text: Lemma 10 Consider a tree with no regular cores and a single small core v, such that v has three ⊆ | | ≥ ⊆ standard legs. If L V contains a local set for v and L 3, then L is a landmark set. If L V | | ≤ contains a local set for v and L 2, then L is a landmark set if and only if L consists of the two vertices of two short legs of v. | | ≥ ∈ Proof. Assume that L 3. By Lemma 8, if v L, or if every leg has a vertex of L, we are ∈ done. Otherwise, consider a pair x, y / L. If d(x, u) = d(y, u), then by Lemma 3 there are two separations between x and y as L contains a local set for the g-legs of u. If d(x, u) < d(y, u), and the two legs that are not the leg of y have at least two vertices of L, we are done too (by Claim 5). | | ≥ As L 3, the remaining case is that the leg of y has at least two vertices in L, while one of the other legs of u has one vertex of L, and it separates x and y. At most one vertex of the leg of y can have equal distances to x and y, and therefore, any other vertex of the leg of y which is in L separates them as well, and we assumed there is at least one such vertex for this case. A local set cannot contain less than two vertices by its properties, and in this case there are ∈ two legs with one vertex of L each, one standard leg without any vertices of L, and v / L. Every ∈ | | ∈ ≥ vertex of L separates every pair x, y / L, as L = 2. For every leg (cid:96), if (cid:96)i L for i 1 then (cid:96) has − ≥ i vertices, as otherwise (cid:96)i does not separate (cid:96)i 1 and (cid:96)i+1 if i 2, and it does not separate v and (cid:96)2 if i = 1. As there is a leg with a type (s, 0) solution, the solution of (cid:96) is of type (s, 2) or (s, 3), and since it has one vertex of L, the solution must be of type (s, 3). Thus, i = 1, and (cid:96) is short. On the other hand, if L consists of two vertices of short legs, the remaining vertices are v and the vertices of one leg (cid:96)˜, each having a diﬀerent distance to the two vertices of L (this distance is 1 for v, and i + 1 for (cid:96)˜i). Consider a tree with two small cores v and u, and no regular cores. The tree consists of a path between u and v, and each of them has two standard legs, a short leg and another standard leg which can be short or long. For each small core, the path to the other small core and its legs can be seen as its modiﬁed leg. ⊆ Lemma 11 Consider a tree with no regular cores and two small cores v and u. A set L V is a landmark set if and only if it contains a local set for the three g-legs of v and it is a local set for the three g-legs of u. A minimal landmark set, with respect to set inclusion, will contain at most one vertex on the path between u and v (excluding u and v), and at most two vertices of each long leg of u and v. Proof. Since the tree can be seen as a core and its three g-legs in two way, L must contain a local set for the g-legs of u and for the g-legs of v. Assume now that L contains local sets for the two sets of g-legs. If for at least one small core, ∈ each of its g-legs has a vertex of L, we are done by the ﬁrst part of Lemma 8. Moreover, if u L ∈ or v L, we are done by the fourth part of Lemma 8. Since a modiﬁed leg has at least one vertex of a local set (by property 2 of local sets) we ﬁnd that each of u and v has a standard leg without any vertex of L, and every modiﬁed leg has at least two vertices of L. If L contains a vertex z on the path between u and v (excluding u and v), for the sake of the proof we see z as having two g-legs (one containing u as a small core, and the other one containing v as a small core). As each of u and v has a standard leg with at least one vertex in L, each g-leg ∈ ∈ of z has at least one vertex of L. Consider two vertices x, y / L. As z L, each of x and y is on a g-leg of z. If x and y are on the same g-leg, assume (without loss of generality) that this is the g-leg containing u. As the solution of one of the standard legs of u is of type (s, 0), its other leg has 9

text: Lemma

text: Consider

text: a

text: tree

text: with

text: no

text: regular

text: cores

text: and

text: a

text: single

text: small

text: core

text: v,

text: such

text: that

text: v

text: has

text: three

text: standard

text: legs.

text: If

text: L

text: V

text: contains

text: a

text: local

text: set

text: for

text: v

text: and

text: L

text: then

text: L

text: is

text: a

text: landmark

text: set.

text: If

text: L

text: V

text: contains

text: a

text: local

text: set

text: for

text: v

text: and

text: L

text: then

text: L

text: is

text: a

text: landmark

text: set

text: if

text: and

text: only

text: if

text: L

text: consists

text: of

text: the

text: two

text: vertices

text: of

text: two

text: short

text: legs

text: of

text: v.

text: Proof.

text: Assume

text: that

text: L

text: By

text: Lemma

text: if

text: v

text: L,

text: or

text: if

text: every

text: leg

text: has

text: a

text: vertex

text: of

text: L,

text: we

text: are

text: done.

text: Otherwise,

text: consider

text: a

text: pair

text: x,

text: y

text: L.

text: If

text: d(x,

text: u)

text: d(y,

text: u),

text: then

text: by

text: Lemma

text: there

text: are

text: two

text: separations

text: between

text: x

text: and

text: y

text: as

text: L

text: contains

text: a

text: local

text: set

text: for

text: the

text: g-legs

text: of

text: u.

text: If

text: d(x,

text: u)

text: d(y,

text: u),

text: and

text: the

text: two

text: legs

text: that

text: are

text: not

text: the

text: leg

text: of

text: y

text: have

text: at

text: least

text: two

text: vertices

text: of

text: L,

text: we

text: are

text: done

text: too

text: (by

text: Claim

text: As

text: L

text: the

text: remaining

text: case

text: is

text: that

text: the

text: leg

text: of

text: y

text: has

text: at

text: least

text: two

text: vertices

text: in

text: L,

text: while

text: one

text: of

text: the

text: other

text: legs

text: of

text: u

text: has

text: one

text: vertex

text: of

text: L,

text: and

text: it

text: separates

text: x

text: and

text: y.

text: At

text: most

text: one

text: vertex

text: of

text: the

text: leg

text: of

text: y

text: can

text: have

text: equal

text: distances

text: to

text: x

text: and

text: y,

text: and

text: therefore,

text: any

text: other

text: vertex

text: of

text: the

text: leg

text: of

text: y

text: which

text: is

text: in

text: L

text: separates

text: them

text: as

text: well,

text: and

text: we

text: assumed

text: there

text: is

text: at

text: least

text: one

text: such

text: vertex

text: for

text: this

text: case.

text: A

text: local

text: set

text: cannot

text: contain

text: less

text: than

text: two

text: vertices

text: by

text: its

text: properties,

text: and

text: in

text: this

text: case

text: there

text: are

text: two

text: legs

text: with

text: one

text: vertex

text: of

text: L

text: each,

text: one

text: standard

text: leg

text: without

text: any

text: vertices

text: of

text: L,

text: and

text: v

text: L.

text: Every

text: vertex

text: of

text: L

text: separates

text: every

text: pair

text: x,

text: y

text: L,

text: as

text: L

text: For

text: every

text: leg

text: if

text: L

text: for

text: i

text: then

text: has

text: i

text: vertices,

text: as

text: otherwise

text: does

text: not

text: separate

text: and

text: if

text: i

text: and

text: it

text: does

text: not

text: separate

text: v

text: and

text: if

text: i

text: As

text: there

text: is

text: a

text: leg

text: with

text: a

text: type

text: (s,

text: solution,

text: the

text: solution

text: of

text: is

text: of

text: type

text: (s,

text: or

text: (s,

text: and

text: since

text: it

text: has

text: one

text: vertex

text: of

text: L,

text: the

text: solution

text: must

text: be

text: of

text: type

text: (s,

text: Thus,

text: i

text: and

text: is

text: short.

text: On

text: the

text: other

text: hand,

text: if

text: L

text: consists

text: of

text: two

text: vertices

text: of

text: short

text: legs,

text: the

text: remaining

text: vertices

text: are

text: v

text: and

text: the

text: vertices

text: of

text: one

text: leg

text: each

text: having

text: a

text: diﬀerent

text: distance

text: to

text: the

text: two

text: vertices

text: of

text: L

text: (this

text: distance

text: is

text: for

text: v,

text: and

text: i

text: for

text: Consider

text: a

text: tree

text: with

text: two

text: small

text: cores

text: v

text: and

text: u,

text: and

text: no

text: regular

text: cores.

