text: 3. MG II H & K LINE MODELING 1000 ergs/cm3/sec of heating in 200 km depths over the pho- tosphere and temperature minimum, similar to Fig. 1 (Grid 5). We have found that energy deposited in the 500 - 700 km The second grid applied the same heating function in the chro- layer resulted in the best match to typically observed EB line mosphere, up to 2 Mm above the photospheric ﬂoor (Grid 6). proﬁles for the Hα and Ca II 8542 Å lines (e.g. Fig. 5 of Vis- We ﬁnd that injecting the energy into the chromosphere now sers et al. (2013)). In this Section, we consider the response of the Mg II h & k lines to similar EB heating. Recent obser- only causes strong line core emission in the Hα, Ca II 8542 Å, and Mg II h & k line proﬁles. This is accompanied by large vations with the IRIS instrument hint at enhancements in the − wings of these lines (see Fig. 10 of (Vissers et al. 2015)). We Doppler shifts due to very strong velocities (>20 km s 1) in the leading shock front. The strong Doppler velocities appear use the RH line synthesis code (Uitenbroek 2001; Pereira & due to the shock front interacting with transition region which Uitenbroek 2015) to calculate the line proﬁles in partial re- is now at higher column masses. This forces the transition distribution (PRD). The 2 grids of models run for Hα and Ca region to shift even lower in the atmosphere. II in the previous section produce no enhancement in the Mg II h & k lines (Grids 1 and 2). In Grid 5, in which we varied the energy deposition location over the photosphere and temperature minimum region, we A further grid of models was created, which increases the ﬁnd a similar atmospheric response to that of Fig. 1 (Grid 2), energy deposited around the temperature minimum exponen- only slightly stronger due to the doubled heating rate. This tially, between 1000 - 1 x 106 ergs/cm3/sec (Grid 3). Again, can be seen in Fig. 3. no enhancement is observed in the Mg II h & k lines, though Fig. 4 shows the corresponding line proﬁles synthesised strong continuum enhancement appears with these high en- for the atmospheres shown in Fig. 3. When the energy is ergy deposition rates. The atmospheric response to this sud- deposited below the temperature minimum region (300 - 500 den energetic injection did not create dramatically higher tem- km above the photospheric ﬂoor), there is very little response peratures. In fact, the peak temperature in the 1 x 106 ergs/cm3 in all synthesised lines, but a noticeable continuum increase, model reached 9500 K, with a corresponding electron density of 1014 cm−3 and a peak shock velocity of 5 km s−1. and slight wing enhancement in Hα and Ca II 8542 Å. Sim- The contribution function of the Mg II h & k lines appeared ilar to the quiet Sun model, the best ﬁtting for the Hα and to show a line core formation height concentrated in the upper Ca II 8542 Å line proﬁles appears when the energy is in- chromosphere, with little contribution from the photosphere. serted around the temperature minimum region (500 - 700 The wings of the lines were formed slightly lower, closer to km). However, the Mg II h & k lines appear to have only their outer wings showing enhancement. The response of Mg the line core formation region of Ca II 8542 Å. If the energy II h & k when the heating is applied between 700 - 900 km deposition occurs in the lower-mid chromosphere, enhanced above the photospheric ﬂoor is most similar to that of strong emission in the Mg II h & k wings can be achieved, though observed IRIS bombs (see Fig. 10 in (Vissers et al. 2015)). this will only cause ﬂare-like proﬁles producing full emission However, the Ca II 8542 Å line proﬁle shows strong core in Ca II 8542 Å and Hα (Grid 4). emission with energy deposited at this height. When halv- It also became apparent that the background Mg II h & k ing the energy deposition rate to the ideal rate identiﬁed in the line proﬁle of our model appeared in absorption rather than previous section, we ﬁnd the enhancements in the lines reduce in emission as is observed. In an attempt to rectify this is- sue, a diﬀerent starting atmosphere was adopted which has slightly in Ca II 8542 Å and Mg II h & k, with a slightly larger it’s transition region pushed to higher column masses, to re- diﬀerential apparent in the Hα line proﬁle. ﬂect a more active, plage-like atmosphere (QS.SL.HT from To investigate if a larger depth range for the energy depo- (Allred et al. 2015)). sition is more appropriate, we also ran two larger models in With this new starting atmosphere, the initial proﬁle of the both the quiet Sun and pre-ﬂare atmospheres (Grids 7 and 8). Mg II h & k lines appear in emission, similar to the back- These models inject 500 ergs/cm3/sec over ranges of 500 - ground proﬁles observed (Peter et al. 2014; Vissers et al. 1000 km and 500 - 1200 km. In all cases, strong core en- 2015; Kim et al. 2015; Tian et al. 2016). However, the line hancements were seen in the synthesised Ca II 8542 Å and Hα cores of Ca II 8542 Å and Hα are also slightly enhanced, with line proﬁles. The Mg II h & k line proﬁles appeared to show Draft version May 29, 2020 ﬂare-like proﬁles with the pre-ﬂare atmosphere, and strong CaPrIeIpr8in5t4ty2peÅset ubseinignLgATEmXostryelepemroumlateianpejnv.t1.2/16/11 enhancement in the line wings in the quiet Sun atmosphere. Again, two grids of models were run, one grid applying SOLAR ELLERMAN BOMBS IN 1-D RADIATIVE HYDRODYNAMICS A. Reid1,4, M. Mathioudakis1, A. Kowalski2,3, J. G. Doyle4, J. C. Allred5 0 1. Astrophysics Research Centre, School of Mathematics and Physics, Queen’s University Belfast, BT7 1NN, Northern Ireland, UK; e-mail: aaron.reid@qub.ac.uk 2. Department of Astrophysical and Planetary Sciences, University of Colorado Boulder, 2000 Colorado Ave, Boulder, CO 80305, USA. 3. National Solar Observatory, University of Colorado Boulder, 3665 Discovery Drive, Boulder, CO 80303, USA 4. Armagh Observatory and Planetarium, College Hill, Armagh, BT61 9DG, UK 5. NASA/Goddard Space Flight Center, Code 671, Greenbelt, MD 20771 Keywords: Sun: Activity — Hydrodynamics — Sun: Photosphere

