text: ∗ ∗ − → − − W γ π form factors in the τ π (cid:96)+(cid:96) ν decays τ Adolfo Guevara Departamento de F´ısica, Centro de Investigacio´n y de Estudios Avanzados, Avenida IPN 2508, Col. San Pedro Zacatenco, 07360 M´exico DF, M´exico E-mail: aguevara@fis.cinvestav.mx − → − − Abstract. We study the disintegration τ π (cid:96)+(cid:96) ν , where l denotes either an electron or ∗ ∗τ a muon. This process requires knowledge about the W γ π interaction, which is modeled using Resonance Chiral Theory. Within this framework we study the sensitivity of the observables of the process, such as the branching ratios and the dilepton mass spectra, to the model dependent contributions. The information of the hadronization of the quark currents is encoded in form factors which are evaluated using the lightest vector and axial-vector multiplets as degrees of freedom. We determine the couplings by matching the asymptotic QCD behavior of the form factors in the limit of an inﬁnite number of colours. The branching ratios we obtain show that the process with l = e should be measured soon in B factories. 1. Introduction It is well known that the interaction between quarks and gluons becomes stronger in low-energy processes [1]. Due to this non-perturbative characteristic of QCD it is advisable to appeal to eﬀective theories. To study this process it is necessary to have precise information about the ∗ ∗ W γ π interaction. The relevant form factors have been studied before with Chiral Perturbation → − Theory (χPT) [2] in the π e+e e ν decays[3, 4, 5], where the photon is almost on-shell. The e knowledge of this interaction can be used to compute radiative corrections and the couplings needed to calculate the lightest pseudoscalar contributions to the hadronic light-by-light piece of the anomalous magnetic moment of the muon[6]. − → − − The diﬀerence between the τ π (cid:96)+(cid:96) ν process and the pion decay is that the process τ we study may have an energy above the chiral symmetry breaking scale, where the χPT is no longer valid; this is why we employ Resonance Chiral Theory (RχT)[7, 8]. RχT introduces meson resonances as dynamical ﬁelds preserving the chiral symmetry properties and carrying out the required calculation by means of an expansion in powers of the inverse of the number of colours, 1 [9]. NC 2. Decay amplitude − → − − The τ π (cid:96)+(cid:96) ν [10] decays have not been measured yet, and in order to compute the τ branching ratios and invariant mass spectra we follow the work done in [11], where the process with a real photon is analyzed and the participating vector and axial-vector form factors are obtained. As a consequence of the virtuality of the photon, a diagram that vanished in [11] ##LTLine##

text: W

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text: form

text: factors

text: in

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text: τ

text: Adolfo

text: Guevara

text: Departamento

text: de

text: F´ısica,

text: Centro

text: de

text: Investigacio´n

text: y

text: de

text: Estudios

text: Avanzados,

text: Avenida

text: IPN

text: Col.

text: San

text: Pedro

text: Zacatenco,

text: M´exico

text: DF,

text: M´exico

text: E-mail:

text: aguevara@fis.cinvestav.mx

label: footer

text: Abstract.

text: We

text: study

text: the

text: disintegration

text: τ

text: π

text: ν

text: where

text: l

text: denotes

text: either

text: an

text: electron

text: or

text: ∗τ

text: a

text: muon.

text: This

text: process

text: requires

text: knowledge

text: about

text: the

text: W

text: γ

text: π

text: interaction,

text: which

text: is

text: modeled

text: using

text: Resonance

text: Chiral

text: Theory.

text: Within

text: this

text: framework

text: we

text: study

text: the

text: sensitivity

text: of

text: the

text: observables

text: of

text: the

text: process,

text: such

text: as

text: the

text: branching

text: ratios

text: and

text: the

text: dilepton

text: mass

text: spectra,

text: to

text: the

text: model

text: dependent

text: contributions.

text: The

text: information

text: of

text: the

text: hadronization

text: of

text: the

text: quark

text: currents

text: is

text: encoded

text: in

text: form

text: factors

text: which

text: are

text: evaluated

text: using

text: the

text: lightest

text: vector

text: and

text: axial-vector

text: multiplets

text: as

text: degrees

text: of

text: freedom.

text: We

text: determine

text: the

text: couplings

text: by

text: matching

text: the

text: asymptotic

text: QCD

text: behavior

text: of

text: the

text: form

text: factors

text: in

text: the

text: limit

text: of

text: an

text: inﬁnite

text: number

text: of

text: colours.

text: The

text: branching

text: ratios

text: we

text: obtain

text: show

text: that

text: the

text: process

text: with

text: l

text: e

text: should

text: be

text: measured

text: soon

text: in

text: B

text: factories.

text: Introduction

text: It

text: is

text: well

text: known

text: that

text: the

text: interaction

text: between

text: quarks

text: and

text: gluons

text: becomes

text: stronger

text: in

text: low-energy

text: processes

text: Due

text: to

text: this

text: non-perturbative

text: characteristic

text: of

text: QCD

text: it

text: is

text: advisable

text: to

text: appeal

text: to

text: eﬀective

text: theories.

text: To

text: study

text: this

text: process

text: it

text: is

text: necessary

text: to

text: have

text: precise

text: information

text: about

text: the

text: W

text: γ

text: π

text: interaction.

text: The

text: relevant

text: form

text: factors

text: have

text: been

text: studied

text: before

text: with

text: Chiral

text: Perturbation

text: Theory

text: (χPT)

text: in

text: the

text: π

text: e+e

text: e

text: ν

text: where

text: the

text: photon

text: is

text: almost

text: on-shell.

text: The

text: e

text: knowledge

text: of

text: this

text: interaction

text: can

text: be

text: used

text: to

text: compute

text: radiative

text: corrections

text: and

text: the

text: couplings

text: needed

text: to

text: calculate

text: the

text: lightest

text: pseudoscalar

text: contributions

text: to

text: the

text: hadronic

text: light-by-light

text: piece

text: of

text: the

text: anomalous

text: magnetic

text: moment

text: of

text: the

text: The

text: diﬀerence

text: between

text: the

text: τ

text: π

text: ν

text: process

text: and

text: the

text: pion

text: decay

text: is

text: that

text: the

text: process

text: τ

text: we

text: study

text: may

text: have

text: an

text: energy

text: above

text: the

text: chiral

text: symmetry

text: breaking

text: scale,

text: where

text: the

text: χPT

text: is

text: no

text: longer

text: valid;

text: this

text: is

text: why

text: we

text: employ

text: Resonance

text: Chiral

text: Theory

text: RχT

text: introduces

text: meson

text: resonances

text: as

text: dynamical

text: ﬁelds

text: preserving

text: the

text: chiral

text: symmetry

text: properties

text: and

text: carrying

text: out

text: the

text: required

text: calculation

text: by

text: means

text: of

text: an

text: expansion

text: in

text: powers

text: of

text: the

text: inverse

text: of

text: the

text: number

text: of

text: colours,

text: NC

text: Decay

text: amplitude

text: The

text: τ

text: π

text: ν

text: decays

text: have

text: not

text: been

text: measured

text: yet,

text: and

text: in

text: order

text: to

text: compute

text: the

text: τ

text: branching

text: ratios

text: and

text: invariant

text: mass

text: spectra

text: we

text: follow

text: the

text: work

text: done

text: in

text: where

text: the

text: process

text: with

text: a

text: real

text: photon

text: is

text: analyzed

text: and

text: the

text: participating

text: vector

text: and

text: axial-vector

text: form

text: factors

text: are

text: obtained.

text: As

text: a

text: consequence

text: of

text: the

text: virtuality

text: of

text: the

text: photon,

text: a

text: diagram

text: that

text: vanished

text: in

text: ##LTLine##


text: − → − − Table 1. The central values of the diﬀerent contributions to the branching ratio of the τ π (cid:96)+(cid:96) ν decays τ are given on the left hand side of the table. The error bands due to variations of the couplings are displayed in the right hand side of the table. The error band of the IB-IB contributions stems from the uncertainties on the F decay constant and the τ lepton lifetime [1]. π (cid:2) (cid:3) (cid:2) (cid:3) l = e l = µ l = e l = µ × − × − (cid:2) ± × − (cid:3) (cid:2) ± × − (cid:3) IB-IB 1.461 10 5 1.600 10 7 0.006 10 5 0.007 10 7 − × − × − (cid:2)− × − × −(cid:3) (cid:2)− × − × −(cid:3) IB-V 2 10 8 1.4 10 8 1 10 7, 1 10 7 4 10 9, 4 10 8 − × − × − (cid:2)− × − × −(cid:3) (cid:2)− × − × −(cid:3) IB-A 9 10 7 1.01 10 7 3 10 6, 2 10 6 2 10 7, 6 10 7 × − × − × − × − × − × − V-V 1.16 10 6 6.30 10 7 4 10 7, 4 10 6 1 10 7, 3 10 6 × − × − (cid:0)× − × −(cid:1) (cid:2) × − × − (cid:3) A-A 2.20 10 6 1.033 10 6 1 10 6, 9 10 6 2 10 7, 6 10 6 × − − × − ∼ − ∼ − V-A 2 10 10 5 10 11 10 10 10 10 × − × − × − × − × − Total 1.710 10 5 1.938 10 6 1.7+−1.1 10 5 3 10 7, 1 10 5 0.3 (cid:18) (cid:19) F 2 − 2G − F 2 F√ F − (cid:48)(cid:48) F (t, k2) = V 1 V D (k2) A D (t) + A V D (t)D (k2)( λ t + λ m2), (7) A Fπ fV ρ Fπ a1 2Fπ a1 ρ 0 π where 1 D (s) = − − . (8) R M2 s iM Γ (s) R R R and −| − F π+π (k2) 1 B(k2) = F V ρ , (9) π k2 −| − where F π+π (k2) is the I = 1 part of the π+π vector form factor. V ρ Given the rather wide widths of the ρ(770) and a (1260) resonances, their oﬀ-shell behavior 1 becomes important and has to be considered in eq. (8). A consistent and formalism independent deﬁnition for the energy-dependent spin-one resonance widths was given in [13], which we follow for the ρ(770) case. The simpliﬁed expression that we use for the a (1260) meson was obtained 1 in [14]. 4. Branching ratios The form factors are completely determined using short distance constraints [7, 15], which − allows us to compute the branching ratio and the (cid:96)+(cid:96) invariant mass spectra. After numerical integration, we obtained the contributions to the total branching ratio of the diﬀerent parts. These diﬀerent contributions come from the squared moduli of the three amplitudes described in the previous section and their interferences. However, in our approach there is an uncertainty (cid:46) in this short distance constraints which is 20%. Taking this into account, the contributions vary as noted in Table 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## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: caption

text: Table

label: paragraph

text: The

text: central

text: values

text: of

text: the

text: diﬀerent

text: contributions

text: to

text: the

text: branching

text: ratio

text: of

text: the

text: τ

text: π

text: ν

text: decays

text: τ

text: are

text: given

text: on

text: the

text: left

text: hand

text: side

text: of

text: the

text: table.