text: The

text: tree

text: consists

text: of

text: a

text: path

text: between

text: u

text: and

text: v,

text: and

text: each

text: of

text: them

text: has

text: two

text: standard

text: legs,

text: a

text: short

text: leg

text: and

text: another

text: standard

text: leg

text: which

text: can

text: be

text: short

text: or

text: long.

text: For

text: each

text: small

text: core,

text: the

text: path

text: to

text: the

text: other

text: small

text: core

text: and

text: its

text: legs

text: can

text: be

text: seen

text: as

text: its

text: modiﬁed

text: leg.

text: Lemma

text: Consider

text: a

text: tree

text: with

text: no

text: regular

text: cores

text: and

text: two

text: small

text: cores

text: v

text: and

text: u.

text: A

text: set

text: L

text: V

text: is

text: a

text: landmark

text: set

text: if

text: and

text: only

text: if

text: it

text: contains

text: a

text: local

text: set

text: for

text: the

text: three

text: g-legs

text: of

text: v

text: and

text: it

text: is

text: a

text: local

text: set

text: for

text: the

text: three

text: g-legs

text: of

text: u.

text: A

text: minimal

text: landmark

text: set,

text: with

text: respect

text: to

text: set

text: inclusion,

text: will

text: contain

text: at

text: most

text: one

text: vertex

text: on

text: the

text: path

text: between

text: u

text: and

text: v

text: (excluding

text: u

text: and

text: v),

text: and

text: at

text: most

text: two

text: vertices

text: of

text: each

text: long

text: leg

text: of

text: u

text: and

text: v.

text: Proof.

text: Since

text: the

text: tree

text: can

text: be

text: seen

text: as

text: a

text: core

text: and

text: its

text: three

text: g-legs

text: in

text: two

text: way,

text: L

text: must

text: contain

text: a

text: local

text: set

text: for

text: the

text: g-legs

text: of

text: u

text: and

text: for

text: the

text: g-legs

text: of

text: v.

text: Assume

text: now

text: that

text: L

text: contains

text: local

text: sets

text: for

text: the

text: two

text: sets

text: of

text: g-legs.

text: If

text: for

text: at

text: least

text: one

text: small

text: core,

text: each

text: of

text: its

text: g-legs

text: has

text: a

text: vertex

text: of

text: L,

text: we

text: are

text: done

text: by

text: the

text: ﬁrst

text: part

text: of

text: Lemma

text: Moreover,

text: if

text: u

text: L

text: or

text: v

text: L,

text: we

text: are

text: done

text: by

text: the

text: fourth

text: part

text: of

text: Lemma

text: Since

text: a

text: modiﬁed

text: leg

text: has

text: at

text: least

text: one

text: vertex

text: of

text: a

text: local

text: set

text: (by

text: property

text: of

text: local

text: sets)

text: we

text: ﬁnd

text: that

text: each

text: of

text: u

text: and

text: v

text: has

text: a

text: standard

text: leg

text: without

text: any

text: vertex

text: of

text: L,

text: and

text: every

text: modiﬁed

text: leg

text: has

text: at

text: least

text: two

text: vertices

text: of

text: L.

text: If

text: L

text: contains

text: a

text: vertex

text: z

text: on

text: the

text: path

text: between

text: u

text: and

text: v

text: (excluding

text: u

text: and

text: v),

text: for

text: the

text: sake

text: of

text: the

text: proof

text: we

text: see

text: z

text: as

text: having

text: two

text: g-legs

text: (one

text: containing

text: u

text: as

text: a

text: small

text: core,

text: and

text: the

text: other

text: one

text: containing

text: v

text: as

text: a

text: small

text: core).

text: As

text: each

text: of

text: u

text: and

text: v

text: has

text: a

text: standard

text: leg

text: with

text: at

text: least

text: one

text: vertex

text: in

text: L,

text: each

text: g-leg

text: of

text: z

text: has

text: at

text: least

text: one

text: vertex

text: of

text: L.

text: Consider

text: two

text: vertices

text: x,

text: y

text: L.

text: As

text: z

text: L,

text: each

text: of

text: x

text: and

text: y

text: is

text: on

text: a

text: g-leg

text: of

text: z.

text: If

text: x

text: and

text: y

text: are

text: on

text: the

text: same

text: g-leg,

text: assume

text: (without

text: loss

text: of

text: generality)

text: that

text: this

text: is

text: the

text: g-leg

text: containing

text: u.

text: As

text: the

text: solution

text: of

text: one

text: of

text: the

text: standard

text: legs

text: of

text: u

text: is

text: of

text: type

text: (s,

text: its

text: other

text: leg

text: has


text: 2 F. Piscitelli: Novel Boron-10-based detectors for Neutron Scattering Science Fig. 1. The ESS long pulse compared to the pulses of existing spallation sources in the world and to the steady ﬂux available at reactor sources [1]. is to expand the detector performance mostly in terms of counting rate capability and spatial resolution, as well as cost-eﬀectiveness. The development of small area detectors is focused on reﬂectometry applications because is where the requirements are more challenging. In tab. 1 are summarized the features of the state of the art detector technology at existing facilities for chopper spectrometers and reﬂectometers. Instrument Facility active area spatial res. eﬃciency global rate local rate × × − − − (m m) (mm mm) (s 1) (s 1cm 2) chopper spectrometers × × ∼ IN5 [8] ILL 10 3 26 26 74% @ 1.8˚A - - reﬂectometers × × ∼ · · FIGARO [9] ILL 0.5 0.25 2 7.5 63% @ 2.5˚A 3 107 23 103 × × INTER [10] ISIS 0.2 0.2 1 1 - - - Table 1. Detectors features on instruments at existing facilities, where ﬁgures are publically available. Table 2 summarizes the requirements at ESS for large area detectors for chopper spectrometers and for small area detector for reﬂectometry applications. The ﬂux at sample must be also interpreted as the maximum rate the detector should tolerate. It is not unlikely that the full beam is completely reﬂected toward the detector in a neutron reﬂec- tometry experiment. ESS instrument type active area wavelength band spatial resolution time resolution max ﬂux at sample × − − (m2) (˚A) (mm mm) (µs) (s 1cm 2) × chopper spectrometers [11] 50 [0.8, 20] 20 20 10 107 ∼ × reﬂectometers [12] 1 [2, 23] 0.5 2 100 109 Table 2. Summary of generalized detector requirements for chopper spectrometers and reﬂetometers at ESS. The Multi-Grid [13,14] prototype has been developed in the framework of the collaboration between ILL and ESS in order to address the problem of the 3He shortage for large area detectors. It is a gaseous detector that contains 30 10B C layers as neutron converters. Each neutron can be converted in one of these layers that are crossed orthogo- 4 nally by the neutron beam. Several prototypes using the Multi-Grid design have been built and tested, showing the feasibility of such a design for large area coverage with a suitable neutron detection eﬃciency. On the other hand, the Multi-Blade [15,16] prototype, already introduced at ILL in 2005 [17], is a small area detector which wants to push the limit of spatial resolution beyond that of 3He-based detectors for high ﬂux applications. In particular it has been conceived to be suitable for neutron reﬂectometry instruments, though potential applications could be broader than that. A detailed study on this prototype has been carried out at ILL showing the improvements in terms of detector features of such a design. The Multi-Grid and the Multi-Blade prototypes are described and the results are illustrated in the following sections. ##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##

label: paragraph

text: F.

text: Piscitelli:

text: Novel

text: detectors

text: for

text: Neutron

text: Scattering

text: Science

text: Fig.

text: The

label: caption

text: ESS

text: long

text: pulse

text: compared

text: to

text: the

text: pulses

text: of

text: existing

text: spallation

text: sources

text: in

text: the

text: world

text: and

text: to

text: the

text: steady

text: ﬂux

text: available

text: at

text: reactor

text: sources

text: is

text: to

text: expand

text: the

text: detector

text: performance

text: mostly

text: in

text: terms

text: of

text: counting

text: rate

text: capability

text: and

text: spatial

text: resolution,

text: as

text: well

text: as

text: cost-eﬀectiveness.