text: MG

text: II

text: H

text: K

text: LINE

text: MODELING

text: of

text: heating

text: in

text: km

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text: the

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text: Ca

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text: of

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text: atmosphere,

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text: in

text: the

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text: the

text: quiet

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text: atmosphere.

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text: two

text: grids

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text: were

text: run,

text: one

text: grid

text: applying

text: SOLAR

text: ELLERMAN

text: BOMBS

text: IN

text: RADIATIVE

text: HYDRODYNAMICS

text: A.

text: M.

text: A.

text: J.

text: G.

text: J.

text: C.

text: Astrophysics

text: Research

text: Centre,

text: School

text: of

text: Mathematics

text: and

text: Physics,

text: Queen’s

text: University

text: Belfast,

text: Northern

text: Ireland,

text: UK;

text: e-mail:

text: aaron.reid@qub.ac.uk

text: Department

text: of

text: Astrophysical

text: and

text: Planetary

text: Sciences,

text: University

text: of

text: Colorado

text: Boulder,

text: Colorado

text: Ave,

text: Boulder,

text: CO

text: USA.

text: National

text: Solar

text: Observatory,

text: University

text: of

text: Colorado

text: Boulder,

text: Discovery

text: Drive,

text: Boulder,

text: CO

text: USA

text: Armagh

text: Observatory

text: and

text: Planetarium,

text: College

text: Hill,

text: Armagh,

text: UK

text: NASA/Goddard

text: Space

text: Flight

text: Center,

text: Code

text: Greenbelt,

text: MD

text: Keywords:

text: Sun:

text: Activity

text: Hydrodynamics

text: Sun:

