout of Transportable Refrigeration Apparatus for CO Investigation (TRACI) 2 Results: Figure 8 shows the temperature variation of the ﬁfth FEC. The ﬁrst ﬂat part of the curve implies that the cooling has been started. Soon after when cooling is stopped, the curve starts to rise ◦ sharply and reaches nearly 65 C. Electronics is then switched oﬀ. After few minutes when the electronics is started again, the temperature also starts to rise. Then cooling is circulated and again temperature goes to a stable value. The variation in temperature agrees with the pictures of the pad-plane taken by a thermal camera [14, 15]. This plot contains both temperature rise and fall. Simulation (section 5) has been carried out to study similar temperature variation for one module and the result is compared with this plot. 4 Two-phase CO cooling at Large Prototype-TPC 2 An attempt is taken to apply the same cooling technique for all seven MM modules, which are commissioned at the LP-TPC end-plane for test beam [15]. Setup: After the experiment with a single module, the cooling structure is updated before it is ap- plied for all seven modules at the LP-TPC. The present setup replaced the old air-cooling radiators by aluminum block radiators. The new shape of the radiator with cooling pipe is shown in ﬁgure 1 and in ﬁgure 9a. The pipe is placed in such a way that it may not rest along the length of the FECs but rather in a transverse direction. This makes the cooling uniform. Seven MM modules are prepared for commissioning at the LP-TPC end-plate. Each of them are equipped with a set of radiators and a cooling pipe. The circulation of CO is divided in parallel to all the modules so that the ﬂow could be 2 uniform for all of the modules and also the heat transfer may happen only during phase change which ensures uniform cooling. The distribution of the parallel ﬂow is accomplished with two clarinets. One distributes the inlets while the other is for the outlets. The clarinets are shown in ﬁgure 9b. Each part of the cooling structure is made up of non-ferromagnetic materials because they would be used under 1 T magnetic ﬁeld. The cooling system is kept far away from the TPC and connected to the bypass box with a long hose. The same setup is applied during test beam in February 2014 and March 2015. It performed consistently each time for more than 80 hours. 7 ##LTFigure## ##LTLine## ##LTLine##

text: Figure

text: The

text: layout

text: of

text: Transportable

text: Refrigeration

text: Apparatus

text: for

text: CO

text: Investigation

text: (TRACI)

text: Results:

text: Figure

text: shows

text: the

text: temperature

text: variation

text: of

text: the

text: ﬁfth

text: FEC.

text: The

text: ﬁrst

text: ﬂat

text: part

text: of

text: the

text: curve

text: implies

text: that

text: the

text: cooling

text: has

text: been

text: started.

text: Soon

text: after

text: when

text: cooling

text: is

text: stopped,

text: the

text: curve

text: starts

text: to

text: rise

text: sharply

text: and

text: reaches

text: nearly

text: C.

text: Electronics

text: is

text: then

text: switched

text: oﬀ.

text: After

text: few

text: minutes

text: when

text: the

text: electronics

text: is

text: started

text: again,

text: the

text: temperature

text: also

text: starts

text: to

text: rise.

text: Then

text: cooling

text: is

text: circulated

text: and

text: again

text: temperature

text: goes

text: to

text: a

text: stable

text: value.

text: The

text: variation

text: in

text: temperature

text: agrees

text: with

text: the

text: pictures

text: of

text: the

text: pad-plane

text: taken

text: by

text: a

text: thermal

text: camera

text: This

text: plot

text: contains

text: both

text: temperature

text: rise

text: and

text: fall.

text: Simulation

text: (section

text: has

text: been

text: carried

text: out

text: to

text: study

text: similar

text: temperature

text: variation

text: for

text: one

text: module

text: and

text: the

text: result

text: is

text: compared

text: with

text: this

text: plot.

text: Two-phase

text: CO

text: cooling

text: at

text: Large

text: Prototype-TPC

text: An

text: attempt

text: is

text: taken

text: to

text: apply

text: the

text: same

text: cooling

text: technique

text: for

text: all

text: seven

text: MM

text: modules,

text: which

text: are

text: commissioned

text: at

text: the

text: LP-TPC

text: end-plane

text: for

text: test

text: beam

text: Setup:

text: After

text: the

text: experiment

text: with

text: a

text: single

text: module,

text: the

text: cooling

text: structure

text: is

text: updated

text: before

text: it

text: is

text: ap-

text: plied

text: for

text: all

text: seven

text: modules

text: at

text: the

text: LP-TPC.