text: The

text: development

text: of

text: small

text: area

text: detectors

text: is

text: focused

text: on

text: reﬂectometry

text: applications

text: because

text: is

text: where

text: the

text: requirements

text: are

text: more

text: challenging.

text: In

text: tab.

text: are

text: summarized

text: the

text: features

text: of

text: the

text: state

text: of

text: the

text: art

text: detector

text: technology

text: at

text: existing

text: facilities

text: for

text: chopper

text: spectrometers

text: and

text: reﬂectometers.

text: Instrument

label: table

text: Facility

text: active

text: area

text: spatial

text: res.

text: eﬃciency

text: global

text: rate

text: local

text: rate

text: (m

text: m)

text: (mm

text: mm)

text: (s

text: (s

text: chopper

text: spectrometers

text: ILL

text: reﬂectometers

text: FIGARO

text: ILL

text: INTER

text: ISIS

text: Table

text: Detectors

text: features

text: on

text: instruments

text: at

text: existing

text: facilities,

text: where

text: ﬁgures

text: are

text: publically

text: available.

text: Table

text: summarizes

text: the

text: requirements

text: at

text: ESS

text: for

text: large

text: area

text: detectors

text: for

text: chopper

text: spectrometers

text: and

text: for

text: small

text: area

text: detector

text: for

text: reﬂectometry

text: applications.

text: The

text: ﬂux

text: at

text: sample

text: must

text: be

text: also

text: interpreted

text: as

text: the

text: maximum

text: rate

text: the

text: detector

text: should

text: tolerate.

text: It

text: is

text: not

text: unlikely

text: that

text: the

text: full

text: beam

text: is

text: completely

text: reﬂected

text: toward

text: the

text: detector

text: in

text: a

text: neutron

text: reﬂec-

text: tometry

text: experiment.

text: ESS

text: instrument

text: type

text: active

text: area

text: wavelength

text: band

text: spatial

text: resolution

text: time

text: resolution

text: max

text: ﬂux

text: at

text: sample

text: (˚A)

text: (mm

text: mm)

text: (µs)

text: (s

text: chopper

text: spectrometers

text: reﬂectometers

text: Table

text: Summary

text: of

text: generalized

text: detector

text: requirements

text: for

text: chopper

text: spectrometers

text: and

text: reﬂetometers

text: at

text: ESS.

text: The

text: Multi-Grid

text: prototype

text: has

text: been

text: developed

text: in

text: the

text: framework

text: of

text: the

text: collaboration

text: between

text: ILL

text: and

text: ESS

text: in

text: order

text: to

text: address

text: the

text: problem

text: of

text: the

text: shortage

text: for

text: large

text: area

text: detectors.

text: It

text: is

text: a

text: gaseous

text: detector

text: that

text: contains

text: C

text: layers

text: as

text: neutron

text: converters.

text: Each

text: neutron

text: can

text: be

text: converted

text: in

text: one

text: of

text: these

text: layers

text: that

text: are

text: crossed

text: orthogo-

text: nally

text: by

text: the

text: neutron

text: beam.

text: Several

text: prototypes

text: using

text: the

text: Multi-Grid

text: design

text: have

text: been

text: built

text: and

text: tested,

text: showing

text: the

text: feasibility

text: of

text: such

text: a

text: design

text: for

text: large

text: area

text: coverage

text: with

text: a

text: suitable

text: neutron

text: detection

text: eﬃciency.

text: On

text: the

text: other

text: hand,

text: the

text: Multi-Blade

text: prototype,

text: already

text: introduced

text: at

text: ILL

text: in

text: is

text: a

text: small

text: area

text: detector

text: which

text: wants

text: to

text: push

text: the

text: limit

text: of

text: spatial

text: resolution

text: beyond

text: that

text: of

text: detectors

text: for

text: high

text: ﬂux

text: applications.

text: In

text: particular

text: it

text: has

text: been

text: conceived

text: to

text: be

text: suitable

text: for

text: neutron

text: reﬂectometry

text: instruments,

text: though

text: potential

text: applications

text: could

text: be

text: broader

text: than

text: that.

text: A

text: detailed

text: study

text: on

text: this

text: prototype

text: has

text: been

text: carried

text: out

text: at

text: ILL

text: showing

text: the

text: improvements

text: in

text: terms

text: of

text: detector

text: features

text: of

text: such

text: a

text: design.

text: The

text: Multi-Grid

text: and

text: the

text: Multi-Blade

text: prototypes

text: are

text: described

text: and

text: the

text: results

text: are

text: illustrated

text: in

text: the

text: following

text: sections.

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text: 11 0 1OrangeWave Innovative Science, LLC, Moncks Corner, SC 29461 3Department of Physics & Space Sciences, Florida Institute of Technology, Melbourne, FL 32901 and Keywords: 4. DISCUSSION ture’s radiative characteristics (Figures 1 and 9) starkly contrasts such descriptions, i.e., those most common of Investigations of temporal radiative emission across “classical” jets (e.g., Allen et al. 1997; Shibata et al. 1997; large scale solar atmospheric temperature gradients and Chandrashekhar et al. 2014). co-spatial LOS magnetograms of our QS transient fea- Similarities of our observations to the chromospheric ture, as well as the CP site (i.e., footpoint shared with jet evolution detailed by Liu et al. (2011a), mainly, rup- two hot coronal loop arcades) provided evidence for turing loops followed by two evolving bright plasma plasma heating as a result of wide-spread magnetic ﬁeld patches (e.g., 14:10 UT – 14:12 UT in Figure 7), favors re-organization events. It was pointed out, e.g., Fig- the fan-spine magnetic topology typical of jets. Partic- ures 2 and 3, the QS transient revealed a dynamic “sub- § ularly, the outer spine region (see Figure 1 of Liu et al. structure” composition. Detailed investigations ( ), 2011a), where continuous injections of heated plasma oc- indicated the QS transient feature sub-structure was cur along newly reconnected ﬁeld lines, and leads to the comprised of episodically heated (Figures 3 and 6) open formation of a diﬀuse bundle of bright emission (Liu et al. and closed magnetic ﬁelds. Of distinct interest was that 2011a; Pariat et al. 2015), e.g., see Figure 9. Observed episodic heating events exhibited a similar trend, which radiative emission enhancements near the conclusion of we summarize as follows. each heating episode, localized to regions out of the Prior to each episode a re-organization of the underly- ing LOS magnetic environment of the QS transient fea- direct line of the diﬀuse bundle of bright 171 ˚A emis- ture (i.e., our CP, NP, and SP sites), such as emergence sion, are additionally considered here as supportive to a and/or cancelations of magnetic ﬂux densities, was ob- magnetic environment typical of jets. In relation to the served (Figure 4). Enhanced radiative emission at vari- fan-spine topology in 3D space, such events could indi- ous localized regions, established in conjunction with the cate the dome structure as streamlines of falling material QS transient feature (i.e.,, CP site, and/or NL and SL (Liu et al. 2011a), i.e., closed ﬁeld structures (Figure 9). features), followed. Commonly the appearance of either It is again emphasized, our QS transient maintains its a co-spatial, or adjacent spire-like feature, relative to radiat≈ive structuring, and visibly bright TR signature either or both the NL or SL features (e.g., Figure 8), over 35 min time-frame, which as previously pointed and/or the “splitting” of heated loop structures into two out is in stark contrast to that typical of “classical” disjoint patches of bright emission (e.g., Figure 7), ac- eruptive jets, as well as latter evolutions of the chromo- companied the localized radiative enhancements. Heat- spheric jet study of Liu et al. (2011a). Thereby, below ing episodes concluded with enhanced radiative emission we postulate to the source of our features unique, and events of the opposing footpoints of either or both the energetically stable radiative nature in a, speculated, NL and SL features. In that respect, below we speculate magnetic ﬁeld environment, where “classical” eruptive to the source of the episodic heating, and its relation to jets most commonly occur. ≈ our QS transient feature’s unique visibly bright radiative Episodic heating events, witnessed here on 3 min – signature. In compliment, discussions are presented to 5 min intervals (Figure 6), and the QS transients visibly the general magnetic ﬁeld environment and its temporal bright radiative signature over our observational time- evolution, as it relates to our QS transient feature. frame favors continuous type heating, with moderate re- Evidence for loop rupturing, and open magnetic ﬁelds connection rates (e.g., Pariat et al. 2010; Liu et al. 2011a; are suggestive to interchange reconnection events (e.g., Pariat et al. 2015). The slower, quasi-steady, recon- Madjarska et al. 2004; Krista et al. 2011), as well as re- nection rates and subsequent heating events would ex- ported wide-spread magnetic ﬁeld emergence and can- plain well the lack of evidence in our observational data celation events. Our observations of dynamically evolv- for high-speed injections, and signiﬁcant torsional mo- ing bright plasma patches and the formation of spire- tion of the TR bundle; consistent with QS expectations like structures following rupturing events are speculated (Tate Arbacher et al. 2015). It is speculated such no- as the occurrence of such reconnections, which leave in tions point to heated plasma injections, in relation to their wake newly opened and closed magnetic ﬁelds (e.g., spire structures with decreasing intensity distributions Pariat et al. 2015). Spire-like structures supportive of an away from the CP site, resulting from secondary ther- evaporating plasma ﬂow (e.g., see 14:38 UT of Figure 8), modynamic processes (Shibata et al. 1997; Shimojo & possibly indicate injections of heated plasma into the Shibata 2000) Such spire-like structure formation, which QS transient feature, likely along newly opened magnetic we emphasize includes observed collimated lower ends ﬁeld lines (e.g., Shibata et al. 1997; Shimojo et al. 2001; to ruptured loops, in that respect result from enhanced Pariat et al. 2015). Note, reported spire structures with pressure and temperature gradients (e.g, Shimojo et al. visibly bright plasma distributions not decreasing away 2001; Miyagoshi & Yokoyama 2003) at the CP site. We from the CP site (see 14:30 UT of Figure 8), and correla- note, as highlighted above, rupturing loop tops that spa- tions with evolving bright plasma patches (see 14:10 UT tially correlate with our QS transient features bundle of of Figure 7), are inconsistent with such interpretations. bright TR plasma (Figure 9), imply the deposition of ad- We emphasize here, additionally the QS transient fea- ditional heated plasma, in the vicinity of the outer spine (Liu et al. 2011a). Therefore, both rupturing loop tops, and small-scale evaporation driven jets are speculated to

