text: − Q2 r − − f(u) = ( 3 + ) h (u u ) + O[(u u )2], (67) r4 r 0 0 h c ≈ − Q2 − r h f(u) (3 )(1 u). (68) r4 r h c Note that the prefactor appearing in the above equation is the same as that which occurs in − → the expression for the temperature and is denoted as δ = (3 Q2 ). When we have δ 0 the → r4 h temperature is low and when δ 3 temperature is high. Using the approximated form of the (cid:90) lapse function given by eq.(68) in the integral in eq.(16) for subsystem length l, we obtain (cid:113) 2√ 1 √ u2 1 l = − − . (69) rc δ 0 1 u4 1 rh u rc Using the expansion given in eq.(76)(cid:88)with x = rh u, the integral can be evaluated rc ∞ √1 Γ(n + 1) Γ(n+3) r lr = 2 4 ( h )n. (70) c 2 δ n=0 Γ(n + 1) Γ(n+4 5) rc ∼ → The series in the above equation goes as xn and therefore diverges as r r . We n c h isolate the divergent term. As the horizon radius is large due to the large charge regime, we set r = r (1 + (cid:15)) and expand in (cid:15) to extract the leading term. We start with c h √ − δlr = log[(cid:15)] + k + O[(cid:15)], h √ ≈ − (cid:15) ε e δlrh, (71) ent (cid:18) (cid:19) (cid:88) where the constants ε and k are given by ent √ ∞ πΓ(3) Γ(n + 1) Γ(n+3) − 1 ε = ek, k = 4 + 2 4 . ent 2Γ(5) 2Γ(n + 1) Γ(n+5) n 4 n=1 4 Now using the approximated form of the lapse function given by eq.(68) in the integral in A (cid:90) eq.(17) for the extremal surface area , we obtain The area (cid:113) A 2√Lr 1 √ 1 1 c = − − . (72) δ 0 u2 1 u4 1 rh u rc The area integral above is divergent. The details of the regularization and the computation are presented in Appendix-C. The ﬁnal result is the ﬁnite part of the extremal area is given by (cid:20) (cid:21) eq.(121) which we reproduce here A 2√Lr (cid:48) (cid:48) finite = Llr2 + h K + K (cid:15) + O[(cid:15)2] , (73) h 1 2 δ (cid:48) (cid:48) where K and K are constants given in Appendix-C. The renormalized entanglement en- 1 A 2 tropy (Sfinite = finite ) for the non-extremal black hole in the large charge regime, therefore A 4G comes out to be as follows 15 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: r

text: f(u)

text: h

text: (u

text: u

text: O[(u

text: u

text: r

text: h

text: c

text: r

text: h

text: f(u)

text: u).

text: r

text: h

text: c

text: Note

text: that

text: the

text: prefactor

text: appearing

text: in

text: the

text: above

text: equation

text: is

text: the

text: same

text: as

text: that

text: which

text: occurs

text: in

text: the

text: expression

text: for

text: the

text: temperature

text: and

text: is

text: denoted

text: as

text: δ

text: When

text: we

text: have

text: δ

text: the

text: h

text: temperature

text: is

text: low

text: and

text: when

text: δ

text: temperature

text: is

text: high.

text: Using

text: the

text: approximated

text: form

text: of

text: the

text: lapse

text: function

text: given

text: by

text: in

text: the

text: integral

text: in

text: for

text: subsystem

text: length

text: l,

text: we

text: obtain

text: 2√

text: l

text: rc

text: δ

text: rh

text: u

text: rc

text: Using

text: the

text: expansion

text: given

text: in

text: x

text: rh

text: u,

text: the

text: integral

text: can

text: be

text: evaluated

text: rc

text: √1

text: Γ(n

text: r

text: lr

text: h

text: )n.

text: c

text: δ

text: Γ(n

text: rc

text: The

text: series

text: in

text: the

text: above

text: equation

text: goes

text: as

text: xn

text: and

text: therefore

text: diverges

text: as

text: r

text: r

text: We

text: n

text: c

text: h

text: isolate

text: the

text: divergent

text: term.

text: As

text: the

text: horizon

text: radius

text: is

text: large

text: due

text: to

text: the

text: large

text: charge

text: regime,

text: we

text: set

text: r

text: r

text: and

text: expand

text: in

text: to

text: extract

text: the

text: leading

text: term.

text: We

text: start

text: with

text: c

text: h

text: δlr

text: k

text: h

text: ε

text: e

text: δlrh,

text: ent

text: where

text: the

text: constants

text: ε

text: and

text: k

text: are

text: given

text: by

text: ent

text: Γ(n

text: ε

text: ek,

text: k

text: ent

text: n

text: Now

text: using

text: the

text: approximated

text: form

text: of

text: the

text: lapse

text: function

text: given

text: by

text: in

text: the

text: integral

text: in

text: A

text: for

text: the

text: extremal

text: surface

text: area

text: we

text: obtain

text: The

text: area

text: A

text: c

text: δ

text: rh

text: u

text: rc

text: The

text: area

text: integral

text: above

text: is

text: divergent.

text: The

text: details

text: of

text: the

text: regularization

text: and

text: the

text: computation

text: are

text: presented

text: in

text: Appendix-C.

text: The

text: ﬁnal

text: result

text: is

text: the

text: ﬁnite

text: part

text: of

text: the

text: extremal

text: area

text: is

text: given

text: by

text: which

text: we

text: reproduce

text: here

text: A

text: finite

text: h

text: K

text: K

text: h

text: δ

text: where

text: K

text: and

text: K

text: are

text: constants

text: given

text: in

text: Appendix-C.

text: The

text: renormalized

text: entanglement

text: en-

text: A

text: tropy

text: (Sfinite

text: finite

text: for

text: the

text: non-extremal

text: black

text: hole

text: in

text: the

text: large

text: charge

text: regime,

text: therefore

text: A

text: comes

text: out

text: to

text: be

text: as

text: follows

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text: 1 Introduction Quantum correlations play an important role in studying various aspects of many-body physics pertaining to condensed matter systems, statistical mechanics, quantum information and quan- tum gravity. In this regard, entanglement entropy is a crucial quantity which provides a measure for quantum correlations in a bipartite quantum system. Such a system is described as the union of a subsystem A that is being studied and which is correlated with the rest of the system de- H scribed by its complement A . For this system the full Hilbert space may be expressed as c H H ⊗ H a tensor product of the Hilbert spaces of A and A as = . At zero temperature, c | (cid:105) HA Ac the whole system can be represented by a pure state ψ (cid:15) , with the density matrix given | (cid:105)(cid:104) | | (cid:105) by ρ = ψ ψ . However, if the subsystems A and A are entangled then the full state ψ c | (cid:105) (cid:54) | (cid:105) ⊗ cannot be expressed as a tensor product of any two states of the subsystems i.e ψ = φ | (cid:105) | (cid:105) | (cid:105) A λ where, φ and λ are states in the bases of A and A respectively. This naturally Ac A Ac c also implies that the density matrix (ρ) of the full system cannot be expressed as the tensor (cid:54) ⊗ product of ρ and ρ i.e. ρ = ρ ρ . The reduced density matrix (ρ ) for the subsystem A Ac A Ac A A is obtained via tracing ρ over the degrees of freedom of A giving ρ = T r ρ and the en- c A Ac tanglement entropy of the subsystem A is deﬁned as the corresponding Von Neumann entropy, − S = T r (ρ log ρ ). Entanglement entropy is know to have several important properties A A A A such as the strong subadditivity condition for two diﬀerent regions A and A of a quantum ≥ 1 2 system (S + S S ∪ + S ∩ ) [1]. A1 A2 A1 A2 A1 A2 To obtain the entanglement entropy for 1+1 dimensional conformal ﬁeld theories (CF T ), 1+1 a method was developed by Calabrese and Cardy using what is known as the replica trick. They showed that the entanglement entropy for a CF T exhibits an universal logarithmic 1+1 behavior [2–7]. The forementioned authors further developed techniques to study quantum entanglement of CF T in various static as well as dynamic scenarios [8–12]. However, very 1+1 little is known about CFTs in higher dimensions, where, only the case of quasi free fermions and bosons has been studied rigorously in [13–18]. Further insight into the entanglement entropy for quantum ﬁeld theories was facilitated through the AdS/CF T correspondence which relates (d)-dimensional strongly coupled boundary quantum ﬁeld theory at its conformal ﬁxed point in a large N limit, to a theory of weakly coupled gravity in (d + 1)-dimensional bulk space-time [19–23]. Using this correspondence, Ryu and Takyanagi proposed a conjecture to holographically evaluate the entanglement entropy of a (d)-dimensional quantum ﬁeld theory [24, 25] which was later generalized for time dependent backgrounds by Hubeny-Rangamani-Takayanagi (HRT) in [26]. This holographic prescription for obtaining the entanglement entropy S for a region A (enclosed by the boundary ∂A) in A the (d)-dimensional boundary ﬁeld theory involves the computation of the area of the extremal surface (denoted by γ ) extending from the boundary ∂A of the region A into the (d + 1)- A dimensional bulk such that S is given by A Area(γ ) A S = , (1) A 4Gd+1 N (d+1) where G is the gravitational constant of the bulk [27]. Besides this, they correctly N reproduced the logarithmic behavior of the entanglement entropy for a (1 + 1)-dimensional boundary CF T in AdS /CF T and also conﬁrmed the validity of the strong subadditivity 2+1 1+1 inequalities in holographic CFTs [27–33]. One of the several applications of the conjecture involves the study of the entanglement entropy for a boundary CF T at ﬁnite temperature which is dual to a AdS-Schwarzschild black hole in the bulk [34–37]. The results of which were subsequently used to study the low and the high temperature behavior of the entanglement entropy. This in turn led to the proposal of an analogous “First law of thermodynamics” like relation for the entanglement entropy by the authors in [38,39]. Another class of boundary ﬁeld theories that have been studied using the Ryu and Takyanagi 2 ##LTLine##

label: section

text: Introduction

text: Quantum

label: paragraph

text: correlations

text: play

text: an

text: important

text: role

text: in

text: studying

text: various

text: aspects

text: of

text: many-body

text: physics

text: pertaining

text: to

text: condensed

text: matter

text: systems,

text: statistical

text: mechanics,

text: quantum

text: information

text: and

text: quan-

text: tum

text: gravity.

text: In

text: this

text: regard,

text: entanglement

text: entropy

text: is

text: a

text: crucial

text: quantity

text: which

text: provides

text: a

text: measure

text: for

text: quantum

text: correlations

text: in

text: a

text: bipartite

text: quantum

text: system.