text: The

text: error

text: bands

text: due

text: to

text: variations

text: of

text: the

text: couplings

text: are

text: displayed

text: in

text: the

text: right

text: hand

text: side

text: of

text: the

text: table.

text: The

text: error

text: band

text: of

text: the

text: IB-IB

text: contributions

text: stems

text: from

text: the

text: uncertainties

text: on

text: the

text: F

text: decay

text: constant

text: and

text: the

text: τ

text: lepton

text: lifetime

text: π

label: table

text: l

text: e

text: l

text: µ

text: l

text: e

text: l

text: µ

text: IB-IB

text: IB-V

text: IB-A

text: V-V

text: A-A

text: V-A

text: Total

text: F

text: F

text: F√

text: F

text: F

text: (t,

text: V

text: V

text: D

text: A

text: D

text: (t)

text: A

text: V

text: D

text: (t)D

text: λ

text: t

text: λ

text: A

text: Fπ

text: fV

text: ρ

text: Fπ

text: ρ

text: π

text: where

text: D

text: (s)

text: R

text: s

text: iM

text: Γ

text: (s)

text: R

text: R

text: R

text: and

text: F

text: π+π

text: F

text: V

text: ρ

text: π

text: where

text: F

text: π+π

text: is

text: the

text: I

text: part

text: of

text: the

text: π+π

text: vector

text: form

text: factor.

text: V

text: ρ

text: Given

text: the

text: rather

text: wide

text: widths

text: of

text: the

text: and

text: a

text: resonances,

text: their

text: oﬀ-shell

text: behavior

text: becomes

text: important

text: and

text: has

text: to

text: be

text: considered

text: in

text: eq.

text: A

text: consistent

text: and

text: formalism

text: independent

text: deﬁnition

text: for

text: the

text: energy-dependent

text: spin-one

text: resonance

text: widths

text: was

text: given

text: in

text: which

text: we

text: follow

text: for

text: the

text: case.

text: The

text: simpliﬁed

text: expression

text: that

text: we

text: use

text: for

text: the

text: a

text: meson

text: was

text: obtained

text: in

text: Branching

text: ratios

text: The

text: form

text: factors

text: are

text: completely

text: determined

text: using

text: short

text: distance

text: constraints

text: which

text: allows

text: us

text: to

text: compute

text: the

text: branching

text: ratio

text: and

text: the

text: invariant

text: mass

text: spectra.

text: After

text: numerical

text: integration,

text: we

text: obtained

text: the

text: contributions

text: to

text: the

text: total

text: branching

text: ratio

text: of

text: the

text: diﬀerent

text: parts.

text: These

text: diﬀerent

text: contributions

text: come

text: from

text: the

text: squared

text: moduli

text: of

text: the

text: three

text: amplitudes

text: described

text: in

text: the

text: previous

text: section

text: and

text: their

text: interferences.

text: However,

text: in

text: our

text: approach

text: there

text: is

text: an

text: uncertainty

text: in

text: this

text: short

text: distance

text: constraints

text: which

text: is

text: Taking

text: this

text: into

text: account,

text: the

text: contributions

text: vary

text: as

text: noted

text: in

text: Table

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text: (cid:2) (cid:3) − → − − × − − BR (τ π µ+µ ν ) = 3 10 7, 10 5 , where the hadronic error dominates our result for τ the (cid:96) = µ case. From these ﬁgures, we conclude that the (cid:96) = e decay mode should be measured soon in the B-factories. This may also be true for the (cid:96) = µ case only if the branching fraction happens to be close to the upper limit of the range we have given. 6. Acknowledgements This work has been partially supported by the Spanish grant FPA2011-25948 and Conacyt (M´exico). References [1] Beringer J et al. (Particle Data Group) 2012 Phys. Rev. D 86 010001. [2] Weinberg, S 1979 Physica A 96 327. Gasser J and Leutwyler H 1984 Annals Phys. 158 142, 1985 Nucl.Phys. B250 465. [3] Bryman D, Depommier P and Leroy C 1982 Phys. Reports 88 151. DeBaenst P and Pestieau J 1968 Nuovo Cim. A53 137. [4] Kersch A and Scheck F 1986 Nucl. Phys. B 263 475. [5] Vaks V and Ioﬀe B 1958 Nuovo Cim. 10 342. [6] Roig P, Guevara A and L´opez Castro G 2013 1401.4099 [hep-ph]. [7] Ecker G et. al. 1989 Nucl.Phys. B 321 311, Phys.Lett. B 223 425. [8] Portol´es J 2010AIP Conf.Proc. 1322. [9] ’t Hoooft G 1974 Nucl.Phys. B 72 461, 75 461. Witten E 1979 Nucl.Phys. B160 57. [10] Roig P, Guevara A and L´opez Castro G 2012Phys. Rev. D88. [11] Guo Z and Roig P 2010 Phys. Rev. D 82 113016. [12] G´omez Dumm D and Roig P 2013 Eur. Phys. Jour. C 73 2528. [13] G´omez Dumm D, Pich A and Portol´es J 2000 Phys. Rev. D62 054014. [14] G´omez Dumm D et. al. 2010 Phys. Lett. B 685 158. [15] Ruiz-Femen´ıa P, Pich A and Portol´es J 2003 JHEP 0307 003. G´omez Dumm D, Pich A and Portol´es J 2004 Phys. Rev. D 69 073002. Cirigliano, V et. al. 2004 Phys. Lett. B 596 96, 2006 Nucl.Phys. B 753 139. Guo Z H 2008 Phys. Rev. D 78 033004. G´omez Dumm D et. al. 2010 Phys. Rev. D 81 034031. Kampf, K and Novotny, J 2011 Phys. Rev. D 84 014036. Roig P and Sanz-Cillero J J Preprint hep-ph/1312.6206.

text: BR

text: (τ

text: π

text: µ+µ

text: ν

text: where

text: the

text: hadronic

text: error

text: dominates

text: our

text: result

text: for

text: τ

text: the

text: µ

text: case.

text: From

text: these

text: ﬁgures,

text: we

text: conclude

text: that

text: the

text: e

text: decay

text: mode

text: should

text: be

text: measured

text: soon

text: in

text: the

text: B-factories.

text: This

text: may

text: also

text: be

text: true

text: for

text: the

text: µ

text: case

text: only

text: if

text: the

text: branching

text: fraction

text: happens

text: to

text: be

text: close

text: to

text: the

text: upper

text: limit

text: of

text: the

text: range

text: we

text: have

text: given.

label: section

text: Acknowledgements

text: This

text: work

text: has

text: been

text: partially

text: supported

text: by

text: the

text: Spanish

text: grant

text: and

text: Conacyt

text: (M´exico).

text: References

text: Beringer

text: J

text: et

text: al.

text: (Particle

text: Data

text: Group)

text: Phys.

text: Rev.

text: D

text: Weinberg,

text: S

text: Physica

text: A

text: Gasser

text: J

text: and

text: Leutwyler

text: H

text: Annals

text: Phys.

text: Nucl.Phys.

text: Bryman

text: D,

text: Depommier

text: P

text: and

text: Leroy

text: C

text: Phys.

text: Reports

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text: P

text: and

text: Pestieau

text: J

text: Nuovo

text: Cim.

text: Kersch

text: A

text: and

text: Scheck

text: F

text: Nucl.

text: Phys.

text: B

text: Vaks

text: V

text: and

text: Ioﬀe

text: B

text: Nuovo

text: Cim.

text: Roig

text: P,

text: Guevara

text: A

text: and

text: L´opez

text: Castro

text: G

text: [hep-ph].

text: Ecker

text: G

text: et.

text: al.

text: Nucl.Phys.

text: B

text: Phys.Lett.

text: B

text: Portol´es

text: J

text: Conf.Proc.

text: ’t

text: Hoooft

text: G

text: Nucl.Phys.

text: B

text: Witten

text: E

text: Nucl.Phys.

text: Roig

text: P,

text: Guevara

text: A

text: and

text: L´opez

text: Castro

text: G

text: Rev.

text: Guo

text: Z

text: and

text: Roig

text: P

text: Phys.

text: Rev.

text: D

text: G´omez

text: Dumm

text: D

text: and

text: Roig

text: P

text: Eur.

text: Phys.

text: Jour.

text: C

text: G´omez

text: Dumm

text: D,

text: Pich

text: A

text: and

text: Portol´es

text: J

text: Phys.

text: Rev.

text: G´omez

text: Dumm

text: D

text: et.

text: al.

text: Phys.

text: Lett.

text: B

text: Ruiz-Femen´ıa

text: P,

text: Pich

text: A

text: and

text: Portol´es

text: J

text: JHEP

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text: Dumm

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text: J

text: Phys.

text: Rev.

text: D

text: Cirigliano,

text: V

text: et.

text: al.

text: Phys.

text: Lett.

text: B

text: Nucl.Phys.

text: B

text: Guo

text: Z

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text: Phys.

text: Rev.