label: section

label: author

text: Innovative

text: Science,

text: LLC,

text: Moncks

text: Corner,

text: SC

text: of

text: Physics

text: Space

text: Sciences,

text: Florida

text: Institute

text: of

text: Technology,

text: Melbourne,

text: FL

text: and

text: Keywords:

text: DISCUSSION

text: ture’s

text: radiative

text: characteristics

text: (Figures

text: and

text: starkly

text: contrasts

text: such

text: descriptions,

text: i.e.,

text: those

text: most

text: common

text: of

text: Investigations

text: of

text: temporal

text: radiative

text: emission

text: across

text: “classical”

text: jets

text: (e.g.,

text: Allen

text: et

text: al.

text: Shibata

text: et

text: al.

text: large

text: scale

text: solar

text: atmospheric

text: temperature

text: gradients

text: and

text: Chandrashekhar

text: et

text: al.

text: co-spatial

text: LOS

text: magnetograms

text: of

text: our

text: QS

text: transient

text: fea-

text: Similarities

text: of

text: our

text: observations

text: to

text: the

text: chromospheric

text: ture,

text: as

text: well

text: as

text: the

text: CP

text: site

text: (i.e.,

text: footpoint

text: shared

text: with

text: jet

text: evolution

text: detailed

text: by

text: Liu

text: et

text: al.

text: mainly,

text: rup-

text: two

text: hot

text: coronal

text: loop

text: arcades)

text: provided

text: evidence

text: for

text: turing

text: loops

text: followed

text: by

text: two

text: evolving

text: bright

text: plasma

text: plasma

text: heating

text: as

text: a

text: result

text: of

text: wide-spread

text: magnetic

text: ﬁeld

text: patches

text: (e.g.,

text: UT

text: UT

text: in

text: Figure

text: favors

text: re-organization

text: events.

text: It

text: was

text: pointed

text: out,

text: e.g.,

text: Fig-

text: the

text: fan-spine

text: magnetic

text: topology

text: typical

text: of

text: jets.

text: Partic-

text: ures

text: and

text: the

text: QS

text: transient

text: revealed

text: a

text: dynamic

text: “sub-

text: ularly,

text: the

text: outer

text: spine

text: region

text: (see

text: Figure

text: of

text: Liu

text: et

text: al.

text: structure”

text: composition.

text: Detailed

text: investigations

text: where

text: continuous

text: injections

text: of

text: heated

text: plasma

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text: indicated

text: the

text: QS

text: transient

text: feature

text: sub-structure

text: was

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text: along

text: newly

text: reconnected

text: ﬁeld

text: lines,

text: and

text: leads

text: to

text: the

text: comprised

text: of

text: episodically

text: heated

text: (Figures

text: and

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text: a

text: diﬀuse

text: bundle

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text: Of

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text: Pariat

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text: al.

text: e.g.,

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text: Observed

text: episodic

text: heating

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text: exhibited

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text: similar

text: trend,

text: which

text: radiative

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text: near

text: the

text: conclusion

text: of

text: we

text: summarize

text: as

text: follows.

text: each

text: heating

text: episode,

text: localized

text: to

text: regions

text: out

text: of

text: the

text: Prior

text: to

text: each

text: episode

text: a

text: re-organization

text: of

text: the

text: underly-

text: ing

text: LOS

text: magnetic

text: environment

text: of

text: the

text: QS

text: transient

text: fea-

text: direct

text: line

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text: the

text: diﬀuse

text: bundle

text: of

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text: emis-

text: ture

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text: CP,

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text: magnetic

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text: jets.

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text: relation

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text: served

text: (Figure

text: Enhanced

text: radiative

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text: fan-spine

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text: ous

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text: regions,

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text: conjunction

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text: the

text: cate

text: the

text: dome

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text: as

text: streamlines

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text: falling

text: material

text: QS

text: transient

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text: CP

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text: and/or

text: NL

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text: SL

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text: emission

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text: Figure

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text: jets,

text: as

text: well

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text: latter

text: evolutions

text: of

text: the

text: chromo-

text: companied

text: the

text: localized

text: radiative

text: enhancements.

text: Heat-

text: spheric

text: jet

text: study

text: of

text: Liu

text: et

text: al.

text: Thereby,

text: below

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text: episodes

text: concluded

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text: enhanced

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text: we

text: postulate

text: to

text: the

text: source

text: of

text: our

text: features

text: unique,

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text: energetically

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text: that

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text: magnetic

text: ﬁeld

text: environment,

text: where

text: “classical”

text: eruptive

text: to

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text: the

text: episodic

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text: relation

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text: unique

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text: reconnection

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text: Pariat

text: et

text: al.

text: The

text: slower,

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text: Madjarska

text: et

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text: et

text: al.

text: as

text: well

text: as

text: re-

text: nection

text: rates

text: and

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text: heating

text: events

text: would

text: ex-

text: ported

text: wide-spread

text: magnetic

text: ﬁeld

text: emergence

text: and

text: can-

text: plain

text: well

text: the

text: lack

text: of

text: evidence

text: in

text: our

text: observational

text: data

text: celation

text: events.

text: Our

text: observations

text: of

text: dynamically

text: evolv-

text: for

text: high-speed

text: injections,

text: and

text: signiﬁcant

text: torsional

text: mo-

text: ing

text: bright

text: plasma

text: patches

text: and

text: the

text: formation

text: of

text: spire-

text: tion

text: of

text: the

text: TR

text: bundle;

text: consistent

text: with

text: QS

text: expectations

text: like

text: structures

text: following

text: rupturing

text: events

text: are

text: speculated

text: (Tate

text: Arbacher

text: et

text: al.