text: The

text: present

text: setup

text: replaced

text: the

text: old

text: air-cooling

text: radiators

text: by

text: aluminum

text: block

text: radiators.

text: The

text: new

text: shape

text: of

text: the

text: radiator

text: with

text: cooling

text: pipe

text: is

text: shown

text: in

text: ﬁgure

text: and

text: in

text: ﬁgure

text: The

text: pipe

text: is

text: placed

text: in

text: such

text: a

text: way

text: that

text: it

text: may

text: not

text: rest

text: along

text: the

text: length

text: of

text: the

text: FECs

text: but

text: rather

text: in

text: a

text: transverse

text: direction.

text: This

text: makes

text: the

text: cooling

text: uniform.

text: Seven

text: MM

text: modules

text: are

text: prepared

text: for

text: commissioning

text: at

text: the

text: LP-TPC

text: end-plate.

text: Each

text: of

text: them

text: are

text: equipped

text: with

text: a

text: set

text: of

text: radiators

text: and

text: a

text: cooling

text: pipe.

text: The

text: circulation

text: of

text: CO

text: is

text: divided

text: in

text: parallel

text: to

text: all

text: the

text: modules

text: so

text: that

text: the

text: ﬂow

text: could

text: be

text: uniform

text: for

text: all

text: of

text: the

text: modules

text: and

text: also

text: the

text: heat

text: transfer

text: may

text: happen

text: only

text: during

text: phase

text: change

text: which

text: ensures

text: uniform

text: cooling.

text: The

text: distribution

text: of

text: the

text: parallel

text: ﬂow

text: is

text: accomplished

text: with

text: two

text: clarinets.

text: One

text: distributes

text: the

text: inlets

text: while

text: the

text: other

text: is

text: for

text: the

text: outlets.

text: The

text: clarinets

text: are

text: shown

text: in

text: ﬁgure

text: Each

text: part

text: of

text: the

text: cooling

text: structure

text: is

text: made

text: up

text: of

text: non-ferromagnetic

text: materials

text: because

text: they

text: would

text: be

text: used

text: under

text: T

text: magnetic

text: ﬁeld.

text: The

text: cooling

text: system

text: is

text: kept

text: far

text: away

text: from

text: the

text: TPC

text: and

text: connected

text: to

text: the

text: bypass

text: box

text: with

text: a

text: long

text: hose.

text: The

text: same

text: setup

text: is

text: applied

text: during

text: test

text: beam

text: in

text: February

text: and

text: March

text: It

text: performed

text: consistently

text: each

text: time

text: for

text: more

text: than

text: hours.

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##


text: Figure 10: Temperature proﬁle of all the FECs and the FEM of a MM module during test beam in March 2015 Test result: During the test beam in 2014 and in 2015, the performance of TRACI and the cooling structure is quite satisfactory. The temperature from the temperature sensors from each FECs and the FEMs of all 7 modules is recorded. When plotted, they show a stable temperature over hours during data taking. In ﬁgure 10, temperature proﬁle (from test beam March 2015) of all the six FECs and the FEM of a MM module is shown. The ﬁgure shows that the setup is successfully implemented to limit the ◦ temperature below 28 C. Temperature of the FEM is seen to be higher than the FECs as it consumes more power. This does not concern too much as the FEM, the outer most component, is relatively far ◦ from the pad plane. It may be noted that the temperature is stable within 0.2 C during the experiment over days. 5 Numerical study for two-phase CO cooling 2 A generalized study of the cooling process is always desirable because it can help to ﬁgure out the inﬂuence of the varying parameters and to justify the necessity of the process as well. Moreover, by carrying out a systematic study of the available parameter space, it is also possible to optimize the conceived process in a comprehensive manner. For this reason, numerical simulation of the whole process has always been appreciated upto an achievable limit of precision. In SINP, Kolkata, an attempt to simulate the basic physics of the cooling process has been carried out by making use of COMSOL Multiphysics [8]. 5.1 Mathematical Modeling The heat transfer processes for a MM module has been mathematically formulated by considering it to be a case of three-dimensional heat ﬂow in a body by the vector method. The MM module has been considered to be a combination of several isotropic bodies having its own thermal conductivity K, density ρ and speciﬁc heat c. 9 ##LTFigure## ##LTLine## ##LTLine##