text: Such

text: a

text: system

text: is

text: described

text: as

text: the

text: union

text: of

text: a

text: subsystem

text: A

text: that

text: is

text: being

text: studied

text: and

text: which

text: is

text: correlated

text: with

text: the

text: rest

text: of

text: the

text: system

text: de-

text: H

text: scribed

text: by

text: its

text: complement

text: A

text: For

text: this

text: system

text: the

text: full

text: Hilbert

text: space

text: may

text: be

text: expressed

text: as

text: c

text: H

text: H

text: H

text: a

text: tensor

text: product

text: of

text: the

text: Hilbert

text: spaces

text: of

text: A

text: and

text: A

text: as

text: At

text: zero

text: temperature,

text: c

text: HA

text: Ac

text: the

text: whole

text: system

text: can

text: be

text: represented

text: by

text: a

text: pure

text: state

text: ψ

text: with

text: the

text: density

text: matrix

text: given

text: by

text: ρ

text: ψ

text: ψ

text: However,

text: if

text: the

text: subsystems

text: A

text: and

text: A

text: are

text: entangled

text: then

text: the

text: full

text: state

text: ψ

text: c

text: cannot

text: be

text: expressed

text: as

text: a

text: tensor

text: product

text: of

text: any

text: two

text: states

text: of

text: the

text: subsystems

text: i.e

text: ψ

text: φ

text: A

text: λ

text: where,

text: φ

text: and

text: λ

text: are

text: states

text: in

text: the

text: bases

text: of

text: A

text: and

text: A

text: respectively.

text: This

text: naturally

text: Ac

text: A

text: Ac

text: c

text: also

text: implies

text: that

text: the

text: density

text: matrix

text: (ρ)

text: of

text: the

text: full

text: system

text: cannot

text: be

text: expressed

text: as

text: the

text: tensor

text: product

text: of

text: ρ

text: and

text: ρ

text: i.e.

text: ρ

text: ρ

text: ρ

text: The

text: reduced

text: density

text: matrix

text: (ρ

text: for

text: the

text: subsystem

text: A

text: Ac

text: A

text: Ac

text: A

text: A

text: is

text: obtained

text: via

text: tracing

text: ρ

text: over

text: the

text: degrees

text: of

text: freedom

text: of

text: A

text: giving

text: ρ

text: T

text: r

text: ρ

text: and

text: the

text: en-

text: c

text: A

text: Ac

text: tanglement

text: entropy

text: of

text: the

text: subsystem

text: A

text: is

text: deﬁned

text: as

text: the

text: corresponding

text: Von

text: Neumann

text: entropy,

text: S

text: T

text: r

text: (ρ

text: log

text: ρ

text: Entanglement

text: entropy

text: is

text: know

text: to

text: have

text: several

text: important

text: properties

text: A

text: A

text: A

text: A

text: such

text: as

text: the

text: strong

text: subadditivity

text: condition

text: for

text: two

text: diﬀerent

text: regions

text: A

text: and

text: A

text: of

text: a

text: quantum

text: system

text: (S

text: S

text: S

text: S

text: To

text: obtain

text: the

text: entanglement

text: entropy

text: for

text: dimensional

text: conformal

text: ﬁeld

text: theories

text: (CF

text: T

text: a

text: method

text: was

text: developed

text: by

text: Calabrese

text: and

text: Cardy

text: using

text: what

text: is

text: known

text: as

text: the

text: replica

text: trick.

text: They

text: showed

text: that

text: the

text: entanglement

text: entropy

text: for

text: a

text: CF

text: T

text: exhibits

text: an

text: universal

text: logarithmic

text: behavior

text: The

text: forementioned

text: authors

text: further

text: developed

text: techniques

text: to

text: study

text: quantum

text: entanglement

text: of

text: CF

text: T

text: in

text: various

text: static

text: as

text: well

text: as

text: dynamic

text: scenarios

text: However,

text: very

text: little

text: is

text: known

text: about

text: CFTs

text: in

text: higher

text: dimensions,

text: where,

text: only

text: the

text: case

text: of

text: quasi

text: free

text: fermions

text: and

text: bosons

text: has

text: been

text: studied

text: rigorously

text: in

text: Further

text: insight

text: into

text: the

text: entanglement

text: entropy

text: for

text: quantum

text: ﬁeld

text: theories

text: was

text: facilitated

text: through

text: the

text: AdS/CF

text: T

text: correspondence

text: which

text: relates

text: (d)-dimensional

text: strongly

text: coupled

text: boundary

text: quantum

text: ﬁeld

text: theory

text: at

text: its

text: conformal

text: ﬁxed

text: point

text: in

text: a

text: large

text: N

text: limit,

text: to

text: a

text: theory

text: of

text: weakly

text: coupled

text: gravity

text: in

text: (d

text: bulk

text: space-time

text: Using

text: this

text: correspondence,

text: Ryu

text: and

text: Takyanagi

text: proposed

text: a

text: conjecture

text: to

text: holographically

text: evaluate

text: the

text: entanglement

text: entropy

text: of

text: a

text: (d)-dimensional

text: quantum

text: ﬁeld

text: theory

text: which

text: was

text: later

text: generalized

text: for

text: time

text: dependent

text: backgrounds

text: by

text: Hubeny-Rangamani-Takayanagi

text: (HRT)

text: in

text: This

text: holographic

text: prescription

text: for

text: obtaining

text: the

text: entanglement

text: entropy

text: S

text: for

text: a

text: region

text: A

text: (enclosed

text: by

text: the

text: boundary

text: ∂A)

text: in

text: A

text: the

text: (d)-dimensional

text: boundary

text: ﬁeld

text: theory

text: involves

text: the

text: computation

text: of

text: the

text: area

text: of

text: the

text: extremal

text: surface

text: (denoted

text: by

text: γ

text: extending

text: from

text: the

text: boundary

text: ∂A

text: of

text: the

text: region

text: A

text: into

text: the

text: (d

text: A

text: dimensional

text: bulk

text: such

text: that

text: S

text: is

text: given

text: by

text: A

text: Area(γ

text: A

text: S

text: A

text: N

text: where

text: G

text: is

text: the

text: gravitational

text: constant

text: of

text: the

text: bulk

text: Besides

text: this,

text: they

text: correctly

text: N

text: reproduced

text: the

text: logarithmic

text: behavior

text: of

text: the

text: entanglement

text: entropy

text: for

text: a

text: boundary

text: CF

text: T

text: in

text: AdS

text: /CF

text: T

text: and

text: also

text: conﬁrmed

text: the

text: validity

text: of

text: the

text: strong

text: subadditivity

text: inequalities

text: in

text: holographic

text: CFTs

text: One

text: of

text: the

text: several

text: applications

text: of

text: the

text: conjecture

text: involves

text: the

text: study

text: of

text: the

text: entanglement

text: entropy

text: for

text: a

text: boundary

text: CF

text: T

text: at

text: ﬁnite

text: temperature

text: which

text: is

text: dual

text: to

text: a

text: AdS-Schwarzschild

text: black

text: hole

text: in

text: the

text: bulk

text: The

text: results

text: of

text: which

text: were

text: subsequently

text: used

text: to

text: study

text: the

text: low

text: and

text: the

text: high

text: temperature

text: behavior

text: of

text: the

text: entanglement

text: entropy.

text: This

text: in

text: turn

text: led

text: to

text: the

text: proposal

text: of

text: an

text: analogous

text: “First

text: law

text: of

text: thermodynamics”

text: like

text: relation

text: for

text: the

text: entanglement

text: entropy

text: by

text: the

text: authors

text: in

text: Another

text: class

text: of

text: boundary

text: ﬁeld

text: theories

text: that

text: have

text: been

text: studied

text: using

text: the

text: Ryu

text: and

text: Takyanagi

text: ##LTLine##


text: (cid:90) − A L√r (cid:20) (cid:90)1 √ u 1 (cid:21) h (cid:88) = − du (110) 1 δ 1 u4 0 ∞ = L√rh (cid:20) 1 1 + √Γ(cid:21)(k + 12 1 δ rc u πΓ(k + 1) 4k rb k=1 L√r r log[4] h b = log[ ] + . δ rc 4 ∼ Just like large charge case of extremal black ho(cid:20)le, the(cid:21)quantity log[rrcb ] 0. This is because both r and r are both large. Therefore we ignore it. b c A L√r log[4] finite = h . (111) 1 δ 4 Substituting the ﬁnite parts of the terms corresponding to n = 0 and n = 1 as given by (109) and eq.(111) in eq.(106) we obtain the ﬁnite part of the area to be (cid:90) (cid:88) ∞ − Afinite = Afini(cid:20)te + Afinite + 2√Lrc √Γ(n + (cid:88)12) 1 √un−2 (rh )n (cid:21) (112) 0 √1 δ πΓ(n + 1) 0 1 u4 rc n=2 ∞ − 2√Lr − πΓ(3) r log[4] 1 Γ(n + 1) Γ(n 1) r = c 4 + h ( ) + 2 4 ( h )n . (113) δ Γ(14) 2rc 4 4 n=2 Γ(n + 1) Γ(n+4 1) rc Using Γ(n + 1(cid:20)) = nΓ(n), we can write the above equation as follows (cid:21) (cid:88) √ ∞ A 2√Lr − πΓ(3) r log[4] 1 2 Γ(n + 1) Γ(n+3) r finite = c 4 + h ( ) + (1 + − ) 2 4 ( h )n . (114) δ Γ(14) 2rc 4 4 n=2 n 1 Γ(n + 1) Γ(n+4 5) rc We can now use eq.(70) for the ﬁrst term of the series in the above equation and simplify to get (cid:20) √ √ √ A 2√Lr − πΓ(3) r log[4] lr δ − πΓ(3) −(cid:21)1 finite = c 4 + h(cid:88)( ) + c 4 (115) δ Γ(14) 2rc∞ 4 2 4Γ(54) 4 1 1 Γ(n + 1) Γ(n+3) r + − 2 4 ( h )n . (116) 2 n 1 Γ(n + 1) Γ(n+5) r n=2 4 c ∼ → The series in the above equation goes as xn for large n and hence converges as r r . n2 c h Therefore the leading term can be found just by putting r = r . c h A L√r finite = Llr2 + h C, (117) h δ (cid:88) where, √ − ∞ −4 πΓ(3) log[4] 2 1 Γ(n + 1) Γ(n+3) C = 4 + + − ) 2 4 . (118) Γ(1) 4 n 1 Γ(n + 1) Γ(n+5) 4 n=2 4 23 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: equation

text: A

text: L√r

text: u

text: h

text: du

text: δ

text: L√rh

text: δ

text: rc

text: u

text: πΓ(k

text: rb

text: L√r

text: r

text: h

text: b

text: log[

text: δ

text: rc

text: Just

text: like

text: large

text: charge

text: case

text: of

text: extremal

text: black

text: log[rrcb

text: This

text: is

text: because

text: both

text: r

text: and

text: r

text: are

text: both

text: large.