text: Photosphere


text: 6 Reid et al. Figure 3. Left: Temperature proﬁle across the temperature minimum region for 1000 ergs/cm3/sec deposited at various heights, using the pre-ﬂare starting atmosphere. Middle: Vertical velocity proﬁles. Right: Electron density proﬁles. The dashed lines indicate runs with the injected energy density halved. Figure 4. Top: Mg II h & k line proﬁles, calculated in PRD from the atmospheres in Fig. 3. Bottom Left: Hα line proﬁles. Bottom Right: Ca II 8542 Å line proﬁles. 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text: Astron. Nachr. / AN (2017) 791 order to systematically discover phenomena that occur, say, ∼ in only 1% of the population, one simply has to follow- up in detail >100 objects. With XMM-Newton and Chandra we have now entered the era where 100s of well-selected objects are indeed being followed up. In the future, given the EPIC-pnfactor ~5larger effective area at 10 keV. new large parent samples provided by eROSITA and Euclid, => XMM-Newtonrequired for temperature measurements of massive clusters. we will be able to ﬁnd striking objects that occur only once in >1,000 clusters! 4 Suggestions for future XMM-Newton observing strategies Based on the above assessment and examples, some suggestions for modiﬁcations (or not) of the current XMM- Merloni+12 (arXiv:1209.3114) Newton observing strategies are outlined below. Fig.2 Comparison of XMM-Newton EPIC-pn and eROSITA ef- – Improve the possibility for very large systematic studies fective area as a function of photon energy. While eROSITA out- of statistically meaningful, representative, and complete performs EPIC-pn at soft energies, XMM-Newton is signiﬁcantly samples further; e.g., >3 Ms programs, maybe split over more efﬁcient at high energies (>5 keV). Adapted from Merloni >2 cycles. et al. (2012). – Keep the possibility to study individual interesting tar- (cid:28) (cid:29) gets with short ( 100 ks) and long exposures ( 100 ks). Such targets may, e.g., be found through systematic ters, therefore, takes about 10 Ms; i.e., less than one year of follow-up observations of large samples. typical XMM-Newton time per Announcement of Opportu- ∼ – Encourage Time Allocation Committees (TACs) to look nity (AO; 14 Ms). More realistically, adding a soft-proton- ∼ more favorably upon√completing “the last 10% of the ﬂare-factor ( 1.4), this comes out to pretty much one year in total. However, this would naturally be spread over multi- sample.” The naive N argument often does not ap- ∼ ply to well-selected samples. For instance, the unob- ple cycles; e.g., for ﬁve cycles, this would amount to 20% served 10% will likely not be drawn from the same of the total time. So, per AO, this would even ﬁt well within parent population as the full sample if the follow-up the existing Very Large Programme category (up to 3 Ms). plan includes a large fraction of independent archival In addition to the speciﬁc cosmology example above, observations (i.e., observations of “interesting” individ- XMM-Newton will play a key role for other galaxy cluster ual objects proposed because of their special properties). science areas. To provide some background for this, let us Therefore, an unknown selection bias would be intro- have a look at previous Large and Very Large Programmes duced that cannot be corrected for. (LPs, VLPs) performed with XMM-Newton in the ﬁeld of galaxy clusters. Fig. 3 lists LPs and VLPs with a relation to – Require a minimum (ﬂare-corrected) exposure of >6 ks, unless speciﬁcally justiﬁed, e.g., for variability studies, galaxy clusters according to the XMM-Newton website. As ∼ since eROSITA will provide 1.6 ks anywhere. is obvious from the titles and number of observations, the – Increase the efﬁciency by observing low-surface bright- large majority of (V)LPs observe samples of clusters, typi- ness objects at low-ﬂare-risk periods. For example, typ- cally selected in a well-deﬁned way from larger parent sam- ically, the exposure for all ﬂared times is lost for low- ples. Such follow-up observations of complete sub-samples surface brightness objects, often resulting in science be- (e.g., eeHIFLUGCS, Fig. 4) often serve two purposes: (i) ing partially or completely lost, while this is not neces- to constrain population properties in a systematic way with sarily the case for very bright point sources. corrections for selection effects, and (ii) to discover rare, new, unexpected, and exciting phenomena. Acknowledgements. THR acknowledges support by the German An example for the latter has been provided Research Association (DFG) through grant RE 1462/6 and through by Schellenberger & Reiprich (2015, see, e.g., the Transregional Collaborative Research Centre TRR33 The Dark http://chandra.harvard.edu/photo/2015/z8338/). In sys- Universe (project B18) as well as by the German Aerospace tematic sample follow-up observations, they discovered a Agency (DLR) with funds from the Ministry of Economy and large galaxy, apparently falling in or passing by a cluster, Technology (BMWi) through grant 50 OR 1608. exhibiting one of the longest X-ray tails ever observed for a single galaxy. Even more surprisingly, the tail has been completely stripped off the galaxy, including the high density head. Further theoretical and observational studies of this system will provide detailed insights into stripping processes and gas microphysics. Naively, one might think one has to be quite lucky to run into such rare systems. That is, however, not the case. In www.an-journal.org (cid:13)c 2017WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim ##LTFigure## ##LTFigure## ##LTLine## ##LTLine##

text: Astron.

text: Nachr.

text: AN

text: order

text: to

text: systematically

text: discover

text: phenomena

text: that

text: occur,

text: say,

text: in

text: only

text: of

text: the

text: population,

text: one

text: simply

text: has

text: to

text: follow-

text: up

text: in

text: detail

text: objects.

text: With

text: XMM-Newton

text: and

text: Chandra

text: we

text: have

text: now

text: entered

text: the

text: era

text: where

text: of

text: well-selected

text: objects

text: are

text: indeed

text: being

text: followed

text: up.

text: In

text: the

text: future,

text: given

text: the

text: EPIC-pnfactor

text: effective

text: area

text: at

text: keV.

text: new

text: large

text: parent

text: samples

text: provided

text: by

text: eROSITA

text: and

text: Euclid,

text: XMM-Newtonrequired

text: for

text: temperature

text: measurements

text: of

text: massive

text: clusters.

text: we

text: will

text: be

text: able

text: to

text: ﬁnd

text: striking

text: objects

text: that

text: occur

text: only

text: once

text: in

text: clusters!

text: Suggestions

text: for

text: future

text: XMM-Newton

text: observing

text: strategies

text: Based

text: on

text: the

text: above

text: assessment

text: and

text: examples,

text: some

text: suggestions

text: for

text: modiﬁcations

text: (or

text: not)

text: of

text: the

text: current

text: XMM-

text: Newton

text: observing

text: strategies

text: are

text: outlined

text: below.

text: Comparison

text: of

text: XMM-Newton

text: EPIC-pn

text: and

text: eROSITA

text: ef-

text: Improve

text: the

text: possibility

text: for

text: very

text: large

text: systematic

text: studies

text: fective

text: area

text: as

text: a

text: function

text: of

text: photon

text: energy.