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text: Structural stability for the splash singularities of the water waves problem Angel Castro, Diego Co´rdoba, Charles Feﬀerman, Francisco Gancedo, Javier Go´mez-Serrano May 26, 2020 Abstract In this paper we show a structural stability result for water waves. The main moti- vation for this result is that we would like to exhibit a water wave whose interface starts as a graph and ends in a splash. Numerical simulations lead to an approximate solution with the desired behaviour. The stability result will conclude that near the approximate solution to water waves there is an exact solution. 1 Introduction The water waves problem models the motion of an incompressible ﬂuid with constant density ρ in a domain Ω(t) with a free boundary ∂Ω(t), which satisﬁes the Euler equation R with the presence of gravity and whose ﬂow in potential. The system, in 2, can be written, after some computations, as an equation for the free boundary, { ∈ R} ∂Ω(t) = z(α, t) = (z (α, t), z (α, t)) : α , (1) 1 2 and an equation for the amplitude of the vorticity, ω(α, t), in the following way z (α, t) = BR(z, ω)(α, t) + c(α, t)z (α, t), (2) t α (cid:16) (cid:17) − · − ω2 ω (α, t) = 2BR (z, ω)(α, t) z (α, t) | | (α, t) + (cω) (α, t) t t α 4 ∂αz 2 α α (3) · − + 2c(α, t)BR (z, ω)(α, t) z (α, t) 2(z ) (α, t), α α 2 α (cid:90) where BR(z, ω) is the classical Birkhoﬀ-Rott integral − ⊥ 1 (z(α, t) z(β, t)) BR(z, ω)(α, t) = P V | − | ω(β, t)dβ. (4) 2π R z(α, t) z(β, t) 2 1 ##LTLine## ##LTLine## ##LTLine##

text: Structural

label: title

text: stability

text: for

text: the

text: splash

text: singularities

text: of

text: the

text: water

text: waves

text: problem

text: Angel

label: author

text: Castro,

text: Diego

text: Co´rdoba,

text: Charles

text: Feﬀerman,

text: Francisco

text: Gancedo,

text: Javier

text: Go´mez-Serrano

text: May

text: Abstract

text: In

label: abstract

text: this

text: paper

text: we

text: show

text: a

text: structural

text: stability

text: result

text: for

text: water

text: waves.

text: The

text: main

text: moti-

text: vation

text: for

text: this

text: result

text: is

text: that

text: we

text: would

text: like

text: to

text: exhibit

text: a

text: water

text: wave

text: whose

text: interface

text: starts

text: as

text: a

text: graph

text: and

text: ends

text: in

text: a

text: splash.

text: Numerical

text: simulations

text: lead

text: to

text: an

text: approximate

text: solution

text: with

text: the

text: desired

text: behaviour.

text: The

text: stability

text: result

text: will

text: conclude

text: that

text: near

text: the

text: approximate

text: solution

text: to

text: water

text: waves

text: there

text: is

text: an

text: exact

text: solution.

text: Introduction

text: The

text: water

text: waves

text: problem

text: models

text: the

text: motion

text: of

text: an

text: incompressible

text: ﬂuid

text: with

text: constant

text: density

text: ρ

text: in

text: a

text: domain

text: Ω(t)

text: with

text: a

text: free

text: boundary

text: ∂Ω(t),

text: which

text: satisﬁes

text: the

text: Euler

text: equation

text: R

text: with

text: the

text: presence

text: of

text: gravity

text: and

text: whose

text: ﬂow

text: in

text: potential.

text: The

text: system,

text: in

text: can

text: be

text: written,

text: after

text: some

text: computations,

text: as

text: an

text: equation

text: for

text: the

text: free

text: boundary,

text: R}

text: ∂Ω(t)

text: z(α,

text: t)

text: (z

text: (α,

text: t),

text: z

text: (α,

text: t))

text: α

text: and

text: an

text: equation

text: for

text: the

text: amplitude

text: of

text: the

text: vorticity,

text: ω(α,

text: t),

text: in

text: the

text: following

text: way

text: z

text: (α,

text: t)

text: BR(z,

text: ω)(α,

text: t)

text: c(α,

text: t)z

text: (α,

text: t),

text: t

text: α

text: ω2

text: ω

text: (α,

text: t)

text: (z,

text: ω)(α,

text: t)

text: z

text: (α,

text: t)

text: (α,

text: t)

text: (cω)

text: (α,

text: t)

text: t

text: t

text: α

text: ∂αz

text: α

text: α

text: t)BR

text: (z,

text: ω)(α,

text: t)

text: z

text: (α,

text: t)

text: (α,

text: t),

text: α

text: α

text: α

text: where

text: BR(z,

text: ω)

text: is

text: the

text: classical

text: Birkhoﬀ-Rott

text: integral

text: (z(α,

text: t)

text: z(β,

text: t))

text: BR(z,

text: ω)(α,

text: t)

text: P

text: V

text: ω(β,

text: t)dβ.

text: 2π

text: R

text: z(α,

text: t)

text: z(β,

text: t)

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:18) (cid:90) − − − ⊥ × (cid:90)1 π (z(α) z(α β)) − − · − − − (cid:19) Q2 | − − | (z(α) z(α β)) (∂4z(α) ∂4z(α β))ω(α β)dβ z π − z(α) z(α β) 4 α α π − − ⊥ (cid:90) 1 π (x(α) x(α β)) − − · − − − + Q2 | − − | (x(α) x(α β)) (∂4x(α) ∂4x(α β))γ(α β)dβ x π − x(α) x(α β) 4 α α π π Q2 − I = | z| σ ∂4D∂4(Q2BR(z, ∂4ω) Q2BR(x, ∂4γ))dα 1,4 − z 2 z α α z α x α π α where l.o.t stands for low order terms, nice terms easier to deal with. (cid:90) I1,1 = l.o.t + I1,1,1 where (cid:90) π Q2 ∇ · − ∇ · I = 2 | z| σ ∂4D( Q(z) ∂4zBR(z, ω) Q(x) ∂4xBR(x, γ))dα 1,1,1 − z 2 z α α α π α (cid:90) π Q2 ∇ · = 2 | z| σ ∂4D Q(z) ∂4DBR(z, ω)dα − z 2 z α α π α (cid:90) π Q2 ∇ · − ∇ · + 2 | z| σ ∂4D( Q(z) ∂4xBR(z, ω) Q(x) ∂4xBR(x, γ))dα − z 2 z α α α π α (cid:124) (cid:123)(cid:122) (cid:125) ≤ π Q2 | | (cid:107)∇ (cid:107) (cid:90)2 | z| σ ∂4D 2d(cid:90)α Q(z)BR(z, ω) ∞ − z 2 z α L π α bounded as for local existence π Q2 | | (cid:124) π Q2 |∇ · (cid:123)(cid:122) − ∇ · | (cid:125) + | z| σ ∂4D 2 + | z| σ Q(z) ∂4xBR(z, ω) Q(x) ∂4xBR(x, γ) 2dα − z 2 z α − z 2 z α α π α π α l.o.t in D and d ≤ CP (E(t)) which means I is done. 1,1 From now on we will denote − − ∆ z(α) = z(α) z(α β) β (cid:90) (cid:90) I = I + I + I + I where 1,2 1,2,1 1,2,2 1,2,3 1,2,4 ⊥ (cid:90) π Q2 1 (cid:90) π ∆ ∂4D (α)(cid:18) − (cid:19) I = | z| σ ∂4DQ2 |β α | ω(α β)dβdα 1,2,1 − z 2 z α z 2π − ∆ z(α) 2 π α π β (cid:90) π Q2 1 (cid:90) π ⊥ 1 − 1 − I = | z| σ ∂4DQ2 ∆ ∂4x (α) | | | | ω(α β)dβdα 1,2,2 − z 2 z α z 2π − β α ∆ z(α) 2 ∆ x(α) 2 π α π β β ⊥ (cid:90) π Q2 1 π ∆ (cid:90)∂4x (α) − I = | z| σ ∂4DQ2 | β α | d(α β)dβdα 1,2,3 − z 2 z α z 2π − ∆ x(α) 2 π α π β ⊥ π Q2 − 1 π ∆ ∂4x (α) − I = | z| σ ∂4D(Q2 Q2) | β α | γ(α β)dβdα 1,2,4 − z 2 z α z x 2π − ∆ x(α) 2 π α π β 21 ##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: π

text: (z(α)

text: z(α

text: β))

text: (z(α)

text: z(α

text: β))

text: β))ω(α

text: β)dβ

text: z

text: π

text: z(α)

text: z(α

text: β)

text: α

text: α

text: π

text: π

text: (x(α)

text: x(α

text: β))

text: (x(α)

text: x(α

text: β))

text: β))γ(α

text: β)dβ

text: x

text: π

text: x(α)

text: x(α

text: β)

text: α

text: α

text: π

text: π

text: I

text: z|

text: σ

text: ∂4ω)

text: z

text: z

text: α

text: α

text: z

text: α

text: x

text: α

text: π

text: α

text: where

text: l.o.t

text: stands

text: for

text: low

text: order

text: terms,

text: nice

text: terms

text: easier

text: to

text: deal

text: with.

text: l.o.t

text: where

text: π

text: I

text: z|

text: σ

text: Q(z)

text: ω)

text: Q(x)

text: γ))dα

text: z

text: z

text: α

text: α

text: α

text: π

text: α

text: π

text: z|

text: σ

text: Q(z)

text: ω)dα

text: z

text: z

text: α

text: α

text: π

text: α

text: π

text: z|

text: σ

text: Q(z)

text: ω)

text: Q(x)

text: γ))dα

text: z

text: z

text: α

text: α

text: α

text: π

text: α

text: π

text: z|

text: σ

text: Q(z)BR(z,

text: ω)

text: z

text: z

text: α

text: L

text: π

text: α

text: bounded

text: as

text: for

text: local

text: existence

text: π

text: π

text: z|

text: σ

text: z|

text: σ

text: Q(z)

text: ω)

text: Q(x)

text: γ)

text: z

text: z

text: α

text: z

text: z

text: α

text: α

text: π

text: α

text: π

text: α

text: l.o.t

text: in

text: D

text: and

text: d

text: CP

text: (E(t))

text: which

text: means

text: I

text: is

text: done.

