text: A porism for cyclic quadrangles Jerzy Kocik Mathematics department, SIU, Carbondale, IL 62901 jkocik@siu.edu Abstract: We present a geometric theorem on porism on cyclic quadrilaterals, namely the existence of an arbitrary number of cyclic quadrilaterals through four fixed points on a line once one exists. Also, a technique of proving such properties with the use of pseudounitary traceless matrices is presented. The result holds for general quadrics as well as the circle. Keywords: geometry, porism, complex matrices, pseudounitary group, reversion calculus, two-dimensional Clifford algebras, duplex numbers, hyperbolic numbers,. MSC: 51M15 (geo constr) 11E88 (Cliff alg, spinor). Notation: P = a point, P = the reversion it defines, p = complex number representing it. 1. Introduction Inscribe a butterfly-like quadrilateral in a circle and draw a line L, see Figure 1. The sides of the quadrilateral will cut the line at some four (not necessarily distinct) points. It turns out that as we deform continuously the inscribed quadrilateral, the points of intersection remain invariant! More precisely, think of the quadrilateral as a path from vertex X through collinear points P, R, S and Q back to X (see Figure 3, left). If we redraw the path starting from another point on the circle but passing through the same points on the line in the same order, the path closes up to a polygon (that is, we shall arrive at the starting point). Jerz y Kocik A porism for cyclic quadrilaterals 2 L from another point on the circle but passing through the same points on the line in the same order, the path closes to form an inscribed polygon, that is, we shall arrive at t he starting point. In spirit, this startling property is similar to Steiner’s famous porism [3, 9], wh ich statesFtihgautroe n1c: ePowriesmﬁ nodn tcwocoyccliircc qleusa,dorannegliensn tehrrotuogthh ceoolltihneear,r spuocinhtst hat a closed cha in of neighbor-wise tangent circles inscribed in the region between them is possi- In spirit, this startling property is similar to Steiner’s famous porism [CG, W1], which states that ble, then an inﬁnite number of such inscribed chains exist (Figure 1.2). One may set once we find two circles, one inner to the other, such that a closed chain of neighbor-wise tangent theciinrcilteisa ilncscirricbleed aint tahney repgoiosni tbieotnweaennd thtehme ics hpaosinsibwlei, ltlhecnlo asne inwfiintihte tnaunmgbeenr coyf .suYceh tinasncroibthede r geocmhaeintsr iecxipsht. e Ononem meanyo sneti tnhet hineitsiaal mcierclcea atte agnoyr pyoissitiPono nancde ltehet’ schpaionr wisimll c[l1os,e8 w, i1th0 t]a.ngency. Yet another geometric phenomenon in the same category is Poncelet’s porism [Po, BP, W2]. Figure 2: Steiner porism. The Figure 1.2ch:aSinte minaeyr bpe oreridsrmaw.n Tstharetincgh ain fromm aayrbitbraery rpeodsritaiwonn of stthaer ting starftriongm ciracrlbei trary position of the starting circle. 1 The property for the cyclic quadrilateral described at the outset may be restated similarly: if four points on a line admit a cyclic quadrilateral then an inﬁnite number of such quadrilaterals inscribed in the same circle exist. Hence the term porism is justiﬁed. In the next sections we restate the theorem and deﬁne a map of reversion through point, which may be represented by pseudo-unitary matrices. The technique devel- oped allows one to prove the theorem as well as a diagrammatic representation of the relativistic addition of velocities, presented elsewhere [5]. A slight modiﬁcation to arbitrary two-dimensional Cliﬀord algebras allows one to modify the result to hold for hyperbolas and provides a geometric realization of trigonometric tangent- like addition. 2. The main result Let us present the result more formally. Reversion of a point A on a circle through point P gives point P(A) on the circle such that points A, P and P(A) are collinear (see Figure 2.1). More precisely: Deﬁnition 2.1. Given a circle K, a reversion through point P (cid:60) K is a map P: → (cid:55)→ K K, A P(A), such that points A, P, P(A) are collinear, and P(A) (cid:44) A. If ∈ ∈ P K, then for any A K we deﬁne P(A) = P. By bold P we denote the reversion map deﬁned by point P (not bold). Clearly, reversion through P (cid:60) K is an involution, P2 = id, and therefore invertible. The product of reversions is in general not commutative, PQ (cid:44) QP. The main result may now be given a concise form: ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: A

text: porism

text: for

text: cyclic

text: quadrangles

text: Jerzy

text: Kocik

text: Mathematics

text: department,

text: SIU,

text: Carbondale,

text: IL

text: jkocik@siu.edu

text: Abstract:

text: We

text: present

text: a

text: geometric

text: theorem

text: on

text: porism

text: on

text: cyclic

text: quadrilaterals,

text: namely

text: the

text: existence

text: of

text: an

text: arbitrary

text: number

text: of

text: cyclic

text: quadrilaterals

text: through

text: four

text: fixed

text: points

text: on

text: a

text: line

text: once

text: one

text: exists.

text: Also,

text: a

text: technique

text: of

text: proving

text: such

text: properties

text: with

text: the

text: use

text: of

text: pseudounitary

text: traceless

text: matrices

text: is

text: presented.

text: The

text: result

text: holds

text: for

text: general

text: quadrics

text: as

text: well

text: as

text: the

text: circle.

text: Keywords:

text: geometry,

text: porism,

text: complex

text: matrices,

text: pseudounitary

text: group,

text: reversion

text: calculus,

text: two-dimensional

text: Clifford

text: algebras,

text: duplex

text: numbers,

text: hyperbolic

text: numbers,.

text: MSC:

text: (geo

text: constr)

text: (Cliff

text: alg,

text: spinor).