text: While

text: eROSITA

text: out-

text: of

text: statistically

text: meaningful,

text: representative,

text: and

text: complete

text: performs

text: EPIC-pn

text: at

text: soft

text: energies,

text: XMM-Newton

text: is

text: signiﬁcantly

text: samples

text: further;

text: e.g.,

text: Ms

text: programs,

text: maybe

text: split

text: over

text: more

text: efﬁcient

text: at

text: high

text: energies

text: keV).

text: Adapted

text: from

text: Merloni

text: cycles.

text: et

text: al.

text: Keep

text: the

text: possibility

text: to

text: study

text: individual

text: interesting

text: tar-

text: gets

text: with

text: short

text: ks)

text: and

text: long

text: exposures

text: ks).

text: Such

text: targets

text: may,

text: e.g.,

text: be

text: found

text: through

text: systematic

text: ters,

text: therefore,

text: takes

text: about

text: Ms;

text: i.e.,

text: less

text: than

text: one

text: year

text: of

text: follow-up

text: observations

text: of

text: large

text: samples.

text: typical

text: XMM-Newton

text: time

text: per

text: Announcement

text: of

text: Opportu-

text: Encourage

text: Time

text: Allocation

text: Committees

text: (TACs)

text: to

text: look

text: nity

text: (AO;

text: Ms).

text: More

text: realistically,

text: adding

text: a

text: soft-proton-

text: more

text: favorably

text: upon√completing

text: “the

text: last

text: of

text: the

text: ﬂare-factor

text: this

text: comes

text: out

text: to

text: pretty

text: much

text: one

text: year

text: in

text: total.

text: However,

text: this

text: would

text: naturally

text: be

text: spread

text: over

text: multi-

text: sample.”

text: The

text: naive

text: N

text: argument

text: often

text: does

text: not

text: ap-

text: ply

text: to

text: well-selected

text: samples.

text: For

text: instance,

text: the

text: unob-

text: ple

text: cycles;

text: e.g.,

text: for

text: ﬁve

text: cycles,

text: this

text: would

text: amount

text: to

text: served

text: will

text: likely

text: not

text: be

text: drawn

text: from

text: the

text: same

text: of

text: the

text: total

text: time.

text: So,

text: per

text: AO,

text: this

text: would

text: even

text: ﬁt

text: well

text: within

text: parent

text: population

text: as

text: the

text: full

text: sample

text: if

text: the

text: follow-up

text: the

text: existing

text: Very

text: Large

text: Programme

text: category

text: (up

text: to

text: Ms).

text: plan

text: includes

text: a

text: large

text: fraction

text: of

text: independent

text: archival

text: In

text: addition

text: to

text: the

text: speciﬁc

text: cosmology

text: example

text: above,

text: observations

text: (i.e.,

text: observations

text: of

text: “interesting”

text: individ-

text: XMM-Newton

text: will

text: play

text: a

text: key

text: role

text: for

text: other

text: galaxy

text: cluster

text: ual

text: objects

text: proposed

text: because

text: of

text: their

text: special

text: properties).

text: science

text: areas.

text: To

text: provide

text: some

text: background

text: for

text: this,

text: let

text: us

text: Therefore,

text: an

text: unknown

text: selection

text: bias

text: would

text: be

text: intro-

text: have

text: a

text: look

text: at

text: previous

text: Large

text: and

text: Very

text: Large

text: Programmes

text: duced

text: that

text: cannot

text: be

text: corrected

text: for.

text: (LPs,

text: VLPs)

text: performed

text: with

text: XMM-Newton

text: in

text: the

text: ﬁeld

text: of

text: galaxy

text: clusters.

text: Fig.

text: lists

text: LPs

text: and

text: VLPs

text: with

text: a

text: relation

text: to

text: Require

text: a

text: minimum

text: (ﬂare-corrected)

text: exposure

text: of

text: ks,

text: unless

text: speciﬁcally

text: justiﬁed,

text: e.g.,

text: for

text: variability

text: studies,

text: galaxy

text: clusters

text: according

text: to

text: the

text: XMM-Newton

text: website.

text: As

text: since

text: eROSITA

text: will

text: provide

text: ks

text: anywhere.

text: is

text: obvious

text: from

text: the

text: titles

text: and

text: number

text: of

text: observations,

text: the

text: Increase

text: the

text: efﬁciency

text: by

text: observing

text: low-surface

text: bright-

text: large

text: majority

text: of

text: (V)LPs

text: observe

text: samples

text: of

text: clusters,

text: typi-

text: ness

text: objects

text: at

text: low-ﬂare-risk

text: periods.

text: For

text: example,

text: typ-

text: cally

text: selected

text: in

text: a

text: well-deﬁned

text: way

text: from

text: larger

text: parent

text: sam-

text: ically,

text: the

text: exposure

text: for

text: all

text: ﬂared

text: times

text: is

text: lost

text: for

text: low-

text: ples.

text: Such

text: follow-up

text: observations

text: of

text: complete

text: sub-samples

text: surface

text: brightness

text: objects,

text: often

text: resulting

text: in

text: science

text: be-

text: (e.g.,

text: eeHIFLUGCS,

text: Fig.