text: From

text: now

text: on

text: we

text: will

text: denote

text: z(α)

text: z(α)

text: z(α

text: β)

text: β

text: I

text: I

text: I

text: I

text: I

text: where

text: π

text: π

text: I

text: z|

text: σ

text: |β

text: α

text: ω(α

text: β)dβdα

text: z

text: z

text: α

text: z

text: 2π

text: z(α)

text: π

text: α

text: π

text: β

text: π

text: π

text: I

text: z|

text: σ

text: (α)

text: ω(α

text: β)dβdα

text: z

text: z

text: α

text: z

text: 2π

text: β

text: α

text: z(α)

text: x(α)

text: π

text: α

text: π

text: β

text: β

text: π

text: π

text: (α)

text: I

text: z|

text: σ

text: β

text: α

text: d(α

text: β)dβdα

text: z

text: z

text: α

text: z

text: 2π

text: x(α)

text: π

text: α

text: π

text: β

text: π

text: π

text: (α)

text: I

text: z|

text: σ

text: β

text: α

text: γ(α

text: β)dβdα

text: z

text: z

text: α

text: z

text: x

text: 2π

text: x(α)

text: π

text: α

text: π

text: β

text: ##LTLine##

text: ##LTLine##

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text: (cid:90) −(cid:124) π Q2 (cid:123)(cid:122) (cid:125) I = | z| σ ∂4D∂4b x dα 2,2,3 − z 2 z α α e α π α Error term We deal with I more carefully. We use that 2,2,1 · · − · − · ∂4D z = ∂4z z ∂4x x ∂4x D α α α α α α α α − · · − · = 3∂3z ∂2z + 3∂3x ∂2x ∂4x D α α α α α α − · − · − · = 3∂3D ∂2z 3∂3x ∂2D ∂4x D α α α α α α to obtain (cid:90) I = I1 + I2 + I3 2,2,1 2,2,1 2,2,1 2,2,1 − (cid:90) π Q2 · − I1 = 3 | z| σ ∂3D ∂2z∂4(c b )dα 2,2,1 − z 2 z α α α s π α − (cid:90) π Q2 · − I2 = 3 | z| σ ∂3x ∂2D∂4(c b )dα 2,2,1 − z 2 z α α α s π α − π Q2 · − I3 = | z| σ ∂4x ∂ D∂4(c b )dα 2,2,1 − z 2 z α α α s π α We can integrate by parts in all of the above terms to get low order terms. We are ﬁnally done with I. − 4.2 Computing the diﬀerence ϕ ψ From the local existence proof we(cid:18)ﬁnd(cid:19)the equ(cid:18)ation for ϕ : (cid:19) t       −  − − Q2 ϕ2 − · z (P 1(z)) ϕ = ϕB (t) | z |∂ Q2 BR | α| + 2| | α  t z 2 z α Q2 z t z z  (cid:90) al z α α  ω − · z − | |Qz − Q3 | | − | |  +Q Qz | | 2cBR | α|Q Qz c2 z α | z| BR 2Qz (c z ) (18) z t z z z α α Q z α α t α α z α 1 π · z B (t) = (Q2BR) | α|dα z 2π − z α z π α We will show how to ﬁnd the equation for ψ . We start from t Q2γ − | | ψ = | x | b x s α 2 x α and therefore | | ψ2 Q2γ2 b2 x 2 − = |x | + s α γb , Q2 4 x Q2 s x α x 30 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: π

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text: we ﬁnd that (cid:18) (cid:19) · ⊥ (Q ) · ∇ · x ∇ · ⊥ x (cid:90) x | x |tα = x 2Q(x) | α| + Q(x) x α| t | α x t x α x 3 α α α ∇ · x ∇ · x 1 π · x + Q(x) | α|B (t) + Q(x) | α| f | α| dα (19) x x x 2π − α x 2 α α π α That yields (cid:18) (cid:19) x (cid:124) · ·(cid:123)(cid:122)∇ · x (cid:125) (7) = 2(Q (Q ) BR) | α| = NICE + 2Q BR x x 2Q(x) | α| x x t α x α t x x α α NICE (at the level of xα,xt,BR) · ⊥ · ∇ · ⊥ x x (cid:124) · ∇(cid:123)(cid:122) · x (cid:125) + 2Q BR x Q(x) x α| t | α + 2Q BR x Q(x) | α|B (t) x α α x (cid:90)3 x α x x α α NICE (at the level of xα,xt,BR) (cid:124) · ∇ · (cid:123)x(cid:122) 1 π · x (cid:125) · x + 2Q BR x Q(x) | α| f | α| dα +2Q (Q ) BR | α| x α x 2π − α x 2 x x t α x α π α α part of error terms Finally: · ⊥ x · ∇ · ⊥ x x · x (7) = 2(Q (Q ) BR) | α| = NICE + 2Q BR x Q(x) x α| t | α + 2Q (Q ) BR | α| x(cid:18) x t α x (cid:19) (cid:18) x α (cid:19) α x 3 x x t α x α α α − − (P 1(z)) − ∇ − · x −(cid:124) ∇ · (cid:123)∇(cid:122) − · x (cid:125) (8) = Q2 2| | α = Q2 P 1(x) | α| = 2Q Qx x P 1(x) | α| x(cid:18) x (cid:19) x 2 x x x α 2 x α α α α α NICE (at the level of xα) · ⊥ −(cid:124) · ∇ (cid:123)(cid:122)− · x (cid:125) − ∇ − · ⊥ x x Q2x 2P 1(x) | α| Q2 P 1(x) x α|α | α x α 2 x x 2 α x 3 α α NICE (at the level of xα) which means (cid:18) (cid:19) − · ⊥ − (P 1(z)) − ∇ − · ⊥ x x (8) = Q2 2| | α = NICE Q2 P 1(x) x α|α | α (cid:18) (cid:19)x x x 2 α x 3 (cid:18) (cid:19) α α α (cid:124) (cid:123)(cid:122) (cid:125) γ γ (Q ) γ (9) = Q (Q ) | | = (Q ) (Q ) | | +Q | x |αt γ + Q (Q ) | | x x t x α x t x x x t x x x x α α α α α α NICE (at the level of xα,xt) 35 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: E We gather all the formulas from (4) to (12) absorbing the error terms by ˜1 whenever we α encounter them. It yields: · ⊥ · ⊥ · ⊥ (cid:124)ψ2 ∇ (cid:123)·(cid:122) ⊥ x x (cid:125) −(cid:124) ·(cid:123)(cid:122)⊥ x x (cid:125) −(cid:124) ∇ − (cid:123)(cid:122)· ⊥ x x (cid:125) ψ = NICE + Q(x) x α|α | α Q2BR x α|α | α Q2 P 1(x) x α|α | α αt Q α x 3 (cid:18)x (cid:19)t α x 3 x 2 α x 3 x α α α (16) (15) (15) · ⊥ · ⊥ (cid:124) ∇ (cid:123)·(cid:122) ⊥ x x (cid:125) (cid:124) (cid:123)(cid:122) γ (cid:125) (cid:124) · ∇ (cid:123)(cid:122) · ⊥ x x (cid:125) + Q γ Q(x) x α| t | α + Q (Q ) | | + 2Q BR x Q(x) x α| t | α x α xα 3 (cid:18) x x t (cid:19)xα α x α α xα 3 (18) (14) (18) · ⊥ (cid:124) (cid:123)(cid:122) · x (cid:125) −(cid:124) · x(cid:123)(cid:122) (cid:125) −(cid:124) · ∇(cid:123)(cid:122) · ⊥ x x (cid:125) + Q (Q ) 2BR | α| 2b BR | α| Q (Q ) 2b BR x Q Q(x) x α|α | α x x t α x x x x x α s α x α x 3 α α α α (14) (17) (16) | | · ⊥ −(cid:124) | (cid:123)|(cid:122) (Q ) (cid:125) −(cid:124) b2 x 2 ∇ (cid:123)(cid:122) · ⊥ x x (cid:125) −(cid:124) Q3 | (cid:123)(cid:122)| (cid:125) (b2 x ) x α s α Q(x) x α|α | α | x|( BR 2) (Q ) s α α Q Q α x 3 x α x α x x α α (17) (16) (17) · ⊥ · ⊥ −(cid:124) | | ∇ (cid:123)(cid:122) · ⊥ x x (cid:125) · ⊥ x x E Q3 BR 2 Q(x) x α|α | α +(Q2BR) x α| t | α + ˜1 x α x 3 x α α x 3 α α α (16) (cid:18) (cid:19) We compute γ(cid:18) (cid:19) · x (14) = Q (Q ) | | + Q (Q ) 2BR | α| x x t x x t α x x α α α (Q ) γ (Q ) · x = 2 x t (Q )2 | | + 2 x t (Q )2BR | α| x x α Q 2 x Q x x α α x α (Q ) − (Q ) γ − (Q ) x − (Q ) | | = 2 x t ψ 2 x t (Q2) | | 2 x t (Q2) BR | α| 2 x t ( x B (t)) Q α Q x α 2 x Q x α α x Q α x x x α x α x The last formula allows us to conclude that (14)=NICE. We reorganize using (15), (16), (17) and (18). · ⊥ (cid:18) − · ⊥ ∇ − · ⊥ x(cid:19) x ψ = NICE Q2(BR x + P 1(x) x ) α|α | α αt x t α 2 α x 3 α | | · · ⊥ − | | b2 x 2 BR x − ψ2 ∇ · ⊥ x x Q3 BR 2 + s α + 2b α Q(x) x α|α | α Q4 s Q2 Q4 α x 3 x x x α · ⊥ · ⊥ (cid:18) · ⊥ x x (cid:18) · (cid:19) ∇ (cid:19) · ⊥ x x + (Q2BR) x α| t | α + (Q γ + 2Q BR x ) Q(x) x α| t | α x α α x 3 x x α α x 3 α α | | | | − Q3( BR 2) (b2 x ) · x E x | | α + s α α + 2b BR | α| Q (Q ) + ˜1 x Q s x x x α α α x α α 37 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: Rayleigh-Taylor

text: function,

text: )(t)

text: min

text: t)σ

text: (α,

text: t),

text: z

text: ∈T

text: z

text: α

text: and

text: ﬁnally

text: m(ql)(t)

text: min

text: z(α,

text: t)

text: ql

text: ∈T

text: α

text: for

text: l

text: √1

text: √1

text: √1

text: √1

text: √1

text: √1

text: √1

text: √1

text: which

text: are

text: the

text: singular

text: points

text: of

text: the

text: transformation

text: P

text: E

text: Remark

text: We

text: can

text: absorb

text: the

text: terms

text: in

text: (t)

text: by

text: E(t)

text: raised

text: to

text: an

text: appropriate

text: power

text: and

text: terms

text: in

text: (x,

text: γ)

text: by

text: performing

text: the

text: splitting

text: z

text: z

text: x

text: x

text: (or

text: the

text: analogous

text: one

text: for

text: a

text: E

text: diﬀerent

text: variable)

text: for

text: any

text: norm

text: or

text: any

text: quantity

text: that

text: appears

text: in

text: (t).