text: Therefore

text: we

text: ignore

text: it.

text: b

text: c

text: A

text: L√r

text: finite

text: h

text: δ

text: Substituting

text: the

text: ﬁnite

text: parts

text: of

text: the

text: terms

text: corresponding

text: to

text: n

text: and

text: n

text: as

text: given

text: by

text: and

text: in

text: we

text: obtain

text: the

text: ﬁnite

text: part

text: of

text: the

text: area

text: to

text: be

text: Afinite

text: Afinite

text: √Γ(n

text: (rh

text: )n

text: √1

text: δ

text: πΓ(n

text: rc

text: r

text: Γ(n

text: Γ(n

text: r

text: c

text: h

text: h

text: )n

text: δ

text: Γ(n

text: rc

text: Using

text: Γ(n

text: nΓ(n),

text: we

text: can

text: write

text: the

text: above

text: equation

text: as

text: follows

text: A

text: r

text: Γ(n

text: r

text: finite

text: c

text: h

text: h

text: )n

text: δ

text: n

text: Γ(n

text: rc

text: We

text: can

text: now

text: use

text: for

text: the

text: ﬁrst

text: term

text: of

text: the

text: series

text: in

text: the

text: above

text: equation

text: and

text: simplify

text: to

text: get

text: A

text: r

text: lr

text: δ

text: finite

text: c

text: c

text: δ

text: Γ(n

text: r

text: h

text: )n

text: n

text: Γ(n

text: r

text: c

text: The

text: series

text: in

text: the

text: above

text: equation

text: goes

text: as

text: xn

text: for

text: large

text: n

text: and

text: hence

text: converges

text: as

text: r

text: r

text: c

text: h

text: Therefore

text: the

text: leading

text: term

text: can

text: be

text: found

text: just

text: by

text: putting

text: r

text: r

text: c

text: h

text: A

text: L√r

text: finite

text: h

text: C,

text: h

text: δ

text: where,

text: −4

text: Γ(n

text: C

text: n

text: Γ(n

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text: If we substitute for r from eq.(26)in eq.(28) and keep terms up to O(r3l3) then we get the c(cid:20) h (cid:21) following approximated form of the ﬁnite part of extremal area − − A L πΓ(3)Γ( 1) l3r 3Γ(1)2(8Γ(3) + Γ( 1)) finite = 4 4 + h 4 4 4 + O[(r l)4 . (29) l Γ(1)2 32Γ(3)3 h 4 4 Since the renormalized entanglement entropy (Sfinite) is related to the ﬁnite part of the A extremal area as A finite Sfinite = . (30) A (3+1) 4G N The explicit expression for the entanglement entropy of subsystem A in a boundary theory (cid:20) (cid:21) dual to the extremal black hole in the small charge regime may be written down as follows 1 L − 4πΓ(3)2 l3r 3Γ(1)2 Sfinite = 4 + h 4 . + O[(r l)4 (31) A 4G3+1 l Γ(1)2 8Γ(3)2 h N 4 4 For extremal black holes we also have, r3 = Mext which follows from eq.(14) and eq.(19). h 4 Thus replacing r3 by Mext (Mass of the extremal black hole) in eq.(31) we obtain following form h of renormalized entanglement entropy ≈ 1 Γ(1)2 Sfinite SAdS + kMextLl2, k = 4 , (32) A A 4G3+1 32Γ(3)2 N 4 where, SAdS is the entanglement entropy of the subsystem (A) when the bulk theory is pure A AdS [37]. We see that when the charge is small, the leading contribution to the entanglement entropy of the subsystem (A) in the boundary ﬁeld theory dual to the extremal black hole arises from the pure AdS spacetime. We will see in later subsections how this sub-leading correction term in the above equation becomes important in deﬁning the ﬁrst law like relation. 4.2 Non-extremal charged black hole (low temperature) We now consider the subsystem A of the boundary ﬁeld theory dual to a non-extremal charged black hole in AdS with a small charge and at a low temperature. It is observed from 4 the extremality bound in eq.(18) that when the temperature and the charge both are small, ∼ the horizon radius is small. This implies that Q/r2 1 and r is small such that, r << r is (cid:18)h h (cid:19) h c satisﬁed. For a non-extremal black hole the form of the lapse function f(u) may be given as − r − Q2 r − r f(u) = 1 ( h )3u3 ( h )3u3 ( h )4u4 . (33) r r4 r r c h c c We deﬁne a new parameter α = Q2 to substitute for black hole charge Q in eq.(33) and then r4 − h Taylor expand the quantity f(u) 1/2 around rh = 0 while keeping the non vanishing terms up rc to O[(rh )3u3] as rc − ≈ 1 + α r f(u) 12 1 + ( h )3u3. (34) 2 r c Using the approximated form of the lapse function given by eq.(34) in the integral (16) for subsystem length l we obtain 9 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: If

text: we

text: substitute

text: for

text: r

text: from

text: and

text: keep

text: terms

text: up

text: to

text: then

text: we

text: get

text: the

text: h

text: following

text: approximated

text: form

text: of

text: the

text: ﬁnite

text: part

text: of

text: extremal

text: area

text: A

text: L

text: Γ(

text: finite

text: h

text: O[(r

text: l

text: h

text: Since

text: the

text: renormalized

text: entanglement

text: entropy

text: (Sfinite)

text: is

text: related

text: to

text: the

text: ﬁnite

text: part

text: of

text: the

text: A

text: extremal

text: area

text: as

text: A

text: finite

text: Sfinite

text: A

text: N

text: The

text: explicit

text: expression

text: for

text: the

text: entanglement

text: entropy

text: of

text: subsystem

text: A

text: in

text: a

text: boundary

text: theory

text: dual

text: to

text: the

text: extremal

text: black

text: hole

text: in

text: the

text: small

text: charge

text: regime

text: may

text: be

text: written

text: down

text: as

text: follows

text: L

text: Sfinite

text: h

text: O[(r

text: A

text: l

text: h

text: N

text: For

text: extremal

text: black

text: holes

text: we

text: also

text: have,

text: Mext

text: which

text: follows

text: from

text: and

text: h

text: Thus

text: replacing

text: by

text: Mext

text: (Mass

text: of

text: the

text: extremal

text: black

text: hole)

text: in

text: we

text: obtain

text: following

text: form

text: h

text: of

text: renormalized

text: entanglement

text: entropy

text: Sfinite

text: SAdS

text: k

text: A

text: A

text: N

text: where,

text: SAdS

text: is

text: the

text: entanglement

text: entropy

text: of

text: the

text: subsystem

text: (A)

text: when

text: the

text: bulk

text: theory

text: is

text: pure

text: A

text: AdS

text: We

text: see

text: that

text: when

text: the

text: charge

text: is

text: small,

text: the

text: leading

text: contribution

text: to

text: the

text: entanglement

text: entropy

text: of

text: the

text: subsystem

text: (A)

text: in

text: the

text: boundary

text: ﬁeld

text: theory

text: dual

text: to

text: the

text: extremal

text: black

text: hole

text: arises

text: from

text: the

text: pure

text: AdS

text: spacetime.

text: We

text: will

text: see

text: in

text: later

text: subsections

text: how

text: this

text: sub-leading

text: correction

text: term

text: in

text: the

text: above

text: equation

text: becomes

text: important

text: in

text: deﬁning

text: the

text: ﬁrst

text: law

text: like

text: relation.

text: Non-extremal

text: charged

text: black

text: hole

text: (low

text: temperature)

text: We

text: now

text: consider

text: the

text: subsystem

text: A

text: of

text: the

text: boundary

text: ﬁeld

text: theory

text: dual

text: to

text: a

text: non-extremal

text: charged

text: black

text: hole

text: in

text: AdS

text: with

text: a

text: small

text: charge

text: and

text: at

text: a

text: low

text: temperature.

text: It

text: is

text: observed

text: from

text: the

text: extremality

text: bound

text: in

text: that

text: when

text: the

text: temperature

text: and

text: the

text: charge

text: both

text: are

text: small,

text: the

text: horizon

text: radius

text: is

text: small.

text: This

text: implies

text: that

text: and

text: r

text: is

text: small

text: such

text: that,

text: r

text: r

text: is

text: h

text: h

text: c

text: satisﬁed.

text: For

text: a

text: non-extremal

text: black

text: hole

text: the

text: form

text: of

text: the

text: lapse

text: function

text: f(u)

text: may

text: be

text: given

text: as

text: r

text: r

text: r

text: f(u)

text: h

text: h

text: h

text: r

text: r

text: r

text: c

text: h

text: c

text: c

text: We

text: deﬁne

text: a

text: new

text: parameter

text: α

text: to

text: substitute

text: for

text: black

text: hole

text: charge

text: Q

text: in

text: and

text: then

text: h

text: Taylor

text: expand

text: the

text: quantity

text: f(u)

text: around

text: rh

text: while

text: keeping

text: the

text: non

text: vanishing

text: terms

text: up

text: rc

text: to

text: O[(rh

text: as

text: rc

text: α

text: r

text: f(u)