text: often

text: serve

text: two

text: purposes:

text: (i)

text: ing

text: partially

text: or

text: completely

text: lost,

text: while

text: this

text: is

text: not

text: neces-

text: to

text: constrain

text: population

text: properties

text: in

text: a

text: systematic

text: way

text: with

text: sarily

text: the

text: case

text: for

text: very

text: bright

text: point

text: sources.

text: corrections

text: for

text: selection

text: effects,

text: and

text: (ii)

text: to

text: discover

text: rare,

text: new,

text: unexpected,

text: and

text: exciting

text: phenomena.

text: Acknowledgements.

text: THR

text: acknowledges

text: support

text: by

text: the

text: German

text: An

text: example

text: for

text: the

text: latter

text: has

text: been

text: provided

text: Research

text: Association

text: (DFG)

text: through

text: grant

text: RE

text: and

text: through

text: by

text: Schellenberger

text: Reiprich

text: see,

text: e.g.,

text: the

text: Transregional

text: Collaborative

text: Research

text: Centre

text: The

text: Dark

text: In

text: sys-

text: Universe

text: (project

text: as

text: well

text: as

text: by

text: the

text: German

text: Aerospace

text: tematic

text: sample

text: follow-up

text: observations,

text: they

text: discovered

text: a

text: Agency

text: (DLR)

text: with

text: funds

text: from

text: the

text: Ministry

text: of

text: Economy

text: and

text: large

text: galaxy,

text: apparently

text: falling

text: in

text: or

text: passing

text: by

text: a

text: cluster,

text: Technology

text: (BMWi)

text: through

text: grant

text: OR

text: exhibiting

text: one

text: of

text: the

text: longest

text: X-ray

text: tails

text: ever

text: observed

text: for

text: a

text: single

text: galaxy.

text: Even

text: more

text: surprisingly,

text: the

text: tail

text: has

text: been

text: completely

text: stripped

text: off

text: the

text: galaxy,

text: including

text: the

text: high

text: density

text: head.

text: Further

text: theoretical

text: and

text: observational

text: studies

text: of

text: this

text: system

text: will

text: provide

text: detailed

text: insights

text: into

text: stripping

text: processes

text: and

text: gas

text: microphysics.

text: Naively,

text: one

text: might

text: think

text: one

text: has

text: to

text: be

text: quite

text: lucky

text: to

text: run

text: into

text: such

text: rare

text: systems.

text: That

text: is,

text: however,

text: not

text: the

text: case.

text: In

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##


text: Automatic Climate Control System Compiled SDF-based C-code S-function SDF- based Code Generation Figure 12: SDF code-generation of the single-rate triggered Climate controller model [17] (a) Model-In-the-Loop Simulation Results (b) Software-In-the-Loop Simulation Results Figure 13: Veriﬁcation results of the Heat control model showing equivalent outputs of the SIL (see Fig. 13b) and the MIL (see Fig. 13a) simulations. ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: Automatic

text: Climate

text: Control

text: System

text: Compiled

text: SDF-based

text: C-code

text: S-function

text: SDF-

text: based

text: Code

text: Generation

text: Figure

text: SDF

text: code-generation

text: of

text: the

text: single-rate

text: triggered

text: Climate

text: controller

text: model

text: (a)

text: Model-In-the-Loop

text: Simulation

text: Results

text: (b)

text: Software-In-the-Loop

text: Simulation

text: Results

text: Figure

text: Veriﬁcation

text: results

text: of

text: the

text: Heat

text: control

text: model

text: showing

text: equivalent

text: outputs

text: of

text: the

text: SIL

text: (see

text: Fig.

text: and

text: the

text: MIL

text: (see

text: Fig.

text: simulations.