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: | | − | | Q2∂3σ = z H(∂3ϕ ) c z H(∂4ϕ) + NICE(z, ω, ϕ) z α z α α t α α This implies (cid:90) (cid:18) (cid:19) II = II + II + II + II 2,3 2,3,1 2,3,2 2,3,3 2,3,4 · ⊥ · ⊥ (cid:90)π D |(cid:18) | z z − | | x x (cid:19) II = H(∂3 ) z H(∂3ϕ ) α|α | α x H(∂3ψ ) α|α | α dα 2,3,1 − α α α t z 3 α α t x 3 π α α · ⊥ · ⊥ −(cid:90) π D(cid:18) | | z z − | | x x (cid:19) II = H(∂3 ) c z H(∂4ϕ) α|α | α b x H(∂4ψ) α|α | α dα 2,3,2 − α α α z 3 s α α x 3 π α α · ⊥ · ⊥ (cid:90)π D z z − x x II = H(∂3 ) NICE(z, ω, ϕ) α|α | α NICE(x, γ, ψ) α|α | α dα 2,3,3 − α z 3 x 3 π α α · ⊥ − π D x x II = H(∂3 ) α|α | α dα + errors 2,3,4 − α x 3 π α It is easy to ﬁnd ≤ II CP (E(t)) + cδ(t), error terms 2,3,4 ≤ II CP (E(t)), l.o.t 2,3,3 In II we split further: 2,3,2 (cid:90) II = II1 + II2 2,3,2 2,3,2 2,3,2 · ⊥ − (cid:90) π D | | (cid:18)D z z (cid:19) II1 = H(∂3 ) z H(∂4 ) α|α | α dα 2,3,2 − α α α z 3 π α · ⊥ · ⊥ − π D | |z z − | |x x II22,3,2 = − H(cid:90)(∂α3 )H(∂α4ψ)(cid:18) c zα α|αz |3(cid:19)α bs xα α|xα |3α dα π α α Then · ⊥ 1 π | D | | |z z ≤ II1 = H(∂3 ) 2 z α|α | α dα CP (E(t)) 2,3,2 2 − α α z 3 π α α For II2 (cid:90) : (cid:18) (cid:18) (cid:19)(cid:19) 2,3,2 · ⊥ · ⊥ − π D | |z z − | |x x ≤ II2 = Λ1/2(∂3ψ)Λ1/2 H(∂3 ) c z α|α | α b x α|α | α dα CP (E(t)) 2,3,2 − α α α z 3 s α x 3 π α α It remains (cid:90) II = II1 + II2 2,3,1 2,3,1 2,3,1 · ⊥ (cid:90) π D D z(cid:18) z (cid:19) II1 = H(∂3 )H(∂3 ) α|α | α dα 2,3,1 − α α t z 2 π (cid:124) (cid:123)(cid:122) (cid:125) α · ⊥ · ⊥ π D z z − x x II2 = H(∂3 ) H(∂3ψ ) α|α | α α|α | α dα 2,3,1 − α α t z 2 x 2 π α α approx. sol. 45 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: z

text: c

text: z

text: NICE(z,

text: ω,

text: ϕ)

text: z

text: α

text: z

text: α

text: α

text: t

text: α

text: α

text: This

text: implies

text: II

text: II

text: II

text: II

text: II

text: D

text: z

text: z

text: x

text: x

text: II

text: z

text: α|α

text: α

text: x

text: α|α

text: α

text: dα

text: α

text: α

text: α

text: t

text: z

text: α

text: α

text: t

text: x

text: π

text: α

text: α

text: π

text: z

text: z

text: x

text: x

text: II

text: c

text: z

text: α|α

text: α

text: b

text: x

text: α|α

text: α

text: dα

text: α

text: α

text: α

text: z

text: s

text: α

text: α

text: x

text: π

text: α

text: α

text: D

text: z

text: z

text: x

text: x

text: II

text: NICE(z,

text: ω,

text: ϕ)

text: α|α

text: α

text: NICE(x,

text: γ,

text: ψ)

text: α|α

text: α

text: dα

text: α

text: z

text: x

text: π

text: α

text: α

text: π

text: D

text: x

text: x

text: II

text: α|α

text: α

text: dα

text: errors

text: α

text: x

text: π

text: α

text: It

text: is

text: easy

text: to

text: ﬁnd

text: II

text: CP

text: (E(t))

text: cδ(t),

text: error

text: terms

text: II

text: CP

text: (E(t)),

text: l.o.t

text: In

text: II

text: we

text: split

text: further:

text: II

text: π

text: D

text: z

text: z

text: z

text: α|α

text: α

text: dα

text: α

text: α

text: α

text: z

text: π

text: α

text: π

text: D

text: |z

text: z

text: |x

text: x

text: c

text: zα

text: α|αz

text: bs

text: xα

text: α|xα

text: dα

text: π

text: α

text: α

text: Then

text: π

text: D

text: |z

text: z

text: z

text: α|α

text: α

text: dα

text: CP

text: (E(t))

text: α

text: α

text: z

text: π

text: α

text: α

text: For

text: π

text: D

text: |z

text: z

text: |x

text: x

text: c

text: z

text: α|α

text: α

text: b

text: x

text: α|α

text: α

text: dα

text: CP

text: (E(t))

text: α

text: α

text: α

text: z

text: s

text: α

text: x

text: π

text: α

text: α

text: It

text: remains

text: II

text: π

text: D

text: D

text: z

text: α|α

text: α

text: dα

text: α

text: α

text: t

text: z

text: π

text: α

text: π

text: D

text: z

text: z

text: x

text: x

text: α|α

text: α

text: α|α

text: α

text: dα

text: α

text: α

text: t

text: z

text: x

text: π

text: α

text: α

text: approx.

text: sol.

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text: ##LTLine##

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text: Theorem ?? was announced in [?]. C If we knew (t), f(t), g(t), k or bounds on them, a priori, then we could provide bounds on E (cid:107) − (cid:107) (t) at any time T . We point out here that E(t) controls the norm ∂ z1(α) ∂ x1(α) ∞. α α L∀ Let T be a time in which the approximate solution is a graph, i.e. ∂ x1(α, T ) > 0 α. g α g Now, if E(T ) < ∂ x1(α, T ) then g α g −(cid:107) − (cid:107) ∂ z1(α, T ) > ∂ z1(α) ∂ x1(α) ∞ + ∂ x1(α, T ) > 0, α g α α L α g and this shows that z is a graph. In other words, the possible set of solutions of the water waves equation is a ball centered at (x, γ, ζ) with the topology given by E. All of the elements of this ball are graphs, therefore the solution is necessarily a graph. Thus, the problem is C reduced to study and ﬁnd bounds for (t), f(t), g(t), k. The recent developments of computer architecture have boosted their use in mathematics, giving birth to a full set of new results only achievable by this enormous power. However, it has the drawback that ﬂoating-point operations can not be performed exactly, resulting in numerical errors. In order to overcome this diﬃculty and be able to prove rigorous results, we use the so-called interval arithmetics, in which instead of working with arbitrary real numbers, we perform computations over intervals which have representable numbers as endpoints. On these objects, an arithmetic is deﬁned in such a way that we are guaranteed that for every ∈ ∈ x X, y Y ∈ x (cid:63) y X (cid:63) Y, for any operation (cid:63). For example, [x, x] + [y, y] = [x + y, x + y] × { } { } [x, x] [y, y] = [min xy, xy, xy, xy , max xy, xy, xy, xy ] We can also deﬁne the interval version of a function f(X) as an interval I that satisﬁes that ∈ ∈ for every x X we have f(x) I. The article is organized as follows: in sections 2 and 3 we give some details about how to control the errors f, g and the constants that arise in Theorem ?? by using the computer. Finally, in section 4 we give a complete proof of Theorem ??. 2 Bounds for f (t) and g(t) 2.1 Representation of the functions and Interpolation The ﬁrst thing one has to decide is how to represent the data and how to pass from the cloud of points in space-time obtained by non-rigorous simulation to a function deﬁned − × everywhere in [ π, π] [0, T ]. We need to interpolate in some way. In our case, we chose to represent the functions x and γ by piecewise polynomials (splines) of high degree (10) in space, and low degree (3) in time. To do so, we ﬁrst interpolate in space for every node in the time mesh. The interpolation is made via B-Splines. Since the interpolation is reduced to solve a linear (interval) system Ac = y, where A is constant in 6 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: Theorem

text: was

text: announced

text: in

text: C

text: If

text: we

text: knew

text: (t),

text: f(t),

text: g(t),

text: k

text: or

text: bounds

text: on

text: them,

text: a

text: priori,

text: then

text: we

text: could

text: provide

text: bounds

text: on

text: E

text: (t)

text: at

text: any

text: time

text: T

text: We

text: point

text: out

text: here

text: that

text: E(t)

text: controls

text: the

text: norm

text: α

text: α

text: L∀

text: Let

text: T

text: be

text: a

text: time

text: in

text: which

text: the

text: approximate

text: solution

text: is

text: a

text: graph,

text: i.e.

text: T

text: α.

text: g

text: α

text: g

text: Now,

text: if

text: E(T

text: T

text: then

text: g

text: α

text: g

text: T

text: T

text: α

text: g

text: α

text: α

text: L

text: α

text: g

text: and

text: this

text: shows

text: that

text: z

text: is

text: a

text: graph.

text: In

text: other

text: words,

text: the

text: possible

text: set

text: of

text: solutions

text: of

text: the

text: water

text: waves

text: equation

text: is

text: a

text: ball

text: centered

text: at

text: (x,

text: γ,

text: ζ)

text: with

text: the

text: topology

text: given

text: by

text: E.

text: All

text: of

text: the

text: elements

text: of

text: this

text: ball

text: are

text: graphs,

text: therefore

text: the

text: solution

text: is

text: necessarily

text: a

text: graph.

text: Thus,

text: the

text: problem

text: is

text: C

text: reduced

text: to

text: study

text: and

text: ﬁnd

text: bounds

text: for

text: (t),

text: f(t),

text: g(t),

text: k.

text: The

text: recent

text: developments

text: of

text: computer

text: architecture

text: have

text: boosted

text: their

text: use

text: in

text: mathematics,

text: giving

text: birth

text: to

text: a

text: full

text: set

text: of

text: new

text: results

text: only

text: achievable

text: by

text: this

text: enormous

text: power.

text: However,

text: it

text: has

text: the

text: drawback

text: that

text: ﬂoating-point

text: operations

text: can

text: not

text: be

text: performed

text: exactly,

text: resulting

text: in

text: numerical

text: errors.

text: In

text: order

text: to

text: overcome

text: this

text: diﬃculty

text: and

text: be

text: able

text: to

text: prove

text: rigorous

text: results,

text: we

text: use

text: the

text: so-called

text: interval

text: arithmetics,

text: in

text: which

text: instead

text: of

text: working

text: with

text: arbitrary

text: real

text: numbers,

text: we

text: perform

text: computations

text: over

text: intervals

text: which

text: have

text: representable

text: numbers

text: as

text: endpoints.