text: Notation:

text: P

text: a

text: point,

text: P

text: the

text: reversion

text: it

text: defines,

text: p

text: complex

text: number

text: representing

text: it.

text: Introduction

text: Inscribe

text: a

text: butterfly-like

text: quadrilateral

text: in

text: a

text: circle

text: and

text: draw

text: a

text: line

text: L,

text: see

text: Figure

text: The

text: sides

text: of

text: the

text: quadrilateral

text: will

text: cut

text: the

text: line

text: at

text: some

text: four

text: (not

text: necessarily

text: distinct)

text: points.

text: It

text: turns

text: out

text: that

text: as

text: we

text: deform

text: continuously

text: the

text: inscribed

text: quadrilateral,

text: the

text: points

text: of

text: intersection

text: remain

text: invariant!

text: More

text: precisely,

text: think

text: of

text: the

text: quadrilateral

text: as

text: a

text: path

text: from

text: vertex

text: X

text: through

text: collinear

text: points

text: P,

text: R,

text: S

text: and

text: Q

text: back

text: to

text: X

text: (see

text: Figure

text: left).

text: If

text: we

text: redraw

text: the

text: path

text: starting

text: from

text: another

text: point

text: on

text: the

text: circle

text: but

text: passing

text: through

text: the

text: same

text: points

text: on

text: the

text: line

text: in

text: the

text: same

text: order,

text: the

text: path

text: closes

text: up

text: to

text: a

text: polygon

text: (that

text: is,

text: we

text: shall

text: arrive

text: at

text: the

text: starting

text: point).

text: Jerz

text: y

text: Kocik

text: A

text: porism

text: for

text: cyclic

text: quadrilaterals

text: L

text: from

text: another

text: point

text: on

text: the

text: circle

text: but

text: passing

text: through

text: the

text: same

text: points

text: on

text: the

text: line

text: in

text: the

text: same

text: order,

text: the

text: path

text: closes

text: to

text: form

text: an

text: inscribed

text: polygon,

text: that

text: is,

text: we

text: shall

text: arrive

text: at

text: t

text: he

text: starting

text: point.

text: In

text: spirit,

text: this

text: startling

text: property

text: is

text: similar

text: to

text: Steiner’s

text: famous

text: porism

text: wh

text: ich

text: statesFtihgautroe

text: ePowriesmﬁ

text: nodn

text: tcwocoyccliircc

text: qleusa,dorannegliensn

text: tehrrotuogthh

text: ceoolltihneear,r

text: spuocinhtst

text: hat

text: a

text: closed

text: cha

text: in

text: of

text: neighbor-wise

text: tangent

text: circles

text: inscribed

text: in

text: the

text: region

text: between

text: them

text: is

text: possi-

text: In

text: spirit,

text: this

text: startling

text: property

text: is

text: similar

text: to

text: Steiner’s

text: famous

text: porism

text: [CG,

text: which

text: states

text: that

text: ble,

text: then

text: an

text: inﬁnite

text: number

text: of

text: such

text: inscribed

text: chains

text: exist

text: (Figure

text: One

text: may

text: set

text: once

text: we

text: find

text: two

text: circles,

text: one

text: inner

text: to

text: the

text: other,

text: such

text: that

text: a

text: closed

text: chain

text: of

text: neighbor-wise

text: tangent

text: theciinrcilteisa

text: ilncscirricbleed

text: aint

text: tahney

text: repgoiosni

text: tbieotnweaennd

text: thtehme

text: ics

text: hpaosinsibwlei,

text: ltlhecnlo

text: asne

text: inwfiintihte

text: tnaunmgbeenr

text: coyf

text: .suYceh

text: tinasncroibthede

text: r

text: geocmhaeintsr

text: iecxipsht.

text: e

text: Ononem

text: meanyo

text: sneti

text: tnhet

text: hineitsiaal

text: mcierclcea

text: atte

text: agnoyr

text: pyoissitiPono

text: nancde

text: ltehet’

text: schpaionr

text: wisimll

text: w,

text: t]a.ngency.

text: Yet

text: another

text: geometric

text: phenomenon

text: in

text: the

text: same

text: category

text: is

text: Poncelet’s

text: porism

text: [Po,

text: BP,

text: Figure

text: Steiner

text: porism.

text: The

text: Figure

text: minaeyr

text: bpe

text: oreridsrmaw.n

text: Tstharetincgh

text: ain

text: fromm

text: aayrbitbraery

text: rpeodsritaiwonn

text: of

text: stthaer

text: ting

text: starftriongm

text: ciracrlbei

text: trary

text: position

text: of

text: the

text: starting

text: circle.

text: The

text: property

text: for

text: the

text: cyclic

text: quadrilateral

text: described

text: at

text: the

text: outset

text: may

text: be

text: restated

text: similarly:

text: if

text: four

text: points

text: on

text: a

text: line

text: admit

text: a

text: cyclic

text: quadrilateral

text: then

text: an

text: inﬁnite

text: number

text: of

text: such

text: quadrilaterals

text: inscribed

text: in

text: the

text: same

text: circle

text: exist.

text: Hence

text: the

text: term

text: porism

text: is

text: justiﬁed.

text: In

text: the

text: next

text: sections

text: we

text: restate

text: the

text: theorem

text: and

text: deﬁne

text: a

text: map

text: of

text: reversion

text: through

text: point,

text: which

text: may

text: be

text: represented

text: by

text: pseudo-unitary

text: matrices.