text: h

text: r

text: c

text: Using

text: the

text: approximated

text: form

text: of

text: the

text: lapse

text: function

text: given

text: by

text: in

text: the

text: integral

text: for

text: subsystem

text: length

text: l

text: we

text: obtain

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text: 10 D.PANYUSHEV AND O.YAKIMOVA is b = 1. Then one can safely remove two central vertices from the graph and conclude t (cid:62) that F − = F − − − as long as n 2m + 2. (The last step is opposite to one that is n,n m n 1,n 1 m used in the proof of Lemma 4.4.) | ∈ (ii) Again, for q = qC (a b) F , we consider b , the last coordinate of b. If 2m+1 2m+1,m+1 t b = 1, then the central pair of vertices in ΓC(q) can be removed, which yields a seaweed in t ∈ { } F . Next, it is easily seen that if b 2, 3, . . . , 2m , then ΓC(q) contains a cycle. Hence 2m,m t this i(cid:124)s im(cid:123)p(cid:122)oss(cid:125)ible. While for b = 2m + 1, one obtains a unique admissible possibility t (cid:3) a = (1, 1, . . . , 1). m (cid:62) Remark. Using a similar analysis, one can show that F = F − − + 3, if m 3. 2m,m 2m 1,m 1 Remark 4.6. Our stabilisation result for F − can be compared with [3], where Duﬂo and n,n m Yu consider a partition of the set of standard Frobenius seaweeds in sl into classes and n study the asymptotic behaviour of the cardinality of these classes as n tends to inﬁnity. Let p(a) be the number of nonzero parts of the composition a and let F˜ be the number of the | ∩ n,p standard Frobenius seaweeds qA(a b) sl such that p(a)+p(b) = p. By [3, Theorem 1.1(b)], n if n is sufﬁciently large, then F˜ − is a polynomial in n of degree [t/2], with positive n,n+1 t rational coefﬁcients. It seems that F is the most interesting part of the symplectic Frobenius seaweeds. Re- n,1 ⊂ call from Section that to any standard seaweed q sp one can associate a “symmetric” ⊂ ∩ 2n ∩ seaweed ˜q gl such that q = ˜q sp . In this context, we also set ˜q = ˜q sl . 2n 2n 0 2n ∈ Proposition 4.7. (i) If q F , then ind ˜q = 1, hence ˜q is a Frobenius seaweed in sl . n,1 → 0 (cid:62) 2n (ii) There is an injective map F F , which is not onto if n 2. n,1 n+1,1 ∈ Proof. (i) If q F , then ΓC(q) and(cid:80)thereby ΓA(˜q) consist(cid:80)s of a sole segment (Lemma 4.3). · n,1 − By Eq. (2 1), we have ind ˜q = 1 and therefore ind ˜q = ind ˜q 1 = 0. | ∈ −0 (ii) If q = qC(a b) F , then s a = n 1 and t b = n. We associate to it ∈n n,1 i=1 i | j=1 j a seaweed ˆq F as follows. Set ˆq = qC (aˆ b), where aˆ = (a , . . . , a , 2). Note that | n+1,1 n+1 1| s ΓC(a b) has one central arc above the horizontal line, while ΓC (aˆ b) has one central arc n (cid:7) (cid:4) (cid:7) (cid:4) (cid:7) n+(cid:4)1 (cid:55)→ below. The following is a graphica(cid:115)l ill(cid:115)ustration of the(cid:115)tra(cid:115)nsfo(cid:115) rm(cid:115) q ˆq: (cid:5) (cid:6) (cid:5) (cid:6) (cid:5) (cid:6) (cid:55)→ . . . . . . . . . . . . This provides a bijection between F and the seaweeds in F whose last part of the n,1 n+1,1 (cid:62) composition that sums up to n + 1 equals 2. If n + 1 3, then there are seaweeds in F n+1,1 (cid:3) such that the above-mentioned last part is bigger than 2. Hence F < F . n,1 n+1,1 Remark 4.8. Another curious observation is that F and F are related to certain Frobe- n,1 n,2 nius seaweeds in sl : n ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: D.PANYUSHEV

text: AND

text: O.YAKIMOVA

text: is

text: b

text: Then

text: one

text: can

text: safely

text: remove

text: two

text: central

text: vertices

text: from

text: the

text: graph

text: and

text: conclude

text: t

text: that

text: F

text: F

text: as

text: long

text: as

text: n

text: (The

text: last

text: step

text: is

text: opposite

text: to

text: one

text: that

text: is

text: n,n

text: m

text: n

text: m

text: used

text: in

text: the

text: proof

text: of

text: Lemma

text: (ii)

text: Again,

text: for

text: q

text: qC

text: (a

text: b)

text: F

text: we

text: consider

text: b

text: the

text: last

text: coordinate

text: of

text: b.

text: If

text: t

text: b

text: then

text: the

text: central

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text: 16 ⊗ φT M C˜ Ah˜ = φT (z c ) i i i i i = cT Xz . (36) i i Using the deﬁnitions of A and C˜ , given in (22) and (26), we write i M − M M C˜ AC˜T = i C˜ C˜T i (C˜ 1 − )(C˜ 1 − )T i i i 2 i i 2D i D 1 i D 1 ⊗ M − ⊗ M = I − i c cT 1 − 1T − i c cT . D 1 2 i i D 1 D 1 2D i i (cid:16)As a result, fr(cid:17)om (30(cid:18)), (cid:19) (cid:18) ⊗ M − − ⊗ M − (cid:19) Tr M C˜ AC˜T Φ = Tr I − i c cT F 1 1 − 1T − i c cT F 1 i i i D 1 2 i i D 1 D 1 2D i i − − (cid:20) ⊗ M − (cid:21) ⊗ M (D 1) Tr 1 − 1T − i c cT ∆ 1 − 1T − i c cT ∆ D 1 D 1 2 i i D 1 D 1 2D i i − − M (D 1)2 − − M (D 1) = cT i F 1 i ∆ c . (37) i 2D 2D i Similarly, using (31) we write (cid:16) (cid:17) (cid:88) (cid:18) (cid:19) (cid:88) (cid:88) (cid:18) (cid:19) − − − D 1 D 1 D 1 Tr M C˜ AC˜T φφT =  Tr Mi c cT x xT − Tr Mi c cT x xT i i i (cid:88)2 i i d d (cid:88) (cid:88) 2D i i d j d=1 d=1 j=1  − − − D 1 D 1 D 1 M − M = cT i x xT i x xT c . (38) i 2 d d 2D d j i d=1 d=1 j=1 Substituting (36), (37) and (38) in (35) we have the following quadratic program − 1 cˆ = arg max cT Γ c + cT γ i ci 2 i i i i i ≥ (cid:88) (cid:88) (cid:88) subject to c 0, k = 1, . . . , K, (39) ik M (D − 1)2 − − M (D − 1) M D−1 − M D−1 D−1 Γ = i F 1 i ∆ + i x xT i x xT i 2D(cid:88) 2D 2 d d 2D d j d=1 d=1 j=1 Ni γ = BT κ w + Xz , i i in i n=1 which can be eﬃciently solved using the interior point method. The resulting algorithm is summarized as Algorithm 1. ##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: The

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text: 23 3.5 Proposed algorithm 3 Separate modeling 2.5 2 1.5 1 0.5 0 R o b i n W illi a mH is n d u L Fe ea rd g e ur s o n m Uo nn rwe es ta lt h GG la omb ea ls I c e B Lu oc ck aelt I c e B u c k ePta u l G e o r g e m C o Fig. 10 Goodness of ﬁt plot for the proposed event detection algorithm and the individual vMF models separately ﬁtted to the geolocation data for each hashtag. Note that the separate modeling does not perform event detection; it is the baseline technique for data ﬁtting. It is seen that the proposed algorithm not only detects events (see the previous ﬁgures), but also ﬁts the geolocation data as well as the baseline technique. For each event, the mean likelihood values are shown with markers (circle and cross), and the 95% conﬁdence intervals are shown with bars. 22. Bramon, R., Boada, I., Bardera, A., Rodriguez, J., Feixas, M., Puig, J., & Sbert, M. (2012). Multimodal Data Fusion Based on Mutual Information. IEEE Transactions on Visualization and Computer Graphics, 18(9), 1574–1587. 23. Wu, Y., Chang, K.C.-C., Chang, E.Y., & Smith, J.R. (2004). Optimal Multimodal Fusion for Multimedia Data Analysis. ACM International Conference on Multimedia. 24. Sui, J., Adali, T., Yu, Q., Chen, J., & Calhoun, V.D. (2012). A review of Multivariate Methods for Multimodal Fusion of Brain Imaging Data. Journal of Neuroscience Methods, 204(1), 68–81. 25. Ngiam, J., Khosla, A., Kim, M., Nam, J., Lee, H., & Ng, A.Y. (2011). Multimodal Deep Learning. International Conference on Machine Learning. 26. Christoudias, C.M., Urtasun, R., & Darrell, T. (2008). Multi-View Learning in the Presence of View Disagreement. Conference on Uncertainty in Artiﬁcial Intelligence. 27. He, J., & Lawrence, R. (2011). A Graph-Based Framework for Multi-Task Multi-View Learning. International Conference on Machine Learning. 28. Sun, S. (2013). A Survey of Multi-View Machine Learning. Neural Computing and Appli- cations, 23(7), 2031–2038. 29. Mardia, K.V., & Jupp, P.E. (2000). Directional Statistics. Chichester: Wiley. 30. Mardia, K.V., & El-Atoum, S.A.M. (1976). Bayesian Inference for the Von Mises-Fisher Distribution. Biometrika, 63(1), 203–206. 31. Harman, H.H., (1976). Modern Factor Analysis. University of Chicago Press. 32. Abramowitz, M., & Stegun, I.A. (1972). Handbook of Mathematical Functions. National Bureau of Standards Applied Mathematics Series, 55. doohilekiL ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

font: default

text: Proposed

text: algorithm

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text: modeling

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text: mH

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text: c

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text: B

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text: G

text: e

text: o

text: r

text: g

text: e

text: m

text: C

text: o

text: Fig.

text: Goodness

text: of

text: ﬁt

text: plot

text: for

text: the

text: proposed

text: event

text: detection

text: algorithm

text: and

text: the

text: individual

text: vMF

text: models

text: separately

text: ﬁtted

text: to

text: the

text: geolocation

text: data

text: for

text: each

text: hashtag.

text: Note

text: that

text: the

text: separate

text: modeling

text: does

text: not

text: perform

text: event

text: detection;

text: it

text: is

text: the

text: baseline

text: technique

text: for

text: data

text: ﬁtting.

text: It

text: is

text: seen

text: that

text: the

text: proposed

text: algorithm

text: not

text: only

text: detects

text: events

text: (see

text: the

text: previous

text: ﬁgures),

text: but

text: also

text: ﬁts

text: the

text: geolocation

text: data

text: as

text: well

text: as

text: the

text: baseline

text: technique.

text: For

text: each

text: event,

text: the

text: mean

text: likelihood

text: values

text: are

text: shown

text: with

text: markers

text: (circle

text: and

text: cross),

text: and

text: the

text: conﬁdence

text: intervals

text: are

text: shown

text: with

text: bars.

text: Bramon,

text: R.,

text: Boada,

text: I.,

text: Bardera,

text: A.,

text: Rodriguez,

text: J.,

text: Feixas,

text: M.,

text: Puig,

text: J.,

text: Sbert,

text: M.

text: Multimodal

text: Data

text: Fusion

text: Based

text: on

text: Mutual

text: Information.

text: IEEE

text: Transactions

text: on

text: Visualization

text: and

text: Computer

text: Graphics,

text: Wu,

text: Y.,

text: Chang,

text: K.C.-C.,

text: Chang,

text: E.Y.,

text: Smith,

text: J.R.

text: Optimal

text: Multimodal

text: Fusion

text: for

text: Multimedia

text: Data

text: Analysis.

text: ACM

text: International

text: Conference

text: on

text: Multimedia.

text: Sui,

text: J.,

text: Adali,

text: T.,

text: Yu,

text: Q.,

text: Chen,

text: J.,

text: Calhoun,

text: V.D.

text: A

text: review

text: of

text: Multivariate

text: Methods

text: for

text: Multimodal

text: Fusion

text: of

text: Brain

text: Imaging

text: Data.