text: On

text: these

text: objects,

text: an

text: arithmetic

text: is

text: deﬁned

text: in

text: such

text: a

text: way

text: that

text: we

text: are

text: guaranteed

text: that

text: for

text: every

text: x

text: X,

text: y

text: Y

text: x

text: y

text: X

text: Y,

text: for

text: any

text: operation

text: For

text: example,

text: [x,

text: x]

text: [y,

text: y]

text: [x

text: y,

text: x

text: y]

text: [x,

text: x]

text: [y,

text: y]

text: [min

text: xy,

text: xy,

text: xy,

text: xy

text: max

text: xy,

text: xy,

text: xy,

text: xy

text: We

text: can

text: also

text: deﬁne

text: the

text: interval

text: version

text: of

text: a

text: function

text: f(X)

text: as

text: an

text: interval

text: I

text: that

text: satisﬁes

text: that

text: for

text: every

text: x

text: X

text: we

text: have

text: f(x)

text: I.

text: The

text: article

text: is

text: organized

text: as

text: follows:

text: in

text: sections

text: and

text: we

text: give

text: some

text: details

text: about

text: how

text: to

text: control

text: the

text: errors

text: f,

text: g

text: and

text: the

text: constants

text: that

text: arise

text: in

text: Theorem

text: by

text: using

text: the

text: computer.

text: Finally,

text: in

text: section

text: we

text: give

text: a

text: complete

text: proof

text: of

text: Theorem

text: Bounds

text: for

text: f

text: (t)

text: and

text: g(t)

text: Representation

text: of

text: the

text: functions

text: and

text: Interpolation

text: The

text: ﬁrst

text: thing

text: one

text: has

text: to

text: decide

text: is

text: how

text: to

text: represent

text: the

text: data

text: and

text: how

text: to

text: pass

text: from

text: the

text: cloud

text: of

text: points

text: in

text: space-time

text: obtained

text: by

text: non-rigorous

text: simulation

text: to

text: a

text: function

text: deﬁned

text: everywhere

text: in

text: π,

text: π]

text: T

text: We

text: need

text: to

text: interpolate

text: in

text: some

text: way.

text: In

text: our

text: case,

text: we

text: chose

text: to

text: represent

text: the

text: functions

text: x

text: and

text: γ

text: by

text: piecewise

text: polynomials

text: (splines)

text: of

text: high

text: degree

text: in

text: space,

text: and

text: low

text: degree

text: in

text: time.

text: To

text: do

text: so,

text: we

text: ﬁrst

text: interpolate

text: in

text: space

text: for

text: every

text: node

text: in

text: the

text: time

text: mesh.

text: The

text: interpolation

text: is

text: made

text: via

text: B-Splines.

text: Since

text: the

text: interpolation

text: is

text: reduced

text: to

text: solve

text: a

text: linear

text: (interval)

text: system

text: Ac

text: y,

text: where

text: A

text: is

text: constant

text: in

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##


text: time and space and y depends on the values of the function at time t since the mesh in space is constant, we precondition by multiplying by the non-rigorous inverse of the midpoints of the entries of A. We remark that the system is interval-based because we need to produce a curve that is a splash (i.e. there have to be two points α , α such that we can guarantee 1 2 x (α ) = x (α ). Finally, the system is solved using a rigorous Gauss-Seidel iterative method. 0 1 0 2 We also remark that the need for interval-based calculations is only strictly necessary at time t = 0 since it is the only point in which we have to guarantee some equality. By working − with multiprecision (1024 bits) we can get widths in the coeﬃcients of the order of 10 300. In order to perform interpolation in time, we ﬁx the values of the function and its time derivative at the mesh points. This gives us lots of systems of 4 equations (the values of the function and its derivative at both endpoints) and 4 unknowns (the 4 coeﬃcients of the degree 3 polynomial) but with an explicit formula for each of them. With this method, our spline will be C1 in time but it might not be C2. 2.2 Rigorous bounds for Singular integrals In this section we will discuss the computational details of the rigorous calculation of some singular integrals. In particular we will focus on the Hilbert transform, but the methods apply to any integral kernel whose main singularity is homogeneous of degree -1. Parts of the computation (the N part) are slightly related to the Taylor models with relative remainder presented in M. Jolde¸s’ thesis [?]. Let us suppose that we have a function f given explicitly by a spline (piecewise polyno- − mial) which is Ck 1 everywhere and Ck except at ﬁnitely many points (the points in which the diﬀerent pieces of the spline are glued together). We need to calculate rigorously the Hilbert Transform of f, that is (cid:90) (cid:0) (cid:1) − P V f(x) f(y) Hf(x) = − dy, π T 2 tan x y 2 and we want to approximate it by a piecewise polynomial function with less regularity, ≤ ≤ ∞ plus an error that can be bounded in Hq, 0 q c < k and in L . Let us assume that the − ∈ knots of the spline are α , i = 0, . . . , N 1 and that we ﬁx x [α , α ] where the indices i i i+1 Z are taken modulo N and the distance between the indices is taken over . We can split our N integral in (cid:90) (cid:90) (cid:88) (cid:0) (cid:1) (cid:0) (cid:1) − − P V f(x)(cid:90) f(y) P V αj+1 f(x) f(cid:90)(y) Hf(x) = (cid:88) − dy = (cid:88) − dy π T 2 tan x2y (cid:0) π (cid:1) j αj 2 tan x2y (cid:0) (cid:1) − − P V αj+1 f(x) f(y) P V αj+1 f(x) f(y) = − dy + − dy π |j−i|>K αj 2 tan x2y π |j−i|≤K αj 2 tan x2y ≡ HfF (x) + HfN(x). Now, if we want to express HfF (x) as a polynomial, it is easy since the integrand does 7 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: time

text: and

text: space

text: and

text: y

text: depends

text: on

text: the

text: values

text: of

text: the

text: function

text: at

text: time

text: t

text: since

text: the

text: mesh

text: in

text: space

text: is

text: constant,

text: we

text: precondition

text: by

text: multiplying

text: by

text: the

text: non-rigorous

text: inverse

text: of

text: the

text: midpoints

text: of

text: the

text: entries

text: of

text: A.

text: We

text: remark

text: that

text: the

text: system

text: is

text: interval-based

text: because

text: we

text: need

text: to

text: produce

text: a

text: curve

text: that

text: is

text: a

text: splash

text: (i.e.

text: there

text: have

text: to

text: be

text: two

text: points

text: α

text: α

text: such

text: that

text: we

text: can

text: guarantee

text: x

text: (α

text: x

text: (α

text: Finally,

text: the

text: system

text: is

text: solved

text: using

text: a

text: rigorous

text: Gauss-Seidel

text: iterative

text: method.

text: We

text: also

text: remark

text: that

text: the

text: need

text: for

text: interval-based

text: calculations

text: is

text: only

text: strictly

text: necessary

text: at

text: time

text: t

text: since

text: it

text: is

text: the

text: only

text: point

text: in

text: which

text: we

text: have

text: to

text: guarantee

text: some

text: equality.

text: By

text: working

text: with

text: multiprecision

text: bits)

text: we

text: can

text: get

text: widths

text: in

text: the

text: coeﬃcients

text: of

text: the

text: order

text: of

text: In

text: order

text: to

text: perform

text: interpolation

text: in

text: time,

text: we

text: ﬁx

text: the

text: values

text: of

text: the

text: function

text: and

text: its

text: time

text: derivative

text: at

text: the

text: mesh

text: points.

text: This

text: gives

text: us

text: lots

text: of

text: systems

text: of

text: equations

text: (the

text: values

text: of

text: the

text: function

text: and

text: its

text: derivative

text: at

text: both

text: endpoints)

text: and

text: unknowns

text: (the

text: coeﬃcients

text: of

text: the

text: degree

text: polynomial)

text: but

text: with

text: an

text: explicit

text: formula

text: for

text: each

text: of

text: them.

text: With

text: this

text: method,

text: our

text: spline

text: will

text: be

text: in

text: time

text: but

text: it

text: might

text: not

text: be

text: Rigorous

text: bounds

text: for

text: Singular

text: integrals

text: In

text: this

text: section

text: we

text: will

text: discuss

text: the

text: computational

text: details

text: of

text: the

text: rigorous

text: calculation

text: of

text: some

text: singular

text: integrals.

text: In

text: particular

text: we

text: will

text: focus

text: on

text: the

text: Hilbert

text: transform,

text: but

text: the

text: methods

text: apply

text: to

text: any

text: integral

text: kernel

text: whose

text: main

text: singularity

text: is

text: homogeneous

text: of

text: degree

text: Parts

text: of

text: the

text: computation

text: (the

text: N

text: part)

text: are

text: slightly

text: related

text: to

text: the

text: Taylor

text: models

text: with

text: relative

text: remainder

text: presented

text: in

text: M.

text: Jolde¸s’

text: thesis

text: Let

text: us

text: suppose

text: that

text: we

text: have

text: a

text: function

text: f

text: given

text: explicitly

text: by

text: a

text: spline

text: (piecewise

text: polyno-

text: mial)

text: which

text: is

text: Ck

text: everywhere

text: and

text: Ck

text: except

text: at

text: ﬁnitely

text: many

text: points

text: (the

text: points

text: in

text: which

text: the

text: diﬀerent

text: pieces

text: of

text: the

text: spline

text: are

text: glued

text: together).

text: We

text: need

text: to

text: calculate

text: rigorously

text: the

text: Hilbert

text: Transform

text: of

text: f,

text: that

text: is

text: P

text: V

text: f(x)

text: f(y)

text: Hf(x)

text: dy,

text: π

text: T

text: tan

text: x

text: y

text: and

text: we

text: want

text: to

text: approximate

text: it

text: by

text: a

text: piecewise

text: polynomial

text: function

text: with

text: less

text: regularity,

text: plus

text: an

text: error

text: that

text: can

text: be

text: bounded

text: in

text: Hq,

text: q

text: c

text: k

text: and

text: in

text: L

text: Let

text: us

text: assume

text: that

text: the

text: knots

text: of

text: the

text: spline

text: are

text: α

text: i

text: N

text: and

text: that

text: we

text: ﬁx

text: x

text: [α

text: α

text: where

text: the

text: indices

text: i

text: i

text: Z

text: are

text: taken

text: modulo

text: N

text: and

text: the

text: distance

text: between

text: the

text: indices

text: is

text: taken

text: over

text: We

text: can

text: split

text: our

text: N

text: integral

text: in

text: P

text: V

text: f(y)

text: P

text: V

text: f(x)

text: Hf(x)

text: dy

text: dy

text: π

text: T

text: tan

text: π

text: j

text: αj

text: tan

text: P

text: V

text: f(x)

text: f(y)

text: P

text: V

text: f(x)

text: f(y)

text: dy

text: dy

text: π

text: |j−i|>K

text: αj

text: tan

text: π

text: |j−i|≤K

text: αj

text: tan

text: HfF

text: (x)

text: HfN(x).