text: The

text: technique

text: devel-

text: oped

text: allows

text: one

text: to

text: prove

text: the

text: theorem

text: as

text: well

text: as

text: a

text: diagrammatic

text: representation

text: of

text: the

text: relativistic

text: addition

text: of

text: velocities,

text: presented

text: elsewhere

text: A

text: slight

text: modiﬁcation

text: to

text: arbitrary

text: two-dimensional

text: Cliﬀord

text: algebras

text: allows

text: one

text: to

text: modify

text: the

text: result

text: to

text: hold

text: for

text: hyperbolas

text: and

text: provides

text: a

text: geometric

text: realization

text: of

text: trigonometric

text: tangent-

text: like

text: addition.

text: The

text: main

text: result

text: Let

text: us

text: present

text: the

text: result

text: more

text: formally.

text: Reversion

text: of

text: a

text: point

text: A

text: on

text: a

text: circle

text: through

text: point

text: P

text: gives

text: point

text: P(A)

text: on

text: the

text: circle

text: such

text: that

text: points

text: A,

text: P

text: and

text: P(A)

text: are

text: collinear

text: (see

text: Figure

text: More

text: precisely:

text: Deﬁnition

text: Given

text: a

text: circle

text: K,

text: a

text: reversion

text: through

text: point

text: P

text: K

text: is

text: a

text: map

text: P:

text: K

text: K,

text: A

text: P(A),

text: such

text: that

text: points

text: A,

text: P,

text: P(A)

text: are

text: collinear,

text: and

text: P(A)

text: A.

text: If

text: P

text: K,

text: then

text: for

text: any

text: A

text: K

text: we

text: deﬁne

text: P(A)

text: P.

text: By

text: bold

text: P

text: we

text: denote

text: the

text: reversion

text: map

text: deﬁned

text: by

text: point

text: P

text: (not

text: bold).

text: Clearly,

text: reversion

text: through

text: P

text: K

text: is

text: an

text: involution,

text: id,

text: and

text: therefore

text: invertible.

text: The

text: product

text: of

text: reversions

text: is

text: in

text: general

text: not

text: commutative,

text: PQ

text: QP.

text: The

text: main

text: result

text: may

text: now

text: be

text: given

text: a

text: concise

text: form:

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Jerzy Kocik A porism for cyclic quadrilaterals 11 [4] Jerzy Kocik, Duplex numbers, diﬀusion systems and generalized quantum me- chanics, International Journal of Theoretical Physics, 38 No. 8 (1999) pp. 2221-30. [5] Jerzy Kocik, Diagram for relativistic addition of velocities, Am. J. Phys. 80 No. 8, (Aug 2012) p. 720. [6] Jerzy Kocik, Interactive diagrams at Lagrange.math.siu.edu/Kocik /geome- try/geometry.htm [7] C. Stanley Ogilvy, Excursions in Geometry. New York: Dover, pp. 51-54, 1990. [8] Jean-Victor Poncelet, Traité des propriétés projectives des ﬁgures: ou- vrage utile à qui s’occupent des applications de la géométrie descriptive et d’opérations géométriques sur le terrain, Vols. 1-2, 2nd ed. Paris: Gauthier- Villars, 1865-66. [9] Eric W. Weisstein, “Steiner Chain.” From MathWorld –A Wolfram Web Re- source. http://mathworld.wolfram.com/SteinerChain.html [10] Eric W. Weisstein, “Poncelet’s Porism.” From MatWorld–A Wolfram Web Re- source. http://mathworld.wolfram.com/PonceletsPorism.html ##LTLine##

text: Jerzy

label: paragraph

text: Kocik

text: A

text: porism

text: for

text: cyclic

text: quadrilaterals

label: reference

text: Jerzy

text: Kocik,

text: Duplex

text: numbers,

text: diﬀusion

text: systems

text: and

text: generalized

text: quantum

text: me-

text: chanics,

text: International

text: Journal

text: of

text: Theoretical

text: Physics,

text: No.

text: pp.

text: Jerzy

text: Kocik,

text: Diagram

text: for

text: relativistic

text: addition

text: of

text: velocities,

text: Am.

text: J.

text: Phys.

text: No.

text: (Aug

text: p.

text: Jerzy

text: Kocik,

text: Interactive

text: diagrams

text: at

text: Lagrange.math.siu.edu/Kocik

text: /geome-

text: try/geometry.htm

text: C.

text: Stanley

text: Ogilvy,

text: Excursions

text: in

text: Geometry.

text: New

text: York:

text: Dover,

text: pp.

text: Jean-Victor

text: Poncelet,

text: Traité

text: des

text: propriétés

text: projectives

text: des

text: ﬁgures:

text: ou-

text: vrage

text: utile

text: à

text: qui

text: s’occupent

text: des

text: applications

text: de

text: la

text: géométrie

text: descriptive

text: et

text: d’opérations

text: géométriques

text: sur

text: le

text: terrain,

text: Vols.

text: ed.

text: Paris:

text: Gauthier-

text: Villars,

text: Eric

text: W.

text: Weisstein,

text: “Steiner

text: Chain.”

text: From

text: MathWorld

text: –A

text: Wolfram

text: Web

text: Re-

text: source.

text: Eric

text: W.

text: Weisstein,

text: “Poncelet’s

text: Porism.”

text: From

text: MatWorld–A

text: Wolfram

text: Web

text: Re-

text: source.

text: ##LTLine##


text: David Garcia, Claudio J. Tessone, Pavlin Mavrodiev, Nicolas Perony: The digital traces of bubbles: Electronic Supplementary Material S7 Detailed results of impulse response functions shock P P S S W W U U t+1 t+2 t+1 t+2 t+1 t+2 t+1 t+2 P 0.94 0.14 (cid:21) -0.139 (cid:21) 0.087 0.175 0.142 t S (cid:21) 0.374 0.845 0.281 (cid:21) (cid:21) (cid:21) (cid:21) t W (cid:21) (cid:21) (cid:21) 0.192 0.86 -0.277 (cid:21) (cid:21) t U (cid:21) 0.175 0.517 (cid:21) (cid:21) (cid:21) 0.818 -0.086 t Table S7: Response function estimates of lags 1 and 2 under 97.5% con(cid:28)dence level (95% after Bonferroni correction). S12/12 ##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##

text: David

font: default

text: Garcia,

text: Claudio

text: J.