text: Journal

text: of

text: Neuroscience

text: Methods,

text: Ngiam,

text: J.,

text: Khosla,

text: A.,

text: Kim,

text: M.,

text: Nam,

text: J.,

text: Lee,

text: H.,

text: Ng,

text: A.Y.

text: Multimodal

text: Deep

text: Learning.

text: International

text: Conference

text: on

text: Machine

text: Learning.

text: Christoudias,

text: C.M.,

text: Urtasun,

text: R.,

text: Darrell,

text: T.

text: Multi-View

text: Learning

text: in

text: the

text: Presence

text: of

text: View

text: Disagreement.

text: Conference

text: on

text: Uncertainty

text: in

text: Artiﬁcial

text: Intelligence.

text: He,

text: J.,

text: Lawrence,

text: R.

text: A

text: Graph-Based

text: Framework

text: for

text: Multi-Task

text: Multi-View

text: Learning.

text: International

text: Conference

text: on

text: Machine

text: Learning.

text: Sun,

text: S.

text: A

text: Survey

text: of

text: Multi-View

text: Machine

text: Learning.

text: Neural

text: Computing

text: and

text: Appli-

text: cations,

text: Mardia,

text: K.V.,

text: Jupp,

text: P.E.

text: Directional

text: Statistics.

text: Chichester:

text: Wiley.

text: Mardia,

text: K.V.,

text: El-Atoum,

text: S.A.M.

text: Bayesian

text: Inference

text: for

text: the

text: Von

text: Mises-Fisher

text: Distribution.

text: Biometrika,

text: Harman,

text: H.H.,

text: Modern

text: Factor

text: Analysis.

text: University

text: of

text: Chicago

text: Press.

text: Abramowitz,

text: M.,

text: Stegun,

text: I.A.

text: Handbook

text: of

text: Mathematical

text: Functions.

text: National

text: Bureau

text: of

text: Standards

text: Applied

text: Mathematics

text: Series,

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text: The beneﬁt of Deﬁnition 2.1 is the following Lemma. Lemma 2.2. Assume an approximate zero x¯ of f deﬁned in (41) to be given. If β(r) < 0 with β deﬁned in (15) for a positive radius r , then T given by (44) is a contraction on B x¯∈ B (x¯). Hence there is a unique zero x˜ with x˜ (x¯). rx¯ rx¯ A proof of this lemma has appeared in many places. See for example [28]. The decisive feature of the condition β(r) < 0 in our context is that after deriving explicit expressions for the bounds in (13) and (14) are it can be checked rigorously by a computer using interval arithmetic. 2.2 Norms and spaces We are interest(cid:40)ed in solutions of PDEs whose spectral representation h(cid:41)ave coeﬃcients with very rapid decay. To be more precise let us consider (cid:88) ∈ R | | | | | | ∞ (cid:96)1 = a = (a ) ≥ , a : a d=ef a + 2 a νk < . (16) ν k k 0 k ν 0 k k=1 | · | ∈ Denote the induced operator norm by (cid:88). Note that if ν > 1 and a (cid:96)1 the function (cid:96)1ν ν ∞ u(x) = a + 2 a cos(kx), 0 k k=1 is real analytic and extends to a periodic and analytic function on the complex strip of width log(ν). Moreover we have that (cid:107) (cid:107) | | ≤ | | u := sup u(x) a , C0([0,2π]) ∈ ν x [0,2π] i.e. the (cid:96)1 norm provides bounds on the supremum norm of the corresponding analytic ν function. (cid:88) ∗ (cid:96)1 is a Banach algebra under the discrete convolution operation a b given by ν ∗ (a b) = a| |b| |. (17) k k1 k2 k1k+ik∈2Z=k (cid:124) (cid:123)(cid:122) (cid:125) ∈ ∗ ∗ · · · ∗ with a, b (cid:96)1. For later use we deﬁne the notation a n for a a. ν n times When we consider unstable manifolds for PDEs we are interested in analytic functions taking their values in (cid:96)1ν as just deﬁned(cid:32). Such functions have co(cid:33)nvergent power series repre- sentations of the form (cid:88) (cid:88) ∞ ∞ P(θ, x) = p + 2 p cos(kx) θm, (18) m0 mk | | m =0 n=1 ∈ N ∈ C where m = (m , . . . , m ) d is a d-dimensional multi-index, θ = (θ , . . . , θ ) d and { } 1 d 1 d p ∈N ∈N is a sequence of Fourier-Taylor coeﬃcients (more speciﬁcally cosine-Taylor mk m dk coeﬃcients in this case). { } ∈ N ∈ We shorten this notation and write p ∈N , where for each m d we have p (cid:96)1. m m d m ν Then we are led to consider the multi-sequence space 12

text: The

text: beneﬁt

text: of

text: Deﬁnition

text: is

text: the

text: following

text: Lemma.

text: Lemma

text: Assume

text: an

text: approximate

text: zero

text: x¯

text: of

text: f

text: deﬁned

text: in

text: to

text: be

text: given.

text: If

text: β(r)

text: with

text: β

text: deﬁned

text: in

text: for

text: a

text: positive

text: radius

text: r

text: then

text: T

text: given

text: by

text: is

text: a

text: contraction

text: on

text: B

text: x¯∈

text: B

text: (x¯).

text: Hence

text: there

text: is

text: a

text: unique

text: zero

text: x˜

text: with

text: x˜

text: (x¯).

text: rx¯

text: rx¯

text: A

text: proof

text: of

text: this

text: lemma

text: has

text: appeared

text: in

text: many

text: places.

text: See

text: for

text: example

text: The

text: decisive

text: feature

text: of

text: the

text: condition

text: β(r)

text: in

text: our

text: context

text: is

text: that

text: after

text: deriving

text: explicit

text: expressions

text: for

text: the

text: bounds

text: in

text: and

text: are

text: it

text: can

text: be

text: checked

text: rigorously

text: by

text: a

text: computer

text: using

text: interval

text: arithmetic.

text: Norms

text: and

text: spaces

text: We

text: are

text: in

text: solutions

text: of

text: PDEs

text: whose

text: spectral

text: representation

text: coeﬃcients

text: with

text: very

text: rapid

text: decay.

text: To

text: be

text: more

text: precise

text: let

text: us

text: consider

text: R

text: a

text: (a

text: a

text: a

text: d=ef

text: a

text: a

text: νk

text: ν

text: k

text: k

text: k

text: ν

text: k

text: Denote

text: the

text: induced

text: operator

text: norm

text: by

text: Note

text: that

text: if

text: ν

text: and

text: a

text: the

text: function

text: ν

text: u(x)

text: a

text: a

text: cos(kx),

text: k

text: is

text: real

text: analytic

text: and

text: extends

text: to

text: a

text: periodic

text: and

text: analytic

text: function

text: on

text: the

text: complex

text: strip

text: of

text: width

text: log(ν).

text: Moreover

text: we

text: have

text: that

text: u

text: sup

text: u(x)

text: a

text: ν

text: x

text: i.e.

text: the

text: norm

text: provides

text: bounds

text: on

text: the

text: supremum

text: norm

text: of

text: the

text: corresponding

text: analytic

text: ν

text: function.

text: is

text: a

text: Banach

text: algebra

text: under

text: the

text: discrete

text: convolution

text: operation

text: a

text: b

text: given

text: by

text: ν

text: (a

text: b)

text: a|

text: |b|

text: k

text: with

text: a,

text: b

text: For

text: later

text: use

text: we

text: deﬁne

text: the

text: notation

text: a

text: n

text: for

text: a

text: a.

text: ν

text: n

text: times

text: When

text: we

text: consider

text: unstable

text: manifolds

text: for

text: PDEs

text: we

text: are

text: interested

text: in

text: analytic

text: functions

text: taking

text: their

text: values

text: in

text: as

text: just

text: Such

text: functions

text: have

text: power

text: series

text: repre-

text: sentations

text: of

text: the

text: form

text: P(θ,

text: x)

text: p

text: p

text: cos(kx)

text: θm,

text: mk

text: m

text: N

text: C

text: where

text: m

text: (m

text: m

text: d

text: is

text: a

text: d-dimensional

text: multi-index,

text: θ

text: (θ

text: θ

text: d

text: and

text: d

text: d

text: p

text: ∈N

text: ∈N

text: is

text: a

text: sequence

text: of

text: Fourier-Taylor

text: coeﬃcients

text: (more

text: speciﬁcally

text: cosine-Taylor

text: mk

text: m

text: dk

text: coeﬃcients

text: in

text: this

text: case).

text: N

text: We

text: shorten

text: this

text: notation

text: and

text: write

text: p

text: ∈N

text: where

text: for

text: each

text: m

text: d

text: we

text: have

text: p

text: m

text: m

text: d

text: m

text: ν

text: Then

text: we

text: are

text: led

text: to

text: consider

text: the

text: multi-sequence

text: space


text: ∈ H Assuming that H (so that H is zero to second order) we have that s − − →lim−∞ e Ast0x(t) = l→im∞ eAst0x( t0) t0 t0 − = l→im∞ eAst0Hs(e λt0θ) t0 = 0 as (cid:16) (cid:17) (cid:107) − (cid:107) ≤ (cid:107) (cid:107)(cid:107) − (cid:107) eAst0H (e λt0θ) eAst0 H (e λt0θ s s ≤ − − ∗ 2 (cid:107) (cid:107) Me µ∗t0 e µ t0 H s ≤ − ∗ (cid:107) (cid:107) Me (µ∗+2µ )t0 H s (cid:107) (cid:107) ∞ (cid:90) → −∞ (cid:90) and H < . Then taking t in Equation (31) and t = 0 gives s 0 1 ∞ 0 − − x(0) = e AstE (eλtθ) dt = eAstE (e λtθ) dt, −∞ s s 0 → − after switching the limits of integration (cid:90)and changing t t. (cid:2) (cid:3) Motivated by this discussion we deﬁne the linear solution operator ∞ − − L 1[E ](θ) := eAstπ E(e λtθ) dt. s s s 0 − Let H := L 1[E ], and note that s s s (cid:107) (cid:107) ≤ M (cid:107) (cid:107) Hs 2µ∗ + µ∗ Es , as the integrand satisﬁes (cid:107) − (cid:107) ≤ − − ∗ (cid:107) (cid:107) eAstE (e λtθ) Me µ∗te 2µ t E , s s i.e. the operator is well deﬁned and bounded. Moreover we see that H is analytic by s Morera’s Theorem. To see that H is zero to second order we diﬀerentiate under the integral s ≤ ≤ (cid:107) (cid:107) ∞ and note that E (0) = 0 and ∂E /∂θ (0) = 0 for 1 j d. To see that H < we s s j − s 1 expand E as a power series inside the formula for L 1[E ] and, after exchanging the sum s (cid:107) (cid:107) (cid:107) (cid:107) s s and the integral, bound H in terms of E . We then check by diﬀerentiating that H s 1 s 1 s so deﬁned solves the desired equation. Proof of Theorem 3.2. Let | | max s = s. ≤ ≤ j 1 j d ∈ H (cid:107) (cid:107) ≤ Then for any H with H r we have that (cid:107) (cid:107) ≤ sup P (θ) + H(θ) s + r. ∈ 1 θ B1 23 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: H

text: Assuming

text: that

text: H

text: (so

text: that

text: H

text: is

text: zero

text: to

text: second

text: order)

text: we

text: have

text: that

text: s

text: →lim−∞

text: e

text: l→im∞

text: l→im∞

text: as

text: (e

text: H

text: (e

text: λt0θ

text: s

text: s

text: Me

text: µ∗t0

text: e

text: µ

text: H

text: s

text: Me

text: (µ∗+2µ

text: H

text: s

text: and

text: H

text: Then

text: taking

text: t

text: in

text: Equation

text: and

text: t

text: gives

text: s

text: e

text: AstE

text: (eλtθ)

text: dt

text: eAstE

text: (e

text: λtθ)

text: dt,

text: s

text: s

text: after

text: switching

text: the

text: limits

text: of

text: integration

text: changing

text: t

text: t.