text: Now,

text: if

text: we

text: want

text: to

text: express

text: HfF

text: (x)

text: as

text: a

text: polynomial,

text: it

text: is

text: easy

text: since

text: the

text: integrand

text: does

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: × T × T trigonometric polynomial on each rectangle I J of some partition of . We suppose A, B, E are smooth enough. (cid:90) (cid:16) (cid:17) T Let H be the Hilbert transform acting on functions on , i.e. P V y − Hf(x) = cot f(x y)dy. 2π T 2 T Assume that A and B have no common zeros(cid:90)on . Let ∈ T Sf(x) = A(x)f(x) + B(x)Hf(x) + E(x, y)f(y)dy, f L2( ). T Thus, S is a singular integral operator. − We hope that S 1 exists and has a not-so-big norm on L2, but we don’t know this yet. Our goal here is to ﬁnd approximate solutions F of the equation SF = f for suitable ∈ T (cid:107) − (cid:107) given f L2( ), and to check that SF f T < δ for suitable δ. Our computation of F L2( ) (cid:107) − (cid:107) will be based on heuristic ideas, but the computation of an upper bound for SF f T L2( ) will be rigorous. In our case, A(x) = 1, B(x) = 1. ⊂ ⊂ T To carry this out, let H H L2( ) be ﬁnite-dimensional subspaces, e.g. with H 0 1 ≥ − i consisting of the span of wavelets (from a wavelet bases) having lengthscale 2 Ni. Here ≥ T N N + 3 (say). Let π be the orthogonal projection from L2( ) to H , and let us solve 1 0 i i the equation π Sπ F = π f. (10) 1 1 0 If f is given explicitly in a wavelet bases, then (??) is a linear algebra problem, since π Sπ is of ﬁnite rank, and its matrix (in terms of some given basis for H ) can be computed 1 1 1 explicitly. • (cid:54)∈ If π f Range(π Sπ ), then our heuristic procedure fails. 0 1 1 • ∈ ∈ If π f Range(π Sπ ), then we ﬁnd F H such that π Sπ F = π f, i.e. π SF = 0 1 1 1 1 1 0 1 π f. 0 We then have (cid:107) − (cid:107) ≤ (cid:107) − (cid:107) (cid:107) − (cid:107) SF f T (I π )SF T + (I π )f T , L2( ) 1 L2( ) 0 L2( ) and both norms on the right-hand side may be estimate(cid:90)d explicitly. Now, our goal is to make a heuristic computation of an operator of the form S˜f(x) = A˜(x)f(f) + B˜(x)Hf(x) + E˜(x, y)f(y)dy T − T such that SS˜ I has small norm on L2( ). 9 ##LTLine## ##LTLine##

text: T

text: T

text: trigonometric

text: polynomial

text: on

text: each

text: rectangle

text: I

text: J

text: of

text: some

text: partition

text: of

text: We

text: suppose

text: A,

text: B,

text: E

text: are

text: smooth

text: enough.

text: T

text: Let

text: H

text: be

text: the

text: Hilbert

text: transform

text: acting

text: on

text: functions

text: on

text: i.e.

text: P

text: V

text: y

text: Hf(x)

text: cot

text: f(x

text: y)dy.

text: 2π

text: T

text: T

text: Assume

text: that

text: A

text: and

text: B

text: have

text: no

text: common

text: Let

text: T

text: Sf(x)

text: A(x)f(x)

text: B(x)Hf(x)

text: E(x,

text: y)f(y)dy,

text: f

text: T

text: Thus,

text: S

text: is

text: a

text: singular

text: integral

text: operator.

text: We

text: hope

text: that

text: S

text: exists

text: and

text: has

text: a

text: not-so-big

text: norm

text: on

text: but

text: we

text: don’t

text: know

text: this

text: yet.

text: Our

text: goal

text: here

text: is

text: to

text: ﬁnd

text: approximate

text: solutions

text: F

text: of

text: the

text: equation

text: SF

text: f

text: for

text: suitable

text: T

text: given

text: f

text: and

text: to

text: check

text: that

text: SF

text: f

text: T

text: δ

text: for

text: suitable

text: δ.

text: Our

text: computation

text: of

text: F

text: will

text: be

text: based

text: on

text: heuristic

text: ideas,

text: but

text: the

text: computation

text: of

text: an

text: upper

text: bound

text: for

text: SF

text: f

text: T

text: will

text: be

text: rigorous.

text: In

text: our

text: case,

text: A(x)

text: B(x)

text: T

text: To

text: carry

text: this

text: out,

text: let

text: H

text: H

text: be

text: ﬁnite-dimensional

text: subspaces,

text: e.g.

text: with

text: H

text: i

text: consisting

text: of

text: the

text: span

text: of

text: wavelets

text: (from

text: a

text: wavelet

text: bases)

text: having

text: lengthscale

text: Ni.

text: Here

text: T

text: N

text: N

text: (say).

text: Let

text: π

text: be

text: the

text: orthogonal

text: projection

text: from

text: to

text: H

text: and

text: let

text: us

text: solve

text: i

text: i

text: the

text: equation

text: π

text: Sπ

text: F

text: π

text: f.

text: If

text: f

text: is

text: given

text: explicitly

text: in

text: a

text: wavelet

text: bases,

text: then

text: is

text: a

text: linear

text: algebra

text: problem,

text: since

text: π

text: Sπ

text: is

text: of

text: ﬁnite

text: rank,

text: and

text: its

text: matrix

text: (in

text: terms

text: of

text: some

text: given

text: basis

text: for

text: H

text: can

text: be

text: computed

text: explicitly.

text: If

text: π

text: f

text: Range(π

text: Sπ

text: then

text: our

text: heuristic

text: procedure

text: fails.

text: If

text: π

text: f

text: Range(π

text: Sπ

text: then

text: we

text: ﬁnd

text: F

text: H

text: such

text: that

text: π

text: Sπ

text: F

text: π

text: f,

text: i.e.

text: π

text: SF

text: π

text: f.

text: We

text: then

text: have

text: SF

text: f

text: T

text: (I

text: π

text: )SF

text: T

text: (I

text: π

text: )f

text: T

text: and

text: both

text: norms

text: on

text: the

text: right-hand

text: side

text: may

text: be

text: explicitly.

text: Now,

text: our

text: goal

text: is

text: to

text: make

text: a

text: heuristic

text: computation

text: of

text: an

text: operator

text: of

text: the

text: form

text: S˜f(x)

text: A˜(x)f(f)

text: B˜(x)Hf(x)

text: E˜(x,

text: y)f(y)dy

text: T

text: T

text: such

text: that

text: SS˜

text: I

text: has

text: small

text: norm

text: on

text: ##LTLine##

text: ##LTLine##


text: C. Rodr´ıguez Lucatero, L. Alarc´on, R. Bernal Jaquez, and A. Schaum. Decision dynamics in complex networks subject to mass media and social contact transmission mechanisms. arXiv preprint arXiv:1210.8193, 2012. F. B. Rogers. Medical subject headings. Bulletin of the Medical Library Association, 51(1): 114–116, 1963. P. Sobkowicz, M. Kaschesky, and G. Bouchard. Opinion formation in the social web: agent-based simulations of opinion convergence and divergence. Agents and Data Mining Interaction, pages 288–303, 2012. J.A. Sokolowski and C.M. Banks. Principles of modeling and simulation: a multidisciplinary approach. Wiley, Hoboken, 2011. P.S. Teng. A comparison of simulation approaches to epidemic modeling. Annual Review of Phytopathology, 23(1):351–379, 1985. Christoph Waldhauser. Revil: Zombie outbreak simulator for the analysis of opinion propagation, 2013. URL http://knutur.at/Revil. R package version 0.1. B. Wessels and W.E. Miller. System characteristics matter: empirical evidence from ten repre- sentation studies. In Miller et al. [1999], pages 137–161. Hadley Wickham. ggplot2: elegant graphics for data analysis. Springer New York, 2009. ISBN 978-0-387-98140-6. URL http://had.co.nz/ggplot2/book. 12

text: C.

label: reference

text: Rodr´ıguez

text: Lucatero,

text: L.

text: Alarc´on,

text: R.

text: Bernal

text: Jaquez,

text: and

text: A.

text: Schaum.

text: Decision

text: dynamics

text: in

text: complex

text: networks

text: subject

text: to

text: mass

text: media

text: and

text: social

text: contact

text: transmission

text: mechanisms.

text: arXiv

text: preprint

text: F.

text: B.

text: Rogers.

text: Medical

text: subject

text: headings.

text: Bulletin

text: of

text: the

text: Medical

text: Library

text: Association,

text: P.

text: Sobkowicz,

text: M.

text: Kaschesky,

text: and

text: G.

text: Bouchard.

text: Opinion

text: formation

text: in

text: the

text: social

text: web:

text: agent-based

text: simulations

text: of

text: opinion

text: convergence

text: and

text: divergence.

text: Agents

text: and

text: Data

text: Mining

text: Interaction,

text: pages

text: J.A.

text: Sokolowski

text: and

text: C.M.

text: Banks.

text: Principles

text: of

text: modeling

text: and

text: simulation:

text: a

text: multidisciplinary

text: approach.

text: Wiley,

text: Hoboken,

text: P.S.

text: Teng.

text: A

text: comparison

text: of

text: simulation

text: approaches

text: to

text: epidemic

text: modeling.

text: Annual

text: Review

text: of

text: Phytopathology,

text: Christoph

text: Waldhauser.

text: Revil:

text: Zombie

text: outbreak

text: simulator

text: for

text: the

text: analysis

text: of

text: opinion

text: propagation,

text: URL

text: R

text: package

text: version

text: B.

text: Wessels

text: and

text: W.E.

text: Miller.

text: System

text: characteristics

text: matter:

text: empirical

text: evidence

text: from

text: ten

text: repre-

text: sentation

text: studies.

text: In

text: Miller

text: et

text: al.

text: pages

text: Hadley

text: Wickham.

text: elegant

text: graphics

text: for

text: data

text: analysis.