text: It

text: is

text: speculated

text: such

text: no-

text: as

text: the

text: occurrence

text: of

text: such

text: reconnections,

text: which

text: leave

text: in

text: tions

text: point

text: to

text: heated

text: plasma

text: injections,

text: in

text: relation

text: to

text: their

text: wake

text: newly

text: opened

text: and

text: closed

text: magnetic

text: ﬁelds

text: (e.g.,

text: spire

text: structures

text: with

text: decreasing

text: intensity

text: distributions

text: Pariat

text: et

text: al.

text: Spire-like

text: structures

text: supportive

text: of

text: an

text: away

text: from

text: the

text: CP

text: site,

text: resulting

text: from

text: secondary

text: ther-

text: evaporating

text: plasma

text: ﬂow

text: (e.g.,

text: see

text: UT

text: of

text: Figure

text: modynamic

text: processes

text: (Shibata

text: et

text: al.

text: Shimojo

text: possibly

text: indicate

text: injections

text: of

text: heated

text: plasma

text: into

text: the

text: Shibata

text: Such

text: spire-like

text: structure

text: formation,

text: which

text: QS

text: transient

text: feature,

text: likely

text: along

text: newly

text: opened

text: magnetic

text: we

text: emphasize

text: includes

text: observed

text: collimated

text: lower

text: ends

text: ﬁeld

text: lines

text: (e.g.,

text: Shibata

text: et

text: al.

text: Shimojo

text: et

text: al.

text: to

text: ruptured

text: loops,

text: in

text: that

text: respect

text: result

text: from

text: enhanced

text: Pariat

text: et

text: al.

text: Note,

text: reported

text: spire

text: structures

text: with

text: pressure

text: and

text: temperature

text: gradients

text: (e.g,

text: Shimojo

text: et

text: al.

text: visibly

text: bright

text: plasma

text: distributions

text: not

text: decreasing

text: away

text: Miyagoshi

text: Yokoyama

text: at

text: the

text: CP

text: site.

text: We

text: from

text: the

text: CP

text: site

text: (see

text: UT

text: of

text: Figure

text: and

text: correla-

text: note,

text: as

text: highlighted

text: above,

text: rupturing

text: loop

text: tops

text: that

text: spa-

text: tions

text: with

text: evolving

text: bright

text: plasma

text: patches

text: (see

text: UT

text: tially

text: correlate

text: with

text: our

text: QS

text: transient

text: features

text: bundle

text: of

text: of

text: Figure

text: are

text: inconsistent

text: with

text: such

text: interpretations.

text: bright

text: TR

text: plasma

text: (Figure

text: imply

text: the

text: deposition

text: of

text: ad-

text: We

text: emphasize

text: here,

text: additionally

text: the

text: QS

text: transient

text: fea-

text: ditional

text: heated

text: plasma,

text: in

text: the

text: vicinity

text: of

text: the

text: outer

text: spine

text: (Liu

text: et

text: al.

text: Therefore,

text: both

text: rupturing

text: loop

text: tops,

text: and

text: small-scale

text: evaporation

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text: jets

text: are

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Recently, many studies [5, 6] have investigated the quantitative Expectedly, the optimal routing obtains the best performance, relationship between user perception of the quality of the providing the maximum available resources in the bottleneck Internet services and different network conditions. In [2, 6], link. The worst results belong to the InvCap and DMetric, by the MOS of the Skype call users is expressed as a logarithmic virtue of being oblivious to the trafﬁc type and routing all function of bitrate: arriving ﬂows through the shortest path between each O-D pair. Since the links belong to each type, wired and wireless, − are homogeneous, having identical capacities and metrics, the MOS = β + γ ln(X α) (4) results of InvCap and DMetric overlap each other. As can be seen in the ﬁgure, the proposed scheme with 40% share of real- where α = 4.091, β = 1.515, γ = 1, and X is the available time trafﬁc (MDelay-40%) achieves close to the optimal value. bitrate in Kbps. Substituting these values, the equation returns In order to analyse the effect of increasing the real-time share the MOS in a single number, ranging from 1 to 5 with a higher of the total trafﬁc, we evaluate the performance of the MDelay score representing a better quality of the service. According to with 60% real-time trafﬁc between each node pair (MDelay- equation (4), the value of MOS is proportional to the available 60%). The results show that even in the case of higher share of bandwidth between an O-D pair. Obtainable resources in each trafﬁc with tight-delay bounds, the proposed scheme outper- path X(P), due to the concave nature of bandwidth, can be forms other algorithms with a signiﬁcant gap. This makes the deﬁned as follows: scheme more competent for future Internet with the real-time trafﬁc (voice and video) being deemed to be dominant. Forcing { − − } ﬂows to chose the longest acceptable paths while keeping the X(P) = Min 1 MLU(s, i), ..., 1 MLU(j, t) (5) shortest ones for the most eligible trafﬁc can alleviate the effect of heterogeneous and bursty nature of multimedia trafﬁc. Next, we evaluate the performance ratio of the algorithms, where MLU(i, j) is the maximum link utilization of link deﬁned as a minimum residual capacity of each algorithm as (i, j). To that end, we ﬁrst deﬁne the MLU of a given algorithm, in each time interval. The results obtained are compared to one obtained by the optimal routing (baseline). shown in ﬁgure 4, which is expectedly the inverse of the The time intervals are sorted in the ascending order of minimum residual capacity represented in ﬁgure 3(a). After the performance ratio. The higher the ratio, the closer the performance of the algorithm to the optimal. The results are ﬁnding X for each time interval, the value is substituted in equation (6), where the output is bounded between 1 and 5: depicted in ﬁgure 3(b). While in the busy hours of a day the widely-used InvCap and DMetric can only gain 30% of the { − } optimal value, the proposed scheme can achieve a ratio more MOS = Max 1, β + γ ln(X α) min { − } (6) than two and three times better for 60% and 40% share of MOS = Min 5, β + γ ln(X α) max ##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## ##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## ##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##