text: Motivated

text: by

text: this

text: discussion

text: we

text: deﬁne

text: the

text: linear

text: solution

text: operator

text: L

text: ](θ)

text: eAstπ

text: E(e

text: λtθ)

text: dt.

text: s

text: s

text: s

text: Let

text: H

text: L

text: and

text: note

text: that

text: s

text: s

text: s

text: M

text: Hs

text: 2µ∗

text: µ∗

text: Es

text: as

text: the

text: integrand

text: satisﬁes

text: eAstE

text: (e

text: λtθ)

text: Me

text: µ∗te

text: 2µ

text: t

text: E

text: s

text: s

text: i.e.

text: the

text: operator

text: is

text: well

text: deﬁned

text: and

text: bounded.

text: Moreover

text: we

text: see

text: that

text: H

text: is

text: analytic

text: by

text: s

text: Morera’s

text: Theorem.

text: To

text: see

text: that

text: H

text: is

text: zero

text: to

text: second

text: order

text: we

text: diﬀerentiate

text: under

text: the

text: integral

text: s

text: and

text: note

text: that

text: E

text: and

text: ∂E

text: /∂θ

text: for

text: j

text: d.

text: To

text: see

text: that

text: H

text: we

text: s

text: s

text: j

text: s

text: expand

text: E

text: as

text: a

text: power

text: series

text: inside

text: the

text: formula

text: for

text: L

text: and,

text: after

text: exchanging

text: the

text: sum

text: s

text: s

text: s

text: and

text: the

text: integral,

text: bound

text: H

text: in

text: terms

text: of

text: E

text: We

text: then

text: check

text: by

text: diﬀerentiating

text: that

text: H

text: s

text: s

text: s

text: so

text: deﬁned

text: solves

text: the

text: desired

text: equation.

text: Proof

text: of

text: Theorem

text: Let

text: max

text: s

text: s.

text: j

text: j

text: d

text: H

text: Then

text: for

text: any

text: H

text: with

text: H

text: r

text: we

text: have

text: that

text: sup

text: P

text: (θ)

text: H(θ)

text: s

text: r.

text: θ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ∈ R | | 0 ( M +1)(K+1). To deﬁne the Newton-like ﬁxed point operator we need an approximate inverse of Df(p¯). Deﬁnition 3.9. 1. The following operator is an approximate inverse of Df(p¯). Let ≈ −  AMK DfMK(p¯) 1 and set  ∈ I (AMKpMK))mk (m, k) MK | | ≤ (Ap) = p m 1 and k > K (43) mk mk ∈ I | | (λ˜·m1)−µk pmk (m, k) / MK and m > 1 2. If A is injective, then ﬁxed points of → − T : Xν Xν, Tp = p Af(p) (44) correspond to zeros of f.  † ≈ − We also specify the operator A A 1: ∈ I (DfMK(p¯)pMK) (m, k) mk MK † | | ≤ (A p) = p m 1 and k > K . (45) mk mk · − ∈ I | | (λ˜ m µ )p (m, k) / and m > 1 k mk MK 4 Applications Consider Fisher’s equation with Neumann boundary conditions as speciﬁed in (4). Be- cause we impose Neumann boundary conditions we expand u(x, t) in a Fourier cosine series (cid:88) and obtain the inﬁnite system of ODEs for the real Fourier coeﬃcients a = (a ) ≥ k k 0 (cid:48) − a (t) = (α k2)a (t) + c| |a| |a| | d=ef g (a) (46) k k k1 k2 k3 k k1+k2∈+Zk3=k ki where c = (c ) ≥ is the sequence of real Fourier coeﬃcients of c(x). k k 0 4.1 Validated computation of the ﬁrst order data In order to build a high order approximation of the unstable manifold of an equilibrium a˜ of (46) we need a validated representation of a˜ and also its eigendata. In the following paragraphs we discuss how this is achieved. All of the computer programs discussed in this Section are available for download at [78]. Equilibrium solution: We look for an equilibrium solution a˜ of Equation (46), that is we demand g(a˜) = 0. The map is well deﬁned as long as 1 < ν¯ < ν, but unbounded for → ν¯ = ν. In fact g is Frechet diﬀerentiable with diﬀerential Dg(a): (cid:96)1 (cid:96)1 given by ν ν¯ − − ∗ ∗ ≥ [Dg(a)h] = (α k2)h 2α(c a h) , k 0, k k k ∈ † for a, h (cid:96)1. Let us specify the operator A and A in this speciﬁc context. We choose ν R → R K > 0, and deﬁne the projection gK : K+1 K+1 by − − ∗ ∗ gK(aK) := (α k2)a α(cK aK aK)K, k k 31 ##LTLine##

text: R

text: M

text: To

text: deﬁne

text: the

text: Newton-like

text: ﬁxed

text: point

text: operator

text: we

text: need

text: an

text: approximate

text: inverse

text: of

text: Df(p¯).

text: Deﬁnition

label: list

text: The

text: following

text: operator

text: is

text: an

text: approximate

text: inverse

text: of

text: Df(p¯).

text: Let

text: AMK

text: DfMK(p¯)

text: and

text: set

text: I

text: (AMKpMK))mk

text: (m,

text: k)

text: MK

text: (Ap)

text: p

text: m

text: and

text: k

text: K

text: mk

text: mk

text: I

text: (λ˜·m1)−µk

text: pmk

text: (m,

text: k)

text: MK

text: and

text: m

text: If

text: A

text: is

text: injective,

text: then

text: ﬁxed

text: points

text: of

text: T

text: Xν

text: Xν,

text: Tp

text: p

text: Af(p)

text: correspond

text: to

text: zeros

text: of

text: f.

text: We

text: also

text: specify

text: the

text: operator

text: A

text: A

text: I

text: (DfMK(p¯)pMK)

text: (m,

text: k)

text: mk

text: MK

text: (A

text: p)

text: p

text: m

text: and

text: k

text: K

text: mk

text: mk

text: I

text: (λ˜

text: m

text: µ

text: )p

text: (m,

text: k)

text: and

text: m

text: k

text: mk

text: MK

text: Applications

text: Consider

text: Fisher’s

text: equation

text: with

text: Neumann

text: boundary

text: conditions

text: as

text: speciﬁed

text: in

text: Be-

text: cause

text: we

text: impose

text: Neumann

text: boundary

text: conditions

text: we

text: expand

text: u(x,

text: t)

text: in

text: a

text: Fourier

text: cosine

text: series

text: and

text: obtain

text: the

text: inﬁnite

text: system

text: of

text: ODEs

text: for

text: the

text: real

text: Fourier

text: coeﬃcients

text: a

text: (a

text: k

text: k

text: a

text: (t)

text: (α

text: (t)

text: c|

text: |a|

text: |a|

text: d=ef

text: g

text: (a)

text: k

text: k

text: k

text: ki

text: where

text: c

text: (c

text: is

text: the

text: sequence

text: of

text: real

text: Fourier

text: coeﬃcients

text: of

text: c(x).

text: k

text: k

text: Validated

text: computation

text: of

text: the

text: ﬁrst

text: order

text: data

text: In

text: order

text: to

text: build

text: a

text: high

text: order

text: approximation

text: of

text: the

text: unstable

text: manifold

text: of

text: an

text: equilibrium

text: a˜

text: of

text: we

text: need

text: a

text: validated

text: representation

text: of

text: a˜

text: and

text: also

text: its

text: eigendata.

text: In

text: the

text: following

text: paragraphs

text: we

text: discuss

text: how

text: this

text: is

text: achieved.

text: All

text: of

text: the

text: computer

text: programs

text: discussed

text: in

text: this

text: Section

text: are

text: available

text: for

text: download

text: at

text: Equilibrium

text: solution:

text: We

text: look

text: for

text: an

text: equilibrium

text: solution

text: a˜

text: of

text: Equation

text: that

text: is

text: we

text: demand

text: g(a˜)

text: The

text: map

text: is

text: well

text: deﬁned

text: as

text: long

text: as

text: ν¯

text: ν,

text: but

text: unbounded

text: for

text: ν¯

text: ν.

text: In

text: fact

text: g

text: is

text: Frechet

text: diﬀerentiable

text: with

text: diﬀerential

text: Dg(a):

text: given

text: by

text: ν

text: ν¯

text: [Dg(a)h]

text: (α

text: a

text: h)

text: k

text: k

text: k

text: k

text: for

text: a,

text: h

text: Let

text: us

text: specify

text: the

text: operator

text: A

text: and

text: A

text: in

text: this

text: speciﬁc

text: context.