text: ##LTFigure##

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text: 11 G. Alkauskas Theorem 3. Let us deﬁne the vector ﬁ(cid:16)eld (cid:17)V =(cid:16)(V , V(cid:17) ), whe(cid:16)re (cid:17) (cid:16) (cid:17) x y √ √ √ − (cid:16)x(cid:17) (cid:16) 3y (cid:17) (cid:16) 3x(cid:17) (cid:16)y (cid:17) V = cos y + 3 sin sin + cos cos , x √ √ 2 2 2 2 √ − y 3x 3y x V = cos x + 3 sin sin + cos cos . y 2 2 2 2 Also, (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) √ √ √ (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) 3(cid:16)x (cid:17) y − (cid:16) (cid:17)3x (cid:16) y(cid:17) Q = 2 cos x + 2 cos y + 2x sin y + 2 3 sin sin x cos sin x √ √ √ 2 2 2 2 √ √ (cid:16) (cid:17)3x (cid:16) y(cid:17) − (cid:16)x (cid:17) (cid:16) 3y (cid:17) (cid:16) x (cid:17) (cid:16) 3(cid:17)y + 3y sin cos 2 cos cos + 3y cos sin √ √ √ 2 2 2 2 2 2 √ − x 3y − x 3y − 3x y 2 3 sin sin 3x sin cos 2 cos cos , 2 2 2 2 2 2 Q = (Q , Q ), Q (x, y) = Q (y, x). Then x y y x ∈ D 1) The vector ﬁeld V and has a dihedral symmetry of order 6: for any ζ , one has − ◦ ◦ 3 ζ 1 V ζ = V; 2) as a Taylor series, V contains only terms with even compound degrees, and it satisﬁes the vector Helmholtz equation and solenoidality: ∇ − ∇ · 2V = V, V = 0; both these properties hold for Q, too; 3) the Taylor series for V and Q start form a degree 2 vector ﬁeld 3M and 3M, resepctively, 8 2 where M = ((cid:37), (cid:36)) is given − − ((cid:36), (cid:37)) = (2xy x2 + y2, 2xy + x2 y2). References [1] G. Alkauskas, The projective translation equation and unramiﬁed 2-dimensional ﬂows with rational vector ﬁelds, Aequationes Math. 89 (3) (2015), 873–913. http://arxiv.org/abs/1202.3958. [2] G. Alkauskas, Algebraic and abelian solutions to the projective translation equation, Aequationes Math. 90 (4) (2016), 727–763. http://arxiv.org/abs/1506.08028. [3] G. Alkauskas, Beltrami vector ﬁelds with an icosahedral symmetry, https://arxiv.org/abs/1706. 09295. [4] G. Alkauskas, Projective and polynomial superﬂows. I. http://arxiv.org/abs/1601.06570. [5] G. Alkauskas, Projective and polynomial superﬂows. II. O(3) and the icosahedral group. http://arxiv. org/abs/1606.05772. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: G.

text: Alkauskas

text: Theorem

text: Let

text: us

text: deﬁne

text: the

text: vector

label: equation

text: x

text: y

text: V

text: cos

text: y

text: sin

text: sin

text: cos

text: cos

text: x

text: y

text: x

text: V

text: cos

text: x

text: sin

text: sin

text: cos

text: cos

text: y

text: Also,

text: y

text: Q

text: cos

text: x

text: cos

text: y

text: sin

text: y

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text: x).

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text: ζ

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text: V;

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text: a

text: Taylor

text: series,

text: V

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text: only

text: terms

text: with

text: even

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text: degrees,

text: and

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text: form

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text: References

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text: G.

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text: and

text: polynomial

text: superﬂows.

text: I.

text: G.

text: Alkauskas,

text: Projective

text: and

text: polynomial

text: superﬂows.

text: II.

text: and

text: the

text: icosahedral

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text: 9 G. Alkauskas (cid:16) (cid:17) − − − P + ( 1)dQ , ( 1)dP Q 2d 2d 2d 2d D has a dihedral symmetry of order 4d + 2, and is solenoidal. Thus, potentially it gives 2d+1 rise to a lambent ﬂow. We will explore the case d = 1, and only make the ﬁrst few steps in a research, since the vector ﬁelds we obtain already contain very rich dynamic and analytic structure. − Thus, let (we multiply the above vector ﬁeld by a factor 1 just for convenience) − − ((cid:36), (cid:37)) = (2xy x2 + y2, 2xy + x2 y2), (cid:18) (cid:19) D (cid:18) (cid:19) which is solenoidal, and has a dihedral group , generated by matrices 3 √ − − 1 √1 3 0 1 α = − , β = , 2 3 1 1 0 as its symmetry group. This gives rise to the dihedral superﬂow. The vector ﬁeld, linearly R − − conjugate (over GL ( ), not over O(2)) to ((cid:36), (cid:37)) is given by (x2 2xy, y2 2xy). This vector 2 ﬁeld is indeed a very fascinating one, and it was explored in detail in [1]. For example, this vector ﬁeld can be explicitly integrated in terms Dixonian elliptic functions [14]. In particular, this ﬂow turns out to be unramiﬁed. The property of the ﬂow to be unramiﬁed is indeed a − − profound one. For example, to prove that a ﬂow with a vector√ﬁeld (x2 xy, y2 2xy) is Q unramiﬁed, requires to employ arithmetic of the number ﬁeld ( 3 ) [2]. The very theory of unramiﬁed ﬂows is still in its initial stages of development. D Returning to Beltrami ﬂows in dimension 2 and to the group (analogously as in the (cid:16) (cid:17) 3 icosahedral case in [3]), let us deﬁne 4 linear forms and 4 vectors (cid:16) (cid:17) √ √ (cid:96) (x, y) = 1x + 3y, j = (cid:16)1, 3(cid:17), x,0 2 2 √ x,0 2 2 √ − − (cid:96) (x, y) = 1x + 3y, j = (cid:16) 1, (cid:17)3 , x,1 √ 2 2 x,1 √ 2 2 (cid:96) (x, y) = 3x + 1y, j = 3, 1 , y,0 √2 2 y,0 √2 2 − − (cid:96) (x, y) = 3x 1y, j = 3, 1 . y,1 2 2 y,1 2 2 We have a vector ﬁeld (G(x, y), G(y, x)), where G = a cos x + b cos y + c cos (cid:96) + d cos (cid:96) + e cos (cid:96) + f cos (cid:96) x,0 x,1 y,0 y,1 + gy sin x + hx sin y (7) + i(cid:96) sin (cid:96) + j(cid:96) sin (cid:96) y,1 x,0 y,0 x,1 + k(cid:96) sin (cid:96) + l(cid:96) sin (cid:96) . x,1 y,0 x,0 y,1 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: G.