text: Springer

text: New

text: York,

text: ISBN

text: URL


text: while convinced advocates of that opinion will be in class I. Class R, those that are removed from the population to halt the spreading of a disease or particular opinion, has no real equivalent in any democratic system. The locking-up of individuals holding any particular, perhaps even iconoclastic opinion, is in essence incompatible with our understanding of liberal democracy. Likewise, it is unlikely that any political opinion will cause its holder’s death. Therefore, opinion propagation models for democracies must do without the R class and are limited to be of type SI or perhaps SIS.1 Another factor that is crucial in the simulation of opinion propagation in human populations is the topology of the underlying network. It has been shown, that human interaction is organized in the fashion of small-world networks [Rodr´ıguez Lucatero et al., 2012, Wessels and Miller, 1999, Eubank et al., 2004, Amaral et al., 2000]. These networks govern how individuals are linked to each other. Most prominently, the small world property states that most individuals are connected to most other individuals via only a hand full of other individuals. This requires some individuals to function as hubs, that are connecting many people. These small world networks are on one hand quite fault tolerant and avoid the breaking down of communication if single individuals are removed. More importantly for the case at hand, however, is that they also facilitate the spreading of disease, once central nodes that connect many individuals are infected. Even though these models are representative of the way humans interact, they add an ad- ditional level of complexity to the modeling of disease and opinion propagation. Therefore, the simplest models of disease propagation focus on a relaxed model that ignores these network properties. Rather, the simpler model assumes a population that is well-mixed. Here, every in- dividual is connected to every other with the same probability. This boils down to a mean-ﬁelds model in the terminology of physics’ parlance. 2.2 Disease model As detailed above and elsewhere [Rahmandad and Sterman, 2008, Patlolla et al., 2006], there are compelling reasons to diverge from canonical epidemiological models when dealing with opinions. We therefore employ a SI-model ignorant to complex network topologies. In our model of opinion propagation, the infected are not being secluded from the population, and every individual has an equal, mean connectedness to every other individual. However, further modiﬁcations apply that we will describe in the following. Here we use the Revil zombie outbreak simulator [Waldhauser, 2013]2. This package allows to model not only the conventional dichotomous disease outcome of healthy versus infected. Rather, it assumes a continuous spectrum of infectiousness. Humans carry parts of the disease within them, at all times. But only once they pass a certain threshold, they will turn into zombies and start infecting others. This notion of a continuously conceptualized disease, aptly named C-Virus, is quite similar to the way opinions work. Take for example xenophobia. While only very convinced xenophobes will start propagating these views, who can honestly say to not have reservations against the unknown? Or while many cannot agree with every point Marx ever made, who would not cherish the overall improvement of labor conditions? Aspects of even the most radical of views are present in everyone of us, to a varying degree. In its design, the disease model is quite simple. A society of a given size is initialized to a random state of infection. This is another point of departure from the traditional model that 1A noteworthy exemption are perhaps the Verbotsgesetze enacted in Austria and Germany under the impres- sion of denaziﬁcation. Here, indeed, prominent advocates of especially despicable opinions, are being held in prison and thus eﬀectively removed from the population. 2All computations were done in R [R Core Team, 2012]; plots were produced using ggplot2 [Wickham, 2009]. 3 ##LTLine##

text: while

text: convinced

text: advocates

text: of

text: that

text: opinion

text: will

text: be

text: in

text: class

text: I.

text: Class

text: R,

text: those

text: that

text: are

text: removed

text: from

text: the

text: population

text: to

text: halt

text: the

text: spreading

text: of

text: a

text: disease

text: or

text: particular

text: opinion,

text: has

text: no

text: real

text: equivalent

text: in

text: any

text: democratic

text: system.

text: The

text: locking-up

text: of

text: individuals

text: holding

text: any

text: particular,

text: perhaps

text: even

text: iconoclastic

text: opinion,

text: is

text: in

text: essence

text: incompatible

text: with

text: our

text: understanding

text: of

text: liberal

text: democracy.

text: Likewise,

text: it

text: is

text: unlikely

text: that

text: any

text: political

text: opinion

text: will

text: cause

text: its

text: holder’s

text: death.

text: Therefore,

text: opinion

text: propagation

text: models

text: for

text: democracies

text: must

text: do

text: without

text: the

text: R

text: class

text: and

text: are

text: limited

text: to

text: be

text: of

text: type

text: SI

text: or

text: perhaps

text: Another

text: factor

text: that

text: is

text: crucial

text: in

text: the

text: simulation

text: of

text: opinion

text: propagation

text: in

text: human

text: populations

text: is

text: the

text: topology

text: of

text: the

text: underlying

text: network.

text: It

text: has

text: been

text: shown,

text: that

text: human

text: interaction

text: is

text: organized

text: in

text: the

text: fashion

text: of

text: small-world

text: networks

text: [Rodr´ıguez

text: Lucatero

text: et

text: al.,

text: Wessels

text: and

text: Miller,

text: Eubank

text: et

text: al.,

text: Amaral

text: et

text: al.,

text: These

text: networks

text: govern

text: how

text: individuals

text: are

text: linked

text: to

text: each

text: other.

text: Most

text: prominently,

text: the

text: small

text: world

text: property

text: states

text: that

text: most

text: individuals

text: are

text: connected

text: to

text: most

text: other

text: individuals

text: via

text: only

text: a

text: hand

text: full

text: of

text: other

text: individuals.

text: This

text: requires

text: some

text: individuals

text: to

text: function

text: as

text: hubs,

text: that

text: are

text: connecting

text: many

text: people.

text: These

text: small

text: world

text: networks

text: are

text: on

text: one

text: hand

text: quite

text: fault

text: tolerant

text: and

text: avoid

text: the

text: breaking

text: down

text: of

text: communication

text: if

text: single

text: individuals

text: are

text: removed.

text: More

text: importantly

text: for

text: the

text: case

text: at

text: hand,

text: however,

text: is

text: that

text: they

text: also

text: facilitate

text: the

text: spreading

text: of

text: disease,

text: once

text: central

text: nodes

text: that

text: connect

text: many

text: individuals

text: are

text: infected.

text: Even

text: though

text: these

text: models

text: are

text: representative

text: of

text: the

text: way

text: humans

text: interact,

text: they

text: add

text: an

text: ad-

text: ditional

text: level

text: of

text: complexity

text: to

text: the

text: modeling

text: of

text: disease

text: and

text: opinion

text: propagation.

text: Therefore,

text: the

text: simplest

text: models

text: of

text: disease

text: propagation

text: focus

text: on

text: a

text: relaxed

text: model

text: that

text: ignores

text: these

text: network

text: properties.

text: Rather,

text: the

text: simpler

text: model

text: assumes

text: a

text: population

text: that

text: is

text: well-mixed.

text: Here,

text: every

text: in-

text: dividual

text: is

text: connected

text: to

text: every

text: other

text: with

text: the

text: same

text: probability.

text: This

text: boils

text: down

text: to

text: a

text: mean-ﬁelds

text: model

text: in

text: the

text: terminology

text: of

text: physics’

text: parlance.

text: Disease

text: model

text: As

text: detailed

text: above

text: and

text: elsewhere

text: [Rahmandad

text: and

text: Sterman,

text: Patlolla

text: et

text: al.,

text: there

text: are

text: compelling

text: reasons

text: to

text: diverge

text: from

text: canonical

text: epidemiological

text: models

text: when

text: dealing

text: with

text: opinions.

text: We

text: therefore

text: employ

text: a

text: SI-model

text: ignorant

text: to

text: complex

text: network

text: topologies.

text: In

text: our

text: model

text: of

text: opinion

text: propagation,

text: the

text: infected

text: are

text: not

text: being

text: secluded

text: from

text: the

text: population,

text: and

text: every

text: individual

text: has

text: an

text: equal,

text: mean

text: connectedness

text: to

text: every

text: other

text: individual.

text: However,

text: further

text: modiﬁcations

text: apply

text: that

text: we

text: will

text: describe

text: in

text: the

text: following.

text: Here

text: we

text: use

text: the

text: Revil

text: zombie

text: outbreak

text: simulator

text: [Waldhauser,

text: This

text: package

text: allows

text: to

text: model

text: not

text: only

text: the

text: conventional

text: dichotomous

text: disease

text: outcome

text: of

text: healthy

text: versus

text: infected.

text: Rather,

text: it

text: assumes

text: a

text: continuous

text: spectrum

text: of

text: infectiousness.

text: Humans

text: carry

text: parts

text: of

text: the

text: disease

text: within

text: them,

text: at

text: all

text: times.

text: But

text: only

text: once

text: they

text: pass

text: a

text: certain

text: threshold,

text: they

text: will

text: turn

text: into

text: zombies

text: and

text: start

text: infecting

text: others.

text: This

text: notion

text: of

text: a

text: continuously

text: conceptualized

text: disease,

text: aptly

text: named

text: C-Virus,

text: is

text: quite

text: similar

text: to

text: the

text: way

text: opinions

text: work.

text: Take

text: for

text: example

text: xenophobia.

text: While

text: only

text: very

text: convinced

text: xenophobes

text: will

text: start

text: propagating

text: these

text: views,

text: who

text: can

text: honestly

text: say

text: to

text: not

text: have

text: reservations

text: against

text: the

text: unknown?

text: Or

text: while

text: many

text: cannot

text: agree

text: with

text: every

text: point

text: Marx

text: ever

text: made,

text: who

text: would

text: not

text: cherish

text: the

text: overall

text: improvement

text: of

text: labor

text: conditions?

text: Aspects

text: of

text: even

text: the

text: most

text: radical

text: of

text: views

text: are

text: present

text: in

text: everyone

text: of

text: us,

text: to

text: a

text: varying

text: degree.

text: In

text: its

text: design,

text: the

text: disease

text: model

text: is

text: quite

text: simple.

text: A

text: society

text: of

text: a

text: given

text: size

text: is

text: initialized

text: to

text: a

text: random

text: state

text: of

text: infection.

text: This

text: is

text: another

text: point

text: of

text: departure

text: from

text: the

text: traditional

text: model

text: that

text: noteworthy

text: exemption

text: are

text: perhaps

text: the

text: Verbotsgesetze

text: enacted

text: in

text: Austria

text: and

text: Germany

text: under

text: the

text: impres-

text: sion

text: of

text: denaziﬁcation.

text: Here,

text: indeed,

text: prominent

text: advocates

text: of

text: especially

text: despicable

text: opinions,

text: are

text: being

text: held

text: in

text: prison

text: and

text: thus

text: eﬀectively

text: removed

text: from

text: the

text: population.

text: computations

text: were

text: done

text: in

text: R

text: [R

text: Core

text: Team,

text: plots

text: were

text: produced

text: using

text: [Wickham,

text: ##LTLine##