text: We

text: choose

text: ν

text: R

text: R

text: K

text: and

text: deﬁne

text: the

text: projection

text: gK

text: by

text: gK(aK)

text: (α

text: α(cK

text: aK

text: aK)K,

text: k

text: k

text: ##LTLine##


text: or as the Kolmogorov-Petrovsky-Piscounov equation, and has applications in mathematical ecology, genetics, and the theory of Brownian motion [40, 1, 65]. 1.2 Methodology of the present work: sketch of the approach Let a˜ be an equilibrium solution of (2) with known Morse index and eigendata. More precisely suppose that Dg(a˜) has exactly d unstable eigenvalues λ˜ . In the present work we j assume that the unstable eigenvalues are real, and that each has multiplicity one. Then let ≤ ≤ ξ˜ , 1 j d denote an associated choice of unstable eigenvectors, i.e. assume that j Dg(a˜)ξ˜ = λ˜ ξ˜ j = 1, . . . , d. (5) j j j In practice the ﬁrst order data is not explicitly given, and we perform a sequence of pre- liminary computer assisted proofs in order to verify that the assumptions are satisﬁed. We refer the reader again to the references mentioned in Remark 1.1 above, and also to Sections and of the present work for more reﬁned discussion of these preliminary considerations. We are now ready to give an informal description of the Parameterization Method for unstable manifolds. See also [34]. Let B { ∈ R | | ≤ ≤ } := (θ , . . . , θ ) d : θ < 1, 1 j d . 1 1 d j We seek solutions of the functional equation ∂ ∂ g(P(θ , . . . , θ )) = λ˜ θ P(θ , . . . , θ ) + . . . + λ˜ θ P(θ , . . . , θ ), (6) 1 d 1 1 ∂θ 1 d d d ∂θ 1 d 1 d ∈ B for all θ = (θ , . . . , θ ) satisfying the linear constraints 1 d 1 P(0) = a˜ (7a) ∂ ≤ ≤ P(0) = ξ˜ , for 1 j d. (7b) ∂θ j j Note that Equation (6) is actually a Banach spaced valued partial diﬀerential equation (or a system of inﬁnitely many scalar partial diﬀerential equations when the Banach space is a sequence space). We refer to Equation (6) as the invariance equation, and note that it is expressed more concisely as ◦ (g P)(θ) = DP(θ)A θ. (8) u × Here A is the d d diagonal matrix of unstable eigenvalues. Equation (8) makes it u clear that the vector ﬁeld g is tangent to the image of P, i.e. P parameterizes an invariant manifold. Indeed we have the following lemma, which makes precise the claim that P recovers the dynamics on the manifold. Lemma 1.3. Assume that P solves (6) and satisﬁes the ﬁrst order constraints of (7). Then ∈ B for every θ the function 1 a(t) = P(exp(A t)θ) (9) u (cid:48) ∈ −∞ solves a = g(a) for all t ( , T(θ)) for a positive time T(θ). In particular lim a(t) = →−∞ t a˜. 5 ##LTLine## ##LTLine## ##LTLine##

text: or

text: as

text: the

text: Kolmogorov-Petrovsky-Piscounov

text: equation,

text: and

text: has

text: applications

text: in

text: mathematical

text: ecology,

text: genetics,

text: and

text: the

text: theory

text: of

text: Brownian

text: motion

text: Methodology

text: of

text: the

text: present

text: work:

text: sketch

text: of

text: the

text: approach

text: Let

text: a˜

text: be

text: an

text: equilibrium

text: solution

text: of

text: with

text: known

text: Morse

text: index

text: and

text: eigendata.

text: More

text: precisely

text: suppose

text: that

text: Dg(a˜)

text: has

text: exactly

text: d

text: unstable

text: eigenvalues

text: λ˜

text: In

text: the

text: present

text: work

text: we

text: j

text: assume

text: that

text: the

text: unstable

text: eigenvalues

text: are

text: real,

text: and

text: that

text: each

text: has

text: multiplicity

text: one.

text: Then

text: let

text: ξ˜

text: j

text: d

text: denote

text: an

text: associated

text: choice

text: of

text: unstable

text: eigenvectors,

text: i.e.

text: assume

text: that

text: j

text: Dg(a˜)ξ˜

text: λ˜

text: ξ˜

text: j

text: d.

text: j

text: j

text: j

text: In

text: practice

text: the

text: ﬁrst

text: order

text: data

text: is

text: not

text: explicitly

text: given,

text: and

text: we

text: perform

text: a

text: sequence

text: of

text: pre-

text: liminary

text: computer

text: assisted

text: proofs

text: in

text: order

text: to

text: verify

text: that

text: the

text: assumptions

text: are

text: satisﬁed.

text: We

text: refer

text: the

text: reader

text: again

text: to

text: the

text: references

text: mentioned

text: in

text: Remark

text: above,

text: and

text: also

text: to

text: Sections

text: and

text: of

text: the

text: present

text: work

text: for

text: more

text: reﬁned

text: discussion

text: of

text: these

text: preliminary

text: considerations.

text: We

text: are

text: now

text: ready

text: to

text: give

text: an

text: informal

text: description

text: of

text: the

text: Parameterization

text: Method

text: for

text: unstable

text: manifolds.

text: See

text: also

text: Let

text: B

text: R

text: (θ

text: θ

text: d

text: θ

text: j

text: d

text: d

text: j

text: We

text: seek

text: solutions

text: of

text: the

text: functional

text: equation

text: g(P(θ

text: θ

text: λ˜

text: θ

text: P(θ

text: θ

text: λ˜

text: θ

text: P(θ

text: θ

text: d

text: ∂θ

text: d

text: d

text: d

text: ∂θ

text: d

text: d

text: B

text: for

text: all

text: θ

text: (θ

text: θ

text: satisfying

text: the

text: linear

text: constraints

text: d

text: a˜

text: ξ˜

text: for

text: j

text: d.

text: ∂θ

text: j

text: j

text: Note

text: that

text: Equation

text: is

text: actually

text: a

text: Banach

text: spaced

text: valued

text: partial

text: diﬀerential

text: equation

text: (or

text: a

text: system

text: of

text: inﬁnitely

text: many

text: scalar

text: partial

text: diﬀerential

text: equations

text: when

text: the

text: Banach

text: space

text: is

text: a

text: sequence

text: space).

text: We

text: refer

text: to

text: Equation

text: as

text: the

text: invariance

text: equation,

text: and

text: note

text: that

text: it

text: is

text: expressed

text: more

text: concisely

text: as

text: (g

text: P)(θ)

text: DP(θ)A

text: θ.

text: u

text: Here

text: A

text: is

text: the

text: d

text: d

text: diagonal

text: matrix

text: of

text: unstable

text: eigenvalues.

text: Equation

text: makes

text: it

text: u

text: clear

text: that

text: the

text: vector

text: ﬁeld

text: g

text: is

text: tangent

text: to

text: the

text: image

text: of

text: P,

text: i.e.

text: P

text: parameterizes

text: an

text: invariant

text: manifold.

text: Indeed

text: we

text: have

text: the

text: following

text: lemma,

text: which

text: makes

text: precise

text: the

text: claim

text: that

text: P

text: recovers

text: the

text: dynamics

text: on

text: the

text: manifold.

text: Lemma

text: Assume

text: that

text: P

text: solves

text: and

text: satisﬁes

text: the

text: ﬁrst

text: order

text: constraints

text: of

text: Then

text: B

text: for

text: every

text: θ

text: the

text: function

text: a(t)

text: P(exp(A

text: t)θ)

text: u

text: solves

text: a

text: g(a)

text: for

text: all

text: t

text: T(θ))

text: for

text: a

text: positive

text: time

text: T(θ).

text: In

text: particular

text: lim

text: a(t)

text: t

text: a˜.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Lower triangular structure To conclude this section let us remark on some implemen- ∗ tation related issues. Recalling (20) to analyze the structure of (Xν,d, ) we see that ∗ (cid:22) TF (p q) does only depend on terms with m˜ m. This entails that the derivative TF ∈mL (cid:22) ∂ f ((cid:96)1, (cid:96)1) (1 < ν˜ < ν) is only non-zero for m˜ m. We refer to this structure ∂pm˜ m ν ν˜ as lower triangular structure. We use this structure in our numerical implementation. To explain the idea let us restrict to the case d = 1, so there is a canoncial ordering on the N indexing set for m. Given truncation dimensions M > 0 and K > 0 the map fMK is a R ∈ R nonlinear map on (M+1)(K+1). Its derivative matrix DfMK(p) (M+1)(K+1),(M+1)(K+1) has the following block structure:      B 0 0   00  B B 0 0 10 11 DfMK(p) = ... ... ... 0 , · · · B B B M0 11 MM ∈ R with B = ∂ fMK(p) K+1,K+1. Thus to solve a linear system of the form DfMK(p)h = ij ∂p∈j Ri b, with h, b (M+1)(K+1) we can implement a block backsubstitution algorithm, that (cid:81) necessitates the solution of M + 1 systems of size (K + 1) instead of the solution of one system of size (M +1)(K+1). So the complexity is O((K+1)3) against O(((M +1)(K+1))3) In the case that d > 1 we see that this translates to the solution of d M systems of size i=1 i (K + 1) making the advantage drawn from this way of implementation even more crucial. 4.3 Computer assisted proof of a heteroclinic connecting orbit In this section we demonstrate how our technique can be used to prove the existence of a connecting orbit. We restrict our attention to the constant case c(x) = 1. In this case the ﬁxed point 1 reduces in Fourier space to a˜1 = (1, 0, 0, . . .). We see directly from the c(x) linearization of g around a˜1 = (1, 0, 0, . . .) that a˜1 is spectrally stable, as all eigenvalues are negative. In addition we can derive nonlinear stability information in terms of an attracting neighborhood of a˜1, see Lemma 4.14 below. Lemma 4.14. The equilibrium point a˜1 of (46) is an attracting ﬁxed point with attracting (cid:8) (cid:9) neighborhood A ∈ | − | = a (cid:96)1 : a a˜1 < 1 . (77) ν ν Proof. See Appendix . This provides us with the ingredient to state the following Theorem proving the existence of a connecting orbit from the equilibrium a˜2 with Morse index 1 to the sink (1, 0, 0, . . .). Note that we can use Lemma 4.2, 4.4 and 4.5 as well as Theorem 4.7 and 4.9 to compute this equilbirium a˜2 together with its stability information. This is achieved in the Matlab script a_validateLinearDatac1_paperVersion.m at the parameter value α = 2.1. Theorem 4.15. Let ν = 1.1. Let P be a parametrization of the unstable manifold of a˜2 | | ± − − ± − with DP(0) = 0.68194897863182 10 16. For θ = 0.505050505050505 10 16 we have ∈ Aν that P(θ) . Hence there is a connecting orbit from a˜2 to a˜1. Proof. By using the inequality | − | ≤ | − | | − | ≤ | − | P(θ) a˜1 P(θ) PMK(θ) + PMK(θ) a˜1 r + PMK(θ) a˜1 ν ν ν P ν 48 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Lower

text: triangular

text: structure

text: To

text: conclude

text: this

text: section

text: let

text: us

text: remark

text: on

text: some

text: implemen-

text: tation

text: related

text: issues.

text: Recalling

text: to

text: analyze

text: the

text: structure

text: of

text: (Xν,d,

text: we

text: see

text: that

text: TF

text: (p

text: q)

text: does

text: only

text: depend

text: on

text: terms

text: with

text: m˜

text: m.

text: This

text: entails

text: that

text: the

text: derivative

text: TF

text: ∈mL

text: f

text: ν˜

text: ν)

text: is

text: only

text: non-zero

text: for

text: m˜

text: m.