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text: By a classical theorem of Bretagnolle, Dacunha-Castelle and Krivine [6] (see also [31, Theo- ∈ (cid:18) (cid:107) − (cid:107)(cid:19) rem 5.11]), for every α (0, 1] the metric space (L , x y α) admits an isometric embedding into 1 1 L . Hence, the metric space 1 F (cid:107) − (cid:107)1 M ( ), x y p (cid:0) n 2 (cid:1) (cid:96)n1((cid:96)n1) admits an isometric embedding into L , and consequently (41) implies that 1 c1 (cid:96)nq (Fn2, (cid:107) · (cid:107)p) (cid:54) np1−1q . (cid:3) Remark 4.1. Arguing similarly to the above discussion also justiﬁes the assertion in Remark 1.9. ∈ N (cid:62) ∈ F Indeed, suppose that m, n and q > p 1. Then by H¨older’s inequality for every x, y (cid:96)m( n), q 2 (cid:107) − (cid:107)1 (cid:107) − (cid:107) x myp1−(cid:96)pm11q ((cid:96)n1) = x myp1−(cid:96)mp1q ((cid:96)np) (cid:54) (cid:107)x − y(cid:107)(cid:96)mq ((cid:96)np) (cid:54) (cid:107)x − y(cid:107)(cid:96)mp ((cid:96)np) = (cid:107)x − y(cid:107)(cid:96)p1m1 ((cid:96)n1). (42) (cid:18) ∈ F (cid:19) ∈ { (cid:20) } (cid:107) − (cid:107) ∈ { (cid:21) } Also, because for every x, y(cid:88)(cid:96)m( n) and j 1, . . . , n we have x y 0, . . . , n , q 2 j j 1 (cid:107) − (cid:107) n (cid:107) − (cid:107)q 1q ∈ (cid:107) − (cid:107)1 1−1 (cid:107) − (cid:107)1 x y (cid:96)mq ((cid:96)np) = xj yj 1p x y (cid:96)qm1 ((cid:96)n1), np q x y (cid:96)qm1 ((cid:96)n1) . (43) (cid:18) j=1 (cid:19) (cid:18) (cid:19) Since the metric spaces F (cid:107) − (cid:107)1 F (cid:107) − (cid:107)1 M ( ), x y p and M ( ), x y q n 2 (cid:96)n((cid:96)n) n 2 (cid:96)n((cid:96)n) 1 1 (cid:110) (cid:111) 1 1 (cid:0) (cid:1) admit an isometric embedding into L , it follows from (42) and (43) that 1 c1 (cid:96)mq (Fn2, (cid:107) · (cid:107)p) (cid:54) min mp1−1q ,(cid:16)np1−1q . (cid:17) (44) (cid:16) (cid:16) (cid:17)(cid:17) Since (cid:96)m (Fn, (cid:107) · (cid:107) ) co(cid:0)ntains an isom(cid:1)etric copy of (cid:96)min{m,n} Fmin{m,n}, (cid:107) · (cid:107) , by (14) and (44), q 2 p q 2 p c (cid:96)m (Fn, (cid:107) · (cid:107) ) (cid:16) c (cid:96)min{m,n} Fmin{m,n}, (cid:107) · (cid:107) , 1 q 2 p 1 q 2 p (cid:0) (cid:1) as required. (cid:54) (cid:54) (cid:38) − We end with a brief justiﬁcation of (15). If 1 p < q and p 2 then c (cid:96)n((cid:96)n) n1/p 1/q, as 1 q p proved by Kwapien´ and Schu¨tt [16]. The reverse inequality follows from the fact that the (cid:96)n((cid:96)n) − q p norm is n1/p 1/q-equivalent to the (cid:96)n((cid:96)n) norm, and from the fact [13] that (cid:96) is isometric to a (cid:54) p p − p subspace of L when p 2. When q > p > 2, the (cid:96)n norm is n1/2 1/q-equivalent to the (cid:96)n norm 1 − p − − 2 and the (cid:96)n norm is n1/2 1/q-equivalent to the (cid:96)n norm. So, the (cid:96)n((cid:96)n) norm is n1 1/p 1/q-equivalent q 2 q p to the (cid:96)n((cid:96)n) norm, which embeds isometrically into L . For the matching lower bound, suppose that T :2(cid:96)n(2(cid:96)n(cid:88)) →(cid:88)L is an injective lin(cid:88)ear ma(cid:13)(cid:13)pp(cid:88)ing(cid:88). Sin1ce L (cid:13)(cid:13)has cotype 2 (see e.g. [19]), q p 1 (cid:13) 1 (cid:13) n2 (cid:54) n n (cid:107) (cid:107) (cid:46) 1 n n 2 (cid:107)T −1(cid:107)2 T ejk 21 2n2 ∈{− } (cid:88)εjkT(cid:13)(cid:13)e(cid:88)jk 1(cid:88) (cid:13)(cid:13) j=1 k=1 ε 1,1 n2 j=1 k=1 (cid:13) (cid:13) (cid:107) (cid:107) (cid:54) 2Tn22 ε∈{−1,1}n2 j=n1 k=n1 ejk 2(cid:96)nq ((cid:96)np) = (cid:107)T (cid:107)2 · np2+2q . (45) 14 ##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: By

text: a

text: classical

text: theorem

text: of

text: Bretagnolle,

text: Dacunha-Castelle

text: and

text: Krivine

text: (see

text: also

text: Theo-

text: rem

text: for

text: every

text: α

text: the

text: metric

text: space

text: (L

text: x

text: y

text: α)

text: admits

text: an

text: isometric

text: embedding

text: into

text: L

text: Hence,

text: the

text: metric

text: space

text: F

text: M

text: x

text: y

text: p

text: n

text: admits

text: an

text: isometric

text: embedding

text: into

text: L

text: and

text: consequently

text: implies

text: that

text: Remark

text: Arguing

text: similarly

text: to

text: the

text: above

text: discussion

text: also

text: justiﬁes

text: the

text: assertion

text: in

text: Remark

text: N

text: F

text: Indeed,

text: suppose

text: that

text: m,

text: n

text: and

text: q

text: p

text: Then

text: by

text: H¨older’s

text: inequality

text: for

text: every

text: x,

text: y

text: n),

text: q

text: x

text: x

text: F

text: Also,

text: because

text: for

text: every

text: x,

text: n)

text: and

text: j

text: n

text: we

text: have

text: x

text: y

text: n

text: q

text: j

text: j

text: n

text: x

text: y

text: xj

text: yj

text: x

text: y

text: np

text: q

text: x

text: y

text: Since

text: the

text: metric

text: spaces

text: F

text: F

text: M

text: x

text: y

text: p

text: and

text: M

text: x

text: y

text: q

text: n

text: n

text: admit

text: an

text: isometric

text: embedding

text: into

text: L

text: it

text: follows

text: from

text: and

text: that

text: min

text: Since

text: (Fn,

text: an

text: copy

text: of

text: Fmin{m,n},

text: by

text: and

text: q

text: p

text: q

text: p

text: c

text: (Fn,

text: c

text: Fmin{m,n},

text: q

text: p

text: q

text: p

text: as

text: required.

text: We

text: end

text: with

text: a

text: brief

text: justiﬁcation

text: of

text: If

text: p

text: q

text: and

text: p

text: then

text: c

text: as

text: q

text: p

text: proved

text: by

text: Kwapien´

text: and

text: Schu¨tt

text: The

text: reverse

text: inequality

text: follows

text: from

text: the

text: fact

text: that

text: the

text: q

text: p

text: norm

text: is

text: to

text: the

text: norm,

text: and

text: from

text: the

text: fact

text: that

text: is

text: isometric

text: to

text: a

text: p

text: p

text: p

text: subspace

text: of

text: L

text: when

text: p

text: When

text: q

text: p

text: the

text: norm

text: is

text: to

text: the

text: norm

text: p

text: and

text: the

text: norm

text: is

text: to

text: the

text: norm.

text: So,

text: the

text: norm

text: is

text: q

text: q

text: p

text: to

text: the

text: norm,

text: which

text: embeds

text: isometrically

text: into

text: L

text: For

text: the

text: matching

text: lower

text: bound,

text: suppose

text: that

text: T

text: is

text: an

text: injective

text: L

text: cotype

text: (see

text: e.g.

text: q

text: p

text: n

text: n

text: n

text: n

text: T

text: ejk

text: ε

text: ejk

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text: [29] M. Ribe. On uniformly homeomorphic normed spaces. Ark. Mat., 14(2):237–244, 1976. [30] I. J. Schoenberg. Metric spaces and positive deﬁnite functions. Trans. Amer. Math. Soc., 44(3):522–536, 1938. [31] J. H. Wells and L. R. Williams. Embeddings and extensions in analysis. Springer-Verlag, New York-Heidelberg, 1975. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 84. Mathematics Department, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544- 1000, USA Email address: naor@math.princeton.edu Department of Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel Email address: gideon@weizmann.ac.il 16

label: reference

text: M.

text: Ribe.

text: On

text: uniformly

text: homeomorphic

text: normed

text: spaces.

text: Ark.

text: Mat.,

text: I.

text: J.

text: Schoenberg.

text: Metric

text: spaces

text: and

text: positive

text: deﬁnite

text: functions.

text: Trans.

text: Amer.

text: Math.

text: Soc.,

text: J.

text: H.

text: Wells

text: and

text: L.

text: R.

text: Williams.

text: Embeddings

text: and

text: extensions

text: in

text: analysis.