text: Figure

text: Temperature

text: proﬁle

text: of

text: all

text: the

text: FECs

text: and

text: the

text: FEM

text: of

text: a

text: MM

text: module

text: during

text: test

text: beam

text: in

text: March

text: Test

text: result:

text: During

text: the

text: test

text: beam

text: in

text: and

text: in

text: the

text: performance

text: of

text: TRACI

text: and

text: the

text: cooling

text: structure

text: is

text: quite

text: satisfactory.

text: The

text: temperature

text: from

text: the

text: temperature

text: sensors

text: from

text: each

text: FECs

text: and

text: the

text: FEMs

text: of

text: all

text: modules

text: is

text: recorded.

text: When

text: plotted,

text: they

text: show

text: a

text: stable

text: temperature

text: over

text: hours

text: during

text: data

text: taking.

text: In

text: ﬁgure

text: temperature

text: proﬁle

text: (from

text: test

text: beam

text: March

text: of

text: all

text: the

text: six

text: FECs

text: and

text: the

text: FEM

text: of

text: a

text: MM

text: module

text: is

text: shown.

text: The

text: ﬁgure

text: shows

text: that

text: the

text: setup

text: is

text: successfully

text: implemented

text: to

text: limit

text: the

text: temperature

text: below

text: C.

text: Temperature

text: of

text: the

text: FEM

text: is

text: seen

text: to

text: be

text: higher

text: than

text: the

text: FECs

text: as

text: it

text: consumes

text: more

text: power.

text: This

text: does

text: not

text: concern

text: too

text: much

text: as

text: the

text: FEM,

text: the

text: outer

text: most

text: component,

text: is

text: relatively

text: far

text: from

text: the

text: pad

text: plane.

text: It

text: may

text: be

text: noted

text: that

text: the

text: temperature

text: is

text: stable

text: within

text: C

text: during

text: the

text: experiment

text: over

text: days.

text: Numerical

text: study

text: for

text: two-phase

text: CO

text: cooling

text: A

text: generalized

text: study

text: of

text: the

text: cooling

text: process

text: is

text: always

text: desirable

text: because

text: it

text: can

text: help

text: to

text: ﬁgure

text: out

text: the

text: inﬂuence

text: of

text: the

text: varying

text: parameters

text: and

text: to

text: justify

text: the

text: necessity

text: of

text: the

text: process

text: as

text: well.

text: Moreover,

text: by

text: carrying

text: out

text: a

text: systematic

text: study

text: of

text: the

text: available

text: parameter

text: space,

text: it

text: is

text: also

text: possible

text: to

text: optimize

text: the

text: conceived

text: process

text: in

text: a

text: comprehensive

text: manner.

text: For

text: this

text: reason,

text: numerical

text: simulation

text: of

text: the

text: whole

text: process

text: has

text: always

text: been

text: appreciated

text: upto

text: an

text: achievable

text: limit

text: of

text: precision.

text: In

text: SINP,

text: Kolkata,

text: an

text: attempt

text: to

text: simulate

text: the

text: basic

text: physics

text: of

text: the

text: cooling

text: process

text: has

text: been

text: carried

text: out

text: by

text: making

text: use

text: of

text: COMSOL

text: Multiphysics

text: Mathematical

text: Modeling

text: The

text: heat

text: transfer

text: processes

text: for

text: a

text: MM

text: module

text: has

text: been

text: mathematically

text: formulated

text: by

text: considering

text: it

text: to

text: be

text: a

text: case

text: of

text: three-dimensional

text: heat

text: ﬂow

text: in

text: a

text: body

text: by

text: the

text: vector

text: method.

text: The

text: MM

text: module

text: has

text: been

text: considered

text: to

text: be

text: a

text: combination

text: of

text: several

text: isotropic

text: bodies

text: having

text: its

text: own

text: thermal

text: conductivity

text: K,

text: density

text: ρ

text: and

text: speciﬁc

text: heat

text: c.

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##