text: We

text: refer

text: to

text: this

text: structure

text: ∂pm˜

text: m

text: ν

text: ν˜

text: as

text: lower

text: triangular

text: structure.

text: We

text: use

text: this

text: structure

text: in

text: our

text: numerical

text: implementation.

text: To

text: explain

text: the

text: idea

text: let

text: us

text: restrict

text: to

text: the

text: case

text: d

text: so

text: there

text: is

text: a

text: canoncial

text: ordering

text: on

text: the

text: N

text: indexing

text: set

text: for

text: m.

text: Given

text: truncation

text: dimensions

text: M

text: and

text: K

text: the

text: map

text: fMK

text: is

text: a

text: R

text: R

text: nonlinear

text: map

text: on

text: Its

text: derivative

text: matrix

text: DfMK(p)

text: has

text: the

text: following

text: block

text: structure:

text: B

text: B

text: B

text: DfMK(p)

text: B

text: B

text: B

text: MM

text: R

text: with

text: B

text: fMK(p)

text: Thus

text: to

text: solve

text: a

text: linear

text: system

text: of

text: the

text: form

text: DfMK(p)h

text: ij

text: ∂p∈j

text: Ri

text: b,

text: with

text: h,

text: b

text: we

text: can

text: implement

text: a

text: block

text: backsubstitution

text: algorithm,

text: that

text: necessitates

text: the

text: solution

text: of

text: M

text: systems

text: of

text: size

text: (K

text: instead

text: of

text: the

text: solution

text: of

text: one

text: system

text: of

text: size

text: (M

text: So

text: the

text: complexity

text: is

text: against

text: O(((M

text: In

text: the

text: case

text: that

text: d

text: we

text: see

text: that

text: this

text: translates

text: to

text: the

text: solution

text: of

text: d

text: M

text: systems

text: of

text: size

text: i

text: (K

text: making

text: the

text: advantage

text: drawn

text: from

text: this

text: way

text: of

text: implementation

text: even

text: more

text: crucial.

text: Computer

text: assisted

text: proof

text: of

text: a

text: heteroclinic

text: connecting

text: orbit

text: In

text: this

text: section

text: we

text: demonstrate

text: how

text: our

text: technique

text: can

text: be

text: used

text: to

text: prove

text: the

text: existence

text: of

text: a

text: connecting

text: orbit.

text: We

text: restrict

text: our

text: attention

text: to

text: the

text: constant

text: case

text: c(x)

text: In

text: this

text: case

text: the

text: ﬁxed

text: point

text: reduces

text: in

text: Fourier

text: space

text: to

text: We

text: see

text: directly

text: from

text: the

text: c(x)

text: linearization

text: of

text: g

text: around

text: that

text: is

text: spectrally

text: stable,

text: as

text: all

text: eigenvalues

text: are

text: negative.

text: In

text: addition

text: we

text: can

text: derive

text: nonlinear

text: stability

text: information

text: in

text: terms

text: of

text: an

text: attracting

text: neighborhood

text: of

text: see

text: Lemma

text: below.

text: Lemma

text: The

text: equilibrium

text: point

text: of

text: is

text: an

text: attracting

text: ﬁxed

text: point

text: with

text: attracting

text: neighborhood

text: A

text: a

text: a

text: ν

text: ν

text: Proof.

text: See

text: Appendix

text: This

text: provides

text: us

text: with

text: the

text: ingredient

text: to

text: state

text: the

text: following

text: Theorem

text: proving

text: the

text: existence

text: of

text: a

text: connecting

text: orbit

text: from

text: the

text: equilibrium

text: with

text: Morse

text: index

text: to

text: the

text: sink

text: Note

text: that

text: we

text: can

text: use

text: Lemma

text: and

text: as

text: well

text: as

text: Theorem

text: and

text: to

text: compute

text: this

text: equilbirium

text: together

text: with

text: its

text: stability

text: information.

text: This

text: is

text: achieved

text: in

text: the

text: Matlab

text: script

text: at

text: the

text: parameter

text: value

text: α

text: Theorem

text: Let

text: ν

text: Let

text: P

text: be

text: a

text: parametrization

text: of

text: the

text: unstable

text: manifold

text: of

text: with

text: For

text: θ

text: we

text: have

text: Aν

text: that

text: P(θ)

text: Hence

text: there

text: is

text: a

text: connecting

text: orbit

text: from

text: to

text: Proof.

text: By

text: using

text: the

text: inequality

text: P(θ)

text: P(θ)

text: PMK(θ)

text: PMK(θ)

text: r

text: PMK(θ)

text: ν

text: ν

text: ν

text: P

text: ν

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: [50] A. Haro and R. de la Llave. A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: explorations and mechanisms for the breakdown of hyperbolicity. SIAM J. Appl. Dyn. Syst., 6(1):142–207 (electronic), 2007. [51] Xiaolong He and Rafael de la Llave. Construction of quasi-periodic solutions of state- dependent delay diferential equations by the parameterization method i: ﬁnitely diﬀer- entiable, hyperbolic case. (Submitted), 2015. [52] Xiaolong He and Rafael de la Llave. Construction of quasi-periodic solutions of state- dependent delay diﬀerential equations by the parameterization method ii: Analytic case. (submitted), 2015. [53] Yasuaki Hiraoka. Rigorous numerics for symmetric homoclinic orbits in reversible dy- namical systems. Kybernetika (Prague), 43(6):797–806, 2007. [54] G. Huguet, R. de la Llave, and Y. Sire. Computation of whiskered invariant tori and their manifolds: new fast algorithms. Discrete Contin. Dyn. Syst., 32(4):1309–1353, 2012. [55] Gemma Huguet and Rafael de la Llave. Computation of limit cycles and their isochrons: fast algorithms and their convergence. SIAM J. Appl. Dyn. Syst., 12(4):1763–1802, 2013. [56] A. Hungria, J.P. Lessard, and J. Mireles-James. Rigorous numerics for analytical so- lutions of diﬀerential equations: the radii polynomial approach. Mathematics of Com- putation, Published Electronically: September 28 2015. [57] Tomas Johnson and Warwick Tucker. A note on the convergence of parametrised non- resonant invariant manifolds. Qual. Theory Dyn. Syst., 10(1):107–121, 2011. [58] Donald E. Knuth. The art of computer programming. Vol. 2. Addison-Wesley, Reading, MA, 1998. Seminumerical algorithms, Third edition [of MR0286318]. [59] Hans Koch, Alain Schenkel, and Peter Wittwer. Computer-assisted proofs in analysis and programming in logic: a case study. SIAM Rev., 38(4):565–604, 1996. [60] Oscar E. Lanford, III. A computer-assisted proof of the Feigenbaum conjectures. Bull. Amer. Math. Soc. (N.S.), 6(3):427–434, 1982. [61] Jean-Philippe Lessard, Jason D. Mireles James, and Christian Reinhardt. Computer assisted proof of transverse saddle-to-saddle connecting orbits for ﬁrst order vector ﬁelds. J. Dynam. Diﬀerential Equations, 26(2):267–313, 2014. [62] J.P. Lessard and J. D. Mireles James. Computer assisted proof in analysis and the radii polynomial approach in ck and analytic categories. (In preperation). [63] J.P. Lessard, J. D. Mireles James, and J. Ransford. Automatic diﬀerentiation for fourier series and the radii polynomial approach. (To appear in Physica D), 2016. [64] Kaname Matsue. Rigorous numerics for stationary solutions of dissipative pdes - exis- tence and local dyanmics. Nonlinear Theory and Its Applications, IEICE, 4(1):62–79, 2013. [65] H. P. McKean. Application of Brownian motion to the equation of Kolmogorov- Petrovskii-Piskunov. Comm. Pure Appl. Math., 28(3):323–331, 1975. 54

label: reference

text: A.

text: Haro

text: and

text: R.

text: de

text: la

text: Llave.

text: A

text: parameterization

text: method

text: for

text: the

text: computation

text: of

text: invariant

text: tori

text: and

text: their

text: whiskers

text: in

text: quasi-periodic

text: maps:

text: explorations

text: and

text: mechanisms

text: for

text: the

text: breakdown

text: of

text: hyperbolicity.

text: SIAM

text: J.

text: Appl.

text: Dyn.

text: Syst.,

text: (electronic),

text: Xiaolong

text: He

text: and

text: Rafael

text: de

text: la

text: Llave.

text: Construction

text: of

text: quasi-periodic

text: solutions

text: of

text: state-

text: dependent

text: delay

text: diferential

text: equations

text: by

text: the

text: parameterization

text: method

text: i:

text: ﬁnitely

text: diﬀer-

text: entiable,

text: hyperbolic

text: case.

text: (Submitted),

text: Xiaolong

text: He

text: and

text: Rafael

text: de

text: la

text: Llave.

text: Construction

text: of

text: quasi-periodic

text: solutions

text: of

text: state-

text: dependent

text: delay

text: diﬀerential

text: equations

text: by

text: the

text: parameterization

text: method

text: ii:

text: Analytic

text: case.

text: (submitted),

text: Yasuaki

text: Hiraoka.

text: Rigorous

text: numerics

text: for

text: symmetric

text: homoclinic

text: orbits

text: in

text: reversible

text: dy-

text: namical

text: systems.

text: Kybernetika

text: (Prague),

text: G.

text: Huguet,

text: R.

text: de

text: la

text: Llave,

text: and

text: Y.

text: Sire.

text: Computation

text: of

text: whiskered

text: invariant

text: tori

text: and

text: their

text: manifolds:

text: new

text: fast

text: algorithms.

text: Discrete

text: Contin.

text: Dyn.

text: Syst.,

text: Gemma

text: Huguet

text: and

text: Rafael

text: de

text: la

text: Llave.

text: Computation

text: of

text: limit

text: cycles

text: and

text: their

text: isochrons:

text: fast

text: algorithms

text: and

text: their

text: convergence.

text: SIAM

text: J.

text: Appl.

text: Dyn.

text: Syst.,

text: A.

text: Hungria,

text: J.P.

text: Lessard,

text: and

text: J.

text: Mireles-James.

text: Rigorous

text: numerics

text: for

text: analytical

text: so-

text: lutions

text: of

text: diﬀerential

text: equations:

text: the

text: radii

text: polynomial

text: approach.

text: Mathematics

text: of

text: Com-

text: putation,

text: Published

text: Electronically:

text: September

text: Tomas

text: Johnson

text: and

text: Warwick

text: Tucker.

text: A

text: note

text: on

text: the

text: convergence

text: of

text: parametrised

text: non-

text: resonant

text: invariant

text: manifolds.

text: Qual.

text: Theory

text: Dyn.

text: Syst.,

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