text: Alkauskas

text: P

text: Q

text: D

text: has

text: a

text: dihedral

text: symmetry

text: of

text: order

text: and

text: is

text: solenoidal.

text: Thus,

text: potentially

text: it

text: gives

text: rise

text: to

text: a

text: lambent

text: ﬂow.

text: We

text: will

text: explore

text: the

text: case

text: d

text: and

text: only

text: make

text: the

text: ﬁrst

text: few

text: steps

text: in

text: a

text: research,

text: since

text: the

text: vector

text: ﬁelds

text: we

text: obtain

text: already

text: contain

text: very

text: rich

text: dynamic

text: and

text: analytic

text: structure.

text: Thus,

text: let

text: (we

text: multiply

text: the

text: above

text: vector

text: ﬁeld

text: by

text: a

text: factor

text: just

text: for

text: convenience)

text: D

text: which

text: is

text: solenoidal,

text: and

text: has

text: a

text: dihedral

text: group

text: generated

text: by

text: matrices

text: √1

text: α

text: β

text: as

text: its

text: symmetry

text: group.

text: This

text: gives

text: rise

text: to

text: the

text: dihedral

text: superﬂow.

text: The

text: vector

text: ﬁeld,

text: linearly

text: R

text: conjugate

text: (over

text: GL

text: not

text: over

text: to

text: is

text: given

text: by

text: This

text: vector

text: ﬁeld

text: is

text: indeed

text: a

text: very

text: fascinating

text: one,

text: and

text: it

text: was

text: explored

text: in

text: detail

text: in

text: For

text: example,

text: this

text: vector

text: ﬁeld

text: can

text: be

text: explicitly

text: integrated

text: in

text: terms

text: Dixonian

text: elliptic

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text: particular,

text: this

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text: out

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text: be

text: unramiﬁed.

text: The

text: property

text: of

text: the

text: ﬂow

text: to

text: be

text: unramiﬁed

text: is

text: indeed

text: a

text: profound

text: one.

text: For

text: example,

text: to

text: prove

text: that

text: a

text: ﬂow

text: with

text: a

text: vector√ﬁeld

text: xy,

text: is

text: Q

text: unramiﬁed,

text: requires

text: to

text: employ

text: arithmetic

text: of

text: the

text: number

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text: ﬂows

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text: in

text: its

text: initial

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text: of

text: development.

text: D

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text: to

text: Beltrami

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text: dimension

text: and

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text: the

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text: as

text: in

text: the

text: icosahedral

text: case

text: in

text: let

text: us

text: deﬁne

text: linear

text: forms

text: and

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text: (x,

text: y)