text: Tessone,

text: Pavlin

text: Mavrodiev,

text: Nicolas

text: Perony:

text: The

text: digital

text: traces

text: of

text: bubbles:

text: Electronic

text: Supplementary

text: Material

text: Detailed

text: results

text: of

text: impulse

text: response

text: functions

text: shock

text: P

text: P

text: S

text: S

text: W

text: W

text: U

text: U

text: P

text: t

text: S

text: t

text: W

text: t

text: U

text: t

text: Table

text: Response

text: function

text: estimates

text: of

text: lags

text: and

text: under

text: level

text: after

text: Bonferroni

text: correction).

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text: D. Garcia, C.J. Tessone, P. Mavrodiev, N. Perony: Feedback cycles between socio-economic signals in the Bitcoin economy 0 A B 3 200 20 5 0 C]2 ce150 --4200 T0 Pri100 -60 B2 D/ 50 3Mar2013 12Apr2013 22May2013 S5 U1 0 [ C e0 28Nov2012 100 c1 200 Pri atio150 500 5 R -50 et100 -100 8Oct2011 e 0 w 50 3Mar2013 12Apr2013 22May2013 T 0 5 10 15 0 Fundamental Value [USD] 23Dec2012 2Apr2013 Date 11Jul2013 19Oct2013 Figure 3: A. Trajectory of weighted mean price and mean fundamental value over a rolling window of width of one week (one point per day, connected in chronological order) between March 1st, 2011 and March 1st, 2013. The red line denoting strict equality between price and fundamental value is hit on two occasions (around October 18th, 2011, and November 28th, 2012), which delimit the characteristic periods of our analysis. The blue line shows values be- fore the ﬁrst hitting event, the green line shows the values after the last hitting event, and the grey line shows the period in between. B, C. Time series of the cumulative price and tweet ra- tio (black dots), and cumulative estimated values by the model for the period since November 30, 2012. Insets: time series of estimates and empirical changes of price and Tweet ratio for the period between March 1 and May 30, 2013. increases, and 9 of the 10 largest price drops during P3 (Table S6 of the ESM). 4 Discussion Due do the decentralised character of the Bitcoin currency, the dynamics of its economy largely depend on the behaviour of its users, who (i) mine new bitcoins and maintain the block chain, and (ii) inﬂuence the exchange rate by trading bitcoins to and from other currencies; these interactions between users form the social backbone of Bitcoin. In this paper, we used the digital traces of user activity in the Bitcoin economy to disentangle the relationships between its diﬀerent variables: user base, information search, information sharing, and price. While each of these four 10/28 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: D.

text: Garcia,

text: C.J.

text: Tessone,

text: P.

text: Mavrodiev,

text: N.

text: Perony:

text: Feedback

text: cycles

text: between

text: socio-economic

text: signals

text: in

text: the

text: Bitcoin

text: economy

text: A

text: B

text: D/

text: C

text: Pri

text: R

text: e

text: w

text: T

text: Fundamental

text: Value

text: [USD]

text: Date

text: Figure

text: A.

text: Trajectory

text: of

text: weighted

text: mean

text: price

text: and

text: mean

text: fundamental

text: value

text: over

text: a

text: rolling

text: window

text: of

text: width

text: of

text: one

text: week

text: (one

text: point

text: per

text: day,

text: connected

text: in

text: chronological

text: order)

text: between

text: March

text: and

text: March

text: The

text: red

text: line

text: denoting

text: strict

text: equality

text: between

text: price

text: and

text: fundamental

text: value

text: is

text: hit

text: on

text: two

text: occasions

text: (around

text: October

text: and

text: November

text: which

text: delimit

text: the

text: characteristic

text: periods

text: of

text: our

text: analysis.

text: The

text: blue

text: line

text: shows

text: values

text: be-

text: fore

text: the

text: ﬁrst

text: hitting

text: event,

text: the

text: green

text: line

text: shows

text: the

text: values

text: after

text: the

text: last

text: hitting

text: event,

text: and

text: the

text: grey

text: line

text: shows

text: the

text: period

text: in

text: between.

text: B,

text: C.

text: Time

text: series

text: of

text: the

text: cumulative

text: price

text: and

text: tweet

text: ra-

text: tio

text: (black

text: dots),

text: and

text: cumulative

text: estimated

text: values

text: by

text: the

text: model

text: for

text: the

text: period

text: since

text: November

text: Insets:

text: time

text: series

text: of

text: estimates

text: and

text: empirical

text: changes

text: of

text: price

text: and

text: Tweet

text: ratio

text: for

text: the

text: period

text: between

text: March

text: and

text: May

text: increases,

text: and

text: of

text: the

text: largest

text: price

text: drops

text: during

text: (Table

text: of

text: the

text: ESM).

text: Discussion

text: Due

text: do

text: the

text: decentralised

text: character

text: of

text: the

text: Bitcoin

text: currency,

text: the

text: dynamics

text: of

text: its

text: economy

text: largely

text: depend

text: on

text: the

text: behaviour

text: of

text: its

text: users,

text: who

text: (i)

text: mine

text: new

text: bitcoins

text: and

text: maintain

text: the

text: block

text: chain,

text: and

text: (ii)

text: inﬂuence

text: the

text: exchange

text: rate

text: by

text: trading

text: bitcoins

text: to

text: and

text: from

text: other

text: currencies;

text: these

text: interactions

text: between

text: users

text: form

text: the

text: social

text: backbone

text: of

text: Bitcoin.