text: Springer-Verlag,

text: New

text: York-Heidelberg,

text: Ergebnisse

text: der

text: Mathematik

text: und

text: ihrer

text: Grenzgebiete,

text: Band

text: Mathematics

text: Department,

text: Princeton

text: University,

text: Fine

text: Hall,

text: Washington

text: Road,

text: Princeton,

text: NJ

text: USA

text: Email

text: address:

text: naor@math.princeton.edu

label: footer

text: Department

text: of

text: Mathematics,

text: Weizmann

text: Institute

text: of

text: Science,

text: Rehovot

text: Israel

text: Email

text: address:

text: gideon@weizmann.ac.il


text: 5 ing loading duality, wherein a liquid column is stable at low loading, but becomes unstable at higher loading. In this L is the distance between the i-th and (i+1)-th i For the case of uniform loading above the critical load- droplets, and all other material properties in the problem ing threshold, a beading instability should be observed have been subsumed into the deﬁnitions of length, volume in which the liquid column breaks up into a string of and time. Note that V > V implies J > 0, so that liq- i+1 i i droplets, which subsequently coarsen by mass transport uid ﬂows from smaller droplets to larger droplets. Given along connecting liquid columns. With a simplifying as- the ﬂuxes, mass conservation dictates that, cf. Eq. (6), sumption about the dependence of the Laplace pressure on the droplet volume, a mean-ﬁeld scaling ansatz in- dV − dicates that the droplet size and line density scale with i = J − J . (A2) non-trivial power laws in time, and simulations show that dt i 1 i this is not destroyed by the one-dimensional nature of the problem. Eqs. (A1) and (A2) are the required set of non-linear Of course, the simplifying assumption about the Laplace kinetic equations. Since they predict that droplets pressure scaling does not hold in reality, and in general I shrink, as well as grow, we need a rule which governs would expect the clean power law behaviour to be mod- how shrinking droplets can disappear. At this point it is ∼ iﬁed by ﬁnite-droplet-size eﬀects, which may be quite convenient to introduce a ﬁducial volume V R3, where 0 0 persistent. Some general predictions should be robust the ﬁducial length R is set by the height of the connect- 0 however, such as the relatively slow droplet growth via ing liquid column (in the simulations, V = R = 1). 0 0 transport along connecting liquid columns, and the neg- A simple rule for shrinking droplets is that they vanish ative correlation between nearest neighbour droplet sizes when V < αV . If this happens, the droplet is removed i 0 shown in Fig. 5. These could perhaps be tested in an and the distance between the remaining droplets is set electrowetting experiment [3, 4]. I should caution that equal to L − +L +βV 1/3, where the third term is an ad i 1 i i the predicted slow coarsening kinetics may be overtaken hoc correction for the length contributed by the vanished by other, ultimately faster, mechanisms. For example droplet (taking β as a free parameter). transport through the vapou∼r phase for a volatile liquid As an initial condition I set Vi = riV0 where ri is ≤a may ultimately lead to a R t1/3 coarsening law, as in random number chosen from a uniform distribution, 1 LSW theory [27]. r < r . The droplets are initially equispaced, with L = i m i Another prediction arising from the loading duality R . Periodic boundary conditions are imposed. 0 is that a droplet deposited in the groove should show a The droplet volumes are evolved according to Eqs. (A1) staged spreading kinetics, starting with the classic Bell- and (A2), alongside the above rule for removing droplets Cameron-Lucas-Washburn law as the droplet initially which become too small. Eqs. (A1) and (A2) are in- acts like a reservoir, slowing when the loading falls ev- tegrated using a simple, adaptive, Euler-type forward erywhere below the critical value, and possibly ending ﬁnite diﬀerence scheme, with a time step ∆t chosen such × ≤ with a new power law in the ﬁnal starved state. Again, dV /dt ∆t/V (cid:15), in other words so that the fractional i i this may perhaps be probed experimentally. change in any droplet size does not exceed (cid:15) in any time Appendix A: Simulation of coarsening kinetics step. For the reported simulations I used α = 0.9 and β = 1 for the vanishing rule, r = 2 for the maximum m I introduce a simpliﬁed model of the coarsening kinet- initial drop size relative to V0, and (cid:15) = 0.02 for the choice ics to test speciﬁcally the mean-ﬁeld ansatz presented of time step. I have checked the results are insensitive to in the main text. In the model, I consider a one- these choices. dimensional string of i = 1 . . . N droplets, connected As time evolves, the larger droplets grow at the ex- by liquid columns as indicated in the lower inset in pense of the smaller droplets, and the smallest droplets (cid:80) Fig. 3. Big droplets grow, and small droplets shrink, shrink and vanish. Eventually the simulation stops when under the inﬂuence of viscous hydrodynamic transport there is one large ﬁnal droplet (N = 1). I monitor the (cid:104) (cid:105) through the liquid columns. This is driven by diﬀerences mean droplet volume V = (1/N) N V as a func- i=1 i in the Laplace pressure between neighbouring droplets. tion of time (note that N changes as droplets disappear), To establish a system of kinetic equations for the droplet and at selected time points record the drop size distri- sizes, I suppose that the Laplace pressure in the i-th bution. Typical results, aggregated from multiple sim- − droplet is proportional to V 1/3, where V is the droplet ulation runs from independent starting points, are sum- i i volume (this is the simplifying assumption mentioned in marised in Figs. 3 and 4. the main text). According to the HP law, the Laplace I also calculate periodically the equal-time correlation pressure diﬀerence drives a mass ﬂux through the con- function necting liquid column as, cf. Eq. (5), (cid:104) (cid:105) | − | ∆V ∆V − − − C( i j ) = (cid:104) i (cid:105)j , (A3) V 1/3 V 1/3 ∆V 2 J = i i+1 . (A1) i L i ##LTLine## ##LTLine## ##LTLine##

text: ing

text: loading

text: duality,

text: wherein

text: a

text: liquid

text: column

text: is

text: stable

text: at

text: low

text: loading,

text: but

text: becomes

text: unstable

text: at

text: higher

text: loading.

text: In

text: this

text: L

text: is

text: the

text: distance

text: between

text: the

text: i-th

text: and

text: i

text: For

text: the

text: case

text: of

text: uniform

text: loading

text: above

text: the

text: critical

text: load-

text: droplets,

text: and

text: all

text: other

text: material

text: properties

text: in

text: the

text: problem

text: ing

text: threshold,

text: a

text: beading

text: instability

text: should

text: be

text: observed

text: have

text: been

text: subsumed

text: into

text: the

text: deﬁnitions

text: of

text: length,

text: volume

text: in

text: which

text: the

text: liquid

text: column

text: breaks

text: up

text: into

text: a

text: string

text: of

text: and

text: time.

text: Note

text: that

text: V

text: V

text: implies

text: J

text: so

text: that

text: liq-

text: i

text: i

text: droplets,

text: which

text: subsequently

text: coarsen

text: by

text: mass

text: transport

text: uid

text: ﬂows

text: from

text: smaller

text: droplets

text: to

text: larger

text: droplets.

text: Given

text: along

text: connecting

text: liquid

text: columns.

text: With

text: a

text: simplifying

text: as-

text: the

text: ﬂuxes,

text: mass

text: conservation

text: dictates

text: that,

text: cf.

text: Eq.

text: sumption

text: about

text: the

text: dependence

text: of

text: the

text: Laplace

text: pressure

text: on

text: the

text: droplet

text: volume,

text: a

text: mean-ﬁeld

text: scaling

text: ansatz

text: in-

text: dV

text: dicates

text: that

text: the

text: droplet

text: size

text: and

text: line

text: density

text: scale

text: with

text: i

text: J

text: J

text: non-trivial

text: power

text: laws

text: in

text: time,

text: and

text: simulations

text: show

text: that

text: dt

text: i

text: i

text: this

text: is

text: not

text: destroyed

text: by

text: the

text: one-dimensional

text: nature

text: of

text: the

text: problem.

text: Eqs.

text: and

text: are

text: the

text: required

text: set

text: of

text: non-linear

text: Of

text: course,

text: the

text: simplifying

text: assumption

text: about

text: the

text: Laplace

text: kinetic

text: equations.

text: Since

text: they

text: predict

text: that

text: droplets

text: pressure

text: scaling

text: does

text: not

text: hold

text: in

text: reality,

text: and

text: in

text: general

text: I

text: shrink,

text: as

text: well

text: as

text: grow,

text: we

text: need

text: a

text: rule

text: which

text: governs

text: would

text: expect

text: the

text: clean

text: power

text: law

text: behaviour

text: to

text: be

text: mod-

text: how

text: shrinking

text: droplets

text: can

text: disappear.