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text: (G(x,

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text: page posting content on the platform. We suppose v and the considered timespan of the employed dataset), 2 v interacted with a given content posted by s, and v re- and its cascade is complete. 4 3 shared the post from v . Node v did not interact with the 2 1 These two scenarios are of particular interest, both from content, hence is not present on the potential propagation a research and a practical points of view. Indeed, be- graph of the post, that we represent in Figure 1b. There, with ing able to issue automated warnings as early as possible edges (s, v ), (s, v ), (v , v ) we represent possible propa- 2 4 2 4 about potential misinformation could help in reducing their gation paths of the content: either v or v could have seen 2 4 spread (Friggeri et al. 2014). On the other hand, still being the content from the seed page, or from previous interaction able to perform such a classiﬁcation when the post already of their friend. Node v only has an edge (v , v ), as we 3 2 3 propagated might serve as a warning in order to prevent its know that the interaction happened thanks to v , and there- 2 further diffusion. Moreover, retroactively ﬂagging old posts fore the content propagated from this node. Additionally, as potentially fake could help training users to discriminate we highlight that some possible propagation edges, repre- between fact-checked and dubious information, a major di- sented with a dashed line, correspond to friendship edges of rection in the ﬁght against misinformation (First Draft News the friendship graph. Coalition 2016). To investigate these two scenarios, we set up two different experiments, modeled as binary classiﬁcation tasks, where the two classes are conspiracy and science. We describe s s them in the following. v v Early Stage Classiﬁcation. In this scenario, we have ac- 4 4 v cess only to users’ interactions with a post up to a certain v v 1 time t + δ, after it has been created at time t, (e.g., up 2 2 to after two hours after creation). With this scenario, we simulate how OSNs such as Facebook could try to continu- ously detect misinformation content during its propagation. v v Therefore, for each post p, given its creation time t, we ex- 3 3 G tract features from the partial propagation graph p , and t+δ from the evolution of properties of the propagation graph be- (a) Facebook Friendship Graph (b) Potential Propagation Graph fore the current time t + δ (discussed further in Section 4.2). We use 30 minutes steps, as this proved to be a good trade- Figure 1: Sample Facebook friendship graph, and potential off between the granularity of the analysis and the number propagation graph of some content. A dashed line repre- of intervals to consider. We stop at two days (2880 min- sents potential propagation edges that also corresponds to a utes, or 96 time steps) after the publication time of a post, friendship edge of the friendship graph. Node v1 does not because we observed that most of the interactions happen interact with the content, and is therefore not present in the in this period (> 95%). Finally, we compare the perfor- potential propagation graph. mance of different well-known classiﬁers, namely Random Forests (Ho 1995), Linear Discriminant Analysis (Izenman 2013), and Multi-Layer Perceptron (Haykin 1998), for each ∈ 4 Experiments δ [30, 60, ..., 2880], in a cross-validation scheme. In this section, we describe the details of the experiment and of the analysis performed on the dataset. We ﬁrst de- Final Stage Classiﬁcation. In this scenario, we suppose scribe the experimental design (Section 4.1). We then thor- that a post already stopped its propagation, and no new in- oughly report and motivate the different features that we ex- teractions with it have been observeGd for a long time. We de- tracted from propagation cascades (Section 4.2). scribe the ﬁnal propagation graph p of post p with a set of 4.1 Experimental Design high-level and topological features, that we describe in more detail in Section 4.2. We then compare the performance of The aim of our experiments is to show that it is difﬁcult to different classiﬁers, in a cross-validation scheme. discriminate between conspiracy theories, and fact-checked scientiﬁc news, by using only the propagation graph of the 4.2 Feature Extraction post, as it would be difﬁcult to manipulate by misinforma- tion creators. In particular, we evaluate this in two speciﬁc To extract information from the different propagation graphs G G moments of content propagation: p and p, we identify three possible categories of features: t (i) high-level properties of the content propagation, (ii) topo- 1. Early Stage, meaning that classiﬁcation of the type logical properties of the propagation graphs, (iii) evolution of post happens as early as possible during its prop- properties of the content propagation. We use feature sets agation phase; (i), (ii), and (iii) considering the evolution after two days, in 2. Final Stage, meaning that classiﬁcation happens the Final Stage Classiﬁcation scenario. We only use feature when the post already stopped propagating (within set (iii), in the Early Stage Classiﬁcation scenario, because ##LTFigure## ##LTFigure##

text: page

text: posting

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text: the

text: platform.

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text: a

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text: content

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text: by

text: s,

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text: and

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text: cascade

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text: complete.

text: shared

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text: post

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text: v

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text: view.

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text: graph

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text: post,

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text: possible

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text: sented

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text: a

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text: correspond

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text: friendship

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text: rection

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text: ﬁght

text: against

text: misinformation

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text: Draft

text: News

text: the

text: friendship

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text: Coalition

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text: investigate

text: these

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text: scenarios,

text: we

text: set

text: up

text: two

text: different

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text: modeled

text: as

text: binary

text: classiﬁcation

text: tasks,

text: where

text: the

text: two

text: classes

text: are

text: conspiracy

text: and

text: science.

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text: creation).

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text: this

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text: OSNs

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text: as

text: Facebook

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text: to

text: continu-

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text: detect

text: misinformation

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text: during

text: its

text: propagation.

text: v

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text: for

text: each

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text: G

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text: the

text: partial

text: propagation

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text: the

text: evolution

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text: (a)

text: Facebook

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text: Potential

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text: time

text: t

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text: minutes

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text: Sample

text: Facebook

text: friendship

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text: the

text: granularity

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text: the

text: analysis

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text: number

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text: some

text: content.

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text: of

text: intervals

text: to

text: consider.

text: We

text: stop

text: at

text: two

text: days

text: min-

text: sents

text: potential

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text: or

text: time

text: steps)

text: after

text: the

text: publication

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text: of

text: a

text: post,

text: friendship

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text: of

text: the

text: friendship

text: graph.

text: Node

text: does

text: not

text: because

text: we

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text: that

text: most

text: of

text: the

text: interactions

text: happen

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text: with

text: the

text: content,

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text: therefore

text: not

text: present

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text: the

text: in

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text: we

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text: the

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text: potential

text: propagation

text: graph.

text: mance

text: of

text: different

text: well-known

text: classiﬁers,

text: namely

text: Random

text: Forests

text: (Ho

text: Linear

text: Discriminant

text: Analysis

text: (Izenman

text: and

text: Multi-Layer

text: Perceptron

text: (Haykin

text: for

text: each

text: Experiments

text: δ

text: in

text: a

text: cross-validation

text: scheme.

text: In

text: this

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text: we

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text: the

text: details

text: of

text: the

text: experiment

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text: of

text: the

text: analysis

text: performed

text: on

text: the

text: dataset.

text: We

text: ﬁrst

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