text: In

text: this

text: paper,

text: we

text: used

text: the

text: digital

text: traces

text: of

text: user

text: activity

text: in

text: the

text: Bitcoin

text: economy

text: to

text: disentangle

text: the

text: relationships

text: between

text: its

text: diﬀerent

text: variables:

text: user

text: base,

text: information

text: search,

text: information

text: sharing,

text: and

text: price.

text: While

text: each

text: of

text: these

text: four

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text: energy condensation state. Interestingly, according to [57], even if the small-scale slow modes are also forced by the external forcing or by the slow-fast coupling, a nonlinear superposition of forward-enstrophy and inverse-energy cascades of slow − modes also appears to support the observed atmospheric k 3 spectra among many other proposals (c.f., Ref. [23]): Ref. [22] showed that the large-scale pumping can be provided by the partial vortex-wave coupling, and the condensation absolute equilibrium spectra may oﬀer an intrinsic mechanism for such coupling to feed the large-scale vortex modes; for more critical readers see below for more. 3.3 Relevant questions on our conjecture and possible solutions One may fairly question: Why the even smaller-Ro (=0.0021) case of Chen et al. [15] does not show inverse transfer of Z − ? And, the 2w w interactions are those responsible for the conservation laws used in the absolute equilibrium calculation; Z − then, why it was observed in [22] that the inverse transfer of was provided by 33 w while the forward transfer was − given there by 2w w? ‘Devils’ are in the dynamical details, which are beyond the general conservation-law argument, and may provide the explanations: When the Rossby number is too small, the coupling would be too weak to be able to trigger the spontaneous mirror symmetry breaking (chirality) through the coupling (dissipation might work jointly as the trigger — note that the equipartitions presented for the small-Ro Galerkin-truncated inviscid simulations of Bourouiba [41] indicate that there is no spontaneous mirror symmetry breaking between that reasonably decoupled subsystems.) And, of course, when the Rossby number is too large, the coupling is too strong to identify the 2D3C subdynamics separately. For the latter question, − though it is true that Bourouiba et al. [22] did not directly show 2w w inverse transfer, one possible consistent dynamical scenario goes as follows. First of all, we need to make it clear that the transfer directions in triadic interactions/relaxation in general depend on the given (relative) amplitudes of the Fourier modes. The simple examples are the cases of the familiar nonhelical initial energy spectra ansatz ka in 3D: if a > 2, the large-k modes’ energies are higher than equipartition (‘too hot’) and that should be transferred to smaller k, contrary to the common forward turbulence cascade, unless that for some special reason spontaneous symmetry breaking, as we just proposed for the ﬁndings of Chen et al. [15] and Bourouiba et al. [22], happens to allow a subsystem with the helical absolute equilibrium to indeed be able to contain the much higher energy at the smallest scales [8]; similarly for pure 2D case, if the initial energy and enstrophy are respectively too high at the two ends of the wave number range (‘too hot’ - ‘hotter’ means ‘more negative’ of the negative temperature, because the more negative the temperature is, the larger is the singular wavenumber, thus the higher is the energy level around there), their transfer directions during the relaxation are contrary to the ones usually observed. Now, as a possibility, suppose ‘initially’ (starting from some time the partial decoupling occurs) the temperatures of the system are somehow set up, say, (mainly) − by just the 2w w interactions due to the distributions of the invariants and the interactions at that moment (assumed to be in quasi-equilibrium), then for some reason, due to the re-distribution of the invariants and the change of the strength of the − coupling, the slow-mode subsystem feels ‘too hot’ (the hotter the more negative is ΓE) so that 2w w interactions want to Z transfer forwardly; but, the coupled waves however may feel the slow-mode subsystem still ‘too cold’ (in the sense of the − full Euler absolute equilibrium) and keep warming it by 33 w interactions: A steady cycle of the energy may be formed by such imbalance, and such ‘too hot’ and ‘too cold’ dissymmetry appears to be in line with the trigger of the ‘spontaneous mirror symmetry breaking’ of exchanging opposite-sign helicities just mentioned. Of course the system is not in equilibrium, but taking the objective absolute equilibria as the aim the system tends to relax is a good way of thinking. If one imposes Z Z large-scale friction, transfers to large scales of in a statistical steady state might ‘naturally’ emerge. The conservation of these quantities was shown to be reasonably accurately valid only in the small Ro regime in Ref. [41], not in the moderate/intermediate-Ro case. How can our argument work for moderate/intermediate Ro, and is it possible for the existence of additional invariant(s) introduced by resonant and near-resonant modes (see, e.g., Smith and Lee[17] and references therein)? The answer lies in that we are talking about the persistent eﬀects of the otherwise self-autonomous 2D3C subdynamics as a conceptual comprehension, not any sort of precise shape or exact dynamics. This is reminiscent of the ﬁnding of the partial thermalization at the end of the inertial range[58], where hyperviscosity or other variants with appropriate parameterization [59] can greatly enhance the bottleneck phenomena as the residue of thermalization: Although theory says about the asymptotic inﬁnite parameter (say, the hyperviscosity in Ref. [58]) limit, the bottleneck appears already in the normal ﬂuid case and is already greatly strengthened for ‘moderate/intermediate’ parameters there. On the other hand, such remarks mean that the relevant argument does not need to consider that complete two-dimensionalization and separation between fast (waves) and slow (vortex) modes by fast rotation are unconditionally ascertained in the large-Ro limit. As for the near-resonant modes (see, e.g., Refs. [23, 17] and references therein), it may be that they, or at least part of them with small | | kz /k, can be included into the slow manifold as if they also obey the same conservation laws and the statistical mechanics, at the appropriate time regime for good approximation: For example, one may further check whether in the data the near- − − | | resonant modes, kicking in ﬁrst with 33 2 and 33 w interactions, are those with small kz /k, i.e., the ‘near-slow’ modes whose dynamical time scales are relatively slow. Of course, another speculation is that the resonant manifold (including the slow one) may present its own special conservation laws and statistical mechanics (favoring the inverse transfers), and that near-resonant modes help the interactions between the slow and fast modes in this manifold. However, the following analysis shows that the resonant condition does not introduce extra generic invariant(s): The already known symmetry relations of the full dynamics in the rotating frame and the slow-mode dynamics are the same, Cˆccpkcq +Cˆccqpck +Cˆcckqcp = 0 and ckkCˆccpkcq +cppCˆccqpck +cqqCˆcckqcp = 0, 8