text: At

text: this

text: point

text: it

text: is

text: iﬁed

text: by

text: ﬁnite-droplet-size

text: eﬀects,

text: which

text: may

text: be

text: quite

text: convenient

text: to

text: introduce

text: a

text: ﬁducial

text: volume

text: V

text: where

text: persistent.

text: Some

text: general

text: predictions

text: should

text: be

text: robust

text: the

text: ﬁducial

text: length

text: R

text: is

text: set

text: by

text: the

text: height

text: of

text: the

text: connect-

text: however,

text: such

text: as

text: the

text: relatively

text: slow

text: droplet

text: growth

text: via

text: ing

text: liquid

text: column

text: (in

text: the

text: simulations,

text: V

text: R

text: transport

text: along

text: connecting

text: liquid

text: columns,

text: and

text: the

text: neg-

text: A

text: simple

text: rule

text: for

text: shrinking

text: droplets

text: is

text: that

text: they

text: vanish

text: ative

text: correlation

text: between

text: nearest

text: neighbour

text: droplet

text: sizes

text: when

text: V

text: αV

text: If

text: this

text: happens,

text: the

text: droplet

text: is

text: removed

text: i

text: shown

text: in

text: Fig.

text: These

text: could

text: perhaps

text: be

text: tested

text: in

text: an

text: and

text: the

text: distance

text: between

text: the

text: remaining

text: droplets

text: is

text: set

text: electrowetting

text: experiment

text: I

text: should

text: caution

text: that

text: equal

text: to

text: L

text: +L

text: +βV

text: where

text: the

text: third

text: term

text: is

text: an

text: ad

text: i

text: i

text: i

text: the

text: predicted

text: slow

text: coarsening

text: kinetics

text: may

text: be

text: overtaken

text: hoc

text: correction

text: for

text: the

text: length

text: contributed

text: by

text: the

text: vanished

text: by

text: other,

text: ultimately

text: faster,

text: mechanisms.

text: For

text: example

text: droplet

text: (taking

text: β

text: as

text: a

text: free

text: parameter).

text: transport

text: through

text: the

text: vapou∼r

text: phase

text: for

text: a

text: volatile

text: liquid

text: As

text: an

text: initial

text: condition

text: I

text: set

text: Vi

text: where

text: ri

text: is

text: ≤a

text: may

text: ultimately

text: lead

text: to

text: a

text: R

text: coarsening

text: law,

text: as

text: in

text: random

text: number

text: chosen

text: from

text: a

text: uniform

text: distribution,

text: LSW

text: theory

text: r

text: r

text: The

text: droplets

text: are

text: initially

text: equispaced,

text: with

text: L

text: i

text: m

text: i

text: Another

text: prediction

text: arising

text: from

text: the

text: loading

text: duality

text: R

text: Periodic

text: boundary

text: conditions

text: are

text: imposed.

text: is

text: that

text: a

text: droplet

text: deposited

text: in

text: the

text: groove

text: should

text: show

text: a

text: The

text: droplet

text: volumes

text: are

text: evolved

text: according

text: to

text: Eqs.

text: staged

text: spreading

text: kinetics,

text: starting

text: with

text: the

text: classic

text: Bell-

text: and

text: alongside

text: the

text: above

text: rule

text: for

text: removing

text: droplets

text: Cameron-Lucas-Washburn

text: law

text: as

text: the

text: droplet

text: initially

text: which

text: become

text: too

text: small.

text: Eqs.

text: and

text: are

text: in-

text: acts

text: like

text: a

text: reservoir,

text: slowing

text: when

text: the

text: loading

text: falls

text: ev-

text: tegrated

text: using

text: a

text: simple,

text: adaptive,

text: Euler-type

text: forward

text: erywhere

text: below

text: the

text: critical

text: value,

text: and

text: possibly

text: ending

text: ﬁnite

text: diﬀerence

text: scheme,

text: with

text: a

text: time

text: step

text: ∆t

text: chosen

text: such

text: with

text: a

text: new

text: power

text: law

text: in

text: the

text: ﬁnal

text: starved

text: state.

text: Again,

text: dV

text: /dt

text: ∆t/V

text: in

text: other

text: words

text: so

text: that

text: the

text: fractional

text: i

text: i

text: this

text: may

text: perhaps

text: be

text: probed

text: experimentally.

text: change

text: in

text: any

text: droplet

text: size

text: does

text: not

text: exceed

text: in

text: any

text: time

text: Appendix

text: A:

text: Simulation

text: of

text: coarsening

text: kinetics

text: step.

text: For

text: the

text: reported

text: simulations

text: I

text: used

text: α

text: and

text: β

text: for

text: the

text: vanishing

text: rule,

text: r

text: for

text: the

text: maximum

text: m

text: I

text: introduce

text: a

text: simpliﬁed

text: model

text: of

text: the

text: coarsening

text: kinet-

text: initial

text: drop

text: size

text: relative

text: to

text: and

text: for

text: the

text: choice

text: ics

text: to

text: test

text: speciﬁcally

text: the

text: mean-ﬁeld

text: ansatz

text: presented

text: of

text: time

text: step.

text: I

text: have

text: checked

text: the

text: results

text: are

text: insensitive

text: to

text: in

text: the

text: main

text: text.

text: In

text: the

text: model,

text: I

text: consider

text: a

text: one-

text: these

text: choices.

text: dimensional

text: string

text: of

text: i

text: N

text: droplets,

text: connected

text: As

text: time

text: evolves,

text: the

text: larger

text: droplets

text: grow

text: at

text: the

text: ex-

text: by

text: liquid

text: columns

text: as

text: indicated

text: in

text: the

text: lower

text: inset

text: in

text: pense

text: of

text: the

text: smaller

text: droplets,

text: and

text: the

text: smallest

text: droplets

text: Fig.

text: Big

text: droplets

text: grow,

text: and

text: small

text: droplets

text: shrink,

text: shrink

text: and

text: vanish.

text: Eventually

text: the

text: simulation

text: stops

text: when

text: under

text: the

text: inﬂuence

text: of

text: viscous

text: hydrodynamic

text: transport

text: there

text: is

text: one

text: large

text: ﬁnal

text: droplet

text: (N

text: I

text: monitor

text: the

text: through

text: the

text: liquid

text: columns.

text: This

text: is

text: driven

text: by

text: diﬀerences

text: mean

text: droplet

text: volume

text: V

text: N

text: V

text: as

text: a

text: func-

text: i

text: in

text: the

text: Laplace

text: pressure

text: between

text: neighbouring

text: droplets.

text: tion

text: of

text: time

text: (note

text: that

text: N

text: changes

text: as

text: droplets

text: disappear),

text: To

text: establish

text: a

text: system

text: of

text: kinetic

text: equations

text: for

text: the

text: droplet

text: and

text: at

text: selected

text: time

text: points

text: record

text: the

text: drop

text: size

text: distri-

text: sizes,

text: I

text: suppose

text: that

text: the

text: Laplace

text: pressure

text: in

text: the

text: i-th

text: bution.

text: Typical

text: results,

text: aggregated

text: from

text: multiple

text: sim-

text: droplet

text: is

text: proportional

text: to

text: V

text: where

text: V

text: is

text: the

text: droplet

text: ulation

text: runs

text: from

text: independent

text: starting

text: points,

text: are

text: sum-

text: i

text: i

text: volume

text: (this

text: is

text: the

text: simplifying

text: assumption

text: mentioned

text: in

text: marised

text: in

text: Figs.

text: and

text: the

text: main

text: text).

text: According

text: to

text: the

text: HP

text: law,

text: the

text: Laplace

text: I

text: also

text: calculate

text: periodically

text: the

text: equal-time

text: correlation

text: pressure

text: diﬀerence

text: drives

text: a

text: mass

text: ﬂux

text: through

text: the

text: con-

text: function

text: necting

text: liquid

text: column

text: as,

text: cf.

text: Eq.

text: ∆V

text: ∆V

text: C(

text: i

text: j

text: i

text: V

text: V

text: ∆V

text: J

text: i

text: i

text: L

text: i

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##