text: energy

text: condensation

text: state.

text: Interestingly,

text: according

text: to

text: even

text: if

text: the

text: small-scale

text: slow

text: modes

text: are

text: also

text: forced

text: by

text: the

text: external

text: forcing

text: or

text: by

text: the

text: slow-fast

text: coupling,

text: a

text: nonlinear

text: superposition

text: of

text: forward-enstrophy

text: and

text: inverse-energy

text: cascades

text: of

text: slow

font: FDQTJF+txsy

text: modes

text: also

text: appears

text: to

text: support

text: the

text: observed

text: atmospheric

text: k

text: spectra

text: among

text: many

text: other

text: proposals

text: (c.f.,

text: Ref.

text: Ref.

text: showed

text: that

text: the

text: large-scale

text: pumping

text: can

text: be

text: provided

text: by

text: the

text: partial

text: vortex-wave

text: coupling,

text: and

text: the

text: condensation

text: absolute

text: equilibrium

text: spectra

text: may

text: oﬀer

text: an

text: intrinsic

text: mechanism

text: for

text: such

text: coupling

text: to

text: feed

text: the

text: large-scale

text: vortex

text: modes;

text: for

text: more

text: critical

text: readers

text: see

text: below

text: for

text: more.

label: section

text: Relevant

text: questions

text: on

text: our

text: conjecture

text: and

text: possible

text: solutions

text: One

text: may

text: fairly

text: question:

text: Why

text: the

text: even

text: smaller-Ro

text: case

text: of

text: Chen

text: et

text: al.

text: does

text: not

text: show

text: inverse

text: transfer

text: of

text: Z

text: And,

text: the

text: w

text: interactions

text: are

text: those

text: responsible

text: for

text: the

text: conservation

text: laws

text: used

text: in

text: the

text: absolute

text: equilibrium

text: calculation;

text: Z

text: then,

text: why

text: it

text: was

text: observed

text: in

text: that

text: the

text: inverse

text: transfer

text: of

text: was

text: provided

text: by

text: w

text: while

text: the

text: forward

text: transfer

text: was

text: given

text: there

text: by

text: w?

text: ‘Devils’

text: are

text: in

text: the

text: dynamical

text: details,

text: which

text: are

text: beyond

text: the

text: general

text: conservation-law

text: argument,

text: and

text: may

text: provide

text: the

text: explanations:

text: When

text: the

text: Rossby

text: number

text: is

text: too

text: small,

text: the

text: coupling

text: would

text: be

text: too

text: weak

text: to

text: be

text: able

text: to

text: trigger

text: the

text: spontaneous

text: mirror

text: symmetry

text: breaking

text: (chirality)

text: through

text: the

text: coupling

text: (dissipation

text: might

text: work

text: jointly

text: as

text: the

text: trigger

text: note

text: that

text: the

text: equipartitions

text: presented

text: for

text: the

text: small-Ro

text: Galerkin-truncated

text: inviscid

text: simulations

text: of

text: Bourouiba

text: indicate

text: that

text: there

text: is

text: no

text: spontaneous

text: mirror

text: symmetry

text: breaking

text: between

text: that

text: reasonably

text: decoupled

text: subsystems.)

text: And,

text: of

text: course,

text: when

text: the

text: Rossby

text: number

text: is

text: too

text: large,

text: the

text: coupling

text: is

text: too

text: strong

text: to

text: identify

text: the

text: subdynamics

text: separately.

text: For

text: the

text: latter

text: question,

text: though

text: it

text: is

text: true

text: that

text: Bourouiba

text: et

text: al.

text: did

text: not

text: directly

text: show

text: w

text: inverse

text: transfer,

text: one

text: possible

text: consistent

text: dynamical

text: scenario

text: goes

text: as

text: follows.

text: First

text: of

text: all,

text: we

text: need

text: to

text: make

text: it

text: clear

text: that

text: the

text: transfer

text: directions

text: in

text: triadic

text: interactions/relaxation

text: in

text: general

text: depend

text: on

text: the

text: given

text: (relative)

text: amplitudes

text: of

text: the

text: Fourier

text: modes.

text: The

text: simple

text: examples

text: are

text: the

text: cases

text: of

text: the

text: familiar

text: nonhelical

text: initial

text: energy

text: spectra

text: ansatz

text: ka

text: in

text: if

text: a

text: the

text: large-k

text: modes’

text: energies

text: are

text: higher

text: than

text: equipartition

text: (‘too

text: hot’)

text: and

text: that

text: should

text: be

text: transferred

text: to

text: smaller

text: k,

text: contrary

text: to

text: the

text: common

text: forward

text: turbulence

text: cascade,

text: unless

text: that

text: for

text: some

text: special

text: reason

text: spontaneous

text: symmetry

text: breaking,

text: as

text: we

text: just

text: proposed

text: for

text: the

text: ﬁndings

text: of

text: Chen

text: et

text: al.

text: and

text: Bourouiba

text: et

text: al.

text: happens

text: to

text: allow

text: a

text: subsystem

text: with

text: the

text: helical

text: absolute

text: equilibrium

text: to

text: indeed

text: be

text: able

text: to

text: contain

text: the

text: much

text: higher

text: energy

text: at

text: the

text: smallest

text: scales

text: similarly

text: for

text: pure

text: case,

text: if

text: the

text: initial

text: energy

text: and

text: enstrophy

text: are

text: respectively

text: too

text: high

text: at

text: the

text: two

text: ends

text: of

text: the

text: wave

text: number

text: range

text: (‘too

text: hot’

text: ‘hotter’

text: means

text: ‘more

text: negative’

text: of

text: the

text: negative

text: temperature,

text: because

text: the

text: more

text: negative

text: the

text: temperature

text: is,

text: the

text: larger

text: is

text: the

text: singular

text: wavenumber,

text: thus

text: the

text: higher

text: is

text: the

text: energy

text: level

text: around

text: there),

text: their

text: transfer

text: directions

text: during

text: the

text: relaxation

text: are

text: contrary

text: to

text: the

text: ones

text: usually

text: observed.

text: Now,

text: as

text: a

text: possibility,

text: suppose

text: ‘initially’

text: (starting

text: from

text: some

text: time

text: the

text: partial

text: decoupling

text: occurs)

text: the

text: temperatures

text: of

text: the

text: system

text: are

text: somehow

text: set

text: up,

text: say,

text: (mainly)

text: by

text: just

text: the

text: w

text: interactions

text: due

text: to

text: the

text: distributions

text: of

text: the

text: invariants

text: and

text: the

text: interactions

text: at

text: that

text: moment

text: (assumed

text: to

text: be

text: in

text: quasi-equilibrium),

text: then

text: for

text: some

text: reason,

text: due

text: to

text: the

text: re-distribution

text: of

text: the

text: invariants

text: and

text: the

text: change

text: of

text: the

text: strength

text: of

text: the

text: coupling,

text: the

text: slow-mode

text: subsystem

text: feels

text: ‘too

text: hot’

text: (the

text: hotter

text: the

text: more

text: negative

text: is

text: ΓE)

text: so

text: that

text: w

text: interactions

text: want

text: to

text: Z

text: transfer

text: forwardly;

text: but,

text: the

text: coupled

text: waves

text: however

text: may

text: feel

text: the

text: slow-mode

text: subsystem

text: still

text: ‘too

text: cold’

text: (in

text: the

text: sense

text: of

text: the

text: full

text: Euler

text: absolute

text: equilibrium)

text: and

text: keep

text: warming

text: it

text: by

text: w

text: interactions:

text: A

text: steady

text: cycle

text: of

text: the

text: energy

text: may

text: be

text: formed

text: by

text: such

text: imbalance,

text: and

text: such

text: ‘too

text: hot’

text: and

text: ‘too

text: cold’

text: dissymmetry

text: appears

text: to

text: be

text: in

text: line

text: with

text: the

text: trigger

text: of

text: the

text: ‘spontaneous

text: mirror

text: symmetry

text: breaking’

text: of

text: exchanging

text: opposite-sign

text: helicities

text: just

text: mentioned.

text: Of

text: course

text: the

text: system

text: is

text: not

text: in

text: equilibrium,

text: but

text: taking

text: the

text: objective

text: absolute

text: equilibria

text: as

text: the

text: aim

text: the

text: system

text: tends

text: to

text: relax

text: is

text: a

text: good

text: way

text: of

text: thinking.

text: If

text: one

text: imposes

text: Z

text: Z

text: large-scale

text: friction,

text: transfers

text: to

text: large

text: scales

text: of

text: in

text: a

text: statistical

text: steady

text: state

text: might

text: ‘naturally’

text: emerge.

text: The

text: conservation

text: of

text: these

text: quantities

text: was

text: shown

text: to

text: be

text: reasonably

text: accurately

text: valid

text: only

text: in

text: the

text: small

text: Ro

text: regime

text: in

text: Ref.

text: not

text: in

text: the

text: moderate/intermediate-Ro

text: case.

text: How

text: can

text: our

text: argument

text: work

text: for

text: moderate/intermediate

text: Ro,

text: and

text: is

text: it

text: possible

text: for

text: the

text: existence

text: of

text: additional

text: invariant(s)

text: introduced

text: by

text: resonant

text: and

text: near-resonant

text: modes

text: (see,

text: e.g.,

text: Smith

text: and

text: and

text: references

text: therein)?

text: The

text: answer

text: lies

text: in

text: that

text: we

text: are

text: talking

text: about

text: the

text: persistent

text: eﬀects

text: of

text: the

text: otherwise

text: self-autonomous

text: subdynamics

text: as

text: a

text: conceptual

text: comprehension,

text: not

text: any

text: sort

text: of

text: precise

text: shape

text: or

text: exact

text: dynamics.

text: This

text: is

text: reminiscent

text: of

text: the

text: ﬁnding

text: of

text: the

text: partial

text: thermalization

text: at

text: the

text: end

text: of

text: the

text: inertial

text: where

text: hyperviscosity

text: or

text: other

text: variants

text: with

text: appropriate

text: parameterization

text: can

text: greatly

text: enhance

text: the

text: bottleneck

text: phenomena

text: as

text: the

text: residue

text: of

text: thermalization:

text: Although

text: theory

text: says

text: about

text: the

text: asymptotic

text: inﬁnite

text: parameter

text: (say,

text: the

text: hyperviscosity

text: in

text: Ref.

text: limit,

text: the

text: bottleneck

text: appears

text: already

text: in

text: the

text: normal

text: ﬂuid

text: case

text: and

text: is

text: already

text: greatly

text: strengthened

text: for

text: ‘moderate/intermediate’

text: parameters

text: there.

text: On

text: the

text: other

text: hand,

text: such

text: remarks

text: mean

text: that

text: the

text: relevant

text: argument

text: does

text: not

text: need

text: to

text: consider

text: that

text: complete

text: two-dimensionalization

text: and

text: separation

text: between

text: fast

text: (waves)

text: and

text: slow

text: (vortex)

text: modes

text: by

text: fast

text: rotation

text: are

text: unconditionally

text: ascertained

text: in

text: the

text: large-Ro

text: limit.

text: As

text: for

text: the

text: near-resonant

text: modes

text: (see,

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text: (a) (b) (c) (d) Fig. 8.— Electron phase space at ω t=700 (a), 1000 (b), 1600 (c) and 1800 (d), showing electron trapping pe between two DLs. The same format is used as Figure 5. initial hot electron temperature. In Sato & Okuda responsible for driving DLs, and results in subse- (1981), since eφ was far less than the applied quent breeding of secondary DLs. The chain of DL potential, which could be considered as the max- DLs occurs over an extended region. Electrons imum available energy in the system, there was are trapped between DLs, suppressing the trans- room for the DLs to grow stronger. In our case, port of electron heat in the trapped regions. The however, eφ already comes close to the initial chain of DLs accelerates ambient ions, creating DL temperature, which can similarly be understood as strong ion ﬂows across the system. The simu- an upper limit on the potential. Further strength- lations therefore reveal the mechanism by which ening beyond this value is unlikely. This may be hot electrons heated in the corona couple to ions why more DLs in larger systems do not result in a to drive the strong outﬂows that make up the so- stronger total potential jump here as they did in lar wind. This in turn separates adjacent DLs, Sato & Okuda (1981). expanding the DL breeding territory. These dy- In conclusion, using large-scale PIC simula- namics provide a more realistic picture of DL oc- tions, we reveal the self-consistent generation of currence in the solar corona that is important for multiple DLs during the transport of ﬂare-heated understanding energetic electron transport and in electrons. Dynamics only observable in multi-DL preparing for future space missions such as Solar systems are investigated for the ﬁrst time. The Probe Plus. primary DL enhances the return current, which is Acknowledgments 11 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## 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text: (a)

text: (b)

text: (c)

text: (d)

text: Fig.

text: Electron

label: caption

text: phase

text: space

text: at

text: ω

text: (a),

text: (b),

text: (c)

text: and

text: (d),

text: showing

text: electron

text: trapping

text: pe

text: between

text: two

text: DLs.

text: The

text: same

text: format

text: is

text: used

text: as

text: Figure

text: initial

text: hot

text: electron

text: temperature.

text: In

text: Sato

text: Okuda

text: responsible

text: for

text: driving

text: DLs,

text: and

text: results

text: in

text: subse-

text: since

text: eφ

text: was

text: far

text: less

text: than

text: the

text: applied

text: quent

text: breeding

text: of

text: secondary

text: DLs.

text: The

text: chain

text: of

text: DL

text: potential,

text: which

text: could

text: be

text: considered

text: as

text: the

text: max-

text: DLs

text: occurs

text: over

text: an

text: extended

text: region.

text: Electrons

text: imum

text: available

text: energy

text: in

text: the

text: system,

text: there

text: was

text: are

text: trapped

text: between

text: DLs,

text: suppressing

text: the

text: trans-

text: room

text: for

text: the

text: DLs

text: to

text: grow

text: stronger.

text: In

text: our

text: case,

text: port

text: of

text: electron

text: heat

text: in

text: the

text: trapped

text: regions.

text: The

text: however,

text: eφ

text: already

text: comes

text: close

text: to

text: the

text: initial

text: chain

text: of

text: DLs

text: accelerates

text: ambient

text: ions,

text: creating

text: DL

text: temperature,

text: which

text: can

text: similarly

text: be

text: understood

text: as

text: strong

text: ion

text: ﬂows

text: across

text: the

text: system.

text: The

text: simu-

text: an

text: upper

text: limit

text: on

text: the

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text: Further

text: strength-

text: lations

text: therefore

text: reveal

text: the

text: mechanism

text: by

text: which

text: ening

text: beyond

text: this

text: value

text: is

text: unlikely.

text: This

text: may

text: be

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text: electrons

text: heated

text: in

text: the

text: corona

text: couple

text: to

text: ions

text: why

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text: DLs

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text: larger

text: systems

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text: make

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text: potential

text: jump

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text: wind.

text: This

text: in

text: turn

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text: adjacent

text: DLs,

text: Sato

text: Okuda

text: expanding

text: the

text: DL

text: breeding

text: territory.

text: These

text: dy-

text: In

text: conclusion,

text: using

text: large-scale

text: PIC

text: simula-

text: namics

text: provide

text: a

text: more

text: realistic

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text: the

text: self-consistent

text: generation

text: of

text: currence

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text: the

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text: corona

text: that

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text: important

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text: DLs

text: during

text: the

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text: of

text: ﬂare-heated

text: understanding

text: energetic

text: electron

text: transport

text: and

text: in

text: electrons.

text: Dynamics

text: only

text: observable

text: in

text: multi-DL

text: preparing

text: for

text: future

text: space

text: missions

text: such

text: as

text: Solar

text: systems

text: are

text: investigated

text: for

text: the

text: ﬁrst

text: time.

text: The

text: Probe

text: Plus.

text: primary

text: DL

text: enhances

text: the

text: return

text: current,

text: which

text: is

text: Acknowledgments

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